# 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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1ce1c-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1cf5c-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1d006-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 0, "chroma:document": "Then 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 [\nG. 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."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1d092-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 0, "chroma:document": "Large 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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1d128-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1d1be-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 1, "chroma:document": "Determination 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\non well XX22. Due to the limit volume of sampler container the cycling time for influx is short: 17 sec"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1d24a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 1, "chroma:document": "- 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).\nFig. 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"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1d2d6-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 1, "chroma:document": "is 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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1d362-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1d3ee-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 2, "chroma:document": "number 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,\nmeasuring pressure using well pressure buildup curve interpretation method for producer after hydraulicfracturing leads to significant oil production losses."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1d47a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 2, "chroma:document": "The 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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1d506-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1d592-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 3, "chroma:document": "linear 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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1d614-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1d6a0-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 4, "chroma:document": "not 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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1d72c-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 4, "chroma:document": "observed 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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1d7b8-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1d84e-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1d8da-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1d966-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 5, "chroma:document": "horizontal 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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1d9f2-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 5, "chroma:document": "hydraulic 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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1da7e-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1db14-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1db96-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/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 with reservoir pressure"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1dc22-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 6, "chroma:document": "cost and expenditures saving connected with reservoir pressure detection. Hydro-dynamic logging is much more expensive."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1dcb8-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1dd44-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/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\nwij \u2013 measurement accuracy at hydrodynamic logging, unit fraction"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1ddd0-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 7, "chroma:document": "oPij \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,\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."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1de5c-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-171175-MS.pdf", "page": 7, "chroma:document": "Engineering 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\nUsing Injection Test After-Closure Analysis: Field Case History in a Depleted Oil Reservoir . SPE 128052.8 SPE-171175-MS"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1dee8-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1df74-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1e000-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/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 fracture tip extension. In"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1e14a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 0, "chroma:document": "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 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"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1e1f4-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 0, "chroma:document": "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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1e280-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1e30c-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/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\nG*dP/dG may appear concave up or down ( Soliman and Kabir, 2012 ;Cramer and Nguyen, 2013 ). In some"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1e3a2-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 1, "chroma:document": "cases 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 (\nMcClure and Horne, 2013 ). The simulations"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1e438-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/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\nal., 2007 ;Soliman and Kabir, 2012 ;Padmakar, 2013 ;Cramer and Nguyen, 2013 ;Wallace et al., 2014 )."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1e4c4-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 1, "chroma:document": "Our 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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1e550-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 1, "chroma:document": "deformation 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)\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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1e5dc-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1e668-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1e6f4-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 2, "chroma:document": "identification 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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1e776-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 2, "chroma:document": "pressure 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\nfollowing shut-in, pressure dependent leakoff, and fluid storage from transverse or dilated naturalfractures."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1e7f8-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 2, "chroma:document": "Barree 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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1e884-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 2, "chroma:document": "conducted 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\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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1e910-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/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"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1e992-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1ea28-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 3, "chroma:document": "(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).\nTherefore, 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"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1eabe-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 3, "chroma:document": "described 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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1eb86-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1ec1c-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/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\nand/H9268n,Eref is the 90% closure stress for void aperture. /H9268n=is the effective normal stress, equal to the"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1eca8-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 4, "chroma:document": "normal 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.\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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1ed3e-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/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\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."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1edd4-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 5, "chroma:document": "The 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\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"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1ee56-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 5, "chroma:document": "Shou 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\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."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1eee2-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 5, "chroma:document": "Figure 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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1ef6e-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1eff0-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 6, "chroma:document": "defined. 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\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"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1f07c-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 6, "chroma:document": "The\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.,\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)."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1f108-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 6, "chroma:document": "timestep).\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 (\nMcClure 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."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1f194-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 6, "chroma:document": "A 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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1f25c-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1f2de-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 7, "chroma:document": "percentage.\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\nbe constant at 20 microns. When fracture elements opened, their hydraulic aperture could grow higherthan e"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1f36a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 7, "chroma:document": "0due 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) .\nWellbore storage"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1f3f6-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1f478-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 7, "chroma:document": "kilograms 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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1f4fa-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 7, "chroma:document": "hydraulic 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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1f586-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/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"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1f612-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 8, "chroma:document": "permeability), 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.\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"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1f694-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 8, "chroma:document": "permeability, 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\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"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1f720-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 8, "chroma:document": "aperture) (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/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1f7ac-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/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)\nwhere his the fracture height, p*is the slope of"}


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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 9, "chroma:document": "(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\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."}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1f8ba-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 9, "chroma:document": "Because 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.\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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id: 9ae1f93c-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/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/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1f9c8-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 10, "chroma:document": "et 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\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"}


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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 10, "chroma:document": "closure 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\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"}


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id: 9ae1fad6-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 10, "chroma:document": "pressure 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\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/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1fb62-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/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"}


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id: 9ae1fbee-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 11, "chroma:document": "Figure 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"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1fcac-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/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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{"source": "drive/MyDrive/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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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 12, "chroma:document": "Effect 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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{"source": "drive/MyDrive/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/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1fed2-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/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\nclosure, and then an abrupt drop in pressure afterclosure. Wellbore storage has an effect when thepressure in the wellbore is changing ("}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae1ff5e-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 13, "chroma:document": "Equation 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\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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id: 9ae1fff4-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/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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id: 9ae20080-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 14, "chroma:document": "percentage 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\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 ("}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae20102-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 14, "chroma:document": "Figure 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\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"}


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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 14, "chroma:document": "residual aperture percentage had been higher (as in Simulation HRAP ), the transition would have been less\nabrupt 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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{"source": "drive/MyDrive/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\nbetween storage and leakoff. In Simulation CMT , the much higher permeability caused the fracture that"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae20292-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 15, "chroma:document": "formed 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\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"}


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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 15, "chroma:document": "permeability, 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\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"}


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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 15, "chroma:document": "plots. 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\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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{"source": "drive/MyDrive/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 (\nFigure 13 ). Because the walls begin to come in contact while"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae204ae-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2053a-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 16, "chroma:document": "0-2 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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae205c6-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/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"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae20648-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 16, "chroma:document": "injection 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 ).\nFigure 16\u2014Comparison of two methods for identifying closure stress."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae20706-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 16, "chroma:document": "Drawing 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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae20878-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/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.\nFigure 12 ,Figure 13 , and Figure"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae20922-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae209ae-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 17, "chroma:document": "closure 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\nthis 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 ("}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae20a3a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 17, "chroma:document": "Fracture 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"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae20abc-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/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 fracture, which may explain"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae20b48-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 17, "chroma:document": "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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae20bca-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/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"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae20c56-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 18, "chroma:document": "G-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:\n1. In low permeability formations, a concave up deviation of the G*dP/dG curve from a linear trend"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae20ce2-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 18, "chroma:document": "can 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\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-"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae20d6e-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 18, "chroma:document": "ance 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\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"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae20df0-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae20e7c-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 19, "chroma:document": "KI : 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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae20f08-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae20f8a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 20, "chroma:document": "hydraulic 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\n22(3): 121\u2013140, doi: 10.1016/0148-9062(85)93227-9.\nBlanton, Thomas. 1982. An experimental study of interaction between hydraulically induced and"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae21016-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 20, "chroma:document": "pre-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\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."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae21098-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 20, "chroma:document": "Davis, 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"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae21124-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/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"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae211b0-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 21, "chroma:document": "and 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\nFORTRAN usage. ACM Transactions on Mathematical Software 5(3): 308\u2013323, doi: 10.1145/\n355841.355847."}


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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 21, "chroma:document": "355841.355847.\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\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."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae21340-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 21, "chroma:document": "Nolte, 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\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"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae213f4-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae21480-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 22, "chroma:document": "Mechanics 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.\nSuarez-Rivera, Roberto, Jeff Burghardt, Sergei Stanchits et al. 2013. Understanding the effect of rock"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae215b6-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 22, "chroma:document": "fabric 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.\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"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2166a-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-170956-MS.pdf", "page": 22, "chroma:document": "17496, 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"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae21700-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae21840-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 0, "chroma:document": "deviation 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\nBackground. DFITs are used for a variety of purposes, including\nestimating the magnitude of the minimum principal stress in a for-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae218e0-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 0, "chroma:document": "mation. 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\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."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae21980-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 0, "chroma:document": "principal 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\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-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae21a34-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 0, "chroma:document": "closure 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\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)."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae21ad4-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 0, "chroma:document": "time 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\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"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae21b74-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 0, "chroma:document": "where 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-\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"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae21c0a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 0, "chroma:document": "ID: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/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae21ca0-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae21d4a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 1, "chroma:document": "and 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-\nsion. These interpretations are based on numerical simulations"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae21dea-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 1, "chroma:document": "sion. 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)\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"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae21e80-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 1, "chroma:document": "Field 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\n(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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae21f20-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/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: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2201a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 2, "chroma:document": "since 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\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"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae220ba-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 2, "chroma:document": "I\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.\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"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae22150-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 2, "chroma:document": "(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\neffective normal stress, or the derivative of aperture with respectto pressure [Chapter 12 from Jaeger et al. (2007)]. The fracture\nstiffness is equal to the reciprocal of the fracture compliance. The\nname \u201cfracture compliance method\u201d is chosen because the\nmethod is based on understanding how fracture compliance (and"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae221f0-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 2, "chroma:document": "method is based on understanding how fracture compliance (and\nequivalently, stiffness) change because of fracture closure.At very early time, there is often a sharp drop in pressure\ncaused by the disappearance of friction in the wellbore and near-\nwellbore. This period should not be considered when picking clo-sure pressure. A linear period on the plot of Pvs.Gshould be\nvisible after the end of this initial period. When closure occurs in\nlow-permeability formations, increasing fracture stiffness causes\npressure to de\ufb02ect downward and G/C2dP/dGto de\ufb02ect upward, as\ndiscussed in the Stiffness Contrast at Closure section. A detailedderivation of the equations describing this process are provided in\nAppendix A.