# migrations

dir: embeddings_queue
filename: 00001-embeddings.sqlite.sql
hash: d3755dfd232be8e8301f4d7fcfb3a486
sql:
CREATE TABLE embeddings_queue (
    seq_id INTEGER PRIMARY KEY,
    created_at TIMESTAMP NOT NULL DEFAULT CURRENT_TIMESTAMP,
    operation INTEGER NOT NULL,
    topic TEXT NOT NULL,
    id TEXT NOT NULL,
    vector BLOB,
    encoding TEXT,
    metadata TEXT
);


dir: sysdb
filename: 00001-collections.sqlite.sql
hash: 38352d725ad1c16074fac420b22b4633
sql:
CREATE TABLE collections (
    id TEXT PRIMARY KEY,
    name TEXT NOT NULL,
    topic TEXT NOT NULL,
    UNIQUE (name)
);

CREATE TABLE collection_metadata (
    collection_id TEXT REFERENCES collections(id) ON DELETE CASCADE,
    key TEXT NOT NULL,
    str_value TEXT,
    int_value INTEGER,
    float_value REAL,
    PRIMARY KEY (collection_id, key)
);


dir: sysdb
filename: 00002-segments.sqlite.sql
hash: 2913cb6a503055a95f625448037e8912
sql:
CREATE TABLE segments (
    id TEXT PRIMARY KEY,
    type TEXT NOT NULL,
    scope TEXT NOT NULL,
    topic TEXT,
    collection TEXT REFERENCES collection(id)
);

CREATE TABLE segment_metadata (
    segment_id TEXT  REFERENCES segments(id) ON DELETE CASCADE,
    key TEXT NOT NULL,
    str_value TEXT,
    int_value INTEGER,
    float_value REAL,
    PRIMARY KEY (segment_id, key)
);


dir: sysdb
filename: 00004-tenants-databases.sqlite.sql
hash: 048867ce8fcdefe4023c7110e4433591
sql:
CREATE TABLE IF NOT EXISTS tenants (
    id TEXT PRIMARY KEY,
    UNIQUE (id)
);

CREATE TABLE IF NOT EXISTS databases (
    id TEXT PRIMARY KEY, -- unique globally
    name TEXT NOT NULL, -- unique per tenant
    tenant_id TEXT NOT NULL REFERENCES tenants(id) ON DELETE CASCADE,
    UNIQUE (tenant_id, name) -- Ensure that a tenant has only one database with a given name
);

CREATE TABLE IF NOT EXISTS collections_tmp (
    id TEXT PRIMARY KEY, -- unique globally
    name TEXT NOT NULL, -- unique per database
    topic TEXT NOT NULL,
    dimension INTEGER,
    database_id TEXT NOT NULL REFERENCES databases(id) ON DELETE CASCADE,
    UNIQUE (name, database_id)
);

-- Create default tenant and database
INSERT OR REPLACE INTO tenants (id) VALUES ('default_tenant'); -- The default tenant id is 'default_tenant' others are UUIDs
INSERT OR REPLACE INTO databases (id, name, tenant_id) VALUES ('00000000-0000-0000-0000-000000000000', 'default_database', 'default_tenant');

INSERT OR REPLACE INTO collections_tmp (id, name, topic, dimension, database_id)
    SELECT id, name, topic, dimension, '00000000-0000-0000-0000-000000000000' FROM collections;
DROP TABLE collections;
ALTER TABLE collections_tmp RENAME TO collections;


dir: metadb
filename: 00001-embedding-metadata.sqlite.sql
hash: 2b4cf52c4bb2676e21d6860a4409f856
sql:
CREATE TABLE embeddings (
    id INTEGER PRIMARY KEY,
    segment_id TEXT NOT NULL,
    embedding_id TEXT NOT NULL,
    seq_id BLOB NOT NULL,
    created_at TIMESTAMP NOT NULL DEFAULT CURRENT_TIMESTAMP,
    UNIQUE (segment_id, embedding_id)
);

CREATE TABLE embedding_metadata (
    id INTEGER REFERENCES embeddings(id),
    key TEXT NOT NULL,
    string_value TEXT,
    int_value INTEGER,
    float_value REAL,
    PRIMARY KEY (id, key)
);

CREATE TABLE max_seq_id (
    segment_id TEXT PRIMARY KEY,
    seq_id BLOB NOT NULL
);

CREATE VIRTUAL TABLE embedding_fulltext USING fts5(id, string_value);


dir: metadb
filename: 00002-embedding-metadata.sqlite.sql
hash: 12a570f7121b3a8ce750a2a7c36da20f
sql:
-- SQLite does not support adding check with alter table, as a result, adding a check
-- involve creating a new table and copying the data over. It is over kill with adding
-- a boolean type column. The application write to the table needs to ensure the data
-- integrity.
ALTER TABLE embedding_metadata ADD COLUMN bool_value INTEGER


dir: metadb
filename: 00003-full-text-tokenize.sqlite.sql
hash: f97ad6334aeaa8f419f01110b648b97a
sql:
CREATE VIRTUAL TABLE embedding_fulltext_search USING fts5(string_value, tokenize='trigram');
INSERT INTO embedding_fulltext_search (rowid, string_value) SELECT rowid, string_value FROM embedding_metadata;
DROP TABLE embedding_fulltext;


