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1. INTRODUCTION  
Diagnostic fracture injection tests ( DFIT) involve 
pumping a fluid (typically water), at a constant rate for a 
short period of time, creating a relatively small hydraulic 
fracture before the well is shut in. The pressure transient 
data after shut -in is analyzed to obtain in -situ stresses 
and reservoir properties. In addition, the net pressure 
trends in a DFIT can be also used to infer the induced 
fracture complexity in different geological settings 
(Potocki 2012) and stress -dependent un -propped fracture 
conductivity can be estimated through forward  modeling 
or inverse calculation ( Wang and Sharma 2018 ). This 
valuable information  obtained from DFITs provides key 
input parameters for modeling hydraulic fracture 
propagation ( Wang 2015; Wang 2016; Wang et al. 2016 ), 
stimulation design (Ramurthy et al. 2011), development 
of reservoir models  (Mirani et al. 2017; Loughry et al. 
2015 ; Wang 2017 ) and post -fracture analysis (Fu et al. 
2017) .


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2017) . 
The advent of fracturing pressure decline analysis was 
pioneered by the work of Nolte (1979 and1986). With 
the assumptions of power -law fracture growth, 
negligible spurt loss, constant fracture compliance (or 
stiffness), constant fracture surface area afte r shut -in and 
Carter’s leak -off (a constant pressure boundary 
condition), a remarkably simple and useful equation for the pressure decline can be obtained based on a G -
function approach. Wang and Sharma (2017b) showed  
that that the traditional “tangent lin e method ( Barree et al. 
2009 ) ” undere stimate s the closure stress while the  
“compliance method ” (McClure et al. 2016 ) estimates 
the onset of closure rather than the  closure stress. Based 
on their analysis, a new “varia ble compliance method” 
was proposed to  estimate the closure stress. Wang and 
Sharma (2018) further confirmed that fracture 
compliance is pressure -dependent during closure , and 
the mechanical ly closed fracture is still subject to leak -


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the mechanical ly closed fracture is still subject to leak -
off. The stress -dependent un -propped fracture 
conductivity c an be estimated by analysis of DFIT data.  
Previous papers  have been mainly focused on planar 
fracture s in homogen eous reservoirs, and relatively little 
work has been done to interpret DFIT data in naturally 
fractured reservoirs with fracture pressure depe ndent 
leak-off (FPDL) and variable fracture compliance. In 
this study, we extend our previous work to investigate 
pressure transient behavior in naturally fractured 
reservoir s.  
2. THE GOVERNING EQUATI ONS  
2.1. 1D Fracture Pressure Dependent Leak -off 
with Variable Compliance              
ARMA 1 8–225                                                                
 
Estimating Fracture Closure Stress in Naturally Fractured  
Reservoirs with Diagnostic Fracture Injection Test s 
 
HanYi Wang  
The University of Texas at Austin, Austin, Texas, USA  
Mukul M. Sharma


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Mukul M. Sharma  
The University of Texas at Austin, Austin, Texas, USA  
 
Copyright 20 18 ARMA, American Rock Mechanics Association  
This paper was prepared for presentation at the 52nd US Rock Mechanics / Geomechanics Symposium held in Seattle , Washington , USA, 17–20 
June  2018. This paper was selected for presentation at the symposium by an ARMA Technical Program Committee based on a technical and 
critical review of the paper by a minimum of two technical reviewers. The material, as pres ented, does not necessarily reflect any position of 
ARMA, its officers, or m embers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the 
written consent of ARMA is prohibited. Permission to reprodu ce in print is restricted to an abstract of not more than 200 words; illus trations may 
not be copied. The abstract must contain conspicuous acknowledgement of where and by whom the paper was presented.


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ABSTRACT:  Diagnostic Fracture Injection Tests (DF IT), which have also been referred to as Injection -Falloff Tests, Fracture 
Calibration Tests  or Mini -Frac Tests in the literature, have evolved into a commonly used technique to obtain in -situ stress. 
Simplifying  assumption s used  in traditional G -function model often lead to an underestimation of the  closure stress, even for a  
planar fracture geometry. When a DFIT  is conducted  in naturally fractured reservoirs, the stimulated natural fractures can either 
alter the effective reservoir permeability within the distance of investigation or interact with  the hydraulic fracture to form a 
complex fracture geometry. In this study, we p resent a new pressure transient model for DFIT analysis in naturally fractured 
reservoirs, and it is shown that using established methods to pick closure stress often lead to significant over or underesti mates,


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regardless of whether the reservoir is heavil y fractured or sparsely fractured. Our proposed “variable compliance method” gives a 
much more accurate and reliable estimation of closure stress in both homogenous and naturally fractured reservoirs.  
 
 
 
 
  
 
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The transient pressure response during fracture closure is 
derived using the following assumptions:  
1. Reservoir  is isotropic and homogeneous and contains 
a single slightly compressible fluid.  
2. The fluid viscosity, formation porosity and total 
compressibility are independent of pressure.  
3. Reservoir permeability is low so that poroelastic 
effects caused by fluid leak -off are negligible  
4. Gravity effects are negligible and pressure gradients 
are small thorough out the reservoir.  
5. Mechanical ly closed fracture has residual fracture 
width so it is still subject to leak -off (Wang and 
Sharma 2018) and the fracture leak -off surface area 
remains unchanged.  
6. Fracture pressure is uniformly distributed inside the 
fracture.  
7. The pore pressure disturbance c aused by fracture 
propagation is negligible. This means that fluid leak -
off during pumping is very small and the duration of 
injection is very short (typically 3 -5 minutes) while


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injection is very short (typically 3 -5 minutes) while 
the total shut -in time can be hours, days or even 
weeks.  
8. Leak -off is linear  and perpendicular to the fracture 
surface and pressure interference among fractures and 
late time radial flow has not been developed yet.  
Assuming 1D linear Darcy flow and a slightly 
compressible, single -phase fluid in the reservoir, the 
differential form of the mass balance can be written as:  
∂P
∂t=1
𝜇𝑓𝜙𝑐𝑡∂
∂x(k∂P
∂x)                                  (1) 
where P is the pressure, k is formation permeability, 
which can be a function of P. μf is fluid viscosity, ϕ is 
formation porosity and ct is total formation 
compressibility. From a material balance perspective 
(fluid compressibility is negligible compared to that of 
the fracture), the total rate of fluid leak -off into the 
formation,  qf , after shut -in equals the rate of shrinkage 
of total f racture volume,  Vf, as pressure declines:  
𝑞𝑓=− 𝑑𝑉𝑓
𝑑𝑡                                      (2)


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𝑞𝑓=− 𝑑𝑉𝑓
𝑑𝑡                                      (2) 
Fracture stiffness, which is the reciprocal of fracture 
compliance, is defined as (Wang and Sharma 2017a; 
Wang et al. 2018):  
𝑆𝑓=𝐴𝑓𝑑𝑃𝑓
𝑑𝑉𝑓                                              (3) 
where 𝑃𝑓 is pressure inside fracture and 𝐴𝑓 is  half of the 
total fracture surface area (only account for 1 wall). 
Substituting  Eq.(3) into Eq.(2), we have  
𝑞𝑓=− 𝐴𝑓𝑑𝑃𝑓
𝑆𝑓𝑑𝑡                                      (4) 
At the boundary of the fracture surface, both Darcy’s 
law and material balance have to be honored:  𝑞𝑓=− 2𝐴𝑓𝑘
𝜇𝑓 𝑑𝑃
𝑑𝑥=−  𝐴𝑓𝑑𝑃𝑓
𝑆𝑓𝑑𝑡 at x=0        (5) 
Rearranging Eq. (5), we get the boundary condition at 
the fracture surface:  
𝑘
𝜇𝑓 𝑑𝑃
𝑑𝑥=1
2𝑆𝑓𝑑𝑃𝑓
𝑑𝑡 at x=0                     (6) 
With initial condition (disregarding the pressure 
disturbance during the short injection period)  
  𝑃=𝑃0 at t=0,x>0                              (7) 
  𝑃=ISIP  at t=0,x=0                           (8) 
where 𝑃0 is the initial pore pressure and ISIP  is the


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where 𝑃0 is the initial pore pressure and ISIP  is the 
instant shut -in pressure. The governing equation  Eq.(1), 
plus the initial condition Eq. (7) -(8) and boundary 
condition Eq.(6) uniquely describe the pressure transient 
behavior during DFITs. To account for wellbore storage 
effects, the fracture stiffness 𝑆𝑓 needs to be replaced by 
the fracture -wellbore system stiffness, which is defined 
as: 
𝑆𝑠=    𝑆𝑓𝐴𝑓
𝐴𝑓+𝑆𝑓𝑉𝑤 𝑐𝑤                                   (9) 
where 𝑉𝑤 is the wellbore volume and 𝑐𝑤 is the water 
(injection fluid in the wellbore) compressibility.   
2.2. Fracture Stiffness of Complex Frac ture 
Geometry  
Because fracture stiffness (or compliance) represents 
surface normalized compressibility, for a fracture with 
complex fracture geometry, the total fracture stiffness is 
controlled by the stiffness of multiple individual fracture 
segments, as illustrated in Figure.1.  
 
Fig. 1 Illustration of complex fracture geometry . 
The complex fracture is formed by individual segments


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The complex fracture is formed by individual segments 
with different orientations and fracture surface area. Let 
the ith fracture surface area of one wall denoted as Af,i 
and its pressure -dependent stiffness denoted as Sf,i, the 
total fracture stiffness  can be determined from the 
stiffness of each individual fracture segment (analogous 
to capacitors in  parallel s): 
𝑆𝑓=𝐴𝑓
𝐴𝑓,1
𝑆𝑓,1+𝐴𝑓,2
𝑆𝑓,2+⋯+𝐴𝑓,𝑛
𝑆𝑓,𝑛                       (10) 
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Where,  
𝐴𝑓=𝐴𝑓,1+𝐴𝑓,2+⋯+𝐴𝑓,𝑛                    (11) 
2.3. Pressure Dependent Permeability  
In previous literature (Barree et al. 2009), the 
phenomenon of pressure -dependent permeability is often 
called as pressure -dependent leak -off ( PDL), however, 
this is not a n appropriate term, because leak -off is 
always pressure -dependent, even  if the permeability is 
constant (Wang and Sharma 2017b). There are many 
equations  that can be used to express the  formation 
effective permeability as a function of pressure, such as 
a linear, power law or hyperbolic relationship. To make 
a general case and use as few  free parameters as 
possible, we can use a general formula to relate 
formation permeability and pressure:  
k(P)=(𝑘𝑒−𝑘0)𝑒𝑐−𝑒𝐼𝑆𝐼𝑃 −𝑃
𝐼𝑆𝐼𝑃 −𝑃𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙𝑐
𝑒𝑐−1+𝑘0      (12) 
where 𝑘𝑒 is the effective reservoir  permeability at the 
end of pumping and 𝑘0 is the original reservoir 
permeability. 𝑃𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙  is the pressure where reservoir 
permeability drops to its original value a nd c a free


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permeability drops to its original value a nd c a free 
parameter controls the evolution of permeability. 𝑘𝑒 is 
larger than 𝑘0 because the surrounding permeability is 
enhanced by stimulated fracture networks during 
pumping. After the pressure drops to or below 𝑃𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 , 
the stimulated f racture networks close back to the 
original in -situ state and the effective permeability 
return s to its original value.  Figure.2 demonstrates the 
pressure -dependent permeability for different values of c.  
 
Fig. 2 Illustration of pressure -dependent permeab ility with 
different values of c  
2.4. Pressure Dependent Fracture Compliance  
Fracture closure is a gradual process with increasing 
fracture stiffness (or decreasing fracture compliance) as 
fracture closes from tip to wellbore or closes from higher 
stress region  to lower stress region in layered formations. 
A method t o obtain a pressure -dependent fracture 
compliance based on fracture geometry, rock properties


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compliance based on fracture geometry, rock properties 
and surface roughness has been well discussed by Wang 
and Sharma (2017a) and Wang et al. (2018). The 
influence of surface roughness on fracture compliance is described by a contact law relat ing the  fracture width to 
the net closure stress for fractured rocks (Willis -Richards 
et al. 1996):  
  σc=σref
9(w0
wf−1)for  wf≤w0     (17) 
where wf is the fracture aperture and, w0 is the contact 
width, which represents the fracture aperture when the 
contact normal stress is equal to zero, σc is the contact 
normal stress on the fracture, and σref is a contact 
reference stress, which denotes the effective normal 
stress at which the aperture is reduced by 90%.  
3. DFIT IN NATURALLY FR ACTURED 
RESERVOIRS  
All reservoirs are fractured to some degree. D epending 
on the density  and dimension s of natural fractures and 
the location where the DFIT is done, the natural 
fractures can impact hydraulic fracture propagation and


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fractures can impact hydraulic fracture propagation and 
the adjacent flow capacity. If the reservoir is heavily 
fractured and these natural fra ctures are well -connected, 
as shown in Figure 3, then the formation can be 
considered as homogenous within the distance of 
investigation and the effective permeability is controlled 
by both fracture and matrix. High -pressure  fluid 
injection  during the pump ing period enhance s the local 
effective permeability because it widens the natural 
fractures. However, as pressure declines during shut -in, 
the effective permeability will gradually decline until it 
reaches the original value. The original effective 
permea bility does not necessar ily represent the matrix 
permeability, because the existence of natural fractures 
may already have enhanced the permeability.  
 
Fig. 3 Heavily fractured reservoir with well -connected natural 
fractures  
 
If the reservoir is sparsely or moderately fractured, and


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If the reservoir is sparsely or moderately fractured, and 
the stimulated natural fractures are discrete and not well -
connected, then these natural fractures or isolated 
fracture networks will not change the overall flow 
capacity in the reservoir  (Wang 2017) , but they can 
interact with  hydraulic fractures, and generate  a complex 
fracture geometry, as shown in Figure  4. Under this 
scenario, the fracture compliance and leak -off surface 
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area are impacted by the sequential closure of individual 
fracture segment s or branch es, where the branc hes with 
the highest normal stress perpendicular to local fracture 
orientation will close first, and the fracture segments that 
open against the minimum principal stress will be the 
last to close. If  the hydraulic fracture does not initiate 
along the maxim um principal stress direction because of 
the local hoop stress around perforations, non -planar 
fracture s can also be formed. In this section, we are 
going to investigate the pressure transient behavior 
under these scenarios and discuss the implications of 
estimating closure stress using  different methods . 
 
Fig. 4 Low to Moderate Fractured Reservoir with Discrete 
Natural Fractures  
3.1. Base Case  
Before we embark on investigating the pressure transient 
behavior with closing  fracture in fractured reservoirs, 
we’ll first examine pressure decline behavior for a planar 
fracture and discuss the causes for the discrepancy for


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fracture and discuss the causes for the discrepancy for 
estimated closure stress using  established methods. 
Table 1  shows the input parameters for the Base Case 
Scenario . 
Table 1. Input parameters for the Base Case scenario  
Fracture type  PKN 
Fracture height  12 m  
Fracture half -length  50 m  
Pumping time  5 min  
ISIP 40 MPa  
Minimum in -situ stress  35 MPa  
Initial pore pressure  20 MPa  
Young's modulus  20 GPa 
Total compressibility  6.3e -4 MPa-1 
In-Situ fluid viscosity  0.3 cP  
Formation original permeability  0.0005 md 
Poisson's Ratio  0.25  
Initial porosity  0.03  
Contact width  of asperities  2 mm  
Contact reference stress  of asperities  5 MPa  
 
With the known fracture geometry, rock modulus, 
minimum in -situ stress and surface roughness 
(represented by two contact parameters 𝑤0 and 𝜎𝑟𝑒𝑓), 
the fracture stiffness and volume evolution can be 
uniquely determined at different fluid pressure s (Wang


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uniquely determined at different fluid pressure s (Wang  
and Sharma 2017a; Wang et al. 2018), the results are shown in Figure 5. As can be seen, when the fluid 
pressure inside the fracture is relatively high, the fracture 
volume declines linearly with pressure under constant 
fracture stiffness (or compliance). However, as the 
pressure declines to a certain level the fracture volume 
and pressure depart from a linear relationship, and 
fracture stiffness start s to increase noticeably. In the end, 
when the fluid pressure drops to a sufficient low level, 
the residual  fracture volume (supported by asperities) 
becomes more or less insensitive to pressure drop with 
extremely high fracture stiffness (low compliance). This 
is because when the pressure inside the fracture is 
relatively high, it is still wide open and its st iffness is a 
constant . However, as pressure continues to drop, the 
fracture will gradually close from its edges to its center, 
which increase s the fracture stiffness gradually.


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which increase s the fracture stiffness gradually.  
 
Fig. 5 Illustration of the fracture stiffness and volume 
evolution from modeling  
Figure.6 shows  the normalized tiltmeter response (its 
slope is proportional to fracture compliance or the 
reciprocal of fracture stiffness) plotted against wellbore 
pressure during the shut -in period of 2B well at three 
different stations from the  GRI/DOE M -site. The 
tiltmeter demonstrates that soon after shut -in, the 
measured displacement declines linearly with pressure 
(roughly constant fracture stiffness). After the pressure 
declines to a certain level, the measured displacement vs 
pressure depa rts from a linear relationship. As pressure 
continues declining, more and more fracture surface area 
comes into contact with the rough fracture surfaces and 
asperities, and the fracture stiffness increases gradually. 
This field measurement is consistent wi th the general 
trend from out modeling results shown in Figure.5. It is


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trend from out modeling results shown in Figure.5. It is 
challenging to estimate closure stress based on 
mechanical closure from tiltmeter measurement. If we 
pick the closure stress using  the “tangent line method” 
on tiltmeter measurement (C raig et al. 2017), the closure 
stress is 15 MPa, and if we pick the closure stress using  
the “compliance method”, the closure stress is 21 MPa. 
There is a 6 MPa discrepancy between these two 
methods. As discussed by Wang and Sharma (2017b), 
the “tangent li ne method” only marks the end of fracture 
storage dominated flow, so it always underestimate s 
closure stress while the “compliance method” can 
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overestimate closure stress,  because it only marks the 
beginning fracture closure on edges or tips. The 
“variable  compliance method” proposed by Wang and 
Sharma (2017b) provides an alternative way to estimate 
closure stress that not only accounts for the difference 
between two endpoints  (fracture compliance start to 
change and fracture compliance become small enough to 
be considered as negligible), but also compensates for 
the impact of leak -off, which  is reflected on G -function 
time or the square root of time.  
 
Fig. 6 Tiltmeter measurement from 2B well of GRI/DOE M -
site project  
To illustrate the benefit  of the “variable compliance 
method” and compare the estimated closure stress from 
different methods, the pressure -dependent fracture 
stiffness from Figure.  5 is used as a boundary condition 
for Eq.(6) to simulate the pressure decline response, all 
other parameter s are provided in Table 1. Figure.7 shows 
the pressure decline and its derivatives on the G -function


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the pressure decline and its derivatives on the G -function 
and square root of time plots. Compared to the input 
closure stress of 35 MPa, the estimated closure stress by  
the “tangent line method” is 32.2 MPa ( -2.8 MPa error), 
the estimated closure stress by the “compliance method” 
is 36.8 MPa (1.8 MPa error), and the estimated closure 
stress by  the “variable compliance method” is 35.2 MPa 
(0.2 MPa error). It is clear that the closure stress 
estimated by the “variabl e compliance method” is more 
reliable. Detailed comparisons with different fracture 
geometry , surface roughness  and reservoir properties 
have been presented by Wang and Sharma (2017b), and 
hence will not be discussed further.  
Figure.8 show s the pressure profiles in the formation 
away from the fracture surface at different shut -in times. 
Together with Fig.6, we can notice that the closure time 
is around 8.85 hr when fluid pressure drops to the close 
stress, and the corresponding distance of i nvestigation is


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stress, and the corresponding distance of i nvestigation is 
less than 7 m at the time of closure. This is very typical 
in low permeability formations.   
Fig. 7 Pressure decline and its derivatives of Base Case 
scenario  
 
Fig. 8 Pressure profile and distance of investigation for Base 
Case Scenario . 
Now the question remains: what unique features will 
emerge from  DFIT data in naturally fractured reservoirs 
and how do we reliably estimate closure stress under 
these contexts? Are these established methods are 
reliable ways to estimate closure stress?  In  the 
following sections, we’ll examine these cases and 
discuss the associated challenges in identify ing closure 
stress in naturally  fractured reservoir.  
3.2. Heavily Fractured Reservoir with Pressure 
Dependent Permeability  
First, let’s investigate the case wher e the reservoir is 
heavily fractured and all the stimulated fractures form 
well-connected networks that alter the effective


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well-connected networks that alter the effective 
permeability around the hydraulic fracture. The effective 
permeability is highest at the end of pumping and 
gradually declines as pr essure drops and fluid leak-offs 
into formation . Assumes the effective permeability 
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follows Eq.(12), with c=3 and 𝑃𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙  is 4.5 MPa 
below the ISIP, all the other input parameters are the 
same as the base case. Figure.9  shows the pressure 
decline and its derivatives with different values of 
enhanced effective permeability 𝑘𝑒 at the end of 
pumping. From the results, we can clearly observe that 
there exists a “bump” on the pressure derivative during 
fracture closure befor e it reaches its final peak. And in 
all cases, the pressure derivatives “bump” end at a 
fracturing pressure that is 4.5 MPa below the ISIP, 
which is the exactly value of 𝑃𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 . So the values of 
𝑃𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙  can be readily obtained from f ield DFIT data if 
it exhibits this pressure -dependent permeability 
signature. We can also observe that the higher the 𝑘𝑒, the 
large the initial slope of pressure derivatives.  
 