\nIn high-permeability formations, closure reduces the leakoff\nrate, and closure can cause pressure to de\ufb02ect upward and G/C2dP/\ndGto de\ufb02ect downward (though continue increasing), as dis-\ncussed in Section 3.1.6. The G/C2dP/dGplot is useful for identify-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae22286-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 2, "chroma:document": "ing closure because it makes it easy to identify these de\ufb02ections.\nThe key difference between our work and most previous work\non this topic is that we consider the effect of residual aperture on\n\ufb02uid storage after mechanical closure. We use a 3D hydraulic-fracture simulator that rigorously solves the hydromechanically\ncoupled problem of fracture initiation and closure. When fracture\nwalls come into contact, the simulator switches to a nonlinearjoint-closure law on the basis of the work of Barton et al. (1985),\nwhich allows the fracture to remain hydraulically conductive\nand continue to store \ufb02uid after mechanical closure has occurred.\nSimilar approaches were used by other authors (Koning and\nNiko 1985; Plahn et al. 1997; van den Hoek 2002; Ribeiro andHorne 2013).\nThere is abundant evidence demonstrating that fractures retain\nstorativity and conductivity after the walls have come into contact\n(mechanical closure). Fractures retain aperture after mechanical"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2231c-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 2, "chroma:document": "closure because of mismatch of asperities in the fracture walls.An extensive literature exists on the relationship between fracture\naperture, conductivity, \ufb02uid pressure, and stress (Gangi 1978;\nWitherspoon et al. 1980; Walsh 1981; Barton et al. 1985; Brown\nand Bruhn 1998; Pyrak-Nolte and Morris 2000). The \u20131/2 slope\npressure transients frequently observed in Bourdet plots of DFITdata indicate the presence of a post-closure in\ufb01nite conductivity\nfracture, demonstrating that the fracture has retained signi\ufb01cant\naperture after closure. In high-permeability formations, the \ufb02uidstorage and conductivity-retained post-closure quickly becomes\nnegligible. But in very-low permeability formations, leakoff into\nthe matrix is very slow, and an extremely narrow fracture aperture\ncan have a dominant effect on the pressure transient.\nThe dimensionless fracture conductivity (or equivalently,\ndimensionless fracture transmissivity) is de\ufb01ned as:\nC\nfD\u00bcT\nkLf; \u00f05\u00de"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae223b2-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 2, "chroma:document": "C\nfD\u00bcT\nkLf; \u00f05\u00de\nwhere Tis the fracture transmissivity, kis the matrix permeability,\nandLfis the fracture half-length. A fracture will behave similar to\nan in\ufb01nite-conductivity fracture as long as Cfd>300. From the\ncubic law (Witherspoon et al. 1980), the fracture transmissivity isde\ufb01ned as\nT\u00bce3\n12; \u00f06\u00de\nwhere eis the hydraulic aperture.\nFor a fracture with half-length of 50 m and a matrix permeabil-\nity of 10/C019m2(/C25100 nd), the fracture will be in\ufb01nitely conduc-\ntive as long as the fracture aperture is greater than 26 mm.\nMethods\nThe numerical simulations are performed with CFRAC (Complex\nFracturing ReseArch Code), which fully couples \ufb02uid \ufb02ow with\nthe stresses induced by fracture deformation in discrete-fracture\nnetworks (McClure and Horne 2013). All simulations described in\nthis paper assume a single planar fracture perpendicular to theminimum principal stress. The simulations are fully 3D, with the\nfractures discretized into square elements in the vertical and hori-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae22452-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 2, "chroma:document": "zontal directions. Complete descriptions of CFRAC are provided..............................\n...................\n...............................\n................................J179725 DOI: 10.2118/179725-PA Date: 2-August-16 Stage: Page: 1323 Total Pages: 19\nID:jaganm Time: 18:39 I Path: S:/J###/Vol00000/160007/Comp/APPFile/SA-J###160007August 2016 SPE Journal 1323"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae224e8-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 3, "chroma:document": "by McClure and Horne (2013) and McClure et al. (2015). Rele-\nvant details are summarized in this section.\nThe \ufb02uid is single phase, single component, constant compres-\nsibility ( cf), constant viscosity ( l), and has initial density of qinit.\nThe simulations are isothermal, and the effect of gravity is\nneglected. The simulations are initialized with homogeneous\nstress and \ufb02uid pressure.\nNonlinear relationships are used to relate pressure, stress, and\ndeformation to aperture. We de\ufb01ne a \u201cclosed\u201d fracture as a frac-\nture in which the \ufb02uid pressure is less than the normal stress and\nthe walls of the fracture are in contact. Roughness of the fracture\nwalls allows closed fractures to retain hydraulic and void aperture.\nWe de\ufb01ne an \u201copen\u201d fracture as a fracture where the \ufb02uid pres-sure has reached the normal stress and the walls are no longer\nin contact.\nThe aperture of closed-fracture elements is estimating with the\nfollowing joint-closure relation, which was written in this form by"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2257e-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 3, "chroma:document": "Willis-Richards et al. (1996), on the basis of the work of Bartonet al. (1985):\nE\u00bc\nE0\n1\u00fe9r0\nn=rn;ref; \u00f07\u00de\nwhere Eis the void aperture (\ufb02uid volume stored in the fracture\nper surface area), E0is the residual aperture (the aperture at effec-\ntive normal stress equal to zero), and rn,refis the 90% closure\nstress (i.e., the effective normal stress that reduces Eto 10% of\nE0). The fracture-effective normal stress, r0\nn, is equal to the nor-\nmal stress, rn,minus \ufb02uid pressure, P. In the simulations for this\nstudy, the void aperture and hydraulic aperture are assumed to\nbe equal.\nFor open elements, the aperture is de\ufb01ned as:\nE\u00bcE0\u00feEopen; \u00f08\u00de\nwhere Eopenis the mechanical separation between the fracture walls.\nA special algorithm is used to de\ufb01ne the value of E0for hy-\ndraulic fractures. When a hydraulic-fracture element \ufb01rst starts, it\nis given a very small aperture, 0.1 mm. But the residual aperture of\na fully open fracture should be on the order of ten or hundred of"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae22614-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 3, "chroma:document": "microns. To handle this issue, an algorithm is used that allows E0\nto increase as the fracture opens. The algorithm mimics the natu-\nral process of fracture-roughness development as the crack starts\nand opens. The algorithm sets E0of each element to be equal to\n90% of the maximum aperture reached by the element, allowingE\n0to increase, but never decrease. A maximum value for E0is\nspeci\ufb01ed, Ehf,max,resid .\nFluid leakoff is calculated, assuming that the reservoir is\nslightly compressible with compressibility of c/,initial porosity\nof/init, and contains the same \ufb02uid as the injection \ufb02uid. Leakoff\nis calculated with the semianalytical method of Vinsome and\nWesterveld (1980), which assumes 1D leakoff, but accounts for\nchanging pressure in the fracture over time.\nPoroelastic-stress changes in the fracture caused by \ufb02ow in the\nmatrix are neglected. Also, poroelastic-pressure response in thematrix caused by fracture deformation is neglected. Poroelastic-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae226b4-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 3, "chroma:document": "stress effects on the fracture are not typically considered for DFIT\nanalysis (Nolte 1979; Barree and Mukherjee 1996). In a typical\nDFIT (very-low permeability), the region of pressure increase\naround the fracture is very narrow, and so the poroelastic effect ofstress on the fracture is slight (Perkins and Gonzalez 1985; Wal-\nlace et al. 2014).\nThe mechanical calculations are performed, assuming that the\nformation is mechanically homogeneous, isotropic, and linearly\nelastic. The calculations are performed with the boundary-element\nmethod of Okada (1992), which uses constant displacement, rec-\ntangular elements in an elastic half-space. This method satis\ufb01es\nthe equations of quasistatic-stress equilibrium, Hooke\u2019s law,and compatibility, giving results that are convergent with\nmesh re\ufb01nement to exact analytical solutions (Babazadeh and\nMcClure 2015).For a mechanically open element, the condition is enforced\nthat (Crouch and Star\ufb01eld 1983)\nr\nr\nn/C0P\u00feDrn\u00bc0; \u00f09\u00de\nwhere rr"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2275e-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 3, "chroma:document": "that (Crouch and Star\ufb01eld 1983)\nr\nr\nn/C0P\u00feDrn\u00bc0; \u00f09\u00de\nwhere rr\nnis the normal stress on the fracture caused by remote\nloading (calculated from the initial conditions) and Drnis the\nchange in normal stress caused by stress induced by the cumula-\ntive deformation of all the fracture elements in the system, as cal-\nculated from the Okada (1992) method.\nClosed elements experience small normal displacements in\nresponse to changes in effective normal stress (Eq. 7). The stress\nchanges caused by these aperture changes are included in the sim-\nulations with the method described in Sections 2.2.3 and 2.2.4 of\nMcClure and Horne (2013).\nFracture initiation occurs at the wellbore when the \ufb02uid pres-\nsure reaches the minimum principal stress. Propagation is handled\nwith linear elastic-fracture mechanics. The stress-intensity factor\nat the crack tips (vertically and horizontally) is calculated numeri-\ncally, and propagation occurs when the stress-intensity factorreaches the fracture toughness."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae227f4-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 3, "chroma:document": "In the simulations for this paper, the fracture height is assumed\nto be limited to no more than 7 m. The only exception is Simula-tion G1, in which there is no height con\ufb01nement and the fracture\npropagates radially.\nThe wellbore and near-wellbore pressure drop is modeled as\nproportional to the square of the \ufb02ow rate from the wellbore to\nthe formation,\nDP\nnw\u00bcCnwQ2; \u00f010\u00de\nwhere Cnwis a constant and Qis the volumetric \ufb02ow rate from the\nwellbore to the formation. This term combines the effect of fric-\ntion in the wellbore, the perforations, and the near-wellbore\nregion. The perforation pressure drop is a function of the numberof \ufb02owing perforations, the perforation diameter, and the perfora-\ntion shape (Cramer 1987). Friction in the wellbore is related to\nwellbore diameter and the wellbore roughness (Moody 1944).\nTypically, the pressure drop associated with injection through\nwell tubulars and perforations during DFIT treatments is verysmall, especially when the injection rate is 2 bbl/min or less. The"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae22934-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 3, "chroma:document": "most-important pressure drop is typically downstream of the per-\nforations: (1) along a tortuous \ufb02ow path from the perforations tothe primary fracture and/or (2) caused by the interaction of multi-\nple (short) fractures of different orientations (longitudinal, multi-\nple transverse orientations, horizontal) which is set up as the\nresult of near-wellbore stress concentration (e.g., hoop stress).\nResults and Discussion\nIn the following sections, various simulation results are presented\nand discussed. In the Generic Simulations subsection, a series ofgeneric simulations are performed to investigate the effect of differ-\nent physical processes on pressure-transient behavior. In the Field\nDFIT #1 subsection, a simulation result is provided that matches\nField DFIT #1. Field DFIT #1 is a straight-forward, uncomplicated\ncase, and so an almost perfect match to the data is possible. In theGas Research Institute/Department of Energy (GRI/DOE) M-site\nTests subsection, data are provided from the GRI/DOE M-site \ufb01eld-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae229d4-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 3, "chroma:document": "demonstration project. At this site, hydraulic-fracture injectionswere performed in conjunction with downhole tiltmeter measure-\nments in an offset well that provide an independent estimate of the\nclosure pressure. Table 1 provides the settings used in all simula-\ntions. Table 2 provides settings that are varied between simulations.\nIn all \ufb01gures showing pressure-transient results, arrows indi-\ncate the identi\ufb01cation of closure from the tangent method (blue),\nthe fracture compliance method (orange), the Mohamed et al.\n(2011) method (green), and the Craig and Blasingame (2006)\nmethod (red).\nGeneric Simulations. In this section, a series of generic DFIT\nsimulations is presented. Superposition time, s,andG-time are........................\n.....................................................\n..........................J179725 DOI: 10.2118/179725-PA Date: 2-August-16 Stage: Page: 1324 Total Pages: 19\nID:jaganm Time: 18:39 I Path: S:/J###/Vol00000/160007/Comp/APPFile/SA-J###1600071324 August 2016 SPE Journal"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae22a74-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 4, "chroma:document": "calculated as functions of time of shut-in, Dt,and the duration of\ninjection, te. Before the leakoff point, injected \ufb02uid is almost\nentirely stored in the wellbore, and leakoff into the formation isnegligible. To account for this, t\neis de\ufb01ned to be equal to the du-ration of injection after the leakoff point is reached, rather than\nthe entire duration of injection.\nSimple Baseline Simulations. Simulation G1 assumes radial-\nfracture geometry (no height containment) with no wellbore stor-\nage, very-small residual aperture, and fairly high fracture tough-ness. Simulation G2 is similar to Simulation G1, but the hydraulic\nfracture is assumed to be con\ufb01ned in height to 7 meters. Figs. 3 and\n4show the results from Simulations G1 and G2. For calculating t\ne,\nthe \u201cleakoff point\u201d in Simulations G1 and G2 occurs immediately at\nthe start of injection because there is not any wellbore storage.\nSimulations G1 and G2 have a very sharp (unrealistic) spike in\nderivative when closure occurred. Closure clearly and unambigu-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae22b14-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 4, "chroma:document": "derivative when closure occurred. Closure clearly and unambigu-\nously occurs at a \ufb02uid pressure equal to 55.1 MPa, precisely the\nsame as the minimum principal stress assumed in the simulation.\nClosure can be identi\ufb01ed consistently in all four plots.\n(a) Stiffness Contrast at Closure. In Appendix A, a detailed\nderivation is provided for the equations governing a DFIT pres-sure transient. After shut-in and the period after shut-in when\n\ufb02ow-rate-dependent pressure gradients dissipate, it is shown thatTable 1 \u2014Simulation settings used in all simulations. Variables are\nde\ufb01ned in the Nomenclature section.\nTable 2 \u2014Simulation settings varied between simulations. Variables are de\ufb01ned in the Nomenclature section.\n60\n57\n5154\n48\n45\n42Pressure (solid) (MPa)\n0 20 40 60 80 100 12030\n25\n20\n15\n10\n5\n0\nG-time\n\u0394t (hours) \u0394t (hours)\u0394t (hours)\u0394P (solid), \u03c4*dP/d\u03c4 (dashed) (MPa)\nI (solid), \u0394t*dI/d\u0394t (dashed)\n(MPa-hr)\u0394P (solid), \u0394t*dP/d\u0394t ((dashed)\n(MPa)G*dP/dG (dashed) (MPa)\n10\u2013210\u20132\n100100\n102\n10\u20132100100100100\n102102103\n102105"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae22baa-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 4, "chroma:document": "10\u2013210\u20132\n100100\n102\n10\u20132100100100100\n102102103\n102105\n101101\n101\nFig. 3\u2014Results from Simulation G1. Upper left: G-function plot. Upper right: Bourdet plot with derivative with respect to time.\nLower left: Bourdet plot with derivative with respect to superposition time. Lower right: Bourdet plot with integrated pressure\n(zoomed in).J179725 DOI: 10.2118/179725-PA Date: 2-August-16 Stage: Page: 1325 Total Pages: 19\nID:jaganm Time: 18:39 I Path: S:/J###/Vol00000/160007/Comp/APPFile/SA-J###160007August 2016 SPE Journal 1325"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae22c86-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 5, "chroma:document": "dP\ndt\u00bc/C0qleakoff\ncfVwl\u00feA\nSf; \u00f011\u00de\nwhere cfis the compressibility of the \ufb02uid, Vwlis the volume of\nthe well, Ais the surface area of the fracture, qleakoff is the volu-\nmetric leakoff rate from the fracture, and Sfis the fracture stiff-\nness. Eq. 11 neglects the effect of the compressibility of the \ufb02uid\nin the fracture and the effect of the changing wellbore volume\nbecause both terms are of negligible magnitude in typical DFITapplications (Appendix A).\nFor a radial fracture, the stiffness of an open fracture (de\ufb01ned\nasS\nf,o) can be written as (Sneddon 1946)\nSf;r\u00bc3pGm\n8\u00f01/C0/C23\u00deRf; \u00f012\u00de\nwhere Gmis the shear modulus, /C23is Poisson\u2019s ratio, and Rfis the\nradius. For a fracture with length much longer than its height, the\nstiffness can be estimated with the Perkins-Kern-Nordgren (PKN)\nassumption of plane strain in vertical cross section (Sneddon1946; Perkins and Kern 1961; Nordgren 1972):\nS\nf;PKN\u00bc4Gm\np\u00f01/C0/C23\u00dehf; \u00f013\u00de\nwhere hfis the fracture height."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae22d26-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 5, "chroma:document": "S\nf;PKN\u00bc4Gm\np\u00f01/C0/C23\u00dehf; \u00f013\u00de\nwhere hfis the fracture height.\nBefore closure, the leakoff rate scales with time according to\ntheG-function (Nolte 1979), and the denominator of Eq. 11 is\nconstant, which is why a plot of pressure vs. G-time forms a\nstraight line.\nIn Appendix A, the stiffness of a closed fracture is derived [in\nanalogy to the discussion in Chapter 12 from Jaeger et al. (2007)]:\ndr0\nn\ndE\u00bcSf;c\u00bcSf;o\u00feSf;cont; \u00f014\u00de\nwhere Sf,cis the stiffness of a closed fracture and Sf,contis the con-\ntribution to the stiffness from the contacting of the fracture walls.\nSf,contcan be calculated from a constitutive law that relates aper-\nture to effective normal stress (such as Eq. 7) by solving for effec-\ntive normal stress and then taking the derivative with respect\nto aperture:Sf;cont\u00bcrn;ref\n9E0\nE2\u00bcrn;ref\n9E01\u00fe9\u00f0rn/C0P\u00de\nrn;ref/C20/C212\n: \u00f015\u00de\nIn Simulation G1, the fracture is circular. The radius at shut-in"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae22dc6-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 5, "chroma:document": "is 21 m, but the fracture extends slightly after shut-in and reachesa \ufb01nal radius of 21.6 m. From Eq. 12, the preclosure stiffness is\n1.09 MPa/mm. From Eq. 13, the preclosure stiffness in Simulation\nG2 (which has fracture height of 7 m) is 3.64 MPa/mm.\nEq. 15 shows that two parameters, E\n0andrn;ref, affect the con-\ntact stiffness at the moment of closure. To avoid a sharp disconti-nuity in stiffness and the pressure derivative, the stiffness at\nmechanical closure cannot be much greater than the stiffness\nbefore closure. In Simulations G1 and G2, the value of E\n0at clo-\nsure is very small, causing Sf,contto be very large (5.7 /C2106MPa/\nmm when Pis equal to rn), which is why there is a very sharp\nchange in the pressure derivative with closure.\n(b) Comparison to Analytical Solutions. As summarized by\nMarongiu-Porcu et al. (2011), the pressure trend before closure\ncan be used to estimate the leakoff coef\ufb01cient and the fracture size"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae22e66-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 5, "chroma:document": "(Nolte 1979; Shlyapobersky et al. 1988; Valko and Economides1999). The technique involves drawing a straight line through the\nplot of pressure vs. G-time during the preclosure period.\nIn Simulation G1, the slope of the line is \u20130.0166 MPa/ Gwith\nan intercept of 56.51 MPa, and the fracture has radial geometry.\nThis yields an estimate of fracture radius of 20.4. In the simula-tion, the actual radius at shut-in is 21 m, and a bit of additional\npropagation occurs so that the \ufb01nal radius is 21.6 m. With the \ufb01nal\nvalue of radius in the simulation, 21.6 m, the equations yield an\nestimate of leakoff coef\ufb01cient of 6.95 /C210\n/C07m/s. From Howard\nand Fast (1957), CL\u00bcDP\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03k/ct\nplr\n, where ctis total compressibil-\nity, the sum of the \ufb02uid compressibility and porosity compressi-\nbility. Assuming a value of DPequal to 56.51 \u2013 35 \u00bc21.51 MPa,\nthe true value of CLis 9.18 /C210/C07m/s, close to the estimated\nvalue, though somewhat larger. The modest discrepancy is per-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae22f06-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 5, "chroma:document": "value, though somewhat larger. The modest discrepancy is per-\nhaps not surprising because the derivation of these G-function-\nbased methods relies on several assumptions, such as Carter\nleakoff and power-law growth of the fracture, that are not exactlysatis\ufb01ed in our more-general numerical calculations. With the true\nvalues of radius at shut-in (21 m) and leakoff coef\ufb01cient, the ana-\nlytical equation predicts an average fracture aperture of 1.12 mm,.........................\n........................\n.......................\n..........................\n60\n57\n5154\n48\n45\n42Pressure (solid) (MPa)\n0 20 40 60 80 100 12030\n25\n20\n15\n10\n5\n0\nG-time\n\u0394t (hours) \u0394t (hours) \u0394t (hours) \u0394P (solid), \u03c4*dP/d\u03c4 (dashed) (MPa)\nI (solid), \u0394t*dI/d\u0394t (dashed)\n(MPa-hr)\u0394P (solid), \u0394t*dP/d\u0394t (dashed)\n(MPa)G*dP/dG (dashed) (MPa)\n10\u2013210\u20132\n100100\n102\n10\u20132100100100100\n102102103\n102105\n104\n102\n101101\n101\nFig. 4\u2014Results from Simulation G2. Upper left: G-function plot. Upper right: Bourdet plot with derivative with respect to time."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae23082-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 5, "chroma:document": "Lower left: Bourdet plot with derivative with respect to superposition time. Lower right: Bourdet plot with integrated pressure\n(zoomed in).J179725 DOI: 10.2118/179725-PA Date: 2-August-16 Stage: Page: 1326 Total Pages: 19\nID:jaganm Time: 18:39 I Path: S:/J###/Vol00000/160007/Comp/APPFile/SA-J###1600071326 August 2016 SPE Journal"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae23136-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 6, "chroma:document": "compared to the actual value from the simulation value of\n1.11 mm. With the actual simulation values of average aperture\nand radius at shut-in, the analytical equation predicts ef\ufb01ciency\n(volume of \ufb02uid in the fracture divided by volume injected) at\nshut-in of 97%, the same as the actual ef\ufb01ciency at shut-in during\nthe simulation.\nIn Simulation G2, the slope of the line is -0.0496 MPa/ Gwith\nan intercept of 58.19 MPa. With the equations for a PKN fracture,\nthis yields an estimate of fracture half-length of 133.7 m. At shut-in, the actual fracture half-length is 108.6 m. The fracture propa-\ngates a bit further after shut-in, reaching 122.6 m. Withthe actual\nvalue of \ufb01nal fracture half-length, the analytical equation predicts\nleakoff coef\ufb01cient of 6.95 /C210/C07m/s. With actual values of half-\nlength at shut-in and leakoff coef\ufb01cient, the analytical expressionspredict average aperture of 1.02 mm, compared to the actual simu-\nlation value of 1.01 mm. With the actual value of average aperture"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae231d6-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 6, "chroma:document": "and fracture half-length at shut-in, the analytical expression pre-\ndicts ef\ufb01ciency at shut-in of 97%, the same as the actual ef\ufb01-\nciency at shut-in during the simulation.\nEffect of Increasing Residual Aperture. Simulation G3 is a\nPKN-like simulation (height con\ufb01ned to 7 m), identical to Simula-tion G2, except that the fracture is given a maximum residual\naperture of 500 mm. The results are shown in Fig. 5.\nA discontinuity in the derivative of pressure still occurs when\nclosure occurs, but it is much less pronounced than in Simulations\nG1 and G2. The 90% closure stress is 51 MPa, and so the stiffnessat closure caused by contact is 11.3 MPa/mm, a factor of 3 greater\nthan the stiffness of the fracture preclosure.\nAs indicated by the orange arrows, the fracture compliance\nmethod indicates closure at the upward deviation of the G/C2dP/dG\ncurve, at G-time of approximately 20 (2.4 hours) and a pressure\nequal to 57 MPa, or at G-time of 25.5 (3.8 hours) and a pressure\nequal to 56.6 MPa."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae23276-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 6, "chroma:document": "equal to 56.6 MPa.\nAs indicated by the blue arrow, the tangent method indicates\nclosure from the G-function plot at G-time equal to approximately\n46, equivalent to approximately 12 hours. This yields an estimate\nof closure pressure equal to approximately 51.8 MPa. In the Bour-\ndet plot with superposition derivative, there appears to be a 1.5\nslope, followed by a disturbance in the derivative at around thetime of closure, followed by what appears to be a resumption of the\n1.5 slope until approximately 12 hours. Picking closure at the end\nof the second 1.5 slope (solid green arrow) yields a closure estimatesimilar to the tangent method, approximately 51.8 MPa. Picking\nclosure at the end of the \ufb01rst 1.5 slope (dashed green arrow) yields\na result similar to the fracture compliance method. In the Bourdet\nplot with integrated pressure, an upward de\ufb02ection of the deriva-\ntive plot can be seen as early as 1.7 hours after shut-in (red arrow),\nequivalent to G-time of 16.7, giving a closure pick of 57.2 MPa."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae23320-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 6, "chroma:document": "From the simulation results, it is possible to identify when and\nhow mechanical closure occurs. Fig. 6 shows the distribution of\naperture and residual aperture in the fracture shortly after shut-in.The red lines represent the residual aperture, and the blue lines\nrepresent the total aperture. The upper part of the \ufb01gure shows the\naperture distribution along the fracture length, down the midpoint.\nThe lower part of the \ufb01gure shows the aperture distribution along\na vertical line running through the center of the fracture.\nFig. 7 shows the aperture distribution after 2.32 hours of shut-\nin. Here, one can see that although the fracture has not closed\nalong its length, a fracture-height recession process has begun to\noccur. The fracture has closed to the residual aperture along the\nupper and bottom 1.5 m of the fracture. The beginning of this frac-ture-height recession process is what causes the initial slight dis-\ncontinuity and increase in the derivative at G-time equal to\napproximately 20."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae233c0-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 6, "chroma:document": "approximately 20.\nThe height recession is not caused by stress layering in the for-\nmation (stress is homogeneous). Instead, height recession iscaused by the residual aperture. As predicted by the PKN approxi-\nmation, the aperture distribution in the vertical direction is ellipti-\ncal, tapering to zero at the edges (though this is imperfectlycaptured in the simulation because of mesh coarseness). The max-\nimum residual aperture is speci\ufb01ed to be a constant value,\n500mm. As the \ufb02uid pressure decreases, aperture gradually\ndecreases, maintaining an elliptical shape and continuing to taper\nto zero at the top and bottom. As aperture decreases, regions de-velop where the aperture has reached the residual aperture (me-\nchanical closure). These regions extend inward from the top and\nbottom of the fracture.\nFig. 8 shows that at approximately 4 hours, the aperture has\nreached the residual aperture everywhere along the fracture. Thisis complete mechanical closure and coincides with the sharp\nincrease in the derivative."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae23460-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 6, "chroma:document": "increase in the derivative.