# embeddings_queue

created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87959c-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 0, "chroma:document": "SPE-171175-MS\nExamples of Mini-Frac Test Data Interpretation in Low-Permeability\nReservoir\nN. Makhota, A. Davletbaev, and A. Fedorov, RN-UfaNIPIneft LLC; R. Asmandiyarov, RN-Yuganskneftegas LLC;\nI. Afanasiev, A. Sergeychev, and I. Yamalov, Rosneft Oil Company\nCopyright 2014, Society of Petroleum Engineers\nThis paper was prepared for presentation at the SPE Russian Oil and Gas Exploration and Production Technical Conference and Exhibition held in Moscow , Russia,\n14\u201316 October 2014.\nThis paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author (s). Contents\nof the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessar ily reflect\nany position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this p aper without the written"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b8796c8-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 0, "chroma:document": "consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; i llustrations may\nnot be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.\nSummary\nWhen dealing with ultra low-permeability reservoirs, one comes across a certain problem of choosing low\ncost way of getting reservoir pressure data. Classical well test implies long-lasting shut down ofproduction wells (up to several months) due to low permeability and presence of extended hydraulicfractures. The paper contains examples of surveys carried out using hydrodynamic logging. These testsshow significant restrictions of the method applied in low permeability resevoirs.\nWell pressure based on minifrac test data was evaluated here for ultra low permeability reservoirs."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b879740-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 0, "chroma:document": "Well pressure based on minifrac test data was evaluated here for ultra low permeability reservoirs.\nThen the data was then compared with the results of average pressure at hydrodynamic logging. Fractureclosure pressure data was used to enrich correlation dependencies for fracture closure pressure (sponta-neous growth of fracture) in injection wells using well pressure fall-off curve interpretation methodsduring minifrac test. Each minifrac test in horizontal wells with multi-stage fracturing was analysed, theresults of injection efficiency ratio were gained that are in agreement with production profile accordingto PLT and allow to test the most productive well influx intervals.\nIntroduction\nWhen dealing with ultra-low permeability reservoir almost all step-out wells are subjected to hydraulicfracturing. As a rule the main frac is preceded by minifrac test which is following stop to monitor wellpressure fall-off curve ["}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b8797b8-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 0, "chroma:document": "G. LeBlanc et. al., 2010 ;V.A. Baikov et.al., 2012 ]. Minifrac test data is used for\nsetting geomechanical models and revising program sheet for the main frac.\nLarge volume fracs carried out in all wells, fractures being as long as 150 meters. Classical well tests\nby means of well pressure build-up curve methods are complicated by the necessity of shutting down thewells (for more than 60 days). As a result it leads to significant production loss and large expenditures forobtaining well data in the regions of drilling new wells. Before shutting off the performance of a wellshould have a significant period of time (up to 6 months) until it has steady state flow. As a result withthe dynamic drilling in new zones being used reservoir parameters and pressure data can be received bymeans of standard well tests in a significant period of time by the moment when data received is not upto date. Thus, there is certain problem of getting low cost and up to date well pressure data in ultra"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87981c-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 1, "chroma:document": "low-permeability fields. The authors convinced that the necessary reservoir data should be received before\nstarting up the producer well.\nThis work contains detailed analyses of initial data for bottom hole pressure change during minifrac\ntest, interpretation in Ecrin, estimation of fracture closure and reservoir pressures. The gained reservoirpressure results through minifrac test were compared with the ones gained through hydrodynamic logging.The results compared showed satisfactory repeatability of measurement data on level of reservoirpressure. The interpretation was carried out during minifrac test with the help of deep pressure gauges.Minifrac tests are commonly used that is why in new or infill drilling zones the reservoir pressure can bereceived without substantial costs and with sufficient precision for all wells.\nDetermination of reservoir pressure according to hydrodynamic logging"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b879880-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 1, "chroma:document": "Determination of reservoir pressure according to hydrodynamic logging\nDetermination of reservoir pressure according to hydrodynamic logging in the zones of drilling new wellsas a rule is carried out in open-hole wells by means of test tools dropped down on cable (XPT\u00abSchlumberger\u00bb, FMT \u00abWeatherford\u00bb packages are usually used for logging of reservoir pressure).During the measurement with the packages in each test interval sampler container collects a small amountof liquid (several cubic centimeters) with pressure curve recording. As a rule, there are two samplingcycles and two pressure buildup curves are recorded. The results achieved are further compared. If thevalues are more or less fit (with sufficient precision), then the sampling is of high integrity. Otherwise,depending on the difference in pressure values the sampling is of low or satisfactory integrity.\nOn\nfig. 1 there is a diagram showing pressure change and fluid influx when testing one of the intervals"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b8798e4-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 1, "chroma:document": "On\nfig. 1 there is a diagram showing pressure change and fluid influx when testing one of the intervals\non well XX22. Due to the limit volume of sampler container the cycling time for influx is short: 17 sec\n- for the first cycle, 13 sec- for the second. After sampling cycles well pressure buildup curves are\nrecorded: 11 minfor the first, 5 minfor the second. It is seen on the diagram (for both ones) that after\nwellbore storage effect logarithmic derivative tend to zero ( fig. 1b ). This can be explained by a short\nperiod of sampling before pressure build-up curve. The pressure on build-up cycle quickly aligns andtends to constant. In this case the difference between the observed values is: ~1.4 atm.\nWhen testing one well up to 30 reservoir pressure values can be achieved through different time\nintervals. Experience of studying highly-compartmentalized low-permeability reservoirs shows that in theinterval of one geological object pressure values can differ (100 atmand even more)."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b879948-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 1, "chroma:document": "Fig. 2 shows test\nreport for well XX39. In object interval C formation pressure values during hydrodynamic logging aredifferent by the value of 190 atm, the depths of the testing intervals is as much as 14 m. The difficulty\nis in data averaging for the object. Only one value should be given to an object when forming intervalpressure map, reservoir models.\nCurrently the engineers from \u201cRosneft\u201d and \u201cSchlumberger\u201d use the following expressions (1) to\nevaluate pressure average value. This approach takes into account the thickness of beds according to\nFigure 1\u2014Dynamics of bottom hole pressure and liquid rate (a) and its logarithmic derivative of build-up cycles (b) on well XX222 SPE-171175-MS"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b8799ac-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 2, "chroma:document": "geophysical logging interpretation and study intervals as well as actuality of each sampling. An average\npressure for the object is expressed as follows:\n(1)\nWhere Priis a reservoir pressure of i-th bed ( i) in which several measurement tests can be carried out,\n(k//H9262)i\u2013mobility of i-th bed. In the first expression the summing up is carried out for all thin beds Lwith\ninformative measurement test. Second and third expressions the summing up is carried out for allmeasurement tests Min each i-th thin bed. It is widely accepted that weight factors take the following\nvalues according to test reliability ( w\nij),wij/H110051 \u2013 for measuring high integrity, wij/H110050.5 \u2013 for measuring\nsatisfactory integrity, wij/H110050.1 - for measuring low integrity.\nAmid the deteorating filtration properties of the wells drilled there is a tendency for low integrity study"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b879a06-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 2, "chroma:document": "Amid the deteorating filtration properties of the wells drilled there is a tendency for low integrity study\nnumber and number of dry measuring growing. In this case the absence of influx during the test can beexplained by the limitations of the measuring instrument (the model doesn\u2019t support obtaining influx fromultra low permeability reservoirs). For example, when measuring well XX39 in the interval of Achimovskireservoir (with average permeability less than 1 mD) 12 measurements were carried out - 8 of them weredry sampling (with no influx from reservoir), 3 samplings were of low integrity and 1 with satisfactoryintegrity. In other case, during XX44 well surveying for object US0 measurements were carried out in 26intervals. All cases showed dry samplings. After hydraulic fracturing the well was put on operation. Thewell is being productive for 3 years.\nThus, tests with the help of hydrodynamic logging have restrictions in low-permeability reservoirs,"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b879a6a-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 2, "chroma:document": "Thus, tests with the help of hydrodynamic logging have restrictions in low-permeability reservoirs,\nmeasuring pressure using well pressure buildup curve interpretation method for producer after hydraulicfracturing leads to significant oil production losses.\nThe examples of reservoir pressure determining according to minifrac test\ndata\nExample 1 . Measuring pressure change curve using bottomhole pressure gauge and with the help of\ninjection fluid flow at the stage of minfra /slovotest during hydraulic fracture for XX45 well is shown on fig.\nFigure 2\u2014Interpretation results of testing by using hydrodynamic logging (a) /izhepressure profile according to the depth (b) for well XX39SPE-171175-MS 3"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b879ace-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 3, "chroma:document": "3a. During the injection for 4.3 min. 12.8 m3of fluid was injected. Well pressure fall-off curve\ninterpretation method was recorded after injection stop. Initial data interpretation (for the well pressurefall-off curve interpretation method) was carried out in Ecrin (Kappa Eng). According to pressurederivative diagram using index with G-function [\nBarree R. et. al., 2009 ] fracture closure time (~3.1 min)\nand closure pressure (361.6 atm) were determined ( fig.3b ). By the shape of the curve GdP/dG fluid loss\ntype from the fracture to reservoir and closure time point were determined [ Craig D. et. al., 2000 ].\nReservoir pressure (using minifrac test) was determined by means of pressure analyses after the\nclosure. Pressure derivative diagram shows flow patterns according to line flow function square ( fig. 3 ).\nFigure 3\u2014Data on bottomhole pressure and flow rate of the liquid injected (a), pressure and pressure derivative chart with G-function (b), square"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b879b28-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 3, "chroma:document": "Figure 3\u2014Data on bottomhole pressure and flow rate of the liquid injected (a), pressure and pressure derivative chart with G-function (b), square\nlinear flow pressure derivative chartFL(/slovo), radial flow function FR(d) during minifrac test for well XX45\nFigure 4\u2014Data on bottomhole pressure and flow rate of the liquid injected (a), pressure and pressure derivative chart with G-function (b), square\nlinear flow pressure derivative chartFL(/slovo), radial flow function FR (d) during minifrac test for well XX714 SPE-171175-MS"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b879b8c-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 4, "chroma:document": "So, the \u00bd slope corresponds to pseudolinear flow pattern, singular slope-to pseudoradial. In this case,\npseudoradial flow pattern can be observed for well XX45 in ~30 minfrom stopping time on well pressure\nfall-off curve interpretation method that is why reservoir pressure is determined according to linear curveapproximation from the last point measured to the beginning of pseudoradial flow pattern on radial flowfunction pressure diagram F\nR(fig.3d ). After minifrac test data analyses was carried out the pressure for\nwell XX45 amounts to 278.9 atm.\nExample 2 . Pressure change curve view during minifrac test for well XX71 is shown on fig.4a . Liquid\ninjection was carried out for 4.2 min, 12.8 m3were injected. At this formation fracturing pressure is 464.7\natm. However, in this case it is seen on pressure derivative diagram where pseudoradial flow pattern can"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b879bf0-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 4, "chroma:document": "atm. However, in this case it is seen on pressure derivative diagram where pseudoradial flow pattern can\nnot be detected on square linear flow, that is why reservoir pressure was measured on pseudolinear flowpattern with \u00bd slope. Estimated value of reservoir pressure is 294, 6 atm(\nfig. 4b ), fracture closure pressure\n\u2013 374.4 atm.(fig. 4b ).\nExample 3 . Minifrac test review with stop at well pressure fall-off curve interpretation method is shown\natfig. 5a . Formation fracturing pressure is 464.7 atm, fracture closure pressure \u2013 321.1 atm(fig.5b ).\nPseudoradial flow pattern is not tested as it was in the previous case ( fig.5c ), thus, reservoir pressure\nassessment is done through pseudolinear flow pattern \u2013 246.4 atm(fig.5d ).\nBefore hydraulic fracturing the measurements were carried with the help of hydrodynamic logging for"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b879c54-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 4, "chroma:document": "Before hydraulic fracturing the measurements were carried with the help of hydrodynamic logging for\nobserved in the examples above wells. Consequently, after averaging the pressure with expressions (1)reservoir pressures at minifrac test and hydrodynamic logging were compared. Well XX45 at pseudoradialflow pattern has the best congruity: reservoir pressure on minifrac test is 278.9 atm, hydrodynamical\nlogging pressure is 272.5 atm. In wells in which pressure estimation was carried out on pseudolinear flow\npattern pressure variation is greater: in well XX71 pressure on minifrac test is 294.6 atm. On hydrody-\nnamic logging - 275.0 atm. In well XX82 at minifrac test 246.4 atm, at hydradynamic logging \u2013 262.7 atm.\nThus, when analyzing bottomhole gauge data on injection and fall-off cycles at minifrac tests one can\nget estimated values of reservoir pressure. The tests carried out showed that the difference betweenpressures at mininfrac test and hydrodynamic logging can reach 20 atm."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b879cae-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 4, "chroma:document": "Figure 5\u2014Bottomhole pressure and fluid flow rate data (a), pressure and pressure derivative chart on G-function (b), pressure derivative chart with\nsquare linear flow function FL(/slovo), pressure change chart at radial flow function FR(d) at minifrac test for well XX82SPE-171175-MS 5"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b879d1c-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 5, "chroma:document": "Correlation relationship of fracture closure pressure vs. reservoir pressure\nIn the paper [ Davletbaev et. al., 2010 and2014 ] correlation relationship between fracture closure pressure\nand reservoir pressure is given. This generalised relationship has sufficient statistics for reservoir pressurerange for more than 350 atm, for the range with less pressure testing points used are not enough. The"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b879d76-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 5, "chroma:document": "reason for this is that well candidates for the study in steady flow regime are chosen with substantialinjection volumes and. as a rule, waterflooding radius in these wells are comparable to welltestinvestigation radius. When analyzing minifrac tests data using downhole gauge one can also get reservoirpressure and fracture closure pressure values. Fracturing is carried out in step-out wells in undisturbedzones with initial reservoir pressure, in infill wells in zones with pressure less than initial reservoirpressure. Consequently, correlation relationship can be complemented with data from the range of missingpressures for the relationship observed. Correlation relationship of fracture closure pressure with reservoirpressure for reservoirs T10, T11, T12 according to study on steady state injection test which is added withthe results of analyses of minifrac tests with bottomhole pressure gauges for the reservoirs.\nData comparison of minifrac test and production profile of PLT for"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b879dd0-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 5, "chroma:document": "Data comparison of minifrac test and production profile of PLT for\nhorizontal well with multi-stage fracturing\nAt this time the number of horizontal wells drilled in low-permeability collectors (at multi-stage\nfracturing) is growing. However, according to reports on fracturing the planned production is notcorrelated to the volumes of proppant agent. The most difficult question is - which fractures in horizontalhole provide influx for the well as the injection of the proppant agent in the fracture do not guarantee it\u2019sinflux to the well. For these purposes in a number of wells PLT logging is carried out with productionprofile record for horizontal well. This is done for determining the influx from intervals with fixedhyradulic fractures.\nInflux/production distribution to frac sleeves recoreded for production profile at PLT logging is shown\nfor well XX46 on\nfig.7 The study helped to learn that the main influx/prodcution (60%) is provided by"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b879e34-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 5, "chroma:document": "for well XX46 on\nfig.7 The study helped to learn that the main influx/prodcution (60%) is provided by\nhydraulic fracture in the interval with frac sleeve P4, low level influx comes from other 4 fractures. Inaddition to this 25% of liquid comes from nonhermetic linear coupling.\nIn all five cases before the main fracturing minifrac test were carried out. The results were analized as\nin\nfig.3\u20135. Apart from fracture closure pressure estimation and etc. fluid injection efficiency kiewas also\nFigure 6\u2014The connection of fracture closure pressure with reservoir pressure for reservoirs T10, T11, T12 according to study results at steady\ninjection patterns and minifrac test data6 SPE-171175-MS"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b879e98-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 6, "chroma:document": "received. Injection efficiency is a parameter showing the difference between the initial and maximum\nfracture with the efflux absence. So is determined by efflux efficiency at hydraulic fracture and reservoircharacteriscics. The difference k\nieat different stages of hydraulic fracturing along horizontal well at equal\nfluid properties indicates the difference between reservoir intervals. The less leakage efficiency is thebetter reservoir characterisctics are and more fluid moves to reservoir at hydraulic fracturing. Injectionefficiency for 4 minifrac tests are shown on\nfig. 7 . There is no data for the sleeve P3. The estimation is"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b879efc-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 6, "chroma:document": "fig. 7 . There is no data for the sleeve P3. The estimation is\nunsuccessful. Comparison of injection efficiency at hydraulic fracturing and according to productiondistribution at PLT logging shows that interval with themain influx (60%) has the lowest injectionefficiency at minifrac tests. Thus, analysis of minifrac tests can reveal the most productive intervals athydraulic fracturing in horizontal wells. This helps to save expenditures for proppant agent volumeinjected to non-commercial intervals and reveal the most productive ones for planning of proppant agentinjection volume.\nConclusion"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b879f60-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 6, "chroma:document": "Recording and interpretation of minifrac test data give the possibility of estimating reservoir pressure withsufficient precision in ultra low-permeability reservoir rocks where it is impossible to get informativemeasurements with the help of hydradynamic logging and there\u2019s no possibility to test operating wellsafter reservoir fracturing due to significant production loss. It should be noted that more or less precisereservoir pressure measurements can be received using well pressure fall-off curve interpretation methodduring minifrac test at pseudoradial flow pattern. Apart from that realization of hydradynamic logging inopen well bore connected with additional technological risks and well construction time increment due tostudy (average time is 60 hours). Along with that minifrac tests are usually the integral part of hydraulicfracturing and additional expenditures will be connected with wireline gage run. Consequently, themethod applicated is cost and expenditures saving connected"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b879fc4-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 6, "chroma:document": "and additional expenditures will be connected with wireline gage run. Consequently, themethod applicated is cost and expenditures saving connected with reservoir pressure detection. Hydro-dynamic logging is much more expensive."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87a028-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 6, "chroma:document": "Minifrac test analyses provide data on fracture closure pressure and can complement correlation\nrelationship on steady state injection test in injection wells with spontaneous growth of fracture.Evaluation of injection efficiency during minifrac test provides an opportunity to receive the mostproductive intervals in horizontal wells with multi-stage fracturing and is corresponding with theinflux/production profile on PLT logging.\nAcknowledgments\nThe authors express thanks to the colleagues Latipov Alfred, Kazakevitch Eduard (Schlumberger) for theparticipation, help, advice and realization of averaging reservoir pressure method according to hydrody-namic logging study, to Serdyuk Alexander Nikolaevich, Valeev Sergei Valerievich (LLC \u201cRN-\nFigure 7\u2014Influx profile comparison with the results of minifrac test for well XX46SPE-171175-MS 7"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87a08c-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 7, "chroma:document": "Yuganskneftegaz\u201d), Yudin Aleksei (Schlumberger) for the help and coordination at planning the mini-frac\nmeasurement, to Ivan Tichonov and Askar Valiullin (LLC \u201cPKF Gis-Nefteservis\u201d) for help and analysesof PLT logging results for horizontal well XX46 with multi-stage fracturing.\nNomenclature\nF\nL \u2013 linear flow time function\nFR \u2013 radial flow time function\nhi \u2013 bed thickness, m\nk \u2013 reservoir permeability, mD\n(k//H9262)i \u2013 mobility of i-th bed, mD/cP\n(k//H9262)HL_ij \u2013 mobility of reservoir fluid according to j-th measurement with hydrodynamic logging in\ni-th bed\nL \u2013 number of beds with measurement results after hydrodynamic loggingM \u2013 measurement number with hydrodynamic logging in one bedP \u2013 pressure, atm\ntdP/dt \u2013 pressure derivatives, /az/tverdo/myslite\nP\nra v \u2013 average reservoir pressure, atm\nPri \u2013 reservoir pressure for i-th bed, atm\nPrH Li j \u2013 reservoir presuure according to j-th measurement with hydrodynamic logging in i-th bed,\natm"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87a0e6-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 7, "chroma:document": "Pri \u2013 reservoir pressure for i-th bed, atm\nPrH Li j \u2013 reservoir presuure according to j-th measurement with hydrodynamic logging in i-th bed,\natm\nwij \u2013 measurement accuracy at hydrodynamic logging, unit fraction\noPij \u2013 hydrostatic pressure diferential between the depth of the measurement and reservoir\npressure calculation, atm\n/H9262 \u2013 reservoir fluid fiscosity, cP\nPi \u2013 reservoir pressure, atm\nPff \u2013 fracture closure pressure, atm\nkie \u2013 injection efficiency at minifrac test\nSSIT \u2013 steady state injection test\nReferences\nBaikov V.A., Burakov I.M., Latypov I.D., Yakovlev A.A., Asmandiyarov R.N. 2012. Waterflood\ninduced hydraulic fracturing control under reservoir pressure maintenance conditions on RN-Yugan-skneftegas oilfields. Neftyanoe khozyaystvo \u2013 Oil Industry . No 11. P. 30\u201333 (Russian)\nBarree R., Mukherjee H. 1996. Determination of Pressure-dependent Leakoff and Its Effects on\nFracture Geometry . SPE 36424,"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87a14a-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 7, "chroma:document": "Barree R., Mukherjee H. 1996. Determination of Pressure-dependent Leakoff and Its Effects on\nFracture Geometry . SPE 36424,\nBarree R., Barree V., Craig D. 2009. Holistic Fracture Diagnostics: Consistent Interpretation of\nPrefrac Injection Tests Using Multiple Analysis Methods . SPE 107877,\nCraig D., Eberhard M., Barree R. 2000. Adapting High Permeability Sands for Estimating Reservoir\nEngineering Parameters . SPE 60291.\nDavletbaev A., Baikov V., Bikbulatova G., Asmandiyarov R., Nazargalin E., Slabetskiy A., Sergey-\nchev A., Nuriev R. 2014. Field Studies of Spontaneous Growth of Induced Fractures in Injection Wells .\nSPE-171232.\nDavletbaev A., Baikov V., Ozkan E., Garipov T., Usmanov T., Asmandiyarov R., Slabetskiy A.,\nNazargalin E. 2010. Multi-Layer Steady-State Injection Test with Higher Bottomhole Pressure than the\nFormation Fracturing Pressure . SPE 136199.\nLeBlanc G., Gunasan E., Boutaud de la Combe J-L., Jaffrezic V. 2010. Reservoir Characterization"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87a1a4-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 7, "chroma:document": "Formation Fracturing Pressure . SPE 136199.\nLeBlanc G., Gunasan E., Boutaud de la Combe J-L., Jaffrezic V. 2010. Reservoir Characterization\nUsing Injection Test After-Closure Analysis: Field Case History in a Depleted Oil Reservoir . SPE 128052.8 SPE-171175-MS"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87a208-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 0, "chroma:document": "SPE-170956-MS\nThe Effect of Changing Fracture Compliance on Pressure Transient\nBehavior During Diagnostic Fracture Injection Tests\nMark W. McClure, Christopher A. J. Blyton, Hojung Jung, and Mukul M. Sharma, The University of Texas at\nAustin, Department of Petroleum and Geosystems Engineering\nCopyright 2014, Society of Petroleum Engineers\nThis paper was prepared for presentation at the SPE Annual Technical Conference and Exhibition held in Amsterdam, The Netherlands, 27\u201329 October 201 4.\nThis paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author (s). Contents\nof the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessar ily reflect\nany position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this p aper without the written"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87a262-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 0, "chroma:document": "consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; i llustrations may\nnot be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.\nAbstract\nDiagnostic Fracture Injection Tests (DFIT) are commonly interpreted with G-function derivative plots. In"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87a2d0-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 0, "chroma:document": "these plots, an upward deviation from linearity is generally attributed to fracture height recession ortransverse storage. This interpretation neglects changing fracture compliance during closure. Resultspresented in this study show that in low permeability formations, an upward deviation of the G-functionplot G*dP/dG is caused by a sharp reduction in fracture compliance due to closure. It is not necessary toinvoke more complicated explanations such as height recession or transverse storage. The specific shapeof the G*dP/dG curve is controlled by factors that affect fracture compliance before and after closure.These factors include the fracture height, fracture stiffness, and the residual aperture (which is affected bythe microscopic and macroscopic roughness of the created hydraulic fracture). Wellbore storage broadensthe post-closure peak in the G*dP/dG curve. A curve of G*dP/dG that constantly increases during theshut-in period may be caused by wellbore storage, rather than"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87a32a-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 0, "chroma:document": "peak in the G*dP/dG curve. A curve of G*dP/dG that constantly increases during theshut-in period may be caused by wellbore storage, rather than fracture tip extension. In higher permea-bility formations (with non-zero wellbore storage), the effect of fracture compliance is less important. Asa result, changing fracture compliance during closure does not have a significant effect on the pressuretransient plots, and closure decreases the rate of pressure decline because it reduces the rate at which fluidcan leak off into the matrix. In all cases, as long as a single planar fracture has formed, the best pick forthe minimum principal stress is at the deviation from linearity on the G*dP/dG plot, whether or not thisdeparture is concave up or concave down. Closure picks via a tangent to near the peak in G*dP/dG (asis commonly performed) may significantly underestimate the minimum principal stress. These picksshould be confirmed with other measurements, such as a subsequent injection period"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87a38e-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 0, "chroma:document": "underestimate the minimum principal stress. These picksshould be confirmed with other measurements, such as a subsequent injection period or a pump-inflow-back test. The fact that an extremely sharp upward trend in G*dP/dG is generally not observed inDFIT tests suggests that the actual geometry of the created fracture is rougher and more complex than istypically assumed by standard models."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87a3f2-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 0, "chroma:document": "Introduction\nOverviewDiagnostic fracture injection tests (DFIT) are a type of minifrac test performed in low permeabilityformations. The procedure includes pumping a small volume of water into a short wellbore interval over"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87a44c-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 1, "chroma:document": "a few minutes, most often out of a perforated cased wellbore. The pressure transient after shut-in is used\nto infer the closure pressure (assumed equal to the minimum principal stress), the formation permeability,the leak-off coefficient, and the initial formation pressure (\nBarree et al., 2007 ;Soliman and Kabir, 2012 ;\nCramer and Nguyen, 2013 ).\nConventionally, closure pressure is estimated from the shut-in pressure transient of a minifrac test by\nplotting pressure versus a function of time called G-function time (or alternatively, versus the square rootof time) and identifying the point where a deviation from linearity occurs (\nNolte, 1979 ;Castillo, 1987 ).\nEquivalently, closure may be identified by plotting G*dP/dG and identifying the departure from linearity(\nBarree et al., 2007 ).\nMinifrac tests in very low permeability formations have proven difficult to interpret because plots of"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87a4ba-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 1, "chroma:document": "Barree et al., 2007 ).\nMinifrac tests in very low permeability formations have proven difficult to interpret because plots of\nG*dP/dG may appear concave up or down ( Soliman and Kabir, 2012 ;Cramer and Nguyen, 2013 ). In some\ncases the trend is concave down with a persistent gradual increase apparent. This makes the departurefrom linearity difficult to identify. These trends are usually attributed to a variety of different non-idealprocesses. In such cases the interpretation is difficult given the non-unique nature of the problem.\nIn this study, we simulated DFIT tests in low permeability formations assuming a single planar\nfracture. We used CFRAC (Complex Fracturing ReseArch Code), a hydraulic fracturing simulator thatperforms a full numerical simulation for fluid flow in the fracture and the matrix, calculates the stressesinduced by fracture opening, includes wellbore storage, and uses a nonlinear joint closure model todescribe the evolution of fracture compliance after closure ("}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87a51e-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 1, "chroma:document": "McClure and Horne, 2013 ). The simulations\nindicated that a concave up trend on a G*dP/dG plot can be explained as a natural consequence of thedecreasing fracture compliance that occurs during closure. A concave downward G*dP/dG trend can becaused by wellbore storage. The simulations also indicated why these behaviors are more likely to beobserved in low permeability formations.\nOur simulations suggest that closure pressure should be picked according to the classical method of\nidentifying the point of departure from linearity on the pressure versus G-function time (or square root oftime) plot or the G*dP/dG versus G-function time plot (\nCastillo, 1987 ), whether or not the G*dP/dG curve\nis concave up or concave down. In the presence of curving G*dP/dG plots, widespread industry practiceis to pick closure much later (at a much lower pressure), near the peak of the G*dP/dG curve (\nBarree et"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87a582-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 1, "chroma:document": "Barree et\nal., 2007 ;Soliman and Kabir, 2012 ;Padmakar, 2013 ;Cramer and Nguyen, 2013 ;Wallace et al., 2014 ).\nOur results suggest that in low permeability formations, closure actually occurs near the bottom of thepeak in the G*dP/dG curve where a subtle departure from linearity is often present.\nLiterature review\nNolte (1979) published the first analysis technique for post fracturing pressure decline. The method\nallowed the overall fluid loss coefficient, efficiency, geometry, and time for the fracture to close to becalculated. The assumptions made were consistent with PKN fracture geometry, presented by\nPerkins and\nKern (1961) and extended by Nordgren (1972) . These include constant fracture height, elastic rock"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87a5e6-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 1, "chroma:document": "Perkins and\nKern (1961) and extended by Nordgren (1972) . These include constant fracture height, elastic rock\ndeformation with no slip at bedding planes, a bi-wing fracture created by a constant fluid injection rateand continuous propagation during pumping. Further, it was assumed that fluid loss is a function of theinverse square root of exposure time. This assumption is often termed Carter leakoff (Howard and Fast,1970). Because of the greater exposure time closer to the wellbore, the leakoff rate from the fracture isgreatest furthest from the wellbore.\nAn analytical approach was used by\nNolte (1979) to develop a function of shut-in time and pumping\ntime that produces a straight line when plotted against pressure. This time function is commonly referredto as G-function time. The overall fluid loss coefficient can be determined from a lumped parameter equalto the gradient of the pressure versus G-function time. The only reference to closure stress by\nNolte (1979)"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87a640-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 1, "chroma:document": "Nolte (1979)\nwas an estimate taken from a pump-in and flow-back test and not from analysis of post fracturing pressuredecline.2 SPE-170956-MS"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87a6a4-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 2, "chroma:document": "Nolte (1986) later extended the earlier work to other geometries, including both KGD ( Khristianovich\nand Zheltov, 1955 ;Geertsma and de Klerk, 1969 ) and radial fracture geometry. Reference to closure stress\nwas only made with regard to corrections needed in the case of its alteration due to the injection processitself and not a determination from the post-fracturing pressure decline.\nMartins and Harper (1985)\npresented a similar analysis for elliptical fracture geometry. They also included a discussion of pressuredecline analysis, noting that the rate of decrease of average pressure depends on both the rate of fluid lossto the formation and the fracture compliance. They noted that for a given fracture length and height, orradius in the case of radial fracture geometry, the fracture compliance remains constant while the fractureis open. The focus of the work was primarily on determination of the overall leakoff coefficient.\nSubsequent work by"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87a712-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 2, "chroma:document": "Subsequent work by\nNolte (1988) covered pressure transient analysis of fracture treatments both during\npumping and after shut-in. An observable increase in the rate of pressure decline upon closure could beidentified from data plotted against time, allowing an inference of closure stress. The author noted that thiswas due to the increasing stiffness of the fracture as it closed on proppant. A departure from linearity ona plot versus the square root of time was also presented. The author suggested that this closure stressestimate could be compared with the results of a pump-in and flow-back test.\nCastillo (1987) suggested an extension in application of the Nolte (1979) G-function time to the"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87a76c-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 2, "chroma:document": "Castillo (1987) suggested an extension in application of the Nolte (1979) G-function time to the\nidentification of fracture closure and hence its use to infer closure stress. This was based on identifyinga departure from linearity of the pressure versus G-function time plot, after a period of post shut-infracture extension, if present. The first derivative of pressure with respect to G-function time was used asan identification aid.\nPlots of G-time multiplied by the derivative of pressure with the respect to G-time are commonly used\nas a diagnostic tool.\nBarree and Mukherjee (1996) used a numerical fracture simulation code to produce"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87a7da-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 2, "chroma:document": "as a diagnostic tool.\nBarree and Mukherjee (1996) used a numerical fracture simulation code to produce\npressure transients of non-ideal fracture cases, potentially allowing for their identification in field data.The simulations included an ideal base case, a case with post shut-in fracture extension, a case withpressure dependent leakoff, a case with both pressure dependent leakoff and pressure dependent fracturecompliance, and a case with fracture height recession from bounding layers. Ad-hoc functions were usedto relate leakoff with Young\u2019s modulus to pressure, and so the results should be considered qualitative.\nIn later research\nBarree et al. (2007) used other diagnostic tools, in addition to G-function time, to\nexamine simulated pressure transients for a number of cases with departures from the basic G-functiontime assumptions of\nNolte (1979) . The phenomena studied included an ideal base case, fracture extension"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87a83e-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 2, "chroma:document": "Nolte (1979) . The phenomena studied included an ideal base case, fracture extension\nfollowing shut-in, pressure dependent leakoff, and fluid storage from transverse or dilated naturalfractures.\nBarree et al. (2007) picked the fracture closure pressure by drawing a straight line from the\norigin tangent to the G*dP/dG curve where it begins to curve down. Barree et al. (2007) interpreted a\nconcave up G*dP/dG curve as representing either fracture height recession or opening of transversefractures. A concave down G*dP/dG curve was interpreted as fracture tip extension.\nBarree et al. (2007)\nconsidered fracture tip extension to be a process that could occur for an extended time after shut-in, oreven for the entire duration of the shut-in period.\nPadmakar (2013) presented results of a coupled geomechanics and reservoir flow simulation study"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87a91a-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 2, "chroma:document": "Padmakar (2013) presented results of a coupled geomechanics and reservoir flow simulation study\nconducted using commercially available software. An advantage of this approach was that it enabled studyof both pre and post fracture closure pressure transient behavior. A Barton-Bandis model was used tomodel the fracture permeability, which included both hysteresis in loading and retention of residualaperture on closure. There was only one simulated pressure transient presented, which showed asignificant departure from the ideal response expected.\nPadmakar (2013) noted that the pick made for\nclosure stress was lower than that input to the simulation.\nWallace et al. (2014) modeled a single fracture case and complex discrete fracture network cases with"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87a974-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 2, "chroma:document": "Wallace et al. (2014) modeled a single fracture case and complex discrete fracture network cases with\nan assumed fracture density and aperture. The complex discrete fracture network was upscaled to acontinuum model for simulation. The effect of fracture closure was captured in a simple and non-hysteretic manner by use of non-linear transmissibility multiplier versus pressure curves. The simulatedSPE-170956-MS 3"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87a9d8-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 3, "chroma:document": "pressure transients produced by their simulations were analyzed by means of the pressure and derivative\nversus G-function time approach. The apparent closure pick made for the single fracture case matched thatinput to the simulator. The complex discrete fracture case showed a pressure transient signature that wasinterpreted to be the result of transverse storage or natural fracture dilation.\nMethodology\nSimulation methodologySimulations were performed with CFRAC (Complex Fracturing ReseArch Code). CFRAC is a two-dimensional discrete fracture network simulator that simultaneously calculates fluid flow and stressesinduced by fracture deformation. Fluid flow in the matrix is also simulated. The full details of CFRACwere summarized by\nMcClure and Horne (2013) , and only a brief overview is given in this section. Some\nmodifications have been made to CFRAC since the version described by McClure and Horne (2013) , and\nthose modifications are described here in detail.\nFluid flow"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87aa32-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 3, "chroma:document": "those modifications are described here in detail.\nFluid flow\nFluid flow is single-phase liquid water (no proppant) and isothermal. CFRAC solves the unsteady statemass balance equation with Darcy\u2019s law:\n(1)\n(2)\nwhere mis the mass density at a location (kg/m3),tis time (s), qfluxis the mass flux vector (kg/(s-m2)),\nsis a source term for a well (kg/(m3-s)),/H9267is fluid density (kg/m3),kis permeability (m2),/H9262is fluid\nviscosity (MPa-s), Pis pressure (MPa), and xiis location in the direction i(m).\nThe mass flux is converted to mass flow rate, q(kg/s), by multiplying by the cross sectional area for\nflow. The cross-sectional area for flow is equal to the height (the out-of-plane dimension in a two-dimensional simulation), h, multiplied by width available for flow. For flow in the fractures, the width for\nflow is defined equal to the hydraulic aperture, e. The permeability of the fracture is defined as equal to:\n(3)"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87aa8c-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 3, "chroma:document": "flow is defined equal to the hydraulic aperture, e. The permeability of the fracture is defined as equal to:\n(3)\nWhen permeability is multiplied by hydraulic aperture, the result is the transmissivity (units of m3),T,\nand the classical cubic law results ( Witherspoon et al., 1980 ):\n(4)\nWe define two different types of fracture aperture. The \u201cvoid\u201d aperture, E, is the volume of fluid stored\nper surface area of fracture. The \u201chydrauli\u201d aperture, e, is related through the cubic law to the\ntransmissivity. In actual practice, the hydraulic aperture may be lower than the mechanical aperture (thephysical apparent thickness of the fracture) or E, the void aperture (the volume of fluid stored per area)."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87aae6-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 3, "chroma:document": "Therefore, while transmissivity is a physically meaningful parameter because of its role in Darcy\u2019s law,hydraulic aperture is an \u201ceffective\u201d parameter that is only meaningful in terms of its relationship totransmissivity. Void aperture, E, is an important physical parameter because it describes the fluid stored\nin the fracture.\nThe equations are solved fully implicitly (backward Euler in time) using the finite volume method\ndescribed by\nKarimi-Fard et al. (2004) . Time step duration is chosen adaptively during the simulation.\nFollowing the method of Karimi-Fard et al. (2004) , fractures are meshed as one-dimensional line\nsegments, and the surrounding matrix is meshed with triangles. Karimi-Fard et al. (2004) described the\nmethod for calculating geometric transmissibility factors for calculating flow between elements. Thesegeometric transmissibility factors calculate an effective \u201carea available for flow between two elements\u201d4 SPE-170956-MS"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87ab90-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 4, "chroma:document": "and \u201cdistance for flow between two elements\u201d in such a way that the method is convergent to the exact\nsolution.\nFor a matrix element, the local mass density is defined as the product of fluid density and porosity. For\na finite volume element, this mass density is converted to mass contained in the element (kg) bymultiplying by element area and formation height. Fluid density and matrix porosity are given by theconstitutive relationships:\n(5)\n(6)\nwhere /H9267initand/H9278initare the fluid density and porosity at the initial pressure Pinit, and cfandc/H9278are the\ncompressibility of the fluid and matrix (assumed constant).\nIn fractures, the local mass density can be defined as simply the fluid density (since the \u201cporosity\u201d\nwithin the fracture can be considered to be unity). To calculate the mass content of a finite volume fractureelement (kg), the fluid density is multiplied by element length, formation height, and the void aperture,\nE."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87abf4-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 4, "chroma:document": "E.\nNonlinear relationships are used to relate pressure, stress, and deformation to void aperture and\nhydraulic aperture. Before describing these relations, it is necessary to define some terminology. Wedefine a \u201cclosed\u201d fracture as a fracture where the fluid pressure is less than the normal stress and the wallsof the fracture are in contact. Roughness of the fracture walls allows closed fractures to retain hydraulicand void aperture. We define an \u201copen\u201d fracture as a fracture where the fluid pressure has reached thenormal stress and the walls are no longer in contact.\nFor calculating the aperture of closed fracture elements, CFRAC uses the following joint closure\nrelation, which was used by\nWillis-Richards et al. (1996) and was based on the work of Barton et al.\n(1985) :\n(7)\nwhere E0and/H9268n,Eref are specified constants that characterize the joint. E0is the residual void aperture"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87ac4e-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 4, "chroma:document": "(1985) :\n(7)\nwhere E0and/H9268n,Eref are specified constants that characterize the joint. E0is the residual void aperture\nand/H9268n,Eref is the 90% closure stress for void aperture. /H9268n=is the effective normal stress, equal to the\nnormal stress minus fluid pressure.\nWe use the same relation with different constants to calculate hydraulic aperture:\n(8)\nIf a fracture element is open, then its aperture is equal to its residual aperture (aperture at normal stress\nof zero) plus the mechanical opening that has occurred. The aperture at zero normal stress is e0andE0.\nIn this case, aperture is defined as:\n(9)\n(10)\nwhere Eopenis mechanical opening, the amount of physical separation between the fracture walls that\nhas occurred during opening. For hydraulic fracture elements, the value of E0is permitted to evolve during\nthe simulation (and is different for each element), as described in Section 2.1.3."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87aca8-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 4, "chroma:document": "the simulation (and is different for each element), as described in Section 2.1.3.\nThe problem is solved using a conforming mesh created using the publicly available program Triangle\n(Shewchuk, 1996 ). Fracture elements consist of line segments and matrix elements consist of triangles.\nThe mesh is created so that the fracture element length is constant, ensuring the accuracy of themechanical calculations.SPE-170956-MS 5"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87ad8e-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 5, "chroma:document": "For the simulations in this study, there was a single hydraulic fracture and a wellbore, represented as\na line segment. The problem domain was 70 m long in the direction of fracture propagation and 15 meterswide in the direction perpendicular to fracture propagation. The domain was meshed into roughly 68,000matrix elements and roughly 250 fracture elements. The path of the propagating hydraulic fracture wasassumed in advance, linear and perpendicular to the minimum principal stress. By specifying thepropagation path in advance, the entire problem domain could be meshed prior to the simulation andre-meshing during fracture propagation was unnecessary.\nAt the beginning of a simulation, the hydraulic fracture elements do not physically exist, but have"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87ade8-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 5, "chroma:document": "At the beginning of a simulation, the hydraulic fracture elements do not physically exist, but have\nalready been meshed and are treated as \u201cinactive.\u201d As the fracture initiates and propagates along thepredetermined path, elements are converted from inactive to active. Inactive elements are treated asnon-existent and omitted from the system of equations.\nThe edges of the domain were assumed to be no-flow boundaries. The simulations were run for a short\nenough duration that the edges of the model did not have an effect on the results. The well was a lineoriented perpendicular to the direction of fracture propagation with a length of four meters. The fracturewas assumed to initiate at the center of the well (which is also the center of the problem domain).\nFigure\n1shows a zoomed in image of the center of the problem domain (the figure shows only a part of the\nproblem domain).\nMechanical calculations"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87ae42-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 5, "chroma:document": "Figure\n1shows a zoomed in image of the center of the problem domain (the figure shows only a part of the\nproblem domain).\nMechanical calculations\nIn addition to solving the fluid flow equations, CFRAC calculates the stresses induced by fracturedeformation - both opening and sliding. In the simulations for this study, there was no fracture slidingbecause there was only a single hydraulic fracture, which was linear and perpendicular to the minimumprincipal stress. Therefore, the treatment of fracture sliding will not be described in this section.\nThe stresses are calculated using the boundary element method of\nShou and Crouch (1995) , which uses"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87ae9c-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 5, "chroma:document": "The stresses are calculated using the boundary element method of\nShou and Crouch (1995) , which uses\nthe displacement discontinuity (boundary element) method using quadratic basis functions assuming anelastically homogeneous and isotropic formation, linear elastic deformation, and small strain. Themechanical boundary condition is that the fracture(s) are embedded in an infinite whole space. Thetechnique satisfies the equations of classical continuum mechanics: quasistatic stress equilibrium, thecompatibility equations, and the linear elastic constitutive relation. Calculation results are fully convergentto analytical solutions for crack deformation (such as\nSneddon, 1946 ). The calculations are fully"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87aeec-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 5, "chroma:document": "Sneddon, 1946 ). The calculations are fully\nnumerical, enabling complex problems to be solved (for example, arbitrarily varying fluid pressure alongthe fracture length). An advantage of the displacement discontinuity method is that it is only necessary tomesh the fractures themselves, not the area around the fractures, to perform stress calculations.\nFigure 1\u2014An image of the center of the problem domain (the full problem domain is larger). The blue triangles are the matrix elements. The\nhorizontal red line is the path traveled by the propagating hydraulic fracture and the vertical red line is the wellbore.6 SPE-170956-MS"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87af50-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 6, "chroma:document": "For an open element, the stress boundary condition is that effective normal stress must be equal to zero:\n(11)\nwhere\n is the normal stress on the fracture due to remote loading and /H9004/H9268nis the change in normal\nstress due to backstress created by the deformation of all the fracture elements in the system.\nIn addition to calculating the stresses induced by the opening deformation of open elements, CFRAC\nhas the capability to calculate the stresses induced due to normal deformations (changes of E) of closed\nelements (the cstress option in McClure and Horne, 2013 ). This capability was used in the simulations\nperformed for this study. For brevity, the numerical details are omitted here, and they can be found in\nMcClure and Horne (2013) . Normal deformations of closed elements are small and the resulting induced\nstresses have only a minor effect on simulation results.\nThe compliance (derivative of aperture with respect to pressure) of open fractures is not explicitly"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87afaa-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 6, "chroma:document": "The compliance (derivative of aperture with respect to pressure) of open fractures is not explicitly\ndefined. Instead, the compliance behavior of the fracture is a property that arises emergently from thecoupling of fluid pressure and deformation in the mechanical network. When fractures are closed, theircompliance is primarily controlled by\nEquation 7 , but also affected by the stresses caused by normal\ndisplacement of the fracture.\nPoroelastic behavior in the matrix is not included, which means that stresses induced by fracture\ndeformation are only calculated on other fractures, not throughout the matrix and that changes in fluidpressure of the matrix do not cause stress changes in the fracture.\nA code called Hmmvp (\nBradley, 2011 ) is used to very accurately and efficiently approximate the"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87b004-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 6, "chroma:document": "A code called Hmmvp (\nBradley, 2011 ) is used to very accurately and efficiently approximate the\nmatrices of interaction coefficients arising from the displacement discontinuity method. This improves theefficiency and scaling of the method and enables much larger fracture networks to be simulated (thoughin this study, only a single, relatively small fracture was simulated).\nThe\nShou and Crouch (1995) method assumes plane strain deformation, implying infinite fracture size\nin the out-of-plane dimension. In this study, the Olson (2004) adjustment was used to adjust the stress\ncalculations to account for the finite fracture height.\nThe nonlinear system of equations is solved using a specially developed iterative method with iterative\ncoupling between (1) opening and fluid flow and (2) sliding displacements. Within the nonlinear solver,the sparse matrix linear solver UMFPACK is used (\nDavis, 2004 ), which also uses BLAS ( Lawson et al.,"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87b072-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 6, "chroma:document": "Davis, 2004 ), which also uses BLAS ( Lawson et al.,\n1979 ;Dongarra et al., 1988 ), AMD ( Amestoy et al., 2004 ), and LAPACK ( Anderson et al., 1999 ).\nFracture propagation is determined using linear elastic fracture mechanics. The stress intensity factor\nat each crack tip, KI,is numerically evaluated using the method of Olson (2007) and if it exceeds the\nfracture toughness, KIc(this is specified material property), the fracture extends by a single element (per\ntimestep).\nThe simulations in this study used arithmetic averaging of transmissivity between adjacent elements.\nFor fluid properties such as viscosity and density, upwinding is used when calculating properties for flowbetween elements.\nResidual aperture of a hydraulic fracture\nThere are unique challenges for defining the aperture of newly forming hydraulic fractures. For this study,different constitutive relations for the aperture of newly formed fractures were used than in past versionsof CFRAC ("}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87b0c2-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 6, "chroma:document": "McClure and Horne, 2013 ). When an element first forms at the tip of the fracture, it is given\na very small residual void aperture, 10 microns. We define \u201cresidual aperture\u201d as the aperture wheneffective normal stress is equal to zero, which is E\n0. Because the elements are initiated with a very small\nresidual aperture, the algorithm does not perfectly conserve mass. However, because the initial apertureis very small, the error is slight.\nA special algorithm is applied to allow residual aperture to increase as a hydraulic fracture opens.\nTherefore, each element may have a different E\n0, and it can evolve during the simulation. As shownSPE-170956-MS 7"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87b11c-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 7, "chroma:document": "below, if the residual aperture of the elements is kept constant at 10 microns, the simulation results during\nclosure are very unrealistic. But the elements cannot be initiated with a larger residual aperture becausethis would create a significant mass balance error. So an algorithm is needed that allows elements to havevery low residual aperture initially, but for their residual aperture to increaes as they open. This algorithmis an attempt to mimic the natural formation of fracture roughness as a crack forms and opens. Thisprocess does not appear to be investigated in detail in the literature, and so we are using an approach thatis somewhat ad-hoc, but physically plausible and necessary for realistic model behavior.\nThe percentage of total opening retained as residual aperture is defined as the \u201cresidual aperture"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87b176-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 7, "chroma:document": "The percentage of total opening retained as residual aperture is defined as the \u201cresidual aperture\npercentage.\u201d It is not clear what value to use for the residual aperture percentage. For most simulations,we used 50%, but this was treated as a sensitivity analysis parameter and turned out to have a very strongeffect on simulation results. Presumably, as a fracture opens more and more, there will be some limitingvalue for which further opening does not increase residual aperture further. We have not defined a limitingresidual aperture, but this was not necessary because the maximum opening in any simulation performedin this study was only 0.6 mm, which is the same order of magnitude as a plausible residual aperture fora fracture.\nHydraulic aperture of fracture elements was treated differently than void aperture. For all hydraulic\nfracture elements, regardless of the residual void aperture, the residual hydraulic aperture, e\n0, was set to"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87b1d0-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 7, "chroma:document": "fracture elements, regardless of the residual void aperture, the residual hydraulic aperture, e\n0, was set to\nbe constant at 20 microns. When fracture elements opened, their hydraulic aperture could grow higherthan e\n0due to the separation of the fracture walls, as defined by Equation 9 .\nThe fluid flow calculations were validated by comparing against analytical solutions to the diffusivity\nequation for one-dimensional flow and radial flow. Also, the DFIT simulations shown in this paper wereperformed to very late time (using a larger problem domain than used for the simulations in this study)so that late time radial flow could be identified in the pressure transient. Plots of pressure versus theinverse of time were used to calculate permeability (using the method of\nGu et al., 1993 ) and to confirm\nthat the calculated permeability was consistent with the permeability specified as a model input.Validation of the mechanical calculations was described by\nMcClure and Horne (2013) ."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87b220-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 7, "chroma:document": "McClure and Horne (2013) .\nWellbore storage\nWellbore storage was newly implemented in CFRAC for this study. During injection, a constant injectionrate was imposed in the simulation. This was accomplished by adding an additional equation to the systemof equations, specifying that the sum of the flow rates between the well and the adjacent fracture elementsmust sum to the specific injection rate. To include wellbore storage, the equation was modified slightly:\n(12)\nwhere qis the specified mass injection rate, fwiis the mass flow rate between the well and an adjacent\nfracture element i, Cwis the wellbore storage coefficient (m3/MPa), /H9267wis the density of the injection fluid,\nandNis the number of elements connected to the wellbore. The equation is expressed in units of"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87b27a-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 7, "chroma:document": "andNis the number of elements connected to the wellbore. The equation is expressed in units of\nkilograms per second. The wellbore storage term accounts for the fluid stored in the wellbore due to fluidcompressibility. The wellbore storage coefficient is equal to the product of wellbore volume and fluidcompressibility. If a surface shut-in is used during the DFIT test, the wellbore storage may be significantenough to affect the pressure transient, even if the well is full of only water, which has low compress-ibility. This is especially true because the volume of fluid injected during a DFIT test is so small, only afew barrels. Wellbore storage may be minimized if a downhole shut-in tool is used (\nCramer and Nguyen,\n2013 ).\nAt the beginning of the simulation, the wellbore fluid pressure is equal to the initial fluid pressure. The"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87b2d4-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 7, "chroma:document": "Cramer and Nguyen,\n2013 ).\nAt the beginning of the simulation, the wellbore fluid pressure is equal to the initial fluid pressure. The\nhydraulic fracture cannot initiate until the fluid pressure has exceeded the minimum principal stress. Ifthere is no wellbore storage, the fluid pressure will rise instantaneously to the minimum principal stress,forcing the hydraulic fracture to initiate. But with wellbore storage, the initial injection into the wellbore8 SPE-170956-MS"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87b360-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 8, "chroma:document": "will be taken up by wellbore storage and fluid pressure will not rise instantaneously. As long as Cwis\nnonzero, Equation 12 can be satisfied even if the hydraulic fractures have not yet formed.\nIn all simulations except one, it was assumed that there is no hydraulic connection between the\nwellbore and the matrix. This setting was intended to mimic a DFIT test in a perforated, cased wellbore.In one of the simulations, fluid flow was allowed directly between the wellbore and the adjacent matrix(as shown in\nFigure 1 ).\nDetails of the simulations\nResults of eleven simulations are presented in this paper. The simulations were designed to mimic a DFITtest. Injection was performed for five minutes at 1 kg/s, and then the well was shut-in. The total simulationduration was ten hours.\nThe simulation parameters used in all the simulations are given in\nTable 1 . The specific differences\nbetween the simulations are given in Table 2 . The simulations are named BC(base case), LP(low"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87b3c4-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 8, "chroma:document": "Table 1 . The specific differences\nbetween the simulations are given in Table 2 . The simulations are named BC(base case), LP(low\npermeability), HP(high permeability), LRAP (low residual aperture percentage), HRAP (high residual\naperture percentage), LCS (low closure stress), HCS (high closure stress), LH(low height), HWS (high\nwellbore storage), HRAP-HWS (high residual aperture percentage and high wellbore storage), and\nCMT (conventional minifrac test).\nSimulation BCwas designed to be representative of a DFIT test in a low permeability formation. As\ndiscussed in Section 3, the shut-in response of Simulation BCdoes not quite appear realistic, and several\nmodifications were investigated to make the result more realistic. The permeability was 10 /H9262D, the initial\nfluid pressure was 18 MPa, and the minimum principal stress was 28 MPa. Estimates of fracture closurepressure should therefore be close to 28 MPa."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87b414-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 8, "chroma:document": "fluid pressure was 18 MPa, and the minimum principal stress was 28 MPa. Estimates of fracture closurepressure should therefore be close to 28 MPa.\nExcept for Simulations CMT andLH, all simulations were the same as Simulation BCexcept for\nvarying a single parameter. Simulation CMT had three parameters varied from the base case: much higher\npermeability, a direct connection between the well and the matrix, and a modest wellbore storagecoefficient. These parameters were chosen to represent a minifrac test in a conventional reservoirTable 1\u2014Simulation settings used in all simulations.\n/H9271p(Poisson\u2019s ratio) 0.25 G(shear modulus) 15 GPa\nPinit(initial fluid pressure) 18 MPa /H9268yy(initial y-direction principal stress) 28 MPa\nh(fracture/formation height) 15 m KIc(fracture toughness) 3 MPa-m1/2\ne0(reference hydraulic aperture, Equation 8 )2 0 /H9262m /H9262(fluid viscosity) 1 cp\n/H9267init(initial fluid density) 1000 kg/m3cf(fluid compressibility) 0.000458 MPa-1"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87b46e-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 8, "chroma:document": "/H9267init(initial fluid density) 1000 kg/m3cf(fluid compressibility) 0.000458 MPa-1\n/H9278init(initial porosity) 0.03 c/H9278(pore volume compressibility) 0.00145 MPa-1\nEhfresid,min (minimum residual aperture) 10 /H9262m\nTable 2\u2014Simulation settings specific to the different simulations.\nSimulation name BC LP HP LRAP HRAP LCS HCS LH HWS HRAP-HWS CMT\n/H9268n,Eref (90% closure stress for void\naperture) (MPa)20 20 20 20 20 5 90 20 20 20 20\n/H9268n,eref (90% closure stress for\nhydraulic aperture) (MPa)20 20 20 20 20 5 90 20 20 20 20\nResidual aperture percentage 50% 50% 50% 0% 90% 50% 50% 50% 50% 90% 50%\nCw(wellbore storage coefficient)\n(m3/MPa)00000000 0.01 0.01 0.003\nk(permeability) ( /H9262D) 10 1 100 10 10 10 10 10 10 10 1000\nh(fracture/formation height) (m) 15 15 15 15 15 15 15 2.5 15 15 15\nHydraulic connection from well to\nmatrixNo No No No No No No No No No Yes\nInjection rate (kg/s) 1111111 0 . 5 111SPE-170956-MS 9"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87b4d2-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 9, "chroma:document": "(relatively high permeability) and with an open-hole\ncompletion. Simulation LHhad a lower fracture\nheight, and also had a lower injection rate (for thesame injection duration).\nResults and Discussion\nSimulation BC\nSimulation BC before closure Figure 3 shows a\nclose view of the G*dP/dG and pressure curvesprior to closure. Following convention, in all G-function plots, the derivative terms dP/dG andG*dP/dG are plotted as positive. At the beginningof the transient, there was an initial slight upwardmovement of G*dP/dG, which was caused by asmall amount of tip extension that occurred afterclosure. Then the curve formed a straight line, asexpected for a DFIT test before closure (\nCastillo,\n1987 ). The total length of the hydraulic fracture that\nformed was 28.75 m.\nBased on the slope of the plot of pressure versus\nG-time prior to closure, it should be possible toestimate the leakoff coefficient, C\nL(Equation 9 -58\nfrom Economides and Nolte, 2000 ):\n(13)"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87b522-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 9, "chroma:document": "G-time prior to closure, it should be possible toestimate the leakoff coefficient, C\nL(Equation 9 -58\nfrom Economides and Nolte, 2000 ):\n(13)\nwhere his the fracture height, p*is the slope of\npressure versus G-time, E=is equal to the Young\u2019s\nmodulus divided by one minus the square of Pois-son\u2019s ratio (equal to 40 GPa), and t\npis the time of\nproduction (in our case, representing the duration ofinjection). We have used the form of the equationappropriate for the PKN model, a Newtonian injection fluid, and a fracture that has uniform leakoff alongits face. From\nFigure 3 , the value of p*was equal to roughly 0.1 MPa/G-time. Using this value and the\nvalues from Table 1 , we calculate that the leakoff coefficient to be 2.2 /H1100310-6m3/(m2-s).\nThe simulation was not strictly a PKN model. It used the Olson (2004) adjustment to account for"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87b57c-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 9, "chroma:document": "The simulation was not strictly a PKN model. It used the Olson (2004) adjustment to account for\nformation height, meaning that if fracture length is much greater than height, it will behave like the PKNmodel and if length is much less than height it will behave like the KGD model. The half-length wasroughly equal to the formation height, so neither the KGD nor PKN model would be expected to perfectlyapply.\nBecause the formation properties are known, we can calculate the leakoff coefficient from the\ndefinition, for a fracture with no filter cake (\nHoward and Fast, 1957 ):\n(14)\nwhere/H9004Pis the increase in fluid pressure above the formation fluid pressure. The values in Equation\n14can be read from Table 1 andTable 2 , except /H9004P, which was taken to be 10.4 MPa. Using Equation\n14, the leakoff coefficient can be calculated as 4.4 /H1100310-6m3/(m2-s). This value is in reasonable agreement\nFigure 2\u2014G-function plot from Simulation BC."}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87b5cc-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 9, "chroma:document": "Figure 2\u2014G-function plot from Simulation BC.\nFigure 3\u2014Zoomed in plot of G*dP/dG and pressure at early time from\nSimulation BC.10 SPE-170956-MS"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87b626-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 10, "chroma:document": "with the estimate from the pressure transient using Equation 13 , considering the variety of approximations\nused in deriving and applying Equation 13 .\nSimulation BC after closure A correct pick of the closure stress should be roughly around the minimum\nprincipal stress, 28 MPa. Figure 2 andFigure 3 show that in Simulation BC, pressure was equal to 28 MPa\nat a G-function time around 2. The most striking feature in Figure 3 is that when closure occurs, the\npressure begins to drop much more rapidly, causing a large spike in G*dP/dG. In field data, the trend inthe decline of pressure with time may accelerate or decelerate following closure (\nNolte, 1988 ). In low\npermeability formations, it is very common to see a concave up G*dP/dG plot, indicating the decline inpressure during shut-in is accelerating (\nSoliman and Kabir, 2012 ;Cramer and Nguyen, 2013 ). The\ncommon interpretation of this data is that closure occurs later, near the top of the peak in G*dP/dG ( Barree"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87b680-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 10, "chroma:document": "common interpretation of this data is that closure occurs later, near the top of the peak in G*dP/dG ( Barree\net al., 2007 ;Soliman and Kabir, 2012 ;Cramer and Nguyen, 2013 ), but as discussed below, this would not\nbe the correct interpretation of the data generated in our simulations.\nIn Simulation BC, the G*dP/dG plot increased more sharply than is typically observed in the field. In\nSections 3.3, 3.4, and 3.6, we discuss how residual aperture, closure stress, and fracture height affect thesharpness of this transition. Despite the rather unrealistically sharp increase in derivative, the result inSimulation BCdemonstrates an important physical process, and later simulations show how the settings\ncould be modified to create a more realistic looking simulation result.\nFigure 3 shows a close view of the preclosure plot of pressure and G*dP/dG in Simulation BC.A s\npredicted by Castillo (1987) based on the solution of Nolte (1979) , pressure makes a straight line (and so"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87b6d0-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 10, "chroma:document": "predicted by Castillo (1987) based on the solution of Nolte (1979) , pressure makes a straight line (and so\ntherefore G*dP/dG should also form a straight line).\nThe sharp increase in G*dP/dG occurred because the fracture compliance dramatically dropped when\nclosure occurred. The effect of changes in fracture compliance on the pressure derivative can be bestexplained through an application of the chain rule:\n(15)\nwhere Vis the volume of fluid in the fracture,\n is the average pressure in the fracture, and tis time.\nFor simplicity, we are assuming that the average fracture pressure is equal to the pressure measured at thewellbore. The first term,\n, is proportional to the reciprocal of the compliance of the fracture. The second\nterm,\n , is the leakoff rate from the fracture. The leakoff rate is proportional to the inverse of the square\nroot of time at early time, but is also a function of pressure, decreasing as fracture pressure decreases. As"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87b72a-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 10, "chroma:document": "root of time at early time, but is also a function of pressure, decreasing as fracture pressure decreases. As\nfracture closure occurs, the compliance drops abruptly, causing a rapid increase in the magnitude of\nand\n . Eventually, the leakoff rate decreases sufficiently (encouraged by the lowering fluid pressure\nin the fracture), and the magnitude of the pressure derivative drops back down. An inflection point in\npressure occurs at the maximum value of the magnitude of the pressure derivative, which occurs when thesecond derivative of pressure is zero, implying:\n(16)\nAccording to classical analytical solutions, the compliance of an open fracture (walls out of contact)\nis very high and proportional to the fracture size. For example, for an open, circular crack with constantfluid pressure, the relationship between maximum aperture and pressure is (\nSneddon, 1946 ):\n(17)\nwhere Ris radius, Pis pressure, /H9271pis Poisson\u2019s ratio, and EYoung\u2019s Modulus is the Young\u2019s modulus. For"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87b7f2-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 10, "chroma:document": "Sneddon, 1946 ):\n(17)\nwhere Ris radius, Pis pressure, /H9271pis Poisson\u2019s ratio, and EYoung\u2019s Modulus is the Young\u2019s modulus. For\na crack with radius of 20 m, Poisson\u2019s ratio equal to 0.25, and Young\u2019s modulus equal to 37.5 GPa(equivalent to shear modulus of 15 GPa, as used in our simulations), the compliance at fracture center,\n,\nis equal to 0.64 mm/MPa.SPE-170956-MS 11"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87b84c-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 11, "chroma:document": "Figure 4 shows plots of aperture and compliance\n(the derivative of aperture with respect to pressure)versus effective normal stress based on\nEquation 7\nforE0equal to 0.5 mm and a value of /H9268n,Eref equal\nto 5 MPa and 20 MPa. Figure 4 makes clear that\nfracture closure results in a sharp drop in fracturecompliance. If\n/H9268n,Eref is low enough, compliance\nmay initially remain high as closure begins, butclosure will drop rapidly after effective normalstress has reached several MPa. If\n/H9268n,Eref is higher,\ncompliance will drop less rapidly as effective nor-mal stress increases, but there will be a more abruptdrop in compliance when closure occurs.\nIn Simulation BC,\n/H9268n,Eref was equal to 20 MPa.\nFigure 4 shows that for this value, the derivative of"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87b8a6-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 11, "chroma:document": "In Simulation BC,\n/H9268n,Eref was equal to 20 MPa.\nFigure 4 shows that for this value, the derivative of\naperture with respect to pressure will be 0.2 mm/MPa, sharply lower than the compliance of the openfracture and continuing to decline rapidly. This ex-plains why the pressure derivative increases sharplywhen fracture closure occurs. As discussed below,the fact that such a sharp increase in pressure de-rivative is not usually observed suggests either thepost-closure compliance must be higher than inSimulation BCor the pre-closure compliance must\nbe lower than in Simulation BC.\nBarree et al. (2007) suggested that a concave up\nG*dP/dG plot could be caused by fracture heightrecession or opening of transverse fractures. Ouranalysis suggests that the upward curving figure is anatural result of the decreasing fracture complianceas the fracture closes, and does not necessarily in-dicate fracture height recession or opening of trans-verse fractures.\nEffect of matrix permeability"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87b8f6-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 11, "chroma:document": "Effect of matrix permeability\nDecreasing matrix permeability delays closure, ex-tending the duration of the pre-closure straight lineon the G*dP/dG curve. However, in both Simula-tionHPandLP, a sharp increase in G*dP/dG oc-\ncurred only after closure (\nFigure 5 andFigure 6 ).\nEffect of residual aperture percentage\nThe residual aperture percentage has a huge effecton the shape of the G*dP/dG curve (\nFigure 7 and\nFigure 8 ). A large spike occurred in the G*dP/dG\nplot for Simulation LRAP. If residual aperture per-\ncentage is zero, then the residual aperture whenclosure occurs is very tiny, 10\n/H9262m (the default initial\nvalue used by the model). Equation 7 implies that\nFigure 4\u2014Aperture (blue lines) and dE/d /H9268n=(red lines) versus effective\nnormal stress for E0equal to 0.5 mm and /H9268n,Eref equal to 5 MPa (solid\nlines) and 20 MPa (dashed lines).\nFigure 5\u2014G-function plot from Simulation LP.\nFigure 6\u2014G-function plot from Simulation HP.12 SPE-170956-MS"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87b950-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 12, "chroma:document": "the derivative of aperture with respect to pressure (for a closed fracture) is directly proportional to the\nresidual aperture. Therefore, if residual aperture is very small, the contrast in compliance that occurs withclosure is huge, causing an extremely sharp spike in the G*dP/dG curve. This behavior is not realistic, butthe result from Simulation LRAP is instructive to illustrate the effect of the compliance contrast during\nclosure.\nSimulation HRAP had a high residual aperture percentage, 90%. The maximum opening was around\n0.64 mm, tapering to around 0.2 mm near the tip, and so the residual aperture in the fracture was 90% ofthose values. Apparently, a residual aperture of 90% was high enough to prevent an abrupt drop incompliance when closure occurred.\nFigure 8 shows that the result from this simulation looked like a\ntypical DFIT curve from a low permeability formation, with the plot of G*dP/dG gently curving up.\nEffect of closure stress"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87b9a0-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 12, "chroma:document": "typical DFIT curve from a low permeability formation, with the plot of G*dP/dG gently curving up.\nEffect of closure stress\nFigure 9 andFigure 10 show the effect of changes in closure stress on the G-function plot. As shown in\nFigure 4 , a lower closure stress means that the fracture has a higher compliance when closure begins. As\na result, when closure starts, the G*dP/dG curve increases less abruptly when closure stress is low. InSimulation HCS, the closure stress was 90 MPa, and so compliance at closure was very low, and the\ncontrast in compliance at closure was extreme, causing a very abrupt increase in the G*dP/dG curve.\nFigure 7\u2014G-function plot from Simulation LRAP.\nFigure 8\u2014G-function plot from Simulation HRAP.\nFigure 9\u2014G-function plot from Simulation LCS.\nFigure 10\u2014G-function plot from Simulation HCS.SPE-170956-MS 13"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87b9fa-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 13, "chroma:document": "Effect of wellbore storage\nThe effect of wellbore storage on the pumping pe-riod can be seen in\nFigure 11 , which shows the\ninjection pressure during pumping for SimulationBCandHWS . In Simulation BC, injection pressure\ninstantaneously rose above the minimum principalstress and initiated a hydraulic fracture. In Simula-tion HWS , injection for nearly two minutes was\nrequired before the fluid pressure has reached theminimum principal stress. For the first two minutesof pumping, the hydraulic fracture had not yetformed and all injected fluid was being stored in thewellbore.\nIn both simulations, a breakdown pressure was"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87ba7c-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 13, "chroma:document": "In both simulations, a breakdown pressure was\nobserved modestly above the minimum principalstress. As a model assumption, the fracture initiatedwhen the fluid pressure reached the minimum prin-cipal stress (to avoid mesh dependence, the fracturewas initiated at a half-length of 1 meter). But inorder for the fracture to propagate, the fluid pressureneeded to rise high enough that the stress intensityfactor at the tips reached the fracture toughness.Once this occurred, injection pressure dropped be-cause as the fracture grew, a lower fluid pressurewas required for the stress intensity factor to reachthe fracture toughness.\nThe effect of wellbore storage was to decrease\nthe rate at which pressure declined (\nFigure 12 and\nFigure 13 ). Other than wellbore storage, Simulation\nHWS was identical to Simulation BC. In Simulation\nBC, there was very little drop in pressure prior to"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87bad6-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 13, "chroma:document": "HWS was identical to Simulation BC. In Simulation\nBC, there was very little drop in pressure prior to\nclosure, and then an abrupt drop in pressure afterclosure. Wellbore storage has an effect when thepressure in the wellbore is changing (\nEquation 12 ),\nso in Simulation HWS , the effect of wellbore storage\nwas felt primarily after closure. It is interesting tonote that in conventional pressure transients, pres-sure changes most rapidly at the beginning of thetransient, and so wellbore storage is only importantat early time. But in a DFIT test, pressure maychange most rapidly after closure, and also the totalvolume of fluid in the formation may not be muchlarger than the volume expelled due to wellborestorage. As a result, wellbore storage can affect theentire pressure transient response, not just the be-ginning.\nWellbore storage causes fluid to flow out of the well into the formation as the pressure draws down.\nA wellbore storage coefficient of 0.01 m"}