 
Fig. 9 Pressure decline and its derivatives with pressure -
dependent permeab ility 
Because the effect of pressure -dependent permeability


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Because the effect of pressure -dependent permeability 
masks the initial phase of fracture closure, so the 
moment when the fracture stiffness start s to change 
cannot be identified. In such cases , the “compliance 
method” is hard to apply . We can, howev er, modify our 
“variable compliance method” approach  as follows : first, 
we identify the dimensionless G -time or the square root 
of time at the end of the first pressure derivative “bump” 
(i.e., end of pressure -dependent permeability) and at the 
intersectio n of pressure derivative and a tangent line 
passing through the origin (i.e., the closure point picked 
by the “tangent line method”). Then average the values 
of these two dimensionless G -time or the square root of 
time, finally extrapolate back to the pressure curve that 
corresponds to the averaged G -time or the square root of time. Table 2 shows the estimated closures by “tangent 
line method” and “variable compliance method”. As can 
be seen, the “tangent line method” consistently


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be seen, the “tangent line method” consistently 
underestimates the closure stress. This stems from the 
fact that the “tangent line method” assumes constant 
fracture stiffness and Carter’s leak -off during closure , 
but these two assumptions are not valid (Wang and 
Sharma 2017b). The closure stress estimated by 
“variable compliance method” gives significantly lower 
errors and is a more reliable approach to estimate closure 
stress.  
Table 2. Comparison of estimated closure stress from different 
methods with pressure -depe ndent permeability  
Scenarios  Tangent Line Method  Variable Compliance Method  
Estimated  
 𝝈𝒉𝒎𝒊𝒏   Absolute 
Error  Estimated  
𝝈𝒉𝒎𝒊𝒏   Absolute 
Error  
𝑘𝑒=𝑘0 32.2 MPa  -2.8 MPa  35.2 MPa  0.2 MPa  
𝑘𝑒=5𝑘0 32.3 MPa  -2.7 MPa  34.4 MPa  0.4 MPa  
𝑘𝑒=10𝑘0 32.1 MPa  -2.9 MPa  34.5 MPa  0.5 MPa  
𝑘𝑒=20𝑘0 32.2 MPa  -2.8 MPa  34.4 MPa  0.4 MPa  
 
3.3. Discrete Natural Fractures  
In the case of sparsely or moderately fractured reservoirs,


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In the case of sparsely or moderately fractured reservoirs,  
even though  the stimulated natural fractures are d iscrete 
and not well -connected, they can interact with 
propagating hydraulic fracture and generate complex 
fracture geometry. To make modeling and analysis 
trackable, we’ll first investigate a simplified version of 
complex fracture geometry that only consist two sets of 
fractures with di fferent orientations, as shown in 
Figure.10, then more general scenarios will be discussed 
at the end. In Figure.10, we can see that the main 
hydraulic fracture (i.e., fracture set 1) is perpendicular to 
the minimum principal stress, while the natural frac tures 
(i.e., fracture set 2) are perpendicular to the maximum 
principal stress. As pressure declines inside fracture after 
shut-in, the natural fracture that open s against the 
maximum principal stress will close first, and the 
hydraulic fracture that open s against the minimum 
principal stress will be the last to closure. Here, there are


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principal stress will be the last to closure. Here, there are 
two possible scenarios are likely to happen during this 
sequential closure process: 1) the closed natural fracture 
remain hydraulic connectivity with the main hydraulic 
fracture. 2) the closed natural fracture lost the hydraulic 
connectivity with the main hydraulic fracture. We’ll 
discuss these two scenarios and their implications in 
interpreting DFIT data separately.  
3.3.1.  Closed Fracture Remain Hydraulic 
Connectivity  
If the inter section of fractures are well aligned (not 
skewed) and supported by asperities and tortuous walls, 
then it is likely that the closed natural fractures can 
maintain  hydraulic connectivity to the open fracture.  
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Fig. 10 Top view  of complex fracture geometry with two sets 
of fracture s. 
In this case, the sequential closure natural fractures or 
branches of a complex fracture will not alter the total 
leak-off surface area,  but will change the overall fracture 
stiffness evolution. Ass ume a complex fracture that 
consists of two sets of fracture, as shown in Figure.10. 
The total surface areas are 𝐴𝑓1 and  𝐴𝑓2 for fracture set 
1 and fracture set 2, respectively. The maximum 
principal stress is 38 MPa and the minimum principal 
stress is 33 MPa. We further assume that all fracture 
closes like a PKN fracture. Figure.11 shows the fracture 
stiffness evolution with different surface area ratio. As 
can be seen, if we only have fracture set 2 (i.e., natural 
fracture), the fracture stiffness w ill increase at 39.5 MPa, 
and if we only have fracture set 1 (i.e., hydraulic 
fracture), the fracture stiffness will increase around 34.5 
MPa. The overall fracture stiffness evolution is


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MPa. The overall fracture stiffness evolution is 
calculated using Eq.(10) with different surface area ratio. 
Because f or complex fracture geometry, the overall 
fracture stiffness is like a parallel system of capacitors, 
so the changes in fracture stiffness is not directly 
proportional to the surface area ratio, but is mostly 
influenced by the fracture set that has the highest ratio of 
surface area to stiffness.  
 
Fig. 11 Fracture stiffness evolution for a  complex fracture 
geometry (hydraulic connectivity remains after closure)  
Figure.12 shows the pressure decline and its derivatives 
of complex fracture geometry when closed  natural 
fractures remain their hydraulic connectivity to the main hydraulic fracture. Because the fracture stiffness 
evolution is smooth and gradual, so the closure of 
natural fracture is undetectable on G -function or the 
square root of time plot. But fro m the pressure 
derivatives, we can notice that the larger the surface area


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derivatives, we can notice that the larger the surface area 
of natural  fracture, the sooner the pressure derivatives 
curve upward  and the faster the pressure decline during 
closure . 
 
 
Fig. 12 Pressure decline and its derivatives of complex  
fracture geometry (remain hydraulic connectivity after 
closure)  
Table 3 summarizes the estimated closure stress 
(minimum principal stress) by different methods. As 
always, the “tangent line method” underestimates the 
closure stress, while “compliance method” overestimates 
closure stress. The “compliance method” gives a much 
larger error if the surface area of natural fracture is large. 
This is because “compliance method” picks the pressure 
when natural fracture edges or tips start to closure, 
which can  be much higher than the minimum principal 
stress. The “variable compliance method” yields the 
least error.  
Table 3. Comparison of estimated closure stress of complex 
fracture geometry (hydraulic connectivity remains after 
closure)  
𝑨𝒇𝟐


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closure)  
𝑨𝒇𝟐
𝑨𝒇𝟏 Tangent Lin e 
Method  Compliance 
Method  Variable Compliance 
Method  
𝝈𝒉𝒎𝒊𝒏  
(MPa)  Error 
(MPa)  𝝈𝒉𝒎𝒊𝒏  
(MPa)  Error 
(MPa)  𝝈𝒉𝒎𝒊𝒏  
(MPa)  Error 
(MPa)  
0.5 31  -2  34.8  1.8  33.4  0.4  
1 31.1   -1.9  35.4  2.4  33.6  0.6  
5 31.6   -1.4  37.8  4.8  33.3   0.3  
10 32  -1  39.2  6.2  33.5   0.5  
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In fact, the information on fracture complexity can’t be 
detected under such cases, and compared to a single 
hydraulic fracture scenario (i.e., only fracture set 1 on 
Fig.10), the impact of natural fractures on overall 
fractures stiffness evolution can be substituted by a 
“rougher” fracture walls with larger contact width and 
higher contact reference stress (Wang and Sharma 
2017b) to obtain earlier and steeper increase in fracture 
stiffness as pressure declines. So based on before -closure 
DFIT data alone, w e may not be able to distinguish the 
causes  to the increase of fracture stiffness from fracture 
complexity or surface roughness or recession in barrier 
layers. Nevertheless, as discussed by Wang and Sharma 
(2017b), the “variable compliance method” is insen sitive 
to fracture geometry, surface roughness and reservoir 
properties, and from the above results, we see that it is 
also much less sensitive to the existence of natural 
fractures, so the “variable compliance method” is more


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fractures, so the “variable compliance method” is more 
reliable way of estimate clos ure stress, especially when 
we are not sure whether the create d fracture geometry is 
complex or planar.  
3.3.2.  Closed Fracture Lost Hydraulic 
Connectivity  
If the intersections of fractures are skewed and not well -
aligned (such as T-shaped  fracture), then it is likely that 
the closed fracture will lose its hydraulic connectivity to 
the open fracture. In such case, the sequential closure 
natural fractures or branches of a complex fracture will 
alter both the total leak -off surface area and t he overall 
fracture stiffness evolution. Assume the same complex 
fracture as discussed in the previous section. The 
maximum principal stress is 37 MPa and the minimum 
principal stress is 33 MPa. We further assume the natural 
fractures will gradually lose t heir hydraulic connectivity 
to the open hydraulic fracture when fluid pressure drops 
around 37 MPa.  Figure.13 shows the fracture stiffness


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around 37 MPa.  Figure.13 shows the fracture stiffness 
evolution with different surface area ratio when natural 
fractures lose their hydraulic connectivity after closure. 
Compared to Figure.11, we can notice that the overall 
fracture stiffness evolution is not monotonically 
increasing with declining pressure anymore . The surface 
area ratio will shape the fracture stiffness evolution 
before the closure of natural fractures, b ut once the 
pressure drops to a certain level and disconnect the 
natural fractures, the overall fracture stiffness becomes 
the fracture stiffness of the remaining hydraulic fracture.  
Figure.14 shows the pressure decline and its derivatives 
of complex fract ure geometry when closed fractures lose 
hydraulic connectivity. It can be clearly observed that 
there are two peaks on pressure derivatives curve, the 
first one is associated with the closure process of natural 
fractures, and the second one is associated w ith the 
closure process of the hydraulic fracture.


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closure process of the hydraulic fracture.  
  
Fig. 13 Pressure decline and its derivatives of complex 
fracture geometry (lose hydraulic connectivity after closure)  
 
 
Fig. 14 Pressure decline and its derivatives of complex 
fracture geometry (lose hydraulic connectivity after closure)  
Table 4. Comparison of estimated closure stress of complex 
fracture geometry (lose hydraulic connectivity after closure)  
𝑨𝒇𝟐
𝑨𝒇𝟏 Tangent Line 
Method  Compliance 
Method  Variable Compliance 
Method  
𝝈𝒉𝒎𝒊𝒏  
(MPa)  Error 
(MPa)  𝝈𝒉𝒎𝒊𝒏  
(MPa)  Error 
(MPa)  𝝈𝒉𝒎𝒊𝒏  
(MPa)  Error 
(MPa)  
0.5 30.5  -2.5 35.4  2.4 33.4  0.4 
1 30.6  -2.4 35.5  2.5 33.4  0.4 
5 30.7  -2.3 35. 2.7 33.6  0.6 
10 30.9  -2.1 35.8  2.8 33.6  0.6 
 
If we only analyzed the portion of data after the first 
peak of pressure derivatives and use it to estimate the 
closure stress associated with the closure of the 
hydraulic fracture, we can compare the results from 
different methods, as shown in Table 4. Ag ain, we can


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different methods, as shown in Table 4. Ag ain, we can 
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notice that “tangent line method” underestimates closure 
stress while “compliance method” overestimates closure 
stress. The “variable compliance method” is more 
reliable and gives the least errors.  
In reality, a complex fracture can consist of many 
fracture sets with different orientations and surface area . 
The approach and analysis presented for our simplified 
cases are still valid for these more general scenarios. 
Assum ing that  a complex fracture has four sets of 
fractures with equal surface area, but open against 
different in -situ stresses (37 MPa, 35 MPa, 33MPa and 
31 MPa, where 31 MPa is the minimum principal stress) 
because of their orientations, Figure.15 shows the 
evolutio n of fracture stiffness as a function of fracturing 
pressure. As can be seen, if natural fractures gradually 
lose their hydraulic connectivity around the normal 
stress that is perpendicular to that fracture surface, the 
overall fracture stiffness will exhi bit three peaks before


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overall fracture stiffness will exhi bit three peaks before 
it finally reflects the fracture stiffness of the last open 
fracture.  
 
Fig. 15 Fracture stiffness evolution of complex fracture 
geometry with four sets of fracture  
Figure.16 shows the corresponding pressure decline and 
its derivatives. As expected, four peaks emerge on the 
pressure derivatives and if we only analyze the portion 
of data after the third peak to estimate closure stress, the 
“tangent line method” underes timate s the closure stress 
by around 1 MPa. The “variable compliance method” 
gives a close r value. In general, estimated closure stress 
by the “tangent line method” can be regarded as the 
lower bound of closure stress and the closure stress  
estimated  by th e “compliance method” can be viewed as 
the upper bound of closure stress.  
4. COMPARISON WITH EXPERIMENT  
Craig et al. (2017) compared and discussed fracture 
closure stress estimated by “Compliance Method” and


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closure stress estimated by “Compliance Method” and 
“Tangent Line Method” using tiltmeter measurement 
and laboratory experiments.   
 
Fig. 16 Pressure decline and its derivatives of complex 
fracture geometry with four sets of fracture  
Figure.17 shows the G -function plots using data 
extracted from laboratory experiments from their study. 
As expected  the pressure decline is affected by tip 
extension and fluid transient behavior along the fracture 
soon after shut -in, so a brief pressure derivative spike 
often immediately follows shut -in. In many cases, this 
pressure derivative spike may mask the “open fract ure 
period ” (with constant stiffness) and make it difficult to 
locate the moment when the fracture stiffness starts to 
increase. If the closure time is not long enough, this 
early time abnormal pressure (large pressure drop during 
a short time interval) c an’t be distingu ished from the 
signature of pressure -dependent permeability, which is


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signature of pressure -dependent permeability, which is 
normally the case in laboratory experiments where the 
fracture closes in less than 1 minute. In the soft rock case 
of Figure.17, the imposed closure stress is 7 MPa, but 
the closure stress estimated by the “Tangent Line 
Method” is only 3 MPa, which is significantly lower. 
Because the early -time pressure derivative spike masks a 
large portion of before closure data, the “compliance 
method” and “variable compliance method” a re not 
applicable. In the moderately hard rock case of 
Figure.17, the imposed closure stress is 16 MPa, the 
closure stress estimated by  “Tangent Line Method” is 
14.32 MPa and the closure stress estimated by  
“compliance method” is 20.16 MPa. The closure s tress 
estimated by “variable compliance method” is 17.1 MPa 
at the averaged G -function time of 1.745. This result is 
consistent to our simulated synthetic cases presented in 
this study and previous  study (Wang and Sharma


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this study and previous  study (Wang and Sharma 
2017b) . Laboratory experiments also  show that  the 
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“variable compliance method” gives the least error and 
“Tangent Line Method” normally underestimates closure 
stress.  
 
 
Fig. 17 Laboratory fracture closure experiment for soft (upper 
figure) and moderately hard rock (lower figure) (Craig et  al. 
2017)  
5. DISCUSSION  
Examining Eq.(1)~Eq.(9) closely, it is evident that the 
pressure decline is governed by a linear Partial 
Differential Equation (PDE) system in the formation that 
is coupled with an Ordinary Differential Equation 
(ODE) at the fracture surface. The minimum in -situ 
stress is just one of many implicit factors that shape the 
evolution of the fracture pressure  in the boundary ODE. 
The causes for the error associated with “tangent line 
method” stems from the traditional G -function plot 
analys is, where the assumptions of Carter’s leak -off and 
constant fracture stiffness during closure (changes 
abruptly to infinite at closure stress) are invalid. The


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abruptly to infinite at closure stress) are invalid. The 
causes for the error associated with “ compliance method” is due to the fact that the increase of  fracture 
stiffness can happen even before fracturing pressure 
drops to minimum in -situ stress.  
For a planar fracture geometry, the fracture closes 
progressively from its edges to the center with a gradual 
increase in stiffness, if both numerical modeling (as 
shown in Fig. 5) and tiltmeter measurement s (as shown 
in Fig.6) only shows  smooth curves without a clear 
indication of closure  stress, then the pressure decline 
signature that reflects the combined effects of pressure -
dependent leak -off and fractures  stiffness evolution is 
not going to yield additional information on closure 
stress. The “variable compliance method”, provides an 
alternative way to estimate the minimum in -situ stress. 
By history matching the DFIT data, using properties of 
fracture surfa ce roughness and formation flow capacity,


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fracture surfa ce roughness and formation flow capacity, 
the ranges of un -propped fracture conductivity and 
formation permeability can be obtained for various 
fracture dimension s (Wang and Sharma 2018).  
In naturally fractured reservoirs, it is difficult to quantify 
such information through a history match. In the case of 
pressure -dependent permeability, the pressure response 
is affected by the declining effective permeability and 
the increase of fracture stiffness. Those two independent 
mechanisms are intertwined and can not be separated 
from the before -closure data, especially in the case 
where the period of pressure -dependent permeability 
spans a large portion of before -closure data. In other 
words, the information on the individual evolution of 
fracture stiffness and ef fective permeability is lost and 
cannot be recovered quantitatively from DFIT data. In 
the case of complex fracture s, if the closed fracture 
segments retain their hydraulic connectivity to the main


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segments retain their hydraulic connectivity to the main 
hydraulic fracture, we will get a smooth pressure decline 
response. In such case s, we do not know what the 
topology of the complex fracture is and we will not be 
able to constrain the fracture dimension using  a simple 
material balance calculation. Without a good estimation 
of the initial fracture geometry to star t with, a 
meaningful history match and uncertainty analysis are 
not possible. If the closed fracture segments lose 
hydraulic connectivity to the main hydraulic fracture, we 
will observe multiple pressure derivative peaks on G -
function or square  root of tim e plots. In such case s, we 
can discern, at least, how many sets of fractures (with 
different orientations) exist. However, we still have no 
knowledge of the fracture t opology, because the 
information on the individual fracture surface area and 
their stiffn ess are smeared together and can’t be 
separated. The situation can get more complicated if


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separated. The situation can get more complicated if 
some of the closed fracture segments retain their 
hydraulic connectivity to the main fracture while other s 
do not. Nevertheless, our proposed “variable compliance 
method” provides an alternative way to estimate in -situ 
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stress, which can be more reliable than established 
methods, even in naturally fractured reservoirs.  
6. CONCLUSIONS  
In this study, we  present a new pressure transient model 
for DFIT analysis in naturally fractured reservoirs, 
which not only preserves the physics of unsteady -state 
reservoir flow behavior, elastic fracture mechanics , 
material balance, varia ble fracture compliance during 
fracture closure,  but also incorporates pressure -
dependent effective permeability and sequential closure 
of complex fracture branches. Conclusions reached from 
the analysis presented in this paper include the following:  
1) Fracture closure stress (i.e., minimum in -situ stress) 
can’t be pinpointed from DFIT data because of the 
natur e of progressive fracture closure behav ior. 
2) Heavily fractured reservoirs that exhibit pressure -
dependent effective permeability can partly mask 
some information on fracture stiffness (or 
compl iance) evolution and the associated mechanical 
closure process .


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closure process . 
3) In sparsely  or more moderately fractured reservoir s 
with complex fracture geometry, if the closed fracture 
maintains  its hydraulic connectivity to the open 
fracture, then fracture complexity cannot be detected 
from before -closure data and the pressure decline 
response resembles a  fracture with  “rougher” fracture 
walls . However, if the closed fracture lose s its 
hydraulic connectivity to the open fracture, then 
fracture complexity can be detected qualitatively by 
the existence of multiple peaks on the pressure 
derivative  plot. 
4) The c onventional “tangent line method” consistently 
underestimates closure stress w hile “compliance 
method” tends to only reflect  the onset of fracture 
closure . Our proposed “variable compliance method” 
gives a much more accurate and reliable estimation of 
closure stress in both homogenous and naturally 
fractured reservoirs.  
 