\nEven though fracture-height recession occurs in this simula-\ntion (and occurs in all the simulations with non-negligible residual\naperture), there is no ambiguity between its effect on the pressure\n60\n54\n48\n42Pressure (solid) (MPa) \u0394P (solid), \u03c4*dP/d\u03c4 (dashed) (MPa)\n\u0394t (hours) \u0394t (hours)\u0394t (hours)I (solid), \u0394t*dI/d\u0394t (dashed)\n(MPa-hr)\n100100102\n102102\n102104103\n102101101\n101G-time0 20 40 60 80 100 12015\n10\n5\n0\n\u0394P (solid), \u0394t*dP/d\u0394t (dashed)\n(MPa)G*dP/dG (dashed) (MPa)100\n10010010010\u20131\n10\u2013210\u20132101\nFig. 5\u2014Results from Simulation G3. Upper left: G-function plot. Upper right: Bourdet plot with derivative with respect to time.\nLower left: Bourdet plot with derivative with respect to superposition time. Lower right: Bourdet plot with integrated pressure\n(zoomed in).J179725 DOI: 10.2118/179725-PA Date: 2-August-16 Stage: Page: 1327 Total Pages: 19\nID:jaganm Time: 18:39 I Path: S:/J###/Vol00000/160007/Comp/APPFile/SA-J###160007August 2016 SPE Journal 1327"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae23500-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 7, "chroma:document": "transient and the effect of closure on the pressure transient. Fig. 8\nshows that the fracture closes fully by G-time of 28, coinciding\nwith the time when G/C2dP/dGbegins to increase rapidly (Fig. 5).\nThe fracture-height recession occurring before complete closure(seen by comparison between Fig. 7 and Fig. 8) during the period\nof approximately 20 G-time to 28 G-time causes a visible upward\nde\ufb02ection of G/C2dP/dG, but this de\ufb02ection is slight compared to\nthe much-larger de\ufb02ection that occurs after complete closure.\nThe true value of the minimum principal stress is 55.1 MPa,\nbut the \ufb02uid pressure at the time of mechanical closure is slightly\nlarger, approximately 57 MPa. This occurs because there is stillsome stress shadow on the fracture at closure as a result of the re-\nsidual aperture. This stress shadow can be estimated by multiply-\ning the residual aperture (500 mm) by the fracture stiffness causedby elastic deformation of the formation, S\nf,o(calculated from Eq."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae235a0-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 7, "chroma:document": "f,o(calculated from Eq.\n13). This yields an estimate of 1.82 MPa, close to the overesti-\nmate that is observed.\nAdding Wellbore Storage. In Simulation G3, several aspects\nof the simulation remain unrealistic. Wellbore storage isneglected, there is not near-wellbore pressure loss, and there is\nstill a rather sharp discontinuity in derivative with closure. In Sim-\nulation G4, these issues are resolved. Simulation G4 is similar toSimulation G3 but includes wellbore storage, wellbore/near-well-\nbore pressure drop, and uses a much lower value of r\nn;ref, 5 MPa,\nto prevent a sharp increase in derivative with closure. From these\nvalues, we can calculate Sf,contto be 1.1 MPa/mm at Pequal to rn\n(Eq. 15). This is less than the stiffness caused by the elastic com-\npliance of the surrounding formation, Sf,o(3.64 MPa/mm). Thus,\nTime since shut-in: 0.0013386 hr, G-time: 0.08962\nDistance (m)Distance (m)"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae23640-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 7, "chroma:document": "Distance (m)Distance (m)\n\u20133 \u20132 \u20131 0 1 2 3Aperture distribution in vertical cross section at the wellboreAperture at Crack Center (mm) Aperture Along Crack Center (mm)1.5\n1.51\n10.5\n0.50\n0\u2013150 \u2013100 \u201350 0 50 100 150\nFig. 6\u2014Distribution of aperture during Simulation G3 soon after shut-in. The upper \ufb01gure shows aperture along the length of the\nfracture, and the lower \ufb01gure shows aperture along a vertical cross section at the center of the crack.\nTime since shut-in: 2.3205 hr, G-time: 19.6252\nDistance (m)Distance (m)\n\u20133 \u20132 \u20131 0 1 2 3Aperture distribution in vertical cross section at the wellboreAperture at Crack Center (mm) Aperture Along Crack Center (mm)1.5\n1.51\n10.5\n0.50\n0\u2013150 \u2013100 \u201350 0 50 100 150\nFig. 7\u2014Distribution of aperture during Simulation G3 at 2.3205 hours after shut-in (19.6252 G-time). The upper \ufb01gure shows aper-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae236e0-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 7, "chroma:document": "ture along the length of the fracture, and the lower \ufb01gure shows aperture along a vertical cross section at the center of the crack.J179725 DOI: 10.2118/179725-PA Date: 2-August-16 Stage: Page: 1328 Total Pages: 19\nID:jaganm Time: 18:39 I Path: S:/J###/Vol00000/160007/Comp/APPFile/SA-J###1600071328 August 2016 SPE Journal"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae23780-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 8, "chroma:document": "when mechanical closure occurs in Simulation G4, the stiffness\nincreases only fractionally, and a sharp discontinuity in pressure\nderivative does not occur.\nThe fracture forms after 36.6 seconds of injection, at a break-\ndown pressure of 61.15 MPa, before dropping down to a fracture-propagation pressure approximately 60 MPa. The fracture-propa-\ngation pressure is considerably higher than the ISIP because of\nthe wellbore and near-wellbore pressure drop (Fig. 9).\nThe pressure transient in this simulation appears realistic and\nis similar to the match to \ufb01eld data described in the Field DFIT #1subsection. The fracture-compliance method gives a closure pick\natG-time equal to 33, pressure of 56.7 MPa, and 4.3 hours after\nshut-in (orange arrow). The closure pick from the tangent methodis at G-time equal to 77, pressure of 52.75 MPa, and 22 hours af-\nter shut-in (blue arrow).\nThe method of Mohamed et al. (2011) gives a closure pick\nsimilar to that of the tangent method (green arrow). The Bourdet"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae23820-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 8, "chroma:document": "plot with integrated pressure gives a rather ambiguous pick forclosure, with the \ufb01rst de\ufb02ection of the curve at 1 hour (15.1 G-\ntime; dashed red arrow), earlier than the interpretation from any\nother method. An in\ufb02ection is also apparent at approximately 20\nhours (73 G-time), close to the timing of the closure pick from the\ntangent method (solid red arrow).\nAs in Simulation G3, fracture-height recession occurs. How-\never, this causes only the very-slight concave upward behavior\nseen in the G/C2dP/dGcurve from approximately G-time equal to\nTime since shut-in: 4.4788 hr, G-time: 27.8749\nDistance (m)Distance (m)\n\u20133 \u20132 \u20131 0 1 2 3Aperture distribution in vertical cross section at the wellboreAperture at Crack Center (mm) Aperture Along Crack Center (mm)1.5\n1.51\n10.5\n0.50\n0\u2013150 \u2013100 \u201350 0 50 100 150\nFig. 8\u2014Distribution of aperture during Simulation G3 at 4.4788 hours after shut-in (27.8749 G-time). The upper \ufb01gure shows aper-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae238c0-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 8, "chroma:document": "ture along the length of the fracture, and the lower \ufb01gure shows aperture along a vertical cross section at the center of the crack.\n60\n54\n48\n42Pressure (solid) (MPa)\u0394P (solid), \u03c4*dP/d\u03c4 (dashed) (MPa)\n\u0394t (hours) \u0394t (hours) \u0394t (hours) I (solid), \u0394t*dI/d\u0394t (dashed)\n(MPa-hr)\n100100102\n102102\n102104\n101G-time0 20 40 60 80 100 12015\n10\n5\n0\n\u0394P (solid), \u0394t*dP/d\u0394t (dashed)\n(MPa)G*dP/dG (dashed) (MPa)100\n10010010010\u20132\n10\u2013210\u20132\nFig. 9\u2014Results from Simulation G4. Upper left: G-function plot. Upper right: Bourdet plot with derivative with respect to time.\nLower left: Bourdet plot with derivative with respect to superposition time. Lower right: Bourdet plot with integrated pressure\n(zoomed in).J179725 DOI: 10.2118/179725-PA Date: 2-August-16 Stage: Page: 1329 Total Pages: 19\nID:jaganm Time: 18:39 I Path: S:/J###/Vol00000/160007/Comp/APPFile/SA-J###160007August 2016 SPE Journal 1329"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae239f6-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 9, "chroma:document": "20 to G-time equal to 38. There is no ambiguity between this\nmodest effect and the sharply increasing G/C2dP/dGcurve from 38\nG-time to 80 G-time, which is interpreted as fracture-height reces-\nsion by the tangent method. As we have shown here, the increas-\ningG/C2dP/dGcurve is caused solely by the increasing fracture\nstiffness after mechanical closure.\nReduction in the wellbore and near-wellbore pressure drop\ncauses the very rapid, brief period of pressure decrease immedi-ately after shut-in.\nFracture Tip-Extension. Simulation G5 is performed with a\nlower value of fracture toughness, 2 MPa /C1m1/2(instead of\n10 MPa /C1m1/2, used in Simulations G1-G4). Simulation G5 is oth-\nerwise identical to Simulation G4. With this lower value of frac-\nture toughness, a large amount of fracture tip-extension occurs\nafter shut-in. As in Simulation G4, the fracture forms after 36.6seconds of injection, at a breakdown pressure of 61.15 MPa,\nbefore dropping down to a fracture-propagation pressure approxi-"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae23a96-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 9, "chroma:document": "mately 60 MPa (Fig. 10).\nThe fracture tip-extension is visible on the G-function plot. As\ndescribed by Nolte (1991) and Barree and Mukherjee (1996), tip-extension causes the G/C2dP/dGcurve to increase sharply and then\ncurve concave down. At shut-in, the fracture half-length is 110 m.\nHowever, after shut-in the fracture reaches a half-length of 204 m.\nFracture growth does not end until the fracture closes. The tip-\nextension accelerates closure because it increases surface area andleakoff. Also, as the fracture grows, the average aperture along\nthe fracture is forced to decrease to accommodate mass balance,\ncausing aperture to reach the residual aperture sooner.\nThe high fracture length from this simulation may seem unre-\nalistic for such a relatively modest volume of \ufb02uid pumped. Simu-\nlations (not shown in this paper) show that with multiple\npropagating-fracture strands, the transient looks similar, but the"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae23b40-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 9, "chroma:document": "overall fracture length is proportionally decreased, according tothe number of fractures. Thus, the transient from a single fracture\nwith a half-length of 200 m will be similar to the transient from\nmultiple-fracture strands, each with length much less than 200 m.\nStress shadowing tends to cause only one or two fractures to prop-\nagate, but near-wellbore pressure drop has the opposite effect,encouraging propagation of a larger number of fractures.\nInvestigation of the simulation results indicates that closure\noccurs at G-time of approximately 3.9 (0.1 hour) and a pressureof 56.5 MPa. The fracture-compliance method indicates that clo-\nsure occurs at a G-time of 12.7 (0.7 hour) and a pressure of 55.5\nMPa (orange arrow). The tangent method indicates that closure\noccurs at G-time of 32.6 (4.2 hours) and pressure of 52.35 MPa\n(blue arrow). The Bourdet plot with superposition-derivativemethod suggests that closure occurs at 3.2 hours (28.3 G-time)\nand a pressure of 53.3 MPa (green arrow). The integrated-pressure"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae23be0-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 9, "chroma:document": "plot suggests that closure occurs at approximately 2.5 hours (24.7G-time) and a pressure of 54 MPa (red arrow).\nHigher Permeability. Simulation G6 is performed with a\npermeability of 10\n/C015m2(/C241 millidarcy), much higher than the\nother simulations. As in Simulations G4 and G5, the fracture\nforms after 36.6 seconds of injection, at a breakdown pressure\nof 61.15 MPa, before dropping down to a fracture-propagation\npressure approximately 60 MPa. The other simulations in thisstudy are performed with a 1D leakoff model. But this simulation\nis performed with a fully numerical treatment of \ufb02ow in the\nmatrix around the fracture, with the method described by\nMcClure (2014).\nFig. 11 shows that before mechanical closure, a straight line\ncan be seen on the G/C2dP/dGplot, as expected. When mechanical\nclosure occurs, at approximately 0.85 G-time (0.012 hours), the\nrate of pressure drop decreases (orange arrow). This behavior is\ndifferent from the previous cases, when closure causes the rate of"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae23c80-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 9, "chroma:document": "pressure drop to increase. The different behavior can be under-stood by considering Eq. 11, which shows that the rate of pressure\ndecrease is equal to the product of system stiffness (the reciprocal\nof total storage coef\ufb01cient, de\ufb01ned in Appendix A) and leakoffrate. In a very-low permeability formation, the fracture remains\nin\ufb01nitely conductive (dimensionless fracture conductivity greater\nthan 300) even after closure. But with higher permeability, the\ndimensionless fracture conductivity rapidly drops below 300 after\nclosure. As a result, the rate at which \ufb02uid can \ufb02ow from the well(where \ufb02uid is stored because of wellbore storage) into the frac-\nture is reduced by closure. If leakoff rate decreases enough to\ncounteract the effect of increasing stiffness, the rate of pressure\ndecline will decrease.\nIn addition, in cases of high leakoff, the fracture length is\nmuch shorter. Eq. 11 shows that with lower fracture area, closure\nhas a relatively smaller impact on the system stiffness, and well-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae23d20-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 9, "chroma:document": "bore storage has a relatively larger effect. Thus, closure has less\n60\n51\n42Pressure (solid) (MPa)\u0394P (solid), \u03c4*dP/d\u03c4 (dashed) (MPa)\n\u0394t (hours) \u0394t (hours) \u0394t (hours) I (solid), \u0394t*dI/d\u0394t (dashed)\n(MPa-hr)\n100100102\n102102\n102104\n101G-time0 20 40 60 80 100 12015\n10\n5\n0\n\u0394P (solid), \u0394t*dP/d\u0394t (dashed)\n(MPa)G*dP/dG (dashed) (MPa)100101\n10010010010\u20131\n10\u2013210\u20132\nFig. 10\u2014Results from Simulation G5. Upper left: G-function plot. Upper right: Bourdet plot with derivative with respect to time.\nLower left: Bourdet plot with derivative with respect to superposition time. Lower right: Bourdet plot with integrated pressure\n(zoomed in).J179725 DOI: 10.2118/179725-PA Date: 2-August-16 Stage: Page: 1330 Total Pages: 19\nID:jaganm Time: 18:39 I Path: S:/J###/Vol00000/160007/Comp/APPFile/SA-J###1600071330 August 2016 SPE Journal"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae23dca-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 10, "chroma:document": "effect on the overall stiffness, which lessens the tendency of clo-\nsure to increase the derivative.\nThe linear \ufb02ow period that corresponds to the straight line on\ntheG/C2dP/dGcurve can be seen from approximately 0.001 hour\nto approximately 0.01 hours on the Bourdet plot (upper right ofFig. 11). After the linear \ufb02ow period, the Bourdet derivative-plot\n\ufb02attens out to a slope of approximately zero (with some gradual\nincrease), corresponding to a radial \ufb02ow period (which begins\nrapidly because of the relatively high permeability and short frac-\nture), before \ufb01nally reaching an impulse radial \ufb02ow period with anegative one slope, starting at approximately 1.00 hour ( G-time of\n15). Comparison of the Bourdet plot with the G-function plot indi-\ncates that the G/C2dP/dGcurve continues increasing after closure,\nthough more slowly.\nIn the Bourdet plot with superposition derivative, the late-time\nimpulse radial appears as a \ufb02at line, as expected. The intermediate"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae23e6a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 10, "chroma:document": "time, approximately radial, period shows a slope of 1.0, which is\nexpected because taking the derivative with respect to superposi-tion time increases the log-log slope by one at shut-in times much\ngreater than the period of injection (Section 1.2.2). Careful\ninspection of the Bourdet plot with respect to integrated pressure\nshows no indication of closure.\nAs in the previous cases, closure should be picked at the devia-\ntion from linearity of the G/C2dP/dGcurve (orange arrow). How-\never, in this relatively high-permeability case, the slope of the\nG/C2dP/dGcurve decreases after closure, rather than increasing.\nThe tangent method provides a closure pick at approximately 8\nG-time, at a pressure far below the minimum principal stress\n(blue line).\nField DFIT #1. In this subsection, we review Field DFIT #1.\nThis DFIT was previously shown on the left side of Figs. 4 and 5\nfrom Cramer and Nguyen (2013). The simulation that we perform\nto match the data is called Simulation F1 in Table 2. The injection"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae23f0a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 10, "chroma:document": "rate over time is shown in Fig. 1.\nThere are several reasons that this is a good example of a\nDFIT transient.\n\u2022Cement integrity is excellent, as evidenced by the cement-\nbond log, assuring adequate interzonal isolation throughout\nthe injection and falloff periods.\n\u2022The test interval is mechanically strong (low-clay content)\nand suf\ufb01ciently thick (7 m) for establishing a dominant verti-cal hydraulic fracture. These mechanical characteristics are\nindicated by openhole log signatures: neutron density, resis-\ntivity, and sonic log responses (Fig. 12).\n\u2022As shown in Fig. 12, the stress contrast to higher-stress\nbounding rock layers is suf\ufb01cient for containing a DFIT-\nscale fracture to the test interval.\n\u2022A conductive drilling-induced tensile fracture is exhibited\nin the center of the test interval by an openhole image log(Fig. 13). This pre-existing fracture facilitates noncomplex,\nplanar fracture initiation and propagation.\n\u2022The vertical orientation of the wellbore ensures that the axial"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae23faa-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 10, "chroma:document": "starter fracture will be in line with the plane normal to the\nminimum principal/horizontal stress, further reducing the\npotential for fracture complexity (Cramer and Nguyen\n2013).\n\u2022The pressure rollover event observed near the beginning of\ninjection and at a relatively low bottomhole pressure is a\nstrong indication that the preexisting fracture is reopened\nduring the DFIT (Fig. 1). Reopening of an existing fracture\nensures that the in\ufb02uence of wellbore hoop-stress zone is\nmitigated or eliminated.\n\u2022The onset of the pressure rollover event, in which the pres-\nsure plot departs from the wellbore-storage trend line (at\n55.3 MPa), is, in effect, the fracture-reopening pressure.\nThis fracture-reopening pressure should be slightly above\nthe fracture-closure pressure.\n\u2022There is a water-hammer event at the end of injection, con-\n\ufb01rming that an excellent connection existed between the\nwellbore and primary, far-\ufb01eld fracture. This is assurancethat there was no distortion of the shut-in pressure response"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2404a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 10, "chroma:document": "caused by near-wellbore fracture complexity or tortuosity.\n\u2022Borehole breakout analysis of adjacent intervals indicates\nthe existence of a large difference between minimum and\nmaximum horizontal stresses. Consequently, the probabilityof a primary hydraulic fracture interacting with and reopen-\ning crossed natural fractures is very-low; transverse fracture\nstorage is extremely unlikely.\n\u2022Permeability in the tested interval is very low, in the range\nof 50 to 150 nd. Consequently, dimensionless fracture con-ductivity should be very high following mechanical-fracture\nclosure, resulting in a stable \ufb02uid leakoff rate during closure.\nThus, the change in fracture compliance/stiffness caused by\n60\n40Pressure (solid) (MPa)\u0394P (solid), \u03c4*dP/d\u03c4 (dashed) (MPa)\n\u0394t (hours) \u0394t (hours) \u0394t (hours) I (solid), \u0394t*dI/d\u0394t (dashed)\n(MPa-hr)102G-time0 246 8 10 125\n0\n\u0394P (solid), \u0394t*dP/d\u0394t (dashed)\n(MPa)G*dP/dG (dashed) (MPa)\n100100100100\n105\n100100\n10\u2013310\u2013110\u2013210\u2013210\u2013210\u20132"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae240ea-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 10, "chroma:document": "100100100100\n105\n100100\n10\u2013310\u2013110\u2013210\u2013210\u2013210\u20132\nFig. 11\u2014Results from Simulation G6. Upper left: G-function plot. Upper right: Bourdet plot with derivative with respect to time.\nLower left: Bourdet plot with derivative with respect to superposition time. Lower right: Bourdet plot with integrated pressure.J179725 DOI: 10.2118/179725-PA Date: 2-August-16 Stage: Page: 1331 Total Pages: 19\nID:jaganm Time: 18:39 I Path: S:/J###/Vol00000/160007/Comp/APPFile/SA-J###160007August 2016 SPE Journal 1331"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2418a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 11, "chroma:document": "fracture closure should dominate the pressure falloff\nresponse when fracture closure occurs.\nThe simulation very closely matches the \ufb01eld data. The simu-\nlation match is accomplished with a minimum principal stress\nequal to 54.3 MPa. This compares well with the fracture-reopen-\ning pressure of 55.3 MPa, which is derived in analyzing Fig. 2.\nThe corresponding net pressure at the ISIP (57.1 MPa) is equal to\n2.8 MPa, which is a reasonable value with the small-injection vol-ume and compatible with simulations performed with commercial\nhydraulic-fracture models.Fig. 2 shows there is a very-brief period of rapid-pressure drop\nat the start, as wellbore friction and near-wellbore friction\ndecrease. Subsequently, the G/C2dP/dGcurve appears to be contin-\nuously concave upward during the shut-in period. However, a\nclose examination of the data indicates that it forms a relatively\ufb02at straight line until approximately G-time of 7 (0.48 hours),\nwhen it begins to increase sharply. Thus, according to fracture-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae24220-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 11, "chroma:document": "compliance method, the closure estimate is at G-time of 7, for a\npressure of 55.2 MPa (orange arrow). Plots of aperture vs. posi-\ntion from the simulation results (similar to those shown in theTVD\nm\n3025.0\n3030.03035.03040.0\n3045.0\nHCGR GAPI0 200 100\nHSGR GAPI0 200 100AT90 \u03a9.m0.10 100\nAT30 \u03a9.m\nAT10 \u03a9.mPDMC dec0.30 \u20130.10 0.1000\nTNPH_LIM V/V0.30 \u20130.10 0.1000DTCO US/F50.00 150 100\nDTSM_SL US/F\nDTSM_FA US/FCorr_Stress MPa40.00            65.00 52.50\nLog_Stress MPa40.00               65.00 52.50Dyn_PR unitless0.15              0.30 0.22\nStatic_E GPa20.00                50.00 35.00\nFig. 12\u2014Openhole log data for Field DFIT #1.\nConductive Fracture\nPerforations: 3035\u20133040 m\nFracture noted from 3037 to 3039 m\nFrom Lane/CoP303930383037\nFig. 13\u2014Wellbore imaging log for Field DFIT #1.J179725 DOI: 10.2118/179725-PA Date: 2-August-16 Stage: Page: 1332 Total Pages: 19\nID:jaganm Time: 18:39 I Path: S:/J###/Vol00000/160007/Comp/APPFile/SA-J###1600071332 August 2016 SPE Journal"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae242c0-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 12, "chroma:document": "Effect of Increasing Residual Aperture subsection) con\ufb01rm that\nclosure occurs at G-time of 7 in the simulation. Fracture closure\noccurs at a pressure approximately 0.9 MPa greater than rhmin\nbecause of the stress shadow caused by the residual aperture.\nThe tangent method yields an estimate of closure pressure\napproximately G-time of 36 (approximately 9.4 hours), which is\nat a pressure approximately 49.3 MPa. The net pressure corre-sponding to this estimate is 7.8 MPa, which is unrealistically high\nwith the small injection volume because it implies exceptionally\nhigh fracture toughness. The Mohamed et al. (2011) method picksclosure at approximately 8 hours ( G-time of 33.2), for a pressure\nof approximately 50 MPa. There is a slight deviation in the Bour-\ndet plot with integrated pressure curve at approximately 4 hours\n(G-time of 23) and at pressure of 52.5 MPa.\nThe GRI/DOE M-Site Tests. The GRI/DOE M-Site project\ninvolved a series of hydraulic-fracturing injections into two inter-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae24360-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 12, "chroma:document": "vals - the \u201cB-sand\u201d and the \u201cC-sand\u201d. In this section, we consider\nthe tests in the B-sand. The B-sand consists of a complexsequence of sandstone, mudstone, siltstone, and shale. The site\nwas unusually well-instrumented, involving a microseismic arrayand downhole tiltmeters in an offset well (Branagan et al. 1996).\nThe tiltmeters provide an independent estimate of the mini-\nmum principal stress. In the B-sand, seven different injection tests\nwere performed, some involving clear water, others involving\ngelled or cross-linked \ufb02uid, and one involving proppant. Pump-in\ufb02uid volume during these tests ranges from 3.0 to 95 m\n3(19 to\n600 bbl). Pressure measurements are available for an extended\ntime after shut-in for two of the tests: the initial breakdown injec-tion, which involved 3.0 m3(19 bbl) of crosslinked gel pumped at\n13.25 L/s (5 bbl/min), and injection 6B, which involved 63.6 m3"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae24400-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 12, "chroma:document": "13.25 L/s (5 bbl/min), and injection 6B, which involved 63.6 m3\n(400 bbl) of linear gel pumped at 58.3 L/s (22 bbl/min). The ma-trix permeability is approximately 0.1 md ( /C2510/C016m2), signi\ufb01-\ncantly higher than Field DFIT #1 (see the Field DFIT #1subsection).\nFig. 14 shows the result from the initial breakdown injection\ntest. Breakdown of the formation occurred 2.7 minutes before\nshut-in at 30.77 MPa before dropping to a fracture-propagation\npressure of approximately 27 MPa.\nThe 6B injection shut-in transient is unusual (showing no clear\nindication of closure at any pressure), was the sixth injection into\nthe interval, and used a much larger volume than is typically used\nin a DFIT. Therefore, the results from that test are not presented.\nTheG-function plot shows a period of fracture tip-extension at\nearly-time, which terminates at G-time /C241. Afterward, the G/C2dP/\ndGcurve \ufb02attens out before beginning to increase sharply after\napproximately G-time of 3.5 (0.16 hours; indicated by orange"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae24518-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 12, "chroma:document": "approximately G-time of 3.5 (0.16 hours; indicated by orange\narrow). According to the fracture-compliance method, this\nincrease in the G/C2dP/dGcurve represents closure, and so the\nminimum principal stress should be picked from the pressure cor-\nresponding to G-time of 3.5, which is 24.3 MPa.\nThe tangent method and the Mohamed et al. (2011) method\nyield a closure pick at approximately 0.85 hours of shut-in ( G-\ntime of 9.5), corresponding to a pressure of 22.75 MPa (blue and\ngreen arrows). In the Bourdet with integrated pressure plot, there\nis no apparent de\ufb02ection indicating closure.\nFig. 15 shows the normalized tiltmeter response plotted\nagainst wellbore pressure during injection 2B, the test immedi-\nately after the breakdown test shown in Fig. 14. A smoothed ver-\nsion of this \ufb01gure is shown in Fig. 9A-4 of Economides and Nolte\n30\n25\n20Pressure (solid) (MPa) \u0394P (solid), \u03c4*dP/d\u03c4 (dashed) (MPa)\n\u0394t (hours) \u0394t (hours)\u0394t (hours)I (solid), \u0394t*dI/d\u0394t (dashed)\n(MPa-hr)G-time0 246 8 100"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae245ae-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 12, "chroma:document": "(MPa-hr)G-time0 246 8 100\n\u0394P (solid), \u0394t*dP/d\u0394t (dashed)\n(MPa)G*dP/dG (dashed) (MPa)\n101101\n100100100100\n10\u2013110\u2013110\u20131\n10\u2013310\u20133\n10\u20132100100\n10\u2013110\u20131\n10\u2013310\u2013210\u2013210\u20132\nFig. 14\u2014Field data from the M-site, B-sand breakdown injection. Upper left: G-function plot. Upper right: Bourdet plot with deriva-\ntive with respect to time. Lower left: Bourdet plot with derivative with respect to superposition time. Lower right: Bourdet plot withintegrated pressure.