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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 13, "chroma:document": "Wellbore storage causes fluid to flow out of the well into the formation as the pressure draws down.\nA wellbore storage coefficient of 0.01 m\n3/MPa is equivalent to 10 kg/MPa. The total mass of fluid injected\nFigure 11\u2014Injection pressure versus time during Simulation BC (blue)\nand Simulation HWS (red).\nFigure 12\u2014G-function plot from Simulation HWS.\nFigure 13\u2014G-function plot from Simulation HRAP-HWS.14 SPE-170956-MS"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87bb80-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 14, "chroma:document": "was 300 kg, and the increase in pressure from in-\njection was a bit over 10 MPa, so in SimulationHWS , roughly 100 kg (one third of the total) was\nconsumed by wellbore storage. Therefore, at thetime of shut-in, only 200 kg had been injected intothe formation. After shut-in, as the pressure contin-ued to decline back to 18 MPa, most of the remain-ing 100 kg flowed out of the well into the formation.Unsurprisingly, the constant resupply of fluid intothe formation from the wellbore (due to storage)during shut-in slowed the rate of pressure decline,causing the G*dP/dG curve to have a graduallyincreasing trend for a long time after closure (\nFigure\n12).\nAfter closure, Simulation HRAP-HWS behaved\nvery similarly to Simulation HWS (Figure 13 ). However, prior to closure, the high residual aperture"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87bbda-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 14, "chroma:document": "12).\nAfter closure, Simulation HRAP-HWS behaved\nvery similarly to Simulation HWS (Figure 13 ). However, prior to closure, the high residual aperture\npercentage caused the fracture compliance to be much lower, preventing a very flat G*dP/dG curve. Thecombination of high residual aperture percentage and wellbore storage resulted in only a very subtle effectin the G*dP/dG curve when closure occurred (at around 29 MPa, a bit higher than the true minimumprincipal stress, due to the high residual aperture, as discussed in Section 3.3).\nAs an example comparison to field data, Simulations HRAP andHRAP-HWS appear similar to the\nDFIT field data in\nFigure 8 from Cramer and Nguyen (2013) . In their Figure 8 , \u201cTest 1\u201d was performed\nwith a surface shut-in (larger wellbore storage). The G*dP/dG curve shows a gradually increasing andthen eventually gradually decreasing trend, just as in Simulation HRAP-HWS (\nFigure 13 ). Their \u201cTest 2\u201d"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87bc2a-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 14, "chroma:document": "Figure 13 ). Their \u201cTest 2\u201d\nwas performed with a downhole shut-in (lower wellbore storage). For this test, the plot of G*dP/dG wasconcave upward and the more sharply peaked, followed by a steady decline. This test was qualitativelysimilar to Simulation HRAP (\nFigure 8 ).\nEffect of fracture height\nIn Simulation LH, the fracture height was reduced from 15 to 2.5 m. To maintain a comparable fracture\nlength, injection rate was reduced to 0.5 kg/s. The compliance of an open fracture fracture scales with thesmallest dimension, which for most simulations in this study (including Simulation LH) was height. Note\nthat for plane strain models, the fracture compliance would scale with length, but we used the\nOlson\n(2004) adjustment, which enabled the 2D model to behave more like a PKN model, and so compliance\nscaled with height. Therefore, because Simulation LHhad a lower height, it had a lower pre-closure"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87bc84-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 14, "chroma:document": "scaled with height. Therefore, because Simulation LHhad a lower height, it had a lower pre-closure\ncompliance. This resulted in a less flat G*dP/dG curve prior to closure ( Figure 14 ). The post-closure\ncompliance was low enough that there was still an abrupt increase in derivative after closure. However,from prior results, we can infer that if the closure stress had been lower (as in Simulation LC)o ri ft h e\nresidual aperture percentage had been higher (as in Simulation HRAP ), the transition would have been less\nabrupt or non-existent.\nConventional minifrac test"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87bcd4-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 14, "chroma:document": "abrupt or non-existent.\nConventional minifrac test\nAs discussed above, in DFIT tests in low permeability formations, an upward curving G*dP/dG plot iscommonly observed, indicating that pressure begins to drop more rapidly after closure. As discussed inSection 3.1, this occurs because the system compliance drops rapidly after shut-in. However, in manyminifrac tests, especially in permeable rocks, pressure drops less rapidly, not more rapidly, after closure.An example is given in Figure 7.3 of\nZoback (2007) .Nolte (1988) noted that after closure, the pressure\ntrend may begin to drop more or less rapidly, depending on conditions.\nFigure 14\u2014G-function plot from Simulation LH.SPE-170956-MS 15"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87bd2e-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 15, "chroma:document": "Simulation CMT was performed to demonstrate\nhow and why pressure may drop less rapidly afterclosure. Simulation CMT was different from the\nbase case because it had: (1) much higher formationpermeability, 1 mD, (2) a small but nonzero well-bore storage, and (3) a direct hydraulic connectionbetween the well and the formation (as if the wellwas completed open-hole).\nFigure 15 shows that for\nSimulation CMT , dP/dG was monotonically de-\ncreasing, with closure occurring rapidly, and afterclosure (which must have occurred around 28 MPabecause of the model setup), pressure dropped lessrapidly than before closure. In the other simulationsin our study, dP/dG increased post closure and thendecreased again, with sometimes a very short periodof high derivative at the beginning of closure due totip extension.\nThe different behavior of Simulation CMT can be understood as being caused by the competition"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87bd7e-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 15, "chroma:document": "The different behavior of Simulation CMT can be understood as being caused by the competition\nbetween storage and leakoff. In Simulation CMT , the much higher permeability caused the fracture that\nformed during pumping to be much smaller than in the other simulations in our study, with a total lengthof 3 m (as opposed to 28.75 m in Simulation BC). The compliance of an open fracture scales with the\nsmallest dimension (length in Simulation CMT ), and so not only was the fracture much smaller, it was also\nmuch less compliant.\nThus, in Simulation CMT , the hydraulic fracture was much smaller and much less compliant than in\nthe other simulations, and as a result the fracture had very little ability to store fluid, even before closure.Even with a rather small value of wellbore storage coefficient, 0.001 m\n3/MPa, the wellbore had greater"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87bdce-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 15, "chroma:document": "3/MPa, the wellbore had greater\nfluid storage capacity than the fracture that formed. As a result, when closure occurred, the overall systemcompliance (which includes the fracture and the wellbore) was not much affected.\nAnother effect of closure is to reduce the transmissivity of the fracture. In a formation with very low\npermeability, even a closed fracture may have high transmissivity relative to the matrix. But in a higherpermeability formation, the transmissivity of the fracture after closure may readily become negligiblerelative to the matrix. As a result, the closure of the hydraulic fracture reduces the rate at which fluid canleak out of the wellbore/hydraulic fracture system.\nOverall, in Simulation CMT , closure had a minor effect of the compliance of the wellbore/hydraulic"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87be28-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 15, "chroma:document": "Overall, in Simulation CMT , closure had a minor effect of the compliance of the wellbore/hydraulic\nfracture system but reduced the rate at which leakoff could occur from the fluid stored in the wellbore.The total effect was that closure reduced the rate at which pressure declined, unlike in the othersimulations.\nEstimating the minimum principal stress in low permeability formations\nBarree et al. (2007) described a variety of methods for interpreting reservoir behavior from G-function\nplots. He suggested that an upward curving G-function plot indicates \u201cstorage\u201d due to opening oftransverse fractures or fracture height recession. Our results indicate that the upward curving of thederivative curve is a natural consequence of the decreasing compliance of the fracture as it closes (thoughthis effect should only be apparent in low permeability formations).\nBarree et al. (2007) also suggested that when an upward curving G-function plot is apparent, the timing"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87be78-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 15, "chroma:document": "Barree et al. (2007) also suggested that when an upward curving G-function plot is apparent, the timing\nof fracture closure should be identified by drawing a tangent straight line from the origin to near the peakof the G*dP/dG curve. Then the pressure at that time should be evaluated to estimate the closure pressure.This method of interpretation would significantly underestimate the closure stress in nearly every\nFigure 15\u2014G-function plot from Simulation CMT. Note that the x-axis\nshows only G-time of 0 to 6, unlike the other figures.16 SPE-170956-MS"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87bec8-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 16, "chroma:document": "simulation we performed. In most simulations, this\nmethod would lead to an interpretation of the min-imum principal stress equal to around 24-25 MPa,lower than the true value of 28 MPa.\nOur simulations indicate that closure would\nproperly be picked at the bottom of the peak in theG*dP/dG curve. Specifically, in the early transient,the G*dP/dG curve makes a straight line (except fora very early feature indicating fracture tip exten-sion). When the curve deviates upward from thestraight line, closure has occurred.\nThe G-function plots from the simulations with\nhigh residual aperture percentage are consistentwith the shape of G-function plots from field data(\nFigure 8 andFigure 13 ). However, in these cases,\nit is less clear where to pick closure, because theupward deviation caused by closure is less sharp.Because of the residual aperture, the upward deviation appears to begin when the fluid pressure is a bitabove the true minimum principal stress ("}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87bf22-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 16, "chroma:document": "Figure 13 ). Because the walls begin to come in contact while\nthere is still a significant amount of residual aperture, the backstress on the fracture has not returnedentirely to zero (which would only entirely occur with full closure if the fracture aperture returned to zero).As a result, closure begins with the pressure modestly above the minimum principal stress.\nFigure 16 demonstrates the different ways to pick closure from the curve, either at the first deviation\nfrom the straight line in the G*dP/dG curve or at the intersection of the G*dP/dG curve with a tangent linedrawn from the origin. The actual minimum principal stress is 28 MPa.\nIn practice, we suspect that the backstress is unlikely to cause closure to begin at a pressure more than"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87bf7c-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 16, "chroma:document": "In practice, we suspect that the backstress is unlikely to cause closure to begin at a pressure more than\n0-2 MPa greater than the minimum principal stress. A more severe difficulty may arise if it is ambiguouswhere to identify where the \u201cupward deviation\u201d of the G*dP/dG curve occurs (as perhaps was the casein Simulation HRAP ). However, a relatively sharp increase in G*dP/dG (or alternatively, an abruptly more\nrapid drop in pressure) is often apparent in field data in unconventional formations.\nCastillo (1987) suggested picking the closure stress by plotting pressure versus the square root of time\nand then identifying the deviation from linearity. This method has been widely adopted, sometimesreplacing the G-function plot with a simpler plot of pressure versus the square root of time (\nZoback, 2007 )."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87bfd6-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 16, "chroma:document": "Zoback, 2007 ).\nIn the absence of tip extension, this method is essentially equivalent to what we have suggested (becausein most cases, G-function plots will give results that are reasonably similar to a square root of time plot).However, care would need to be taken to avoid picking closure at the end of the tip-extension period. Forexample, in\nFigure 3 , fracture closure would best be picked at G-time of 2, not at the end of the very early\ntime linear feature at G-time of roughly 0.1.\nA simple method for checking picks of the closure stress would be to perform a second, rather low rate"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87c026-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 16, "chroma:document": "time linear feature at G-time of roughly 0.1.\nA simple method for checking picks of the closure stress would be to perform a second, rather low rate\ninjection test at the end of the shut-in period. Because of hoop stress and the tensile strength of theformation, the fracture may not initiate at exactly the minimum principal stress during the first injectionperiod, making the leakoff point unreliable for estimating the minimum principal stress. However, oncethe fracture has formed, it should be relatively easy to reopen it, and fracture initiation effects will not beapparent. Even in the second test, hoop stress effects could theoretically cause a slight breakdown pressureto be apparent, but shortly after that breakdown, the fluid pressure should stabilize at a fracturepropagation pressure close to the minimum principal stress. The value of repeated injection periods forestimating the minimum principal stress has been emphasized by other authors (\nHickman and Zoback,\n1983 )."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87c08a-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 16, "chroma:document": "Hickman and Zoback,\n1983 ).\nFigure 16\u2014Comparison of two methods for identifying closure stress.\nDrawing a line tangent to G*dP/dG gives a closure stress around 25MPa. Identifying closure at the beginning of the upward deviation in theG*dP/dG curve gives a closure a bit less than 29 MPa.SPE-170956-MS 17"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87c102-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 17, "chroma:document": "Estimating the minimum principal stress in high permeability formations\nSimulation CMT was an example of a conventional minifrac test in a relatively higher permeability\nformation. In this test, the G*dP/dG curve was initially straight and then began to bend downward whenclosure occurred (indicating a more slowly decreasing fluid pressure). For this curve, the methodsuggested by\nCastillo (1987) , to pick closure at the deviation of linearity of the plot of pressure versus\nG-time, would give the correct closure, though it may be ambiguous exactly where to identify the\u201cdeviation from linearity.\u201d\nBarree et al. (2007) interprets a concave down G*dP/dG curve as indicating tip extension. Our results\nare consistent with that interpretation, as the very early time deviation in the G-function plot in Figure 3\ndid correspond to a limited degree of tip extension. However, our results suggest that care should be takenin interpreting tip-extension based on a concave down plot of G*dP/dG."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87c15c-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 17, "chroma:document": "Figure 12 ,Figure 13 , and Figure\n15show downward curving G*dP/dG curves. This behavior was caused by wellbore storage in the first\ntwo cases and the effect of closure in a higher permeability formation in the latter case. In the case offracture tip extension,\nBarree et al. (2007) cautions against picking the closure stress at the first break in\nlinearity of the G*dP/dG curve, suggesting this may lead to an early pick of the closure stress (see hisFigures 17 and 18). However, in Simulations HWS (\nFigure 12 ),HRAP-HWS (Figure 13 ), and CMT (Figure\n15), the correct pick for closure stress would be at the break in linearity of the G*dP/dG curve, and a later\npick would yield a substantial underestimate. Simulations HWS andHRAP-HWS show that wellbore\nstorage can cause the G*dP/dG curve to slowly trend upward for long after closure has occurred.\nImplications for actual fracture geometry\nIt is notable that in the base case simulation, Simulation BC, and in several of the other simulations,"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87c1ac-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 17, "chroma:document": "Implications for actual fracture geometry\nIt is notable that in the base case simulation, Simulation BC, and in several of the other simulations,\nclosure caused an extremely abrupt upward deviation in the G*dP/dG curve. While an upward trendingG*dP/dG curve is common in low permeability formations, such an abrupt deviation seems unrealistic andartificial. In some of our simulations, abrupt deviations did not occur, or the deviations were less abrupt.It is worth considering what this behavior suggests about the geometry of the fractures that are actuallycreated during DFIT tests in low permeability formations.\nAs described in Section 3.1, the analytical solution for the relation between pressure and aperture for\nan open crack suggests that cracks on the scale of meters or larger have very significant compliance. Thisis why such an abrupt increase in G*dP/dG is observed during closure, such as in Simulation BC. Because"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87c206-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 17, "chroma:document": "this feature is not typically observed in field data, an implication may be that open hydraulic fractures maynot be as compliant as classical fracture mechanics models suggest.\nFracture mineback (\nWarpinski and Teufel, 1987 ;Jeffrey et al., 2009 ), coring ( Warpinski et al., 1993 ),\nand laboratory experiments ( Blanton, 1982 ;Renshaw and Pollard, 1995 ;Bahorich et al., 2012 ;Suarez-\nRivera et al., 2013 ) suggest that real hydraulic fractures are complicated features with branching,\nnon-planarity, and termination against natural fractures ( Mahrer, 1999 ; Cipolla et al., 2008). All of these"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87c260-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 17, "chroma:document": "features would tend to decrease fracture compliance. If fluid is actually distributed into a larger numberof smaller features, the overall system compliance will be lower because compliance scales with fracturesize. The back stress on each fracture would tend to increase normal stress on adjacent fractures, furtherdecreasing system compliance. Non-planar fracture formation may result in the normal stress not beinghomogeneous along the newly formed fracture(s). As a result, closure could occur more gradually,because the fracture(s) must close against a distribution of different closure stresses. Stress heterogeneitymay also exist because of either the prior geological loading or heterogeneity in elastic properties in theformation. Stress heterogeneity would tend to smear out closure in the pressure transient response bycausing closure to occur across a distribution of normal stresses along the fracture faces. All of thesebehaviors may manifest themselves as an effectively \u201crougher\u201d"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87c2ba-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 17, "chroma:document": "to occur across a distribution of normal stresses along the fracture faces. All of thesebehaviors may manifest themselves as an effectively \u201crougher\u201d fracture, which may explain why ourmodel using a high residual aperture percentage appeared quite realistic.18 SPE-170956-MS"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87c30a-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 18, "chroma:document": "In short, the DFIT transient pressure response suggests a more complicated geometry than is classically\nassumed in hydraulic fracture modeling (including in this study). The idealized, homogeneous problemgeometry of our base case simulation created an unrealistically abrupt transition at closure, but introducingfeatures associated with heterogeneity and complexity into the model smooths the transition and causesit to appear more realistic. From the perspective of hydraulic fracture modeling, simplification isnecessary to make the problem tractable, but it is important to consider to what degree simplificationactually changes the model predictions and behaviors in a meaningful way.\nOther potential complications\nThere are several other potentially complicating issues that could cloud DFIT interpretation and deservegreater study in future work.\nDFITs involve injection of water and are frequently done in hydrocarbon bearing formations. The"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87c364-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 18, "chroma:document": "DFITs involve injection of water and are frequently done in hydrocarbon bearing formations. The\nG-function analysis is dependent on the assumption that leakoff scales with the square root of time. Thisscaling relationship is based on a single-phase, one-dimensional solution to the diffusivity equation.Nonlinear multiphase flow effects may have an unpredictable effect on the scaling of leakoff rate withtime, affecting the shut-in transient, especially in shale formations, where leakoff behavior is notwell-understood.\nAs noted in Section 3.10, direct hydraulic fracture observations have generally indicated complex\nfracture geometry. If multiple fractures form, if propagating fractures terminate and branch, or if otherforms of complex geometry develop, this could complicate DFIT interpretation significantly, potentiallycausing the minimum principal stress to be overestimated (\nMcClure, 2014 ).\nConclusion\nOur analysis leads to the following conclusions:"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87c3be-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 18, "chroma:document": "McClure, 2014 ).\nConclusion\nOur analysis leads to the following conclusions:\n1. In low permeability formations, a concave up deviation of the G*dP/dG curve from a linear trend\ncan be explained by decreasing fracture compliance due to fracture closure. It is not necessary toinvoke more complicated explanations such as height recession or transverse storage.\n2. Since we have shown that the concave up deviation is caused by a drop in fracture compliance,\nthe specific shape of the post fracture closure curve is affected by factors that affect fracturecompliance before and after shut-in, including the fracture height, the fracture 90% closure stress,and the residual aperture percentage (equivalent to the microscopic or macroscopic roughness ofthe created hydraulic fracture).\n3. Wellbore storage broadens the post-closure peak in the G*dP/dG curve. A curve of G*dP/dG that"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 18, "chroma:document": "3. Wellbore storage broadens the post-closure peak in the G*dP/dG curve. A curve of G*dP/dG that\nconstantly increases for a long time during the shut-in period may be caused by wellbore storage,rather than fracture tip extension.\n4. In higher permeability formations (with nonzero wellbore storage), the effect of fracture compli-\nance is less important. Closure decreases the rate of pressure decline because it reduces the rateat which fluid can leak off into the matrix from the wellbore.\n5. In all cases, the best pick for the minimum principal stress is at the deviation from linearity on the\nG*dP/dG plot. Closure picks based near the peak in G*dP/dG (as is commonly performed) maysignificantly underestimate the minimum principal stress. These picks should be confirmed withother measurements, such as a subsequent injection period.\n6. The fact that an extremely sharp upward trend in G*dP/dG is generally not observed in DFIT tests"}


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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 18, "chroma:document": "6. The fact that an extremely sharp upward trend in G*dP/dG is generally not observed in DFIT tests\nsuggests that the actual geometry of the created fracture is rougher and more complex than istypically assumed by standard models such as PKN and KGD.SPE-170956-MS 19"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 19, "chroma:document": "Acknowledgements\nThe authors would like to acknowledge the support provided by the corporations supporting our research\nthough the Hydraulic Fracturing and Sand Control JIP at the University of Texas at Austin.\nNomenclature\nA : matrix of interaction coefficients (MPa/m)\nCL : leakoff coefficient (m3/(m2-s))\nCw : wellbore storage coefficient (m3/MPa)\ncf : fluid compressibility (MPa-1)\nc/H9278 : pore volume compressibility (MPa-1)\nD : vector of displacement discontinuities (m)\nE : void aperture (m)\nE= : Young\u2019s moduls divided by one minus the square of Poisson\u2019s ratio (MPa)\nEhfresid,min : minimum residual aperture (m)\nEYoung\u2019s Modulus : Young\u2019s modulus (MPa)\nE0 : residual void aperture (m)\nEopen : mechanical separation between walls for an open element (m)\ne : hydraulic aperture (m)\ne0 : residual hydrualic aperture (m)\nfwi : mass flow rate between the well and an adjacent element (kg/s)\nG : shear modulus (MPa)\nh : fracture/formation height (m)\nKI : stress intensity factor (MPa-m1/2)"}


created_at: 2023-11-16 17:06:54
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 19, "chroma:document": "G : shear modulus (MPa)\nh : fracture/formation height (m)\nKI : stress intensity factor (MPa-m1/2)\nKIc : fracture toughness (MPa-m1/2)\nk : permeability (m2)\nm : mass density at a location (kg/m3)\nN : number of elements connected to the wellbore\nP : pressure (MPa)\n: average pressure (MPa)\nPinit : initial pressure (MPa)\np*: slope of the pressure versus G-time curve (MPa/G-time)\nq : mass flow rate (kg/s)\nqflux : mass flux (kg/(m2-s))\nR : radius of a circular crack (m)\nS : matrix of induced stresses (MPa)\ns : mass source term (kg/(m2-s))\nT : transmissivity (m3)\nt : time (s)\ntp : time of production (s)\nx : location (m)\nV : volume of fluid in the fracture (m)\n/H9004P : change in pressure above initial pressure (MPa)\n/H9004/H9268n : induced normal stress (MPa)\n/H9262 : fluid viscosity (MPa-s)\n/H9267 : fluid density (kg/m3)\n/H9267init : initial fluid density (kg/m3)\n/H9267w : density of fluid in the wellbore (kg/m3)20 SPE-170956-MS"}