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1. INTRODUCTION  
Diagnostic fracture injection tests (DFIT), also referred to 
as a minifrac  test, have been a standard stimulation 
technique in unconventional reservoirs to determine the 
reservoir characteristics. DFIT analysis can estimate key 
reservoir parameters such as reservoir permeability and 
initial reservoir pressure to aid in future frac ture designs. 
A typical DFIT test consists of pumping small amounts of 
fluid, usually water, down the wellbore to initiate the 
stimulation test in the reservoir. Enough water is pumped 
down hole to initiate a small fracture to breakdown the 
formation. The fracture  will continue to propagate until 
pumping is completed and the well is shut in. Once shut -
in occurs, the pressure falloff  response is recorded over a 
long period of time which can be from a few days to 
weeks. An analysis on the pressure data recorded is do ne 
to estimate several reservoir characteristics.  
The pressure response from a DFIT analysis can be


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The pressure response from a DFIT analysis can be 
separated into two parts: before -closure analysis (BCA) 
and after -closure analysis (ACA). The BCA region of a DFIT test is considered the period when water is injected 
into the wellbore to create a small fracture  until the 
created fracture is mechanically closed. Key parameters 
that can be determined from BCA are fracture closure 
pressure, instantaneous shut -in pressure (ISP), closure 
gradient, net fracture p ressure, fluid efficiency and leak -
off coefficient. The ACA region of the DFIT test is the 
time after the fracture is closed to the end of the shut -in 
period.  Fig. 1 shows a typical pressure response during a 
DFIT test.  
Key parameters that can be determined from ACA are 
flow regime, permeability, and transmissibility. Several 
numerical and analytical models are available to analyze 
the pressure response recorded during BCA and ACA.  
Traditional DFIT before closure  analysi s does not


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Traditional DFIT before closure  analysi s does not 
consider natural fractures. The Perkins -Kern -Nordgren 
(PKN) fracture geometry model created by Perkins and 
Kern and later improved by Nordgren ( Nordgren , 1972; 
Nolte , 1979; Perkins and Kern , 1961 ) derived several key 
parameters: fluid loss coeff icient, fluid efficiency,             
ARMA 21–1387                                                                 
 
New Diagnostic Fracture Injection Test Model with  
Complex Natural Fractures using Embedded Discrete Fracture Model  
 
Bruno , Jonathan and Sun, He 
The University of Texas at Austin, Austin, TX, USA  
Yu, Wei  
Sim Tech LLC, Houston, Texas, USA  
Sepehrnoori, Kamy  
The University of Texas at Austin, Austin, TX, USA  
 
Copyright 20 21 ARMA, American Rock Mechanics Association  
This paper was prepared for presentation at the 55th US Rock Mechanics/Geomechanics Symposium held in Houston, Texas, USA, 20-23 June


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2021. This paper was selected for presentation at the symposium by an ARMA Tech nical Program Committee based on a technical and critical 
review of the paper by a minimum of two technical reviewers. The material, as presented, does not necessarily reflect any pos ition of ARMA, its 
officers, or m embers. Electronic reproduction, distrib ution, or storage of any part of this paper for commercial purposes without the written consent 
of ARMA is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 200 words; illus trations may not be copied. The 
abstract must contain conspicuous acknowledgement of where and by whom the paper was presented.   
 
 
 ABSTRACT:  Application of diagnostic fracture injection tests (DFIT) are used in unconventional reservoirs to determine rese rvoir 
characteristics for future fracture design and production. The presence of natural fractures can heavily influence the pre - and post -


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shut-in pressure data obtained from DFIT. Traditional DFIT models do not capture the complexity of hydraulic and natu ral fracture 
networks into an integrated workflow. This work uses an embedded discrete fracture model (EDFM) to add natural fracture 
complexity into a new DFIT model in conjunction with any reservoir simulator to simulate the pressure response. Incorporati ng 
fracture modeling tools will enhance the existing reservoir model by creating a more accurate representation of an unconventi onal 
reservoir for future development strategies. EDFM can easily incorporate complex fractures and accurately be embedded in st ructed 
matrix block. Pressure data from the DFIT analysis can be separated into two parts: before -closure analysis (BCA) and after -closure 
analysis (ACA). Past models have analyzed BCA and ACA independently, which can lead to misinterpretation. The new DFI T model


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combines the BCA and ACA while including complex natural fractures into a single workflow. Data from the Lower Eagle Ford 
Formation was used to validate the model by accurately matching the same field data with the new workflow. The workflow consi sts 
of identifying random natural factures connected to the generated small fracture  to observe the true pressure response during DFIT. 
The workflow in this study can be applied to any DFIT data. Understanding the impact complex natural fractures have on pr essure 
response through an integrated DFIT model with EDFM is important to determine reservoir characteristic and geomechanics for 
future fracture designs.  
 
 
  
 
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fracture closure pressure, fracture length, and fracture 
width. Nolte, in his G -function model, was able to easily 
distinguish the closure pressure using a plot of a derived 
G-function and pressure.  Nolte’s derivation did not 
consider the complexity of natural fractures. His work 
only considered a simple planar fracture  with no natural 
fractures. The effects of natural fractures cannot be 
characterized or evaluated.  
 
Fig. 1. Typical DFIT test pressure response diagram  (Wang 
and Sharma, 2017) . 
Traditional DFIT A fter closure analysis also does not 
consider natural fractures. Soliman ( Soliman et al. , 2005 ) 
has worked on the pressure fall off analysis to estimate 
the formation permeability and pressure. From the 
pressure r esponse, each flow regime can be identified. 
Formation permeability can be estimated based on the 
transient pressure behavior.  ACA analysis fails to 
consider the fracture complexity.  
In this work, n atural fractures can heavily influence the


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In this work, n atural fractures can heavily influence the 
before closure  and after closure pressure response during 
a DFIT test. Traditional BCA and ACA DFIT models do 
not capture the complexity of the natural fracture network 
connected to a small fracture . The pressure transient 
behavior will be influenced by the geometry and density 
of the natural fractures. Implementing natural fractures 
would give a more accurate representation of 
unconventional reservoirs. The pressure data during a 
DFIT test can be co herently interpreted using a single 
workflow.  This work will show the new workflow 
considering complex natural fractures coherently with a 
case study. The objective of this work is to understand and 
implement a new DFIT workflow with natural fractures. 
This work is important to understand the impact of natural 
fracturs on reservoir pressure and permeability.  
2. THEORY AND MODEL DEVELOPMENT  
Several analytical models have been developed by Nolte


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Several analytical models have been developed by Nolte 
that are available for DFIT analysis (Nolte , 1979, 1990; Nolte et al., 1993; Nolte et al. , 1997). Nolte pioneered the 
G-function technique for BCA and ACA analysis based 
off the Carter leak -off model (Nolte , 1979).  In 2014, 
McClure developed a fracture -compliance method for 
estimating the closure pressure for DFIT test a ssuming a 
variable fracture compliance (McClure et al. , 2014). The 
G-function derivative plot was analyzed with a changing 
fracture compliance during closure (McClure et al. , 
2014). The variable compliance method was created by 
Wang et al. (2017) where the  fracture closure point is 
determined to be average of the fracture closure points 
picked using the fracture -compliance method and tangent 
line method. The tangent line method is considered the 
upper bound while the fracture -compliance method is 
considered  the lower bound for the fracture closure point.  
In this study, the G -function technique was used  to


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In this study, the G -function technique was used  to 
determine the closure point . 
The application of G -function technique across multiple 
leak-off factors validated Nolte’s model. The original G -
function mode l from Nolte is based on the following 
assumptions:  
(i) constant fracture compliance  
(ii) constant fracture height  
(iii) constant closure pressure  
(iv) constant fracture surface after shut -in 
(v) leak-off coefficient independent of pressure 
change  
(vi) material balance between leak -off fluid volume 
and fracture volume  
The analytical solution for Nolte’s G -function time is the 
following:  
 
𝑔(∆𝑡𝐷)=
{(1+∆𝑡𝐷)sin−1(1+∆𝑡𝐷)−1
2+∆𝑡𝐷1
2    𝛼=1
2
4
3((1+∆𝑡𝐷)3
2−∆𝑡𝐷3
2)                             𝛼=1          (1) 
𝐺(∆𝑡𝐷)=4
𝜋[𝑔(∆𝑡𝐷)−𝑔0]   (2) 
𝑔0=𝑔(∆𝑡𝐷=0)    (3) 
∆𝑡𝐷=𝑡−𝑡𝑝
𝑡𝑝    (4) 
 
 
where t is the shut -in time and t p is the total pump time. A 
α value of 1/2 refers to a formation with low fluid leak -
off. A α value of 1 refers to a formation with high fluid


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off. A α value of 1 refers to a formation with high fluid 
leak-off. The G -function time solution starts at the shut -
in or when the instantaneous shut -in pressure (ISIP) is 
zero. From Nolte’s G -function time model, Barree used a 
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diagnostic approach to identify the fracture closure 
pressure (Barree , 1998; Barree et al. , 2009). The 
composite derivative of G -function, G *dp/dG, is used to 
show where the pressure change occurs. A s traight line 
can be drawn from the origin from the plot of G *dp/dG 
against G -function time. The deviation point from the 
G*dp/dG curve from the straight line is used as the 
fracture closure point. The closure point was estimated to 
be 25.30 G -function time . Fig. 2 shows the of G *dp/dG 
against G -function time.  
 
 
Fig. 2. Closure point selection using G*dp/dG versus G time . 
The closure point is important to estimate the closure 
stress of the formation from the DFIT data. Using Nolte’s 
G-function time model, the closure pressure is estimated 
to be 7100 psi. Fig . 3 shows the corresponding pressure 
profile using Nolte’s analytic al solution for G -function 
time.  
 
 
Fig. 3. Closure pressure corresponding to the previously 
established closure point.


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established closure point.  
Once the fracture closure point is found, the simulation 
model can be developed. Fig . 4 shows the steps of the new 
DFIT workflow used in this case study.  
  
Fig. 4. New DFIT workflow . 
The new workflow consists of starting with a single 
fracture to represent the small fracture  created  in a DFIT 
test. With our new DFIT model fracture height , half -
length, conductivity  and closure pressure can be 
manipulated to satisfy your model setup. The new DFIT 
workflow allows the user more flexibility when setting up 
a DFIT model which previous workflows do not allow 
for. The base case model in thi s paper was set up using 
the following parameters in Table 1.  
 
Table 1. Base case parameters for DFIT reservoir model  
Base Case  
Generated  Fracture  Height (ft)  50 
Generated  Fracture  Half-length (ft)  200 
Generated  Fracture  Conductivity (mD -ft) 10 
Number of Natural Fractures  60 
Natural Fracture Height (ft)  30-50 
Natural Fracture Half -length (ft)  30-70


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Natural Fracture Half -length (ft)  30-70 
Natural Fracture Conductivity (mD -ft) 2.4-5.6 
Matrix Permeability (mD)  0.001  
 
An embedded discrete fracture model (EDFM) is used to 
implement a small fracture  in the reservoir model .  ED FM 
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can represent fractures with a grid outside normal models 
and uses special connections to model fluid and pressure 
transport.  This method is easy  and efficient to model any 
complex fracture geometries using structured gridding 
(Moinfar et al. , 2014; Xu , 2015; Xu et al. , 2017a, 2017b). 
The complex fractures are discretized into fracture 
segments based on the interaction between the fracture 
geometry  and the structured matrix grid boundary. The 
fluid flow between the matrix and fractures can be 
simulated using a commercial reservoir simulator such as 
CMG by using non -neighboring connections (NNCs) ( Yu 
et al, 2018).  Fig. 5 shows a visualization of the EDFM 
methodology.  
 
 
Fig. 5. Explanation of EDFM method with commercial 
reservoir simulato rs to handle 3D complex fractures (Xu et al., 
2017a)  
There are four types of NNCs: fluid flow between matrix 
and fractures, between fracture segments within a single 
fracture, between intersecting fracture segments within


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fracture, between intersecting fracture segments within 
different fractures, and between fractures and wellbores. 
More information on the EDFM methodology is available 
in the work by Xu et al. (2017a, 12017b).  This paper 
introduces the use of EDFM  to implement random  natural 
fractures  into a DFIT model . The model flexibility is 
demonstrated by characterizing d ifferent complex natural 
fracture cas es.  
Validation of the DFIT model is shown through a DFIT 
case study.  The pressure transient behavior is captured 
throughout the fracture network. The new DFIT model 
can predict the pressure decline response. Evaluation of 
the natural fractures and a gener ated small fracture  is done to show the effects of changing characterization with 
varying geometry, density, half-length , and conductivity. 
The new DFIT model can increase and decrease the 
density of natural fractures with ease.   
Once a small fracture  is generated, random natural 
fractures are implemented into the model using EDGS.


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fractures are implemented into the model using EDGS. 
EDGS allows you to generate random natural fractures 
with various geometries and density. Fig . 6 shows the 
generated  small fracture  along with the random natural 
fractures.  
 
 
Fig. 6. Generated base case for the new DFIT workflow with 
complex natural fractures.  
The pressure response is initialized in the generated a 
small fracture  using a natural fracture connectivity code. 
The connectivity code can identify the natural fractures 
that are connected to the generated small fracture . Fig. 7 
shows the DFIT model  after running the connectivity 
code. The areas in red are the natural frac tures that have 
been identified that are not connected to the generated  
small fracture . 
 
 
Fig. 7. Connectivity code evaluation of natural fractures not 
connected to the  generated  small fracture . 
A closer look at the complex fracture network can be 
seen in Fig. 8. The natural fracture connections to the


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seen in Fig. 8. The natural fracture connections to the 
small fracture  are important to analyze the pressure 
behavior using EDFM.  
 
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Fig. 8. Closer look of the complex fracture network . 
A reservoir model is thus created to represent a DFIT test 
with complex natural fractures . The pressure response 
from the generated  small  fracture  is simulated using 
computer modeling group (CMG) reservoir simulator. 
The new workflow was able to estimate the  pressure 
response for a DFIT test.  
A closer look at the complex fracture network can be seen 
in Fig . 9. The pressure response was initialized in the 
generated  small fracture  and the distribution of the 
pressure  response is seen  throughout the complex nat ural 
fracture network.  
 
 
Fig. 9. Closer look of the complex fracture network . 
The pressure response throughout the 1st, 3rd, and 6th day 
of the new DFIT model can be seen in  Fig 10. (a), (b), and 
(c), respectively. The pressure in the fracture network 
continues to decrease the longer the well is shut -in until 
the reservoir pressure is reached.  
 
 
(a) Pressure distribution after 1 day


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(a) Pressure distribution after 1 day  
(b) Pressure distribution after 3 days 
 
 
(c) Pressure distribution after 6 days 
Fig. 10. Pressure distribution changes shown from the (a) 1st 
day, (b) 3rd day, and (c) 6th day. 
 
The new DFIT workflow is capability of replicating the 
ACA pressure response using complex natural fractures.  
A visualization of the pressure distribution is captured 
throughout the complex fracture network.  To test the 
impact of changing various fracture  characterization  
parameters, a sensitivity analysis study is presented in the 
next section.  
3. SENSITIVITY ANALY SIS 
Sensitivity analysis was done in this study to test t he 
effects of varying the characterization of complex natural 
fractures  using our new workflow . The model was set up 
using the same generated small fracture for every case. 
The complexity of the smal l fracture depends on the 
connected natural fracture which has a great impact on the 
pressure response.  The pressure is observed for after the


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pressure response.  The pressure is observed for after the 
fracture closure point. Therefore, the generated small 
fracture is considered mechanically closed. The following 
parameters were adjusted  in this study: varying the 
density of natural fractures, natural fracture  height, 
natural fracture  half-length, and natural fracture  
conductivity.  Fig 11. (a), (b), and (c) show the DFIT 
model with varying density of natural fracture from 30, 
60, and 90, respectively.  
 
(a) DFIT model with 30 natural fractures  
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(b) DFIT model with 60 natural fractures  
 
(c) DFIT model with 90 natural fractures  
Fig. 11. DFIT model with varying densit ies of natural 
fractures from (a) 30 natural fracture s, (b) 60 natural fractures, 
and (c) 90 natural fractures.  
 
Fig. 12 shows the pressure falloff  response when varying 
the number of natural fractures  using  the new DFIT 
workflow.  
 
 
Fig. 12. Sensitivity analysis on the number of natural 
fractures.  
As the number of natural fractures increase s, the pressure 
falloff  becomes less severe. The phenomenon that is 
happening is due to the number of natural fractures 
connected to the small fracture  becomes greater when 
more natural fractures are added. This would allow for a 
more even distribution of pressure throughout the fracture 
network and therefore a less severe pressure falloff . 
 
Fig. 13 shows the pressure falloff  response  of varying the 
height of the natural fractures  in the new DFIT model.


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height of the natural fractures  in the new DFIT model.  
  
Fig. 13. Sensitivity analysis on the natural fracture  height.  
There is not a significant difference between the pressure 
falloff  response as you increase the height of the natural 
fractures . 
 
Fig. 14 shows the pressure falloff  response as you 
increase the half -length of the natural fractures .  
 
 
Fig. 14. Sensitivity analysis on the natural fracture  half-length.  
As you increase the half -length of the natural fractures , 
the pressure falloff response  becomes less severe. This 
could be du e to the larger surface area that the natural 
fractures are covering as it gets larger. The natural 
fractures are  creating a larger fracture network as you 
increase the half -length. A larger fracture network proves 
to have a less severe pressure falloff  behavior.  
 
Fig. 15 shows the pressure falloff  response as you 
increase the conductivity of the natural fractures .


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increase the conductivity of the natural fractures . 
 
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Fig. 15. Sensitivity analysis on the natural fracture  
conductivity.  
As you increase the conductivity of the natural fractures , 
the pressure falloff response becomes less severe.  
4. SUMMARY AND CONCLUSION  
This paper  implemented a new DFIT workflow with 
complex natural fractures.  Including complex natural 
fractures with a DFIT test enhanc es the understanding of 
the characteristics for unconventional reservoirs. The new 
DFIT workflow allows for characterization of complex 
natural fractures with varying geometry, density, 
aperture, and conductivity.  The impact of varying fracture 
parameters is evident in this study.  The number of natural 
fractures, natural fracture  half-length and natural fracture  
conductivity had an impact on the pressure falloff 
response. The pressure falloff response became less 
severe with the increase in natural fractures, the generated  
small fracture  half-length and the generated  small fracture  
conductivity. This could be due to the larger surface area


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conductivity. This could be due to the larger surface area 
that the small fracture  is covering. The new DFIT 
workflow was able to estimate the pressure falloff  
behavior during a DFIT test  using the various reservoir 
and natural fracture parameter . The impact of natural 
fractur es on the pressure falloff behavior is evident in this 
study.  
Compared to traditional DFIT analysis, the new DFIT 
workflow was able to implement natural fractures using 
EDFM. The fracture network proved to h ave an impact on 
the pressure falloff  response. Characterization of the 
fracture network was captured and ensured an efficient 
analysis to estimate the reservoir properties. The 
generated fracture during a DFIT test becomes more 
complex when natural fractu res are added to the model. 
The pressure transient behavior can be captured 
through out the fracture network. The density of natural 
fractures can be easily added and reduced depending on 
the model demand s. The new DFIT workflow


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the model demand s. The new DFIT workflow 
demonstrated the ability to handl e complex natural 
fractures while providing a sufficient analysis on the 
reservoi r properties.  DFIT analysis with complex natural fracture geometries will aid in future reservoir evaluation 
and production design.  
5. REFERENCES  
Barree, R.D. 1998. Applications of Pre -Frac Injection/Falloff 
Tests in Fissured Reservoirs - Field Examples. Paper presented 
at the SPE Rocky Mountain Regional/Low -Permeability 
Reservoirs Symposium, Denver, Colorado , USA,  5-8 April 
1998.  
Barree, R.D., Barr ee, V.L., and Craig, D. 2009. Holistic 
Fracture Diagnostics: Consistent Interpretation of Prefrac 
Injection Tests Using Multiple Analysis Methods. SPE 
Production & Operations  24 (03): 396 -406.  
McClure, M.W., Blyton, C.A., Jung, H., and Sharma, M.M. 
2014. The Effect of Changing Fracture Compliance on Pressure 
Transient Behavior During Diagnostic Fracture Injection Tests. 
Paper presented at the SPE Annual Technical Conference and


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Paper presented at the SPE Annual Technical Conference and 
Exhibition, Amsterdam, The Netherlands, 27 -29 October 2014.   
Moinfar, A., Varav ei, A., Sepehrnoori, K. et al. 2014. 
Development of an Efficient Embedded Discrete Fracture 
Model for 3D Compositional Reservoir Simulation in Fractured 
Reservoirs. SPE Journal  19 (02): 289 -303. 
Nolte, K.G. 1979. Determination of Fracture Parameters from 
Fracturing Pressure Decline. Paper presented at the SPE 
Annual Technical Conference and Exhibition, Las Vegas, 
Nevada,  USA,  23-26 September 1979.  
Nolte, K.G. 1990. Fracturing -Pressure Analysis fo r Nonideal 
Behavior . Journal of Petroleum Technology  43 (02): 210 -218. 
Nolte, K.G., Mack, M.G., and Lie, W.L. 1993. A Systematic 
Method for Applying Fracturing Pressure Decline: Part I. Paper 
presented at the Low Permeability Reservoirs Symposium, 
Denver, Colorado , USA, 26-28 April 1993.  
Nolte, K.G., Maniere, J.L., and Owens, K.A. 1997. After -
Closure Analysis of Fracture Calibration Tests. Paper


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Closure Analysis of Fracture Calibration Tests. Paper 
presented at the SPE Annual Technical Conference and 
Exhibition, San Antonio, Texas , USA,  5-8 October 1997.  
Nordg ren R .P. 1972. Propagation of a Vertical Hydraulic 
Fracture. Society of Petroleum Engineers Journal 12 (04): 306 -
314. 
Perkins T .K, Kern L .R. 1961. Widths of Hydraulic Fractures.  
Journal of Petroleum Technology 13 (09): 937 -949. 
Soliman M .Y., Craig D .P., Bartko K .M., Rahim Z ., Adams 
D.M. 2005.  Post-Closure Analysis to Determine Formation 
Permeability, Reservoir Pressure, Residual Fracture Properties. 
Paper presented at the SPE Middle East Oil and Gas Show and 
Conference, Kingdom of Bahrain, Bahrain, 12 -15 March 2005.  
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Wang, H., and Sharma, M.M. 2017. New Variable Compliance 
Method for Estimating In -Situ Stress and Leak -Off from DFIT 
Data. Paper presented at the SPE Annual Technical Conference 
and Exhibition, San Antonio, Texas, USA, 9 -11 October 2017.  
Xu, Y., 2015. Implementation and Application of the 
Embedded Discrete Fracture Model (EDFM) for Reservoir 
Simulation in Fractured Reservoirs. Master Thesis, The 
University of Texas at Austin, Austin, Texas, USA.  
Xu, Y., Cavalcante Filho, J.S., Yu, W., and Sepe hrnoori, K. 
2017a. Discrete -Fracture Modeling of Complex Hydraulic 
Fracture Geometries in Reservoir Simulators. SPE Reservoir 
Evaluation and Engineering  20 (02): 403 -433. 
Xu, Y., Yu, W., and Sepehrnoori, K. 2017b. Modeling Dynamic 
Behaviors of Complex Frac ture in Conventional Reservoir 
Simulators. Paper presented at the SPE/AAPG/SEG 
Unconventional Resources Technology Conference , Austin, 
Texas, USA, 24 -26 July 2017.  
Yu, W., Xu, Y., Weijermars, R., Wu, K., and Sepehrnnori, K.,