\n1\n0.9\n0.8\n0.7\n0.6\n0.5\n0.4\n0.3\n0.2\n0.1\n0\n20 22 26\nPressure (MPa)4628\n4558\n4487\n44184348\n4274Normalized Tilt\nFig. 15\u2014Normalized tiltmeter data from Injection 2B from the\nGRI/DOE M-site. Data courtesy of Norm Warpinski.J179725 DOI: 10.2118/179725-PA Date: 2-August-16 Stage: Page: 1333 Total Pages: 19\nID:jaganm Time: 18:39 I Path: S:/J###/Vol00000/160007/Comp/APPFile/SA-J###160007August 2016 SPE Journal 1333"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae24694-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 13, "chroma:document": "(2000). Each tiltmeter response is normalized by dividing by the\nmaximum tilt measured at that instrument during the test. The tilt-\nmeters demonstrate that the fracture began opening at approxi-mately 23.9 MPa, very close to the closure pressure estimated by\nthe fracture-compliance method from the shut-in transient after\nthe breakdown injection (which immediately preceded the 2B\ninjection). At 23.9 MPa, the derivative of tilt with respect to pres-\nsure becomes much greater, indicating an abrupt drop in fracturestiffness at that pressure corresponding to reopening.\nFig. 15 shows only the data during injection, not shut-in. The\nshut-in tiltmeter response is much smoother and shows a more-\ngradual relationship between tilt and wellbore pressure. This dif-\nference is caused by the contribution to the tiltmeter responsefrom the poroelastic effect of the \ufb02uid leaked off into the forma-\ntion. During injection, the amount of leakoff is relatively minimal,"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2472a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 13, "chroma:document": "which is why the tiltmeter response during injection (reopening)\nis more appropriate for estimating the minimum principal stress.\nBranagan et al. (1996) report substantial residual deformation\n(20% of the maximum de\ufb02ection) many weeks after the \ufb01rst four\ninjections into the B-sand, which they interpret as suggesting that\nthe fractures retained signi\ufb01cant residual aperture after closure.\nNonuniqueness in Simulation Results. The simulation matches\nto the \ufb01eld data are subject to nonuniqueness. Lower shear modu-\nlus, higher fracture toughness, and greater fracture height (or ra-\ndial geometry) all result in lower fracture surface area, which can\nbe compensated by using a higher permeability or lower \ufb02uid vis-\ncosity. The selection of \ufb02uid viscosity is complicated because inmost cases clear water is being injected into a gas- or oil-saturated\nformation. Leakoff will be affected by multiphase effects, and the\ninjection \ufb02uid has a different viscosity and compressibility than"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae247ca-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 13, "chroma:document": "the reservoir \ufb02uid. In the simulations in this paper, we have not\nmade an attempt to choose a realistic viscosity and have merelyset the \ufb02uid viscosity to 1 cp. Adjustments to viscosity require\nproportional adjustments to the permeability to achieve a simula-\ntion match.\nBecause of this nonuniqueness, many of the values used in the\nsimulation matches should not necessarily be interpreted as beingactual estimates for those parameters in the formation. However,\nthere are certain features of the transients that allow unique esti-\nmation of formation parameters. For example, the wellbore-stor-age coef\ufb01cient can be estimated with high con\ufb01dence from the\nslope of a plot of pressure vs. volume injected before the leakoff\npoint or breakdown pressure. Additional data sources can be used\nto constrain parameters. For example, the shear moduli can be\nestimated from core or geophysical logs.\nRegardless of these uncertainties, the estimates for closure\npressure that are based on the simulations matches are always"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae24874-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 13, "chroma:document": "pressure that are based on the simulations matches are always\nunique. In every case, closure causes the G/C2dP/dGcurve to\nde\ufb02ect upward (except the high-permeability case, when it\nde\ufb02ected downward). Therefore, the location of the de\ufb02ection intheG/C2dP/dGcurve can be taken as a unique estimate for the clo-\nsure pressure. Because the residual aperture creates a stress\nshadow, the closure-pressure estimate could be up to 1 to 2 MPa\ngreater than the actual value of r\nhmin(depending on the magni-\ntude of the residual aperture and the geometry of the fracture),which creates some uncertainty (see the Effect of Increasing Re-\nsidual Aperture subsection).\nThe net pressures implied by the tangent method of G-function\ninterpretation and the Mohamed et al. (2011) method are much\nhigher than the values derived from our proposed method. For\nexample, for Field DFIT #1, the net pressure at the ISIP is 2.8\nMPa according to the pick from the fracture-compliance method,"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae24982-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 13, "chroma:document": "MPa according to the pick from the fracture-compliance method,\nand 7.8 MPa according to the pick from the tangent method.Matching these very-high net pressures requires very-high values\nof fracture toughness or poorly constrained adjustments for a\n\ufb02uid-lag zone or inelastic deformation ahead of the crack tip (Jef-\nfrey 1989; Yew and Liu 1993; Mack and Warpinski 2000). This is\nnot inconceivable, but in the context of the results from this study,the more-likely explanation is that net pressure is overestimated\nbecause of the method of DFIT interpretation. The lower net pres-\nsures implied by the fracture-compliance method are consistentwith values of fracture toughness derived from laboratory\nmeasurements.\nConclusions\n1. Using a hydraulic-fracturing simulator that includes the effect\nof residual-fracture aperture, simulations of DFITs can be per-\nformed that closely match \ufb01eld data, including the preclosure,\nclosure, and post-closure periods.\n2. The results suggest that the tangent method for identifying clo-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae24a0e-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 13, "chroma:document": "sure from G-function plots (Barree and Mukherjee 1996; Bar-\nree et al. 2007) tends to underestimate the minimum principal\nstress in low-permeability formations because it neglects the\neffect of changing residual fracture aperture after closure.\n3. The method proposed by Mohamed et al. (2011) for picking\nclosure from a Bourdet plot with superposition time derivativetends to give results similar to the tangent method, underesti-\nmating closure pressure in low-permeability intervals.\n4. The method of Craig and Blasingame (2006) for identifying\nclosure from a Bourdet plot with integrated pressure is based\non picking closure from a change in storage and thus identi\ufb01esthe correct closure pressure in some cases. However, the\nde\ufb02ection on the plot is often ambiguous or impossible to\nidentify.\n5. Closure can best be picked by identifying the departure from\nlinearity on the plot of pressure vs. G-time or the square root of\ntime (after the very-early time period after shut-in when dissi-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae24b26-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 13, "chroma:document": "pating wellbore and near-wellbore friction causes the pressure\nto drop rapidly). Identi\ufb01cation of this departure from linearitycan be assisted by the G/C2dP/dGcurve. In low-permeability\nformations, the G/C2dP/dGcurve should de\ufb02ect upward at clo-\nsure. We refer to this method as the \u201cfracture compliance\nmethod\u201d for picking closure.\n6. In high-permeability cases, the G/C2dP/dGcurve may de\ufb02ect\ndownward (with pressure de\ufb02ecting upward) when closure\noccurs because closure can signi\ufb01cantly reduce the rate of\nleakoff.\n7. The residual fracture aperture upon closure can cause a stress\nshadow of up to 1 to 2 MPa. As a result, the closure pressurecan be as much as 1 to 2 MPa above the value of the minimum\nprincipal stress.\n8. In a fracture-injection test from the GRI/DOE M-site project,\nthe closure pick can be independently estimated from down-\nhole tiltmeters and is consistent with the pick from the frac-ture-compliance method presented in this paper.\nNomenclature\nA\u00bcfracture area, m\n2\ncf\u00bc\ufb02uid compressibility, MPa/C01"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae24bb2-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 13, "chroma:document": "A\u00bcfracture area, m\n2\ncf\u00bc\ufb02uid compressibility, MPa/C01\nc/\u00bcporosity compressibility, MPa/C01\nCac\u00bcafter-closure storage coef\ufb01cient, m3/MPa\nCfD\u00bcdimensionless fracture conductivity, unitless\nCL\u00bcCarter leakoff model leakoff coef\ufb01cient, m3/(m2/C1s)\nCnw\u00bcconstant for wellbore and near-wellbore pressure\ndrop, MPa/(m3/s)2\nCw\u00bcwellbore-storage coef\ufb01cient, m3/MPa\nCt\u00bctotal storage coef\ufb01cient for fracture and wellbore\nsystem, m3/MPa\ne\u00bchydraulic aperture, related to the transmissivity by\nthe cubic law, m\nE\u00bcvoid aperture, volume of \ufb02uid stored per fracture\nsurface area, m\nE0\u00bcresidual fracture aperture, m\nEhf,max,resid \u00bcmaximum residual aperture, m\nEopen\u00bcmechanical fracture opening, m\ng\u00bcdimensionless variable used in de\ufb01ning G,de\ufb01ned\nin Eq. 2, unitless\nG\u00bcG-time, de\ufb01ned in Eq. 1, unitless\nGm\u00bcshear modulus, MPa\nhf\u00bcfracture height, m\nI\u00bcintegrated pressure, MPa /C1s\nk\u00bcpermeability, m2J179725 DOI: 10.2118/179725-PA Date: 2-August-16 Stage: Page: 1334 Total Pages: 19"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae24c34-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 13, "chroma:document": "ID:jaganm Time: 18:39 I Path: S:/J###/Vol00000/160007/Comp/APPFile/SA-J###1600071334 August 2016 SPE Journal"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae24cb6-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 14, "chroma:document": "KIc\u00bcfracture toughness, MPa /C1m1/2\nLf\u00bcfracture half-length, m\nm\u00bcmass of \ufb02uid in the wellbore and fracture, kg\nP\u00bc\ufb02uid pressure, MPa\nP0\u00bcreference pressure used to calculate \ufb02uid density,\nMPa\nPinit\u00bcinitial \ufb02uid pressure, MPa\nqinj\u00bcvolumetric surface-injection rate, m3/s\nqm,inj\u00bcsurface mass injection rate, kg/s\nqleakoff \u00bctotal volumetric leakoff rate across entire fracture,\nm3/s\nqm,leakoff \u00bctotal-mass leakoff rate across entire fracture, kg/s\nQ\u00bcvolumetric \ufb02ow rate, m3/s\nRf\u00bcfracture radius, m\nSf\u00bcfracture stiffness, MPa/m\nSf,c\u00bcstiffness of a closed fracture, MPa/m\nSf,cont \u00bccontribution to stiffness of a closed fracture from\ncontact stress, MPa/m\nSf,o\u00bcstiffness of open fracture, or equivalently, contribu-\ntion to stiffness of a closed fracture from the defor-\nmation of the surrounding formation, MPa/m\nSf,PKN\u00bcstiffness of an open PKN fracture, MPa/m\nSf,r\u00bcstiffness of an open radial fracture, MPa/m\nt\u00bctime, seconds\nte\u00bcduration of injection, seconds\nT\u00bctransmissivity, m3\nVfr\u00bcvolume of \ufb02uid in fracture, m3"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae24d42-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 14, "chroma:document": "T\u00bctransmissivity, m3\nVfr\u00bcvolume of \ufb02uid in fracture, m3\nVwl\u00bcvolume of \ufb02uid in wellbore, m3\na\u00bcpower-law exponent for fracture growth used in the\nG-time derivation, unitless\nDP\u00bcchange in pressure since the start of the transient,\nMPa\nDPnw\u00bcwellbore and near-wellbore pressure drop, MPa\nDt\u00bctime from start of transient, seconds\nDtD\u00bcdimensionless time from start of transient, unitless\nDrn\u00bcdeformation-induced change in normal stress, MPa\nl\u00bc\ufb02uid viscosity, MPa /C1s\n/C23\u00bcPoisson\u2019s ratio, unitless\nq\u00bc\ufb02uid density, kg/m3\nq0\u00bc\ufb02uid density at the reference pressure, kg/m3\nqfr\u00bcaverage density of \ufb02uid in fracture, kg/m3\nqinit\u00bcinitial \ufb02uid density, kg/m3\nqinj\u00bcdensity of injected \ufb02uid, kg/m3\nqleakoff \u00bcaverage density of \ufb02uid leaking off from the frac-\nture, kg/m3\nqwl\u00bcaverage density of \ufb02uid in wellbore, kg/m3\nrhmin\u00bcminimum principal stress, MPa\nr0\nn\u00bceffective normal stress, MPa\nrn\u00bcnormal stress, MPa\nrr\nn\u00bcremote-loading normal stress, MPa\nrn,ref\u00bcreference normal stress, MPa\n/init\u00bcinitial porosity, unitless\n/\u00bcporosity, unitless"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae24dce-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 14, "chroma:document": "/init\u00bcinitial porosity, unitless\n/\u00bcporosity, unitless\ns\u00bcsuperposition time, unitless\nAcknowledgments\nThe \ufb01nancial support provided by the Fracturing and Sand Con-\ntrol JIP at the University of Texas at Austin is gratefully acknowl-\nedged. The authors would like to thank Norm Warpinski for\nsupplying data from the GRI/DOE M-site project used in the pa-per and Priscila Ribeiro and Bertrand Rohmer for their useful\ncomments. Thank you to our friends and colleagues who encour-\naged us in pursuing this research.\nReferences\nBabazadeh, M. and McClure, M. 2015. Coupling Fluid Flow and Geome-\nchanics in a Three-Dimensional Discrete Fracture Network Simulator.\nPaper presented at the 49th US Rock Mechanics/Geomechanics Sym-\nposium, San Francisco, 28 June\u20131 July.Barree, R. D. and Mukherjee, H. 1996. Determination of Pressure Depend-\nent Leakoff and Its Effect on Fracture Geometry. Paper presented atthe 71st Annual Technical Conference and Exhibition, Denver, 6\u20139\nOctober. SPE-36424-MS. http://dx.doi.org/10.2118/36424-MS."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae24e50-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 14, "chroma:document": "October. SPE-36424-MS. http://dx.doi.org/10.2118/36424-MS.\nBarree, R. D. 1998. Applications of Pre-Frac Injection/Falloff Tests in Fis-\nsured Reservoirs\u2014Field Examples. Paper presented at the SPE Rocky\nMountain Regional/Low Permeability Reservoirs Symposium, Denver,\n5\u20138 April. SPE-39932-MS. http://dx.doi.org/10.2118/39932-MS.\nBarree, R. D., Barree, V. L., and Craig, D. P. 2007. Holistic Fracture Diag-\nnostics. Paper presented at the SPE Rocky Mountain Oil and Gas\nTechnology Symposium, Denver, 16\u201318 April. SPE-107877-MS.http://dx.doi.org/10.2118/107877-MS.\nBarree, R. D., Miskimins, J. L., and Gilbert, J. V. 2014. Diagnostic Frac-\nture Injection Tests: Common Mistakes, Mis\ufb01res, and Misdiagnoses.\nSPE Prod & Oper 30(2): 84\u201398. SPE-169539-PA. http://dx.doi.org/\n10.2118/169539-PA.\nBarton, N., Bandis, S., and Bakhtar, K. 1985. Strength, Deformation and\nConductivity Coupling of Rock Joints. International Journal of Rock\nMechanics and Mining Sciences & Geomechanics Abstracts 22(3):"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae24ed2-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 14, "chroma:document": "Mechanics and Mining Sciences & Geomechanics Abstracts 22(3):\n121\u2013140. http://dx.doi.org/10.1016/0148-9062(85)93227-9.\nBourdet, D., Ayoub, J. A., and Pirard, Y. M. 1989. Use of Pressure Deriva-\ntive in Well-Test Interpretation. SPE Form Eval 4(2): 293\u2013302. SPE-\n12777-PA. http://dx.doi.org/10.2118/12777-PA.\nBranagan, P. T., Warpinski, N. R., Engler, B. et al. 1996. Measuring the\nHydraulic Fracture-Induced Deformation of Reservoir and Adjacent\nRocks Employing a Deeply Buried Inclinometer Array: Gri/Doe\nMulti-Site Project. Paper presented at the SPE Annual Technical Con-ference and Exhibition, Denver, 6\u20139 October. SPE-36451-MS. http://\ndx.doi.org/10.2118/36451-MS.\nBrown, S. R. and Bruhn, R. L. 1998. Fluid Permeability of Deformable\nFracture Networks. J. Geophysical Research 103(B2): 2489\u20132500.\nhttp://dx.doi.org/10.1029/97JB03113.\nCastillo, J. L. 1987. Modi\ufb01ed Fracture Pressure Decline Analysis Includ-\ning Pressure-Dependent Leakoff. Paper presented at the SPE/DOE"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae24f54-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 14, "chroma:document": "ing Pressure-Dependent Leakoff. Paper presented at the SPE/DOE\nLow Permeability Reservoir Symposium, Denver, 18\u201319 May. SPE-\n16417-MS. http://dx.doi.org/10.2118/16417-MS.\nCraig, D. P. and Blasingame, T. A. 2006. Application of a New Fracture-\nInjection/Falloff Model Accounting for Propagating, Dilated, andClosing Hydraulic Fractures. Paper presented at the SPE Gas Technol-\nogy Symposium, Calgary, 15\u201317 May. SPE-100578-MS. http://\ndx.doi.org/10.2118/100578-MS.\nCraig, D. P. 2014. New Type Curve Analysis Removes Limitations of Con-\nventional After-Closure Analysis of D\ufb01t Data. Paper presented at the SPEUnconventional Resources Conference-USA, The Woodlands, Texas,\n1\u20133 April. SPE-168988-MS. http://dx.doi.org/10.2118/168988-MS.\nCramer, D. D. 1987. The Application of Limited-Entry Techniques in\nMassive Hydraulic Fracturing Treatments. Paper presented at the SPE\nProduction Operations Symposium, Oklahoma City, Oklahoma, 8\u201310\nMarch. SPE-16189-MS. http://dx.doi.org/10.2118/16189-MS."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae24fe0-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 14, "chroma:document": "March. SPE-16189-MS. http://dx.doi.org/10.2118/16189-MS.\nCramer, D. D. and Nguyen, D. H. 2013. Diagnostic Fracture Injection\nTesting Tactics in Unconventional Reservoirs. Paper presented at the\nSPE Hydraulic Fracturing Technology Conference, The Woodlands,Texas, 4\u20136 February. SPE-163863-MS. http://dx.doi.org/10.2118/\n163863-MS.\nCrouch, S. L. and Star\ufb01eld, A. M. 1983. Boundary Element Methods in\nSolid Mechanics: With Applications in Rock Mechanics and Geologi-\ncal Engineering . London: Allen & Unwin.\nEconomides, M. J. and Nolte, K. G. (eds.) 2000. Reservoir Stimulation ,\nthird edition. New York, New York: John Wiley.\nGangi, A. F. 1978. Variation of Whole and Fractured Porous Rock Perme-\nability With Con\ufb01ning Pressure. International J. Rock Mechanics and\nMining Sciences 15: 249\u2013257. http://dx.doi.org/10.1016/0148-\n9062(78)90957-9.\nGodbey, J. K. and Hodges, H. D. 1958. Pressure Measurements During\nFormation Fracturing Operations. In Petroleum Transactions , AIME,"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2506c-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 14, "chroma:document": "Vol. 213, 65\u201369. SPE-889-G. Richardson, Texas: Society of Petroleum\nEngineers.\nHaimson, B. and Fairhurst, C. 1967. Initiation and Extension of Hydraulic\nFractures in Rocks. SPE J .7(3): 310\u2013318. SPE-1710-PA. http://\ndx.doi.org/10.2118/1710-PA.J179725 DOI: 10.2118/179725-PA Date: 2-August-16 Stage: Page: 1335 Total Pages: 19\nID:jaganm Time: 18:39 I Path: S:/J###/Vol00000/160007/Comp/APPFile/SA-J###160007August 2016 SPE Journal 1335"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae250ee-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 15, "chroma:document": "Horne, R. N. 1995. Modern Well Test Analysis: A Computer-Aided\nApproach , second edition. Palo Alto, California: Petroway, Inc.\nHoward, G. and Fast, C. R. 1957. Optimum Fluid Characteristics for Frac-\nture Extension. Drilling and Production Practice 24:261\u2013270. API-\n57-261. New York: American Petroleum Institute.\nHubbert, M. K. and Willis, D. G. 1957. Mechanics of Hydraulic Fracturing.\nInPetroleum Transactions , AIME, Vol. 210, 153\u2013168. SPE-686-G.\nRichardson, Texas: Society of Petroleum Engineers.\nJ a e g e r ,J .C . ,C o o k ,N .G .W . ,a n dZ i m m e r m a n ,R . .2 0 0 7 . Fundamentals of Rock\nMechanics , fourth edition. Malden, Massachusetts: Blackwell Publications.\nJeffrey, R. G. 1989. The Combined Effect of Fluid Lag and Fracture\nToughness on Hydraulic Fracture Propagation. Paper presented at the\nLow Permeability Reservoirs Symposium, Denver, 6\u20138 March. SPE-\n18957-MS. http://dx.doi.org/10.2118/18957-MS.\nKoning, E. J. L. and Niko, H. 1985. Fractured Water-Injection Wells: A"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae25170-857d-11ee-bb1c-0242ac1c000c
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metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 15, "chroma:document": "Pressure Falloff Test for Determining Fracture Dimensions. Paper pre-sented at the SPE Annual Technical Conference and Exhibition, Las\nVegas, Nevada, 22\u201325 September. SPE-14458-MS. http://dx.doi.org/\n10.2118/14458-MS.\nMack, M. G. and Warpinski, N. R. 2000. Mechanics of Hydraulic Fractur-\ning. In Reservoir Stimulation , ed. M. J. Economides and K. G. Nolte,\nChap. 6.1\u20136.48. New York, New York: John Wiley & Sons.\nMarongiu-Porcu, M., Ehlig-Economides, C. A., and Economides, M. J.\n2011. Global Model for Fracture Falloff Analysis. Paper presented at\nthe SPE North American Unconventional Gas Conference and Exhibi-tion, The Woodlands, Texas, 14\u201316 June. SPE-144028-MS. http://\ndx.doi.org/10.2118/144028-MS.\nMarongiu-Porcu, M., Retnanto, A., Economides, M. J. et al. 2014. Compre-\nhensive Fracture Calibration Test Design. Paper presented at the SPE\nHydraulic Fracturing Technology Conference, The Woodlands, Texas,\n4\u20136 February. SPE-168634-MS. http://dx.doi.org/10.2118/168634-MS."}


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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 15, "chroma:document": "McClure, M. W. and Horne, R. N. 2013. Discrete Fracture Network Mod-\neling of Hydraulic Stimulation: Coupling Flow and Geomechanics .\nSpringerBriefs in EarthSciences, Springer.\nMcClure, M. W. 2014. The Potential Effect of Network Complexity on\nRecovery of Injected Fluid Following Hydraulic Fracturing. Paper pre-\nsented at the SPE Unconventional Resources Conference-USA, The\nWoodlands, Texas, 1\u20133 April. SPE-168991-MS. http://dx.doi.org/10.2118/168991-MS.\nMcClure, M. W., Christopher, A. J., Blyton, H. J. et al. 2014. The Effect of\nChanging Fracture Compliance on Pressure Transient Behavior During\nDiagnostic Fracture Injection Tests. Paper presented at the SPE Annual\nTechnical Conference and Exhibition, Amsterdam, 27\u201329 October.\nSPE-170956-MS. http://dx.doi.org/10.2118/170956-MS.\nMcClure, M. W., Babazadeh, M., Shiozawa, S. et al. 2015. Fully Coupled\nHydromechanical Simulation of Hydraulic Fracturing in Three-Dimen-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae252ba-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 15, "chroma:document": "sional Discrete Fracture Networks. Paper presented at the SPE Hydrau-lic Fracturing Technology Conference, The Woodlands, Texas, 3\u20135\nFebruary. SPE-173354-MS. http://dx.doi.org/10.2118/173354-MS.\nMeng, C., Wei, L., and Yuan, R. 2014. Coupled Fluid Flow and Geome-\nchanics Modeling for DFIT Analysis in Unconventional Gas Develop-\nment. Paper presented at the International Petroleum Technology\nConference, Kuala Lumpur, 10\u201312 December. IPTC-18159-MS.\nhttp://dx.doi.org/10.2523/IPTC-18159-MS.\nMohamed, I. M., Nasralla, R. A., Sayed, M. A. et al. 2011. Evaluation of\nafter-Closure Analysis Techniques for Tight and Shale Gas Forma-tions. Paper presented at the SPE Hydraulic Fracturing Technology\nConference, The Woodlands, Texas, 24\u201326 January. SPE-140136-MS.\nhttp://dx.doi.org/10.2118/140136-MS.\nMoody, L. F. 1944. Friction Factors for Pipe Flow. Trans. of the ASME 66\n(8): 671\u2013684.\nNolte, K. 1979. Determination of Fracture Parameters From Fracturing\nPressure Decline. Paper presented at the SPE Annual Technical Con-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2533c-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 15, "chroma:document": "ference and Exhibition, Las Vegas, Nevada, 23\u201326 September. SPE-\n8341-MS. http://dx.doi.org/10.2118/8341-MS.\nNolte, K. G. and Smith, M. B. 1981. Interpretation of Fracturing Pressures.\nJ Pet Technol 33(9): 1767\u20131775. SPE-8297-PA. http://dx.doi.org/\n10.2118/8297-PA.\nNolte, K. G. 1991. Fracturing-Pressure Analysis for Nonideal Behavior. J\nPet Technol 43(2): 210\u2013218. SPE-20704-PA. http://dx.doi.org/\n10.2118/20704-PA.Nolte, K. G., Mack, M. G., and Lie, W. L. 1993. A Systematic Method for\nApplying Fracturing Pressure Decline: Part I. Paper presented at theSPE Rocky Mountain Regional/Low Permeability Reservoirs Sympo-\nsium, Denver, 26\u201328 April. SPE-25845-MS. http://dx.doi.org/10.2118/\n25845-MS.\nNordgren, R. P. 1972. Propagation of a Vertical Hydraulic Fracture. SPE\nJ.12(4): 306\u2013314. SPE-3009-PA. http://dx.doi.org/10.2118/3009-PA.\nOkada, Y. 1992. Internal Deformation Due to Shear and Tensile Faults in\na Half-Space. Bull. of the Seismological Society of America 82(2):\n1018\u20131040."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae253be-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 15, "chroma:document": "1018\u20131040.\nPerkins, T. K. and Kern, L. R. 1961. Widths of Hydraulic Fractures. J Pet\nTechnol 13(9): 937\u2013949. SPE-89-PA. http://dx.doi.org/10.2118/89-\nPA.\nPerkins, T. K. and Gonzalez, J. A. 1985. The Effect of Thermoelastic\nStresses on Injection Well Fracturing. SPE J. 25(1): 78\u201388. SPE-\n11332-PA. http://dx.doi.org/10.2118/11332-PA.\nPlahn, S. V., Nolte, K. G., Thompson, L. G. et al. 1997. A Quantitative\nInvestigation of the Fracture Pump-in/Flowback Test. SPE Prod & Fac\n12(1): 20\u201327. SPE-30504-PA. http://dx.doi.org/10.2118/30504-PA.\nPyrak-Nolte, L. J. and Morris, J. P. 2000. Single Fractures Under Normal\nStress: The Relation Between Fracture Speci\ufb01c Stiffness and Fluid\nFlow. International J. Rock Mechanics and Mining Sciences 37:\n245\u2013262. http://dx.doi.org/10.1016/S1365-1609(99)00104-5.\nRaaen, A. M., Skomedal, E., Kjorholt, H. et al. 2001. Stress Determination\nFrom Hydrualic Fracturing Tests: The System Stiffness Approach.International J. for Rock Mechanics and Mining Sciences 38(4):"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2544a-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 15, "chroma:document": "529\u2013541. http://dx.doi.org/10.1016/S1365-1609(01)00020-X.\nRibeiro, P. and Horne, R. 2013. Pressure and Temperature Transient Anal-\nysis: Hydraulic Fractured Well Application. Paper presented at the\nSPE Annual Technical Conference and Exhibition, New Orleans, 30\nSeptember\u20132 October. SPE-166222-MS. http://dx.doi.org/10.2118/\n166222-MS.\nShlyapobersky, J. W., Walhaug, W. W. Shef\ufb01eld, R. E. et al. 1988. Field\nDetermination of Fracturing Parameters for Overpressure CalibratedDesign of Hydraulic Fracturing. Paper presented at the SPE Annual\nTechnical Conference and Exhibition, Houston, 2\u20135 October. SPE-\n18195-MS. http://dx.doi.org/10.2118/18195-MS.\nSneddon, I. N. 1946. The Distribution of Stress in the Neighborhood of a\nCrack in an Elastic Solid. Proc., Royal Society of London Series A 187\n(1009): 229\u2013260. http://dx.doi.org/10.1098/rspa.1946.0077.\nSoliman, M. Y. and Gamadi, T. 2012. Testing Tight Gas and Unconven-\ntional Formations and Determination of Closure Pressure. Paper pre-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae254cc-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 15, "chroma:document": "sented at the SPE/EAGE European Unconventional Resources\nConference and Exhibition, Vienna, Austria, 20\u201322 March. SPE-\n150948-MS. http://dx.doi.org/10.2118/150948-MS.\nSoliman, M. Y. and Kabir, C. S. 2012. Testing Unconventional Forma-\ntions. J. Petroleum Science and Technology 92\u201393: 102\u2013109. http://\ndx.doi.org/10.1016/j.petrol.2012.04.027.\nValko, P. P. and Economides, M. J. 1999. Fluid-Leakoff Delineation in\nHigh-Permeability Fracturing. SPE Prod & Fac 14(2): 117\u2013130. SPE-\n56135-PA. http://dx.doi.org/10.2118/56135-PA.\nvan den Hoek, P. J. 2002. Pressure Transient Analysis in Fractured Pro-\nduced Water Injection Wells. Paper presented at the SPE Asia Paci\ufb01cOil and Gas Conference and Exhibition, Melbourne, Australia, 8\u201310\nOctober. SPE-77946-MS. http://dx.doi.org/10.2118/77946-MS.\nVinsome, P. K. and Westerveld, J. 1980. A Simple Method for Predicting\nCap and Base Rock Heat Loses in Thermal Reservoir Simulators. J\nCan Pet Technol 19(3): 87\u201390. SPE-80-03-04-PA. http://dx.doi.org/\n10.2118/80-03-04."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2554e-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 15, "chroma:document": "10.2118/80-03-04.\nWallace, J., Kabir, C. S., and Cipolla, C. L. 2014. Multiphysics Investiga-\ntion of Diagnostic Fracture Injection Tests in Unconventional Reser-\nvoirs. Paper presented at the SPE Hydraulic Fracturing TechnologyConference, The Woodlands, Texas, 4\u20136 February. SPE-168620-MS.\nhttp://dx.doi.org/10.2118/168620-MS.\nWalsh, J. B. 1981. Effect of Pore Pressure and Con\ufb01ning Pressure on Frac-\nture Permeability. International J. of Rock Mechanics and Mining Sci-\nences 18:429\u2013435. http://dx.doi.org/10.1016/0148-9062(81)90006-1.\nWillis-Richards, J., Watanabe, K., and Takahashi, H. 1996. Progress To-\nward a Stochastic Rock Mechanics Model of Engineered Geothermal\nSystems. J. Geophysical Research 101 (B8): 17481\u201317496. http://\ndx.doi.org/10.1029/96JB00882.J179725 DOI: 10.2118/179725-PA Date: 2-August-16 Stage: Page: 1336 Total Pages: 19\nID:jaganm Time: 18:40 I Path: S:/J###/Vol00000/160007/Comp/APPFile/SA-J###1600071336 August 2016 SPE Journal"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae255d0-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 16, "chroma:document": "Witherspoon, P. A., Wang, J. S. Y., Iwa i, K. et al. 1980. Validity of Cubic Law\nfor Fluid Flow in a Deformable Rock Fracture. Water Resources Research\n16(6): 1016\u20131024. http://dx.doi.org/10.1029/WR016i006p01016.