created_at: 2023-11-16 17:06:54
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 20, "chroma:document": "/H9268n : normal stress (MPa)\n/H9268n= : effective normal stress (MPa)\n/H9268n,Eref : 90% closure stress for void aperture (MPa)\n/H9268n,eref : 90% closure stress for hydraulic aperture (MPa)\n: remote loading normal stress (MPa)\n/H9268yy : initial principal stress in the y-direction (MPa)\n/H9271p : Poisson\u2019s ratio (unitless)\n/H9278 : porosity (unitless)\n/H9278init : initial porosity (unitless)\nReferences\nAmestoy, Patrick R., Timothy A. Davis, Iain S. Duff. 2004. Algorithm 837: AMD, an approximate\nminimum degree ordering algorithm. ACM Transactions on Mathematical Software 30(3):\n381\u2013388, doi: 10.1145/1024074.1024081.\nAnderson, E., Z. Bai, C. Bischof et al. 1999. LAPACK Users\u2019 Guide, Third edition. Philadelphia, PA,\nSociety for Industrial and Applied Mathematics.\nBahorich, B., J. E. Olson, J. Holder. 2012. Examining the effect of cemented natural fractures on"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87c5c6-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 20, "chroma:document": "Society for Industrial and Applied Mathematics.\nBahorich, B., J. E. Olson, J. Holder. 2012. Examining the effect of cemented natural fractures on\nhydraulic fracture propagation in hydrostone block experiments. Paper SPE 160197 presented atthe SPE Annual Technical Conference and Exhibition, San Antonio, TX, doi: 10.2118/160197-MS.\nBarree, R. D., H. Mukherjee. 1996. Determination of pressure dependent leak-off and its effect on\nfracture geometry. Paper SPE 36424 presented at the SPE Annual Technical Conference andExhibition, Denver, CO, doi: 10.2118/36424-MS.\nBarree, R. D., V. L. Barree, D. P. Craig. 2007. Holistic fracture diagnostics. Paper SPE 107877\npresented at the SPE Rocky Mountain Oil & Gas Technology Symposium, Denver, CO, doi:10.2118/107877-MS.\nBarton, N., S. Bandis, K. Bakhtar. 1985. Strength, deformation and conductivity coupling of rock\njoints. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87c616-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 20, "chroma:document": "joints. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts\n22(3): 121\u2013140, doi: 10.1016/0148-9062(85)93227-9.\nBlanton, Thomas. 1982. An experimental study of interaction between hydraulically induced and\npre-existing fractures. Paper SPE 10847 presented at the SPE Unconventional Gas RecoverySymposium, Pittsburgh, Pennsylvania, doi: 10.2118/10847-MS.\nBradley, Andrew M. 2011. H-matrix and block error tolerances. arXiv:1110.2807, source code\navailable at\nhttps://pangea.stanford.edu/research/CDFM/software/index.html , paper available at\nhttp://arxiv.org/abs/1110.2807 .\nCastillo, J.L. 1987. Modified fracture pressure decline analysis including pressure-dependent leakoff.\nPaper SPE 164417 presented at the SPE/DOE Low Permeability Reservoir Symposium, Denver,CO, doi: 10.2118/16417-MS.\nCramer, D. D., D. H. Nguyen. 2013. Diagnostic fracture injection testing tactics in unconventional"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87c670-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 20, "chroma:document": "Cramer, D. D., D. H. Nguyen. 2013. Diagnostic fracture injection testing tactics in unconventional\nreservoirs. Paper SPE 163863 presented at the SPE Hydraulic Fracturing Technology Conference,Woodlands, TX, doi: 10.2118/163863-MS.\nCrouch, Steven L., A. M. Starfield. 1983. Boundary Element Methods in Solid Mechanics: with\nApplications in Rock Mechanics and Geological Engineering . London; Boston, Allen & Unwin.\nDavis, T. A. 2004. Algorithm 832: UMFPACK, an unsymmetric-pattern multifrontal method. ACM\nTransactions on Mathematical Software 30(2): 196\u2013199, doi: 10.1145/992200.992206.\nDongarra, J. J., J. Du Croz, S. Hammarling et al. 1988. An extended set of FORTRAN Basic Linear\nAlgebra Subprograms. ACM Transactions on Mathematical Software 14(1): 1\u201317, doi: 10.1145/\n42288.42291.SPE-170956-MS 21"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 21, "chroma:document": "Economides, Michael J., Kenneth G. Nolte. 2000. Reservoir Stimulation, 3rd edition. Chichester,\nEngland; New York, John Wiley.\nGeertsma, J., F. de Klerk. 1969. A rapid method of predicting width and extent of hydraulically\ninduced fractures. Journal of Petroleum Technology 21(12): 1571\u20131581, doi: 10.2118/2458-PA.\nGu, H., J. L. Elbel, K. G. Nolte et al. 1993. Formation permeability determination using impulse-\nfracture injection. Paper SPE 25425 presented at the Production Operations Symposium, Okla-homa City, OK, doi: 10.2118/25425-MS.\nGulrajani, S. N., K. G. Nolte. 2000. Appendix: background for hydraulic fracturing pressure analysis\ntechniques. In Reservoir Stimulation, ed. Michael J. Economides and Kenneth G. Nolte. Chich-\nester, England; New York, John Wiley.\nHickman, Stephen H., Mark D. Zoback. 1983. The interpretation of hydraulic fracturing pressure-time\ndata for in-situ stress determination. In Hydraulic Fracturing Measurements, ed. M. D. Zoback"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87c724-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 21, "chroma:document": "data for in-situ stress determination. In Hydraulic Fracturing Measurements, ed. M. D. Zoback\nand B. C. Haimson, 44\u201354. Washington D. C., National Academy Press.\nHoward, G., C.R. Fast. 1957. Optimum fluid characteristics for fracture extension. Drilling and\nProduction Practice 24: 261\u2013270.\nJeffrey, Robert, Andrew Bunger, Brice Lecampion et al. 2009. Measuring hydraulic fracture growth\nin naturally fractured rock. Paper SPE 124919 presented at the SPE Annual Technical Conferenceand Exhibition, New Orleans, Louisiana.\nKarimi-Fard, M., L. J. Durlofsky, K. Aziz. 2004. An efficient discrete-fracture model applicable for\ngeneral-purpose reservoir simulators. SPE Journal 9(2): 227\u2013236, doi: 10.2118/88812-PA.\nKhristianovich, S. A., Y. P. Zheltov. 1955. Formation of vertical fracture by means of highly viscous\nliquid. Paper presented at the Fourth World Petroleum Congress, Rome, Italy, 579\u2013586.\nLawson, C. L., R. J. Hanson, D. Kincaid et al. 1979. Basic Linear Algebra Subprograms for"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87c79c-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 21, "chroma:document": "Lawson, C. L., R. J. Hanson, D. Kincaid et al. 1979. Basic Linear Algebra Subprograms for\nFORTRAN usage. ACM Transactions on Mathematical Software 5(3): 308\u2013323, doi: 10.1145/\n355841.355847.\nMahrer, Kenneth D. 1999. A review and perspective on far-field hydraulic fracture geometry studies.\nJournal of Petroleum Science and Engineering 24(1): 13\u201328, doi: 10.1016/S0920-\n4105(99)00020-0.\nMartins, J.P., T.R. Harper. 1985. Mini-frac pressure decline analysis for fractures evolving from long\nperforated intervals and unaffected by confining strata. Paper SPE 13869 presented at theSPE/DOE Low Permeability Gas Reservoirs Symposium, Denver, CO, doi: 10.2118/13869-MS.\nMcClure, Mark W. 2014. Diagnostic fracture injection tests with complex fracture networks. The\nLeading Edge 33(5): 546\u2013548, 550\u2013552, doi: 10.1190/tle33050546.1.\nMcClure, Mark W., Roland N. Horne. 2013. Discrete Fracture Network Modeling of Hydraulic"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87c7f6-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 21, "chroma:document": "McClure, Mark W., Roland N. Horne. 2013. Discrete Fracture Network Modeling of Hydraulic\nStimulation: Coupling Flow and Geomechanics : SpringerBriefs in EarthSciences, Springer.\nNolte, Kenneth. 1979. Determination of fracture parameters from fracturing pressure decline. Paper\nSPE 8341 presented at the SPE Annual Technical Conference and Exhibition, Las Vegas, Nevada.\nNolte, K. G. 1986. A general analysis of fracturing pressure decline with application to three models.\nSPE Formation Evaluation Journal 1(6): 571\u2013583, doi: 10.2118/12941-PA.\nNolte, K. G. 1988. Application of fracture design based on pressure analysis. SPE Production\nEngineering 3(1): 31\u201342, doi: 10.2118/13393-PA.\nNordgren, R. P. 1972. Propagation of a vertical hydraulic fracture. Society of Petroleum Engineers\nJournal 12(4): 306\u2013314, doi: 10.2118/3009-PA.\nOlson, Jon E. 2004. Predicting fracture swarms \u2014 the influence of subcritical crack growth and the"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 21, "chroma:document": "Journal 12(4): 306\u2013314, doi: 10.2118/3009-PA.\nOlson, Jon E. 2004. Predicting fracture swarms \u2014 the influence of subcritical crack growth and the\ncrack-tip process zone on joint spacing in rock. In The Initiation, Propagation, and Arrest of Joints\nand Other Fractures, ed. J. W. Cosgrove and T. Engelder, 73\u201388. Geological Society, London,\nSpecial Publications, Geological Society of London.22 SPE-170956-MS"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87c896-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 22, "chroma:document": "Olson, J. E. 2007. Fracture aperture, length and pattern geometry development under biaxial loading:\na numerical study with applications to natural, cross-jointed systems. In The Relationship between\nDamage and Localization, ed. H. Lewis and G. D. Couples, 123\u2013142. Geological Society, London,\nSpecial Publications, The Geological Society of London.\nPadmakar, A.S. 2013. Geomechanics coupled reservoir flow simulation for diagnostic fracture\ninjection test design and interpretation in shale reservoirs. Paper 166201 presented at the SPEAnnual Technical Conference and Exhibition, New Orleans, LA, doi: 10.2118/166201-MS.\nPerkins, T. K., L. R. Kern. 1961. Widths of hydraulic fractures. Journal of Petroleum Technology 13\n(9): 937\u2013949, doi: 10.2118/89-PA.\nRenshaw, C. E., D. D. Pollard. 1995. An experimentally verified criterion for propagation across\nunbounded frictional interfaces in brittle, linear elastic materials. International Journal of Rock"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87c8f0-84a2-11ee-9ca1-0242ac1c000c
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metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 22, "chroma:document": "unbounded frictional interfaces in brittle, linear elastic materials. International Journal of Rock\nMechanics and Mining Sciences & Geomechanics Abstracts 32(3): 237\u2013249, doi: 10.1016/0148-\n9062(94)00037-4.\nShewchuk, Jonathan Richard. 1996. Triangle: Engineering a 2D Quality Mesh Generator and Delau-\nnay Triangulator. Lecture Notes in Computer Science 1148 : 203\u2013222.\nShou, K. J., S. L. Crouch. 1995. A higher order displacement discontinuity method for analysis of\ncrack problems. International Journal of Rock Mechanics and Mining Sciences & Geomechanics\nAbstracts 32(1): 49\u201355, doi: 10.1016/0148-9062(94)00016-V.\nSneddon, I. N. 1946. The distribution of stress in the neighborhood of a crack in an elastic solid.\nProceedings of the Royal Society of London Series A 187 (1009): 229\u2013260, doi: 10.1098/\nrspa.1946.0077\nSoliman, M. Y., C.S. Kabir. 2012. Testing unconventional formations. Journal of Petroleum Science\nand Technology 92-93 : 102\u2013109, doi: 10.1016/j.petrol.2012.04.027."}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87c940-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 22, "chroma:document": "and Technology 92-93 : 102\u2013109, doi: 10.1016/j.petrol.2012.04.027.\nSuarez-Rivera, Roberto, Jeff Burghardt, Sergei Stanchits et al. 2013. Understanding the effect of rock\nfabric on fracture complexity for improving completion design and well performance. Paperpresented at the International Petroleum Technology Conference, Beijing, China.\nWallace, J., C.S. Kabir, C. L. Cipolla. 2014. Multiphysics investigation of diagnostic fracture injection\ntests in unconventional reservoirs. Paper 168620 presented at the SPE Hydraulic FracturingTechnology Conference, The Woodlands, TX, doi: 10.2118/168620-MS.\nWarpinski, N. R., J. C. Lorenz, P. T. Branagan et al. 1993. Examination of a cored hydraulic fracture\nin a deep gas well, SPE 22876. SPE Production & Facilities 8(3): 150\u2013158, doi: 10.2118/22876-\nPA.\nWarpinski, N. R., L. W. Teufel. 1987. Influence of geologic discontinuities on hydraulic fracture\npropagation. Journal of Petroleum Technology 39(2): 209\u2013220, doi: 10.2118/13224-PA."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87c99a-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 22, "chroma:document": "propagation. Journal of Petroleum Technology 39(2): 209\u2013220, doi: 10.2118/13224-PA.\nWillis-Richards, J., K. Watanabe, H. Takahashi. 1996. Progress toward a stochastic rock mechanics\nmodel of engineered geothermal systems. Journal of Geophysical Research 101(B8): 17481\u2013\n17496, doi: 10.1029/96JB00882.\nWitherspoon, P. A., J. S. Y. Wang, K. Iwai et al. 1980. Validity of cubic law for fluid flow in a\ndeformable rock fracture. Water Resources Research 16(6): 1016\u20131024, doi: 10.1029/\nWR016i006p01016.\nZoback, Mark D. 2007. Reservoir Geomechanics . Cambridge, Cambridge University Press.SPE-170956-MS 23"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87c9ea-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 0, "chroma:document": "SPE-187348-MS\nNew Variable Compliance Method for Estimating In-Situ Stress and Leak-Off\nfrom DFIT Data\nHanYi Wang and Mukul M. SharmaThe University of Texas at Austin\nCopyright 2017, Society of Petroleum Engineers\nThis paper was prepared for presentation at the SPE Annual Technical Conference and Exhibition held in San Antonio, Texas, USA, 9-11 October 2017.\nThis paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents\nof the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect\nany position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87ca44-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 0, "chroma:document": "consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may\nnot be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.\nAbstract\nOver the past two decades, Diagnostic Fracture Injection Tests (DFIT) or Injection-Fall-off Fracture\nCalibration Tests have evolved into a commonly used and reliable technique to evaluate reservoir properties,\nfracturing parameters and obtain in-situ stresses. Since the introduction of DFIT analysis based on G-\nfunction and its derivative, this method has become standard practice for quantifying minimum in-situ stress\nand leak-off coefficient. However, the pressure decline model that underlies the G-function plot makes two\ndistinct and important assumptions: (1) leak-off is not pressure-dependent and, (2) fracture stiffness (or"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87ca9e-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 0, "chroma:document": "distinct and important assumptions: (1) leak-off is not pressure-dependent and, (2) fracture stiffness (or\ncompliance) is assumed to be constant during fracture closure. Fracture closure is a gradual process that\nstarts when asperities at the fracture tip first come into contact with each other. As pressure declines due\nto leak-off, more and more of the fracture wall comes into contact. It is important to model this process\nquantitatively to obtain good estimates of in-situ stresses and leak-off. In this paper, we present a model that\naccounts for changes in fracture stiffness/compliance as the fracture closes, with leak-off that is dependent\non fracture pressure. The model is, therefore, capable of analyzing DFIT data from the end of pumping to\ndays or even weeks after shut-in.\nWe first review Nolte's original G-function model and examine the assumptions inherent in the model.\nWe then present a new global pressure transient model for pressure decline after shut-in which not only"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87caee-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 0, "chroma:document": "We then present a new global pressure transient model for pressure decline after shut-in which not only\npreserves the physics of unsteady-state reservoir flow behavior, elastic fracture mechanics and material\nbalance, but also incorporates the gradual changes of fracture stiffness/compliance due to the contact of\nrough fracture walls during closure. This global model allows us to analyze the whole spectrum of DFIT data\nby bridging before closure and after closure data seamlessly. Analysis of synthetic cases, along with field\ndata are presented to demonstrate how the coupled effects of fracture geometry, fracture surface asperities,\nformation properties, pore pressure and wellbore storage can impact fracturing pressure decline and the\nestimation of minimum in-situ stress.\nIt is shown that so-called \"normal leak-off\" behavior that is modeled using Carter leak-off is an\noversimplification that leads to significant errors in the interpretation of the data. All the before"}


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id: 6b87cb48-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 0, "chroma:document": "oversimplification that leads to significant errors in the interpretation of the data. All the before\nclosure analysis conducted under a \"normal leak-off\" assumption should be reexamined cautiously. Most\nimportantly, this article reveals that previous methods of estimating minimum in-situ stress often lead to\nsignificant over or underestimates, because of their failure to account for changes in fracture stiffness/"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87cb98-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 1, "chroma:document": "2 SPE-187348-MS\ncompliance correctly as the fracture closes progressively from the edge to the center. Based on our modeling\nand simulation results, we propose a much more accurate and reliable method to estimate the minimum in-\nsitu stress, fracture pressure dependent leak-off rate and evaluate the compliance of the un-propped fracture.\nIntroduction\nDiagnostic fracture injection tests (DFIT), which have also been referred to as Injection-Falloff Tests,\nFracture Calibration Tests, Mini-Frac Tests in the literature, involve pumping a fluid (typically water), at\na constant rate for a short period of time, creating a relatively small hydraulic fracture before the well\nis shut in. The pressure transient data after shut-in is analyzed to obtain in-situ stresses and reservoir\nproperties. A typical pressure trend is qualitatively shown in Fig. 1. Conventionally, DFIT analysis has\nfocused on acquiring fracturing treatment design parameters, such as fluid efficiencies, leak-off coefficient"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87cbf2-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 1, "chroma:document": "focused on acquiring fracturing treatment design parameters, such as fluid efficiencies, leak-off coefficient\nand fracture closure pressure (which is normally interpreted as minimum in-situ stress). However, in recent\nyears, DFIT analysis has been extended to obtain reservoir properties such as reservoir pore pressure, and\npermeability in unconventional reservoirs, where traditional pressure transient tests are impractical. The\nreservoir properties determined by DFIT are representative, because the created fracture can pass through\nnear-wellbore damage zone and provide a large volume of investigation for true formation properties. In\naddition, the net pressure trends in a DFIT can be also used to infer the induced fracture complexity in\ndifferent geological settings (Potocki 2012). Thoese valuable information obtained from DFITs provides\nkey input parameters for modeling hydraulic fracture propagation (Wang 2015; Wang 2016; Wang et al."}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87cc4c-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 1, "chroma:document": "key input parameters for modeling hydraulic fracture propagation (Wang 2015; Wang 2016; Wang et al.\n2016), stimulation design (Ramurthy et al. 2011), development of reservoir models (Mirani et al. 2016;\nLoughry et al. 2015; Wang 2017) and post-fracture analysis (Fu et al. 2017). Without a realistic estimation\nof the fracturing parameters and reservoir properties, it would not be possible to optimize hydraulic fracture\ndesign and evaluate the economic viability of producing hydrocarbons from unconventional reservoirs.\nFigure 1\u2014Diagram showing sequence of events observed in a DFIT\nThe pressure decline data after shut-in can be divided into two distinct regions for analysis: before closure\npressure data and after closure pressure data (separated by the fracture closure pressure). Analysis models are\ndeveloped differently for these regimes. Before closure analysis (BCA) assumes normal leak-off behavior"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87cc9c-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 1, "chroma:document": "developed differently for these regimes. Before closure analysis (BCA) assumes normal leak-off behavior\nor uses the portion of normal leak-off data (a straight line on G function and square root of time plots) to\nestimate fracture geometry, leak-off coefficient or formation permeability (Mayerhofer et al. 1995; Nolte\n1979 and 1986; Valko\u2019 and Economides 1999). A departure from the straight line indicates fracture closure\n(Castillo1987). After closure analysis (ACA) models have been developed by several authors (Benelkadi\nand Tiab 2004; Chipperfield 2006; Nolte et al. 1997; Soliman et al. 2005; Soliman et al. 2010) to estimate"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87ccf6-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 2, "chroma:document": "SPE-187348-MS 3\npore pressure and infer formation permeability. Because both BCA and ACA only use some portion of the\nDFIT data, global consistency of interpretation can't be guaranteed, even if the individual analyses match\na certain portion of the data well. Liu et al. (2016) and Marongiu-Porcu et al. (2014) attempted to bridge\nBCA and ACA on a log-log plot and the closure pressure is picked at the end of 3/2 slope of the pressure\nderivative. However, their work is still based on G-function model's assumptions and the 3/2 slope just\narises from a spatial integration of Carter's leak-off (McClure 2017). Currently, no model is available to\nsimulate DFIT pressure decline under a single, coherent mathematic framework, and correctly capture the\ndynamic evolution of fracture stiffness/compliance during closure. In the following section, we are going to\nreview and examine the fundamental limitations of the G-function model, show that closure does not occur"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87cd50-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 2, "chroma:document": "review and examine the fundamental limitations of the G-function model, show that closure does not occur\nat a single point in time but is a continuous and gradual process, and derive a fully coupled global model\nthe fits the entire pressure curve during the entire fracture closure process.\nThe advent of fracturing pressure decline analysis was pioneered by the work of Nolte (1979 and 1986).\nWith the assumptions of power law fracture growth, negligible spurt loss, constant fracture surface area after\nshut-in and Carter's leak-off model (one-dimensional leak-off of fluid from a constant pressure boundary,\nthe solution to the diffusivity equation predicts that the leak-off rate will scale with the inverse of the square\nroot of time), a remarkably simple and useful equation for the pressure decline can be obtained:\n(1)\nHere, is the instantaneous shut-in pressure at the end of pumping, Pf is the fracture pressure at"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87cdaa-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 2, "chroma:document": "(1)\nHere, is the instantaneous shut-in pressure at the end of pumping, Pf is the fracture pressure at\ndimensionless time \u0394t D.tp is the total pump time.r p is the productive fracture ratio, which is the ratio of\nfracture surface area that is subject to leak-off to the total fracture surface area. For low permeability,\nunconventional reservoirs, r p \u22481. To increase the readability of this article, r p will be assumed to be 1 for the\nrest of the discussion. CL is Carter's leak-off coefficient which is a constant. Sf is the fracture stiffness, which\ncan be calculated using Table 1 for different fracture geometries. It is assumed that Sf is a constant until the\nfracture closes instantaneously when the fluid pressure in the fracture reaches the closure pressure. Because\nfracture stiffness is the reciprocal of fracture compliance, they will be used interchangeably throughout this\narticle.\nTable 1\u2014Fracture stiffness expressions for different fracture geometry models"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87ce22-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 2, "chroma:document": "article.\nTable 1\u2014Fracture stiffness expressions for different fracture geometry models\nE\u2032 is the plane strain Young's modulus and can be calculated using Young's Modulus, E, and Poisson's\nRatio, v:\n(2)\nThe dimensionless time \u0394t D is defined by:\n(3)\nWhere t is the generic time and t e is the time at the end of pumping. G-function is defined as"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87ce7c-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 3, "chroma:document": "4 SPE-187348-MS\n(4)\nWhere the g-function of time is approximated by,\n(5)\nFrom Eq. (1), we can infer that for normal leak-off behavior (constant Carter's leak-off coefficient,\nconstant leak-off area and constant fracture stiffness during fracture closure), the pressure declines\nlinearly with G(\u0394t D). Castillo (1987) usedNolte's G-function for modeling the pressure declinebehavior\nand developed the straight-line plotof the G-function vs. pressure. The slope of this curve is used for the\ncomputation of the leak-off coefficient that is independent of pressure. Any departure from this straight-\nline is interpretedasclosure of the fracture.\nUnfortunately, plots of pressure versus G-function often yield curves with multiple points of inflectionthat\nhave been attributed to abnormal leak-off behavior (such as pressure dependent leak-off, fracture height\nrecession, closing of secondary transverse fractures, and fracture tip extension), which makes it difficult"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87cecc-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 3, "chroma:document": "recession, closing of secondary transverse fractures, and fracture tip extension), which makes it difficult\nto interpretthe changes in slope and identify fracture closure. So identification of fracture closure pressure\nand non-ideal behavior is usually done using plots of pressure and GdP/dG versus G-function (Barree and\nMukherjee, 1996; Barree, 1998, Barree et al., 2014), where the closure is picked at the tangential point\nbetween a straight line that passes the origin and the GdP/dG curve. This prevailing method of determining\nminimum in-situ stress (although has been widely accepted, but has never been theoretically proved) will be\nclassified and discussed as \"tangent line method\" in the following text. The fracturing pressure decline model\nthat underlying G-function plot suffers from two distinct and important issues: (1) leak-off is not pressure-\ndependent, i.e. a constant pressure boundary is assumed (2) fracture compliance/stiffness is assumed to be"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87cf26-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 3, "chroma:document": "dependent, i.e. a constant pressure boundary is assumed (2) fracture compliance/stiffness is assumed to be\nconstant during fracture closure. This is why G-function based models are only used for before closure\nanalysis and are not capable of analyzing DFIT data from the end of pumping to days or even weeks after\nshut-in, which requires bridging both before and after closure data seamlessly.\nModel Formulation\nThere are two essential components of the general DFIT model formulated in this paper. In this section, we\nwill present a generalization of the pressure transient model that couples the fluid pressure in the fracture\nwith the leak-off rate without making any assumptions about the pressure in the fracture that acts as a\nboundary condition for leak-off. In addition, special cases will be demonstrated to link previous models and\nour fully coupled approach.\nGlobal DFIT Model\nIn unconventional reservoirs, fluid leak-off during pumping is very small and no filter-cake is formed on the"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87cf80-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 3, "chroma:document": "Global DFIT Model\nIn unconventional reservoirs, fluid leak-off during pumping is very small and no filter-cake is formed on the\nfracture surface because only water is injected. In addition, the duration of injection is very short (typically\n3-5 minutes) while the total shut-in time may be days or even weeks. In order to correctly capture fracturing\npressure response during a DFIT in unconventional reservoirs, the pressure dependent leak-off at the fracture\nsurface and the dynamic changes of fracture compliance have to be accounted for. Fig. 2 illustrates one-\ndimensional leak-off into a semi-infinite formation."}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87cfda-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 4, "chroma:document": "SPE-187348-MS 5\nFigure 2\u2014Illustration of one-dimensional leak-off\nAssuming linear Darcy flow and a slightly compressible, single phase fluid in the reservoir, the\ndifferential form of the mass balance can be written as:\n(6)\nwhere P is the pressure, k is formation permeability, \u03bc f is fluid viscosity, \u03d5 is formation porosity and ct is\ntotal formation compressibility. Inside the fracture, the average fracture width \n  and fracturing net pressure\nPnet is related by the fracture stiffness:\n(7)\nFrom a material balance perspective (fluid compressibility is negligible compared to that of the fracture),\nthe rate of fluid leak-off(one wing of the fracture) into the formation, qf, after shut-in equals the rate of\nshrinkage of fracture volume (one wing of the fracture), V f, as pressure declines:\n(8)\nAnd\n(9)\nwhere Af is the fracture surface area of one face of one wing of the fracture. So Eq. (8) can be re-written as\n(10)"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87d034-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 4, "chroma:document": "(8)\nAnd\n(9)\nwhere Af is the fracture surface area of one face of one wing of the fracture. So Eq. (8) can be re-written as\n(10)\nAt the boundary of the fracture surface, both Darcy's flow and material balance have to be honored\n(11)\nRearranging Eq. (11), we get the boundary condition at the fracture surface\n(12)\nIn the above derivation, it is assumed that the whole fracture surface area is subject to leak-off, which is\nthe norm in low permeabity reservoirs. If only a portion of fracture surface is considered permeable, then"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87d084-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 5, "chroma:document": "6 SPE-187348-MS\none just need to multiply a leak-off surface area ratio (leak-off surface area devided by total fracture surface\narea) on the left side of Eq. (12).\nWith initial condition (disregarding the pressure disturbance during the short injection period)\n(13)\n(14)\nwhere P 0 is the initial pore pressure.\nThe governing Eq. (6), plus the initial condition of Eq. (13)-(14) and boundary condition of Eq. (11)\nuniquely describes the pressure transient behavior during DFITs. In the above derivations, the influence of\nwellbore storage (WBS) is not included. For typical DFITs, untreated water is normally used as injection\nfluid. Since water is slightly compressible, the wellbore fluid will expand after shut-in with the declining\npressure. Ignoring the early time fluid transient behavior along the wellbore after shut-in and assuming\nfracturing pressure and wellbore pressure have reached quasi-equilibrium, leak-off volume into formation"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87d0de-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 5, "chroma:document": "fracturing pressure and wellbore pressure have reached quasi-equilibrium, leak-off volume into formation\nshould equal the shrinkage of fracture volume plus the expansion of wellbore fluid. We can modify the leak-\noff rate from Eq. (10) to account for wellbore fluid expansion as follows:\n(15)\nWhere V W is half the wellbore volume (only one wing of the fracture needs to be modeled) and C W\nis the compressibility of water. Using Eq. (15) with Eq. (10), we can define the fracture-wellbore system\nstiffness as:\n(16)\nUnder extreme circumstances, when wellbore storage is negligible (V w Cw \u2248 0), fracture stiffness\ndominates the system stiffness, and when Vw CW are large and Af is small, WBS dominates the system\nstiffness. If WBS has to be included in the model, one just need simply replace Sf with SS in Eq. (12). It should\nbe emphasized there are two stages of fracture closure: mechanical closure, where the fracture surfaces come"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87d12e-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 5, "chroma:document": "be emphasized there are two stages of fracture closure: mechanical closure, where the fracture surfaces come\ninto contact at asperities, and the other is hydraulic closure, where fracture pressure in the closed section\nis disconnected from the open section or wellbore pressure. In the above derivations, we assume that the\nmechanically closed fracture still retains hydraulic conductivity because of its residual fracture width.\nSpecial Case 1: Constant Fracture Compliance and Carter's Leak-off.   If we assume that a constant\nshut-in pressure (ISIP) is applied at the fracture surface, the solution of Eq. (6) to obtain the leak-off or\nthe velocity Uf across fracture surface into an intact, homogenous formation can be found as ( Economides\nand Nolte 2000):\n(17)\nIf we take \u0394t out of Eq. (17), then the R.H.S of Eq. (17) essentially becomes Carter's leak-off coefficient\nCL, which is a constant parameter. The total leak-off rate qf from one wing of the fracture starting from\nshut-in is:\n(18)"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87d188-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 6, "chroma:document": "SPE-187348-MS 7\nCombining Eq. (18) and Eq. (10), we have:\n(19)\nEq. (19) is a simple first order differential equation, with the initial condition of Pf (\u0394t = 0) = ISIP, the\nsolution of Eq. (19) leads to the pressure declining proportionally to the square root of time:\n(20)\nEq. (20) is the fundamental basis for the square root of time plot. From Eq. (1) and (20), we can infer that\nfor normal-leak-off behavior with constant fracture stiffness and assuming Carter's leak-off, the pressure\nand its derivative will be straight lines on both G-function and the square root of time plots. The fracture\npressure continues declining even when it reaches the initial reservoir pore pressure. This non-physical\nprediction indicates that the assumption of Carter's leak-off and constant stiffness made in the derivation\nprevent Notle's G-function model and the square root of time model from capturing the actual pressure\nresponse during fracture closure and late time pressure transient behavior."}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87d1e2-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 6, "chroma:document": "response during fracture closure and late time pressure transient behavior.\nSpecial Case 2: Constant Fracture Compliance and Fracture Pressure Dependent Leak-off.   When the\nleak-off rate is fracture pressure dependent, a closed analytic form of pressure decline is not available,\nbut the solution can be obtained using a superposition time. A detailed derivation of the pressure decline\nsolution is presented in Appendix A. And from Eq. (A.4), the fracture pressure at the nth time interval, Pf,n,\ncan be calculated explicitly:\n(21)\nwhere Pf,0 = P0, Pf,1 = ISIP, \u0394t 1 = 0. It is important to note that in this article, the term \"fracture pressure\ndependent leak-off\" (FPDL) specifically refers to a leak-off rate that depends on the declining fluid pressure\ninside the fracture, not to how the leak-off rate changes with net stress in the rock surrounding the fracture,\nwhich is often denoted as pressure dependent leak-off or PDL ( Barree et al. 2009). In some cases, excessive"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87d264-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 6, "chroma:document": "which is often denoted as pressure dependent leak-off or PDL ( Barree et al. 2009). In some cases, excessive\npressure drop can be observed during early-time of shut-in in low permeability reservoirs, but it is the result\nof high apparent ISIP, wellbore storage and additional friction away from the wellbore, and not caused by\nPDL, which is unlikely to happen in nanodarcy permeability systems.\nVariable Fracture Compliance Due to Closure on Rough Surfaces and Asperities\nAs pressure declines inside the fracture after shut-in, the fracture will gradually close and the fracture\naperture will approach the scale of the surface roughness. If the fracture faces are perfectly parallel and\nsmooth, they will come into contact all at once when the fluid pressure inside the fracture declines to the\nfar field stress, and the fracture is then mechanically and hydraulically closed. However, there is abundant"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87d2c8-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 6, "chroma:document": "far field stress, and the fracture is then mechanically and hydraulically closed. However, there is abundant\nevidence to suggest that fractures retain their conductivity after the walls have come into contact (mechanical\nclosure). Fractures retain a finite aperture after mechanical closure due to a mismatch of asperities on the\nfracture walls. van Dam et al. (2000) presented scaled laboratory experiments on hydraulic fracture closure\nbehavior. They observed up to a 15% residual aperture (compared to the maximum aperture during fracture\npropagation) long after shut-in. Fredd et al. (2000) demonstrated fracture surface asperities can provide\nresidual fracture width and sufficient conductivity in the absence of proppants. Using sandstone cores from\nthe East Texas Cotton Valley formation, sheared fracture surface asperities that had an average height of"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87d318-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 7, "chroma:document": "8 SPE-187348-MS\nabout 2.286 mm were observed. Warpinski et al. (1993) reported hydraulic fracture surface asperities of\nabout 1.016 mm and 4.064 mm for nearly homogeneous sandstones and sandstones with coal and clay-\nrich bedding planes, respectively. Sakaguchi et al. (2008) created tensile fractures in large rock blocks and\nmeasured the asperity height and distribution. Their work shows that the fracture surfaces can be assumed\nto be a fractal object and most of the asperities fall within 1 to 2 mm in height. Wells and Davatzes (2015)\nconducted topographic measurement on dilated fractures from core samples and found the asperity height\nranges from hundreds to thousands of micrometers. Bhide et al. (2014) created X-ray microtomographic\nimages from shear induced fractures and the roughness values obtained varied from 1.8 to 1.95 mm along\nthe length of the rock samples. Zou et al. (2015) conducted experiments on 20 fractured shale samples and"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87d372-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 7, "chroma:document": "the length of the rock samples. Zou et al. (2015) conducted experiments on 20 fractured shale samples and\nfound the average asperity height to be 1.88 mm. An experimental study ( Zhang et al. 2014) on Barnett Shale\nsamples reveals that the surface topography of the displaced fracture can be altered because of rock failure,\nand the fracture surface exhibited parallel strips of crushed asperities. Field measurements ( Warpinski et al.\n2002) using a down-hole tiltmeter array indicated that the fracture closure process is a smooth, continuous\none which often leaves 20%-30% residual fracture width, regardless of whether the injection fluid is\nwater, linear-gel or cross-linked-gel. The microscopic measurement and modeling of surface roughness and\nmechanical properties of asperities can often be up-scaled to macroscopic contact laws that relate fracture\nwidth and the associated contact stress. Willis-Richards et al. (1996) proposed contact law to relate fracture"}