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Yu, W., Xu, Y., Weijermars, R., Wu, K., and Sepehrnnori, K., 
2018. A Numerical Model for Sim ulating Pressure Response of 
Well Interference and Well Performance in Tight Oil 
Reservoirs with Complex -Fracture Geometries Using the Fast 
Embedded -Discrete -Fracture -Model Method . SPE Reservoir 
Evaluation and Engineering  21 (02): 489 -502. Downloaded from http://onepetro.org/ARMAUSRMS/proceedings-pdf/ARMA21/All-ARMA21/ARMA-2021-1387/2479604/arma-2021-1387.pdf/1 by University of Houston user on 30 October 2023


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URTEC-208276-MS
Can ISIP Really Tell Closure Pressure?—Case Study of Consecutive DFITs in
the Eagle Ford
Shintaro Kuroda , Shunichi Watanabe, and Kenji Ueda , INPEX CORPORATION
Copyright 2021, Unconventional Resources Technology Conference (URTeC) DOI 10.15530/AP-URTEC-2021-208276
This paper was prepared for presentation at the SPE/AAPG/SEG Asia Pacific Unconventional Resources Technology Conference to be held virtually on 16–18
November, 2021.
The URTeC Technical Program Committee accepted this presentation on the basis of information contained in an abstract submitted by the author(s). The contents
of this paper have not been reviewed by URTeC and URTeC does not warrant the accuracy, reliability, or timeliness of any information herein. All information is the
responsibility of, and, is subject to corrections by the author(s). Any person or entity that relies on any information obtained from this paper does so at their own


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risk. The information herein does not necessarily reflect any position of URTeC. Any reproduction, distribution, or storage of any part of this paper without the written
consent of URTeC is prohibited.
Abstract
The objective of this paper is to investigate how to acquire fracture closure pressure data accurately and cost-
effectively in shale development activities. If the injection volume is small during a hydraulic fracturing
job, it is a common practice to consider the ISIP (Instantaneous Shut-In Pressure) as a proxy of fracture
closure pressure, which generally requires a long falloff period to evaluate. However, ISIP should consist
of variable parameters such as the original closure pressure, net pressure, stress shadow effect, and mid-
field fracture complexity.
In this paper, we introduce the consecutive DFITs (Diagnostic Fracture Injection Test) approach. Closure
pressure is evaluated by falloff pressure analysis after a micro fracturing injection operation, and pore


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pressure could also be evaluated if the falloff period is long enough. This micro fracturing operation is
generally called DFIT. In order to evaluate how the ISIP, closure pressure, and pore pressure change from
stage to stage, we performed DFITs consecutively at the sequential hydraulic fracturing stages in a horizontal
well drilled in the Eagle Ford shale.
The consecutive DFITs revealed that the ISIP gradually increased up to certain level from fracturing stage
to stage as expected. However, the observed closure pressure was almost constant in the sequential stages,
which was against our expectations. In addition, the evaluated pore pressure was also almost constant.
Initially we expected that closure pressure would increase because of the uplift due to the stress shadow
effect. Since the consecutive DFITs showed the same closure pressure in each stage, we concluded that stress
uplift could disappear before the fracture closure in next stage or that the stress shadow had little impact on


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the closure pressure and the pore pressure in next stage under the current fracture design. On the other hand,
the ISIP could be affected by the stress shadow in the short term or by the mid-field fracture complexity
becoming higher than the previous stage. The correlation between the ISIP and the closure pressure was
established with these consecutive DFITs results. Although the gap between the ISIP and closure pressure
varies from stage to stage, it was confirmed that the correlation, with some uncertainties, could be used to
estimate the closure pressure within an acceptable range.
This paper presents the Eagle Ford case study, which confirmed how accurately ISIP can determine
closure pressure considering multiple factors. There are hydraulic fracturing operations in huge number of


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horizontal wells in the shale development. Therefore, the correlation built by consecutive DFITs is usefulDownloaded from http://onepetro.org/URTECAP/proceedings-pdf/21APUR/2-21APUR/D021S008R003/2528605/urtec-208276-ms.pdf/1 by University of Houston user on 30 October 2023


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2 URTEC-208276-MS
because that correlation can provide operators with the confidence to optimize the completion design based
on the ISIP which can be obtained at a low cost.
Introduction
Many hydraulic fracturing stages are routinely performed in shale development activities, but it is rare to
spend additional time and money for data acquisition. ISIP (Instantaneous Shut-In Pressure) is an important
parameter by which operators can obtain subsurface information during the hydraulic fracturing job without
extra effort. This ISIP equals the pressure in the fracture just after pump shut-in, and after friction forces in
the wellbore, perforations, and near-wellbore region dissipate. Although many operators often use the ISIP
at the wellhead rather than at the bottom hole for its ease, we use ISIP at bottom hole in this paper, calculating
its value from the ISIP measured by the wellsite supervisor at wellhead pressure and the fluid density.


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If the injection volume is small during the hydraulic fracturing job, it is common practice to consider
ISIP as a proxy of fracture closure pressure, which generally requires a long falloff period to evaluate. Craig
(2005)  showed the injection volume ranged from less than 200 to greater than 10,000 gal, and our case
was 15,000 gal considering the multiple perforation clusters. Smaller injection volume can create smaller
fracture volume, which means lower net pressure. If the net pressure is so low as to be negligible, the
obtained ISIP can be close to the fracture closure pressure. However, Bazan and Meyer (2015)  showed
that there are other factors to consider, including mid-field fracture complexity, back stress, also called
stress shadow effect, as well as the net pressure between the ISIP and the closure pressure. Considering
the existence of several parameters between the ISIP and the closure pressure, the objective of this paper


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is to investigate how to acquire fracture closure pressure data accurately and cost-effectively in shale
development activities.
The study field in this paper is the Eagle Ford shale play located in South Texas, in the United States as
shown in Fig. 1 . The Eagle Ford shale is the second most productive oil play in the United States as of 2020
and it is a Late Cretaceous package of organic-rich marl, limestone, and mudstone overlain by the Austin
chalk and bounded below by the Buda limestone ( Engle et al. 2020 ).
Figure 1—Location of the project.
Methods
A Diagnostic Fracture Injection Test (DFIT) is a small pump-in treatment performed to help design the
follow-up hydraulic fracturing treatment of wells so as to characterize the subject reservoir ( Barree et al.


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2015 ). It is a micro fracturing injection test for data acquisition, and it is also called a mini frac, fractureDownloaded from http://onepetro.org/URTECAP/proceedings-pdf/21APUR/2-21APUR/D021S008R003/2528605/urtec-208276-ms.pdf/1 by University of Houston user on 30 October 2023


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URTEC-208276-MS 3
calibration test, or fracture injection and falloff test. Through the DFIT operation and analysis, we expect to
get several parameters such as the ISIP, the fracture closure pressure, the net pressure, the fluid efficiency, the
reservoir pore pressure, and the permeability, depending on the pressure behavior. The DFIT analysis period
can be divided into two portions: before closure and after closure. The closure pressure is evaluated by a
falloff pressure analysis in the period before closure. Barree and Mukherjee (1996)  described several types
of normal leakoff phenomena including pressure dependent leakoff, tip extension after shut-in, variable
fracture compliance, height recession or transverse storage. Liu and Economides (2015) considered variable
compliance and many other abnormal leakoff behaviors, and McClure et al. (2014 , 2016 ) introduced the
fracture compliance method. In this paper, we did not apply the fracture compliance method but applied the


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conventional tangent line method in a G-function plot for the simplified analysis workflow. Soliman et al.
(2005)  have shown that after-closure analysis can provide reservoir pore pressure and permeability if the
flow regime reaches pseudo linear flow or pseudo radial flow. A flow identification plot is used to identify
these flow regimes after closure, then a linear flow time function or a radial flow time function are used to
evaluate the reservoir pore pressure or permeability.
A DFIT is generally performed in the toe stage of horizontal wells before starting the main fracturing
treatment and to perform DFIT in the middle stages is not common for shale development because it requires
a certain shut-in period leading to non-productive time during the completion operation. However, there are
still DFIT challenges in the toe stage. In the author’s past DFIT practice, using a toe-opened sleeve after
the toe preparation job in a toe stage has not been used because any potential near wellbore damage by


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the wellbore clean-up job may make it difficult to analyze data.. Therefore, a new, fresh perforated entry
point is used to enhance data quality. Also, operators don’t know how much these parameters differ from
the middle and heel of the horizontal section, assuming geological and geomechanical heterogeneity in the
lateral. The ISIP obtained in each stage provides useful information to estimate the subsurface stress and
pressure along the lateral sections, other than the toe stage.
There is also the stress shadow effect from the previous fracturing stages. Roussel (2017)  showed ISIP
escalation stage by stage due to the stress shadowing. This stress interference between fracturing stages is
one of the uncertain factors that can affect the fracture propagation. Another consideration is the mid-field
tortuosity. Weijers et al. (2002) states that mid-field tortuosity is recognized by a high apparent ISIP and


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rapidly declining pressure during the first few minutes of shut-in. This mid-field tortuosity could be caused
by a pressure choke beyond the near-wellbore region of the fracture (Mayer and Bazan 2011; Bazan and
Meyer 2015 ). Considering these parameters as shown in Fig. 2 , our challenge was how to utilize the ISIP
in order to understand the subsurface stress condition at a low cost.
Figure 2—Pressure behavior image in falloff after hydraulic fracturing.Downloaded from http://onepetro.org/URTECAP/proceedings-pdf/21APUR/2-21APUR/D021S008R003/2528605/urtec-208276-ms.pdf/1 by University of Houston user on 30 October 2023


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4 URTEC-208276-MS
In this paper, we introduce "the consecutive DFITs" approach. DFITs were performed consecutively at
the selected sequential hydraulic fracturing stages in a horizontal well targeting the productive zone in the
Lower Eagle Ford shale as shown in Fig. 3 . The goal of the consecutive DFITs is to understand the pressure
changes such as the ISIP, the closure pressure, and the pore pressure from stage to stage. Five stages were
carefully selected from the total of 41 stages to avoid the subsurface fault locations identified by seismic
interpretation and a geo-steering evaluation. The tested well was the first development well in the pad.
Therefore, there were uncertainties how the faults would be affected, but no depletion was assumed from
a parent well.
Figure 3—Location of the consecutive DFITs in the horizontal well overlayed on a geo-steering plot.
Before performing these consecutive DFITs, conventional DFITs were performed at the toe section drilled


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in another Eagle Ford area with one-month falloff period. The experience with the one-month falloff DFITs
showed that the created fracture would close within one hour which was much faster than we expected.
Based on the previous experience, we decided on a one-hour falloff for the consecutive DFITs.
In hydraulic fracturing operation workflows in the Eagle Ford development wells, the initial breakdown
injection is routinely performed before the main fracturing treatment. A short period falloff (for example, a
few minutes) after the initial breakdown injection provides the ISIP. For the consecutive DFITs, the falloff
period after the initial breakdown was extended to one hour as shown in Fig. 4 . In addition to the falloff
period extension, we modified some operation procedures such as constant pump rate, hard shut-in, and
used the same density fluid in the wellbore for better quality of the closure pressure analysis. One fluid type


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in the wellbore can make it easier to calculate the bottomhole pressure from the wellhead pressure.
Figure 4—Falloff period after initial breakdown was extended for DFIT.Downloaded from http://onepetro.org/URTECAP/proceedings-pdf/21APUR/2-21APUR/D021S008R003/2528605/urtec-208276-ms.pdf/1 by University of Houston user on 30 October 2023


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URTEC-208276-MS 5
Results and Discussion
The proposed consecutive DFITs could be performed without any operational issues. The consecutive DFITs
were performed at the five stages from stage 16 to stage 20 in a single-well fracturing operation, not in a
zipper-frac operation. There were three stages where the main fracturing jobs were cancelled because of
abnormally low ISIP caused by the fault zone. As an example of the consecutive DFIT stage results, the
analysis plots in stage 18 are shown in Fig. 5  with the overall treatment, the before-closure analysis with
a G-function plot and a square root time plot, and the after-closure analysis with a flow identification plot.
Fracture closure pressure can be determined in all the five stages successfully in the before-closure analysis
as shown in Fig. 6 . In this paper, we use the pressure gradient calculated from an equation of the pressure
divided by the true vertical depth for comparison. In addition, reservoir pore pressure can also be evaluated


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in four stages based on the pseudo linear flow regime after closure except for stage 16 which is close to a
fault zone. The after-closure analysis in stage 16 showed the pressure derivative increment after a short-time
pseudo linear flow regime in the flow identification plot. The evaluated pore pressure with the short-time
pseudo linear flow regime in stage 16 is higher than in the other stages. It is interpreted as SRV (Stimulated
Reservoir Volume) pressure, meaning temporarily higher pore pressure and higher permeability than the
original at the limited near wellbore area.
Figure 5—Treatment plot and G-function plot for one of the consecutive DFIT stages.Downloaded from http://onepetro.org/URTECAP/proceedings-pdf/21APUR/2-21APUR/D021S008R003/2528605/urtec-208276-ms.pdf/1 by University of Houston user on 30 October 2023


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6 URTEC-208276-MS
Figure 6—Results of consecutive DFITs. Observed stable closure pressure gradient in five stages.
The observed ISIP gradually increased up to a certain level from fracturing stage to stage as expected.
However, the consecutive DFITs revealed that the observed closure pressure was almost constant in the
sequential stages, which went against our expectations. In addition, the evaluated pore pressure was also
almost constant except for stage 16, which was affected by SRV. Initially we expected that the closure
pressure would increase because of the uplift due to the stress shadow effect as shown in Fig 7 . Since the
consecutive DFITs showed the same closure pressure in each stage, it was concluded that stress uplift could
dissipate before the fracture closure in the next stage or that the stress shadow had little impact on the closure
pressure and the pore pressure in the next stage under the current fracture design. In this field case study, it


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takes approximately two hours for plug and perforation operation between main fracturing treatments. On
the other hand, the ISIP could be affected by the stress shadow in the short term or by the mid-field fracture
complexity becoming higher than the previous stage. Permeability in the SRV area also can affect the ISIP
because the leakoff volume affects the net pressure.Downloaded from http://onepetro.org/URTECAP/proceedings-pdf/21APUR/2-21APUR/D021S008R003/2528605/urtec-208276-ms.pdf/1 by University of Houston user on 30 October 2023


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URTEC-208276-MS 7
Figure 7—Expected image of the closure pressure before and after consecutive DFIT operations.
The consecutive DFITs showed the variable differences between the ISIP and the evaluated closure
pressure from stage to stage. To establish the correlation between the ISIP and the closure pressure, we
attempted to increase the number of closure pressure data points at stages other than the five consecutive
stages where we conducted DFITs. We did this by using the falloff period from normal operations, which is
shorter than the falloff after the initial breakdown for the consecutive DFIT stages. This attempt succeeded
and gave us eleven additional points though there were five low-reliability stages. The correlation between
the ISIP and the closure pressure was established as shown in Fig. 8 . The correlation shows that uncertainties
of the estimated closure pressure are within +/- 0.07 psi/ft and that coefficient of determination, r2, is 0.77 in


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this case. The closure pressure is estimated by the correlation as shown in Fig. 9 . Although the gap between
the ISIP and the closure pressure varies from stage to stage, it was confirmed that the correlation, with some
uncertainties, could be used to estimate the closure pressure within an acceptable range.
Figure 8—Correlation between ISIP gradient and closure pressure gradient. In addition to five
consecutive DFIT stages, eleven stages were added. The pre-Frac ISIP was evaluated at the wellsite.Downloaded from http://onepetro.org/URTECAP/proceedings-pdf/21APUR/2-21APUR/D021S008R003/2528605/urtec-208276-ms.pdf/1 by University of Houston user on 30 October 2023


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8 URTEC-208276-MS
Figure 9—Closure pressure profile estimated by ISIP-closure pressure correlation.
There are hydraulic fracturing operations in huge number of horizontal wells in shale development.
Particularly in multi-well pad situations or first areas to develop, cost-effective data acquisition from the
first well is critical to understanding the local geological and geomechanical setting and heterogeneity along
lateral sections. Selection of consecutive DFIT locations needs to be discussed among the members of a
multi-disciplinary team before conducting this operation. The correlation built by consecutive DFITs shown
in this paper provides operators with more insight and confidence to optimize the completion design based
on ISIP which can be obtained at a relatively low cost.
Conclusions
This paper presents the consecutive DFITs approach at the sequential hydraulic fracturing stages in a
horizontal well. These are the findings from consecutive DFIT attempts in the Eagle Ford case study:


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1.The proposed consecutive DFITs could be performed without any operational issues. This approach
allows operators to conduct DFIT in stages other than the toe stage and to interpret how closure
pressure and pore pressure look along a lateral section.
2.The evaluated closure pressure was constant in the sequential stages while ISIP gradually increased
up to a certain level, fracturing stage to stage.
3.The evaluated pore pressure was also almost constant in the sequential stages except for one stage
showing higher pressure. This is assumed to be the SRV pressure.
4.The correlation between the ISIP and the closure pressure showed that the closure pressure can be
estimated from the ISIP within an acceptable range although there were some uncertainties.
Acknowledgements
We wish to express our appreciation to INPEX CORPORATION and INPEX Eagle Ford, LLC for
permission to present this paper. We also acknowledge the operation team at INPEX Eagle Ford, LLC for


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execution of this operation to acquire the data for this study.Downloaded from http://onepetro.org/URTECAP/proceedings-pdf/21APUR/2-21APUR/D021S008R003/2528605/urtec-208276-ms.pdf/1 by University of Houston user on 30 October 2023


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URTEC-208276-MS 9
SI Metric Conversion Factors
bbl × 1.589 874 E-01 = m3
ft × 3.048 * E-01 = m
lbm × 4.535 9237 * E-01 = kg
lbm/gal × 1.198 264 E+02 = kg/m3
psi × 6.894 757 E+03 = Pa
psi/ft × 2.262 059 E+04 = Pa/m
US gallon × 3.785 412 E-03 = m3
*Conversion factor is exact.
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Roussel, N. P. 2017 . Analyzing ISIP Stage-by-Stage Escalation to Determine Fracture Height and Horizontal-Stress
Anisotropy . Paper presented at the SPE Hydraulic Fracturing Technology Conference and Exhibition held in The
Woodlands, Texas, USA, 24-26 January 2017. https://doi.org/10.2118/184865-MS.
Soliman, M. Y., Craig, D. , Bartko, K.  et al. 2005 . After-Closure Analysis to Determine Formation Permeability, Reservoir
Pressure, and Residual Fracture Properties . Paper presented at the SPE Middle East Oil and Gas Show and Conference
held in Bahrain International Centre, Bahrain, 12-15 March 2005. SPE-93419-MS. https://doi.org/10.2118/93419-MS.
Weijers, L., Griffin, L. G., Sugiyama, H.  et al. 2002 . The First Successful Fracture Treatment Campaign Conducted in
Japan: Stimulation Challenges in a Deep, Naturally Fractured Volcanic Rock. Paper presented at the SPE Annual
Technical Conference and Exhibition, San Antonio, Texas, USA, 29 September-2 October 2002. SPE-77678-MS.