\nYew, C. H. and Liu, G. H. 1993. The Fracture Tip and Critical Stress In-\ntensity Factor of a Hydraulically Induced Fracture. SPE Prod & Fac 8\n(3): 171\u2013177. SPE-22875-PA. http://dx.doi.org/10.2118/22875-PA.\nZoback, M. D. 2007. Reservoir Geomechanics . Cambridge: Cambridge\nUniversity Press.\nAppendix A: Derivation of the Equations\nGoverning Pressure-Transient BehaviorDuring a DFIT\nHere, we derive the equations governing pressure-transient behav-\nior during a DFIT.\nWe describe the entire fracture and wellbore system with a\n\u201clumped parameter model,\u201d in which we assume thatdP\ndtis uni-\nform throughout the entire system. This approach is isothermal\nand implicitly neglects pressure gradient caused by \ufb02ow in thewell, perforation, near-wellbore region, and along the fracture."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae25652-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 16, "chroma:document": "Considering these effects would not fundamentally change the\nconclusions drawn from the analysis. Related approaches are used\nby Raaen et al. (2001) and Craig and Blasingame (2006), though\nthe derivation we provide here has important differences.\nThe pressure derivative with time can be decomposed with the\nchain rule as\ndP\ndt\u00bcdP\ndmdm\ndt; \u00f0A-1\u00de\nwhere mis the total mass of \ufb02uid in the wellbore and in the frac-\nture and Pis the pressure that is either measured at the bottom of\nthe well or calculated from surface-pressure measurements.dm\ndtis\nequal to the surface mass-injection rate, qm,inj, minus the rate at\nwhich \ufb02uid mass \ufb02ows out of the well and fracture into the ma-\ntrix,qm,leakoff . We can expand these terms by saying that\ndm\ndt\u00bcqm;inj/C0qm;leakoff \u00bcqinjqinj/C0qleakoff qleakoff ; \u00f0A-2\u00de\nwhere qinjis the volumetric surface injection rate, qinjis the den-\nsity of the \ufb02uid injected, qleakoff is the volumetric rate of leakoff\ninto the matrix, and qleakoff is the average density of the \ufb02uid leak-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae256ca-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 16, "chroma:document": "ing off into the matrix.dP\ndmcan be expanded by starting with\nthe de\ufb01nition\nm\u00bcqwlVwl\u00feqfrVfr; \u00f0A-3\u00de\nwhere qwlis the average density of the \ufb02uid in the wellbore, Vwlis\nthe volume of the wellbore, qfris the average density of the \ufb02uid\nin the fracture, and Vfris the volume of the fracture.\nTo simplify further, we assume that the \ufb02uid is slightly com-\npressible and has constant compressibility, cf. In this case, we\ncan write\nq\u00bcq0exp\u00bd\u00f0P/C0P0\u00decf/C138; \u00f0A-4\u00de\nwhere P0is a reference pressure and q0is the density of the \ufb02uid\nat that pressure. Taking the Taylor-series expansion and discard-\ning higher-order terms provides the approximation\nq\u00bcq0\u00bd1\u00fecf\u00f0P/C0P0\u00de/C138: \u00f0A-5\u00de\nWe can take the derivative of this linearized equation with\nrespect to pressure and determine that\ndq\ndP\u00bcq0cf: \u00f0A-6\u00de\nNow, we take the derivative of Eq. A-3 with respect to\npressure:dm\ndP\u00bcdqwl\ndPVwl\u00feqwldVwl\ndP\u00fedqfr\ndPVfr\u00feqfrdVfr\ndP: \u00f0A-7\u00de\nCombining Eqs. A-1, A-2, and A-7, we arrive at\ndP\ndt\u00bcqinjqinj/C0qleakoff qleakoff\ndqwl\ndPVwl\u00feqwldVwl\ndP\u00fedqfr"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae25c38-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 16, "chroma:document": "dP\ndt\u00bcqinjqinj/C0qleakoff qleakoff\ndqwl\ndPVwl\u00feqwldVwl\ndP\u00fedqfr\ndPVfr\u00feqfrdVfr\ndP: \u00f0A-8\u00de\nEq. A-8 can be simpli\ufb01ed by relying on the assumption that\nthe \ufb02uid is slightly compressible. If we assume that qinj,qleakoff ,\nqwl, and qfrare all approximately equal (for isothermal fresh\nwater, they would be within a few percentages), and these quanti-\nties are also equal to the q0value used to calculate the derivatives\nof \ufb02uid density with respect to pressure in the wellbore and the\nfracture, then we can cancel them out of the numerator and the de-nominator of Eq. A-8 and arrive at\ndP\ndt\u00bcqinj/C0qleakoff\ncfVwl\u00fedVwl\ndP\u00fecfVfr\u00fedVfr\ndP\u00bcqinj/C0qleakoff\nCt;\n/C1/C1/C1/C1/C1/C1/C1/C1/C1/C1/C1/C1/C1/C1/C1/C1/C1/C1/C1 \u00f0 A-9\u00de\nwhere Ctis the total storage coef\ufb01cient of the entire system. The\nde\ufb01nition of total-storage coef\ufb01cient used in Eq. A-9 is more gen-\neral than the expressions used by Raaen et al. (2001) and by Craig\nand Blasingame (2006).\nCraig and Blasingame (2006) de\ufb01ne the after-closure storage"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae25ce2-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 16, "chroma:document": "Craig and Blasingame (2006) de\ufb01ne the after-closure storage\ncoef\ufb01cient as (with slightly different variable de\ufb01nitions)\nCac\u00bccfVwl\u00fecfVfr; \u00f0A-10 \u00de\nwhere Vfris assumed constant. Comparing this expression with\nEq. A-9, we can see that Craig and Blasingame (2006) implicitly\nassume that qwldVwl\ndPandqfrdVfr\ndPare zero. The former term, the\nchanging wellbore volume, is routinely assumed negligible. How-\never, the latter term, for the changing fracture volume after clo-\nsure, is large, as shown next. Craig and Blasingame (2006) permitthat the \ufb02uid in the wellbore and fracture may have different com-\npressibility, whereas we assume that their compressibility is the\nsame. The distinction should have a negligible effect in a DFIT,where clear water is injected.\nRaaen et al. (2001) use an approach similar to that shown in\nEq. A-9. However, they assume that \u201cwhen the fracture is closed,\nthere are three main contributions to the system stiffness: from"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae25d6e-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 16, "chroma:document": "the compliance of the wellbore \ufb02uid, from the deformation of thecasing and top-side tubes, and from the deformation of the open-\nhole below the casing shoe.\u201d Therefore, they implicitly assume\nthat the\ndqfr\ndPVfrandqfrdVfr\ndPterms in Eq. A-9 are zero.\nWith the assumptions we describe, Eq. A-9 applies during all\nphases of a fracturing test: during injection, the before-closureshut-in period, during closure, and the after-closure shut-in period.\nIn reality, during injection, pressure gradient caused by \ufb02ow\nthrough the well, perforations, near-wellbore region, and fracture\ncausedP\ndtto vary spatially throughout the system, but after the\nearly shut-in period, these pressure gradients should be minimal.\nThe total storage coef\ufb01cient has four parts: compressibility of\n\ufb02uid in the wellbore, changes in the wellbore volume, compressi-\nbility of the \ufb02uid in the fracture, and changes in fracture volume.Conventionally, changes in wellbore volume are assumed negligi-\nble relative to c\nfVwland thedVwl"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae25df0-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 16, "chroma:document": "ble relative to c\nfVwland thedVwl\ndPterm is neglected (Horne 1995).\ndVfr\ndPcan be written asA\nSf, where Ais the surface area of the\nfracture and Sfis the fracture stiffness, the derivative of effective\nnormal stress with respect to aperture (Jaeger et al. 2007).dVfr\ndP\ndepends on fracture geometry and mechanical characteristics of\nthe formation. In practice, the fracture geometry is usually not...........................\n....\n......................\n.....................\n.....................\n.................................\n.....\n. ......................J179725 DOI: 10.2118/179725-PA Date: 2-August-16 Stage: Page: 1337 Total Pages: 19\nID:jaganm Time: 18:40 I Path: S:/J###/Vol00000/160007/Comp/APPFile/SA-J###160007August 2016 SPE Journal 1337"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae25e72-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 17, "chroma:document": "well-constrained. Two limiting cases for fracture geometry are\ncircular and PKN (Perkins and Kern 1961; Nordgren 1972). If\nthere is no fracture-height restraint caused by layering, then thefracture will be approximately circular (though not completely cir-\ncular because the trends in stress with depth will cause it to prefer-\nentially propagate upward). If the fracture is con\ufb01ned between\nhigher stress intervals, then the fracture length will be much\ngreater than its height, and stiffness can be approximated from a2D plane-strain solution, the key assumption of the PKN model.\nFor a radial fracture, the stiffness of the fracture (when open),\nS\nf,o, can be written as (Sneddon 1946):\nSf;r\u00bc3pGm\n8\u00f01/C0/C23\u00deRf; \u00f0A-11 \u00de\nwhere Gmis the shear modulus, /C23is Poisson\u2019s ratio, and Rfis the\nradius. For a fracture with length much longer than its height, the\nstiffness can be estimated as (Sneddon 1946; Perkins and Kern\n1961; Nordgren 1972):\nSf;PKN\u00bc4Gm\np\u00f01/C0/C23\u00dehf: \u00f0A-12 \u00de\nThe magnitudes of theA"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae25ef4-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 17, "chroma:document": "Sf;PKN\u00bc4Gm\np\u00f01/C0/C23\u00dehf: \u00f0A-12 \u00de\nThe magnitudes of theA\nSf,cfVwl, and cfVfrterms can be\nroughly assessed (for example, in the fracture simulation shown\nin the Effect of Increasing Residual Aperture subsection). The\n\ufb01nal fracture length is approximately 250 m, with a height of 7 m,\nfor a surface area of 1750 m2. Fig. 6 shows that the average frac-\nture aperture is roughly 1 mm, which means that Vfris roughly\n1.75 m3andcfVfris roughly (with the value of water compressi-\nbility given in Table 1) 0.0008 m3/MPa. Vwldepends on wellbore\nlength and diameter. A typical value is approximately 100 m3, and\nsocfVfrcan be estimated as 0.0458 m3/MPa. The fracture shown\nin Fig. 6 has length much greater than height, 7 m. Therefore, its\npreclosure stiffness can be assessed from Eq. A-12 and is equal to\n3.638 MPa/mm. TheA\nSfterm is equal to 0.48 m3/MPa. This analy-\nsis indicates that the dominant effect in the total-storage coef\ufb01-\ncient is likely to be the fracture stiffness, the wellbore storage has"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae25f80-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 17, "chroma:document": "minor but nonnegligible effect, and the compressibility of the\nwater in the fracture is negligible.\nAfter shut-in, the pressure derivative can be written as (assum-\ning constant wellbore volume and that the cfVfrterm is negligible)\ndP\ndt\u00bc/C0qleakoff\ncfVwl\u00feA\nSf: \u00f0A-13 \u00de\nIf the fracture continues propagating after closure, Amay\nchange signi\ufb01cantly and Sfmay change (depending on the geome-\ntry). If the fracture is no longer propagating, Sfwill be constant\nbefore fracture closure, but will change when the fracture closes.\nUsing the Carter model to calculate leakoff rate yields a value\nofqleakoff that scales with the inverse of the square root of time\n(Howard and Fast 1957). This is why plots of pressure vs. thesquare root of time during shut-in yield a straight line. If leakoff\nrate scales with time according to the G-function introduced by\nNolte (1979), then a plot of pressure vs. G-time will yield a\nstraight line. The straight line will only occur if the denominator"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae26002-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 17, "chroma:document": "in Eq. A-13 is constant, which is why closure causes a deviationfrom the straight line. Closure also decreases the fracture trans-\nmissivity. If this causes the fracture to no longer be effectively in-\n\ufb01nite conductivity, then, pressure gradients can develop along the\nfracture, and this can cause the lumped parameter approach used\nin this section to no longer be applicable.\nTo understand the pressure trend after closure, it is necessary\nto investigate how stiffness is affected by closure. The stiffness\nafter closure, S\nf,c, includes the effect of the elastic response of the\nformation (as shown in Eqs. A-11 and A-12), but also includes a\ncontribution from the contacting fracture walls (Section 12.3 fromJaeger et al. 2007).As a simplifying assumption, we assume that both the stress\nand aperture are constant along the fracture. This is not strictly\ncorrect, but is suf\ufb01ciently accurate for providing a conceptual ex-planation. Under this simplifying assumption, we can write\nr\n0\nn\u00bcrr\nn\u00feESf;o/C0P: \u00f0A-14 \u00de"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae260ca-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 17, "chroma:document": "r\n0\nn\u00bcrr\nn\u00feESf;o/C0P: \u00f0A-14 \u00de\nBecause the initial aperture of a hydraulic fracture is zero\n(because it does not initially exist), backstress is created by its\nentire aperture, E.\nTo \ufb01nd the fracture stiffness, Eq. A-14 can be solved for Pand\ndifferentiated with respect to E:\nSf;c\u00bcdE\ndP/C18/C19/C01\n\u00bcdP\ndE\u00bcSf;o/C0dr0\ndE\u00bcSf;o/C0dE\ndr0n/C18/C19/C01\n\u00bcSf;o\u00fedE\ndP/C18/C19/C01\n\u00bcSf;o\u00feSf;cont;/C1/C1/C1/C1/C1/C1/C1/C1/C1 \u00f0 A-15 \u00de\nwhere Sf,contis the contribution to the stiffness from the contact\nstress. Sf,contcan be calculated from any constitutive equation that\nspeci\ufb01es Eas a function of effective normal stress. For example,\nEq. 12 yields\nSf;cont\u00bc/C0dE\ndr0\nn/C18/C19/C01\n\u00bc9E0\nrn;ref1\u00fe9r0\nn\nrn;ref/C18/C19/C02\"#/C01\n\u00bcrn;ref\n9E0\nE2: \u00f0A-16 \u00de\nThus, with Eq. 12, the stiffness of a closed fracture is\nSf;c\u00bcSf;o\u00fern;ref\n9E0\nE2: \u00f0A-17 \u00de\nThe value of Sf,contincreases as the fracture closes. At closure\nSclosure\nf;cont\u00bcrn;ref\n9E0: \u00f0A-18 \u00de\nReferring to the simulation described in the Effect of Increas-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae26156-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 17, "chroma:document": "Referring to the simulation described in the Effect of Increas-\ning Residual Aperture subsection, we can calculate Sclosure\nf;cont to be\n11.333 MPa/mm (for a total Sfof 14.971 MPa/mm), signi\ufb01cantly\ngreater than the value of Sf,oof 3.638 MPa/mm. Closure causes a\nsharp increase in Sffrom Sf,otoSf,c, which (from Eq. A-13) causes\nan increase in the pressure derivative. The derivative increasesvery sharply because stiffness increases so much with closure,\nincreasing by roughly a factor of 4, and continuing to increase fur-\nther as aperture drops further below E\n0. In the Adding Wellbore\nStorage subsection, a simulation is shown with a smaller value of\nrn;ref, 5 MPa instead of 51 MPa, and so Sclosure\nf;contis 1.111 MPa/mm\n(for a total Sfof 4.749 MPa/mm), causing only a modest increase\ncompared to the stiffness before closure. As a result, the deriva-\ntive increases less abruptly.\nMark McClure is the founder and chief executive officer of"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae261e2-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 17, "chroma:document": "Mark McClure is the founder and chief executive officer of\nMcClure Geomechanics and is a consulting professor at Stan-ford University. He was an assistant professor in the Departmentof Petroleum and Geosystems Engineering at the University of\nTexas at Austin from 2013 to 2015. McClure\u2019s research focuses\non modeling and characterization of hydraulic fracturing,induced seismicity, and enhanced geothermal systems. Hewas honored with the 2015 SPE Regional Completions Optimi-\nzation and Technology Award from the Southwest North Amer-\nican Region, Outstanding Service as a Technical Editor for SPE\nReservoir Evaluation & Engineering in 2014, the Hank Ramey\nAward for Outstanding Research and Service to the Depart-\nment from the Department of Energy Resources Engineering at\nStanford University in 2012, and Best Paper in Geophysics from\nthe Society of Exploration Geophysicists (SEG) in 2011. McClureholds a BS degree in chemical engineering, an MS degree in\npetroleum engineering, and a PhD degree in energy-resources"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2626e-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 17, "chroma:document": "petroleum engineering, and a PhD degree in energy-resources\nengineering, all from Stanford University.\nHojung Jung is a PhD degree student at the University of Texas\nat Austin. Her research interests include experiments and.......................\n.....................\n.............................................\n........................\n.....................\n.........................J179725 DOI: 10.2118/179725-PA Date: 2-August-16 Stage: Page: 1338 Total Pages: 19\nID:jaganm Time: 18:40 I Path: S:/J###/Vol00000/160007/Comp/APPFile/SA-J###1600071338 August 2016 SPE Journal"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae262f0-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 18, "chroma:document": "simulations of geomechanical problems in the context of hy-\ndraulic fracturing and carbon dioxide storage. Jung holds an\nMS degree in petroleum engineering from the University ofTexas at Austin and a BS degree (cum laude) in energy-resour-ces engineering from Seoul National University, South Korea.\nShe is a member of SPE.\nDave Cramer is a senior engineering fellow on the ConocoPhil-\nlips Global Completions Engineering staff in Houston and spe-\ncializes in hydraulic-fracturing applications. He has published\n48 technical papers and holds two US patents. Cramer is aregistered professional engineer in Colorado. Industry recogni-tions include the Henry Mattson Technical Achievement\nAward by the SPE Denver Chapter in 1993 and the SPE Interna-\ntional Completions Optimization and Technologies Award in2011. Cramer was an SPE Distinguished Lecturer during2003\u20132004 and the SPE Region Director for the US and Canada\nRocky Mountain Region from 2004 to 2007.Mukul M. Sharma is a professor and holds the \u201cTex\u201d Moncrief"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2637c-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 18, "chroma:document": "Chair in the Department of Petroleum and Geosystems Engineer-\ning at the University of Texas at Austin, where he has been for thepast 30 years. He served as chair of the Department from 2001 to2005. Sharma\u2019s current research interests include improved oil re-\ncovery, injection-water management, hydraulic fracturing, for-\nmation damage, and petrophysics. He has published more than350 journal articles and conference proceedings and holds 15patents. Sharma has a bachelor of technology degree in chemi-\ncal engineering from the Indian Institute of Technology and MS\nand PhD degrees in chemical and petroleum engineering,respectively, from the University of Southern California. Amonghis many awards, he is the recipient of the 2009 Anthony Lucas\nGold Medal, the 2004 SPE Faculty Distinguished Achievement\nAward, the 2002 SPE Lester C. Uren Award, and the 1998 SPE For-mation Evaluation Award. Sharma served as an SPE DistinguishedLecturer in 2002, has served on the editorial boards of many jour-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae26408-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-PA-Fracture Compliance Method.pdf", "page": 18, "chroma:document": "nals, and has taught and consulted for industry worldwide.J179725 DOI: 10.2118/179725-PA Date: 2-August-16 Stage: Page: 1339 Total Pages: 19\nID:jaganm Time: 18:40 I Path: S:/J###/Vol00000/160007/Comp/APPFile/SA-J###160007August 2016 SPE Journal 1339"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2648a-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 0, "chroma:document": "The Spurious Deflection on Log-Log\nSuperposition-Time Derivative Plots\nof Diagnostic Fracture-Injection Tests\nMark McClure, McClure Geomechanics\nSummary\nLog-log superposition-time derivative plots are used to identify \ufb02ow regimes in well tests with variable rate. The use of superposition\ntime adjusts for the effect of the prior rate history, and (under some conditions) shows what the transient would have looked like if the\ntest had been performed at a constant rate. In this report, I show that if these plots are used to interpret shut-in transients from diagnostic\nfracture-injection tests (DFITs), the superposition-time derivative has an upward de\ufb02ection that does not represent actual reservoir or\ntransient behavior. I review mathematical properties of the superposition-time derivative. I derive equations for the pressure transient ina simpli\ufb01ed model DFIT in which closure does not occur. I show that the onset of late-time impulse \ufb02ow is controlled by injection vol-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae26516-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 0, "chroma:document": "ume and formation, wellbore, and fracture properties, not the duration of injection (as implied by the de\ufb01nition of superposition time).\nLog-log superposition-time derivative plots of DFITs exhibit a slope of 3/2 at intermediate time. However, pressure change never scales\nwith a 3/2 power of time. One form of the G-function super\ufb01cially resembles a superposition-time function constructed by summing\nconstant-rate solutions with 3/2 power scaling. However, this is not a mathematically or physically valid interpretation. The 3/2 powerarises from a spatial integration of the Carter leakoff solution. There is not a mathematical, physical, or practical justi\ufb01cation for plot-\nting DFIT pressure-time data in a way that creates a 3/2 slope. I conclude by providing a \ufb01eld example and practical recommendations\nfor DFIT interpretation.\nIntroduction\nThe log-log derivative plot is widely used in well-test analysis. The slope of the derivative curve, d p/d[ln( t)], is used to diagnose"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae26598-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 0, "chroma:document": "\ufb02ow regimes\u2013it is more precise to call d p/d[ln( t)] the semilog derivative, but it is common convention in well-test analysis to call\nit simply the \u201cderivative curve.\u201d In tests that involve varying rate, such as buildup tests and falloff tests, the derivative is oftencalculated with respect to an effective time function, sometimes called superposition time. In in\ufb01nite-acting radial \ufb02ow, the superposi-\ntion-time derivative curve shows what the transient would have looked like if the test had been performed at a constant rate (Ehlig-\nEconomides 1988).\nIn recent years, log-log derivative plots have been used for the interpretation of DFITs. In a DFIT, 1\u2013100 bbl of water is injected for\nseveral minutes, creating a small hydraulic fracture. The term \u201cDFIT\u201d speci\ufb01cally refers to a fracturing test performed in a low-perme-\nability formation in a completed well. After shut-in, the transient is monitored to estimate the fracture-closure pressure, formation per-\nmeability, formation pressure, and other properties."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae26624-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 0, "chroma:document": "meability, formation pressure, and other properties.\nThe calculation of formation properties requires the identi\ufb01cation of \ufb02ow regimes, which is assisted by log-log derivative plots.\nPractitioners often interpret DFITs with log-log plots with derivative taken with respect to superposition time (Mohamed et al. 2011;\nMarongiu-Porcu et al. 2011; Soliman and Kabir 2012; Bachman et al. 2012; Hawkes et al. 2013; Wallace et al. 2016; Ibrahim and\nPieprzica 2016; Liu et al. 2016). However, unlike in a conventional well test, the superposition-time derivative curve does not show\nwhat the transient would have looked like if the test had been performed at a constant rate. The early-time transient is controlled by the\ninterplay of fracture compliance and leakoff in a way that has no analogy to a conventional well test. The late-time transient is similarto an impulse (or slug) test, in which a volume of \ufb02uid is instantaneously placed into the well, causing an immediate pressure response"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae266a6-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 0, "chroma:document": "and then a decay back to the formation pressure (Craig and Blasingame 2006). The onset of the impulse \ufb02ow regime causes a downward\nde\ufb02ection in the derivative curve.\nAs shown in the Properties of the Superposition-Time Derivative section, taking the derivative with respect to superposition time,\ninstead of shut-in time, causes an upward de\ufb02ection on the log-log plot at the point when shut-in time is equal to injection time. This isa mathematical property of how the derivative is calculated. In a conventional well test; the de\ufb02ection adjusts for the effect of the prior\nrate history because it occurs at approximately the same time as the transition to impulse \ufb02ow. In a DFIT transient, there is no relation-\nship between the timing of the superposition-time derivative de\ufb02ection and the onset of impulse \ufb02ow. In the Simpli\ufb01ed Model Problem\nsection, I show that the timing of the transition to late-time impulse behavior is controlled by injection volume and the formation, well-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae26732-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 0, "chroma:document": "bore, and fracture properties, and is unrelated to injection time. To be precise, the timing of the onset of impulse \ufb02ow actually has aweak (but nonzero) relationship with injection time, but this is a minor effect that does not need to be examined in this report. van den\nHoek (2016) also discusses issues with interpreting DFITs with superposition-time derivative plots.\nAs discussed in the Properties of Nolte\u2019s G-Function section, the superposition-time derivative de\ufb02ection occurs at the same time as\na transition from a one slope to a 1/2 slope in an ideal DFIT shut-in transient. However, the superposition-time derivative de\ufb02ection\nadds unity to the slope, whereas the ideal transient behavior experiences a decrease in slope of 1/2, not unity. In any case, it would not\nassist the interpretation to calculate the derivative in a way that masks the \u201cideal\u201d transition from unit slope to 1/2 slope. In actual \ufb01eld\ndata, the ideal early-time slope of unity is not typically observed."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae267be-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 0, "chroma:document": "In the Properties of Nolte\u2019s G-Function section, I review the properties of Nolte\u2019s G-function, which describes the scaling of pres-\nsure with time in an idealized DFIT shut-in transient. One form of the G-function contains terms with 3/2 powers of time and super\ufb01-\ncially resembles conventional well-test analysis solutions constructed by applying time superposition. However, I show that the 3/2\npower terms arise from a spatial integration of Carter\u2019s leakoff equation, not from using time superposition to sum constant-rate\nsolutions.\nCopyright VC2017 Society of Petroleum Engineers\nOriginal SPE manuscript received for review 5 November 2016. Revised manuscript received for review 1 March 2017. Paper (SPE 186098) peer approved 15 March 2017.REE186098 DOI: 10.2118/186098-PA Date: 26-October-17 Stage: Page: 1045 Total Pages: 11\nID:jaganm Time: 13:19 I Path: //chenas03/cenpro/ApplicationFiles/Journals/SA/SPE/REE#/Vol00000/170018/Comp/APPFile/SA-REE#170018November 2017 SPE Reservoir Evaluation & Engineering 1045"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae26840-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 1, "chroma:document": "In the Practical Implications section, I provide practical recommendations for interpreting DFIT transients. In the Field Example\nsection, I provide a \ufb01eld example. I conclude that there is no physical, mathematical, or practical reason to plot DFIT transients in a\nway that implies 3/2 power scaling of pressure with time.\nProperties of the Superposition-Time Derivative\nSuperposition time can be de\ufb01ned in a variety of ways. For interpretation of DFITs, it is common to de\ufb01ne superposition time as\n(Mohamed et al. 2011; Marongiu-Porcu et al. 2011; Liu et al. 2016)\ns\u00bcte\u00feDt\nDt; \u00f01\u00de\nwhere teis duration of injection and Dtis the time elapsed since the end of injection. This de\ufb01nition of superposition time is identical to\nthe de\ufb01nition of Horner time. A related parameter, Agarwal time, is de\ufb01ned as (Agarwal 1980)\nta\u00bcteDt\nte\u00feDt\u00bc1\n1\nte\u00fe1\nDt\u00bcte\ns: \u00f02\u00de\nThe \ufb01nal equality in Eq. 2 shows that there is a simple relationship between Agarwal time and superposition time (or Horner time)."