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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 7, "chroma:document": "width and the associated contact stress. Willis-Richards et al. (1996) proposed contact law to relate fracture\nwidth and the net closure stress for fractured rocks, based on the work of Barton et al. (1985):\n(22)\nwhere wf is the fracture aperture and, w0 is the contact width, which represents the fracture aperture when\nthe contact normal stress is equal to zero, \u03c3 c is the contact normal stress on the fracture, and \u03c3 ref is a contact\nreference stress, which denotes the effective normal stress at which the aperture is reduced by 90%. it should\nbe emphasized that the contact width w0 is determined by the tallest asperities, and the strength, spatial and\nheight distribution of asperities are reflected by the contact reference stress \u03c3 ref (e.g., if the tallest asperities\non two fracture samples are the same, then they should have the same W0, but the one with a higher median\nasperity height will have higher value of \u03c3 ref, provided other properties are the same)."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87d41c-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 7, "chroma:document": "asperity height will have higher value of \u03c3 ref, provided other properties are the same).\nWith known fracture geometry, rcok properties and surface roughness, the question now is how to\nestimate fracture stiffness/compliance as a function of pressure, if it continuously changes during fracture\nclosure. Wang and Sharma (2017) presented a method and general algorithms to model the dynamic behavior\nof hydraulic fracture closure on rough fracture surfaces and asperities, using linear elastic solutions that\ncoupled with contact law for three different fracture models (PKN, KGD and radial fracture geometry).\nGiven the fracture geometry, rock properties, contact parameters and minimum principal stress, their\napproach can predict the evolution of fracture aperture profile, total fracture volume and fracture stiffness\nas fracturing pressure declines. A brief discussion of modeling non-local dynamic fracture closure behavior\ncan be found in the Appendix B."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87d46c-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 7, "chroma:document": "as fracturing pressure declines. A brief discussion of modeling non-local dynamic fracture closure behavior\ncan be found in the Appendix B.\nAssuming a Young's modulus of 20 GPa, Poisson's ratio of 0.25, W0 of 2 mm, \u03c3 ref of 5 MPa for a\nPKN fracture geometry with 10 m fracture height, 35 MPa minimum in-situ stress, the evolution of the\nfracture width profile and contact stress distribution can be determined, as the fluid pressure inside the\nfracture gradually declines. The results are shown in Fig. 3 and Fig. 4. To demonstrate the impact of fracture\nroughness and surface asperities on fracture closure behavior, the case without surface asperities (fracture\nsurface is completely planar and smooth) is also included. The result shows that at relative high fracturing\nfluid pressure, the fracture asperities have negligible impact on fracture width distribution, and the contact\nstress is always concentrated at the tip of the fracture, where the contact stress is much higher than in the"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87d4ee-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 7, "chroma:document": "stress is always concentrated at the tip of the fracture, where the contact stress is much higher than in the\nmiddle of the fracture. We can also see that the fracture surfaces do not contact each other like parallel\nplates. In fact, the fracture closes on rough surfaces starting from the tip, and closes progressively all the"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87d548-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 8, "chroma:document": "SPE-187348-MS 9\nway from the edges to the center of fracture. As fluid pressure continues to decline, more and more of the\nfracture surfaces come into contact and these changes the subsequent fracture closure behavior. At lower\nfluid pressures, contact stresses start to counter-balance the in-situ stress and the fracture becomes stiffer\nand less compliant. If the fracture faces were perfectly parallel and smooth, the fracture width would have\ncollapsed to zero when the fluid pressure dropped to 35 MPa. The moment when all fracture surfaces come\ninto contact on asperities and the contact stress becomes non-zero on the entire fracture surface, the fracture\nis mechanically closed and this mechanical closure stress is higher than the minimum in-situ stress.\nFigure 3\u2014Fracture width evolution with and without asperities at different fluid pressure for a PKN geometry\nFigure 4\u2014Fracture width and the corresponding contact stress distribution at different fluid pressure for a PKN geometry"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87d5a2-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 8, "chroma:document": "Figure 4\u2014Fracture width and the corresponding contact stress distribution at different fluid pressure for a PKN geometry\nThe simulated fracture volume evolution as a function of fluid pressure is shown in Fig. 5. As can be\nseen, when the fluid pressure inside the fracture is relatively high, the fracture volume declines linearly\nwith pressure (indicates roughly constant fracture compliance/stiffness). However, as the pressure declines\nto a certain level (2 MPa higher than the input minimum in-situ stress of 35 MPa), the fracture volume\nand pressure departs from a linear relationship. The traditional method of estimating fracture stiffness using\nTable 1 can only be used when the fracture pressure is still relatively high and the asperities are only in\ncontact at the edges of the fracture. However, as pressure declines further and more and more fracture surface\narea comes into contact, the effective fracture stiffness becomes pressure dependent:\n(23)"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87d5fc-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 9, "chroma:document": "10 SPE-187348-MS\nFigure 5\u2014Fracture volume and fracture stiffness evolution as fluid pressure declines for a PKN geometry\nThe relationship between Pf and Vf can be used to calculate the pressure dependent fracture stiffness\nand the result is also shown in Fig. 5. This pressure dependent fracture compliance/stiffness obtained\nfrom non-local fracture closure modeling (Wang and Sharma 2017) can be incorporated into Eq. (12) to\nmake Sf pressure dependent. The \"tangent line method\" for picking up minimum in-situ stress is based\non the underlying assumptions of G-function model that assumes perfectly smooth fracture surfaces, and\nthe stiffness remains constant until fluid pressure drops to minimum in-situ stress. This leads to a sudden\nincrease in fracture stiffness when all fracture surfaces get into contact simultaneously. Fracture stiffness\nonly remains constant at high fracture pressures. It starts to increase long before the fracturing pressure"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87d656-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 9, "chroma:document": "only remains constant at high fracture pressures. It starts to increase long before the fracturing pressure\ndeclines to the minimum in-situ stress (35 MPa in this case). McClure et al. (2016) proposed a \"compliance\nmethod\" for picking closure pressure on the G-function plot based on numerical simulations, where closure\npressure is picked at the point where the fracture stiffness starts to increase. Their method can overestimate\nminimum in-situ stress if the mechanical closure pressure is interpreted as the minimum in-situ stress.\nThere is compelling field evidence to indicate that this occurs in the field. Fig. 6 shows the normalized\ntiltmeter response plotted against wellbore pressure during the shut-in period of 2B well at three different\nstations from the GRI/DOE M-site. Each tiltmeter response is normalized by dividing by the maximum\ntilt (displacement, also an indication of fracture width) measured at that instrument during the test. The"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87d6b0-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 9, "chroma:document": "tilt (displacement, also an indication of fracture width) measured at that instrument during the test. The\ntiltmeter demonstrates that soon after shut-in, the measured displacement declines linearly with pressure.\nIf the fracture surface area remains constant during shut-in, then the fracture volume is proportional to the\naverage displacement and should also decline linearly with the pressure within this period. After the wellbore\npressure declines to a certain level, the measured displacement vs pressure departs from a linear relationship.\nThis field measurement is consistent with the general trend shown in Fig. 5. As pressure continues declining,\nmore and more fracture surface area comes into contact with the rough fracture surfaces and asperities, and\nthe fracture stiffness increases gradually. Even though these data were measured from the end of injection to\nweeks of shut-in, we can still observe the existence of a residual fracture width that is supported by surface"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87d70a-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 9, "chroma:document": "weeks of shut-in, we can still observe the existence of a residual fracture width that is supported by surface\nasperities and mismatches even after the fracturing fluid pressure drops below the minimum in-situ stress.\nSo considering the nature of mineralogical heterogeneity and the existence of thin laminated rock layers,\nthe formation of asperities and tortuous fracture walls are inevitable during hydraulic fracture propagation."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87d75a-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 10, "chroma:document": "SPE-187348-MS 11\nFigure 6\u2014Normalized tiltmeter data from shut-in of 2B from the GRI/DOE M-site. Data Courtesy of Norm Warpinski\nSimulation Results and Analysis\nThe pressure decline behavior during DFITs in low permeability formations is governed by Eq. (6) (a PDE)\nwith Eq. (12) (an ODE) acting as the boundary condition. To solve this system of PDE with an ODE\nboundary condition, the method of lines (MOL) is used, by replacing the spatial derivative in the PDE with\nalgebraic approximations. The PDE can then be transformed into a system of ODEs, which can be solved\nsimultaneously and efficiently by well-established numerical methods (Schiesser and Griffiths 2009). Since\nthe MOL essentially replaces the problem PDEs with systems of approximating ODEs, the addition of other\nODEs is easily accomplished. In this section, our fully-coupled global DFIT model is used to investigate how\ndifferent fracture geometry, contact parameters and reservoir properties impact fracturing pressure response"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87d7b4-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 10, "chroma:document": "different fracture geometry, contact parameters and reservoir properties impact fracturing pressure response\nand our interpretation. It should be mentioned that the discussion presented in this section is restricted to\nlow permeability formations where the leak-off rate and poroelastic effects are small.\nNormal leak-off behavior\nNormal leak-off behavior is a special case of our model that satisfies all the assumptions of Nolte's G-\nfunction model, and it can be clearly identified with straight lines of pressure and its derivatives on G-\nfunction and the square root of time plots. All previous BCA models assume that these straight lines indicate\nnormal leak-off behavior and hence, leak-off coefficient or formation permeability can be determined based\non this portion of the data. However, one key assumption in all BCA is Carter's leak-off. To examine the\nimpact of incorrectly assuming Carter's leak-off for the duration of the DFIT, we extend the concept of"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87d804-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 10, "chroma:document": "impact of incorrectly assuming Carter's leak-off for the duration of the DFIT, we extend the concept of\n\"normal leak-off behavior\" to account for different fracture boundary conditions, while other assumptions\nremain the same (e.g., constant fracture compliance/stiffness and constant surface area during closure). The\nsimulation results from three different leak-off models are compared using input parameters from Table\n2. Fig. 7 and Fig. 8 show the simulated pressure and its derivatives on G-function and square root of\ntime plots for different formation permeability. As expected, Carter's leak-off (using Eq. (1) or Eq. (20),\nthey are interchangeable) always produce straight lines on diagnostic plots. But when pressure dependent\nleak-off at the fracturing surface is considered (using Eq. (21) and our fully coupled simulation), we no\nlonger obtain straight lines. This is because as fracturing pressure declines, leak-off rate departs from the\nlinear relationship with"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87d85e-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 10, "chroma:document": "longer obtain straight lines. This is because as fracturing pressure declines, leak-off rate departs from the\nlinear relationship with \n , so the assumption of Carter's leak-off is violated. Carter's leak-off model\noverestimates the leak-off rate and pressure drop by assuming a constant ISIP at the fracturing surface and\nleads to larger errors as shut-in time gets longer."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87d8b8-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 11, "chroma:document": "12 SPE-187348-MS\nFigure 7\u2014Comparison of \"normal leak-off\" behavior for different models, k=0.0001 md\nFigure 8\u2014Comparison of \"normal leak-off\" behavior for different models, k=0.001 md\nTable 2\u2014Input parameters for DFIT simulation with constant fracture compliance\nThe most important observation here is that the so-called \"normal leak-off behavior\" with straight lines\non G-function and the square root of time plots is not \"normal\" at all. For example, the closure time for\ntypical DFIT in a formation with 0.0001 md permeability is around 2~4 days. And in the above case shown\nin Fig. 7, the fracturing pressure and its derivative depart from the straight line after 4 hours of shut-in,\neven with constant fracture compliance/stiffness, surface area and reservoir properties. Many field data sets\ndo show \"normal leak-off behavior\" with straight lines that extend all the way to the point of downward"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87d908-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 11, "chroma:document": "do show \"normal leak-off behavior\" with straight lines that extend all the way to the point of downward\ndeflection. This occurs if some mechanisms that accelerate pressure drop (e.g., the gradual increase of\nfracture stiffness) happen to offset the effects of declining pressure at the fracture surface, so the pressure\ndata fits the straight lines predicted by Carter's leak-off model coincidentally. So any DFIT analysis that is\nbased on the assumption of Carter's leak-off and straight lines on G-function and the square root of time\nplots should be re-examined."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87d958-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 12, "chroma:document": "SPE-187348-MS 13\nVariable Fracture Compliance on Pressure Decline and In-Situ stress Estimation\nAs discussed earlier, G-function and previous BCA models assume Carter's leak-off behavior as if the\npressure in the fracture is a constant value (i.e., ISIP). In addition, these models also assume fracture\ncompliance/stiffness is constant and only changes abruptly at the moment of fracture closure, when\nfracturing pressure drops to the minimum in-situ stress. The failure to account for the coupled mechanisms\nof pressure dependent leak-off at the fracture surface and the gradual increase of fracture stiffness during\nclosure explains why these models are only useful on a small portion of the before closure data, and are\nnot able to simulate the fracturing response for the entire duration of the DFIT test. These simplifying\nassumptions also make their interpretation of DFIT data and estimation of minimum in-situ stress inaccurate."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87d9b2-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 12, "chroma:document": "assumptions also make their interpretation of DFIT data and estimation of minimum in-situ stress inaccurate.\nIn this section, our global DFIT model is used to investigate how each factor impacts the pressure decline\ntrend and our estimation of minimum in-situ stress and fracture compliance.\nPKN Fracture Geometry.   We first examine a PKN geometry using parameters listed in Table 2 and\nassuming w0 is 2 mm, \u03c3 ref is 5 MPa and reservoir permeability is 0.0005 md as the Base Case. Fig. 9 shows\nthe contact stress at different fracture width. As expected, when the fracture width is larger than the contact\nwidth w0, the contact stress is zero. However, when the fracture width is smaller than the contact width,\nthe contact stress and fracture width follows a hyperbolic relationship. Fig. 10 shows the corresponding\nfracture stiffness evolution for different W0 with the given fracture geometry and rock properties. The results"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87da02-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 12, "chroma:document": "fracture stiffness evolution for different W0 with the given fracture geometry and rock properties. The results\nindicate that as the fracture width decreases, the rough fracture walls will come into contact sooner if the\ncontact width is larger, so the noticeable changes of fracture stiffness occur earlier when the contact width\nis larger. We can also observe that when the contact width is larger, the increase in fracture stiffness is more\ngradual and smooth. In the extreme case of W0 close to zero, as would be the case for relative flat fracture\nsurfaces, we can infer that fracture stiffness will increase abruptly as soon as the fluid pressure drops to the\nminimum in-situ stress (35 MPa in this case). This pressure dependent fracture stiffness obtained from non-\nlocal fracture modeling (Wang and Sharma 2017) can be directly put into the boundary ODE of Eq. (12) to\npredict fracture pressure decline for certain reservoir properties."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87da5c-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 12, "chroma:document": "predict fracture pressure decline for certain reservoir properties.\nFigure 9\u2014The relationship between contact stress and fracture width for different W 0"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87dab6-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 13, "chroma:document": "14 SPE-187348-MS\nFigure 10\u2014Fracture stiffness evolution for different W 0 with a PKN geometry\nFig. 11 shows the fracturing pressure and its derivatives for different W0 on G-function and square root\nof time plots. Relate with Fig. 9 and Fig. 10, we can see that the contact width impacts pressure decline\nresponse significantly, because it alters the evolution of fracture stiffness. Large contact width leads to a\nsmooth pressure decline trend while small contact width leads to steep changes in pressure decline rate and\npressure derivatives. We can also infer that if the contact width is close to zero and all fracture walls come\ninto contact simultaneously, then a sudden change in pressure decline rate and the pressure derivative spikes\non both G-function and square root of time plots are inevitable, which is unrealistic and never observed in\nfield cases. So the conventional assumption that a fracture closes on flat, smooth fracture surfaces where W0"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87db10-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 13, "chroma:document": "field cases. So the conventional assumption that a fracture closes on flat, smooth fracture surfaces where W0\n= 0 does not reflect reality. In addition, different contact widths lead to different derivative slopes, so any\nBCA models that attempt to match this portion of data and estimate reservoir properties without considering\nthe variable fracture compliance is questionable. Fig. 11 also shows G-function and square root of time plots\ngive the same quantitative information only slightly different in scales. More importantly, the estimated\nminimum in-situ stress from pressure derivatives using the \"convention method\" and \"compliance method\"\ngive inconsistent results for different contact widths, with the same input parameters of fracture surface\narea, minimum in-situ stress and reservoir properties. A detailed comparison and a new approach to estimate\nminimum in-situ stress will be discussed later.\nFigure 11\u2014Pressure decline response for different W 0 with a PKN geometry"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87db92-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 13, "chroma:document": "minimum in-situ stress will be discussed later.\nFigure 11\u2014Pressure decline response for different W 0 with a PKN geometry\nNext, we examine how contact reference stress affects the pressure decline response. Fig. 12 and Fig. 13\nshow the relationship between contact stress and fracture width for different contact reference stress and the\ncorresponding fracture stiffness evolution at different fracturing pressure. For the same contact width, the\nhigher the contact reference stress, the more rapid the increase of contact stress as the fracture width shrinks.\nPhysically, the contact reference stress represents how hard and strong the fracture surface asperities are.\nThe lower the contact reference stress, the more gradual the change in fracture stiffness as pressure declines."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87dbe2-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 14, "chroma:document": "SPE-187348-MS 15\nEven though the contact reference stress does not have much impact on the pressure at which the fracture\nstiffness starts to changes noticeably, it does impact the fracture stiffness evolution, as shown in Fig. 13.\nFig. 14 shows the fracturing pressure and its derivatives for different contact reference stress on G-function\nand the square root of time plots. It can be observed that the contact reference stress has negligible influence\non early time pressure decline when fracturing pressure is still high. This is because only the fracture edges\nstart to come into contact during this period and the fracture stiffness remains roughly constant. However,\nafter this period as the fracture pressure declines further, more and more fracture surfaces come into contact\nand the contact reference stress begins to affect the pressure decline trend and the peak value of the pressure\nderivatives.\nFigure 12\u2014The relationship between contact stress and fracture width for different \u03c3 ref"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87dc3c-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 14, "chroma:document": "derivatives.\nFigure 12\u2014The relationship between contact stress and fracture width for different \u03c3 ref\nFigure 13\u2014Fracture stiffness evolution for different \u03c3 ref with a PKN geometry\nFigure 14\u2014Pressure decline response for different \u03c3 ref with a PKN geometry\nBesides the contact parameters, fracture geometry also affects fracture stiffness and its evolution during\nclosure. Fig. 15 shows fracture stiffness evolution for different fracture height while all the other parameters"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87dc8c-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 15, "chroma:document": "16 SPE-187348-MS\nremain the same as the Base Case. Similar to what Table 1 implies, for a PKN fracture geometry, the smaller\nthe fracture height, the higher the initial fracture stiffness. We can also observe that smaller fracture height\nleads to noticeable changes in fracture stiffness at higher fracturing pressure. This is because smaller fracture\nheight results in smaller fracture width at the same net pressure, so the more fracture surfaces can come into\ncontact at a higher fracture pressure. When fracture height is 5m, the effective fracture stiffness is already\naffected by surface contact when fluid pressure is 39 MPa.\nFigure 15\u2014Fracture stiffness evolution for different fracture height with a PKN geometry\nFig. 16 shows the fracturing pressure and its derivatives for different fracture height on G-function and\nthe square root of time plots. The results indicate that fracture height impacts the pressure decline trend"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87dce6-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 15, "chroma:document": "the square root of time plots. The results indicate that fracture height impacts the pressure decline trend\nsignificantly. Larger fracture height leads to later occurrence of the peak of the pressure derivative and this\nalso increases the peak value of the pressure derivatives.\nFigure 16\u2014Pressure decline response for different fracture height with a PKN geometry\nFig. 17 shows the fracturing pressure and its derivatives for different reservoir permeability on G-function\nand the square root of time plots. As expected, the pressure declines more rapidly when the reservoir\npermeability is large and the decline rate slows down as the difference between fracturing pressure and\ninitial reservoir pressure becomes smaller and smaller."}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87dd36-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 16, "chroma:document": "SPE-187348-MS 17\nFigure 17\u2014Pressure decline response for different reservoir permeability with a PKN geometry\nNext, we examine the impact of wellbore storage on the fully coupled pressure decline trend. The\nwater compressibility is assumed to be 4.35e-4 MPa\u22121. Fig. 18 shows the fracture-wellbore system stiffness\nevolution for different wellbore volume. As can be seen, when fracturing pressure is high, fracture stiffness\ndominates the system stiffness. However, as fracturing pressure continues declining, the fracture become\nless and less compressible and the role of wellbore storage becomes apparent. In general, the larger the\nwellbore volume, the more gradual and slower the increase in system stiffness will be.\nFigure 18\u2014Fracture-wellbore system stiffness evolution for different wellbore volume with a PKN geometry\nFig. 19 shows the corresponding fracturing pressure and its derivatives for different wellbore volumes"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87dd90-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 16, "chroma:document": "Fig. 19 shows the corresponding fracturing pressure and its derivatives for different wellbore volumes\non G-function and square root of time plots. It can be observed that larger wellbore volume leads to more\ngradual pressure decline trends. A larger wellbore volume also delays the occurrence of fracture closure\nand lowers the peak of the pressure derivative curve. It can be seen that WBS has a small impact during\nearly time of shut-in when the system stiffness is still dominated by fracture stiffness, however, as more\nand more of the fracture surface comes into contact and the fracture becomes stiffer, WBS effects become\napparent, and the after-flow of fluid from wellbore to fracture long after shut-in decelerates the pressure\ndecline rate and extends the tail of the pressure derivative after its reaches the peak. So the WBS effect on\npressure decline response must be accounted for when interpreting and analyzing DFIT data."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87de3a-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 17, "chroma:document": "18 SPE-187348-MS\nFigure 19\u2014Pressure decline response for different wellbore volume with a PKN geometry\nFig. 20 shows the fracturing pressure and its derivatives for different initial reservoir pressure on G-\nfunction and square root of time plots. It can be observed when the initial reservoir pressure is low; the\npressure declines more rapidly. This is also reflected by Eq. (18), a lower reservoir pressure leads to a higher\nleak-off rate. However, when the reservoir pressure is over pressurized with high initial pore pressure, the\npressure decline trend resembles a \"normal-leak-off behavior\". In this case (P 0=28MPa), one can still notice\nthat there exist a subtle \"bump\" in the pressure derivatives that indicates an increase in fracture stiffness as\nreflected in Fig. 10. For small values of \u03c3 ref, this increase can be too gradual to be noticeable on G-function\nand square root of time plots. This example deomonstrates that long closure time not only can be attributed"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87de94-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 17, "chroma:document": "and square root of time plots. This example deomonstrates that long closure time not only can be attributed\nto low formation permeability, but also can be the result of small pressure difference between ISIP and\ninitial pore pressure.\nFigure 20\u2014Pressure decline response for different initial reservoir pressure with a PKN geometry\nIn all the above synthetic simulations, the input minimum in-situ stress remains 35 MPa while other\nparameters are modified from the Base Case. The pressure decline trend varies substantially for different\nfracture geometry, contact parameters, reservoir permeability, wellbore volume and reservoir pore pressure.\nAny interpretation of the pressure data to determine the minimum in-situ stress must properly account for\nall these parameters. Choosing minimum in-situ stress where the pressure or its derivative departs from a\nstraight line (Nolte 1979 and 1986) is not reliable, because it ignores the changes in fracture compliance/"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87deee-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 17, "chroma:document": "straight line (Nolte 1979 and 1986) is not reliable, because it ignores the changes in fracture compliance/\nstiffness during closure and the pressure dependent leak-off at the fracture surface. In addition, pressure does\nnot decline linearly with G-function or \n  in most cases. The most widely used \"tangent line method\" to\npick up minimum in-situ stress using a tangential line with pressure derivatives ( Barree et al., 2009; Barree\net al., 2014) always underestimates minimum in-situ stress, especially in under-pressure reservoirs. Even\nthough this method has been the common practice in the industry, it is an emprical method and is subject to\nthe assumptions discussed earlier. The \"compliance method\" identifies closure pressure as the point at which\nfracture stiffness starts to change (McClure et al. 2016). As we have discussed earlier, the fracture closes"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87df3e-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 18, "chroma:document": "SPE-187348-MS 19\nprogressively from the edges to the center, As a result the fracture stiffness begins to increase long before\nthe fracture pressure drops to the minimum in-situ stress. A comparison of different methods of estimating\nminimum in-situ stress (closure pressure is interpreted as minimum in-situ stress) for all previous synthetic\ncases is presented in Table 3.\nTable 3\u2014Comparison of different methods of estimating minimum in-situ stress for PKN fracture geometry\nExamining Eq. (6)~Eq. (16) closely, it is evident that the pressure decline is governed by a linear PDE\nsystem in the formation that is coupled with an ODE at the fracture surface. The minimum in-situ stress is\njust one of many implicit factors that shape the evolution of the fracture pressure. Previous DFIT models\nsimplified this PDE-ODE system assuming Carter's leak-off and constant fracture stiffness during fracture"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87df98-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 18, "chroma:document": "simplified this PDE-ODE system assuming Carter's leak-off and constant fracture stiffness during fracture\nclosure, so that a closed form of an analytic solution could be obtained. Since our model shows minimum\nin-situ stress implicitly coupled with pressure, it is not possible to analytically estimate the minimum in-situ\nstress from a single diagnostic plot. Based on the analysis of our synthetic cases, it is clear that the \"tangent\nline method\" underestimates minimum in-situ stress and the \"compliance method\" overestimates closure in\nall cases if the closure pressure is interpreted as the minimum in-situ stress.\nOur analysis shows that the answer lies between these two limiting cases, regardless of parameters\nused in simulation. To keep the analysis pratical and not resorting to a complex inverse problem, we\npropose a \"variable compliance method,\" where the minimum in-situ stress can be obtained by averaging"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87dff2-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 18, "chroma:document": "propose a \"variable compliance method,\" where the minimum in-situ stress can be obtained by averaging\nthe dimensionless G-time or the square root of time of these two methods and extrapolating back to the\npressure curve that corresponds to the averaged G-time or the square root of time. An exception is when\npressure derivative plot resembles the so-called \"normal leak-off behavior,\" the changes in fracture fracture/\ncompliance may not be detectable on diagnostic plots and the \"tangent line method\" gives an estimation of\nminimum in-situ stress within an acceptable error.\nThe estimated minimum in-situ stress using \"variable compliance method\" is also included in Table 3.\nWhen we compare the estimated minimum in-situ stress obtained from the different methods with the input\nminimum in-situ stress, Table 3 shows that the \"variable compliance method\" gives significantly lower\nerrors. The maximum error does not exceed a few 100 KPa, while the \"tangent line method\" yield errors"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87e04c-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 18, "chroma:document": "errors. The maximum error does not exceed a few 100 KPa, while the \"tangent line method\" yield errors\nthat are an order of magnitude larger.\nEven though we have focused on the PKN fracture geometry in this section, these conclusions are also\napplicable to a KGD fracture geometry. We can obtain the results for a KGD geometry if we replace fracture\nheight with fracture length in calculating the evolution of fracture stiffness, as discussed in the Appendix B."}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87e0a6-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 19, "chroma:document": "20 SPE-187348-MS\nPKN Fracture Geometry with Multiple Layers.   In previous section, we have analyzed pressure decline\nresponse of DFIT in a single layer; however, it is possible that fracture can penetrate into adjacent layers\nwhere the minimum in-situ stresses are different. Fig. 21 illustrates the examples where fracture geometry\nspan across multiple layers. In a single layer case, fracture gradually closes from edges to center on rough\nfracture wall, while in multilayer cases; fracture can close first within the interval where the minimum in-\nsitu stress is the largest.\nFigure 21\u2014Illustration of multilayer cases with different minimum in-situ stresses\nAssuming the input parameters are the same as the base case in the previous section (provided in Table\n2), and fracture geometry and minimum in-situ stresses are illustrated in Fig. 21, the fracture stiffness\nevolution can be calculated using non-local fracture modeling (Wang and Sharma 2017), as shown in Fig."}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87e0f6-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 19, "chroma:document": "evolution can be calculated using non-local fracture modeling (Wang and Sharma 2017), as shown in Fig.\n22. \"Multilayer Case 1\" is the most common scenario where height recession can happen during fracture\nclosure process where fracture closes first within formation barriers (e.g., Layer 1 and Layer 3) where\nstresses are higher. Compared to the single layer case in Fig. 22, height recession (i.e., Multilayer Case 1)\nleads to earlier and steeper increase in fracture stiffness. The fracture stiffness evolution of the single layer\ncase and \"Multilayer Case 2\" are similar, because they have the same average stress.\nFigure 22\u2014Stiffness evolution for single and multilayer cases\nFig. 23 shows the fracturing pressure and its derivatives for single and multilayer cases on G-function\nand square root of time plots. It can be observed that the trend of pressure derivatives generally reflects"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87e150-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 19, "chroma:document": "and square root of time plots. It can be observed that the trend of pressure derivatives generally reflects\nthe evolution of fracture stiffness. The case of height recession (i.e., Multilayer Case 1) leads to earlier and\nsteeper increase in pressure derivatives on both G-function and square root of time plots."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87e1a0-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 20, "chroma:document": "SPE-187348-MS 21\nFigure 23\u2014Pressure decline response for single and multilayer cases\nA comparison of different methods of estimating minimum in-situ stress for above single and multilayer\ncases is presented in Table 4. Again, we can oberve that the tangent line method under estimates while\ncompliance method overestimates minimum in-situ stress (i.e., 35 MPa in the target layer), and our \"variable\ncompliance method\" gives the least errors for all cases.\nTable 4\u2014Comparison of different methods of estimating minimum in-situ stress for single and multilayer cases\nBased on the disscussion by McClure et al. (2016), the \"compliance method\" gives an estimate of the\nmecanical closure pressure, and this closure pressure can be 1-2 MPa above the actual minimum in-situ\nstress, depending on fracture geometry and contact width. Our anlaysis in Table 3 shows that the differrence\nbetween the infered closure pressure and input minimum in-situ stress that is constrained within 1-2 MPa"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87e1fa-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 20, "chroma:document": "between the infered closure pressure and input minimum in-situ stress that is constrained within 1-2 MPa\nis only limited to a few cases where fracture height is large or contact width is small. For a typical DFIT,\nthe injeciton volume is normally only a few cubic meters and the resulting fracture height or radius is often\nlimited. In addition, depending on rock mineralogy and heterogeneity, the height of asperities can range\nfrom hundreds to thousands of micrometers (Bhide et al. 2014; Fredd et al. 2000; Sakaguchi et al. 2006;\nWells and Davatzes 2015; Warpinski et al. 1993; Zou et al. 2015), and, as shown by the field tiltimeter\nsurvey, the residual fracture width can be as large as 20-30% of the maximum fracture width during pumping\n(Warpinski et al. 2002). Remember is determined by the tallest asperities, so it is challenging to calibrate\nthe minimum in-situ stress using the \"compliance method\". In addtion, if the fracture has penetetrated"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87e27c-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 20, "chroma:document": "the minimum in-situ stress using the \"compliance method\". In addtion, if the fracture has penetetrated\ninto barrier layers with higher minimum pricinpal stress (e.g., Multilayer Case1 in Table 4), the fracture\nwalls will come into contact at a higher pressure within the barrier layers and increase the overall fracture\nstiffness, so the \"compliance method\" can furthur overstimate the minimun in-situ stress. However, in all\nsimulated cases the presented \"variable compliance method\" provides a much more consistent estimation\nof minimum in-situ stress. Most importantly, the results are insensitive to input parameters such as fracture\nsurface roughness and fracture geometry, which significantly reduces the uncertainties associated with the\n\"compliance method\".\nRadial Fracture Geometry.   Next, we investigate the pressure decline response for radial fracture geometry,\nusing parameters from Table 2 and assuming W0 is 2 mm, \u03c3 ref is 5 MPa, fracture radius is 10 m and reservoir"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87e344-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 21, "chroma:document": "22 SPE-187348-MS\npermeability is 0.0005 md as the Base Case. Fracturing pressure decline response for different W0 and are\nshown in Fig. 24. The results and trends are similar to those obtained for the PKN geometry, a small contact\nwidth leads to steeper changes in pressure decline rate and the slope of pressure derivative, and increases\nthe peak value of pressure derivative.\nFigure 24\u2014Pressure decline response for different W 0 with a radial geometry\nFig. 25 shows the pressure decline response on G-function and the square root of time plots for different\ncontact reference stress. As in the PKN cases, a larger value of contact reference stress increases the pressure\ndecline rate after the fracturing pressure drops to a certain level and reduces the peak value of pressure\nderivatives.\nFigure 25\u2014Pressure decline response for different \u03c3 ref with a radial geometry\nFig. 26 shows the corresponding pressure decline response on G-function and the square root of time"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87e394-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 21, "chroma:document": "Fig. 26 shows the corresponding pressure decline response on G-function and the square root of time\nplots. A larger fracture radius leads to lower fracture stiffness and this, in turn, results in a lower pressure\ndecline rate.\nFigure 26\u2014Pressure decline response for different fracture radiiwith a radial geometry."}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87e3e4-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 22, "chroma:document": "SPE-187348-MS 23\nFig. 27 shows the fracturing pressure and its derivatives for different reservoir permeability on G-\nfunction and the square root of time plots. As expected, pressure declines more rapidly when the reservoir\npermeability is large. At early time, the pressure decline rate increases because an increase in the fracture\nstiffness dominates changes in leak-off rate. At late time when the fracture becomes less compliant, the\npressure decline rate decreases due to smaller and smaller differences between the fracture pressure and the\ninitial reservoir pressure. Eventually, the leak-off rate will drop to zero as the fracture pressure becomes\nequal to the reservoir pressure.\nFigure 27\u2014Pressure decline response for different reservoir permeability with a radial geometry\nFig. 28 shows the pressure decline response for different wellbore volumes on G-function and the square\nroot of time plots. Compared to Fig. 19, it can be observed that the WBS effect has stronger influence on"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87e434-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 22, "chroma:document": "root of time plots. Compared to Fig. 19, it can be observed that the WBS effect has stronger influence on\npressure decline trend in our radial fracture geometry for the same wellbore volume. This is because the\nfracture volume and fracture surface are smaller than that in previous case of PKN fracture geometry, which\nintensifies the impact of WBS effect.\nFigure 28\u2014Pressure decline response for different wellbore volume with a radial geometry\nFig. 29 shows the fracturing pressure and its derivatives for different initial reservoir pressure on G-\nfunction and square root of time plots. Similar to Fig. 20, pressure declines rapidly in depleted reservoirs and\ndeclines gradually with much lower peak values of pressure derivatives in over-pressurized reservoirs. Also\nthe pressure decline trend resembles a \"normal-leak-off behavior\" when pore pressure is abnormally high."}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87e484-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 23, "chroma:document": "24 SPE-187348-MS\nFigure 29\u2014Pressure decline response for different initial reservoir pressure with a radial geometry\nTable 5 shows a comparison of the different methods of estimating minimum in-situ stress (closure\npressure is interpreted as the minimum in-situ stress) for the above synthetic cases for a radial fracture\ngeometry. Again, the results indicate that the \"tangent line method\" always underestimates minimum in-\nsitu stress while the \"compliance method\" overestimates the minimum in-situ stress (if the closure stress\nis assumed to be equal to the minimum in-situ stress) if no correction is applied (except for the so-\ncalled \"normal leak-off behavior\" where the changes in fracture stiffness/compliance are undetectable). The\nestimated minimum in-situ stress using our proposed \"variable compliance method\" give the smallest errors\nin all simulated cases.\nTable 5\u2014Comparison of different methods of estimating minimum in-situ stress for radial fracture geometry."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87e4d4-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 23, "chroma:document": "in all simulated cases.\nTable 5\u2014Comparison of different methods of estimating minimum in-situ stress for radial fracture geometry.\nFrom our synthetic analysis presented in this section, we can conclude that the pressure decline response\nduring DFIT is affected by many parameters, such as fracture geometry, reservoir properties, fracture surface\nroughness, etc. So history match DFIT data can be non-unique if we don't have good knowledge (from other\nindependent analysis and measurements) of the constraints on these parameters. Nevertheless, our proposed\n\"variable compliance method\" gives the most reliable estimation of minimum in-situ stress regardless of\nthese parameters, so the uncertainties that associated with fracture geometry, surface roughness and reservoir\nproperties do not impact our estimation of minimum in-situ stress."}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87e524-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 24, "chroma:document": "SPE-187348-MS 25\nField Data Interpretation\nField Case 1\nThe first field case to be analyzed comes from a horizontal well-A drilled through a shale formation. The\nmeasured depth is around 5500 m and a diagnostic fracture injection test is conducted at the toe of the\nhorizontalwellbore, with 2.35 m3 of water injection in 3 minutes, then the well was shut-in for 27 days. Fig.\n30 shows the pressure decline response on G-function and the square root of time plots. As in the previous\nsynthetic cases, even though G-function and the square root of time plots give exactly the same quantitative\ninformation, it is more intuitive to infer the shut-in time on the square root of time plot, while the shut-in\ntime is implicit in the value of G-function, as reflected by Eq. (3)~Eq. (5). This field case also exhibits linear\npressure derivatives and low peak value of pressure derivative with a long extended tail after it reaches the"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87e574-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 24, "chroma:document": "pressure derivatives and low peak value of pressure derivative with a long extended tail after it reaches the\npeak, we can infer that the impact of wellbore storage should be substantial and this reservoir is potentially\nover-pressurized.\nFigure 30\u2014Pressure decline response from Well-A on G-function and square root of time plots\nFrom the signature of pressure derivatives, the changes in fracture stiffness/compliance are undetectable,\nso the minimum in-situ stress is picked at 31.8 MPa as the \"tangent line method\" suggests. A closer look at\nthe square root of time plot, shows that the pressure drops significantly during the first one hour of shut-in.\nBy examining the pressure derivative curve, whose extrapolated value is not zero when \u0394t=0, we can identify\nthat the excessive pressure drop at the beginning is caused by tip extension or near-wellbore tortuosity. The\napparent ISIP is 45.5 MPa while the true ISIP is 34.5MPa. Remember net pressure is the pressuredifference"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87e5ba-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 24, "chroma:document": "apparent ISIP is 45.5 MPa while the true ISIP is 34.5MPa. Remember net pressure is the pressuredifference\nbetween thefluid pressure inside the fracture and closure pressure. In reality, the pressure measured at the\nwellhead can be much higher than expected during fracture propagationduring theearly time of shut-in\nperiod, because significant pressure drop can happen in the near-wellbore region and along the horizontal\nwellbore, due to friction and tortuous/complex fracture path that initiated from perforation clusters. So\npressure data from this period cannot be used to infer fracturing parameters and reservoir properties. If a\nrate step down test can be performed before complete shut-in, the relationship between early-time excessive\npressure drop and flow rate can be quantified.\nLog-log plots of pressure derivatives are often used to determine flow regimes for after closure analysis"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87e60a-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 24, "chroma:document": "Log-log plots of pressure derivatives are often used to determine flow regimes for after closure analysis\nand assist the interpretation of closure stress (Barree et al. 2019; Hawkes et al. 2012; Soliman et al. 2015).\nEven though Well-A was shut-in for nearly four weeks, pseudo-radial flow is still absent, as shown in\nFig. 31. This is a clear evidence of extremely low formation permeability. Without pseudo-radial flow,\nformation pore pressure and permeability cannot be determined independently with enough confidence\nusing just after closure analysis alone. One has to analyze the whole spectrum of DFIT data to reach a\nconsistent interpretation. Using late time data from the linear flow regime, and plotting Pf vs \n  on\nCartesian coordinates, the intercept on the y-axis when \u0394t \u2192 \u221e is the estimated initial pore pressure, which\nis 29.1 MPa. This indicates it is indeed an over-pressurized reservoir. From geological and petrophysical"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87e65a-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 25, "chroma:document": "26 SPE-187348-MS\nstudies, it is known that the thickness of the target formation is 24.4 m with an average porosity of 0.07\nand in-situ fluid viscosity of 0.257 cp, The Young's modulus is 38.9 GPa, the Poisson's ratio is 0.2 and\nformation total compressibility is 3e-3 MPa\u22121. Hydraulic fracture modeling indicates that the fracture is\nwell contained within the target formation with penny-shaped fracture geometry, and the fracture radius is\nroughly 12m. Based on this information, the pressure decline response can be matched globally using our\nfully coupled DFIT model (Eq. (6)-Eq. (16) with pressure dependent fracture stiffness obtained from our\nnon-local fracture closure model (Wang and Sharma 2017)), from the end of tip extension/near-wellbore\ntortuosity dominated period to the end of the test.\nFigure 31\u2014Log-log pressure derivative plot of Well-A\nFig. 32 shows the pressure decline response predicted by our fully coupled DFIT model and field data"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87e6aa-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 25, "chroma:document": "Fig. 32 shows the pressure decline response predicted by our fully coupled DFIT model and field data\non G-function and the square root of time plots. The results indicate that our simulated pressure matches\nextremely well with the field data for the entire duration of the test, excluding the first hour of shut-in. Our\nmatched reservoir permeability is 220 nd, which is within the range of other petrophysical measurements.\nThe matched contact width and contact reference stress are 0.7 mm and 3 MPa, respectively, and Fig. 33\nshows the corresponding fracture and fracture-wellbore system stiffness, based on the matched fracture\ngeometry, wellbore volume and contact parameters. Because the DFIT was conducted at the toe of a\nhorizontal well, it is no surprise that the wellbore storage effect is significant, considering such a large\ncontrast between the wellbore and fracture volume. We can also observe that the fracture stiffness starts"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87e6fa-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 25, "chroma:document": "contrast between the wellbore and fracture volume. We can also observe that the fracture stiffness starts\nto increase noticeably when pressure drops to 34 MPa, over 2 MPa above the minimum in-situ stress. It\nis apparent that the so-called \"normal leak-off behavior\" is just a coincidence when the effect of pressure\ndependent leak-off at the fracture surface and variable fracture compliance counterbalance each other during\nthe fracture closure process.\nFigure 32\u2014Matched pressure decline response for Well-A on G-function and square root of time plots"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87e740-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 26, "chroma:document": "SPE-187348-MS 27\nFigure 33\u2014Matched fracture and fracture-wellbore system stiffness for Well-A\nField Case 2\nThe second field case to be analyzed comes from a vertical well-B drilled through a shale formation. The\ntotal wellbore length is around 2000 m and a diagnostic fracture injection test is conducted with 4.7 m3 of\nwater injection in6 minutes, then the well was shut-in for 11 days. Fig. 34 shows the pressure decline response\non G-function and the square root of time plots. If we pick the fracture closure pressure using the \"tangent\nline method,\" both G-function and the square root of time plots give the same minimum in-situ stress of\n50.4 MPa. One can also notice that there is a slight \"bump\" in the pressure derivatives soon after shut-in. For\nshale formations, this early-time accelerated pressure drop is not likely to be caused by PDL, but attributed\nto tip extension and near wellbore tortuosity (continued after-flow of fluid from wellbore to fracture after"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87e790-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 26, "chroma:document": "to tip extension and near wellbore tortuosity (continued after-flow of fluid from wellbore to fracture after\nshut-in). It should be observed that in the early-time pressure transient response, the pressure derivative\nstarts to deviate upwards at around 55 MPa. Using \"variable compliance method\" through averaging the G-\nfunction or the square root of time that corresponding to a pressure of 50.4 MPa and 55 MPa, respectively,\nwe can obtain the estimated minimum in-situ stress of 52.8 MPa by picking the pressure that corresponds\nto the averaged G-function or square root of time.\nFigure 34\u2014Pressure decline response from Well-B on G-function and square root of time plots\nFrom Fig. 35, we can observe that only linear flow emerged after closure. Using late time data from the\nlinear flow regime, the estimated initial pore pressure is 33.7MPa. Geological and petrophysical studies\nindicate that the thickness of the target formation is 5 m with an average porosity of 0.03 and in-situ fluid"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87e7ea-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 26, "chroma:document": "indicate that the thickness of the target formation is 5 m with an average porosity of 0.03 and in-situ fluid\nviscosity of 0.28 cp. The Young's modulus is 39.5 GPa, the Poisson's ratio is 0.25 and formation total\ncompressibility is 1.9e-3 MPa\u22121. Hydraulic fracture modeling shows that the fracture is well contained within\nthe target formation with roughly 34 m fracture half length. Based on this information and assuming PKN\ngeometry, the pressure decline response can be matched globally using our global DFIT model by adjusting\nreservoir permeability and contact parameters."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87e83a-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 27, "chroma:document": "28 SPE-187348-MS\nFigure 35\u2014Log-log pressure derivative plot of Well-B\nFig. 36 shows the predicted pressure decline response and field data on G-function and the square root of\ntime plots. The results indicate that our simulated pressure agrees well with field data and it even captures\nthe downward curve of pressure derivatives before closure. Traditionally, this phenomenon is interpreted\nas \"height recession\" or \"transverse storage,\" where the top and bottom barrier formations close first and\nsqueeze fluid into the target formation during the closure process. This can be regarded as a special case of\n\"variable compliance,\" as demonstrated in Fig. 23. Because fractures always close from their edges to the\ncenter on rough walls, this constantly alters fracture compliance/stiffness. This variable fracture compliance\ncoupled with fracture pressure dependent leak-off dictates our pressure decline trend for a given formation."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87e88a-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 27, "chroma:document": "coupled with fracture pressure dependent leak-off dictates our pressure decline trend for a given formation.\nIn this field case, our matched reservoir permeability is 210 nd, and the matched contact width and contact\nreference stress are 1.2 mm and 1.1 MPa, respectively.\nFigure 36\u2014Matched pressure decline response for Well-B on G-function and square root of time plots\nFig. 37 shows the corresponding fracture and fracture-wellbore system stiffness as a function of fluid\npressure inside fracture. It can be observed that both fracture and system stiffness increase gradually as\npressure declines. Even though this test was conducted in a vertical well with relative moderate wellbore\nvolume, the wellbore storage effect on the system stiffness is still significant. Re-examining Eq. (16), we\ncan see that the relative influence of wellbore storage not only depends upon the ratio of wellbore to fracture"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87e902-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 27, "chroma:document": "can see that the relative influence of wellbore storage not only depends upon the ratio of wellbore to fracture\nvolume, but it also depends on fracture stiffness itself. For the same wellbore volume, the higher the fracture\nstiffness, the less compressible the fracture is, so the compressed fluid in the wellbore plays a bigger role to\ncompensate leak-off and pressure drop. In this field case, with fracture height contained in 5 m, the fracture\nstiffness should be at least 5.36 MPa/mm (From Table 1) even without considering the contact of fracture\nsurfaces. If the fracture geometry and reservoir properties can be constrained with enough confidence, the\nestimated contact parameters and fracture stiffness evolution can be used to reconstruct the dynamic fracture"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87e952-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 28, "chroma:document": "SPE-187348-MS 29\nclosure process and the evolution of fracture width distribution, thus, the un-propped fracture conductivity\nand its pressure/stress dependence can be inferred.\nFigure 37\u2014Matched fracture and fracture-wellbore system stiffness for Well-B\nConclusions and Discussion\nSince the introduction of Nolte's pioneering work, diagnostic fracture injection test (DFIT) has evolved into\na commonly used technique and are generally accepted as a reliable way to obtain fracturing parameters\n(e.g., minimum in-situ stress, leak-off behavior) and reservoir properties (e.g., initial pore pressure and\nrepresentative formation permeability). These key parametersare needed to run fracture models, post-\nfracture productionprediction and economical evaluations. In this article, we first reviewed Nolte's G-\nfunction model and the associated assumptions (adopted by many subsequent works), and derived the"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87e998-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 28, "chroma:document": "function model and the associated assumptions (adopted by many subsequent works), and derived the\nequations that underpin the interpretation of the square root of time plot in low permeability reservoirs. We\ndemonstrated that the square root of time plot gives the same quantitative information obtained from the\nG-function plot.\nA new global DFIT model is proposed, which for the first time, has the capability to predict pressure\ndecline response across the entire duration of a DFIT test using a single, coherent mathematic framework in\nlow permeability formations. We show how each factor impacts the pressure decline response and the effects\nof previous overlooked coupled mechanisms are examined and discussed, along with synthetic and field\ncase studies. The content of this article promotes our conceptual understanding of pressure transient and\ncoupled behaviors during DIFT and enables us to revisit DFIT analysis from a new perspective. Conclusions"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87e9e8-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 28, "chroma:document": "coupled behaviors during DIFT and enables us to revisit DFIT analysis from a new perspective. Conclusions\nreached from the analysis presented in this paper include the following.\n1.Carter's leak-off, (leak-off rate is proportional to \n ), is applicable during stable fracture\npropagation, when net pressure is roughly constant and injection time is limited. However, during the\nfracture closure peroid, the fluid pressure in the fracture declines significantly, and the leak-off rate\nbecomes fracture pressure dependent and ceases to be proportional to \n .\n2. The so-called \"normal leak-off,\" with a straight line of pressure and its derivatives on G-function and\nthe square root of time plots, is not \"normal\" at all. This is sometimes observed in data sets when\nmechanisms that accelerate pressure decline (e.g., increase in fracture stiffness during closure) offset\nthose that decelerate pressure decline (e.g., pressure dependent leak-off at the fracture surface and"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87ea38-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 28, "chroma:document": "those that decelerate pressure decline (e.g., pressure dependent leak-off at the fracture surface and\nwellbore storage effects). So any \"before closure\" analysis based on this conventional \"normal leak-\noff\" assumption to estimate fracture geometry and formation permeability should be re-examined.\n3. Hydraulic fractures do not close as two smooth parallel plates, but rather close progressively from\nedges to the center on asperities and roughness surfaces, as confirmed by our modeling results and\nfield tiltmeter measurements. The fracture stiffness increases (compliance decreases) gradually as"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87ea88-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 29, "chroma:document": "30 SPE-187348-MS\nmore and more fracture surfaces come into contact, even before the fracture pressure drops to the\nminimum in-situ stress.\n4. It is shown that the pressure decline response during fracture closure is governed by a PDE-ODE\nsystem, where the pressure at the fracture surface should be solved simultaneously with the pressure\ninside the formation. Previous DFIT models assumed a constant fracture compliance/stiffness and\nCarter's leak-off oversimplified and decoupled this system, which prevents them from predicting\npressure decline across the entire duration of the DFIT test.\n5. The conventional \"tangent line method\" underestimates the minimum in-situ stress, especially\nin depleted reservoirs. The \"compliance method\" picks the mechanical closure pressure but\noverestimates the minimum in-situ stress (if the two are equated), especially when height recession\noccurs. Height recession (or transverse storage) is a just special type of variable compliance\nphenomenon."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87ead8-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 29, "chroma:document": "occurs. Height recession (or transverse storage) is a just special type of variable compliance\nphenomenon.\n6. The \"variable compliance method\" proposed in this study suggests that picking an average of the\ndimensionless G-time (or the square root of time) of the \"tangent line\" and \"compliance\" methods\nprovides a much more robust and reliable way of estimating the minimum in-situ stress when concave\npressure derivative exist. The uncertainties that associated with fracture geometry, surface roughness\nand reservoir properties do not impact our estimation of minimum in-situ stress using \"variable\ncompliance method\". Results using this method match well with field data.\n7. Fracture closure on asperities and rough fracture surfaces is a gradual process starting at the fracture\ntip. The model presented here can be used to define the relationship between fracture width and\ncontact stress using DFIT pressure data. If other matching parameters can be constrained with enough"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87eb28-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 29, "chroma:document": "contact stress using DFIT pressure data. If other matching parameters can be constrained with enough\ncertainty, this information can be used to infer unpropped fracture conductivity as a function of\neffective stress. This adds tremendous value to DFIT analysis for completion design and estimates\nof well productivity and EUR.\nNomenclature\na = Half fracture height (PKN model) or half fracture length (KGD model), L, m\nAf =Fracture surface area (one face of one wing),L2, m2\nct =Formation total compressibility, Lt2/m, 1/Pa\ncW =Water compressibility, Lt 2/m, 1/Pa\nCL =Carter's leak-off coefficient, \n , \nE = Young's modulus, m/Lt 2, Pa\nE\u2032 = Plane strain Young's modulus, m/Lt 2, Pa\ng(\u0394t D) =Dimensionless g-function of time\nG(\u0394t D) =Dimensionless G-function of time\nhf =Fracture height, L, m\nISIP = Instant shut-in pressure, m/Lt2, Pa\nk = Formation permeability, L2, m2\nP = Pressure, m/Lt2, Pa\nPf =Fracturing pressure, m/Lt2, Pa\nPnet =Fracturing net pressure, m/Lt 2, Pa"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87eb78-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 29, "chroma:document": "k = Formation permeability, L2, m2\nP = Pressure, m/Lt2, Pa\nPf =Fracturing pressure, m/Lt2, Pa\nPnet =Fracturing net pressure, m/Lt 2, Pa\nP0 =Initial reservoir pressure, m/Lt2, Pa\nqf =Leak-off rate (one wing), L3/t, m3/s\nrD =Normalized radius\nrp =Productive surface ratio\nrf =Local fracture radius, L, m\nRf =Fracture radius, L, m"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87ebc8-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 30, "chroma:document": "SPE-187348-MS 31\nSf =Fracture stiffness, which is the reciprocal of fracture compliance, m/L2t2, Pa/m\nSS =Fracture-wellbore system stiffness, m/L2t2, Pa/m\nt =Generic time,t, s\ntD =Dimensionless time\nte =Generic time at the end of pumping, t, s\ntp =Pumping time, t, s\n\u0394t = Total shut-in time, t, s\n\u0394tD =Dimensionless shut-in time, t, s\nxf =Fracture half-length, L, m\nUf =Flux velocity through fracture surface with constant fracturing pressure, L/t, m/s\nVf =Fracture volume (one wing), L3, m3\nVW =Half of wellbore volume, L3, m\nW0 =Contact width, L,m\nWf =Local fracture width, L, m\n =Averagefracture width, L, m\n\u03bcf =Fluid viscosity, m/Lt, Pa\u00b7s\nv = Poisson's ratio\n\u03c3c =Contact stress, m/Lt2, Pa\n\u03c3hmin =Minimum in-situ stress, m/Lt2, Pa\n\u03c3ref =Contact reference stress, m/Lt2, Pa\n\u03d5 = Formation porosity\nSubscript\nn = Variables at nth time interval\nj =Variables at jth time interval\ni =Variables at jth time interval\nReferences"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87ec22-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 30, "chroma:document": "\u03d5 = Formation porosity\nSubscript\nn = Variables at nth time interval\nj =Variables at jth time interval\ni =Variables at jth time interval\nReferences\nBarr, D.T., 1991. Leading-edge analysis for correct simulation of interface separation and hydraulic fracturing. PHD\ndissertation, MIT, Cambridge, MA.\nBarree, R. D. 1998. Applications of pre-frac injection/falloff tests in fissured reservoirs - field examples. Paper SPE 39932\npresented at the SPE Rocky Mountain Regional/Low Permeability Reservoirs Symposium, Denver, CO, 5-8 April.\nhttp://dx.doi.org/10.2118/39932-MS.\nBarree, R. D., Mukherjee.H. 1996. Determination of pressure dependent leakoff and its effect on fracture geometry.\nPaper SPE 36424 presented at the 71st Annual Technical Conference and Exhibition, Denver, 6-9 October. http://\ndx.doi.org/10.2118/36424-MS.\nBarree, R. D., Miskimins,J. L. Gilbert.J. V . 2014. Diagnostic fracture injection tests: common mistakes, misuses, and"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87ec68-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 30, "chroma:document": "dx.doi.org/10.2118/36424-MS.\nBarree, R. D., Miskimins,J. L. Gilbert.J. V . 2014. Diagnostic fracture injection tests: common mistakes, misuses, and\nmisdiagnoses. SPE Production & Operations 30 (2):84\u201398, http://dx.doi.org/10.2118/169539-PA.\nBarree, R.D., Barree, V .L. and Craig, D., 2009. Holistic fracture diagnostics: consistent interpretation of prefrac\ninjection tests using multiple analysis methods. SPE Production & Operations, 24(03), pp.396\u2013406. http://\ndx.doi.org/10.2118/107877-PA\nBarton, N., Bandis,S. Bakhtar.K. 1985. Strength, deformation and conductivity coupling of rock joints. International\nJournal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts  22 (3): 121\u2013140, http://\ndx.doi.org/10.1016/0148-9062(85)93227-9.\nBenelkadi, S., and Tiab, D. 2004. Reservoir Permeability Determination Using After-Closure Period Analysis of\nCalibration Tests. SPE Reservoir Evaluation & Engineering, V ol (7)03: 230\u2013237. http://dx.doi.org/10.2118/88640-PA"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 30, "chroma:document": "Calibration Tests. SPE Reservoir Evaluation & Engineering, V ol (7)03: 230\u2013237. http://dx.doi.org/10.2118/88640-PA\nBhide, R., Gohring, T., McLennan, J. and Moore, J., 2014. Sheared Fracture Conductivity. In Proceedings of 39th\nWorkshop on Geothermal Reservoir Engineering"}