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SPE-199990-MS
Field-Wide Review of Diagnostic Fracture Injection Tests DFIT in Montney
and Duvernay Formations – Part 1: Stress and Pressure Distributions
Behjat Haghshenas , Reservoir Diagnostics Inc. ; Farhad Qanbari, Seven Generations Energy
Copyright 2020, Society of Petroleum Engineers
This paper was prepared for presentation at the SPE Canada Unconventional Resources Conference originally scheduled to be held in Calgary, Alberta, Canada, 18 –
19 March 2020. Due to COVID-19 the physical event was postponed until 29 September – 2 October 2020 and was changed to a virtual event. The official proceedings
were published online on 24 September 2020.
This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents
of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect


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any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written
consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may
not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.
Abstract
A straightforward test with debatable analysis methods, the diagnostic fracture injection test (DFIT) is
a pressure-transient test extensively used for reservoir and geomechanical characterization of tight/shale
formations. The test provides some key data and information to reservoir and well completion engineers
including instantaneous shut-in pressure (ISIP), pore pressure, closure stress, fluid efficiency, reservoir flow
capacity, and fracture leak-off regime. A comprehensive regional or field-wide study of the DFITs is deemed


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very useful for operators at any stage of exploration and development.
In this study, DFITs from a group of 174 Montney and Duvernay wells operated by more than 30
producing companies are quality checked and analyzed consistently. The results from DFITs are then
compared against the standard poroelasticity equation.
The data set of the current study covers a wide range of DFIT conditions and reservoir and geomechanical
properties which helps the authors develop reliable correlations for reservoir characterization purposes.
True vertical depth (TVD) of the wells ranges from less than 1000m to more than 4000m leading to a wide
range of pressure and stress conditions. The study also covers all the geographical areas of Montney and
Duvernay. The results ofthe DFITs are used to develop usefulcross-plot betweenpore pressure and closure
pressure.A relationship between closure pressure from compliance method and G-function analysis method


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for Montney and Duvernay is also provided. Results of example DFITs from depleted areas were also
provided which gives an idea about the depletion level and the associated well performance degradation.
The current study gives field-wide understanding of the variations and distributions of reservoir and
geomechanical properties in Montney and Duvernay based on DFIT analysis of a sizable population of
wells.
Introduction – Reservoir Initialization and Characterization
Reservoir characterization includes a series of direct and indirect methods that are applied to different
reservoir scales spanning from pore to field. Clarkson et al. (2016) reviewed advances in shale/tight reservoir
characterization at different scales using drill cuttings, cores, well tests, and production data.Pressure


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transient analysis (PTA) methods have been historically used as effective tools for characterization of nearDownloaded from http://onepetro.org/SPEURCC/proceedings-pdf/20URCC/3-20URCC/D033S003R005/2367089/spe-199990-ms.pdf/1 by University of Houston user on 30 October 2023


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2 SPE-199990-MS
wellbore and far-field reservoir and estimation of average reservoir pressure. The methods are used to
interpret well bottom-hole pressure data which is influenced by reservoir properties and limits.
DFIT serves as a PTA tool for unconventional reservoirs which provides estimates of some of the
key reservoir and geomechanical properties of the sampled system as pressure transient during DFIT is
controlled by the hydraulically-induced fractures and native reservoir properties.
Operating companies in unconventional reservoirs usually require an overall understanding of reservoir
conditions and parameters across these giant continuous hydrocarbon systems (in addition to local detailed
studies) using field-wide analysis of reservoir characterization tests such as DFIT. In this study, 134 DFITs
from Montney and 40 DFITs from Duvernay formations are analyzed.The key observations and data trends
are presented withsome examples.The focus of this paper is pressure and stress distribution and leak-off


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types in Montney and Duvernay. Permeability estimation from DFIT and fracture treatment analysis during
DFIT will be presented in Part 2 of the paper.
Diagnostic Fracture Injection Test (DFIT)
A DFIT consists of injection of a small volume of liquid or gas at low rate followed by extended shut-in
period when the pressure falloff is recorded ( Craig and Blasingame, 2006). Three key processes occur during
the test including hydraulic fracturing (during injection), early-time fluid leak-off, fracture closure, and late-
time fluid leak-off.DFIT data analysis includes interpretation of the shut-in pressure data usingreservoir
engineering and geomechanicstools.Two parameters for reservoir initialization(pressure and stress) and
two reservoir properties(permeability and leak-off type) are obtained by analyzing the DFIT data ( Fig.
1). Furthermore, treatment pressure during injection period can be used to tune fracture propagation and
tortuosity models.
Figure 1—Parameters and information derived from DFIT analysis.


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Figure 1—Parameters and information derived from DFIT analysis.
Like other well test methods, uncertainty sources in DFIT results include measurements, input data (fluid
properties, petrophysical properties, etc.), DFIT analysis models, and interpretations methods.Different
DFIT pre-closure analysis models are available and have been used in literature and industry. G-function
derivative analysis or Holistic method (Nolte, 1979; Barree et al., 2009) and type curves (Craig and
Blasingame, 2005 , 2006 ) have been widelyused to identify fracture closure, estimate closure pressure
(which is considered as an estimate of minimum horizontal stress), and characterize leak-off type. Fracture
compliance method (McClure et al., 2016) has also been used for identification of fracture closure
whichprovides higher closure pressure compared to G-function derivative analysis.The closure pressures
for all of the DFITs of this study are estimated using both theG-function derivative and compliance methods.


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Examples of different leak-off types including pressure-dependent leak-off (PDL), normal leak-off, and
storage from Montney formation are provided in Fig. 2  (the red and yellow stars show closure pressures from
G-function derivative analysis and compliance methods, respectively). The common methods for estimation
of permeability and reservoir pressure from DFIT after-closure datainclude Gu et al. (1993), Soliman et al.,
(2005)  Craig and Blasingame (2006) .Downloaded from http://onepetro.org/SPEURCC/proceedings-pdf/20URCC/3-20URCC/D033S003R005/2367089/spe-199990-ms.pdf/1 by University of Houston user on 30 October 2023


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SPE-199990-MS 3
Figure 2—Montney DFIT examples with closure pressures estimated from G-function analysis (red stars) and
compliance method (yellow stars) (a) pressure-dependent leak-off (PDL) (b) normal leak-off and (c) storage.
Montney and Duvernay
Montney and Duvernay are the two key Canadian unconventional plays with significant resource potential
(Fig. 3 ). The black dots and lines on the maps illustrate the location of the current wells which are
mostly on production. Uneven distributions of wells in the playsare mostly due to reservoir development
considerationsand variability ofreservoir properties.
Figure 3—Lithofacies distribution map of Montney formation, on the left side (after Edwards
et al. 1994 ) and Duvernay shale map, on the right side (modified from Switzer et al. 1994).
Reservoir parametersincluding pressure, temperature, stress, porosity, permeability etc. across Montney
and Duvernay change geographically and vertically.TVD of the wells drilled in Montney to date range from


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800 m to 4000 m ( Fig. 4a ). Unconventional part of Montney is locatedat TVDgreater than 1500 m whereas
the conventional Montney resides at shallower depth. TVD of the wells drilled in Duvernay to date range
from 2000 m to more than 4000 m ( Fig. 4b ).Downloaded from http://onepetro.org/SPEURCC/proceedings-pdf/20URCC/3-20URCC/D033S003R005/2367089/spe-199990-ms.pdf/1 by University of Houston user on 30 October 2023


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4 SPE-199990-MS
Figure 4—Histogram of TVD for the wells drilled in (a) Montney and (b) Duvernay to date.
Pressure and Stress Variability in Montney and Duvernay
Closure pressure is estimated using the G-function plots (as shown in Fig. 2 ). To estimate reservoir pressure,
the flow regimes areidentified,and then well shut-in pressure is extrapolated using a proper pressure vs.
time plot.Half-slope and zero-slope lines on the log-log diagnostics plots (Fig. 5a and 5b) indicate linear
and radial flow regimes, respectively. All the Montney and Duvernay wells of this study exhibit linear flow.
Radial flow was barely observed for these wells.For consistency, the estimated reservoir pressures from
linear flow are reported. Estimated reservoir pressure from radial flow is preferred if available.The well with
radial flow shown in Fig. 5b  is from Dunvegan formation which has much higher permeability compared
to Montney and Duvernay. Extrapolation of well shut-in pressures corresponding to linear flow of Fig. 5a


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and radial flow of Fig. 5b  are illustrated in Figs. 5c  and 5d.
Figure 5—Flow regime identification and reservoir pressure estimation (a) linear flow (half-slope line) for a Montney well, (b)
linear (half-slope line) and radial flow (zero-slope line) from a Dunvegan well, (c) reservoir pressure estimation corresponding
to the half-slope line in Fig. 5a, and (d) reservoir pressure estimation corresponding to the zero-slope line in Fig. 5b.
Fig. 6  provides the estimates of closure and pore pressure for some of the wells along the blue arrow
shown on the map (TVD increases moving away from the reference point). Similar results are provided
for other regions of Montney as shown in Fig. A1 - A3 in Appendix. As shown in Fig. 6 , reservoirDownloaded from http://onepetro.org/SPEURCC/proceedings-pdf/20URCC/3-20URCC/D033S003R005/2367089/spe-199990-ms.pdf/1 by University of Houston user on 30 October 2023


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SPE-199990-MS 5
pressure, closure pressure (minimum horizontal stress) and the corresponding gradients increase with depth.
Furthermore, minimum stress (closure pressure) and reservoir pressure are strongly correlated. This is
explained in more detail in "Equation of Poroelasticity" section.
Figure 6—Distribution of closure and pore pressure along the blue arrow on the
Montney map.Similar results for other regions of Montney are provided in Appendix.
Fig. 7  provides the data for some of the Duvernay wells along the blue arrow shown on the map. Similar
results are provided for another region of Duvernay as shown in Fig. A4  in Appendix.Downloaded from http://onepetro.org/SPEURCC/proceedings-pdf/20URCC/3-20URCC/D033S003R005/2367089/spe-199990-ms.pdf/1 by University of Houston user on 30 October 2023


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6 SPE-199990-MS
Figure 7—Distribution of closure and pore pressure along the blue arrow on the
Duvernay map.Similar results for another region of Duvernay is provided in Appendix.
Leak-off Types in Montney and Duvernay
Fluid leak-off type in DFIT depends of the relative contribution of matrix and natural fractures/fissures in
dissipation of injected fluid and pressure after the pumping period of the DFIT. "Normal leak-off" happens
in the case of a fully contributing matrix system exposed to the hydraulic fracture whereas "pressure-
dependent leak-off (PDL)" occurs in the case of enhanced surface area and permeability in the vicinity of
hydraulic fracture at high injection pressures. "Storage" occurs in cases with partially contributing matrix
and/or storage of excess water in open features with high storage capacity at elevated pressures.
Leak-off type is an important reservoir signature which might impact the efficiency of hydraulic fracture


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treatment. Leak-off types for the wells of this study (134 Montney wells and 40 Duvernay wells) were
identified using the G-function plot (as shown in Fig. 2 ). The results are summarized in Fig. 8. This graphDownloaded from http://onepetro.org/SPEURCC/proceedings-pdf/20URCC/3-20URCC/D033S003R005/2367089/spe-199990-ms.pdf/1 by University of Houston user on 30 October 2023


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SPE-199990-MS 7
shows that 89% of Montney wells and 52% of Duvernay wells exhibit storage signature. Normal leak-offis
observed in only one Montney well.
Figure 8—Leak-off types in (a) Montney and (b) Duvernay.
Fracture closure time and fluid efficiency distributions in Montney also differ from Duvernay as shown
in Fig 9 . Generally, Montney fractures close faster and therefore fluid efficiency, 100 Gc/(G c+2), is lower
compared to Duvernay. This might be attributed to higher permeability value in Montney.
Figure 9—Distributions of closure parameters in Montney and Duvernay (a) closure time (b) fluid efficiency.
Equation of Poroelasticity
Based on the classic equation of poroelasticity (Terzaghi 1943), the minimum horizontal stress is a function
of pore pressure (p ), vertical and tectonic stresses (σ v and σ t), and rock elastic properties (Poisson's ratio,
v, and Biot's coefficient, α).
(1)
This model is based on some simplifying assumptions and used to explain high-level trends between


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minimum horizontal stress and pore pressure and elastic properties on relative basis. Plot of closure pressure
(approximately equal to minimum horizontal stress) from Montney and Duverany DFITs is provided in Fig.
10a. As shown in Fig 10a , (1) there is a reasonably strong correlation between closure and pore pressure
and (2) the poroelasticity plot from both Montney and Duvernay follow the same line. Furthermore, closure
pressure is estimated using Eq. 1 with assumed v = 0.25, bulk density of 2500 kg/m3 and σ t = 0. Fig. 10bDownloaded from http://onepetro.org/SPEURCC/proceedings-pdf/20URCC/3-20URCC/D033S003R005/2367089/spe-199990-ms.pdf/1 by University of Houston user on 30 October 2023


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8 SPE-199990-MS
shows that the results from Eq. 1 and DFITs are reasonably close. The variability in the results is due to the
variation ofthe reservoir properties across these plays. The results from Eq. 1 can be tuned further by using
measured values of the rock elastic properties.
Figure 10—Poroelasticity in Montney and Duvernay (a) Pore pressure and closure from
Montney and Duvernay DFITs (b) comparison of closure pressure from DFIT against Eq. 1 .
Closure pressures are also estimated using the compliance method. The closure pressure from the
compliance method is usually higher than the G-function analysis method. Fig. 11 provides a cross-plot of
the closure pressure from compliance vs. closure pressure from the G-function analysis for Montney and
Duvernay DFITs of this study. As shown in the plot, the closure pressure from compliance method is on
average 9% greater than the closure pressure from the G-function analysis method.


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Figure 11—Closure pressure from compliance method vs. closure pressure from G-function analysis.Downloaded from http://onepetro.org/SPEURCC/proceedings-pdf/20URCC/3-20URCC/D033S003R005/2367089/spe-199990-ms.pdf/1 by University of Houston user on 30 October 2023


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SPE-199990-MS 9
Pressure Depletion
Prediction of infill/child well performance (early- and late-time) is an important task in reservoir
engineering, development, and management of conventional and unconventional systems. Infill/child well
performance primarily depends on reservoir pressure distribution/depletion controlled by production from
the existing/parent wells. Pressure depletion may impact infill/child performance in two ways: fracture
geometry (geomechanics) and material balance (reservoir engineering).
Reservoir simulation can be used to estimate reservoir pressure at the infill locations based on the
reservoir properties and the historical production data from the existing wells. The pressure forecast from
the reservoir model can be compared against direct measurement of reservoir pressure (from DFIT) at the
infill locations. DFITs from different infill/child wells are very useful in constructing a pressure distribution


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line which in turn can be used to analyze performance of infill/child wells or tune the reservoir models.
Four pressure depletion examples diagnosed by DFIT are provided in the following subsections including
three Montney cases and a Duvernay case.
Example 1
A group of vertical wells from a higher-permeability area of Montney (the wells in orange in Fig. 12) were
completed in single hydraulic fracture and put on production mostly in 1996-1997. Pressure buildup test
data from the Vertical Well V1 (shown in Fig. 12 ) gives initial reservoir pressure gradient of ~8.6 kPa/m
and reservoir permeability of ~5 md. The operator completed a few infill multi-fractured horizontal wells
(wells in green color in Fig. 12 ) in 2011 and converted some of the vertical wells into injectors. DFITs
were run on the horizontal wells prior to completion. Most of the tests were ended before linear/radial flow
is observed. The measured downhole pressure gradient at the end of DFIT from Well H1 and estimated


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reservoir pressure from linear flow are 5.4 kPa/m and 5 kPa/m, respectively. That is, reservoir pressure
gradient at the DFIT location is depleted by ~40%. The toe of Well H1 is in a region surrounded by five
vertical wells with total oil production and gas productions of ~330 MSTB and 2.2 Bscf, respectively. This
is a critical piece of information which helps the reservoir development team tune their reservoir models
and improve production forecasts and estimated recoveries.
Figure 12—Depletion Example 1 – horizontal infill wells in high permeability Montney.
Example 2
Eight Montney wells on an infill pad (wells in red in Fig. 13) were completed after four years of production
form the neighbour pads (wells in blue and purple in Fig. 13) in a gas condensate region of Montney. The
wells were completed in three benches (each bench with ~50m thickness as shown in Fig. 13). Based on


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DFITs taken from the non-depleted wells in this region, initial reservoir pressure gradient is expected to beDownloaded from http://onepetro.org/SPEURCC/proceedings-pdf/20URCC/3-20URCC/D033S003R005/2367089/spe-199990-ms.pdf/1 by University of Houston user on 30 October 2023


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10 SPE-199990-MS
12-13 kPa/m. DFITs were also performed on all the infill wells to estimate pressure across the toe of the
wells on the infill pad. The results are provided in Table 1. Well C1 shows the highest depletion as expected
because it is the closest well to the parent well on the the neighbor pad (both horizontally and diagonally)
and the toe of the well is located along the parent well. Well C2 is less depleted than C1 as it is farther from
the parent well and located at a different layer. On the other side of the infill pad, toe of Well C8 is more
depleted than C7 and C6 due to proximity to the parent well on neighbor pad. However, Well C8 is less
depleted than Well C1; one reason is that the toe of C8 is located at the heel of the parent well whereas toe
of C1 is located along the parent well. This observation implies that pressure from DFIT of a child well may
not represent the whole wellbore because the conditions may change along the wellbore due to the relative


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position of the well with respect to the parent well.
Figure 13—Depletion Example 2 –an infill pad in Montney.
Table 1—DFIT results for the the wells on infill pad of Fig. 13.
Well C1 C2 C3 C4 C5 C6 C7 C8
True Vertical Depth, TVD (m) 2313 2362 2254 2308 2351 2240 2296 2344
Closure Pressure (MPa) 35.3 41.1 37.3 36.2 40 38 36.2 38
Closure Pressure Gradient (kPa/m) 15.3 17.4 16.5 15.7 17.0 17.0 15.8 16.2
Pressure (MPa) 19.7 26.6 24.7 25.5 27 26.6 26
Pressure Gradient (kPa/m) 8.5 11.3 10.7 10.8 12.1 11.6 11.1
Example 3
Two multi-fractured horizontal wells (Wells H1 and H2 in Fig. 14 ) were completed in cased-hole plug and
perf and put on production in 2009. Both wells were completed on the same bench in Montney. In 2017, Well
H3 was drilled toeing toward H1 and H2 at different bench which is 100 m deeper than those wells and DFITDownloaded from http://onepetro.org/SPEURCC/proceedings-pdf/20URCC/3-20URCC/D033S003R005/2367089/spe-199990-ms.pdf/1 by University of Houston user on 30 October 2023


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SPE-199990-MS 11
was run shortly after (see well configuration in Fig. 14). DFIT from Wells H1 and H2 give initial reservoir
pressure gradient of ~14 kPa/m whereas DFIT from H3 gives pressure gradient of ~12 kPa/m (i.e. ~2 kPa/
m pressure depletion) at the toe of the well. This provides an insight about how far pressure front can move
vertically which can help with optimizing the development plan regarding the lower target in Montney.
Figure 14—Depletion Example 3 – depletion in a well toeing to existing wells in Montney.
Example 4
A pad including Wells H2-H5 (as shown in Fig. 15 ) were completed after 15 months of production from a
pad including Well H1 ( Fig. 15). Cumulative production from Well H1 at the time of completing Wells H2-
H5 was ~1.0 Bcf. DFITs from Wells H2 and H5 (located at distance of 270 m and 1150 m away from Well
H1) showed that Well H2 was depleted by ~7MPa (~1.7 kPa/m) whereas Well H5 was not depleted.
Figure 15—Depletion Example 4 – estimation of pressure distribution


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across a new pad completed beside an older pad in Duvernay using DFIT.
Conclusions
A total of 174 DFITs across Montney and Duverany were analyzed in this study. The study gives field-wide
understanding of the variations and distributions of reservoir and geomechanical properties in MontneyDownloaded from http://onepetro.org/SPEURCC/proceedings-pdf/20URCC/3-20URCC/D033S003R005/2367089/spe-199990-ms.pdf/1 by University of Houston user on 30 October 2023


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12 SPE-199990-MS
and Duvernay.Closure pressure and pore pressure from the DFITs show a strong correlation which reads
with standard poroelasticity relationship. The results from DFITs are comparable against the standard
poroelasticity equation. Different DFITs from depleted parts of Montney are also provided which provide
insights regarding the risks associated with depletion in both horizontal and vertical directions.
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Appendix
Figure A-1—Distribution of closure and pore pressure along the blue arrow on the Montney map.Downloaded from http://onepetro.org/SPEURCC/proceedings-pdf/20URCC/3-20URCC/D033S003R005/2367089/spe-199990-ms.pdf/1 by University of Houston user on 30 October 2023


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14 SPE-199990-MS
Figure A-2—Distribution of closure and pore pressure along the blue arrow on the Montney map.Downloaded from http://onepetro.org/SPEURCC/proceedings-pdf/20URCC/3-20URCC/D033S003R005/2367089/spe-199990-ms.pdf/1 by University of Houston user on 30 October 2023


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SPE-199990-MS 15
Figure A-3—Distribution of closure and pore pressure along the blue arrow on the Montney map.Downloaded from http://onepetro.org/SPEURCC/proceedings-pdf/20URCC/3-20URCC/D033S003R005/2367089/spe-199990-ms.pdf/1 by University of Houston user on 30 October 2023


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16 SPE-199990-MS
Figure A-4—Distribution of closure and pore pressure along the blue arrow on the Duvernay map.Downloaded from http://onepetro.org/SPEURCC/proceedings-pdf/20URCC/3-20URCC/D033S003R005/2367089/spe-199990-ms.pdf/1 by University of Houston user on 30 October 2023


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SPE-194348-MS
DFIT Considering Complex Interactions of Hydraulic and Natural Fractures
Amirhossein Kamali and Ahmad Ghassemi, Reservoir Geomechanics and Seismicity Research Group, The
University of Oklahoma
Copyright 2019, Society of Petroleum Engineers
This paper was prepared for presentation at the SPE Hydraulic Fracturing Technology Conference and Exhibition held in The Woodlands, Texas, USA, 5-7 February
2019.
This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents
of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect
any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written


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consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may
not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.
Abstract
Currently, the closure stress is often predicted using the conventional tangent method (i.e., G-function) or
the variable compliance method. Both methods use several restrictive assumptions such as a single planar
fracture. However, the hydraulic fracture often intersects rock fabric features such as bedding planes and/
or natural fractures causing the pressure transient behavior to become drastically different compared to that
of a single planar hydraulic fracture. Closure of the intersected natural fractures might precede that of the
created HF which impacts the interpretation of the pressure derivate plots and also the closure stress. In this
paper we present and use an advanced fracture diagnostic model that can help recognize the signatures of


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rock fabric features and their impact on estimation of the closure stress. An example field data is used to
illustrate the potential impact on closure stress.
The new DFIT model consists of a fully coupled 3D hydraulic fracture simulator with the ability to
handle the opening, propagation, and closure of natural factures so that the pre- and post-closure stress/
deformation of both the hydraulic and natural fractures can be captured. Fracture propagation, HF-NF
interaction, fracture intersection, and DFIT model are integrated into one simulator to provide a more
realistic view of HF propagation and fracture diagnostics in naturally fractured reservoirs. The current
model is developed without any major assumptions concerning the fluid flow, fracture deformation, and
propagation path. Rock/fracture deformation is calculated using a boundary element formulation whereas
the transport processes are solved using finite elements method.