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae268c2-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 1, "chroma:document": "Agarwal time is equal to one-half the harmonic average of teandDt. Agarwal time has the property that it is approximately equal to Dt\nwhenDt/C28teand approximately equal to tewhenDt/C29te.I fDtis much larger than te,taking the derivative with respect to Agarwal\ntime is equivalent to adding unity to the slope of the d P/d[ln(Dt)] curve on the log-log plot. This is true because\ntadp\ndta\u00bctadp\ndDtdDt\ndta\u00bctadp\ndDt1\nt2\na1\nta/C01\nte/C18/C192\u00bctadp\ndDtDt2\nt2\na\u00bcDt2\ntadP\ndDt/C25Dt2\ntedp\ndDt; \u00f03\u00de\nwhere the \ufb01nal step is taken because taasymptotically approaches teforDt>>te. Therefore, for Dt>>te,\nd log tadp\ndta/C18/C19/C20/C21\nd\u00bdlog\u00f0Dt\u00de/C138/C25d logDt2\ntedp\ndDt/C18/C19/C20/C21\nd\u00bdlog\u00f0Dt\u00de/C138\u00bcd log Dtdp\ndDt/C18/C19/C20/C21\nd\u00bdlog\u00f0Dt\u00de/C138\u00fe1: \u00f04\u00de\nOn a log-log plot, the derivative curve with derivative taken with respect to Agarwal time is the same as the derivative curve with\nderivative taken with respect to superposition time because teis a constant and (from Eq. 2)\ndp\nd\u00bdlog\u00f0ta\u00de/C138\u00bc/C0dp\nd\u00bdlog\u00f0s\u00de/C138: \u00f05\u00de"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2694e-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 1, "chroma:document": "dp\nd\u00bdlog\u00f0ta\u00de/C138\u00bc/C0dp\nd\u00bdlog\u00f0s\u00de/C138: \u00f05\u00de\nThe change in sign does not affect the log-log plot because the derivative is commonly plotted as positive on the log-log plot.\nFig. 1 shows a log-log plot of the function DP\u00bc\ufb03\ufb03tp,along with derivatives taken with respect to time and superposition time.\nSuperposition time is calculated assuming teis equal to 0.0833 hours (5 minutes), which is a typical injection duration for a DFIT.\nWhen time is approximately equal to te,the superposition semilog derivative curves upward to a slope of 3/2; the pressure plot and the\nderivative with respect to time continue with a slope of 1/2..............................................................................\n..................................................................\n...................................\n..........................................\n.....................................................................\n1\u00d710\u20133\n1\u00d710\u201341\u00d710\u201331\u00d710\u201321\u00d710\u20131\nHours\u0394P (blue), d \u0394P/d(ln( t)) (orange), d \u0394P/d(ln( \u03c4)) (grey)"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae269d0-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 1, "chroma:document": "Hours\u0394P (blue), d \u0394P/d(ln( t)) (orange), d \u0394P/d(ln( \u03c4)) (grey)\n1\u00d71001\u00d71011\u00d71021\u00d710\u201321\u00d710\u201311\u00d71001\u00d71011\u00d71021\u00d71031\u00d7104\nFig. 1\u2014Log-log diagnostic plot showing the function DP5\ufb03\ufb03 \ufb03\ntp\n, along with derivative taken with respect to time and superposition\ntime. The superposition-time derivative curves upward when time teis approximately equal to 0.0833 hours.REE186098 DOI: 10.2118/186098-PA Date: 26-October-17 Stage: Page: 1046 Total Pages: 11\nID:jaganm Time: 13:19 I Path: //chenas03/cenpro/ApplicationFiles/Journals/SA/SPE/REE#/Vol00000/170018/Comp/APPFile/SA-REE#1700181046 November 2017 SPE Reservoir Evaluation & Engineering"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae26a5c-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 2, "chroma:document": "Simplified Model Problem\nProblem Definition. I analyze an idealized DFIT model problem with the following assumptions:\n1. A single fracture forms.\n2. The fracture forms effectively instantaneously before the start of the shut-in transient.\n3. The volume of \ufb02uid stored in the fracture is linearly proportional to the difference between formation pressure and fracture\npressure.\n4. The fracture is effectively in\ufb01nite-conductivity so that pressure gradient caused by \ufb02ow along the fracture can be neglected.5. Thermal effects can be neglected.\n6. Leakoff is linear away from the fracture, and can be modeled assuming single-phase, single-component \ufb02ow.\n7. The fracture-surface area is constant during shut-in.\nEven though these assumptions are simpli\ufb01ed relative to actual DFITs, they are suf\ufb01cient to analyze the properties of the superposi-\ntion-time derivative and to investigate certain aspects of real DFIT transients."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae26ade-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 2, "chroma:document": "Assumption 3 neglects the effect of fracture closure. In reality, when the fracture mechanically closes (asperities in contact), the\nstorage coef\ufb01cient changes because fracture compliance becomes controlled by a nonlinear joint-closure process related to the rough-\nness and stiffness of fracture asperities (Barton et al. 1985; McClure et al. 2016).\nAssumptions 4 and 7 are idealizations, but they are reasonable approximations of real DFIT transients in very-low-permeability for-\nmations, suitable for the purposes of this analysis. Fractures are in\ufb01nite-conductivity as long as they have dimensionless fracture con-\nductivity greater than 300. In nanodarcy permeability, fractures retain dimensionless conductivity greater than 300 even if they have\nvery small aperture. This is supported by the consistent observation of post-closure linear \ufb02ow in \ufb01eld data sets (linear \ufb02ow requires an"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae26b60-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 2, "chroma:document": "in\ufb01nite-conductivity fracture). The fracture may cease to be in\ufb01nite-conductivity after closure in formations that are not very low per-meability, or late in the transient when the effective normal stress on the fracture becomes high because of pressure decline. Also, there\nmay be fracture-tip extension immediately after shut-in.\nAnalytical Solution of the Model Problem. From Assumption 2, we assume the fracture forms instantaneously. At the start of the\ntransient, the fracture contains a volume of \ufb02uid equal to C\nt\u00f0Pf;init/C0Pinit\u00de, where Pinitis the initial formation pressure, Pf,initis the initial\npressure in the fracture, and Ctis a total storage coef\ufb01cient equal to\nCt\u00bcCw\u00feA\nSf; \u00f06\u00de\nwhere Cwis the wellbore-storage coef\ufb01cient and Sfis the fracture stiffness. The compressibility of the \ufb02uid in the fracture has a very\nsmall contribution that is neglected in Eq. 6 for simplicity.\nBefore the fracture pressure has changed signi\ufb01cantly, the rate of \ufb02uid \ufb02ow out of the fracture into the matrix can be described"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae26bec-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 2, "chroma:document": "approximately as 1D \ufb02ow from a constant-pressure boundary condition (equivalent to Carter leakoff). The cumulative \ufb02ow of \ufb02uid\nfrom the fracture into the matrix is therefore (Howard and Fast 1957)\nVLs\u00bc4A\ufb03\ufb03\ufb03\ufb03\ufb03\nDtp\n\u00f0pf;init/C0pinit\u00de\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\n/ctk\npls\n; \u00f07\u00de\nwhere VLsis cumulative volume of \ufb02uid leaked off during shut-in, Dtis shut-in time, pf,initis the initial pressure in the fracture, pinitis\nthe initial formation pressure, /is the formation porosity, ctis the formation total compressibility, kis the formation permeability, and\nlis the \ufb02uid viscosity. Note that Eq. 7 is valid only because the fracture is assumed to form instantaneously.\nEqs. 6 and 7 can be combined to write the overall material balance on the system:\nCt\u00f0p/C0pinit\u00de\u00bcCt\u00f0pf;init/C0pinit\u00de/C04A\ufb03\ufb03\ufb03\ufb03\ufb03\nDtp\n\u00f0pf;init/C0pinit\u00de\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\nkct/\npls\n; \u00f08\u00de\nwhere pis the \ufb02uid pressure in the fracture at time Dt:\nIf we de\ufb01ne dimensionless time as\ntD\u00bc16A\nCt/C18/C192kct/\nplDt; \u00f09\u00de\nand we de\ufb01ne dimensionless pressure as\npD\u00bcp/C0pinit\npf;init/C0pinit; \u00f010\u00de"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae26c6e-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 2, "chroma:document": "pD\u00bcp/C0pinit\npf;init/C0pinit; \u00f010\u00de\nthen we can write the material-balance equation as a simple expression giving the evolution of pressure with respect to time,\npDE\u00bc1/C0\ufb03\ufb03\ufb03\ufb03 \ufb03tDp: \u00f011\u00de\nEq. 8 can also be written as,\nCt\u00f0p/C0pf;init\u00de\u00bc/C0 VLs: \u00f012\u00de\nEq. 12 shows that if Ctis constant, then the change in pressure during shut-in is proportional to the cumulative leakoff volume dur-\ning shut-in. If the fracture forms instantaneously, then VLsscales with\ufb03\ufb03\ufb03\ufb03\ufb03\nDtp\n, as shown in Eq. 11. As discussed in the Properties of Nolte\u2019s\nG-Function section, Nolte\u2019s G-function is based on a generalization of Eq. 11 that accounts for the leakoff that occurs as the fracture\npropagates during injection. If fracture-propagation time is considered, then VLsscales with the G-function during shut-in. This is why\nplots of pressure vs. Gare (ideally) linear before closure and can be used to infer leakoff coef\ufb01cient if Ctis assumed to be known (Nolte..........................................................................."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae26cf0-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 2, "chroma:document": ".............................................................\n..........................................\n......................................................................\n........................................................................\n..........................................................................\n.....................................................................REE186098 DOI: 10.2118/186098-PA Date: 26-October-17 Stage: Page: 1047 Total Pages: 11\nID:jaganm Time: 13:19 I Path: //chenas03/cenpro/ApplicationFiles/Journals/SA/SPE/REE#/Vol00000/170018/Comp/APPFile/SA-REE#170018November 2017 SPE Reservoir Evaluation & Engineering 1047"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae26d72-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 3, "chroma:document": "1979; Nolte and Smith 1981). When closure occurs, this affects both the leakoff rate and Ct, which causes pressure to deviate from line-\narity. Whether pressure begins to increase more rapidly or less rapidly after closure depends on the relative effect of closure on leakoff\nrate and the Ct(discussed in the Appendix of McClure et al. 2016). A variety of other \u201cnonideal\u201d processes may affect the pressure\ntrend during a real DFIT transient.\nThe square-root-of-time behavior in Eq. 11 is not the same as the well-test solution for linear \ufb02ow (Gringarten et al. 1974). The\nGringarten et al. (1974) linear-\ufb02ow solution is derived for constant-rate injection and neglects wellbore/fracture storage. Eq. 11 is\nderived on the basis of constant-pressure leakoff from a high-storage wellbore/fracture system. The square root of time appears in both"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae26dfe-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 3, "chroma:document": "solutions because diffusivity has units of length squared per time, and so square-root-of time scaling is frequently present in mathemati-cal relationships involving the diffusivity equation (such as the expression for radius of investigation).\nAfter a period of time, the pressure in the fracture drops suf\ufb01ciently so that leakoff is no longer well-approximated by Carter leakoff,\nwhich assumes constant pressure in the fracture (Howard and Fast 1957). The late-time pressure transient can be found by taking the\ndimensionless time derivative of the nondimensionalized solution for constant-rate injection (Houze et al. 1988; Ayoub et al. 1988; Gu\net al. 1993). The solution for constant-rate injection into an in\ufb01nite-conductivity fracture, neglecting the \ufb02uid storage of the wellbore, is(Gringarten et al. 1974)\np\u00bcp\ninit\u00feq\nA\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03l\npkct/r\ufb03\ufb03\ufb03\ufb03\ufb03\nDtp\n; \u00f013\u00de\nwhere qis the volumetric injection rate. Taking the derivative of the nondimensionalized form of this equation, we can \ufb01nd the solution"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae26e8a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 3, "chroma:document": "for the injection of a volume of \ufb02uid Vsluginto the system:\np\u00bcpinit\u00feVslug\n2A\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03l\npkct/r1\ufb03\ufb03\ufb03\ufb03\ufb03\nDtp : \u00f014\u00de\nFor our problem,\nVslug\u00bcCt\u00f0pf;init/C0pinit\u00de: \u00f015\u00de\nTherefore,\np\u00bcpinit\u00feCt\n2A\u00f0pf;init/C0pinit\u00de\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\n1\npkct/s\n1\ufb03\ufb03\ufb03\ufb03\ufb03\nDtp : \u00f016\u00de\nWith the same dimensionless variables as before, Eq. 16 can be written as\npDL\u00bc2\np1\ufb03\ufb03\ufb03\ufb03 \ufb03tDp : \u00f017\u00de\nEq. 11 describes the early-time behavior of the transient, and Eq. 17 describes the late-time behavior of the transient. There is not a\nsimple expression to describe the evolution of pressure during the transition between these two limiting cases. However, we can \ufb01nd asimple estimate for the time at which the transition occurs.\nEnforcing equality of the derivatives at the transition time,\ndpDE\ndtd\u00bcdpDL\ndtdfortD\u00bctD;trans; \u00f018\u00de\nresults in the expression\ntD;trans\u00bc2\np; \u00f019\u00de\nwhich in dimensional form is\nDttrans\u00bc1\n8Ct\nA/C18/C192l\nkct/: \u00f020\u00de\nThe transition to impulse \ufb02ow marks the peak in the d p/d[ln(Dt)] curve, which occurs at the same time as the peak in the d P/"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae26f0c-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 3, "chroma:document": "d[ln( G)] curve and the transition from 3/2 slope to 1/2 slope on the log-log plot of d p/d[ln( s)]. This provides a physical interpretation\nfor the peak in the d p/d[ln( G)] curve in G-function plots: the onset of impulse \ufb02ow. This occurs as the leakoff rate deviates from Carter\nleakoff (which is derived from the assumption that pressure in the fracture is constant). In real transients, Ctchanges strongly when the\nfracture closes, and so Eq. 20 cannot be applied directly. However, Eq. 20 demonstrates that the onset of the peak in d p/d[ln( G)] is con-\ntrolled by the interplay of formation and fracture properties in causing the onset of impulse \ufb02ow.\nNumerical Solution to the Model Problem. To con\ufb01rm the results in the Analytical Solution of the Model Problem section, I solve"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae26f98-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 3, "chroma:document": "the model problem with a numerical simulator. The simulator calculates leakoff rate from the fracture by solving the diffusivity equa-tion with a 1D \ufb01nite-difference grid with logarithmically spaced elements. On one side of the model, a special \u201cfracture\u201d element is\nplaced with \ufb02uid-storage coef\ufb01cient equal to C\nt. The fracture element is initialized at an elevated pressure pf,init, and the matrix elements\nare initialized with pressure pinit. Away from the fracture element, the grid is suf\ufb01ciently large that it is effectively in\ufb01nite for the dura-\ntion of the simulation.\nFig. 2 shows a log-log diagnostic plot of pressure and the derivative taken with respect to shut-in time. The numerical solution is\nshown with solid lines, and the two approximate analytical solutions (Eqs. 11 and 17) are shown with dashed lines. The comparison\nshows that the analytical results are very accurate at early and late time. The transition between the two solutions occurs at approxi-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2701a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 3, "chroma:document": "mately when the derivatives of the two analytical solutions are equal, consistent with Eq. 19...................................................................\n................................................................\n...................................................................\n.......................................................\n...........................................................................\n.............................................................\n............................................................................\n...................................................................REE186098 DOI: 10.2118/186098-PA Date: 26-October-17 Stage: Page: 1048 Total Pages: 11\nID:jaganm Time: 13:19 I Path: //chenas03/cenpro/ApplicationFiles/Journals/SA/SPE/REE#/Vol00000/170018/Comp/APPFile/SA-REE#1700181048 November 2017 SPE Reservoir Evaluation & Engineering"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae270a6-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 4, "chroma:document": "Fig. 3 shows the numerical solution to the model problem with derivative taken with respect to shut-in time and superposition time.\nAs discussed previously, the model problem neglects the effect of fracture closure and assumes that the fracture formed instantaneously\nduring injection. To calculate superposition time, it is necessary to de\ufb01ne a value for te. In Fig. 1, superposition time is calculated\nassuming teis equal to 0.0833 hours, or 5 minutes, which is a typical injection duration in a DFIT. The superposition-time plot curves\nupward when Dtis equal to te. The derivative with respect to time continues with a slope of 1/2 (indicating square-root-of-time scaling)\nuntil the onset of impulse \ufb02ow at approximately 15 hours. The onset of impulse \ufb02ow causes a transition from a 1/2 slope to a \u20131/2 slope\non the actual-time derivative curve and a 3/2 slope to a 1/2 slope on the superposition-time derivative curve. The superposition-time de-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae27128-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 4, "chroma:document": "rivative plot does not quite reach a full slope of 3/2 because the pressure trend begins deviating downward from pure square-root-of-\ntime scaling as it approaches the onset of impulse \ufb02ow.\nProperties of Nolte\u2019s G-Function\nIn this section, I review properties of Nolte\u2019s G-function (Nolte 1979; Nolte and Smith 1981; Castillo 1987; Gulrajani and Nolte 2000)\nto show that there is not a physical or mathematical justi\ufb01cation for plotting it in a way that implies 3/2 power scaling with respect\nto time.\nThe G-function is de\ufb01ned so that it is proportional to the cumulative leakoff volume during shut-in, given certain simplifying\nassumptions: (a) the leakoff rate at a given location along the fracture is described by Carter leakoff (leakoff rate is proportional to the\ninverse of the square root of leakoff duration); (b) during injection, the fracture length is equal to the duration of injection to the powera; and (c) the fracture immediately stops propagating after shut-in."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae271b4-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 4, "chroma:document": "If the fracture is assumed to grow linearly with time during injection ( a\u00bc1), then Nolte (1979) showed that the G-function\nscales as\nG\u00f0a\u00bc1\u00de/ 1\u00feDt\nte/C18/C191:5\n/C0Dt\nte/C18/C191:5\n/C01: \u00f021\u00de .........................................................\n\u0394P (blue), d \u0394P/d(ln( t)) (orange),\nd\u0394P/d(ln( \u03c4)) (grey)\n1\u00d710\u201321\u00d710\u201311\u00d71001\u00d71011\u00d71021\u00d71031\u00d71041\u00d7105\n1\u00d710\u201341\u00d710\u201331\u00d710\u201321\u00d710\u20131\nHours1\u00d71001\u00d71011\u00d71021\u00d71031\u00d7104\nFig. 3\u2014Log-log diagnostic plot showing the numerical solution to the model problem described in the Simpli\ufb01ed Model Problem\nsection. The model problem is based on the simplifying assumption that the fracture does not mechanically close. The plot showschange in pressure (blue) and the derivative curves with derivative taken with respect to shut-in time (orange) and superpositiontime (gray).\n\u0394P (blue), d \u0394P/d(ln( t)) (orange),\nd\u0394P/d(ln( \u03c4)) (grey)\n1\u00d710\u201321\u00d710\u201311\u00d71001\u00d71011\u00d71021\u00d7103\n1\u00d710\u201341\u00d710\u201331\u00d710\u201321\u00d710\u20131\nHours1\u00d71001\u00d71011\u00d71021\u00d71031\u00d7104"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae27236-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 4, "chroma:document": "1\u00d710\u201341\u00d710\u201331\u00d710\u201321\u00d710\u20131\nHours1\u00d71001\u00d71011\u00d71021\u00d71031\u00d7104\nFig. 2\u2014Log-log diagnostic plot with derivative taken with respect to time showing the numerical solution to the model problem\n(blue lines) and the early-time and late-time analytical solutions to the model problem (green lines). The solid lines show change in\npressure, and the dashed lines show the derivative curves.REE186098 DOI: 10.2118/186098-PA Date: 26-October-17 Stage: Page: 1049 Total Pages: 11\nID:jaganm Time: 13:19 I Path: //chenas03/cenpro/ApplicationFiles/Journals/SA/SPE/REE#/Vol00000/170018/Comp/APPFile/SA-REE#170018November 2017 SPE Reservoir Evaluation & Engineering 1049"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae272b8-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 5, "chroma:document": "Even though Eq. 21 contains terms with a 3/2 power, it never scales according to a 3/2 power. At early time, Eq. 21 scales linearly\nwithDt;and at late time, it scales with the square root of Dt:Derivations of these properties are provided in Appendix A, and they are\nshown graphically in Fig. 4.\nNolte (1979) also de\ufb01ned a function gthat is proportional to the total volume of \ufb02uid leakoff from the start of injection (rather than\ntotal leakoff volume after shut-in). As shown by Eq. 12, pressure after shut-in scales with leakoff volume during shut-in, not total leak-off volume (Gulrajani and Nolte 2000). Nolte (1979) derived the G-function by subtracting gat shut-in from gat a point in time after\nshut-in.\nIn Eq. 21, the transition between linear and square-root scaling occurs at Dt\u00bct\ne:This is the same time at which the spurious de\ufb02ec-\ntion occurs in the superposition-time derivative plot (Eq. 2). It might be argued that the de\ufb02ection in the superposition-time plot adjusts"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2733a-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 5, "chroma:document": "for this downward bend in the G-function that occurs near the same time (though it is unclear why this would assist in an analysis).\nHowever, taking the derivative with respect to superposition time adds unity to the slope of the curve, whereas the transition in Eq. 21\ncauses the slope of the line to reduce by 1/2, not unity. In real DFIT transients, various complex processes cause deviation from ideality\nin the minutes after shut-in. As a result, the early-time unit slope predicted by Eq. 21 is not typically observed.\nFig. 4 shows G\u00f0a\u00bc1\u00de(Eq. 21) and the derivative curves taken with respect to time and superposition time (for teis equal to 0.0833\nhours). All three curves show a slope of unity at early time, indicating linear scaling. At approximately 0.0833 hours, the G-function\ncurve bends toward a slope of 1/2. The derivative curve with respect to time also bends to a slope of 1/2. The superposition-time deriva-\ntive curve bends upward to a slope of 3/2."}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae273c6-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 5, "chroma:document": "tive curve bends upward to a slope of 3/2.\nIf the fracture is assumed to grow according to the square root of time ( a\u00bc0:5), then the G-function scales (Nolte 1979) as\nG\u00f0a\u00bc0:5\u00de/ 1\u00feDt\nte/C18/C19\narcsin 1 \u00feDt\nte/C18/C19/C00:5\"#\n\u00feDt\nte/C18/C190:5\n/C0p\n2: \u00f022\u00de\nIn this form of the G-function, there are not terms with a 3/2 power. G\u00f0a\u00bc0:5\u00dehas the same asymptotic scaling as G\u00f0a\u00bc1:0\u00de: lin-\near at early-time and square root of Dtat late time.\nEqs. 21 and 22 scale linearly with time at early time because shut-in does not cause a discontinuous change in leakoff rate. By de\ufb01ni-\ntion, continuous functions have a constant derivative when evaluated over a suf\ufb01ciently short interval. This implies that leakoff rate is\napproximately constant at early time after shut-in. Eq. 12 shows that if leakoff rate is constant, then pressure change will be linear with\nshut-in time.\nEqs. 21 and 22 scale with the square root of time at late time because the difference in leakoff duration at each point along the frac-"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 5, "chroma:document": "ture becomes negligible. As a result, the leakoff rate asymptotically becomes the same everywhere along the fracture. This is analogousto St. Venant\u2019s principle of mechanical loading, which states that complex applied loads are asymptotically equivalent to point loading\nat suf\ufb01cient distance.\nEq. 21 appears super\ufb01cially similar to superposition-time solutions constructed by addition of constant-rate solutions. However, it is\nnot mathematically or physically valid to interpret Eq. 21 in this way. For example, the linear-\ufb02ow buildup-test transient can be con-\nstructed from superposition by use of the Gringarten et al. (1974) constant-rate solution and scales as\nDP/1\u00feDt\nte/C18/C190:5\n/C0Dt\nte/C18/C190:5\n: \u00f023\u00de\nEq. 23 is a speci\ufb01c case of the more general method of performing a time-convolution integral of a constant-rate analytical solution"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 5, "chroma:document": "multiplied by the derivative of \ufb02ow rate with respect to time. Implicit in the construction of Eq. 23 is the requirement that the constant-.........................................\n................................................................\n\u0394P (blue), d \u0394P/d(ln( t)) (orange), d \u0394P/d(ln( \u03c4)) (grey)\n1\u00d710\u20132\n1\u00d710\u201331\u00d710\u201311\u00d71001\u00d71011\u00d71021\u00d71031\u00d71041\u00d7105\n1\u00d710\u201341\u00d710\u201331\u00d710\u201321\u00d710\u20131\nHours1\u00d71001\u00d71011\u00d7102\nFig. 4\u2014Log-log diagnostic plot showing G(a51) (blue) and the derivative curve taken with respect to time (orange) and superposi-\ntion time (gray).REE186098 DOI: 10.2118/186098-PA Date: 26-October-17 Stage: Page: 1050 Total Pages: 11\nID:jaganm Time: 13:19 I Path: //chenas03/cenpro/ApplicationFiles/Journals/SA/SPE/REE#/Vol00000/170018/Comp/APPFile/SA-REE#1700181050 November 2017 SPE Reservoir Evaluation & Engineering"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2756a-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 6, "chroma:document": "rate solution is not itself changing over time. For example, the line source in the Gringarten et al. (1974) solution is \ufb01xed in size, shape,\nand location. All conventional well-test solutions (in\ufb01nite-acting radial \ufb02ow, bilinear, and others) assume that the source does not\nchange geometry over time. In contrast, in the hydraulic-fracturing problem, the fracture grows during the injection period prior.\nvan den Hoek (2002) and Sections 3.2.3 and 4.2.2 from Koning (1987) analyze the propagation of a hydraulic fracture with very\nlong injection time and very high leakoff. Under these conditions, the fracture-propagation rate is slow, and the injection rate is approx-\nimately equal to the leakoff rate. Under these conditions, they \ufb01nd that it is a reasonable approximation to use time-convolution inte-\ngrals of the constant-rate analytical solution to analyze fracture propagation. This is valid because they restrict their analysis to consider"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae275ec-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 6, "chroma:document": "only a very-high-leakoff, slow-propagation scenario. DFITs are the opposite: very low leakoff, rapid propagation, and short injectiontime. In a DFIT, until Dtis much greater than t\ne,the changing fracture geometry during injection cannot be neglected.