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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 31, "chroma:document": "32 SPE-187348-MS\nCastillo, J.L. 1987. Modified fracture pressure decline analysis including pressure-dependent leakoff. Paper SPE\n164417 presented at the SPE/DOE Low Permeability Reservoir Symposium, Denver, CO, 18-19 May. http://\ndx.doi.org/10.2118/16417-MS.\nChipperfield, S. T. 2006. After-Closure Analysis to Identify Naturally Fractured Reservoirs. SPE Reservoir Evaluation &\nEngineering, V ol (9)01:50\u201360. http://dx.doi.org/10.2118/90002-PA\nEconomides, Michael J., Kenneth G. Nolte. 2000. Reservoir Stimulation, 3rd edition. Chichester, England; New York,\nJohn Wiley.\nEngland, A.H. and Green, A.E., 1963, April. Some two-dimensional punch and crack problems in classical elasticity. In\nProc. Camb. Phil. Soc (V ol. 59, No. 2, p. 489). http://www.doi.org/10.1017/S0305004100037099\nFredd, C. N., McConnell, S. B., Boney, C. L., & England, K. W. 2000. Experimental Study of Hydraulic Fracture\nConductivity Demonstrates the Benefits of Using Proppants. Paper SPE 60326 presented at the SPE Rocky"}


created_at: 2023-11-16 17:06:54
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 31, "chroma:document": "Conductivity Demonstrates the Benefits of Using Proppants. Paper SPE 60326 presented at the SPE Rocky\nMountain Regional/Low-Permeability Reservoirs Symposium and Exhibition, Denver, Colorado, 12-15 March. http://\ndx.doi.org/10.2118/60326-MS\nFu, Y ., Dehghanpour, H., Ezulike, D. O., and Jones, R. S. 2017. Estimating Effective Fracture Pore V olume From Flowback\nData and Evaluating Its Relationship to Design Parameters of Multistage-Fracture Completion. SPE Production &\nOperations (PrePrint). https://doi.org/10.2118/175892-PA\nGringarten, A. C., Ramey, H. J., and Raghavan, R. 1974. Unsteady-State Pressure Distributions Created by a Well with a\nSingle Infinite-Conductivity Vertical Fracture. SPE Journal. V ol(14)04:347\u2013360. http://dx.doi.org/10.2118/4051-PA\nHawkes, R., Bachman, R. C., Walters, D. A., Toussaint, F. L., and Settari, T. 2012. Reappraisal of the G Time Concept\nin Mini-Frac Analysis. Paper SPE 160169 presented at the SPE Annual Technical Conference and Exhibition, San"}


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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 31, "chroma:document": "in Mini-Frac Analysis. Paper SPE 160169 presented at the SPE Annual Technical Conference and Exhibition, San\nAntonio, Texas, USA, 8-10 October. https://dx.doi.org/10.2118/160169-MS\nLiu, G., Ehlig-Economides, C. and Sun, J. 2016. Comprehensive Global Fracture Calibration Model. Paper SPE\n181856 presented at the SPE Asia Pacific Hydraulic Fracturing Conference, Beijing, China, August 24-26. http://\ndx.doi.org/10.2118/181856-MS\nLoughry, D., Epps, D. and Forrest, J., 2015. Using Pad ISIP, DFIT, and ESP Data to Generate a Pore Pressure Model for\nthe Midland Basin. Paper presented at the Unconventional Resources Technology Conference, San Antonio, Texas,\nUSA, 20-22 July. https://doi.org/10.15530/URTEC-2015-2162973\nMarongiu-Porcu, Matteo, Albertus Retnanto, Michael J. Economides, Christine Ehlig-Economides. 2014. Comprehensive\nfracture calibration test design. Paper SPE 168634 presented at the SPE Hydraulic Fracturing Technology Conference,"}


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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 31, "chroma:document": "fracture calibration test design. Paper SPE 168634 presented at the SPE Hydraulic Fracturing Technology Conference,\nThe Woodlands, TX, February 4-6. http://dx.doi.org/10.2118/168634-MS\nMayerhofer, M. J., and Economides, M. J. 1997. Fracture-Injection-Test Interpretation: Leakoff Coefficient vs.\nPermeability. SPE Production & Facilities. V ol (12) 04:231\u2013236. http://dx.doi.org/10.2118/28562-PA\nMayerhofer, M. J., Ehlig-Economides, C. A., and Economides, M. J. 1995. Pressure Transient Analysis of Fracture\nCalibration Tests. Journal of Petroleum Technology. V ol(47)03:229\u2013234. http://dx.doi.org/10.2118/26527-PA\nMcClure, M.W. 2017. The Spurious Deflection on Log-Log Superposition Time Derivative Plots of Diagnostic Fracture.\nSPE Reservoir Evaluation & Engineering (In Press). https://doi.org/10.2118/186098-PA\nMcClure, M.W., Jung, H., Cramer, D.D. and Sharma, M.M., 2016. The Fracture Compliance Method for Picking Closure"}


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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 31, "chroma:document": "McClure, M.W., Jung, H., Cramer, D.D. and Sharma, M.M., 2016. The Fracture Compliance Method for Picking Closure\nPressure from Diagnostic Fracture-Injection Tests. SPE Journal 21(04):1321\u20131339. http://dx.doi.org/10.2118/179725-\nPA\nMirani, A., Marongiu-Porcu, M., Wang, H., and Philippe, E. Production Pressure Drawdown Management for Fractured\nHorizontal Wells in Shale Gas Formations. 2016. Paper SPE 181365 presented at the SPE Annual Technical Conference\nand Exhibition held in Dubai, UAE, 26-28 Sep. http://dx.doi.org/10.2118/181365-MS\nNolte, K. G. 1979. Determination of Fracture Parameters from Fracturing Pressure Decline. Paper 8341 presented\nat the SPE Annual Technical Conference and Exhibition, Las Vegas, Nevada, 23-26 September. http://\ndx.doi.org/10.2118/8341-MS\nNolte, K. G. 1986. A General Analysis of Fracturing Pressure Decline with Application to Three Models. SPE Formation\nEvaluation. V ol (1) 06:571\u2013583. http://dx.doi.org/10.2118/12941-PA"}


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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 31, "chroma:document": "Evaluation. V ol (1) 06:571\u2013583. http://dx.doi.org/10.2118/12941-PA\nNolte, K. G., Maniere, J. L., and Owens, K. A. 1997. After-Closure Analysis of Fracture Calibration Tests. Paper SPE\n38676 presented at the SPE Annual Technical Conference and Exhibition, San Antonio, Texas, 5-8 October. http://\ndx.doi.org/10.2118/38676-MS\nPotocki, D. J. 2012. Understanding Induced Fracture Complexity in Different Geological Settings Using DFIT Net\nFracture Pressure. Paper SPE 162814 presented at the SPE Canadian Unconventional Resources Conference, Calgary,\nAlberta, Canada, 30 October-1 November. http://dx.doi.org/10.2118/162814-MS\nRamurthy, M., Barree, R.D., Kundert, D.P., Petre, J.E. and Mullen, M.J., 2011. Surface-area vs. conductivity-type fracture\ntreatments in shale reservoirs. SPE Production & Pperations, 26(04), pp.357\u2013367. https://doi.org/10.2118/140169-PA"}


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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 32, "chroma:document": "SPE-187348-MS 33\nSakaguchi, K., Tomono, J., Okumura, K., Ogawa, Y . and Matsuki, K., 2008. Asperity height and aperture of an artificial\ntensile fracture of metric size. Rock Mechanics and Rock Engineering, 41(2), pp.325\u2013341. http://dx.doi.org/10.1007/\ns00603-005-0102-3\nSchiesser, W.E. and Griffiths, G.W., 2009. A compendium of partial differential equation models: method of lines analysis\nwith Matlab. Cambridge University Press.\nSneddon, I. N. and Lowengrub, M., Crack Problems in the Classical Theory of Elasticity. John Wiley & Sons, Inc., New\nYork, 1969\nSoliman, M. Y ., Craig, D. P., Bartko, K. M., Rahim, Z., and Adams, D. M. 2005. Post-Closure Analysis to Determine\nFormation Permeability, Reservoir Pressure, Residual Fracture Properties. Paper SPE 93419 presented at the SPE\nMiddle East Oil and Gas Show and Conference, Kingdom of Bahrain,12-15 March. http://dx.doi.org/10.2118/93419-\nMS"}


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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 32, "chroma:document": "Middle East Oil and Gas Show and Conference, Kingdom of Bahrain,12-15 March. http://dx.doi.org/10.2118/93419-\nMS\nSoliman, M. Y ., Shahri, M., and Lamei, H. 2013. Revisiting the Before Closure Analysis Formulations in Diagnostic\nFracturing Injection Test. Paper SPE 163869 presented at the SPE Hydraulic Fracturing Technology Conference, The\nWoodlands, Texas, USA, 4-6 February. http://dx.doi.org/10.2118/163869-MS\nSoliman, M.Y ., Miranda, C., and Wang, H.M. 2010. Application of After-Closure Analysis to a Dual-Porosity\nFormation, to CBM, and to aFractured Horizontal Well. SPE Prod & Oper 25 (4): 472\u2013483. SPE-124135-PA. http://\ndx.doi.org/10.2118/124135-PA.\nValko', P. P., and Economides, M. J. 1999. Fluid-Leakoff Delineation in High-Permeability Fracturing. SPE Production\n& Facilities. V ol (14)02:110\u2013116. http://dx.doi.org/10.2118/56135-PA\nvan Dam, D.B., de Pater, C.J. and Romijn, R., 2000. Analysis of Hydraulic Fracture Closure in Laboratory Experiments."}


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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 32, "chroma:document": "van Dam, D.B., de Pater, C.J. and Romijn, R., 2000. Analysis of Hydraulic Fracture Closure in Laboratory Experiments.\nSPE Production & Facilities,15(03), pp.151\u2013158. http://dx.doi.org/10.2118/65066-PA\nWang, H. and Sharma, M.M. 2017. A Non-Local Model for Fracture Closure on Rough Fracture Faces and Asperities.\nJournal of Petroleum Science and Engineering, 154, 425\u2013437. https://dx.doi.org/10.1016/j.petrol.2017.04.024\nWang, H. 2015. Numerical Modeling of Non-Planar Hydraulic Fracture Propagation in Brittle and Ductile Rocks using\nXFEM with Cohesive Zone Method. Journal of Petroleum Science and Engineering, V ol (135):127\u2013140. http://\ndx.doi.org/10.1016/j.petrol.2015.08.010\nWang, H. 2016. Numerical Investigation of Fracture Spacing and Sequencing Effects on Multiple Hydraulic Fracture\nInterference and Coalescence in Brittle and Ductile Reservoir Rocks. Engineering Fracture Mechanics, V ol (157):\n107\u2013124. http://dx.doi.org/10.1016/j.engfracmech.2016.02.025"}


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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 32, "chroma:document": "107\u2013124. http://dx.doi.org/10.1016/j.engfracmech.2016.02.025\nWang, H. 2017. What Factors Control Shale Gas Production and Production Decline Trend in Fractured Systems: A\nComprehensive Analysis and Investigation. SPE Journal, V ol 22(02): 562\u2013581. http://dx.doi.org/10.2118/179967-PA\nWang, H., Marongiu-Porcu, M., and Economides, M. J. 2016. SPE-168600-PA. Poroelastic and Poroplastic Modeling\nof Hydraulic Fracturing in Brittle and Ductile Formations, Journal of SPE Production & Operations, 31(01): 47\u201359.\nhttp://dx.doi.org/10.2118/168600-PA\nWarpinski, N.R., Branagan, P.T., Engler, B.P., Wilmer,R., and Wolhart, S.L. 2002. Evaluation of a downhole tiltmeter\narray for monitoring hydraulic fractures. International Journal of Rock Mechanics and Mining Sciences, V ol (34)3-4:\n329.e1\u2013329.e13. http://dx.doi.org/10.1016/S1365-1609(97)00074-9\nWarpinski, N.R., Lorenz, J.C., Branagan, P.T., Myal, F.R. and Gall, B.L., 1993. Examination of a Cored Hydraulic Fracture"}


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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 32, "chroma:document": "Warpinski, N.R., Lorenz, J.C., Branagan, P.T., Myal, F.R. and Gall, B.L., 1993. Examination of a Cored Hydraulic Fracture\nin a Deep Gas Well (includes associated papers 26302 and 26946). SPE Production & Facilities, 8(03), pp.150\u2013158.\nhttp://dx.doi.org/10.2118/22876-PA\nWells, O.L. and Davatzes N.C., 2015. The History of Dilation Across Natural Fractures Due to Evolving Surface\nRoughness. In Proc. 40th Workshop on Geothermal Reservoir Engineering\nWillis-Richards, J., Watanabe,K. Takahashi.H. 1996. Progress toward a stochastic rock mechanics model of engineered\ngeothermal systems. Journal of Geophysical Research 101 (B8): 17481\u201317496, http://dx.doi.org/10.1029/96JB00882.\nWu, W. and Sharma, M.M., 2017. Acid Fracturing in Shales: Effect of Dilute Acid on Properties and Pore Structure of\nShale. SPE Production & Operations, 32(01):51\u201363. http://dx.doi.org/10.2118/173390-PA\nZhang, J., Kamenov, A., Hill, A.D. and Zhu, D., 2014. Laboratory Measurement of Hydraulic-Fracture Conductivities in"}


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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 32, "chroma:document": "Zhang, J., Kamenov, A., Hill, A.D. and Zhu, D., 2014. Laboratory Measurement of Hydraulic-Fracture Conductivities in\nthe Barnett Shale. SPE Production & Operations, 29(03), pp.216\u2013227. http://dx.doi.org/10.2118/163839-PA\nZou, Y ., Ma, X., Zhang, S., Zhou, T., Ehlig-Economides, C. and Li, H., 2015. The Origins of Low-Fracture Conductivity in\nSoft Shale Formations: An Experimental Study. Energy Technology, 3(12), pp.1233\u20131242. https://dx.doi.org/10.1002/\nente.201500188"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 33, "chroma:document": "34 SPE-187348-MS\nAppendix A\nSolutions of Pressure Decline under Fracture Pressure Dependent Leak-off\nIn order to overcome the limitations of Nolte's G-function model and circumvent the use of a constant leak-\noff coefficient, Mayerhofer et al. (1995) presented a model that describes unsteady-state fluid flow from\nfractures of varying area into the formation, with the filter cake considered as a time and rate dependent\nskin effect, allowing for superposition of the injection history, filter cake deposition and associated rate\nconvolution. Valko\u2019 and Economides (1999) simplified Mayerhofer et al. (1995) model by assuming\nconstant leak-off rate during fracture propagation and constant fracture area during closure. A special before\nclosure analysis (BCA) plot with a straight line can be constructed to determine reservoir permeability and\nfracture face resistance, without acquiring the initial guess of permeability and fracture area. However, the"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 33, "chroma:document": "fracture face resistance, without acquiring the initial guess of permeability and fracture area. However, the\nprocess to generate the special BCA plot and predict fall-off pressure is complicated and cumbersome,\nprone to mistakes that can lead to dimension inconsistency ( Soliman et al. 2013). In addition, their approach\nused a constant-rate solution of an infinite-conductivity well introduced by Gringarten et al. (1974). The\nresults obtained by use of the constant-rate solution are questionable to account for the fracture pressure\ndependent leak-off behavior. In this appendix, we derive the solutions of pressure decline under fracture\npressure dependent leak-off using superposition based on a constant-pressure solution.\nThe basic principle of superposition is that the response of a system to a number of perturbations is equal\nto the sum of the responses to each of the perturbation as if they were present by themselves if the responses"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 33, "chroma:document": "to the sum of the responses to each of the perturbation as if they were present by themselves if the responses\nare linearly related to the perturbations. From Eq. (18), we can calculate the leak-off rate under a constant\nfracture pressure P f:\n(A.1)\nDivide the shut-in time \u0394t into n time steps, and the leak-off rate at the nth time step can be determined\nbased on superposition:\n(A.2)\nThe pressure difference P f,j - P f,j-1 is the pressure in the fracture at time step j minus the pressure in the\nformation at the fracture-formation interface, which equals to the pressure in the fracture at the previous\ntime step. Combine Eq. (A.2) with Eq. (10) via material balance, we can get:\n(A.3)\nlet P f,0 = P 0, Pf,1 = ISIP, \u0394t 1 = 0 then for n \u2265 2, integrate Eq. (A.3) we can obtain:\n(A.4)\nEq. (A.4) can be solved explicitly with marching time interval, for example, when n = 2\n(A.5)"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 34, "chroma:document": "SPE-187348-MS 35\nOn closer observation of Eq. (A.5), we can realize that it is exactly Carter's leak-off solution to Eq. (19),\nbecause during the first time step, no superposition is needed. The fracture pressure dependent leak-off\ncoefficient can be recovered as:\n(A.6)\nIt should be mentioned that Eq. (A.4) is only valid when Sf is constant, where the linear response system\nis the prerequisite of using superposition principal."}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 35, "chroma:document": "36 SPE-187348-MS\nAppendix B:\nModel Non-Local Fracture Closure on Rough Surfaces and Asperities\nEngland and Green (1963) showed that the width of a 2D fracture with any arbitrary pressure/stress profile\nis given by:\n(B.1)\nwhere \u03b3 is a dummy variable, y is the distance to the middle of the fracture, a is the half fracture height\n(PKN model) or half fracture length (KGD model). Let PNet(y) be the net pressure/stress distribution along\nthe y-direction, which is the total pressure/stress acting on the fracture surfaces minus the far field stress\nin the fracture opening direction. To obtain F(\u03b3) and G(\u03b3), PNet(y) has to be divided into even and odd\nfunctions, such that PNet(y) = -f(y) - g(y), with f(y) being even and g(y) being odd. Then F(\u03b3) and G(\u03b3) are\ndetermined by:\n(B.2)\n(B.3)\nwhere \u03b2 is a dummy variable. If the P Net(y) is symmetric, then g(y) and G(\u03b3) = 0. For example, assuming a\nuniform pressure P Net inside the fracture, f(y) =-P Net, g(y) = 0, From Eq. (B.2):\n(B.4)"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 35, "chroma:document": "uniform pressure P Net inside the fracture, f(y) =-P Net, g(y) = 0, From Eq. (B.2):\n(B.4)\nSubstituting Eq. (B.4) into Equation (B.1), we have,\n(B.5)\nEquation (B.5) is the well-known equation for a 2D static pressurized crack. If a is the fracture half-\nheight, then Eq. (B.5) describes the fracture width distribution forthe PKN fracture geometry, and if a is\nthe fracture half-length, Eq. (B.5) describes the fracture width distribution for the KGD fracture geometry.\nWhenan analytic form of P Net(y) is not available a priori (e.g., coupled with other physics) or P Net(y) is\nnot acontinuous function, numerical integration to calculate the fracture width has to be implemented in a\ndiscretized manner along the fracture height or length. Fig. B1 shows a fracture that is discretized into a\nnumber of n segments. In each segment, there exists a uniform net pressure/stress that acts on the fracture"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 35, "chroma:document": "number of n segments. In each segment, there exists a uniform net pressure/stress that acts on the fracture\nsurfaces. Any net pressure/stress distribution can be approximated by representing it in this manner given\nenough number of n."}