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Our results indicate that natural fractures affect the pre- and post- shut-in response of the hydraulic
fracture in a number of ways. For example, the fracture propagation path, the pumping pressure profile,
and interfering with the post shut-in pressure response. These factors, indeed, impact the estimation of the
minimum horizontal stress which is a key parameter obtained from DFIT. Moreover, our results show how
the normal stiffness of the fracture surface asperities can impact the minimum stress estimation. Closure
of natural fractures is reflected in the slope of the pressure derivative and G-function plots so that correct
interpretation of these signatures is essential to accurate extraction of the Shmin.
Closure of natural fractures is often viewed as a pressure depdendent leakoff mechanism that is reflected
on the Gdp/dG curve. The closure behavior of HF-NF sets is, however, not explicitly modeled in the context


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of pressure transient analysis. Therefore, it is our objective to study the closure behavior of HF-NF sets usingDownloaded from http://onepetro.org/SPEHFTC/proceedings-pdf/19HFTC/1-19HFTC/D012S001R003/1143681/spe-194348-ms.pdf/1 by University of Houston user on 30 October 2023


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2 SPE-194348-MS
a 3D coupled simulator. This novel model is applied to actual field data to illustrate the potential impact on
closure stress and to shed light on the subject of fracture diagnostics in naturally fractured reservoirs. Our
results indicate that the closure behavior of hydraulic and natural fractures in a HF-NF set differs from that
of an isolated fracture due to the effect of stress shadowing. Although the system stiffness method results
in distinct signatures on the diagnostic curves, these signatures are not commonly observed in the field
data. The absence of stiffness signatures in the field cases could be interpreted in two ways: 1) the closure
mechanism assumed in the stiffness/compliance method differs from the actual fracture closure mechanism
or 2) the stiffness of the hydraulic fracture is too low to cause any significant changes in the system stiffness
after closure.
Introduction
Fracture injection/fall-off tests have been extensively used over the past few decades to obtain fracture


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closure pressure and several reservoir properties such as permeability and initial reservoir pressure. The
pioneering work of Nolte (1979)  provided a rigorous means for determining the fracture closure pressure,
leakoff coefficient, and fluid efficiency by introducing the G-function. Nolte (1979) derived the pressure
decline function based on material balance assuming a single planar fracture and constant leakoff from
the fracture surfaces. Following the Nolte's methodology, Castillo (1987) introduced a pressure decline
function where the pressure decline formed a straight line when plotted against the G-function during
the closure. Deviation from the linear portion of the plot was interpreted as the fracture closure. Castillo
(1987)  also improved the constant leakoff assumption in the Nolte's approach by considering other leakoff
mechanisms such as reservoir dominated and filtrate viscosity dominated leakoff modes and formulated the


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required parameters to use in each case. Recognizing that the use of pressure versus G-fucntion could lead
to ambiguous conclusions about the nonlinear leakoff behvaiors, Mukherjee et al. (1991) and Barree and
Mukherjee (1996)  started using the superposition derivative function (i.e., Gdp/dG) along with the pressure
decline plots. Using the combination plots including the P, dP/dG, and GdP/dG plots helps to identify the
cause of the non-linearity in the leakoff behavior. Tip extension, height recession, fissure opening, and
the change of system stiffness can be detected on the Barree and Mukherjee's combination plot. Pivoting
on the fact that neither one of the assumptions in the leakoff formulation (filter cake dominance and the
square root of time decay) is really true, Mayerhofer and Economides (1997) advocated using the simple
log-log diagnostic plots instead of the widely used G-function analysis. To account for the varying leakoff


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rate, Mayerhofer and Economides (1997) used a rate-normalized plot (RNP) and its derivative to detect the
fracture closure and departure from the half-slope line. Craig et al. (2005) conducted a statistical analysis
on the leakoff type and the reservoir permeability of more than 1000 diagnostic fracture injection tests in
the Rocky Mountain Basin, and concluded that the normal leakoff behvaior is very rarely seen. In fact, the
great majority of the tests in their study showed pressure dependent leakoff and other abnormal leakoff
behaviors which makes the G-function interpretation challenging. As an alternative to the conventional
tangent method, several studies have focused on the use of system stiffness or compliance method ( Barree
and Mukherjee, 1996 ; Raaen et al., 2001; Raaen et al., 2005; McClure et al. 2016) to determine the fracture
closure pressure. Raaen et al., (2001) pointed out that there is a change in the system stiffness upon


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closure which is reflected on the pressure and pressure derivative plots. The signature associated with the
change of system stiffness/compliance can therefore help identifying the instance of fracture closure and
the corresponding closure stress. McClure et al., (2016) considered the stiffness change due to the joint
closure as well. On the other hand, the signature of changes caused by the fracture normal stiffness has not
been widely observed in field data and moreover, the interpretation of stiffness (compliance) changes can
be ambiguous. In particular, HF interaction with natural fractures can result is similar signatures that would
complicate the interpretation.Downloaded from http://onepetro.org/SPEHFTC/proceedings-pdf/19HFTC/1-19HFTC/D012S001R003/1143681/spe-194348-ms.pdf/1 by University of Houston user on 30 October 2023


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SPE-194348-MS 3
Traditionally, the impact of natural fractures on the post-shutin pressure decline is implicitly studied
in the context of leakoff behavior. However, the interaction of hydraulic and natural fractures and its
effect on the closure behavior and the pressure decline is not explicitly considered in previous works. In
this work we use a fully 3D hydraulic/natural fracture simulator to investigate the pressure decline in a
system including intersected hydraulic and natural fractures. The mechanical interaction between fractures
is explicitly included in the model to rigorously account for the closure of natural fractures in the diagnostic
fracture injection tests.
Model Development
A fully coupled 3D hydraulic/natural fracture simulator is used in this study to investigate fracture injection
tests and closure stress estimation in the presence of natural fractures. This section describes the constitutive


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equations that are used to construct this model and the numerical methodology to implement the model.
Constitutive Equations
Hydraulic fracturing is a coupled process that involves rock deformation, fracture propagation, fluid
flow, leakoff, thermal, and poroelastic stress components. Proper constitutive equations are required to
describe and capture this multi-physics problem. Unlike fluid-driven hydraulic fractures, pre-exisitng
natural fractures are closed fractures that carry normal and shear stresses. Therefore, the contact status of
the natural fractures should be carefully addressed.
The solution to the problem of an elastic body deformation must satisfy the governing equations for a
given set of stress/displacement boundary conditions. The formulation of this problem relies on the stress
tensor PDE and strain-displacement equations. The solution is completed through a constitutive equation
that relates the stresses to the strains. Using Hooke's law in 3D, this constitutive equation is expressed as


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follows ( Love, 2013 ; Timoshenko and Goodier, 1970):
(1)
Where E is the formation's Young's modulus, v is the Poisson's ratio, σij is the stress tensor components, εij
is the strain tensor components, εkk denotes trace of the strain tensor, and δij is the Kronecker delta function.
Another integral part of the hydraulic fracturing is the fluid flow inside the fracture. Assuming small
variations of the fracture width in time and space, the continuity equation can be expressed as:
(2)
Where q is the volumetric flow rate at location s and time t, w is the local fracture aperture, VL is the leakoff
velocity across the fracture surfaces, and Qinj is the pumping rate. The continuity equation can be formulated
in terms of the fluid pressure and the fracture aperture using the Darcy's law ( Witherspoon et al., 1980):
(3)
In which, μ is the viscosity of the fracturing fluid, and ∇P is the pressure gradient along the fracture.


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Contact elements are used to represent closed natural fractures. The constitutive equations simply relates
the changes of the normal (shear) stress to that of the normal (shear) displacement discontinuity as shown
below ( Goodman et al., 1968 ):
(4)Downloaded from http://onepetro.org/SPEHFTC/proceedings-pdf/19HFTC/1-19HFTC/D012S001R003/1143681/spe-194348-ms.pdf/1 by University of Houston user on 30 October 2023


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4 SPE-194348-MS
Where Kn and Ks are the normal and shear stiffness, and Dn and Ds are the normal and shear displacement
discontinutites for each element. The contact status of the elements may change during the simulation in
both the normal and shear mode. Mohr-Coulomb criterion is used to determine the contact status of the
elements in the shear mode (i.e., ′′ stick′′ and ′′ slip′′ ):
(5)
Where c and Ø are the natural fracture's cohesion and friction angle, respectively. More details about the
contact elements and its implementation can be found elsewhere (Kamali and Ghassemi, 2018; Kamali and
Ghassemi, 2017 ; Safari and Ghassemi, 2015; Tao et al., 2011 )
Numerical Methodology
Fully 3D fracture modeling coupled with the fluid flow results in a set of nonlinear partial differential
equations that are often not amenable to analytical treatments. Therefore, numerical modeling is used to
study problems involving complex fracture simulations. Fracture deformations and stresses are solved


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using displacement discontinuity (Crouch and Starfield, 1983; Curran and Vandamme, 1989) which is an
indirect boundary element method. Displacement discontuity (DD) is proved to be a powerful tool to model
hydraulic fractures and problems involving contact elements such as natural fracutres, and faults (Asgian,
1988 ; Ghassemi and Tao, 2016; Safari and Ghassemi, 2015 ). Stresses at each point on the fracture is obtained
using the following integral form:
(6)
Where Γ is the displacement discontiuty boundary, 
  is the unit induced stress component ′′ij′′ due to
the instantaneous point source DD component ′′kn′′, and 
  is the in-situ stress component at location
X. Figure 1  shows a discplacement discontinuity elements and is displacement components. Numerical
discretization and implementation is provided in Kumar and Ghassemi (2015).


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Figure 1—Schematic showing a displacement discontinuity element and its displacement componentsDownloaded from http://onepetro.org/SPEHFTC/proceedings-pdf/19HFTC/1-19HFTC/D012S001R003/1143681/spe-194348-ms.pdf/1 by University of Houston user on 30 October 2023


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SPE-194348-MS 5
The fluid flow equations are solved using the finite element method by discretizing the fracture surface
into 4-node quadrilateral elements. A linear variation of the fluid pressure is assumed over the elements.
The details of the finite element discretization is discussed in Kumar and Ghassemi (2016).
Results
The post-shutin pressure transient of HF-NF pairs is analyzed in this section to identify the signatures in the
pressure and pressure derivative plots. A combination of log-log diagnostic plot and G dP/dG plot is used
to pick the closure pressure more confidently. Several analyses are conducted to study the impact of in-situ
stress state, intersection angle, and fracture stiffness on the pressure decline curves and diagnostic plots.
The first example shows a typical HF-NF system with a relatively simple intersection in which the
hydraulic fracture intersects the natural fracture with an intersection angle of 90°. The length of the natural


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fracture is three times larger than that of the hydraulic fracture (see Figure 1  and Table 1).
Table 1—Input parameters (assumed values) used in the base case example.
The post-shuting pressure decline is studied for this example using the G-function and also the log-log
diagnostic plot. Both the hydraulic and natural fractures are pressurized above the maximum horizontal
stress prior to the shutin as shown in Figure 2. Therefore, both fractures are open against the in-situ stresses.
It shoud be noted that the pressure can easily rise above the maximum horizontal stress level and open the
natural fracture given that the stress anistorpy is not high in this example. The pump is then shut in and
the pressure is monitored.
Figure 2—Hydraulic and natural fracture pressure before shutin. The fluid pressure is above


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the maximum horizontal stress causing both hydraulic and natural fractures to be open.Downloaded from http://onepetro.org/SPEHFTC/proceedings-pdf/19HFTC/1-19HFTC/D012S001R003/1143681/spe-194348-ms.pdf/1 by University of Houston user on 30 October 2023


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6 SPE-194348-MS
The post-shutin pressure is recorded and plotted against the G-function along with the semilog derivative
G dp/dG in Figure 3-a . The G dp/dG plot shows that there are two sudden changes of slope before the hump
corresponding to the change of the system stiffness; the first due to the natural fracture closure and the
second due to hydraulic fracture closure. The first change of slope could also be detected on the pressure
curve at approximately 4300 psi which is regarded as the closure pressure for the natural fracture (marked
by the blue arrow). The second change of the system stiffness occurs when the pressure falls below the
minimum stress and the hydraulic fracture closes (marked by a circle on the G dp/dG and green arrow on
the pressure plot). The closure pressure obtained from the second change of system stiffness is 4000 psi.
The closure pressure is also picked using the conventional tangent method. This method yields a closure


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pressure of 3950 psi which is a good estimate of the actual input for the minimum stress (i.e., 4000 psi).
Alternatively, the closure pressure is picked from the log-log diagnostic plot in Figure 3-b . The closure
stress obtained from this method (departure from the ½ slope line) is 3950 psi (i.e., Δp = 450 psi) which
is in a good agreement with the tangent method and also the stiffness approach. It is important to note that
although using the stiffness/compliance method provides clear signatures on the G dp/dG plot, one can
misinterpret the fracture closure pressure simply by choosing the first signature which actually corresponds
to the natural fracture closure.
Figure 3—a) Pressure profile and the semilog derivative G dp/
dG for the HF-NF system. b) log-log diagnostic plot for the HF-NF pair.
Angle of Intersection
The angle of intersection between the hydraulic and natural fractures affects their mechanical interaction and


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therefore impacts the coupled hydro-mechanical response. Moreover, the normal stress acting on the natural
fracture plane varies based on its orientation with respect to in-situ stresses. In this example, the pressure
decline and the semilog G derivative is investigated for different angles of intersections. Considering that
the hydraulic fracture is initiated and extended in the direction of the maximum horizontal stress, the angle
of intersection represents the angle between the natural fracture and the maximum stress. The base case
example is chosen such that the HF and NF are orthogonal meaning that the intersection angle is 90°. The
post shutin pressure decline and G dp/dG plots are shown in Figure 4 for different values of intersection
angle. It can be observed in this figure that the first signature on the G dp/dG curve has shifted to the right
indicating a larger closure time for the natural fracture and a lower closure pressure. This is, in fact, due to


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the lower normal stresses on the natural fracture in the second and third cases. It is also interesting to notice
that the second signature has disappeared in the last case with the lowest intersection angle. It appears that
because of the small difference between the closure pressure of the HF and NF in the third case (4000 andDownloaded from http://onepetro.org/SPEHFTC/proceedings-pdf/19HFTC/1-19HFTC/D012S001R003/1143681/spe-194348-ms.pdf/1 by University of Houston user on 30 October 2023


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SPE-194348-MS 7
4072 psi, respectively) the hydraulic fracture signature is overshadowed by that of the natural fracture or
they are both combined to form a larger signature.
Figure 4—Comparison between the pressure response the the G
dp/dG plots for HF-NF pairs with different angle of intersection.
Fracture stiffness
This example shows the impact of normal fracture stiffness on the post-shutin pressure profile and the
pressure derivative Gdp/dG. Figure 5  compares the pressure profile and its semilog derivate for a HF-NF
set with different stiffness values. The angle of intersection is 90° and the rest of the parameters are similar
to those used in the base case example. It can be seen in this figure that the higher stiffness results in a higher
rate of pressure drop once the natural fracture closes (i.e., p < σ n, NF). Hence, the set with higher stiffness
has a more pronounced stiffness signatures as compared to the softer set. Identifying the stiffness signatures


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could become challenging for softer hydraulic or natural fractures (see the HF signature for the softer case
marked by a circle) which limits the application of the system compliance/stiffness method.
Figure 5—Pressure and derivative comparison for HF-NF sets with different normal stiffness.
Natural fracture sets
The pressure transiet could be different in the presence of multiple natural fractures. This example explores
the characteristics of the pressure decline for a hydraulic fracture intersecting two unparallel natural
fractures. To help envision this problem further, the sequence of fracture closure is shown in Figure 6.
The first plot shows that all fractures are open before shutin. The upper right figure shows that the natural
fracture with 90° intersection angle closes first while the HF and the second NF remain open. This is due to


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the higher in-situ stress on its plane causing it to close earlier as the pressure declines. The second naturalDownloaded from http://onepetro.org/SPEHFTC/proceedings-pdf/19HFTC/1-19HFTC/D012S001R003/1143681/spe-194348-ms.pdf/1 by University of Houston user on 30 October 2023


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8 SPE-194348-MS
fracture undergoes closure partially on one wing as shown in the lower left figure. This wing which makes a
sharper angle with the HF experiences more induced compression from the HF and therefore closes earlier
than the rest of second NF. Finally, being subject to minimum horizontal stress, the hydraulic fracture is
the last one to close.
Figure 6—Closure sequence for a set of unparallel natural fractures and a hydraulic fracture. (1) all
fractures are open before shutin. (2) the natural fracture which makes a 90° angle with the HF closes first
due to the higher normal stress on its plane. (3) the second NF closes partially on one wing as a result
of higher compression (stress shadow) caused by the HF, and (4) the HF is the last fracture to close.
Figure 7  shows the post-shutin pressure decline and G dp/dG for this example. It can be observed in
this figure that the derivative shows an early jump and change of slope (blue circle and arrow) which is


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associated with the change of the system stiffness upon the closure of the first natural fracture. The second
signature (green circle and arrow) shows the partial closure of one wing of the second natural fracture where
higher compression is induced by the HF. The last signature which corresponds to the closure of the HF is
not as pronounced as the previous ones considering that the HF normal stiffness is set to a lower value than
that of the NFs. The tangent method yields a closure pressure of 3900 psi which is 100 psi below its input
value for this example whereas the stiffness approach yields a closure pressure of 4100 psi.Downloaded from http://onepetro.org/SPEHFTC/proceedings-pdf/19HFTC/1-19HFTC/D012S001R003/1143681/spe-194348-ms.pdf/1 by University of Houston user on 30 October 2023


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SPE-194348-MS 9
Figure 7—Pressure and semilog G derivative for the set of unparallel natural fractures.
Field example
A field example is shown in this section to highlight the charactaristics of the pressure decline and the
semilog pressure derivate obtained from a field test. The fracture injection test is conducted in shale by
pumping 22 BBLs of water. The injection test is followed by an extended shutin period. Figure 8-a shows
the pressure profile for this test. Pressure decline and its semilig derivative are plotted against G-time for
this test in Figure 8-b . It can be observed in this figure that the fracture closure process does not show signs
of pressure dependent leakoff (PDL) behavior and the before-hump portion of the Gdp/dG curve follows
a linear trend. Moreover, there is not any noticeable changes of slope or spikes before the hump to signify
changes in the system stiffness. The field data is used to obtain the pressure decline and Gdp/dG using


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the compliance/stiffness approach. A relatively low normal stiffness of 0.09 Mpsi/ft is assumed for this
example. It can be seen in Figure 9 that Gdp/dG shows a sudden change of slope even for a soft fracture.
This signature is, however, not observed in the field data. This, in fact, indicates that the fracture closure
mechanism may differ from what is assumed in the system stiffness/compliance method. Alternatively, the
absence of stiffness signatures in the field case shows that the closure of the hydraulic fracture, in this case,
does not contribute to the changes of system stiffness.
Figure 8—(a) Post-shutin pressure decline, and (b) G dp/dG plot for the field example.Downloaded from http://onepetro.org/SPEHFTC/proceedings-pdf/19HFTC/1-19HFTC/D012S001R003/1143681/spe-194348-ms.pdf/1 by University of Houston user on 30 October 2023


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10 SPE-194348-MS
Figure 9—Gdp/dG obtained from the compliance/stiffness method using field data.
Discussion and Conclusions
A fully 3D coupled HF-NF simulator is used in this paper to shed light on the pressure transient of hydraulic
fracture in the presence of natural fractures. Our model is based on a 3D displacement discontinuity method
for the rock/fracture deformation and finite elements method for the fluid flow. Our results indicate that
the pressure profile and the semilog G derivatives differ from that of conventional hydraulic fractures. The
change of slope in the pressure profile as a result of the changes of the system stiffness translates into
distinct signatures on the G-function plot. Having said that, care should be taken when interpreting the
closure pressure as these jumps and changes of slope on the Gdp/dG could cause erronesous interpretations.
As outlined in the simulations, the signature for the natural fracture closure could be wrongly picked as


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the closure pressure. Morever, it is often a difficult task to distinguish between actual stiffness signatures
and the noise in the data in field cases. It was shown in the field example that Gdp/dG shows nearly ideal
leakoff behavior and follows a straightline before the hump. The Discrepency between the simulation results
and the actural field example (absence of stiffness signatures in the field case) could be interpreted in two
ways: 1) the closure mechanism assumed in the stiffness/compliance method differs from the actual fracture
closure mechanism or 2) the stiffness of the hydraulic fracture is too low to cause any significant changes in
the system stiffness after closure. That said, the stiffness signatures that exist in the diagnostic plots might
be caused by the closure of other features such as intersected natural fracture. Using conventional tangent
method or log-log diagnostic plots along with the stiffness/compliance method could help identifying the


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closure pressure more confidently when it comes to naturally fractured reservoirs.
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fracture . Water resources research , 16(6), 1016 –1024 .Downloaded from http://onepetro.org/SPEHFTC/proceedings-pdf/19HFTC/1-19HFTC/D012S001R003/1143681/spe-194348-ms.pdf/1 by University of Houston user on 30 October 2023


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See discussions, st ats, and author pr ofiles f or this public ation at : https://www .researchgate.ne t/public ation/313878526
A New Analysis Technique for Interpreting Injection/Shut-in Tests
Article  · Mar ch 2016
CITATIONS
5READS
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3 author s:
Fahd Siddiqui
Illumina
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Mohamed Y Soliman
Univ ersity of Houst on
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Akif Ibr agimo v
Texas T ech Univ ersity
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All c ontent f ollo wing this p age was uplo aded b y Fahd Siddiqui  on 21 April 2017.
The user has r equest ed enhanc ement of the do wnlo aded file.