\nAccording to the assumptions underlying G\u00f0a\u00bc1:0\u00de, the leakoff duration at each point along the fracture (the duration of time since\nthe fracture formed at that location) is equal to Dt\u00fex\nv, where vis the fracture-propagation velocity (assumed constant during injection\nbecause a\u00bc1) and xis position along the fracture (where the tip of the fracture is located at xequal to 0 and the center is at xequal to\na). Using Carter leakoff, the cumulative leakoff volume (including \ufb02uid leaked off during injection) is equal to\nVL\u00f0Dt\u00de\u00bc8h\u00f0pf;init/C0pinit\u00de\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\n/ctk\npls\u00f0a\n0\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\nDt\u00fex\nvr\ndx/\u00f0a\n0\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\nDt\u00fex\nvr\ndx/1\u00feDt\nte/C18/C191:5\n/C0Dt\nte/C18/C191:5\n; \u00f024\u00de\nwhere his the fracture height and teis equal toa"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae27678-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 6, "chroma:document": "; \u00f024\u00de\nwhere his the fracture height and teis equal toa\nv. Eq. 24 is proportional to the g-function for total leakoff volume (for a\u00bc1:0). The cu-\nmulative leakoff volume during the shut-in period is equal to VLminus the cumulative leakoff volume at shut-in ( Dt\u00bc0\u00de, which recov-\nersG\u00f0a\u00bc1\u00de:\nVLs\u00f0Dt\u00de\u00bcVL\u00f0Dt\u00de/C0VL\u00f00\u00de/ 1\u00feDt\nte/C18/C191:5\n/C0Dt\nte/C18/C191:5\n/C01: \u00f025\u00de\nThe origin of the 3/2 power in G\u00f0a\u00bc1\u00de(Eq. 21) is a spatial integration of the Carter leakoff solution over the length of the fracture.\nIt is not derived in a way that is analogous to Eq. 23, and Eq. 21 cannot be interpreted as being constructed from the sum of constant-rate analytical solutions.\nPractical Implications\nLog-log superposition-time derivative plots can be used to diagnose \ufb02ow regimes in DFIT transients, but the interpreter must be aware\nthat the use of superposition time causes the plot to curve upward when Dtis equal to t\ne. The G-function experiences a downward bend"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae276fa-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 6, "chroma:document": "e. The G-function experiences a downward bend\nat the same time, but the downward bend is a decrease in slope of 1/2, and the superposition-time bend causes an increase in slope of\nunity. The interpreter must understand the complex relationship between the slope of the curve and the scaling of pressure with time,and must be aware that there is no physical signi\ufb01cance to the 3/2 slope on the plot.\nTo avoid confusion, I recommend interpreting DFIT transients from the basic pressure-transient interpretation plot\u2014the log-log plot\nwith derivative taken with respect to actual-time. To assist the identi\ufb01cation of closure and other preclosure trends, plots of pressure\nandG/C2dG/dPvs.Gcan be used.\nWhen the derivative is taken with respect to shut-in time in a log-log plot, there is a simple mathematical interpretation of the slope\nof the derivative curve. If the pressure change scales with respect to t\nb, then the slope of the derivative curve on a log-log plot is equal"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2777c-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 6, "chroma:document": "tob. If the pressure change scales with respect to logarithmic time, then the slope is zero. In contrast, if the derivative is taken with\nrespect to superposition time, then after the spurious de\ufb02ection, the superposition-time derivative curve has a slope of b\u00fe1. Before the\nonset of impulse \ufb02ow, the slope of b\u00fe1 does not have a physical interpretation. In conventional (nonfracturing) buildup/falloff transi-\nents, the onset of impulse \ufb02ow occurs at approximately the same time as the de\ufb02ection, which is why the superposition derivativemethod can be used to show what the transient \u201cwould have been\u201d if it was a constant-rate test. But in DFIT transients, the de\ufb02ection\noccurs much earlier than the onset of impulse \ufb02ow, and the superposition-time derivative curve does not have the same interpretation.\nInterpreters are tempted to identify a change in \ufb02ow regime at the spurious de\ufb02ection. If the de\ufb02ection occurs in a part of the tran-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae27808-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 6, "chroma:document": "sient where the derivative is curving because of an actual pressure-transient response, the real response is distorted by the overprint of\nthe spurious de\ufb02ection and will be more dif\ufb01cult to interpret.\nLate-time impulse \ufb02ow regimes can be identi\ufb01ed easily on derivative plots, regardless of whether the derivative is taken with respect\nto shut-in time or superposition time. The late-time impulse solution is the derivative of the constant-rate solution (Houze et al. 1988).\nIf pressure change in a constant-rate solution scales with respect to t\nbor the logarithm of time, then, for the impulse solution, the slope\nof the derivative curve (taken with respect to shut-in time) is the slope of the constant-rate solution minus unity. For example, impulse\nradial \ufb02ow has a slope of /C01, and impulse linear has a slope of /C01/2.\nField Example\nFig. 5 shows a log-log diagnostic plot for the \ufb01eld data set described by McClure et al. (2016). The data are from a DFIT in a vertical"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2789e-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 6, "chroma:document": "well in a formation with strong stress contrast between layers (causing fracture-height con\ufb01nement), high stress anisotropy (precluding\nthe possibility of \u201copening transverse fractures\u201d), drilling-induced axial tensile fractures visible in an image log perpendicular to the min-\nimum principal stress (indicating that fractures will not be forced to rotate into the far-\ufb01eld orientation as they propagate away from the\nwellbore), and very low permeability (approximately 100 nd). The injection volume was 11.5 bbl. These conditions are as close as practi-cally possible to being an \u201cideal\u201d DFIT in very tight rock. Nevertheless, there is strong deviation from the ideal G-function response.\nTypically, the transient would be interpreted in conjunction with a plot of G/C2dP/dGvs.G. However, for simplicity, I provide only the\nlog-log diagnostic plot. Fig. 2 from McClure et al. (2016) provides a G-function plot of these data. For calculating superposition time, the\nvalue of t"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae27920-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 6, "chroma:document": "value of t\neis determined by dividing the total injection volume by the \ufb01nal injection rate and is equal to 0.0383 hours (138 seconds).\nThe \ufb01rst 0.01 hours (36 seconds) is dominated by water hammer in the wellbore, causing sharp pressure oscillations. The superposi-\ntion-time derivative curve de\ufb02ects upward at approximately te(0.0383) hours of shut-in. As discussed in the Properties of the Superpo-\nsition-Time Derivative section, the upward curve of the superposition-time derivative curve is a consequence of how the derivative is\ncalculated and should not be interpreted as representing a reservoir process. The actual-time derivative curve is nearly \ufb02at from 0.01 to.......................\n.............................................REE186098 DOI: 10.2118/186098-PA Date: 26-October-17 Stage: Page: 1051 Total Pages: 11\nID:jaganm Time: 13:19 I Path: //chenas03/cenpro/ApplicationFiles/Journals/SA/SPE/REE#/Vol00000/170018/Comp/APPFile/SA-REE#170018November 2017 SPE Reservoir Evaluation & Engineering 1051"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae279ac-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 7, "chroma:document": "0.1 hours. The early-time slope of unity predicted by the G-function is not apparent. A variety of processes, with con\ufb02icting effects,\nmay be occurring during the early-time transient: reheating of the \ufb02uid in the wellbore caused by conduction from the surrounding for-\nmation, fracture-tip extension (continued propagation after shut-in), and many other possible effects. Because of the variety of complexprocesses that may be occurring in the early minutes of shut-in, it is dif\ufb01cult to draw conclusions from this part of the transient. The\nnearly \ufb02at actual-time derivative from 0.01 to 0.1 hours should not be interpreted as representing radial \ufb02ow.\nThe actual-time derivative curve gently curves upward to a slope of 1/2, centered at approximately 1.0 hours. However, it subse-\nquently curves up to a slope approaching unity, indicating that the pressure is decreasing signi\ufb01cantly faster than square-root-of-time"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae27a2e-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 7, "chroma:document": "scaling. The superposition-time plot curves upward in tandem, approaching a slope of two. In the superposition-time plot, it would be\ndif\ufb01cult to identify that the slope has deviated from 3/2 slope without the help of the dashed yellow guide line. It is visually easier to\ndistinguish between a slope of 1/2 and unity than a slope of 3/2 and two.\nA curving-upward pressure derivative, indicating pressure scaling more rapid than square root of time, has been the subject of dis-\ncussion in the literature. On a G-function plot, this phenomenon manifests as curving upward G/C2dP/dG. Barree and Miskimins (2016)\ncall this behavior \u201cdelayed leakoff\u201d and hypothesize that it could be caused by wellbore effects, near-wellbore fracture complexity, clo-\nsure of transverse fractures, or poroelastic effects. McClure et al. (2016) provide a different explanation that is based on analytical and"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae27aba-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 7, "chroma:document": "numerical calculations that provide a match to the \ufb01eld data shown in Fig. 5. Their explanation is that fracture compliance decreases(stiffness increases) as the fracture walls come into contact, accelerating the rate of pressure decline. Eventually, the leakoff rate devi-\nates suf\ufb01ciently from Carter leakoff that the rate of pressure decline decelerates, causing the onset of impulse \ufb02ow.\nConventionally, fracture closure is picked from G-function plots by drawing a tangent to the G/C2dP/dGcurve from the origin\n(Barree and Miskimins 2016). This has the effect of picking closure very close to the peak in G/C2dP/dG. The peak in G/C2dP/dGoccurs\nat exactly the same time as the peak in the derivative plot and at the same time as the transition from 3/2 to 1/2 slope on the superposi-tion-time derivative plot. Consequently, picking closure at the start of the 1/2 slope on a superposition-time derivative plot is equivalent"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae27b3c-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 7, "chroma:document": "to the tangent method of picking closure from a G/C2dP/dGcurve. The Simpli\ufb01ed Model Problem section demonstrates that this transi-\ntion occurs at the onset of impulse \ufb02ow. Thus, the conventional closure pick is equivalent to picking closure at the onset of impulse\ufb02ow. In contrast, McClure et al. (2016) suggest picking closure earlier, at the deviation from square-root-of-time scaling. In Fig. 5, the\ntwo different methods provide substantially different closure picks, either at approximately 0.5\u20131.5 hours or at approximately 10 hours.\nIf the transient does not curve upward past square-root-of-time scaling, the methods described by McClure et al. (2016) and Barree\nand Miskimins (2016) yield similar closure picks. McClure et al. (2016) explain that if permeability is not very low, closure can reduce"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae27bc8-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 7, "chroma:document": "dimensionless fracture conductivity suf\ufb01ciently that closure causes a decrease in leakoff rate (as \ufb02uid continues to \ufb02ow into the fracturebecause of wellbore storage) and weakening of the hydraulic connection between the fracture and the wellbore. The decrease in leakoff\nrate can overwhelm the effect of closure on compliance, resulting in a decrease in the rate of leakoff when closure occurs, causing\nDt/C2dP/dDtandG/C2dP/dGto curve downward and peak, rather than curve upward.\nIn real DFITs, there is often a period of rapid pressure drop at the start of the transient (e.g., Fig. 3 from Liu et al. 2016). Fracture-tip\nextension could explain this phenomenon, but sometimes the pressure drop occurs over thousands of psi. It is unlikely that tip-extensioncould explain such large pressure changes. An alternative explanation is that fractures start at the wellbore with orientation different"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae27c4a-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 7, "chroma:document": "from their far-\ufb01eld orientation and then are forced to rotate as they propagate away from the well. After shut-in, pressure drops as \ufb02uid\n\ufb02ows out of the misoriented fractures near the wellbore (which require high pressure to remain open during injection) into the far-\ufb01eld\nfracture (which remains open at a lower pressure). Figs. 13 and 17 from Jung et al. (2016) show numerical simulations of this process.\nConclusions\nThe following conclusions are drawn regarding the use of superposition-time derivative plots for DFIT shut-in transients:\n\u2022Superposition-time derivative plots of DFIT data commonly exhibit a 3/2 slope. This slope does not re\ufb02ect the actual scaling of pres-\nsure with time and does not have a physical or theoretical interpretation. For the particular case of constant propagation velocity, the\nG-function contains terms with a 3/2 power, but these terms arise from an integration of the Carter leakoff solution along the length"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae27d30-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 7, "chroma:document": "of the fracture, and are not analogous to buildup/falloff solutions constructed from superposition of constant-rate solutions.\n1\u00d710\u20133 1\u00d710\u20132 1\u00d710\u20131\nHours1\u00d7100 1\u00d7101 1\u00d7102 1\u00d7103\u0394P (blue), d \u0394P/d(ln( t)) (orange),\nd\u0394P/d(ln( \u03c4)) (grey)\n1\u00d71011\u00d71021\u00d71031\u00d71041\u00d71051\u00d71061\u00d7107\nFig. 5\u2014Log-log diagnostic plot showing the \ufb01eld data set described by McClure et al. (2016). The plot shows change in pressure\n(blue), and the derivative curve taken with respect to time (orange) and superposition time (gray). The dashed yellow lines are pro-vided for reference to show slopes of 1/2 and 3/2.REE186098 DOI: 10.2118/186098-PA Date: 26-October-17 Stage: Page: 1052 Total Pages: 11\nID:jaganm Time: 13:19 I Path: //chenas03/cenpro/ApplicationFiles/Journals/SA/SPE/REE#/Vol00000/170018/Comp/APPFile/SA-REE#1700181052 November 2017 SPE Reservoir Evaluation & Engineering"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae27dc6-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 8, "chroma:document": "\u2022The superposition-time derivative plot de\ufb02ects upward, adding unity to the slope, when shut-in time is equal to injection time. The ideal\nG-function response transitions from a slope of unity to a slope of 1/2 (decrease of 1/2) at approximately the same time as the de\ufb02ection.\n\u2022The transition from 3/2 to 1/2 slope on a superposition-time plot (equivalent to the transition from 1/2 to \u20131/2 slope if the derivative\nis taken with respect to time) indicates the onset of impulse \ufb02ow. This occurs even in an idealized model problem in which closure\ndoes not occur. Closure and the onset of impulse \ufb02ow are related, but they should not be regarded as the same thing, and they do not\nnecessarily occur at the same time.\n\u2022The slope of log-log plots with the derivative taken with respect to actual shut-in time has a simple and intuitive relationship with the\npressure transient. Although superposition-time derivative curves are useful for conventional buildup/falloff tests, they complicate"}


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id: 9ae27e48-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 8, "chroma:document": "the interpretation of DFIT transients and do not aid in the interpretation.\n\u2022I recommend interpreting DFIT transients with log-log diagnostic plots with the derivative taken with respect to actual shut-in time\nand plots of pressure and G/C2dG/dPvs.G.\nNomenclature\na\u00bcfracture half-length, m\nA\u00bcfracture area, m2\nct\u00bcformation total compressibility, MPa\u20131\nCt\u00bctotal-storage coef\ufb01cient, m3/MPa\nCw\u00bcwellbore-storage coef\ufb01cient, m3/MPa\ng\u00bcg-time function (dimensionless \ufb02uid-loss volume function), dimensionless\nG\u00bcG-time function (dimensionless \ufb02uid-loss time function), dimensionless\nh\u00bcfracture height, m\nk\u00bcformation permeability, m2\np\u00bcpressure, MPa\npD\u00bcdimensionless pressure, dimensionless\npDE\u00bcsolution for early-time dimensionless pressure, dimensionless\npDL\u00bcsolution for late-time dimensionless pressure, dimensionless\npf,init\u00bc\ufb01nal fracture pressure, MPa\npinit\u00bcinitial pressure, MPa\nq\u00bcvolumetric injection rate, m3/s\nSf\u00bcfracture stiffness, MPa/m\nt\u00bctime, seconds\nta\u00bcAgarwal time, dimensionless\ntD\u00bcdimensionless time, dimensionless"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae27ec0-857d-11ee-bb1c-0242ac1c000c
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metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 8, "chroma:document": "tD\u00bcdimensionless time, dimensionless\ntD,trans\u00bctime at the transition between early-time and late-time transient, seconds\nte\u00bcduration of injection, seconds\nv\u00bcfracture-propagation velocity, m/s\nVf\u00bcvolume of \ufb02uid in the fracture, m3\nVL\u00bctotal cumulative leakoff volume from the start of injection, m3\nVLs\u00bctotal cumulative leakoff volume from shut-in, m3\nVslug\u00bctotal injection volume, m3\nx\u00bcdistance from the location of the \ufb01nal fracture tip, m\na\u00bcfracture power-law growth exponent, dimensionless\nb\u00bcexponent, dimensionless\nDp\u00bcchange in pressure since the start of shut-in, seconds\nDt\u00bctime elapsed since the start of shut-in, seconds\n/\u00bcformation porosity, dimensionless\nl\u00bc\ufb02uid viscosity, MPa /C1s\ns\u00bcsuperposition time (identical to Horner time), dimensionless\nAcknowledgments\nGratitude is expressed to Roland Horne for informally reviewing this manuscript, and the journal editor and two anonymous reviewers"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae27f38-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 8, "chroma:document": "deserve thanks for their thoughtful comments. Dave Cramer and ConocoPhillips deserve appreciation for providing permission to repro-\nduce the \ufb01eld data shown in Fig. 5.\nReferences\nAgarwal, R. G. 1980. A New Method to Account for Producing Time Effects When Drawdown Type Curves Are Used to Analyze Pressure Buildup and\nOther Test Data. Presented at the SPE Annual Technical Conference and Exhibition, Dallas, 21\u201324 September. SPE-9289-MS. https://doi.org/\n10.2118/9289-MS.\nAyoub, J. A., Bourdet, D. P., and Chauvel, Y. L. 1988. Impulse Testing. SPE Form Eval 3(3): 534\u2013546. SPE-15911-PA. https://doi.org/10.2118/15911-\nPA.\nBachman, R. C., Walters, D. A., Hawkes, R. A. et al. 2012. Reappraisal of the G Time Concept in Mini-Frac Analysis. Presented at the SPE Annual\nTechnical Conference and Exhibition, San Antonio, Texas, USA, 8\u201310 October. SPE-160169-MS. https://doi.org/10.2118/160169-MS."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae27fba-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 8, "chroma:document": "Barree, R. D. and Miskimins, J. L. 2016. Physical Explanation of Non-Linear Derivatives in Diagnostic Fracture Injection Test Analysis. Presented a t\nthe SPE Hydraulic Fracturing Technology Conference, The Woodlands, Texas, USA, 9\u201311 February. SPE-179134-MS. https://doi.org/10.2118/\n179134-MS.\nBarton, N., Bandis, S., and Bakhtar, K. 1985. Strength, Deformation, and Conductivity Coupling of Rock Joints. International Journal of Rock Mechan-\nics and Mining Sciences & Geomechanics Abstracts 22(3): 121\u2013140. https://doi.org/10.1016/0148-9062(85)93227-9.\nCastillo, J. L. 1987. Modi\ufb01ed Fracture Pressure Decline Analysis Including Pressure-Dependent Leakoff. Presented at the SPE/DOE Low-Permeabilit y\nReservoir Symposium, Denver, 18\u201319 May. SPE-16417-MS. https://doi.org/10.2118/16417-MS.REE186098 DOI: 10.2118/186098-PA Date: 26-October-17 Stage: Page: 1053 Total Pages: 11"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae28046-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 8, "chroma:document": "ID:jaganm Time: 13:19 I Path: //chenas03/cenpro/ApplicationFiles/Journals/SA/SPE/REE#/Vol00000/170018/Comp/APPFile/SA-REE#170018November 2017 SPE Reservoir Evaluation & Engineering 1053"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae280c8-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 9, "chroma:document": "Craig, D. P. and Blasingame, T. A. 2006. Application of a New Fracture-Injection/Falloff Model Accounting for Propagating, Dilated, and Closing Hy-\ndraulic Fractures. Paper presented at the SPE Gas Technology Symposium, Calgary, 15\u201317 May. SPE-100578-MS. https://doi.org/10.2118/100578-MS.\nEhlig-Economides, C. 1988. Use of the Pressure Derivative for Diagnosing Pressure-Transient Behavior. J Pet Technol 40(10): 1280\u20131282. SPE-18594-\nPA. https://doi.org/10.2118/18594-PA.\nGringarten, A. C., Ramey, H. J., and Raghavan, R. 1974. Unsteady-State Pressure Distributions Created by a Well With a Single In\ufb01nite Conductivity\nVertical Fracture. SPE J. 14(4): 347\u2013360. SPE-4051-PA. https://doi.org/10.2118/4051-PA.\nGu, H., Elbel, J. L., Nolte, K. G. et al. 1993. Formation Permeability Determination Using Impulse-Fracture Injection. Presented at the Production O pera-\ntions Symposium, Oklahoma City, Oklahoma, USA, 21\u201323 March. SPE-25425-MS. https://doi.org/10.2118/25425-MS."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae28154-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 9, "chroma:document": "Gulrajani, S. N. and Nolte, K. G. 2000. Appendix: Background for Hydraulic Fracturing Pressure Analysis Techniques. In Reservoir Stimulation , ed. M.\nJ. Economides and K. G. Nolte, Chichester, England: John Wiley.\nHawkes, R. V., Anderson, I., Bachman, R. C. et al. 2013. Interpretation of Closure Stress in the Montney Shale Using PTA Based Techniques. Presented\nat the SPE Hydraulic Fracturing Technology Conference, The Woodlands, Texas, USA, 4\u20136 February. SPE-163825-MS. https://doi.org/10.2118/\n163825-MS.\nHouze, O. P., Horne, R. N., and Ramey, H. J. 1988. Pressure-Transient Response of an In\ufb01nite Conductivity Vertical Fracture in a Reservoir With Dou-\nble-Porosity Behavior. SPE Form Eval 3(3): 510\u2013518. SPE-12778-PA. https://doi.org/10.2118/12778-PA.\nHoward, G. and Fast, C. R. 1957. Optimum Fluid Characteristics for Fracture Extension. Drilling and Production Practice 24:261\u2013270. API-57-261."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae281d6-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 9, "chroma:document": "Ibrahim, M. and Pieprzica, C. 2016. Appraising Unconventional Play From Mini-Frac Test Analysis, Actual Field Case. Presented at the SPE Annual\nTechnical Conference and Exhibition, Dubai, 26\u201328 September. SPE-181465-MS. https://doi.org/10.2118/181465-MS.\nJung, H., Sharma, M. M., Cramer, D. D. et al. 2016. Re-examining Interpretations of Non-Ideal Behavior During Diagnostic Fracture Injection Tests.\nJournal of Petroleum Science and Engineering 145: 114\u2013136. https://doi.org/10.1016/j.petrol.2016.03.016.\nKoning, E. J. L. 1987. Water\ufb02ooding Under Fracturing Conditions. PhD thesis, Delft University of Technology.\nLiu, G., Ehlig-Economides, C., and Sun, J. 2016. Comprehensive Global Fracture Calibration Model. Presented at the SPE Asia Paci\ufb01c Hydraulic Frac-\nturing Conference, Beijing, 24\u201326 August. SPE-181856-MS. https://doi.org/10.2118/181856-MS.\nMarongiu-Porcu, M., Ehlig-Economides, C. A., and Economides, M. J. 2011. Global Model for Fracture Falloff Analysis. Presented at the SPE North"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae28262-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 9, "chroma:document": "American Unconventional Gas Conference and Exhibition, The Woodlands, Texas, USA, 14\u201316 June. SPE-144028-MS. https://doi.org/10.2118/\n144028-MS.\nMcClure, M. W., Jung, H., Cramer, D. D. et al. 2016. The Fracture Compliance Method for Picking Closure Pressure From Diagnostic Fracture Injection\nTests. SPE J. 21(4): 1321\u20131339. SPE-179725-PA. https://doi.org/10.2118/179725-PA.\nMohamed, I. M., Nasralla, R. A., Sayed, M. A. et al. 2011. Evaluation of After-Closure Analysis Techniques for Tight and Shale Gas Formations. Pre-\nsented at the SPE Hydraulic Fracturing Technology Conference. The Woodlands, Texas, USA, 24\u201326 January. SPE-140136-MS. https://doi.org/\n10.2118/140136-MS.\nNolte, K. 1979. Determination of Fracture Parameters From Fracturing Pressure Decline. Presented at the SPE Annual Technical Conference and Exhibi -\ntion, Las Vegas, Nevada, USA, 23\u201326 September. SPE-8341-MS. https://doi.org/10.2118/8341-MS."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae282ee-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 9, "chroma:document": "Nolte, K. G. and Smith, M. B. 1981. Interpretation of Fracturing Pressures. J Pet Technol 33(9): 1767\u20131775. SPE-8297-PA. https://doi.org/10.2118/\n8297-PA.\nSoliman, M. Y. and Kabir, C. S. 2012. Testing Unconventional Formations. Journal of Petroleum Science and Engineering 92\u201393: 102\u2013109. https://\ndoi.org/10.1016/j.petrol.2012.04.027.\nvan den Hoek, P. J. 2002. Pressure Transient Analysis in Fractured Produced Water Injection Wells. Presented at the SPE Asia Paci\ufb01c Oil and Gas Con-\nference and Exhibition, Melbourne, Australia, 8\u201310 October. SPE-77946-MS. https://doi.org/10.2118/77946-MS.\nvan den Hoek, P. 2016. A Simple Uni\ufb01ed Pressure Transient Analysis Method for Fractured Water\ufb02ood Injectors and Minifracs in Hydraulic Fracture\nStimulation. Presented at the SPE Annual Technical Conference and Exhibition, Dubai, 26\u201328 September. SPE-181593-MS. https://doi.org/10.2118/\n181593-MS."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2837a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 9, "chroma:document": "181593-MS.\nWallace, J., Kabir, C. S., and Cipolla, C. 2016. Probing Diagnostic Fracture Injection Tests in Conventional Reservoirs. Journal of Petroleum Science &\nEngineering 143: 245\u2013257. https://doi.org/10.1016/j.petrol.2016.01.039.\nAppendix A\nHere, I derive equations for the asymptotic behavior of G\u00f0a\u00bc1\u00defor small Dtand large Dt:G\u00f0a\u00bc1\u00deis de\ufb01ned in Eq. 21. To calculate\nthe behavior at small Dt;take the Taylor-series expansion of the 1 \u00feDt\nte/C18/C191:5\nterm centered at Dt\u00bc0:\nG\u00f0a\u00bc1;Dt!0\u00de//C0Dt\nte/C18/C191:5\n/C01\u00fe1\u00fe0\nte/C18/C191:5\n\u00fe1:5\nte1\u00fe0\nte/C18/C190:5\nDt\u00fe0:75\nt2\ne1\u00fe0\nte/C18/C19/C00:5\nDt2\u00fe\u2026 \u00f0A-1\u00de\nThe two constant terms cancel out. The lowest-order term in Eq. A-1 (the term that decreases most slowly as Dtapproaches zero) is\nlinear with Dt:Therefore,\nlimDt!0G\u00f0a\u00bc1\u00de/1:5Dt\nte: \u00f0A-2\u00de\nTo calculate the behavior at large Dt;note that\nlimDt!1Dt\nte\u00fe1/C18/C191:5\n/C0Dt\nte/C18/C191:5\n\u00bclimDt!1Dt\nte\u00fe1/C18/C191:5\n/C0Dt\nte/C18/C191:5\n1\u00bclimD!0Dt\nte\u00feD/C18/C191:5\n/C0Dt\nte/C18/C191:5\nD\u00bc1:5\ufb03\ufb03\ufb03\ufb03\ufb03\nDt\ntes\n: \u00f0A-3\u00de"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae28406-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 9, "chroma:document": "te\u00feD/C18/C191:5\n/C0Dt\nte/C18/C191:5\nD\u00bc1:5\ufb03\ufb03\ufb03\ufb03\ufb03\nDt\ntes\n: \u00f0A-3\u00de\nThe second equality is true because 1.0 becomes in\ufb01nitesimally small relative toDt\nteasDtapproaches in\ufb01nity. The third equality is\ntrue from the de\ufb01nition of a derivative and the power rule of differentiation. Finally, as Dtapproaches in\ufb01nity, the \u201c\u20131\u201d term in Eq. 21\nbecomes negligibly small; therefore,................\n................................................................