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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 36, "chroma:document": "SPE-187348-MS 37\nFigure B1\u2014DiscretizePKN/KGD fracture into n segments with different net pressure/stress\nDefine integration operator I:\n(B.6)\nthe corresponding fracture width profile at the mth segment can be obtained by integrating Eq. (B.1) across\neach segment to account for non-local elastic behavior (Wang and Sharma 2017):\n(B.7)\nRepeat this calculation from the 1st segment to the nth segment, the entire fracture width profile under\narbitrary normal loading can be obtained. Fig. B2 shows an example where with a fracture half-height\ndivided into three segments with equal length, and a certain net pressure/stress is imposed on each segment\nindividually. As we can see, when the net pressure/stress is the same on all segments, an elliptical fracture\nwidth profile is obtained, which is also validated by Eq. (B.5). When the net pressure/stress varies along\ndifferent segments, the fracture width profile will depart from the smooth, elliptical curve. And P Netn has"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87f370-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 36, "chroma:document": "different segments, the fracture width profile will depart from the smooth, elliptical curve. And P Netn has\nmore impact on the overall width distribution when it is imposed close to fracture center."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87f3c0-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 37, "chroma:document": "38 SPE-187348-MS\nFigure B2\u2014Fracture width distribution along fracture half-height across three segments: a = 30 m, E=20 GPa, v =0.25\nThe radial crack opening due to an excess pressure distribution in a linearly elastic material of infinite\nextent can be expressed as (Sneddon and Lowengrub 1969):\n(B.8)\nwhere Rf is the length of fracture radius, u and s are dummy variables and r D is the normalized radius, which\nis defined as\n(B.9)\nwhere rf is the local fracture radius. For a radius crack with a uniform pressure distribution P Net, Eq. (B.8)\ncan be integrated directly to get an analyticalsolutionfor the fracture width profile:\n(B.10)\nBarr (1991) proved that the double integral of Eq. (B.8) can be transformed into a single integral with\nthe aid of elliptic integrals:\n(B.11)\nF(\u03c6, m) is the elliptic integral of the first kind, which is defined as:\n(B.12)\nwhere \u03c6 are input parameters for elliptic integral, and \u03b8 is a dummy variable."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87f410-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 37, "chroma:document": "(B.12)\nwhere \u03c6 are input parameters for elliptic integral, and \u03b8 is a dummy variable.\nSimilar to Fig. B1, a radial fracture can be discretized into n segments, as shown in Fig. B3. In each\nsegment, there exists a uniform net pressure/stress that acts on the opposite fracture surface. If n is large\nenough, the net pressure/stress distribution can be approximated to any distribution of P Net(r)."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87f460-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 38, "chroma:document": "SPE-187348-MS 39\nFigure B3\u2014Discretized radial fracture (n segments) with different net pressure/stress acting along the radial direction\nDefine integration operators R1 and R2 as:\n(B.13)\n(B.14)\nand the corresponding fracture width profile at the mth segment under arbitrary P Net(r) can be obtained by\nintegrating Eq. (B.11) across each segment (Wang and Sharma 2017):\n(B.15)\nwhere\n(B.16)\n(B.17)\nRepeat this calculation from the 1st segment to the nth segment, the entire fracture width profile can be\nobtained. Fig. B4 shows an example where with a fracture radius divided into three segments with equal\nlength, and a certain net pressure/stress is imposed on each circular segment individually. When the net\npressure/stress is the same on all segments, it reproduces the same fracture width distribution as predicted\nby Eq. (B.10). And when the net pressure/stress varies along different segments, the fracture width within"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87f4b0-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 38, "chroma:document": "by Eq. (B.10). And when the net pressure/stress varies along different segments, the fracture width within\na certain segment can be substantially impacted by the load imposed on other segments."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87f51e-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 39, "chroma:document": "40 SPE-187348-MS\nFigure B4\u2014Fracture width distribution along fracture radius across three segments: R f = 30 m, E=20 GPa, v =0.25\nIn the above discussion, we have presented the method and associated algorithms to determine the\nfracture width profile for PKN, KGD and Radial fracture geometries, given an arbitrary distribution of net\npressure/stress. Now, if the fracture closes on rough surfaces and asperities, then the net pressure/stressis\nimpacted by the contact stress, which itself is a function of fracture width distribution. Therefore, the net\npressure/stressdistribution imposed on fracture faces and the entire fracture width distribution have to be\nsolved simultaneously. Because both the net pressure/stress distribution and fracture width distribution are\ndiscretized, they can be only solved iteratively. Fig. B5 shows the iterative scheme to solve the net pressure/\npressureand fracture width distribution."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87f578-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 39, "chroma:document": "pressureand fracture width distribution.\nFigure B5\u2014Solution flow diagram for solving coupled linear fracture mechanics with contact problem\nIn this Picard iteration scheme, \u03b1 r is a relaxation factor that is less than or equal to 1.0. It controls how\nmuch the net pressure/stress should be corrected within iterations. Smaller \u03b1 r can avoid over-correction\nand improve convergence stability, but more iterations may be required with small value of \u03b1 r. When the\nchanges in net pressure/stress between iterations is less than the tolerance value, the estimation of contact\nstress across the entire fracture faces is converged and the final net pressure/stress distributionwith the\ncorresponding fracture width profile can be obtained. So given initial fracture geometry, rock properties\nand contact law parameters, the fracture width profile can be uniquely determined under certain fracturing"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87f5c8-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 39, "chroma:document": "and contact law parameters, the fracture width profile can be uniquely determined under certain fracturing\npressure. By discretizing fracturing pressure within a certain range, the relationship between fracture volume\nvs. pressure and fracture stiffness vs. pressure can also be obtained."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87f618-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 0, "chroma:document": "SPE-185649-MS\nReinterpretation of Fracture Closure Dynamics During Diagnostic Fracture\nInjection Tests\nB. Zanganeh and C. R. Clarkson, University of Calgary; R. V. Hawkes, Trican Well Service Ltd.\nCopyright 2017, Society of Petroleum Engineers\nThis paper was prepared for presentation at the 2017 SPE Western Regional Meeting held in Bakersfield, California, USA, 23 April 2017.\nThis paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents\nof the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect\nany position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87f668-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 0, "chroma:document": "consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may\nnot be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.\nAbstract\nA fit-for-purpose, fully coupled stress-pore pressure simulation model (Abaqus\u00ae) is used to simulate\ndiagnostic fracture injection tests (DFITs) and generate before closure pressure responses. The simulated\nresponses are used to explain field observations, and to propose a new concept: progressive fracture closure.\nThe cohesive zone model (CZM) is used to model fracture propagation and closure associated with\nDFITs. The customized model in Abaqus\u00ae is capable of modeling all the physical processes involved\nin a typical DFIT including: porous media deformation? fluid flow inside the reservoir? hydraulic\nfracture initiation, propagation and closure? compliance change before and after closure; residual fracture"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87f6ae-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 0, "chroma:document": "fracture initiation, propagation and closure? compliance change before and after closure; residual fracture\nconductivity? and fluid flow inside the fracture and fluid interaction between the fracture and reservoir\n(leakoff).\nA key result obtained is that the previously-introduced fracture compliance method is demonstrated to be\nthe most reliable approach to identify fracture closure. Depending on the pressure distribution around the\nfracture, the closure pressure is usually found to be higher than the minimum principal stress. Based on the\ncontinuity equation, the compliance method is expanded to include the progressive fracture closure (PFC)\nconcept. PFC refers to the scenario where fracture closure occurs gradually along the length of fracture, from\nthe tip of the fracture to near the wellbore. Different estimates of closure pressure will be obtained early and\nlate in this process. Several field cases are presented which exhibit progressive fracture closure. A consistent"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87f6fe-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 0, "chroma:document": "late in this process. Several field cases are presented which exhibit progressive fracture closure. A consistent\nclosure signature can be identified for these cases using the primary pressure derivative. This study further\nsuggests that wellbore storage can mask the closure signature. Therefore, reducing the wellbore storage\neffect by using downhole shut-in makes it easier to identify fracture closure, in addition to accelerating the\ntest.\nA common DFIT fracture closure referred to as \"fracture height recession\" is reinterpreted to be caused\nby the PFC phenomenon. This finding has tremendous implications for interpretation of before closure flow\nregimes and associated reservoir behavior. This study also confirms that the compliance method, which\ncorresponds to fracture tip closure, provides more of a true measure of closure pressure than conventional\napproaches which correspond to fracture closure near the well."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87f74e-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 1, "chroma:document": "2 SPE-185649-MS\nIntroduction\nThe DFIT, also known as a \"minifrac,\" has become the standard pre-stimulation method for determining\nfracturing treatment properties, effective reservoir permeability and initial reservoir pressure in\nunconventional resources. A DFIT is an injection-falloff test conducted before the main hydraulic fracturing\ntreatment. The DFIT procedure involves breakdown of the rock and creation of a short fracture during the\ninjection of high pressure fluid (usually water), and recording of the pressure response during the shut-\nin period. The pressure response is analyzed to obtain fracturing treatment parameters such as breakdown\npressure, fracture closure pressure and leakoff characteristics. If the falloff period is sufficient, good\nestimates of formation permeability and reservoir pressure can be obtained.\nBefore-closure analysis (BCA) was pioneered by Nolte (1979), who introduced the G-function based"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87f79e-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 1, "chroma:document": "Before-closure analysis (BCA) was pioneered by Nolte (1979), who introduced the G-function based\non material balance and the Carter leakoff model (Howard and Fast 1957); Nolte noted that the shut-in\npressure has a linear relationship with G-function during closure. Barree and Mukherjee (1996) and Barree\n(1998) recommended using the G-function and its diagnostic derivatives (\n  and \n ) to identify fracture\nclosure. Using these diagnostics, closure can be picked as the deviation from a straight line on a plot of\npressure, and its derivatives, versus the G-function. Barree (1998) and Barree et al. (2009) also presented\nsignatures for non-ideal leakoff behaviors such as tip extension, pressure-dependent leakoff and height\nrecession/transverse storage.\nMohamed et al. (2011) and Marongiu-Porcu et al. (2011) presented a model to predict the falloff pressure\ntrend of an idealized DFIT (normal leakoff), and identified fracture closure using standard pressure transient"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87f7ee-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 1, "chroma:document": "trend of an idealized DFIT (normal leakoff), and identified fracture closure using standard pressure transient\n(PTA) log-log diagnostic plots. They identified normal leakoff as a \n  slope on the Bourdet-derivative (with\nrespect to superposition time), and picked closure as the deviation from the \n  slope. Bachman et al. (2012)\npresented a systematic PTA approach, and a combination of diagnostic plots, to identify various flow\nregimes before and after fracture closure. Van Den Hoek (2016) and McClure et al. (2016) demonstrated\nlimitations of superposition time and Agarwal's time (1980) for analyzing DFITs where the pumping time\nis small. They stated that the Bourdet-derivative (with respect to superposition time and Agarwal's time)\nstarts to exceed one at shut-in times equal to roughly one-tenth of the pump time and that the \n  slope is\nnot related to fracture closure.\nVan Dam et al. (2002) studied closure mechanisms in elastic and plastic rocks by conducting experiments"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87f83e-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 1, "chroma:document": "slope is\nnot related to fracture closure.\nVan Dam et al. (2002) studied closure mechanisms in elastic and plastic rocks by conducting experiments\non different rock types. They observed that fracture closure happens at the tip first, and then towards the\nwellbore. They reported a break on a plot of pressure versus G-function due to the decrease of fracture\ncompliance at closure, and argued that, instead of a deviation from a straight line on the pressure derivative\n(with regard to G-function), the local minimum should be selected as closure pressure. Further, they stated\nthat plastic deformation can cause a fracture to remain open in the vicinity of the wellbore. McClure et\nal. (2014) and McClure et al. (2016) investigated the effect of fracture compliance on pressure behavior\nusing a numerical simulator. Their modeling of after-closure compliance behavior was based on the Barton-\nBandis (1985) model, which is an empirical relationship. They demonstrated that the previous \"fracture"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87f88e-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 1, "chroma:document": "Bandis (1985) model, which is an empirical relationship. They demonstrated that the previous \"fracture\nheight recession\" signature presented by Barree et al. (2009) is the natural behavior of closure caused by\na change in fracture compliance. Based the \"fracture compliance method,\" closure is picked as the start of\nupward deviation on the \n  curve. McClure et al. (2016) argued that previously-presented methods tend\nto underestimate the closure pressure.\nVan Den Hoek (2016) presented a PTA approach for modeling pressure behavior in DFITs and\nwaterflood-induced fractures based on simplified numerical and analytical solutions. In his work, fracture\ngrowth rate was a predefined input into the simulator, the Carter model was used as the leakoff model for\nthe DFIT, and closure was modeled as a gradual decline of fracture compliance representing a combination\nof \"hinge\" (constant length) and \"zipper\" (length recession) closure."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87f8de-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 2, "chroma:document": "SPE-185649-MS 3\nThe cohesive zone model (CZM) for modeling crack propagation was originally proposed by Dugdale\n(1960) and Barenblatt (1962). Recently, this method has been successfully used for modeling hydraulic\nfracture initiation and propagation during fracturing treatment (Yao 2012; Shen and Cullick 2012; Shin and\nSharma 2012; Chen 2012; Zielonka et al. 2014; Haddad and Sepehrnoori 2015). Some of the advantages\nof the CZM compared to conventional linear elastic fracture modeling are that it avoids a singularity at the\ncrack tip, the location of crack tip is not an input and is a direct output of the solution, and it is capable\nof modeling fracture tip material softening in quasi-brittle rocks such as shale ( Chen 2012; Haddad and\nSepehrnoori 2015). The CZM has been implemented in the Abaqus\u00ae finite element program and validated\nanalytically and experimentally by Zielonka et al. (2014) and Searless et al. (2016)."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87f92e-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 2, "chroma:document": "analytically and experimentally by Zielonka et al. (2014) and Searless et al. (2016).\nIn this study, the CZM implemented in the Abaqus\u00ae finite element program is used to model fracture\ninitiation, propagation and closure for DFITs. A pressure-dependent leakoff model is coupled with the\nprogram through user subroutines. The model is used to generate synthetic pressure responses, which in\nturn are compared to field observations and used to study fracture closure dynamics.\nTheory and Methods\nIn the following, key components of the simulation model used in this study are described, as is the model\nsetup used for generating synthetic DFITs.\nCohesive Zone Model (CZM)\nWith the CZM, the fracture is modeled as a gradual separation between two material (rock) surfaces. This\nseparation is modeled as a progressive degradation of cohesive strength along the cohesive layer, which is\na pre-defined surface embedded in the rock and follows a traction-separation law ( Abaqus Analysis User's"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87f9b0-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 2, "chroma:document": "a pre-defined surface embedded in the rock and follows a traction-separation law ( Abaqus Analysis User's\nGuide 2016; Chen 2012). With traction-separation behavior (Fig. 1(a)), prior to damage initiation, cohesive\nelements exhibit a reversible linear elastic response (characterized by the cohesive layer stiffness, Kn) until\nthe traction reaches the maximum tensile strength (T n0) with a separation of \u03b40. After damage initiation,\nthe cohesive traction evolves from the maximum strength (T n0) down to zero where the element is fully\ndamaged with the separation of \u03b4f. The damage evolution is governed by the critical energy release rate ( Gc)\nthat is equal to the area under traction-separation curve. Fig. 1(b) is a schematic of fracture propagation\nin a cohesive layer, demonstrating the fully damaged cohesive zone (filled with fracturing fluid) and the\nfracture process zone (damage initiation and evolution)."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87fa00-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 2, "chroma:document": "fracture process zone (damage initiation and evolution).\nFigure 1\u2014(a) Typical traction-separation behavior; (b) Damage evolution in cohesive elements.\nDifferent guidelines have been presented for selecting the cohesive element mesh sizes and stiffness.\nTuron et al. (2007) suggested the following equation for the cohesive layer stiffness:"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87fa5a-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 3, "chroma:document": "4 SPE-185649-MS\n(1)\nwhere Kn is the stiffness of the cohesive layer, E is the Young's modulus of the material and tadjacent  is\nthickness of the adjacent sub-laminate. They recommended \u03b1 values much larger than 1 (\u03b1>>1). Zielonka\net al. (2014) and Searless (2016) used \u03b1=100, and presented close matches with analytical solutions and\nexperimental results. Haddad and Sepehrnoori (2015) derived an optimum \u03b1 value of 60 (assuming tadjacent  =\n1) by conducting a sensitivity analysis. As will be shown in the next section, we have observed that cohesive\nlayer stiffness is a major parameter affecting fracture compliance behavior before and after closure.\nFluid Flow within the Fracture\nFig. 2 is a plan view schematic of a wing of a fracture showing the components of fluid flow in the fracture.\nThere are two components of fluid flow inside the fracture: tangential flow within the fracture gap (q frac)"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87faaa-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 3, "chroma:document": "There are two components of fluid flow inside the fracture: tangential flow within the fracture gap (q frac)\nand normal flow (leakoff) from the fracture to the surrounding rock (q leak). The continuity equation for fluid\nflow inside the fracture is:\n(2)\nwhere w is the fracture opening, and q inj is the injection rate into the fracture.\nFigure 2\u2014Plan view schematic of one wing of the fracture.\nTangential Flow\nThe tangential flow is formulated based on Poiseuille's law:\n(3)\nwhere w is the fracture opening, \u03bc is the fluid viscosity and Pf is the fluid pressure along the fracture length.\nLeakoff Model\nThe Carter leak-off model (Howard and Fast 1957) has been used extensively for modeling of hydraulic\nfracture propagation and in the analytical solutions for DFIT analysis. It is a one dimensional pressure-\nindependent leak-off model, applicable to high-viscosity fracturing fluids causing the formation of a low\npermeability filter cake on the fracture walls and is described by:\n(4)"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87fafa-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 3, "chroma:document": "permeability filter cake on the fracture walls and is described by:\n(4)\nwhere qleak(x) is leakoff rate at point x, Ccarter is the Carter leakoff coefficient, t is the time elapsed since the\nstart of injection and \u03c4(x) is the time when the fracture is created at point x (exposure time). The Carter leakoff\ncoefficient depends on filter cake created on fracture walls, and it is not related to formation properties such\nas reservoir permeability and pressure."}