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A New Analysis Technique for Interpreting Injection/Shut-in Tests
This paper presents an analysis technique for interpreting fracture treating pressures. The literature contains various models that give rise to analysis 
techniques, all of which rely on material balance. Fundamentally the models are different in their assumption of leak-off phenomenon. There are mainly two schools of thought concerning fluid leak-off. 
The first school of thought considers the fluid leak-off to be a property of the fluid-rock interaction system. It relies on Carter’s equation (Howard 
and Fast 1957, Carter 1957) to describe leak-off and is based on Howard and Fast’s experiments. The disadvantage is its inadequacy to describe the phenomenon if the conditions of Howard and Fast’s experiment are not strictly met during the injection process.
Nordgren (Nordgren 1972), using mass balance, defined the following differential equation for the flow down a fracture:
(1)


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(1)
It should be noted that the fracture area considered in the above equation is different from the area considered in this paper. The above equation uses 
the fluid loss coefficient defined as: 
(2)
Most of the models existing in the literature rely on the experiments of Howard and Fast (Howard and Fast 1957, Carter 1957), which yield the generally accepted relation: 
(3)
However, the conditions of their experiments may not match the field conditions. As noted by Nordgren (Nordgren 1972) the pressure difference depends on location and time. This was neglected by the original experimental study. The other well-known correction to Equation 3 is the so called 
spurt loss.
Based on the above equations, Nolte (K. G. Nolte 1979, K. G. Nolte 1986) presented the well-known G-function method.The second school of thought considers the leak-off as being governed by the flow in porous media. The advantage of this approach is that it describes


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the phenomena adequately. However it is more complicated. Even though the solutions of the linear flow problem are available, the problem becomes complicated once the change in area at each time step is considered.
Mayerhofer,  et al., (1991) and Mayerhofer & Economides (1993) described the leak-off based on the square root of time solution as will be shown later 
in the paper. They considered the pressure drop in the filter cake based on filter cake resistance. They combined the pressure drop due to leak-off and through the filter cake and developed the equation:IntroductionFahd Siddiqui, Mohamed Y . Soliman, Akif Ibragimov, Texas Tech University 
Abstract
The problem of diagnostic fracture injection fall-off tests is considered. A new model is constructed from basic fluid flow and material balance 
equations and making some assumptions about fracture propagation. The resulting equations are solved analytically.


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A thorough analysis methodology has been developed from these solutions to interpret fracture injection pressure tests. Presented method can 
estimate the fracture half-length and height product ( xf.hf) and the closure pressure. The proposed analysis technique is limited to after shut-in and 
before fracture closure period of the test.
The developed model has been validated against two actual field injection data. The datasets serve to augment the demonstration of the application 
steps. Peer Reviewed


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(4)
In the above equation, the fluid leak-off rates depend on the changes in area. Since the equation in its current form is not easy to apply for interpretation, 
Valko and Economides, (1999) offered a graphical form of the Mayerhofer’s method by defining the variables x1,n, x2,n and yn. Since it is not easy to 
establish the leak-off rates ( qloss,j) at each time step, Valko and Economides considered them to be constant and equal to an “apparent” leak-off rate. 
In fact, the authors chose to work with width loss rather than the average apparent leak-off rate. The application procedure still remains complicated, especially as compared to the well-known G-function analysis.
Lamei and Soliman (2014) relied on the linear diffusivity equation to determine leak-off, instead of Carter’s (1957) equation. However, they presented 
a long time solution which does not work for all cases. This fact becomes evident when applying their technique. 
Problem Analysis


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Problem Analysis 
This study subscribes to the second school of thought and instead of relying on the discrete steps of the Mayerhofer’s method, the whole model is developed from first principles and yields equations that are more continuous (without discrete summation terms). The fluid loss is modeled by the governing partial differential equation (Equation 5 ).
Equation 5  is the well-known partial differential equation used to describe the linear leak-off. This equation governs how the leaking fluid will move 
through the porous media and gives the resulting pressure distribution versus time.
(5)
I
nner Boundary Condition
The inner boundary condition is derived starting from mass balance and accounts for injection followed by shut-in. This model relies on Equation 7 
(K. G. Nolte 1986) to describe the width change due to consequent pressure changes. The complete derivation of this equation is given in Appendix 
A. The final form of the fluid loss equation is obtained as follows: 
(6)
W


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(6)
W
here U(t - t i) is the unit step-function to account for injection rate and shut-in, at time t i. 
(7)
(8)
E
quation 6 essentially states that the fluid loss (pressure gradient term) is equal to the injection rate (q) less the change in fracture volume (change in 
width due to fracture compliance) at each time, while the fracture area remains constant. Considering the problem accurately from this perspective accounts for the fluid loss based on the physics of fluid flow through porous media. It should be noted that fracture face pressure drop is not considered here and can be introduced as skin factor.
For verification, two extreme cases were tested. First case is that of no fracture volume change; i.e. all the injected fluid is lost to the formation. In this 
case setting the change in fracture volume to zero (or c
f = 0) results in the conventional boundary condition of constant rate injection followed by


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shut-in (Equation 9 ) This results in the traditional square root of time solution. In the next section, it is showed that the general solution reduces to 
square root of time equation for this case too.
(9)
Th
e second case is that of no leak-off. In this case the diffusivity equation cannot be used since there is no leak-off into the porous media. Hence the 
mobility ( λ) is zero. Based on this, the boundary condition (Equation 6 ) reduces to the following:
(10)
F
rom the definition of width (Equation 7 ), and integrating, the above equation can be written as:
(11)


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Equation 11 is consistent with material balance and shows that the cumulative injected volume (left-hand side) is equal to the created fracture volume 
(right-hand side).
The extreme cases validate the model by obtaining expected results and therefore the problem needs to be solved for the intermediate case.Inner Boundary Condition Assumptions
1.  The fracture area is already in place and injection and shut-in processes tend to cause a change in fracture width only and lead to fluid leak off  
into the formation (Equation 6 ). This assumption limits the applicability to the shut-in period.
2.  After stopping injection (shut-in) the fracture begins to close along the hinge at the tip. This implies a reduction in width (according to fracture
stiffness) while the fracture area remains constant, which is a reasonable assumption.
3.  Since the fracture volume is small and the fluid compressibility is small, their product can be neglected. Hence the fluid storage/compressibility


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effects in the wellbore and the fracture can be neglected.
4. The width is a function of net pressure defined by Equation 7 .
Model Assumptions
Following assumptions were considered for using the linear diffusivity equation and to develop the overall model:
1. Homogeneous and isotropic porous medium.
2. Negligible gravity effects.
3. The fluid is leaking off linearly into the porous media.
4. Isothermal conditions prevail throughout the injection period within the porous medium.
5. Fluid has small, constant compressibility in the porous media.
6. Fracturing fluid has a comparable mobility with the formation fluid.
7. Fracturing fluid behaves like a Newtonian fluid.
8. The fracture has an infinite conductivity, i.e. no pressure drop inside the fracture.
Analytical SolutionThe analytical solution to the problem described by Equations 6 and 5 was obtained using the Laplace transform. Analytical inversion was carried out


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by convolution theorem. Appendix B gives the essential steps required to arrive at the solution in Equations 12 and 13. Equation 13 is the definition 
of the F-function.
(12)
(13)
M
athematical and Physical Inspection
Mathematical inspection of the solution suggests that the expected solution would show a rise in pressure (during injection time) and then begin to decline during the shut-in period. The boundary condition does not effectively model the injection part because it does not account for the propagation of fracture but only the change in width (assumptions 1 and 2). Therefore the injection part of the solution is not useful. However since the fracture area is assumed not to change significantly during the shut in period, the boundary condition effectively models the after shut-in and before closure part. Because of this, the rest of this study will concentrate on this portion and develop an analysis methodology to analyze this part.
Because setting the fracture stiffness to zero ( c


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Because setting the fracture stiffness to zero ( c
f = 0) results in a conventional boundary condition of constant rate injection followed by shut-in, setting 
cf = 0 should also reduce the solution presented in Equations 12 and 13 to conventional square root time equation. Careful examination shows that in 
fact that is the case and setting c f = 0 results in Equation 14.
(14)
C
onsequently, this leads to a deeper understanding of the solution in Equations 12 and 13. The first part of the F-function (Equation 13) describes 
the change in pressure due to rate (square root of time solution) and the second part describes the correction to that pressure due to changes in width


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(fracture stiffness). Mathematical inspection of the solution reveals that the solution makes intuitive sense and calculates a pressure response which is coherent with the model description and actual phenomenon. Because we used the partial differential equation for obtaining pressure distribution due to leak-off into the porous media, the model is physically rigorous.


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Solution Validation
The solution was validated by a three prong test.
1. The Laplace space solution was numerically inverted using Stehfest’s algorithm (Stehfest 1970) and found to be matching the analytical solution.
2.  The equation along with its boundary condition was solved numerically using a commercial finite element solver and found to be matching the 
analytical solution.
3. The resulting analysis methodology was applied to real field data and yields reliable results.
Analysis Methodology
A simplified methodology is presented and aided with example data to demonstrate the power and effectiveness of this method. As a first step, the analysis methodology depends on the prior knowledge of fracturing fluid mobility (ratio of permeability to viscosity) and fracture compliance ( c
f). 
Knowing these two parameters, the process described below will aid estimating the fracture half-length times height product (A = 4x fhf). The mobility


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of the fracturing fluid can be estimated by the after closure analysis (Soliman, et al. 2005) if it is not known a priori. 
From the solution in Equations 12 and 13 it may be observed that a plot of (p - pr) vs. F and F.dp/dF vs. F would be a straight line with an intercept of 
zero. Based on this, the methodology can be summarized in the following steps:Step 1 : Convert the time to F-function described by Equation 13
Step 2
:  Plot ( p - p r) vs. F and F.dp/dF vs. F on the same graph paper. It should be noted that F decreases as time increases, and it is suggested to plot F 
in reverse direction.
Step 3 :  B ecause both plots would have the same slope and would be passing through the origin, the straight line can be easily identified. The straight 
line manifests itself when the F.dp /dF vs. F plot superimposes on (p - p r) vs. F. 
Step 4 : Find the slope (m) of the straight line.
Step 5 : Using the slope find the area of the fracture:
(15)


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Step 5 : Using the slope find the area of the fracture:
(15)
Step 6 : Closure pressure can be obtained from the late time deviation of data from the straight line. 
Uncertainty in Analysis
Like all pressure transient analysis problems, the quality of analysis depends on the quality and accuracy of the input parameters: fracture height, Y oung’s modulus, reservoir pressure, and fluid/rock properties. Pressure transient analysis is an inverse problem and therefore there is always a chance of non-unique interpretation, some of which can be mitigated by knowing permeability in advance, thereby diverting the uncertainty towards half-length. The capability of the proposed analysis technique of superimposing the pressure and pressure derivative for obtaining the straight line is a device which mitigates much of the uncertainty associated with finding the correct straight line and hence the fracture half-length.
Application to field data


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Application to field data
The above method was applied to two real field injection shut-in tests dubbed A-1 and A-2.
Example A-1
Figure 1 shows an almost constant rate injection of 20 bpm (0.053 m3/s) for about 22 minutes. Table 1 summarizes the essential input parameters 
needed for analysis.
Figure 1 . Injection pressure and rate data for Example A-1.
The F-function was computed for time larger than shut-in time. The plot of pressure difference ( p - pr) and F.dp/dF was plotted against the F-function 
(Figure 2). The F-function is plotted in reverse direction so that real time is increasing to the right. A straight line was identified based on the conditions that it should pass through the origin and that the pressure difference and pressure derivative will align with each other in that region. Even


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though the pressure derivative is not smooth, it is evident that it begins to oscillate around the pressure difference plot. A least square straight line, fit 
on the pressure difference, through this region, and passing through the origin yields an excellent R2 of 0.999. Qualitatively it can be observed that the 
same line fits through the derivative as well. A G-function plot is also demonstrated in Figure 3.
Figure 2 . F-function plot for A-1.
Analysis RecommendationsThe late deviation from this straight line gives the closure stress. The early deviation is due to the assumption that area is not changing during shut-in, which is being violated initially mainly because of the phenomenon of height recession. This is confirmed by looking at Equation 15. A straight line


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fit through the early time region would yield a gentle slope (larger area). The slope begins to increase (which in turns means a decrease in area) until it becomes constant in the middle time region, after which the fracture closes. Another way of viewing this is by observing that a lower pressure than the straight line in the early time region shows that the pressure is not “sufficient” to keep the area constant, and therefore the area needs to recede to reach equilibrium with the pressure. Once this equilibrium is attained, a straight line is visible in the middle time region. The slope of the line (33 MPa) may now be used to calculate fracture area using Equation 15. Table 2 summarizes the analysis results.
Figure 3 . G-function plot for A-1.Table 1 . Input parameters for Example A-1.
k/μ 2.5E-6 m2/MPa.s 2.5 md.cp-1
cf 1.3E-3 m.MPa-13e-5 ft.psi-1
φ.ct 1.4E-3 MPa-19.5E-6 psi-1
pr 25 MPa 3630 psi
Table 2 . Results of analysis for Example A-1.
Slope  MPa 0 psi
Area 620 m26720 ft2
xf.hf 155 m21680 ft2


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Slope  MPa 0 psi
Area 620 m26720 ft2
xf.hf 155 m21680 ft2
Closure Stress 47 MPa 6815 psi
p-pr=66 F


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The G-function analysis plot confirms these results. The closure stress found from the G-function plot was around 45 MPa (6525 psi) which shows 
close agreement within 5% of this technique.
Example A-2
Figure 4 shows the injection pressure data for A-2. Injection was carried out at almost constant injection rate of 10 bpm (0.0253 m3/s) for about 6 
minutes. Table 1 summarizes the essential input parameters needed for analysis.
The F-function was computed for times larger than shut-in time. Figure 5 shows the F-function plot for A-2. The straight line was identified fairly 
easily from the alignment of the derivative plot with the pressure difference plot. A straight line fit on the pressure difference, through this region, and 
passing through the origin yields an excellent R2 of 0.999. A G-function plot is also demonstrated in Figure 6.
Figure 4 . Injection pressure and rate data for Example A-2.


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Figure 4 . Injection pressure and rate data for Example A-2.
Analysis RecommendationsThe deviations from the straight line are quite different from Example A-1. The authors attribute these deviations to violation of different assumptions. The late deviation from this straight line gives the closure stress. However, in this case, the early deviation is attributed to the phenomenon of after growth. A higher (excess) pressure than the straight line causes the fracture to keep growing after shut in. The growth stops when the pressure dissipation reaches equilibrium with the width reduction and leak-off. This is confirmed with the observations earlier based on Equation 15. The slope


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begins to decrease from the early-time region to the mid-time region. The decrease in slope equates to an increase in area. The increase ceases during the mid-time region, when the slope becomes constant until fracture closure. The slope of the mid-time region was found to be 101 MPa which is used to calculate fracture area using Equation 15. Table 4 summarizes the analysis results.
Figure 5 . F-function plot for A-2.Table 3 . Input parameters for Example A-2.
k/μ 1E- m2/MPa.s 0.1 md.cp-1
cf 1.3E-3 m.MPa-13e-5 ft.psi-1
φ.ct 2.8E-3 MPa-11.9E-5 psi-1
pr 8 MPa 1190 psi


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The G-function plot confirms the early-time deviation, however fracture closure is not unambiguously evident. An approximate value of 85 MPa was 
picked from the plot which matches the F-function analysis result.
Figure 6 . G-function plot for A-2.
Conclusion
The problem of fluid leak-off after shut-in and before closure was solved rigorously based on the linear flow diffusivity equation instead of relying on Carter’s leak-off equation. The resulting solution was used to develop an analysis methodology which gives reliable results for analyzing the before-closure data. This technique yields reliable values for fracture area and closure stress. Application of the technique was demonstrated with two real field data sets.
Nomenclature
A  = F racture Area, m2
Ac  = Cr oss-sectional area of fracture, m2
cf  =  F racture stiffness or the inverse of fracture 
compliance, m.Pa-1
cfluid-loss   = Fl uid loss coefficient, m.s-1/2
cfluid  = Fl uid compressibility, Pa-1
ct  = T otal compressibility, Pa-1


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ct  = T otal compressibility, Pa-1
E'  = P lane strain modulus, Pa [ E'= E/(1-ν2)]
F  = F -Function defined by Equation 13
hf  = F racture height, m
k  = P ermeability, m2 [1 md=9.86923x10-15 m2]
L  = L aplace transform
L-1  = I nverse Laplace transform
ṁ  = M ass flow rate (kg/s)
p  =  M easured pressure at wellbore, Pa  
[1 psi = 6894.76 Pa]
pc  = C losure stress, Pa [1 psi = 6894.76 Pa]
pD  = Dim ensionless pressure
pr  = R eservoir pressure, Pa [1 psi = 6894.76 Pa]
q  = I njection rate, m3/s [1 stb/d = 1.84x10-6]
qloss  = R ate of fluid leak-off, m3/s
R0  = Fi lter-cake resistance, Pa.s/m
RD  = Dim ensionless filter-cake resistancerf  = R adius of fracture, m [Radial Geometry]
t  = T ime, s
tD  = Dim ensionless time
ti  = I njection time, s
tiD  = Dim ensionless injection time
u  = L aplace parameter
U  = U nit step-function
v  = V elocity, m/s
w  = F racture width, m [1 ft = 0.3048 m]
x  =  Fl uid flow direction orthogonal to fracture area, 
m [1 ft = 0.3048 m]
xD  = Dim ensionless distance


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m [1 ft = 0.3048 m]
xD  = Dim ensionless distance
xf  = F racture half-length, m
y  =  Fl uid flow direction along the fracture length,  
m [1 ft = 0.3048 m]
λ  = M obility, m2/Pa.s [ λ = k/μ]
μ  = V iscosity, Pa.s [1 cp = 0.001 Pa.s]
ν  = P oisson’s ratio
ρ  = Fl uid density, kg/m3 [1 lb/ft3 = 16 kg/m3]
τ  = A lternate time, s
τD  = D ummy integration variable
φ  = P orosity
 
 Table 4 . Results of analysis for Example A-2.
Slope  MPa 4500 psi
Area 1 m212 ft2
xf.hf  m25 ft2
Closure Stress 85 MPa 12335 psi


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References
Carter, R. D. 1957. “”Derivation of the General Equation for Estimating the Extent of the Fractured Area” Appendix to: “Optimum Fluid Characteristics  
 f or Fracture Extension” by G. C. Howard and C. R. Fast. ” Mid-Continent District, Division of Production. Tulsa: API-57-261. 261-270.
Howard, G. C., and C. R. Fast. 1957. “Optimum Fluid Characteristics for Fracture Extension. ” Mid-Continent District, Division of Production. Tulsa:  
 AP I-57-261. 261-270.
Lamei, C. H., and M. Y . Soliman. 2014. “New Before Closure Analysis Model for Unconventional Reservoirs. ” SPE/EAGE European Unconventional  
 C onference. Vienna: SPE 167790-MS. doi:http://dx.doi.org/10.2118/167790-MS.
Mayerhofer, M. J., and M. J. Economides. 1993. “Permeability Estimation From Fracture Calibration Treatments. ” SPE Western Regional Meeting.  
 A laska: SPE 20639-MS. doi:10.2118/26039-MS.