\n.......REE186098 DOI: 10.2118/186098-PA Date: 26-October-17 Stage: Page: 1054 Total Pages: 11\nID:jaganm Time: 13:19 I Path: //chenas03/cenpro/ApplicationFiles/Journals/SA/SPE/REE#/Vol00000/170018/Comp/APPFile/SA-REE#1700181054 November 2017 SPE Reservoir Evaluation & Engineering"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae28488-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 10, "chroma:document": "limDt!1G\u00f0a\u00bc1\u00de/1:5\ufb03\ufb03\ufb03\ufb03\ufb03\nDt\ntes\n: \u00f0A-4\u00de\nMark McClure is the founder and chief executive officer of McClure Geomechanics and is a consulting professor at Stanford Uni-\nversity. He was an assistant professor in the Department of Petroleum and Geosystems Engineering at the University of Texas atAustin from 2013 to 2015. McClure\u2019s research focuses on modeling and characterization of hydraulic fracturing, induced seismic-\nity, and enhanced geothermal systems. He was honored with the SPE Regional Completions Optimization and Technology\nAward from the Southwest North American Region in 2015 and an Outstanding Service as a Technical Editor award for SPE Res-\nervoir Evaluation & Engineering in 2014, the Hank Ramey Award for Outstanding Research and Service to the Department from\nthe Department of Energy Resources Engineering at Stanford University in 2012, and the Best Paper in Geophysics award from"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2850a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf", "page": 10, "chroma:document": "the Society of Exploration Geophysicists in 2011. McClure holds a BS degree in chemical engineering, an MS degree in petroleum\nengineering, and a PhD degree in energy-resources engineering, all from Stanford University................................................................REE186098 DOI: 10.2118/186098-PA Date: 26-October-17 Stage: Page: 1055 Total Pages: 11\nID:jaganm Time: 13:19 I Path: //chenas03/cenpro/ApplicationFiles/Journals/SA/SPE/REE#/Vol00000/170018/Comp/APPFile/SA-REE#170018November 2017 SPE Reservoir Evaluation & Engineering 1055"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae28596-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-DISC Fracture Compliance Discussions.pdf", "page": 0, "chroma:document": "Discussion of The\nFracture-Compliance Method for\nPicking Closure Pressure From\nDiagnostic Fracture-Injection Tests\nMatteo Marongiu Porcu, Schlumberger; Christine Ehlig-Economides, University of Houston;\nand Robert D. Barree, Barree & Associates\nWe would like to open this discussion on SPE-179725-PA\n(McClure et al. 2016) directly from its title. The \u201cfracture-compli-\nance method\u201d does not allow determination of the closure pres-\nsure associated with minimum principal in-situ stress. What\nMcClure et al. refer to as \u201cclosure pressure\u201d is in fact the manifes-\ntation of known physical phenomena that are generally addressed\nas nonlinear leakoff, departing from Carter\u2019s ideal leakoff\n(Howard and Fast 1957), which all occur at higher to much higher\npressures than the minimum principal in-situ stress.\nMcClure et al. do cite the essential published work that\naddresses these nonlinear-leakoff phenomena, but they do so with\na very dismissive tone. Barree and Miskimins (2016) de\ufb01ned"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae28618-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-DISC Fracture Compliance Discussions.pdf", "page": 0, "chroma:document": "a very dismissive tone. Barree and Miskimins (2016) de\ufb01ned\nthese as \u201caccelerated leakoff\u201d and \u201cdelayed leakoff,\u201d and pro-\nvided a very clear description of all physical causes of these pres-sure-falloff behaviors.\nIn particular, Barree and Miskimins (2016) dedicated a consid-\nerable effort to describing features of diagnostic fracture-injection\ntests (DFITs) in multistage/multicluster hydraulically fractured\nhorizontal wells in naturally fractured/laminated shale formationsthat are not expected in vertical-well DFITs. In such settings, it is\nhighly unrealistic to expect ideal planar biwing propagation of\ninduced hydraulic fractures. Instead, it is possible to have highlynonideal con\ufb01gurations, including sublongitudinal coalescence of\ndiscrete cracks along the wellbore, strong near-wellbore tortuosity,\nand hydraulic-fracture complexity (Weng et al. 2011) because of\nthe severe interaction with near-wellbore stresses possibly altered"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae286a4-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-DISC Fracture Compliance Discussions.pdf", "page": 0, "chroma:document": "by drilling, pre-existing natural fractures, joints, and genericplanes of weakness. These nonideal fracture features (near-well-\nbore and far-\ufb01eld) are arbitrarily oriented, and, at the moment of\ninjection shut-in, are held open against a higher stress than the\nminimal principal stress. Consequently, they will be the \ufb01rst ones\nto gradually close during the pressure falloff, and, during theirown closure process, they will keep transmitting their higher pres-\nsure to the primary fracture (orthogonal to the minimum principal\nin-situ stress), causing the slower or delayed leakoff behavior\nsymptomatic of the upward concavity in a Cartesian ( G-function)\nplot of the GdP/dGderivative function vs. the G-time function.\nThus, application of the \u201cfracture-compliance method\u201d might\nlead to identi\ufb01cation of a closure event ascribable to the nonideal\ncon\ufb01gurations described above, but not to a \ufb01nal primary-fractureclosure that would re\ufb02ect the minimum principal in-situ stress."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae28726-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-DISC Fracture Compliance Discussions.pdf", "page": 0, "chroma:document": "The authors confuse this point in their Conclusions 2 through\n4 by claiming that the picks by other cited analysis methods\nunderestimate minimum stress, while stating in their Conclusion 7\nthat the closure pressure (picked from their method) can be asmuch as 1 to 2 MPa (145 to 290 psi) greater than the minimum-\nprincipal-stress value. The other analysis methods cited in Con-\nclusions 2 through 4 will all pick the same closure event, correctly\nassociated to the minimum principal in-situ stress.\nThe numerical model used by the authors to present the variable\ncompliance simulations exposes signi\ufb01cant limitations, the \ufb01rst ofwhich is that the model is incapable of handling such features as\nnormal leakoff behavior; faster or accelerated leakoff scenariossuch as pressure-dependent leakoff that produce downward con-\ncavities in the GdP/dGderivative function; and multiple closure\nscenarios consistent with multiple straight-line portions that appearin DFIT data illustrated in Mohamed et al. (2011) and others."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae287e4-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-DISC Fracture Compliance Discussions.pdf", "page": 0, "chroma:document": "A second model limitation, which is really a serious model\n\ufb02aw, is shown by obvious discontinuities occurring immediately\nafter what the authors pick as the closure pressure. The authors\nacknowledge that such discontinuities are not seen in DFIT\nresponses. They attempt to avoid the model discontinuity by mod-eling a fracture with a residual aperture exceeding 0.5 mm at the\nclosure pressure claimed to correspond to the minimum principal\nstress. Few if any analysts would envision such a fracture width\nas closed. Even so, to completely remove the model discontinuity,\nthey add wellbore-storage and tip-extension model features. Onlywith all this (including residual fracture aperture at minimum\nprincipal stress exceeding 0.5 mm) do they produce simulations\nremotely similar to typical DFIT data.\nFrom our experience from falloff-test analyses performed on\nEagle Ford, Wolfcamp, Haynesville, Fayetteville, Marcellus, andHorn River shale wells, the real closure events indicating the min-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae28866-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-DISC Fracture Compliance Discussions.pdf", "page": 0, "chroma:document": "imum principal in-situ stresses systematically occur at equivalent\nstress gradients 0.05 to 0.1 psi/ft lower than the gradients associ-ated with nonideal-con\ufb01gurations events (described above) that\none would pick using the \u201cfracture-compliance method.\u201d\nFor instance, the extra 0.05 psi/ft (and often higher) in the mini-\nmum principal in-situ-stress gradient that we systematically obtain\nwhen we attempt to pick closure events using the \u201cfracture-com-pliance method\u201d in a vast number of Eagle Ford and Wolfcamp\ncalibration tests is impossible to reconcile within state-of-the-art\nmechanical Earth models [refer to, among others, the example of\nmechanical Earth models built and used in Berard et al. (2015) and\nMarongiu Porcu et al. (2016)] that were built using anisotropicacoustic borehole measurements calibrated with actual core-\nderived mechanical properties and pore pressures either deter-\nmined from after-closure analysis or measured with MDT tools.\nReferences"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae288f2-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-DISC Fracture Compliance Discussions.pdf", "page": 0, "chroma:document": "References\nBarree, R. D. and Miskimins, J. L. 2016. Physical Explanation of Non-\nLinear Derivatives in Diagnostic Fracture Injection Test Analysis. Pa-\nper presented at the SPE Hydraulic Fracturing Technology Confer-\nence, The Woodlands, Texas, 9\u201311 February. SPE-179134-MS.\nhttps://doi.org/10.2118/179134-MS.\nBerard, T., Desroches, J., Yang, Y. et al. 2015. High-Resolution 3D Struc-\ntural Geomechanics Modeling for Hydraulic Fracturing. Paper pre-\nsented at the SPE Hydraulic Fracturing Technology Conference, The\nWoodlands, Texas, 3\u20135 February. SPE-173362-MS. https://doi.org/10.2118/173362-MS.\nHoward, G. C. and Fast, C. R. 1957. Optimum Fluid Characteristics for\nFracture Extension. Drilling and Production Practices, New York, 1\nJanuary, API-57-261, 261-270 (Appendix by Carter, E. D.). American\nPetroleum Institute.\nMarongiu Porcu, M., Lee, D., Shan, D. et al. 2016. Advanced Modeling of\nInterwell-Fracturing Interference: An Eagle Ford Shale-Oil Study."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae28e06-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-DISC Fracture Compliance Discussions.pdf", "page": 0, "chroma:document": "SPE J. 21(5): 1567\u20131582. SPE-174902-PA. https://doi.org/10.2118/\n174902-PA.\nCopyright VC2017 Society of Petroleum EngineersJ179725 DOI: 10.2118/179725-PA Date: 11-May-17 Stage: Page: 1 Total Pages: 3\nID:jaganm Time: 19:06 I Path: S:/J###/Vol00000/170027/Comp/APPFile/SA-J###1700272017 SPE Journal 1"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae28eba-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-DISC Fracture Compliance Discussions.pdf", "page": 1, "chroma:document": "McClure, M. W., Jung, H., Cramer, D. D. et al. 2016. The Fracture-Com-\npliance Method for Picking Closure Pressure From Diagnostic Frac-ture-Injection Tests. SPE J. 21(4): 1321\u20131339. SPE-179725-PA.\nhttps://doi.org/10.2118/179725-PA.\nMohamed, I. M., Nasralla, R. A., Sayed, M. A. et al. 2011. Evaluation of\nAfter-Closure Analysis Techniques for Tight and Shale Gas Forma-\ntions. Paper presented at the SPE Hydraulic Fracturing TechnologyConference, The Woodlands, Texas, 24\u201326 January. SPE-140136-MS.\nhttps://doi.org/10.2118/140136-MS.\nWeng, X., Kresse, O., Cohen, C. -E. et al. 2011. Modeling of Hydraulic-\nFracture-Network Propagation in a Naturally Fractured Formation.\nSPE Prod & Oper 26(4): 368\u2013380. SPE-140253-PA. https://doi.org/\n10.2118/140253-PA.J179725 DOI: 10.2118/179725-PA Date: 11-May-17 Stage: Page: 2 Total Pages: 3\nID:jaganm Time: 19:06 I Path: S:/J###/Vol00000/170027/Comp/APPFile/SA-J###1700272 2017 SPE Journal"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae28f50-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-DISC Fracture Compliance Discussions.pdf", "page": 2, "chroma:document": "Reply to the Discussion of The\nFracture-Compliance Method for\nPicking Closure Pressure From\nDiagnostic Fracture-Injection Tests\nMark McClure, McClure Geomechanics; Hojung Jung and Mukul Sharma, University of Texas at Austin;\nand Dave Cramer, ConocoPhillips\nMcClure et al. (2016) provides a mathematically consistent\ndescription of an ideal diagnostic fracture-injection test (DFIT)\npressure transient. We set out the governing and constitutive\nequations of the coupled \ufb02uid-\ufb02ow and fracture-mechanics prob-lem, use realistic input parameters, and solve the equations. The\nresult is a close match to a real DFIT pressure transient. Our work\nis the \ufb01rst to provide a single continuous calculation that realisti-\ncally describes an entire DFIT pressure transient\u2014before, during,\nand after closure. Previous investigators have focused on onlyparts of the transient and/or have interpolated between before-clo-\nsure and after-closure solutions without providing a mathematical\ndescription of the closure process itself."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae28fd2-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-DISC Fracture Compliance Discussions.pdf", "page": 2, "chroma:document": "description of the closure process itself.\nThe appendix in McClure et al. (2016) derives equations\ndemonstrating that the effect of closure on the pressure derivative\ndepends on the relative effect on system compliance (which\ndecreases as the asperities in the fracture come into contact) and\non the leakoff rate from wellbore/fracture storage (which is\nrelated to the dimensionless fracture conductivity as closure\noccurs). If the compliance effect dominates, then the pressure\ndecreases more rapidly in response to closure. If the fracture con-\nductivity effect dominates, the pressure decreases less rapidly in\nresponse to closure. These principles should form the basis\nfor DFIT interpretation, whether or not more-complicated (and\ndif\ufb01cult to validate) \u201cnonideal\u201d processes are believed to be tak-\ning place.\nA key aspect of our model (and difference from most prior\nwork) is that we allow the fracture to have residual aperture after\nthe aperities come into contact. Is this reasonable? The strongest"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2905e-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-DISC Fracture Compliance Discussions.pdf", "page": 2, "chroma:document": "proof is that linear \ufb02ow regimes are routinely observed after clo-\nsure in DFITs. Linear \ufb02ow regimes require an in\ufb01nite conductiv-\nity fracture. In McClure et al. (2016), we provide a brief\ncalculation showing that residual aperture of only tens of microns\nis suf\ufb01cient for a fracture to maintain in\ufb01nite conductivity in a\nformation with very low permeability. There is a large scienti\ufb01c\nliterature demonstrating the ability of mechanically closed frac-tures to store and conduct \ufb02uid. We used a fracture closure law\nand input values based on this literature.\nIn their discussion, Marongiu Porcu et al. state that our paper\ndoes not address the wide range of \u201cnonideal\u201d processes that are\npossible during DFITs. In a follow-up paper (Jung et al. 2016),we performed simulations of a variety of nonideal processes,\nincluding near-wellbore tortuosity and opening of transverse frac-\ntures. These results add signi\ufb01cant new insight, but none of them\ncontradict the basic \ufb01ndings of McClure et al. (2016). The simula-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae290ea-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-DISC Fracture Compliance Discussions.pdf", "page": 2, "chroma:document": "tions demonstrate that near-wellbore tortuosity causes a rapiddrop in pressure at the start of the shut-in period, before the tran-\nsient settles into a \u201cnormal\u201d preclosure trend (Jung et al. 2016,\nFig. 17). This causes an early peak in G/C2dP/dGbefore the main\nclosure [as noted in McClure et al. (2016) and by Barree and\nMiskimins (2016)]. Jung et al. (2016) showed that mechanical\nopening/closing of transverse fractures away from the well is\nunlikely to occur unless there is very low stress anisotropy. If it\ndoes occur, the transverse fractures will close early in the tran-sient. This \ufb01nding contradicts the conventional DFIT interpreta-\ntion that \u201cclosure of transverse fractures\u201d causes G/C2dP/dG to\ncurve upward later in the transient (Barree and Miskimins 2016).\nHowever, review of the literature indicates that this conventional\ninterpretation is based on heuristic arguments, not a rigorous\nmathematical solution to the equations governing the problem.\nMarongiu P"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2916c-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-DISC Fracture Compliance Discussions.pdf", "page": 2, "chroma:document": "Marongiu P\norcu et al. note in their discussion that we did not\nconsider nonideal leakoff processes and cite discussion from\nBarree and Miskimins (2016). Barree and Miskimins (2016) state\nthat they will not \u201cdwell on derivative calculation methods and\ntheoretical elastic model derivations\u201d and instead \u201cpresent some\nreal physical mechanisms.\u201d They provide a list of nonideal proc-\nesses that could hypothetically occur during a DFIT. Merely pro-\nviding a list of alternative hypotheses does not refute our analysis;\nthey do not provide mathematical analysis or validating data.\nMarongiu P et al. also state in their discussion that our\nnumerical model is incapable of handling pressure-dependent\nleakoff or multiple closures. The model can handle all these proc-\nesses, as addressed in the follow-up paper Jung et al. (2016).\nMarongiu Porcu et al. note that some of our simulations had a"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae291f8-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-DISC Fracture Compliance Discussions.pdf", "page": 2, "chroma:document": "Marongiu Porcu et al. note that some of our simulations had a\ndiscontinuity in the pressure derivative at closure. As stated in thepaper, those simulations intentionally used unrealistic input pa-rameters and were included for sensitivity analysis and to demon-strate the effect of compliance on the pressure trend; they were notintended to be realistic. We also provided simulations that appearvery realistic and do not have discontinuity in the pressure deriva-tive, including one simulation that closely matches \ufb01eld data.\nFinally, Marongiu Porcu et al. state that our closure picks can-\nnot be reconciled with \u201cstate-of-the-art mechanical Earth models\u201dand cite two papers. We reviewed those papers and did not identify\nany discussion that refuted our results.\nA convincing theory of DFIT transients should lay out the gov-\nerning and constitutive equations of \ufb02uid \ufb02ow and fracture\nmechanics, solve them, and provide a match to \ufb01eld data with a\nsingle, continuous calculation (before, during, and after closure)."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae29284-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-179725-DISC Fracture Compliance Discussions.pdf", "page": 2, "chroma:document": "Our work is the \ufb01rst to have demonstrated this. We stand by our\nconclusions.\nReferences\nBarree, R. D. and Miskimins, J. L. 2016. Physical Explanation of Non-\nLinear Derivatives in Diagnostic Fracture Injection Test Analysis.\nPaper presented at the SPE Hydraulic Fracturing Technology Con-\nference, The Woodlands, Texas, 9\u201311 February. SPE-179134-MS.\nhttps://doi.org/10.2118/179134-MS.\nJung, H., Sharma, M. M., Cramer, D. D. et al. 2016. Re-examining Inter-\npretations of Non-Ideal Behavior During Diagnostic Fracture InjectionTests. Journal of Petroleum Science and Engineering 145: 114\u2013136.\nhttps://doi.org/10.1016/j.petrol.2016.03.016.\nMcClure, M. W., Jung, H., Cramer, D. D. et al. 2016. The Fracture-Com-\npliance Method for Picking Closure Pressure From Diagnostic Frac-\nture-Injection Tests. SPE J. 21(4): 1321\u20131339. SPE-179725-PA.\nhttps://doi.org/10.2118/179725-PA.J179725 DOI: 10.2118/179725-PA Date: 11-May-17 Stage: Page: 3 Total Pages: 3"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
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created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae29392-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 0, "chroma:document": "396 August 2009 SPE Production & OperationsHolistic Fracture Diagnostics: Consistent \nInterpretation of Prefrac Injection Tests \nUsing Multiple Analysis Methods\nR.D. Barree,  SPE, and V .L. Barree,  Barree & Associates LLC; and D.P . Craig,  SPE, Halliburton\nCopyright \u00a9 2009 Society of Petroleum Engineers\nThis paper (SPE 107877) was accepted for presentation at the 2007 SPE Rocky Mountain \nOil & Gas Technology Symposium, Denver, 16 \u201318 April, and revised for publication. Original manuscript received for review 12 February 2007. Revised manuscript received for review 2 April 2008. Paper peer approved 4 April 2008.Summary\nSince the introduction of the G-function derivative analysis, prefrac"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2941e-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 0, "chroma:document": "diagnostic injection tests have become a valuable and commonly used technique. Unfortunately, the technique is frequently misapplied or misinterpreted, leading to confusion and misdiagnosis of fracturing parameters. This paper presents a consistent method of analysis of the G-function, its derivatives, and its relationship to other diagnostic techniques including square-root(time) and log( /H9004p\nwf)-log( /H9004t) plots \nand their appropriate diagnostic derivatives.\nFour field test examples are given for the most common diag-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae294aa-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 0, "chroma:document": "Four field test examples are given for the most common diag-\nnostic curve signatures. These show how multiple analysis methods can be applied to consistently interpret closure pressure and time, as well as pre- and post-closure flow regimes and reservoir proper-ties from the test data. The cases include normal constant-area and constant permeability leakoff, pressure dependent fissure leakoff, fracture tip extension, and variable fracture storage. In some cases conventionally accepted analysis methods, such as the Sqrt(time) plot, can lead to misleading interpretations. A single consistent approach to analysis is described for each case. The example cases can be used to build a foundation for consistent and less ambiguous analysis of any complex fracture injection/falloff test.\nIntroduction\nPrefrac diagnostic injection test analysis provides critical input data"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2952c-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 0, "chroma:document": "for fracture design models, and reservoir characterization data used to predict post-fracture production. An accurate post-stimulation production forecast is necessary for economic optimization of the fracture treatment design. Reliable results require an accurate and consistent interpretation of the test data. In many cases closure is mistakenly identified through misapplication of one or more analy-sis techniques. In general, a single unique closure event will satisfy all diagnostic plots or methods. All available analysis methods should be used in concert to arrive at a consistent interpretation of fracture closure.\nRelationship of the pre-closure analysis to after-closure analysis"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae295b8-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 0, "chroma:document": "results must also be consistent. To correctly perform the after-closure analysis the transient flow regime must be correctly identified. Flow regime identification has been a consistent problem in many analyses. There remains no consensus regarding methods to identify reservoir transient flow regimes after fracture closure. The method presented here is not universally accepted but appears to fit the gen-erally assumed model for leakoff used in most fracture simulators.\nFour examples are presented to show the application of mul-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2963a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 0, "chroma:document": "Four examples are presented to show the application of mul-\ntiple diagnostic analysis methods. The first illustrates the expected behavior of normal fracture closure dominated by matrix leakoff with a constant fracture surface area after shut-in. The second example shows pressure dependent leakoff (PDL) in a reser-voir with pressure-variable permeability or flow capacity, usually caused by natural or induced secondary fractures or fissures. The third example shows fracture tip extension after shut-in. These cases generally show definable fracture closure. The fourth example shows what has been commonly identified as fracture height recession during closure, but which can also indicate vari-\nable storage in a transverse fracture system.\nFor each example the analysis will be demonstrated using the"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae296c6-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 0, "chroma:document": "For each example the analysis will be demonstrated using the \nG-function and its diagnostic derivatives, the sqrt(time) and its derivatives, and the log-log plot of pressure change after shut-in and its derivatives (Nolte 1979; Castillo 1987; Barree and Mukherjee 1996; Barree 1998). When appropriate, the after-closure analy-sis is presented for each case, as is an empirical correlation for permeability from the identified G-function closure time (Talley et al. 1999). A critical part of the analysis is the realization that there is a common event indicating closure that should be consis-tently identified by all diagnostic methods. To reach a conclusion all analyses must give consistent results. \nThe goal of this paper is to provide a method for consistent"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae29752-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 0, "chroma:document": "The goal of this paper is to provide a method for consistent \nidentification of after-closure flow regimes, an unambiguous frac-ture closure time and stress, and a reasonable engineering estimate of reservoir flow capacity from the pressure falloff data, without requiring assumptions such as a known reservoir pressure. Other methods, based on sound transient test theory, require pressure dif-ference curves based on the observed bottomhole pressure during falloff minus the \u201cknown\u201d reservoir pressure (Talley et al. 1999; Craig and Blasingame 2006). While these methods are technically correct they can lead to confusing results at times, especially in low permeability reservoirs when pore pressure is difficult to determine accurately prior to stimulation.\nThis is not a transient test analysis paper but is intended to"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae297d4-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 0, "chroma:document": "This is not a transient test analysis paper but is intended to \npresent a practical approach to analysis of real, and frequently marginal-quality, prefracture field test data. The techniques applied are based on some transient test theory. Some of the results pre-sented here are still under debate and development. The methods shown have been tested and, we believe, proven in the analysis of hundreds of tests. Application of these methods provides consis-tent analysis that helps to avoid misinterpretation of falloff data and gives the most useful information available from diagnostic injection tests.\nStep-rate injection tests and their analysis are not included in the \nscope of this paper. Determination of the pressure-dependent leakoff coefficient is also not described here, because it has been previ-ously reported (Barree and Mukherjee 1996; Barree 1998). Only the analysis of pressure decline following shut-in of a fracture-rate injection test is considered.\nTransient Flow Regimes During and After"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae29860-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 0, "chroma:document": "Transient Flow Regimes During and After \nFracture Closure\nSeveral transient flow regimes may occur during a falloff test after \ninjection at fracture rate. The major flow regimes are graphically illustrated in the classic paper by Cinco-Ley and Samaniego (1981). \nImmediately after shut-in the pressure gradient along the length \nof the fracture dissipates in a short-duration linear flow period. In a long fracture in low permeability rock the initial fracture linear flow can be followed by a bi-linear flow period with the linear flow transient persisting in the fracture while reservoir linear flow occurs simultaneously. After the fracture transient dissipates, the reservoir linear flow period can continue for some time, depend-ing on the permeability of the reservoir and the volume of fluid stored in the fracture and subsequently leaked off during closure. After closure the pressure transient established around the fracture propagates into the reservoir and transitions into elliptical, then"}