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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 4, "chroma:document": "SPE-185649-MS 5\nIn this study, a rigorous leakoff model based on the solution of the 1-D diffusion equation in a half-space\nwith an imposed pressure history at the boundary is used (Sarvaramin and Garagash 2015):\n(5)\nwhere qleak(x) is leakoff rate at point x, S is the storage coefficient and \u03b1l is the hydraulic diffusivity. Unlike the\nCarter model, this model (Sarvaramin and Garagash 2015) is pressure-dependent, and it is directly related\nto formation properties such as permeability, porosity and total compressibility. The final approximation\nof this model is similar to Howard and Fast (1957)'s solution for leakoff of low viscosity fracturing fluids,\nexcept that the pressure difference term is not constant, and it is a function of time and location.\nModel Setup and Description\nIn this study, a 2D plane-strain model is used to model hydraulic fracture initiation, propagation and closure"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87fbc2-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 4, "chroma:document": "Model Setup and Description\nIn this study, a 2D plane-strain model is used to model hydraulic fracture initiation, propagation and closure\nin a DFIT. Fig. 3 provides a schematic of the simulation model. Minimum and maximum horizontal stresses\nare acting in the Y and X directions, respectively. In the Base Case model, wellbore storage is neglected,\nand the fluid is injected directly at the injection point (perforation). In a more complex scenario, a block\nrepresenting the wellbore storage is considered. The fluid is injected into the wellbore block, which is\nconnected to the model using pipe elements. The pipe elements also model possible pressure drops during\nthe test.\nFigure 3\u2014Plan view schematic of the simulation model setup.\nIn order to consider the appropriate far-field boundary conditions and model propagation of the fracture in\nan infinite medium, the model is surrounded by infinite elements. Infinite elements are used in cases where"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87fc08-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 4, "chroma:document": "an infinite medium, the model is surrounded by infinite elements. Infinite elements are used in cases where\nthe modeled region is small in size compared to the surrounding medium (formation), and they provide\nquiet boundaries in the simulation (Abaqus Analysis User's Guide 2016). Chen (2012) and Haddad and\nSepehrnoori (2015) have discussed the advantages of using infinite elements in improving the accuracy\nand runtime of the simulation. For well-testing applications, we have observed improvements in the\nrepresentation of pressure transient behaviors, particularly in after-closure flow regimes and removal of\nunrealistic boundary effects, using infinite elements.\nThe cohesive elements are embedded in the formation rock, and act as the potential pathway for hydraulic\nfracture growth. During the injection of high pressure fluid, once the required criteria (as discussed in the\nprevious section) are reached, the cohesive elements can break and act as a hydraulic fracture. During shut-"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87fc58-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 4, "chroma:document": "previous section) are reached, the cohesive elements can break and act as a hydraulic fracture. During shut-\nin, the fracture aperture reduces to closure aperture (w closure), which is a predefined input into the simulator.\nAfter closure, the aperture can further decrease below the wclosure, but it will always be a positive value,"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87fca8-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 5, "chroma:document": "6 SPE-185649-MS\nmeaning the fracture retains a finite aperture and conductivity even after closure. In low permeability\nformations, even a few microns of residual aperture results in an infinite conductive fracture relative to\nmatrix permeability.\nTable 1 lists properties of the Base Case simulation model. In order to compare simulation results with\nfield observations, four other models with the same reservoir and rock mechanical properties, but different\nsettings, are considered (Table 2). In Model 2, the cohesive layer stiffness is reduced to observe its effect on\nfracture compliance and pressure behavior. In Model 3, the first cohesive element connected to the injection\npoint remains open during shut-in and closure. In Model 4, wellbore storage is included that is comparable\nwith real field data.\nTable 1\u2014Base Case model simulation properties\nInput parameter Value Input parameter Value\nPermeability, md 0.025 Initial porosity, % 10"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87fcf8-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 5, "chroma:document": "Table 1\u2014Base Case model simulation properties\nInput parameter Value Input parameter Value\nPermeability, md 0.025 Initial porosity, % 10\nYoung's modulus, GPa 20.7 Fracture fluid viscosity, cp 1\nPoisson's ratio 0.25 Shmin, MPa 38.6\nTotal compressibility, KPa 4.68\u00d710-6 Shmax, MPa 44.1\nPumping time, sec 120 Initial pore pressure, MPa 24.8\nPumping rate (per unit thickness),\nm3.sec-11.5\u00d710-4 Maximum tensile strength, MPa 1.25\nCohesive layer stiffness, GPa.m-1 2070\nTable 2\u2014Differences in model setup and simulation settings\nBase Case Model 2 Model 3 Model 4\nCohesive layer stiffness,\nGPa.m-12070 20.7 2070 2070\nDisplacement boundary\ncondition applied to the first\ncohesive elementNo No Yes No\nWellbore storage and pipe\nelementsNo No No Yes\nResults and Discussion\nIn the following, synthetic before-closure signatures are generated for the Base Case and Models 2-4 and\ninterpreted for various physical phenomena. Field cases are then analyzed in the context of the simulation"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87fd48-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 5, "chroma:document": "interpreted for various physical phenomena. Field cases are then analyzed in the context of the simulation\nmodel results to understand the signatures that commonly occur in the field.\nBase Case\nFig. 4(a) illustrates the pressure behavior during injection for the Base Case simulation model. Because no\nwellbore is included in the Base Case model, pressure increases quickly during injection, until it reaches the\nbreakdown point pressure. At this point, damage initiates (fracture initiation), and then propagates along\nthe cohesive element (fracture propagation). The smooth pressure behavior during propagation indicates\nproper mesh size of the model and cohesive elements. Fig. 4(b) illustrates the fracture aperture profile during\npropagation and early shut-in time. Fracture tip extension occurs for approximately 1 meter after shut-in."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87fd98-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 6, "chroma:document": "SPE-185649-MS 7\nFigure 4\u2014Base Case model: (a) pressure profile during injection and fracture propagation; (b)\naperture profile during injection and illustration of tip extension during the early shut-in period.\nFig. 5(a) illustrates fracture opening at a typical element along the length of the fracture (e.g. x=1)\nduring pressure falloff. As the fluid inside the fracture leaks off to the surrounding formation and pressure\ninside the fracture decreases, the fracture opening gradually reduces to the closure aperture. The fracture\naperture further reduces after the closure, but will always be a positive value, meaning the fracture retains\na finite aperture and conductivity even after closure. As demonstrated in Fig. 5(a), the rate of aperture\nchange with time before closure, \n , is significantly different from after closure behavior,\n. The simulated response is similar to the experimental results of Van Dam et al. (2002)."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87fdde-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 6, "chroma:document": ", is significantly different from after closure behavior,\n. The simulated response is similar to the experimental results of Van Dam et al. (2002).\nBased on the continuity equation of fluid flow in the fracture ( Eq. 2), the change of \n  at the time of closure\ncauses an abrupt change in pressure. The same behavior is observed for change of the aperture with pressure\n(Fig. 5 (b)) in our simulated case. This is in agreement with the concept of compliance contrast at closure\npresented previously by Van Dam et al. (2002) and McClure et al. (2014).\nFigure 5\u2014Base Case model: fracture aperture behavior at x=1 during falloff with (a) time and (b) pressure.\nProgressive Fracture Closure.   Fig. 6 illustrates the accumulated leakoff volume from the fracture face\nalong the fracture length for a typical time during falloff (e.g. 15 minutes). The accumulated leakoff volume\nis higher at the perforation compared to at the fracture tip. Due to the low permeability of the formation, this"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87fe2e-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 6, "chroma:document": "is higher at the perforation compared to at the fracture tip. Due to the low permeability of the formation, this\nresults in a non-uniform pressure profile and effective stress (on fracture walls) along the fracture length\n(Fig. 6). The effective stress is maximum at the fracture tip (x=11) and gradually decreases along the fracture"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87fe7e-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 7, "chroma:document": "8 SPE-185649-MS\n(up to x=5) where it remains nearly constant up to the perforation (x=0). Therefore, fracture closure first\noccurs at the tip and then moves toward the wellbore, which is referred to herein as \"progressive fracture\nclosure\". Fig. 7 illustrates the fracture aperture over time during progressive closure. As expected, closure\nfirst occurs at the tip (x=11) moving toward x=5, where fracture suddenly closes up to perforation (x=0).\nFigure 6\u2014Accumulated leakoff volume and effective stress profile at fracture face along the fracture.\nFigure 7\u2014Progressive fracture closure from the tip of the fracture to the perforation.\nThe progressive closure causes an increase in the pressure derivative. Fig. 8 illustrates the pressure falloff\ntrend, and its semi-log derivative \n , on a G-function plot. Using the Carter leakoff model, we would\nexpect a linear trend on the \n  curve; however, due to the pressure-dependency included in the leakoff\nmodel used herein (Eq. 5), the"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87fece-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 7, "chroma:document": "expect a linear trend on the \n  curve; however, due to the pressure-dependency included in the leakoff\nmodel used herein (Eq. 5), the \n  curve slightly deviates from the straight line with a concave downward\ntrend. The start of the upward deviation (derivative spike) on the \n  plot is correlated with tip closure, and\nthe end of derivate spike indicates a fully closed fracture (closure at the wellbore)."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87ff1e-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 8, "chroma:document": "SPE-185649-MS 9\nFigure 8\u2014Base Case model: before-closure analysis using G-function and derivative plots.\nThe PTA plot in Fig. 9 includes the Primary Pressure Derivative (PPD), Bourdet-derivative with respect\nto Agarwal's time (\n ) and Bourdet-derivative with respect to shut-in time (\n ). The early time\nfluctuations on the derivative plots are caused by fracture tip extension. According to Mattar and Zaoral\n(1992), the PPD for any type of reservoir effect or flow regime will always either be a constant or decreasing,\nand any non-reservoir effect will appear as an increase in the PPD. The non-reservoir effects in the Base\nCase model are fracture tip extension and fracture closure. After the early time tip extension signature\n(fluctuations), the start of PPD violation (upward trend) is caused by fracture tip closure. The end of PPD\nviolation corresponds to a fully closed fracture. Closure pressure may be estimated at the start and end"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87ff6e-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 8, "chroma:document": "violation corresponds to a fully closed fracture. Closure pressure may be estimated at the start and end\nof the PPD violation as 38.7 MPa and 38.3 MPa, respectively. In real field tests, it is likely that stress\nconditions near wellbore area are significantly affected during drilling and completion. Therefore, the\ndifference between closure pressure at the perforation and fracture tip can be larger.\nFigure 9\u2014Base Case model: before-closure analysis using PTA plot.\nConsistent with previous claims (Van Den Hoek 2016; McClure et al. 2016) it is observed that \nstarts to deviate from \n  at \u0394t = 0.1 tp, and the closure time is not long enough for the \n  slope to establish."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b87ffbe-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 9, "chroma:document": "10 SPE-185649-MS\nModel 2\nThe main difference between this model and the Base Case is that the \n  ratio used in the cohesive layer\nstiffness calculation (Eq. 1) is reduced from 100 to 1. As shown in Fig. 10, even after closure, the fracture\naperture continues to decrease from the closure aperture to a residual aperture. The difference between\n and \n  is also less than for the Base Case.\nFigure 10\u2014Model 2: fracture aperture behavior during falloff with (a) time and (b) pressure.\nSimilar to the Base Case, progressive closure from the tip of the fracture to near the wellbore causes a\nclear break in pressure, and a spike in the pressure derivative on the G-function and PTA plot; however, its\nmagnitude is smaller than for the Base Case model (see Fig. 11). Fracture tip closure starts at 39.4 MPa,\nwellbore closure occurs at 38.4 MPa, and the progressive closure period lasts for 5 minutes.\nFigure 11\u2014Model 2: before-closure analysis using (a) G-function (b) PTA plot."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b88000e-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 9, "chroma:document": "Figure 11\u2014Model 2: before-closure analysis using (a) G-function (b) PTA plot.\nLow values of cohesive layer stiffness are not realistic. But even for low cohesive layer stiffness, a small\nchange in aperture behavior along the length of fracture caused an abrupt change in pressure behavior. While\nthe progressive fracture closure signature (or previously known as height recession/transverse storage) is\nusually observed in field data, a very sharp increase in derivative for a fully closed fracture is not common.\nWith the next two models, a few scenarios are presented that can affect this sudden pressure change."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b88005e-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 10, "chroma:document": "SPE-185649-MS 11\nModel 3\nIn this model, the first cohesive element connected to the injection point remains open during closure.\nThis was achieved by applying appropriate boundary conditions to the first element. Fig. 12 presents\nBCA analysis plots for Model 3. The overall pressure behavior (Fig. 12(a)) is smooth and similar to field\nobservations. Also, the progressive closure signature (upward deviation in derivative plots) is smooth.\nFigure 12\u2014Model 3: before-closure analysis using (a) G-function (b) PTA plot.\nModel 4\nIn this case, wellbore storage is included in the model by addition of a well block and pipe elements. Fig.\n13 provides pressure transient analysis for this model, without any smoothing applied to the derivative.\nAs demonstrated by the schematics included in the figure, at early time, the afterflow caused by wellbore\nstorage increases both the fracture aperture and length identified by a unit slope on Bourdet-derivative."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b8800ae-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 10, "chroma:document": "storage increases both the fracture aperture and length identified by a unit slope on Bourdet-derivative.\nThe transition from the unit slope occurs when the fracture walls start to close (the so-called hinge closure\nsuggested by Van Den Hoek 2016). Due to low leakoff, this can cause additional tip extension during the\nearly closure (abrupt changes and fluctuations) as fluid is pushed toward the tip of fracture. After this\ntransition, leakoff becomes the dominant factor affecting the pressure transient and similar flow regimes\nto previous models appear."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b8800fe-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 11, "chroma:document": "12 SPE-185649-MS\nFigure 13\u2014Model 4: before-closure analysis using PTA plots.\nIn the presence of wellbore storage, the first fracture element connected to the wellbore can stay open\nwithout applying any boundary conditions (Model 3). This can be explained based on the continuity equation\n(Eq. 1). Initially it was assumed that the qinj term is zero during shut-in. In the presence of afterflow caused by\nwellbore storage, the qinj term is no longer zero and can act to reduce the abrupt change in the \n  term during\nclosure. Therefore, progressive fracture closure stops at the wellbore without causing an abrupt change in\npressure and derivatives (as with the Base Case model and Model 2). Also, as mentioned earlier, Van Dam\net al. (2002) demonstrated that plastic deformation can cause the fracture to remain open in vicinity of\nthe wellbore. While plastic deformation is not modeled in this study, it can be another reason why a very"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b880220-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 11, "chroma:document": "the wellbore. While plastic deformation is not modeled in this study, it can be another reason why a very\nsharp spike in derivative plots is not observed in field data. Nonetheless, the start of progressive closure (tip\nclosure) can be used to estimate closure pressure and minimum horizontal stress.\nAs suggested by Cramer and Nguyen (2013), using downhole shut-in accelerates fracture closure and\ntime to reach radial flow after closure. These advantages can be explained with Model 4. In the presence\nof wellbore storage, fracture length and volume increase significantly during the early falloff period. With\nlarger fracture length, a much longer time is required for radial flow to be established in the formation.\nFurther, re-supply of fluid into the fracture due to afterflow, coupled with a larger volume, delays fracture\nclosure significantly. In addition to the benefits suggested by Cramer and Nguyen (2013), reducing afterflow"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b88027a-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 11, "chroma:document": "closure significantly. In addition to the benefits suggested by Cramer and Nguyen (2013), reducing afterflow\nby using downhole shut-in results in a clearer signature of fracture closure (abrupt change in pressure and\nderivatives).\nIn many field cases, a unit slope on the Bourdet-derivative is observed. Hawkes et al. (2013) introduced\na composite permeability concept to explain after-closure behavior. Bachman et al. (2015) expanded this\nidea and stated this behavior can happen before or after closure. A simpler explanation for unit slope after\nclosure is the effect of wellbore storage. Depending on the re-supply rate of fluid from the wellbore, and its\nmagnitude relative to leakoff, a unit slope after closure can exist.\nField Example 1\nThe analysis for this dataset is provided in Fig. 14 and Fig. 15. The pressure values are recorded using\nsurface gauges. No smoothing is used for the purpose of comparison with Model 4. The overall trend and"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b8802ca-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 12, "chroma:document": "SPE-185649-MS 13\nsignatures are very similar to Model 4 except that the progressive closure period is longer. The early unit\nslope (Fig. 14) and fluctuations indicate fracture volume increase and tip extension due to afterflow. The\ntransition from a unit slope represents the start of hinge closure. Similar to Model 4, tip extension occurs\nduring this period as hinge closure pushes the fluid to the tip of the fracture. Progressive fracture closure\nstarts with PPD violation (or upward deviation on derivative plots) at 22.7 MPa and ends at 19.5 MPa. The\nprogressive closure period lasts around 5 hours and the difference between closure pressure estimates is\nsignificant.\nFigure 14\u2014Field Example 1: before-closure analysis using PTA plot.\nFigure 15\u2014Field Example 1: before-closure analysis using the G-function plot.\nField Example 2\nThis well has been analyzed previously by Hawkes et al. (2013; Well 058L). Again, the overall trend is very"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b880324-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 12, "chroma:document": "Field Example 2\nThis well has been analyzed previously by Hawkes et al. (2013; Well 058L). Again, the overall trend is very\nsimilar to Model 4 (Fig. 16). The upward deviation at the end of the \n  slope was of particular interest in the\nprevious analysis, and no explanation was provided at the time for this feature."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b880374-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 13, "chroma:document": "14 SPE-185649-MS\nFigure 16\u2014Field Example 2: before-closure analysis using PTA plot (modified from Hawkes et al., 2013; Well 058L).\nThe early time unit slope indicates fracture expansion and tip extension due to wellbore storage. At the\nend of the unit slope, hinge closure starts and is accompanied by tip extension. After this period (considering\nthat falloff time in this case is large enough), the \n  slope appears on the Bourdet-derivative. Consistent\nwith our modeling results, the upward deviation at the end of the \n  slope, that is consistent with the start\nof the PPD violation, corresponds to the progressive fracture closure period. The start of the PPD violation\nindicates tip closure (52.8 MPa) and the end of PPD violation represents a fully closed fracture (50.9 MPa).\nThe progressive closure period lasts 7.5 hours with the pressure at tip closure interpreted as closure pressure.\nField Example 3"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b8803ce-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 13, "chroma:document": "The progressive closure period lasts 7.5 hours with the pressure at tip closure interpreted as closure pressure.\nField Example 3\nIn this test, pumping time was relatively long (19 minutes). Combined with the low permeability of the\nformation, this probably resulted in a long fracture requiring more time to close. The pressure values\nare recorded using surface gauges and significant ambient temperature variations between day and night\noccurred during the falloff period. As shown in Fig. 17(a), there are severe fluctuations in the G-function\nderivative plot. These functions can result in misinterpretation of the falloff period (e.g. near wellbore\ncomplexities, tip extension, pressure dependent leakoff, etc.); however, when ambient temperature is\nplotted, it is observed that pressure fluctuations are a direct effect of ambient temperature variation. To\nremove the temperature effect from the surface pressure data, the following equation is used to account for"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b88041e-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 13, "chroma:document": "remove the temperature effect from the surface pressure data, the following equation is used to account for\nfluid volume change with pressure and temperature:\n(6)\nwhere Preading is the recorded pressure at surface, T is the current ambient temperature, T0 is the ambient\ntemperature at time of shut-in, \u03b2 is the thermal expansion coefficient of wellbore fluid and Cf is the\ncompressibility of wellbore fluid. In this equation, temperature dependency of \u03b2 and Cf is neglected which\ncan further increase its accuracy."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b88046e-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 14, "chroma:document": "SPE-185649-MS 15\nFigure 17\u2014Field Example 3: (a) effect of ambient temperature variation on derivative plot (b) removing\nthe effect of ambient temperature using Eq. 6 and BCA based on height recession signature.\nAs demonstrated in Fig. 17(b), applying Eq. 6 to recorded pressure values reduces the ambient\ntemperature effect significantly without using a larger derivative window. Closure pressure is estimated to\nbe 21.5 MPa, based on conventional interpretation of fracture height recession/transverse storage.\nFig. 18 provides before-closure analysis based on PTA plot and using the progressive closure concept.\nAlthough closure time is much longer compared to pumping time, a \n  slope is not clear in this test. After the\ntip extension period, the start of the PPD violation at 25.2 MPa represents tip closure. Due to the very low\npermeability of the formation, and large fracture created during pumping, the progressive closure period"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b8804c8-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 14, "chroma:document": "permeability of the formation, and large fracture created during pumping, the progressive closure period\nis very long (around 6 days). The end of the PPD violation at 23.3 MPa indicates a fully closed fracture.\nThe closure pressure estimate based on the conventional interpretation of height recession (Fig. 17(b))\nunderestimates the pressure by 3.7 MPa.\nFigure 18\u2014Field Example 3: before-closure analysis using PTA plots.\nConclusions\nThe concept of compliance change at the time of closure is validated in this study. We demonstrated that\neven for a soft material with low cohesive layer stiffness (not realistic) this feature exists. Therefore, the\nfracture compliance method (McClure et al. 2016) is a reliable approach to identify fracture closure."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b88052c-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 15, "chroma:document": "16 SPE-185649-MS\nFor planar fractures, closure is a transient process, starting from the tip of the fracture to the vicinity of\nthe wellbore. We refer to this process as \"progressive fracture closure\". The common signature referred to as\n\"fracture height recession/transverse storage\" is reinterpreted to be caused by this phenomenon. The start of\nprogressive closure (tip closure) is picked as the closure pressure, which is consistent with the compliance\nmethod. Progressive fracture closure explains the increase of Bourdet-derivative after the 3/2 slope. It can\nalso provide information about the leakoff characteristics, fracture dimensions and stress distribution around\nthe fracture. The application of progressive fracture closure to evaluate these properties will be discussed\nin a follow up paper.\nThe primary pressure derivative (PPD) is a powerful tool for identifying progressive fracture closure."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b88057c-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 15, "chroma:document": "in a follow up paper.\nThe primary pressure derivative (PPD) is a powerful tool for identifying progressive fracture closure.\nThe start of a PPD violation corresponds to tip closure, and the end of the violation indicates full closure.\nOther PTA derivative plots are useful in the identification of flow regimes before the start of progressive\nclosure and explaining other PPD violations. For instance, PPD violations happening before closure during\nthe unit slope on the Bourdet-derivative, and subsequent transition to zero slope during hinge closure, are\ncaused by fracture expansion and tip extension due to wellbore storage.\nIn previous studies, the magnitude and duration of the tip extension period has been underestimated.\nThe use of the cohesive zone model in this study enabled us to capture this process accurately and the\nsimulations are consistent with field observations. The model results illustrate that tip extension can occur"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b88066c-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 15, "chroma:document": "simulations are consistent with field observations. The model results illustrate that tip extension can occur\nduring early fracture expansion period due to wellbore storage and during the hinge closure (and can last\nfor several hours).\nThe afterflow caused by wellbore storage can mask the \n  change at closure, and its transition from\nthe tip of the fracture to near the wellbore. Further, it may result in excessive tip extension and a larger\nfracture which delays after closure radial flow. Therefore, removing wellbore storage using downhole shut-\nin not only accelerates closure and radial flow, but also provides a clear signature of compliance change\nand progressive closure.\nA simple method is presented for removing the effect of ambient temperature change on pressure values\nwithout applying large derivative windows for smoothing. The temperature effect may be interpreted\nincorrectly as tip extension or pressure dependent leakoff. Further, using large derivative windows results"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b8806c6-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 15, "chroma:document": "incorrectly as tip extension or pressure dependent leakoff. Further, using large derivative windows results\nin loss of accuracy and removal of signatures such as tip extension and PPD violation.\nAcknowledgements\nThe authors would like to thank Dassault Systemes Simulia Corporation for providing Abaqus\u00ae program\nand Seven Generations Energy Corporation for providing field data. The authors would also like to\nacknowledge Dr. Jack R. Jones for his valuable advice and comments. The sponsors of Tight Oil Consortium\nare acknowledged for their support. Partial funding for this project was obtained through an NSERC\nCollaborative Research and Development grant (CRDPJ 452752-13) led by Clarkson.\nNomenclature\nCcarter Carte leakoff coefficient, m3/sec0.5\nCfWellbore fluid compressibility, Pa-1\nEYoung's modulus, Pa\nGG-function time, dimensionless\nGcCritical energy release rate, Pa.m\nKnStiffness of the cohesive layer, Pa/m\nmSlope of straight line, dimensionless\nPPressure, Pa"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b880720-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 15, "chroma:document": "GcCritical energy release rate, Pa.m\nKnStiffness of the cohesive layer, Pa/m\nmSlope of straight line, dimensionless\nPPressure, Pa\nPfFluid pressure in the cohesive element, Pa"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b880770-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 16, "chroma:document": "SPE-185649-MS 17\nqfracGap flow rate within the cohesive element, m3/sec\nqinjInjection rate into the cohesive element, m3/sec\nqleakLeakoff rate from cohesive element, m3/sec\nSStorage coefficient, Pa-1\nShmaxMaximum horizontal stress, Pa\nShminMinimum horizontal stress, Pa\ntaAgarwal's time, dimensionless\ntadjacent Thickness of the adjacent sub-laminate, m\nTn0Maximum tensile strength, MPa\ntpPumping time, sec\nwFracture opening (aperture), m\nwclosure Closure aperture, m\nGreek Variables\n\u03b1Coefficient for calculation of cohesive layer stiffness, dimensionless\n\u03b1lHydraulic diffusivity, m2/sec\n\u03b2Thermal expansion coefficient, \u00b0C-1\n\u03b40Separation at damage initiation, m\n\u03b4fSeparation at failure, m\n\u0394tShut-in time, sec\n\u03c4(x) Exposure time, sec\n\u03bcFracturing fluid viscosity, cp\nReferences\nAbaqus Analysis User's Guide 2016. Dassault Systemes Simulia Corp., Providence, RI.\nAgarwal, R.G. 1980. A New Method to Account for Producing Time Effects when Drawdown Type Curves are Used to"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b8807d4-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 16, "chroma:document": "Agarwal, R.G. 1980. A New Method to Account for Producing Time Effects when Drawdown Type Curves are Used to\nAnalyze Pressure Buildup and Other Test Data. Presented at the SPE Annual Technical Conference and Exhibition,\nDallas, Texas, 21-24 September. SPE-9289-MS. http://dx.doi.org/10.2118/9289-MS\nBachman, R.C., Walters, D.A., Hawkes, R.A. et al. 2012. Reappraisal of the G Time Concept in Mini-Frac Analysis.\nPresented at the SPE Annual Technical Conference and Exhibition, San Antonio, Texas, 8-10 October. SPE-160169-\nMS. http://dx.doi.org/10.2118/160169-MS\nBachman, R., Afsahi, B., and Walters, D. 2015. Mini-Frac Analysis in Oilsands and their Associated Cap Rocks Using\nPTA Based Techniques. Presented at the SPE Canada Heavy Oil Technical Conference, Calgary, Alberta, Canada, 9-11\nJune. SPE-174454-MS. http://dx.doi.org/10.2118/174454-MS\nBarenblatt, G. 1962. The Mathematical Theory of Equilibrium Cracks in Brittle Fracture. Adv. Appl. Mech. 7: 55\u2013129."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b880824-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 16, "chroma:document": "Barenblatt, G. 1962. The Mathematical Theory of Equilibrium Cracks in Brittle Fracture. Adv. Appl. Mech. 7: 55\u2013129.\nhttp://dx.doi.org/10.1016/S0065-2156(08)70121-2\nBarree, R. D. 1998. Applications of Pre-Frac Injection/Falloff Tests in Fissured Reservoirs-Field Examples. Presented\nat the SPE Rocky Mountain Regional/Low Permeability Reservoirs Symposium, Denver, Colorado, 5-8 April.\nSPE-39932-MS. http://dx.doi.org/10.2118/39932-MS.\nBarree, R. D. and Mukherjee, H. 1996. Determination of Pressure Dependent Leakoff and Its Effect on Fracture Geometry.\nPresented at the SPE Annual Technical Conference and Exhibition, Denver, Colorado, 6-9 October. SPE-36424-MS.\nhttp://dx.doi.org/10.2118/36424-MS.\nBarree, R.D., Baree, V .L., and Craig, D.P. 2009. Holistic Fracture Diagnostics: Consistent Interpretation of Prefrac\nInjection Tests Using Multiple Analysis Methods. SPE Production & Operations 24(3): 396\u2013406. SPE-107877-PA.\nhttp://dx.doi.org/10.2118/107877-PA"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b88087e-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 16, "chroma:document": "Injection Tests Using Multiple Analysis Methods. SPE Production & Operations 24(3): 396\u2013406. SPE-107877-PA.\nhttp://dx.doi.org/10.2118/107877-PA\nBarton, N., Bandis, S., and Bakhtar, K. 1985. Strength, Deformation and Conductivity Coupling of Rock Joints.\nInternational Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts 22(3): 121\u2013140. http://\ndx.doi.org/10.1016/0148-9062(85)93227-9.\nChen, Z.R. 2012. Finite Element Modeling of Viscosity-Dominated Hydraulic Fractures. J. Petrol. Sci. Eng. 88\u201389:\n136\u2013144. http://dx.doi.org/10.1016/j.petrol.2011.12.021"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b8808ce-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 17, "chroma:document": "18 SPE-185649-MS\nCramer, D. D. and Nguyen, D. H. 2013. Diagnostic Fracture Injection Testing Tactics in Unconventional Reservoirs.\nPresented at the SPE Hydraulic Fracturing Technology Conference, The Woodlands, Texas, 4\u20136 February.\nSPE-163863-MS. http://dx.doi.org/10.2118/163863-MS\nDugdale, D. 1960. Yielding of Steel Sheets Containing Slits. J. Mech. Phys. Solids 8(2): 100\u2013104. http://\ndx.doi.org/10.1016/0022-5096(60)90013-2\nHaddad, M., and Sepehrnoori, K. 2015. Simulation of Hydraulic Fracturing in Quasi-Brittle Shale Formations Using\nCharacterized Cohesive Layer: Stimulation Controlling Factors. Journal of Unconventional Oil and Gas Resources 9:\n65\u201383. http://dx.doi.org/10.1016/j.juogr.2014.10.001\nHoward, C.C., and Fast, C.R. 1957. Optimum Fluid Characteristics for Fracture Extension. API Drilling and Production\nPractice 24(1): 261\u2013270. API-57-261.\nMarongiu-Porcu, M., Ehlig-Economides, C.A. and Economides, M.J. 2011. Global Model for Fracture Falloff Analysis."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b880950-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 17, "chroma:document": "Marongiu-Porcu, M., Ehlig-Economides, C.A. and Economides, M.J. 2011. Global Model for Fracture Falloff Analysis.\nPresented at the SPE North American Unconventional Gas Conference and Exhibition, The Woodlands, Texas, 14-16\nJune. SPE-144028-MS. http://dx.doi.org/10.2118/144028-MS\nMattar, L. and Zaoral, K. 1992. The Primary Pressure Derivative (PPD) A new Diagnostic Tool in Well Test Interpretation,\nJ. Can. Pet. Technol. 31(4). http://dx.doi.org/10.2118/92-04-06\nMcClure, M.W., Blyton, C.A.J., Jung, H. et al. 2014. The Effect of Changing Fracture Compliance on Pressure Transient\nBehavior During Diagnostic Fracture Injection Tests. Presented at the SPE Annual Technical Conference and\nExhibition, Amsterdam, The Netherlands, 27\u201329 October. SPE-170956-MS. http://dx.doi.org/10.2118/170956-MS\nMcClure, M.W., Jung, H., Cramer, D.D. et al. 2016. The Fracture-Compliance Method for Picking Closure"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b8809a0-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 17, "chroma:document": "McClure, M.W., Jung, H., Cramer, D.D. et al. 2016. The Fracture-Compliance Method for Picking Closure\nPressure From Diagnostic Fracture-Injection Tests. Soc. Pet. Eng. 21(4): 1321\u20131339. SPE-179725-PA. http://\ndx.doi.org/10.2118/179725-PA\nMohamed, I.M., Nasralla, R.A., Sayad, M.A. et al. 2011. Evaluation of After-closure Analysis Techniques for Tight\nand Shale Gas Formations. Presented at the SPE Hydraulic Fracturing Technology Conference and Exhibition, The\nWoodlands, Texas, 24-26 January. SPE-140136-MS. http://dx.doi.org/10.2118/140136-MS\nNolte, K.G. 1979. Determination of Fracture Parameters From Fracturing Pressure Decline. Presented at the SPE\nAnnual Technical Conference and Exhibition, Las Vegas, Nevada, 23-26 September. SPE-8341-MS. http://\ndx.doi.org/10.2118/8341-MS\nSarvaramini, E., Garagash, D.I. 2015. Breakdown of a Pressurized Fingerlike Crack in a Permeable Solid. J. Appl. Mech\n82(6). http://dx.doi.org/10.1115/1.4030172"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 17, "chroma:document": "82(6). http://dx.doi.org/10.1115/1.4030172\nSearles, K.H., Zielonka, M.G., Ning, J. et al. 2016. Fully-Coupled 3D Hydraulic Fracture Models: Development,\nValidation, and Application to O&G Problems. Presented at the SPE Hydraulic Fracturing Technology Conference,\nThe Woodlands, Texas, 9-11 February. SPE-179121-MS. http://dx.doi.org/10.2118/179121-MS\nShen, X., Cullick, S. 2012. Numerical Modeling of Fracture Complexity with Application to Production Stimulation.\nPresented at the SPE Hydraulic Fracturing Technology Conference and Exhibition, The Woodlands, Texas, 6\u20138\nFebruary. SPE-151965-MS. http://dx.doi.org/10.2118/151965-MS\nShin, D.H., Sharma, M.M. 2014. Factors Controlling the Simultaneous Propagation of Multiple Competing Fractures in a\nHorizontal Well. Presented at the SPE Hydraulic Fracturing Technology Conference and Exhibition, The Woodlands,\nTexas, 4\u20136 February. SPE-168599-MS. http://dx.doi.org/10.2118/168599-MS."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b880a4a-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 17, "chroma:document": "Texas, 4\u20136 February. SPE-168599-MS. http://dx.doi.org/10.2118/168599-MS.\nTuron, A., Davila, C.G., Camanho, P.P. et al. 2007. An Engineering Solution for Mesh Size Effects in The Simulation\nof Delamination Using Cohesive Zone Models. Eng. Fract. Mech. 74: 1665\u20131682. http://dx.doi.org/10.1016/\nj.engfracmech.2006.08.025\nVan Dam, D.B., Papanastasiou, P., and de Pater, C.J. 2002. Impact of Rock Plasticity on Hydraulic Fracture Propagation\nand Closure. SPE Production & Facilities, 17(3): 149 \u2013 159. SPE-78812-PA. http://dx.doi.org/10.2118/78812-PA\nVan Den Hoek, P. 2016. A Simple Unified Pressure Transient Analysis Method for Fractured Waterflood Injectors and\nMinifracs in Hydraulic Fracture Stimulation. Presented at the SPE Annual Technical Conference and Exhibition,\nDubai, UAE, 26-28 September. SPE-181593-MS. http://dx.doi.org/10.2118/181593-MS\nYao, Y . 2012. Linear Elastic and Cohesive Fracture Analysis to Model Hydraulic Fracture in Brittle and Ductile Rocks."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b880a9a-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 17, "chroma:document": "Yao, Y . 2012. Linear Elastic and Cohesive Fracture Analysis to Model Hydraulic Fracture in Brittle and Ductile Rocks.\nRock Mech. Rock Eng. http://dx.doi.org/10.1007/s00603-011-0211-0\nZielonka, M. G., Searles, K. H., Ning, J., and Buechler, S. R. Development and Validation of Fully-Coupled Hydraulic\nFracturing Simulation Capabilities. 2014 Simulia Community Conference, Providence RI 17."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b880af4-84a2-11ee-9ca1-0242ac1c000c
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metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 0, "chroma:document": "The Fracture-Compliance Method for\nPicking Closure Pressure From Diagnostic\nFracture-Injection Tests\nMark W. McClure and Hojung Jung, University of Texas at Austin; Dave D. Cramer, ConocoPhillips; and\nMukul M. Sharma, University of Texas at Austin\nSummary\nIn this paper, we present the fracture-compliance method, a tech-\nnique for estimating the closure pressure from diagnostic fracture-\ninjection tests (DFITs). The method is based on the observation\nthat fractures retain a \ufb01nite aperture after asperities come into\ncontact (mechanical closure). An empirical, nonlinear joint-clo-sure law is used to relate the after-closure fracture aperture and\nstiffness (the reciprocal of compliance) to effective normal stress.\nFracture closure increases fracture stiffness, which, in low-perme-\nability formations, causes an increase in the pressure derivative.\nOn the basis of these insights, we propose the fracture-compliancemethod, which consists of picking closure at the \ufb01rst point of"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b880b44-84a2-11ee-9ca1-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 0, "chroma:document": "On the basis of these insights, we propose the fracture-compliancemethod, which consists of picking closure at the \ufb01rst point of\ndeviation from linearity on a plot of pressure or G/C2dP/dGvs.G-\ntime (after the end of the very-early-time transient associated withwellbore and near-wellbore friction and fracture tip-extension).\nThe contribution of this paper is to provide theoretical justi\ufb01ca-\ntion for why closure is best picked with the fracture-compliance\nmethod, and not with other widely used methods. We provide a\nseries of numerical DFIT simulations to demonstrate the sensitiv-ity of the pressure transient to input parameters. Governing equa-\ntions are derived and used to demonstrate the effect of changing\nfracture aperture after closure. A \ufb01eld DFIT data set is analyzed\nwith the new method. Finally, a \ufb01eld example is presented in\nwhich downhole tiltmeter measurements provide an independentestimate of the minimum principal stress.\nIntroduction"}


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id: 6b880b9e-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 0, "chroma:document": "which downhole tiltmeter measurements provide an independentestimate of the minimum principal stress.\nIntroduction\nBackground. DFITs are used for a variety of purposes, including\nestimating the magnitude of the minimum principal stress in a for-\nmation. Fluid, usually water, is injected at a low-to-moderate rate\nfor a short period of time, creating a relatively small hydraulic\nfracture, and then the well is shut in. The pressure transient fol-\nlowing shut-in is analyzed to identify the closure pressure, de\ufb01nedas the \ufb02uid pressure at which the fracture walls come into contact.\nThe closure pressure is taken to be an estimate of the minimum\nprincipal stress. In recent years, there has been a growing interestin using DFITs to evaluate very-low permeability formations,\nincluding shale reservoirs. DFITs belong to a family of fracture-\ninjection/falloff tests that include minifracture and extended leak-\noff tests. These fracturing tests were interpreted with a variety of"}


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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 0, "chroma:document": "injection/falloff tests that include minifracture and extended leak-\noff tests. These fracturing tests were interpreted with a variety of\ndifferent methods and used for many decades (Hubbert and Wil-lis 1957; Godbey and Hodges 1958; Haimson and Fairhurst\n1967). Nevertheless, there is still not universal agreement on\nbest practices for interpreting DFITs to estimate the minimum\nprincipal stress.\nFig. 1 shows a \ufb01eld example of pressure data from a DFIT,\nwith different stages of the test numbered. This test was original\npresented by Cramer and Nguyen (2013). Initially, pressure rises\nlinearly with time, because relatively little \ufb02uid leaks out into the\nformation and the transient is dominated by wellbore storage (1).\nAt the leakoff point, the transient deviates from linearity, indicat-ing that a hydraulic fracture has formed (or reopened) at the well-bore (2). The pressure \ufb02attens out to the fracture-propagation\npressure (3). When the injection rate increases between minutes 4"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 0, "chroma:document": "pressure (3). When the injection rate increases between minutes 4\nand 5, the fracture-propagation pressure increases, primarily\nbecause of increased wellbore and near-wellbore pressure drop,\nand then \ufb02attens out again. When injection is stopped, there is animmediate drop in pressure to the initial shut-in pressure (ISIP) as\nwellbore and near-wellbore frictional pressure drop reduces (4).\nSubsequently, the pressure continues to fall off, and the fracture-\nclosure pressure is reached when the walls of the fracture come\ninto contact (5). Further discussion is provided in the Field DFIT#1 subsection.\nMethods for Interpreting DFITs. Fig. 2 shows plots of the\nshut-in pressure transient from the DFIT shown in Fig. 1 in four\ndifferent types of plots: a G-function plot, a Bourdet plot with de-\nrivative taken with respect to shut-in time, a Bourdet plot with de-\nrivative taken with respect to superposition time, and a Bourdet\nplot with time-integrated pressure (Bourdet et al. 1989; Barree"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b880c98-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 0, "chroma:document": "rivative taken with respect to superposition time, and a Bourdet\nplot with time-integrated pressure (Bourdet et al. 1989; Barree\net al. 2007; Marongiu-Porcu et al. 2011; Craig 2014). Conven-\ntional applications of these plots for diagnosing the closure pres-sure are described in this section. Fig. 2 shows both \ufb01eld data and\na numerical-simulation match to the data, as described in the Field\nDFIT #1 section.\nFracture-closure pressure is most-often diagnosed from plots\nof pressure vs. the square root of time or a specialized function of\ntime called the G-function (Nolte 1979; Nolte and Smith 1981).\nFor 1D leakoff of \ufb02uid from a constant-pressure boundary, the so-\nlution to the diffusivity equation predicts that the leakoff rate willscale with the inverse of the square root of time (Howard and Fast\n1957). When coupled with a wellbore/fracture system of constant\ncompliance, the result is pressure decay scaling with the square"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b880ce8-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 0, "chroma:document": "1957). When coupled with a wellbore/fracture system of constant\ncompliance, the result is pressure decay scaling with the square\nroot of time. The G-function is a generalization of this concept\nthat includes the effect of fracture propagation, which causes theduration of leakoff to be different at each point along the fracture.\nNolte (1979) postulated that fracture-growth scales in time\naccording to a power law given by an exponent a. The G\n-function\nis a complex function of a, but it is only weakly dependent on a\nbetween the reasonable bounds of [1/2] to 1.0. If ais assumed to\nbe 1.0, Gcan be written in a particularly convenient form:\nG\u00f0DtD;a\u00de\u00bc4\npg\u00f0DtD;a\u00de/C0g\u00f00;a\u00de \u00bd/C138 ; \u00f01\u00de\nwhere DtD\u00bcDt=te,Dtis time because shut-in teis duration of\ninjection, and\ng\u00f0DtD;a\u00bc1\u00de\u00bc4\n3\u00f01\u00feDtD\u00de1:5/C0Dt1:5\nDhi\n: \u00f02\u00de\nAllG-function plots in this paper use the convention of assum-\ning that aequals 1.0, and use the de\ufb01nition of Ggiven in Eqs. 1\nand 2. The speci\ufb01c choice of adoes not affect the method of pick-"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b880d38-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 0, "chroma:document": "ing that aequals 1.0, and use the de\ufb01nition of Ggiven in Eqs. 1\nand 2. The speci\ufb01c choice of adoes not affect the method of pick-\ning closure from the transient.\nOn either plot, pressure is expected to form a straight line\nbefore closure and deviate from the straight line when closureoccurs. Therefore, the deviation of pressure from the straight line\ncan be used to identify the fracture-closure pressure, an estimate\nof the minimum principal stress (Castillo 1987; Zoback 2007). In..............\n..........\nCopyright VC2016 Society of Petroleum Engineers\nOriginal SPE manuscript received for review 26 August 2014. Revised manuscript received\nfor review 17 November 2015. Paper (SPE 179725) peer approved 8 December 2015.J179725 DOI: 10.2118/179725-PA Date: 2-August-16 Stage: Page: 1321 Total Pages: 19\nID:jaganm Time: 18:38 I Path: S:/J###/Vol00000/160007/Comp/APPFile/SA-J###160007August 2016 SPE Journal 1321"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b880d92-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 1, "chroma:document": "addition, G/C2dP/dGvs.Gis often plotted {equivalent to d P/\nd[ln(G)] vs. G}, where Pis pressure. As long as pressure forms a\nstraight line, G/C2dP/dGshould also form a straight line, and so\nclosure can be picked at the deviation from the straight line (Bar-\nree and Mukherjee 1996). In many cases, the Gfunction is\napproximately equal to Dt0:5, and the plots can be made with Dt0:5\nsubstituted for G.\nUnfortunately, plots of pressure vs. G-time often yield curving\nlines with multiple points of in\ufb02ection. Efforts were made to inter-\npret and explain these nonideal observations. Wellbore and near-\nwellbore pressure drop, pressure-dependent leakoff, fracture-\nheight recession, closing of secondary transverse fractures, andfracture tip-extension are some of the most commonly cited sour-\nces of nonideal behavior (Nolte 1991; Nolte et al. 1993; Barree\nand Mukherjee 1996; Barree 1998; Barree et al. 2007). In high-"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b880de2-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 1, "chroma:document": "ces of nonideal behavior (Nolte 1991; Nolte et al. 1993; Barree\nand Mukherjee 1996; Barree 1998; Barree et al. 2007). In high-\npermeability formations, leakoff limits fracture length, and clo-sure occurs rapidly. But in very-low permeability formations, clo-\nsure can occur hours or days after shut-in, and nonideal transientsare nearly always observed.\nG-function plots of DFITs are conventionally interpreted\naccording to the method described by Barree and Mukherjee\n(1996) and Barree et al. (2007), which we refer to as the \u201ctangent\nmethod.\u201d According to the tangent method, an ideal DFIT has aG/C2dP/dGcurve that is straight until closure, at which time the\ncurve bends, decreasing in slope; eventually, the curve crests,\nthen turns downward. A concave upward G/C2dP/dGcurve is inter-\npreted as being caused by fracture-height recession or transverse-\nfracture storage. A concave downward G/C2dP/dGis interpreted as\nbeing caused by pressure-dependent leakoff or fracture tip-exten-"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b880e32-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 1, "chroma:document": "fracture storage. A concave downward G/C2dP/dGis interpreted as\nbeing caused by pressure-dependent leakoff or fracture tip-exten-\nsion. These interpretations are based on numerical simulations\n(Nolte 1991; Barree and Mukherjee 1996; Barree 1998) and are\nused widely (Economides and Nolte 2000; Barree et al. 2007;\n01 2345678 9 1 0\nElapsed Time (minutes)\u20131.40\n\u20131.20\n\u20131.00\n\u20130.80\n\u20130.60\u20130.40\n\u20130.20\n0.0065\n60\n55\n50\n45\n40\n35\n30Wellbore-storage trendline\nPressure rollover\nindicates fracture reopeningat 55.3 MPaISIP = 57.1 MPa\nWater hammer\nevent at shut in\npdata\nqwaterCalculated Bottomhole Pressure (MPa)Water-Injection Rate (m3/min) 123\n4\n5?\nFig. 1\u2014Example of rate and pressure measurements from a DFIT. This test was original presented by Cramer and Nguyen (2013).\n0 50 100 150 200\nG-time\n10\u2013210\u2013110\u20131\n100100\n100100100\n102102104\n101101101\n10\u20132100102\n102102\n\u0394t (hours) \u0394t (hours)\u0394t (hours)I (solid), \u0394t*dI/d\u0394t (dashed)\n(MPa-hr)\u0394P (solid), \u0394t*dP/d\u0394t (dashed)\n(MPa)"}


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topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b880e8c-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 1, "chroma:document": "102102104\n101101101\n10\u20132100102\n102102\n\u0394t (hours) \u0394t (hours)\u0394t (hours)I (solid), \u0394t*dI/d\u0394t (dashed)\n(MPa-hr)\u0394P (solid), \u0394t*dP/d\u0394t (dashed)\n(MPa)\nField data (red)Field data (red), simulation data (blue) Field data (red), simulation data (blue)\nField data (red), simulation data (blue)60\n55\n50\n45\n40\n35Pressure (solid) (MPa)\u0394P (solid), \u03c4*dP/d\u03c4 (dashed) (MPa)\nG*dP/dG (dashed) (MPa)10\n8\n6\n4\n20\nFig. 2\u2014Field DFIT #1. Field data are plotted in in red. A simulation match to the data is plotted in blue (described in the Field DFIT\n#1 subsection). The transient was originally shown in Fig. 4 of Cramer and Nguyen (2013). Upper left: G-function plot. Upper right:\nBourdet plot with derivative with respect to time. Lower left: Bourdet plot with derivative with respect to superposition time. Lowerright: Bourdet plot with integrated pressure (zoomed in). Arrows indicate the identi\ufb01cation of closure from the tangent method"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b880ee6-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 1, "chroma:document": "(blue), from the fracture-compliance method (orange), from the Mohamed et al. (2011) method (green), and from the Craig and Bla-\nsingame (2006) method (red).J179725 DOI: 10.2118/179725-PA Date: 2-August-16 Stage: Page: 1322 Total Pages: 19\nID:jaganm Time: 18:38 I Path: S:/J###/Vol00000/160007/Comp/APPFile/SA-J###1600071322 August 2016 SPE Journal"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b880f36-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 2, "chroma:document": "Soliman and Gamadi 2012; Soliman and Kabir 2012; Cramer\nand Nguyen 2013; Wallace et al. 2014; Meng et al. 2014; Barree\net al. 2014).\nWhen the G/C2dP/dGcurve is concave upward, closure is\npicked according to the tangent method by drawing a straight line\nfrom the origin, tangent to the G/C2dP/dGcurve (Barree et al.\n2007). The point of tangency is the time of closure, and the frac-\nture-closure pressure is identi\ufb01ed as the pressure at that time. Thisis typically equivalent to picking closure at the end of the second\nstraight line on a plot of pressure vs. G-time. When the G/C2dP/dG\ncurve is concave downward, closure is picked at approximately\nthe time the G/C2dP/dGcurve stops increasing. The closure pick\nfrom the tangent method is labeled with the blue arrow in theupper-left panel of Fig. 2.\nBourdet plots are log-log plots that show the change in pres-\nsure since the start of the transient, DP, and d P/d(ln(Dt)) (which\ncan be equivalently written as Dt/C2dP/dDt) vs. the time elapsed"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b880fae-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 2, "chroma:document": "sure since the start of the transient, DP, and d P/d(ln(Dt)) (which\ncan be equivalently written as Dt/C2dP/dDt) vs. the time elapsed\nsince the start of the transient, Dt(Bourdet et al. 1989). Mohamed\net al. (2011) and Marongiu-Porcu et al. (2011; 2014) advocate\ninterpreting DFITs with Bourdet plots in which the derivative is\ntaken with respect to superposition time, s. For the method pro-\nposed by Mohamed et al. (2011), superposition time is de\ufb01ned as\ns\u00bc\nte\u00feDt\nDt; \u00f03\u00de\nwhere teis duration of injection. If Dt<<te, taking the derivative\nwith respect to superposition time has no effect, compared to tak-\ning the derivative with respect to Dt.I fDt>>te, taking the deriva-\ntive with respect to superposition time adds 1.0 to the slope of thederivative curve. These authors advocate picking closure at the\nbeginning of the [1/2] slope on the derivative plot (labeled with a\ngreen arrow in the lower-left panel of Fig. 2).\nCraig and Blasingame (2006) propose interpreting DFITs by"}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b881008-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 2, "chroma:document": "green arrow in the lower-left panel of Fig. 2).\nCraig and Blasingame (2006) propose interpreting DFITs by\nintegrating the pressure data after shut-in to \ufb01nd the effectivepressure response, I. This can be written as\nI\u00f0DP\u00de\u00bc\u00f0\nte\u00feDt\n0\u00bdP\u00f0t\u00de/C0Pinit/C138dt; \u00f04\u00de\nwhere Pinitis the initial \ufb02uid pressure. This transformed pressure\nresponse can then be plotted on a Bourdet plot. For application to\na DFIT test, the \u201cwellbore storage\u201d region on the Bourdet plot ofintegrated pressure has a very long duration because matrix per-\nmeability is very low and the storage includes both the wellbore\nand the fracture. Craig and Blasingame (2006) suggest identifying\nclosure at the point where the fracture/wellbore storage changes\n(caused by the changing fracture stiffness at closure). This corre-sponds to the time when a de\ufb02ection can be identi\ufb01ed during the\nwellbore storage period in the Bourdet plot with respect to inte-\ngrated pressure. This is labeled with a red arrow in the lower-rightpanel of Fig. 2."}


created_at: 2023-11-16 17:06:54
topic: persistent://default/default/c01518eb-579a-4a9d-8cdf-22ca77573fa8
id: 6b881062-84a2-11ee-9ca1-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 2, "chroma:document": "grated pressure. This is labeled with a red arrow in the lower-rightpanel of Fig. 2.\nWe advocate picking closure from a plot of pressure vs. G(or\nthe square root of time), but in contrast to the tangent method\n(Barree and Mukherjee 1996; Barree et al. 2007), we recommend\npicking closure at the initial deviation from linearity (after the endof the very early-time transient associated with wellbore and near-\nwellbore friction and fracture tip-extension). We refer to this\nmethod as the \u201cfracture compliance method\u201d for picking closure.\nThe discussion in this paper elaborates on concepts presented by\nMcClure et al. (2014).\nThe fracture compliance can be de\ufb01ned in several equivalent\nways: the derivative of fracture closure (fracture closure is equal\nto negative change in aperture) with respect to effective normal\nstress, the negative of the derivative of aperture with respect to"}