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A laska: SPE 20639-MS. doi:10.2118/26039-MS.
Mayerhofer, M. J., M. J. Economides, and K. G. Nolte. 1991. “ An Experimental and Fundamental Interpretation of Fracturing Filtercake Fluid Loss. ”  
 S PE 68th ATCE. Dallas: SPE 22873-MS. doi:http://dx.doi.org/10.2118/22873-MS.
Nolte, K G. 1979. “Determination Of Fracture Parameters From Fracturing Pressure Decline. ” SPE 54th Annual Technical Conference. Las Vegas: SPE  
 8341-MS. do i:http://dx.doi.org/10.2118/8341-MS.
Nolte, K. G. 1986. “ A General Analysis of Fracturing Pressure Decline With Application to Three Models. ” SPE Formation Evaluation (SPE 12941-PA)  
 571-583. do i:http://dx.doi.org/10.2118/12941-PA.
Nordgren, R. P . 1972. “Propagation of a Vertical Hydraulic Fracture. ” SPE 45th Annual Conference. Houston: SPE 3009-PA. doi:http://dx.doi.  
 org/10.2118/3009-PA.
Soliman, M. Y ., D. P . Craig, K. M. Bartko, Z. Rahim, and Z. M. Adams. 2005. “Post-Closure Analysis to Determine Formation Permeability, Reservoir


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P ressure, Residual Fracture Properties. ” SPE Middle East Oil and Gas Show and Conference. Bahrain: SPE 93419-MS. doi:10.2118/93419-MS.
Stehfest, H. 1970. “Numerical Inversion of Laplace Transforms. ” Communication of the ACM 13 47-48. doi:http://dx.doi.org/10.1145/361953.361969.
Valko, P . P ., and M. J. Economides. 1999. “Fluid-leakoff Delineation in High-permeability Fracturing. ” SPE Journal (SPE 56135-PA). doi:http://dx.doi.  
 org/10.2118/56135-PA.
Appendix A: Derivation of Inner Boundary Condition
For x = 0,
  
    
 Recognizing:


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Since fracture volume is small relative to the total injected volume, and compressibility is small, the compressibility term can be neglected.
For the case of shut-in at ti assuming that only width of the fracture is changing due to compliance and area remains constant as the fracture closes 
along the hinge at the tip. By using the step-function U:
 
Appendix B: Solution of Equations 5 and 6
Inner Boundary Condition: 
Initial Condition:  
 
Consider the following dimensionless variables:
Writing the equations in dimensionless variables:
Inner Boundary Condition: 
Initial Condition:  
Outer Boundary Condition:


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Applying Laplace Transform to the inner boundary condition:
Applying Laplace Transform to the partial differential equationSolving the above differential equation and using the initial and outer boundary conditions
  
After substituting from outer boundary condition
Using convolution theorem to invert the above expression
After simplification and removing the step-function:
Carrying out the integrals and simplifying:
Substituting the definitions of dimensionless variables:


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Good afternoon, everyone, and welcome to our webinar. Thank you for joining us this afternoon. My name is Jomana Tarekaptarazic. I'm studying for a master's degree in petroleum engineering at University of Houston, and I will be your moderator this afternoon. Today, we're pleased to welcome Dr. Mohamed Soliman, a very special guest from University of Houston. Professor Solomon is the chairman of the Petroleum Engineering Department at University of Houston. Dr. Solomon has a long industrial and academic experience. He holds 35 patents on hydraulic fracturing, operations, analysis, testing, and conformance applications. operations, analysis, testing, and conformance applications. He is also the author and co-author of over 250 technical papers and articles in various areas of petroleum engineering. Dr. Solomon is a registered professional engineer in the state of Texas. He is a certified petroleum engineer in the state of Texas. He is also a distinguished member of the Society of


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engineer in the state of Texas. He is a certified petroleum engineer in the state of Texas. He is also a distinguished member of the Society of Petroleum Engineers and a fellow of the National Academy of Inventors. He has authored several books and several chapters on stimulation and testing. Dr. Suleiman is the primary editor of Fracturing Horizontal Wells, published by McGraw-Hill in 2016. He is the co-author of Optimization of Hydraulic Fracturing Stages and Sequences in Unconventional Formations, published in 2018 by CRC. Dr. Solomon holds a BSc with top honors from Cairo University and a master's degree and a PhD from Stanford University, all in petroleum engineering. And today he will share with us his expert opinion on hydraulic fracturing. If you haven't already, please make sure to check out our previous sessions on Pyropetro YouTube channel and join our Facebook page Arab Oil and Gas Academy for our latest updates on our upcoming courses. Please feel free to send your


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YouTube channel and join our Facebook page Arab Oil and Gas Academy for our latest updates on our upcoming courses. Please feel free to send your questions in the chat box and we will answer as many questions as possible. With that I ask you to give your full attention to Dr. Mohamed Suleiman and help me in welcoming today's very honorable guest. Please welcome Dr. Mohamed Solomon. Hi, Dr. Solomon. Thanks for joining us today. We're so glad to have you on Pipe Petro. Thank you very much for the kind introduction. Welcome, everyone. My name is Mohamed Solomon. As was mentioned, I want to talk to you today and for three more lectures in October, about Hydraulic Fracturing. It is ... the course is, is an introductory course, ... so we don't go into a lot of details, but we do have in University of Houston, obviously, courses that we, we teach in more details in Hydraulic Fracturing. I personally taught both undergraduate classes, as well as very the involved graduate course. Here we're


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in more details in Hydraulic Fracturing. I personally taught both undergraduate classes, as well as very the involved graduate course. Here we're just giving you an... an outline of Hydraulic fracturing. Maybe in the future we go in some specific areas of hydraulic fracturing. Basically I'm going to cover four lectures, about an hour, hour and 15 minutes each. The first one will be, which is today, we'll talk about the history of fracturing, and I'm going to also talk about fluid flow aspects when you have a hydraulic fracture intersecting the water. Next lecture will be talking about in-situ stresses, then we'll talk about fracture design, and then we'll talk about this thing. There is a lot more that one can discuss later on. ... A lot of material that you are going to see in ... in today, and also in the... later in, in the course, in those lectures, are taken out of these two books. The history, if you want to learn more about the history of hydraulic fracturing, we wrote a very


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those lectures, are taken out of these two books. The history, if you want to learn more about the history of hydraulic fracturing, we wrote a very interesting chapter in that second book, Optimization of Hydraulic Fraction Stages, Sequencing in Unconventional Formations. The rest of the chapters, the rest of the lectures, you're going to find a lot in, in fractional and horizontal well, ... wells. I also plan to, to post those ... a PDF copy of those lectures on, on, online for you to, to see, and you can download. Basically on Researcher Gates, and if you get on Researcher Gates, type my name, you're going to find a lot of material. I will be putting the... those lectures one by one on ... on Research Gates. So, let us talk first about ... history of fracturing. First, the first thing you wanna learn is, what is the basic principles behind the hydraulic fracturing? All you are doing, and we'll ... we'll look at equations later on, hopefully we'll have time to do that. You are


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behind the hydraulic fracturing? All you are doing, and we'll ... we'll look at equations later on, hopefully we'll have time to do that. You are injecting fraction fluid at high enough rate, that the formation cannot take all that fluid. And so what is happening is the wellbore itself balloons a little bit, it gets a little bigger and the formation breaks down. It breaks down under tension. It is similar, in a way, if you blow into a balloon it gets bigger and eventually it pops, it breaks. This is the same thing what's happening inside the well. It balloons, it gets a little bigger, and we have done a lot of experiments, both in the lab, as well as in the field, where we could see that. The well balloons a little bit, and then it breaks in tension, and we're going to talk about, when we get about, talk about stresses, we'll see how this happens. Once it breaks, you are forming a hydraulic fracture. Hydraulic fracture is because as the word mean is you have liquid going in and now is


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happens. Once it breaks, you are forming a hydraulic fracture. Hydraulic fracture is because as the word mean is you have liquid going in and now is usually liquid is water-based as an aqueous phase. Once you propagate the fracture eventually you're going to stop the fracture or close again in order to keep it open. You want to, during the injection process, you want to put something in there that keeps the fracture open after the fluid all leaks off. And that something is your proponent. People used to use all kinds of things before, including walnut shells, if you go back to the 40s and late 40s and early 50s, they used to say walnut shells. Now, then we talk about sand, actual natural sand, then you use artificial propant, and these days use all kind of things and as we're going to discuss later it depends on the formation how deep it is how what stresses and so on and so forth we'll get into that the important thing for us to know now is you're using propant to keep these


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deep it is how what stresses and so on and so forth we'll get into that the important thing for us to know now is you're using propant to keep these fractures open and we're going to see why we want to keep it open when we get into the fluid flow part of it later this morning. So let us talk a little bit, let us talk a little bit about the birth of the process. The idea of hydraulic fracturing, ... came up several years ago, almost 70 plus years ago. And it was conceived by a gentleman called Floyd Farris. It was a small oil company, that later on became Amaco. I... you probably don't even recognize the name, small oil company, that later on became Amaco. I, you probably don't even recognize the name, because Amaco was taken over by, by BP many years ago. Maybe not, when I talk about many, many years ago, I believe it was in sometimes in the 90s. So it is still, it is still 20 plus years ago. The first job was successful, at least mechanically. They fractured the weld, they created


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in the 90s. So it is still, it is still 20 plus years ago. The first job was successful, at least mechanically. They fractured the weld, they created the fracture that they wanted, however there was no production whatsoever. One of the problems they had is in order to carry sand or propant you need to, and in order for the fluid not to leak off into the formation, you want to make it a little thicker, a little higher viscosity, so you gel the fluid. But you, at the end of the job, you need to break that gel again, otherwise it will damage the formation. So these days, at the time, let me go back, at the time, the ferrite job, they did not have a gel breaker included in the fluid. And so what happened, you have no gel breaker, so there's no production, it's damaged well. So obviously, jobs after that, they added the gel breaker, very much like what we do right now, where it start acting, it has delayed reaction. So delayed reaction, very much like when you, one takes a Tylenol and it


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like what we do right now, where it start acting, it has delayed reaction. So delayed reaction, very much like when you, one takes a Tylenol and it releases slowly with time. This is very similar thing. Now you have the breakers released slowly with time, it breaks the gel at the end of the job and you can clean it up, it will be no problem. So that the second and third job worked very well. And, the company, Stan Lloyd Oil and Gas Company, got a patent back in 1948. That's probably one of the most valuable patent any company had got. So, 1948 was a very good year. There was some experiments, let's talk about some of the experiments that happened before. Early on, they were not using, some of the cases were not using water, they used napalm. Napalm is a very nasty fluid, basically it is gel decrucine and gasoline. So you can see it can catch a fire. It actually, napalm was used before and after that time in wars. And so if you can Google napalm, you're going to find out where it was


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catch a fire. It actually, napalm was used before and after that time in wars. And so if you can Google napalm, you're going to find out where it was used. It actually caused a fire in one of the early jobs and the inventor of hydraulic fracturing died in that fire. So it is after that people obviously started shying away from using fluids like that. We'll talk about fluids in a minute. Okay, the revenue from that patent that was issued was actually is the contributed very much to construction of Amaco production research lab in Tulsa, Oklahoma, which has been given to one of the universities now since Amaco moved out or was taken over by BP. So let us talk about how it progressed. So, you started the first job, it was in Hugerton, the first job was in Hugerton Field in Kansas, United States. By late 1948, ... we... there was 23 wells that were fractured, so it was very successful. No other patent was applied as fast and as high impact as the patent on hydraulic fracturing. At the


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that were fractured, so it was very successful. No other patent was applied as fast and as high impact as the patent on hydraulic fracturing. At the time, they called it HydroFrac. And in 1949, a company, a service company called the Hauko got exclusive license for that hydrofrag process. Hauko is actually a Haliburton company. It is, as you can see, it is a Haliburton oil well cementing company. Haliburton company started as a cementing ... company and then ... it got the license for hydraulic fracturing, so it became mostly a hydraulic fracturing company. I'm not talking about it, by the way, just for your information, I worked for almost 32 years for Haliburton company, before going to Universities ... about 10 years ago, actually almost exactly 10 years ago. The commercial application increased very quickly, you have 322 wells that were fractured in one year, success rate was very high. Even offshore application were applied. The first company that did the offshore fracturing was


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fractured in one year, success rate was very high. Even offshore application were applied. The first company that did the offshore fracturing was Kermagui. It doesn't exist anymore. It was taken over by Anadarko and Anadarko was also taken over by another company. So as you can see, there is a lot of consolidation in the industry. It was offshore, although it's only 18 feet of water, but it is still, it was 12 miles offshore. And now we drill and we fracture wells that are thousands of feet underwater. By 53, we had 20,000 jobs. Other companies were licensed to perform hydrofrack, you can see Slumber J, Liberty, Baker Hughes and all those companies came in, and by mid 50s were the high rate waterfracks and large jobs. The, one interesting evolution is, initially people thought that hydraulic fractures are horizontal, basically along the bidding planes. Till there is a paper by Hubert and Willis, in 56 that talked about the fraction should be vertical, unless you are in a very shallow


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bidding planes. Till there is a paper by Hubert and Willis, in 56 that talked about the fraction should be vertical, unless you are in a very shallow situation. Most of our applications are deeper than 1500 feet or so, they are vertical fractions. We are going to actually look at the equation from Hubert and Willis paper, is a extremely important paper that affected the development of hydraulic fracturing. Another paper that was very influential too, was one by Anderson and Stahl, 1967. He basically looked at hydraulic fracturing in the field and at the time there was a start of running experimental experiments in the lab, in rock mechanics labs, and it did confirm that fractures are indeed usually vertical. Does not mean you are not going to see horizontal fractures. They are fairly rare and under certain conditions. usually vertical. Does not mean you are not going to see horizontal fractures. They are fairly rare and under certain condition. We'll get into it very little later on.


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not mean you are not going to see horizontal fractures. They are fairly rare and under certain condition. We'll get into it very little later on. But majority of your fractures will be vertical. We'll talk about when we get in stresses next week, we'll talk more about this issue. Let me show you a couple of pictures so you see what is, how it used to be done. You see this is very early frack and you can see they were just dumping sand by hand and I can tell you from personal experience that this is usually a very bad idea. Now we don't do that, now everything is mixed properly in a blender. I had to run an experiment, a field experiment actually, not exactly, it was not exactly a field, but without going into detail, it was an experiment that we're using an actual fracturing truck. So we're injecting at fairly high rate, about one barret per minute. So for a short time, we were looking at specific effect without, I don't want to go into that specific research, but when we try to do it


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per minute. So for a short time, we were looking at specific effect without, I don't want to go into that specific research, but when we try to do it by hand, things did not really work well, because you can't keep constant sand concentration. So, this is probably one reason they were having some cases that were successful, and some that were not. By the way the experiment that I ran I've never written a paper on that, there was reasons for that, but they were very successful I can tell you. Historical based fluid trends, you can see that although it started with oil-based fluid, quickly people moved into water-based fluid, so it became the dominant application. And also, back in the late 70s and 80s, people started getting into using foam. Foam is a very, very good carrier of problems. However foam does not give you, since it's mostly air, it does not give you a larger hydrostatic head. So it works very well in shallower formations, formation that does not have very high pressure. If


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it does not give you a larger hydrostatic head. So it works very well in shallower formations, formation that does not have very high pressure. If you have high pressure formation, you have deformation, foam may not be exactly the best option. People shy away from oil-based fluid because of potential danger, you know. Although there was some recent, fairly recent work where propane was used, but many major service companies shied away from it. Actually the patent for that fluid was done, if I remember correctly, by a Halliburton company and they sold the patent because they did not want to be involved in potential risk. In the 60s, now people knew that fractures are mostly vertical, thus we have more viscous gel defluid and became a very high problem to concentration in late 70s. When I say high problem concentration, I remember some of the cases we were involved in were all in excess of six pounds per gallon. Some of them were going to 12, 14 and 16 pounds per gallon. That's very


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of the cases we were involved in were all in excess of six pounds per gallon. Some of them were going to 12, 14 and 16 pounds per gallon. That's very high, that's... you don't do that anymore. You do it only if you have a very high probability formation. And, we're going to talk a little about fracturing high probability formation. So, things will to work well and people started looking at stimulation by using, having a computer van on the job collecting information. Pressure, rate, analysis of fluid to get the N prime and K prime the properties of the fluid. Okay, so this is basic summary of how the fluid, fraction fluid went on. If I'm talking too fast, please, you know, let me know. But you can listen to it again. Then there was a the increase in oil price really did change things. So let us look at at prices and this price is if you look at it and looks a little different is because we took all the prices to 2015 base and so if so if you look at the prices in 60 and 70, they were


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look at it and looks a little different is because we took all the prices to 2015 base and so if so if you look at the prices in 60 and 70, they were two or three dollars per barrel. But when you take it to 2015, it was close to $2015. It would be close to $20. And you can see back in 73, there was a sharp jump in price of oil. If I'm sure most of you are too young to remember or to even know, unless you read history books. That's when there was a war between Israel and its neighboring countries, the Arab countries, and it was an oil embargo. And oil prices went up and it was very difficult to go back to that original price. And that's actually when it goes up to $60 per, the oil goes up to $60 per barrel, or 50 goes to 60. That means that you want to, you can spend more money to bring in that oil, and that actually affected fracturing. So, started in the late 70's, in mid, mid to late 70's, there was a lot of effort done into fracturing, a lot of technology been developed. I was


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So, started in the late 70's, in mid, mid to late 70's, there was a lot of effort done into fracturing, a lot of technology been developed. I was lucky, I actually got my PhD in 78, and you can see in 78, 79 prices were going up, companies were hiring, activities were just unbelievable. In 1980, 1981, the price of oil was equivalent to $100, with today's dollars. But, in mid-80s, there was a sharp decline in price. 86 was a crash in, in the whole business, ... different from what we see right now, but still there was a crash in, in prices, and that of course affected, affected level of activities. The prices stayed a kind of low for some time and it picked up again in years, in about 2000s. I will get into that in a bit. You're going to see that graph again. Right now I just want to focus on that early part. So, in mid-80s we had the very first crash and the recount went down from over 4,500 down to, I'm talking about in the United States. Internationally they had 1,700 went down to


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first crash and the recount went down from over 4,500 down to, I'm talking about in the United States. Internationally they had 1,700 went down to 800, so it was not as bad. Fracturing, however, was more concentrated in the United States, although we were involved in fracturing even at the time. There was some in South America, I was involved very much in that, some in Brazil, some in Canada, very much in that, some in Brazil, some in Canada, but was focused mostly in the western hemisphere. There was not a whole lot in the eastern hemisphere. And at the time, there was companies starting to do downsizing and they called it right-, there was salary freeze, there was furlough. I remember working several weeks, six weeks and then take a week off without pay, that was a common thing to save jobs for everybody. And then we got, when we in the late 80s, although some people think it is very recent, in the late 80s we although some people think it is very recent, in the late 80s we got into


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late 80s, although some people think it is very recent, in the late 80s we although some people think it is very recent, in the late 80s we got into fracturing horizontal wells. Actually the first project was a research project. I was personally, I was involved in that project. I was in, we looked at a well, actually a small oil company came up with the idea, they contacted Halliburton company, Halliburton formed a team, we were looking, some people were looking at logs, some people were looking at cementing. I was involved in it from the rock mechanics, fluid flow and refraction site. Actually I wrote a paper on the subject. We presented it in 1988. It was published in Holland, not in Holland, in JPT, Journal of Petroleum Technology, back in 1990. So that's 30 years ago. And if you read it, you're going to see some of the stuff that we still talk about. It's so amazing, the things just to go in. Except we were looking at fracturing wells, horizontal wells, but we're looking at


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we still talk about. It's so amazing, the things just to go in. Except we were looking at fracturing wells, horizontal wells, but we're looking at creating four to six fractures per well. First, the horizontal wells were not very long, and the number of fractures, because we had to create every fracture independently, and so, we did not create a huge number of fractures. We fractured the wells here, in the late 80's we fractured wells in Canada, of course in United States first, we fractured wells in United States, in Canada, South America, and we had a very interesting project in Europe, in Germany, where we fractured one well. I actually, I was involved in that. I ended traveling. I found about it the day before the meeting. I had to fly to Germany. We had our meeting. We talked about how to design it, what to do. And we flew back. It was an interesting trip. interesting trip. And then we got into, eventually, we fractured the, we fractured chalk. That was very interesting. We got


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was an interesting trip. interesting trip. And then we got into, eventually, we fractured the, we fractured chalk. That was very interesting. We got into fracturing Austin chalk. It was very interesting. We got into fracturing Austin Chalk, it was very successful. However, it was hit and miss sometimes because if you don't intersect natural fractures in the chalk, it is not very successful. Then a small company in the Bernet shield wanted to try fracturing Mitchell energy, I should say. Mitchell energy wanted to try fracturing horizontal wells in shale. And boom, it worked very, very well. So, wells that were not very productive all of a sudden, they were extremely productive. productive all of a sudden, they were extremely productive. And so that is started the increase in Shale, or the Shale revolution as people call it. The Barnett Shale works very well for reasons we can discuss later on. Barnet shield works very well for reasons we can discuss later on. So as we started working


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works very well for reasons we can discuss later on. Barnet shield works very well for reasons we can discuss later on. So as we started working with other shields we wanted to do something to make those shields similar to the Barnet shield. We can discuss that later on. Although it is more of a research area, but it's very interesting to talk about. And so, the reason you look at the 2000s now, prices are going higher and increases, and now you end the fracturing shale, and as you fracture shale, you're producing more oil and more gas and that affects the price of ... of oil and gas, so it ends going down. So you, you... industry is really successful in doing things but in a, in a way it hurts, it hurts a little, because it reduces price. So that, you can see recently the prices were going up to $100 per barrel of oil, and then boom suddenly it declines, because you, there's a lot of shale in, in the world and not only the Western Hemisphere was fracturing, but also the Eastern


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suddenly it declines, because you, there's a lot of shale in, in the world and not only the Western Hemisphere was fracturing, but also the Eastern Hemisphere. So, we... most people think of ... of fracturing as only for low permeability reservoirs. That is... that is in a way true, but is also not exactly true. ... We, we fractured low probability reservoirs, we get into fraction share, we'll talk about that in next two, next few weeks as we work. But it, we have also fractured the high probability formation. We, we are not going to get much in that, but if you're interested, there are a bunch of papers, we wrote a paper on the subject back in 1993-94, we brought a couple of papers on that subject. It's a very important subject by the way, because we have been fracturing our formation in the Gulf of Mexico, the formation that have not one or two millidarcy and we talk about low permittivity reservoirs like in shale that is nano-darcy, macro-darcy and nano-darcy, but in the Gulf we


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one or two millidarcy and we talk about low permittivity reservoirs like in shale that is nano-darcy, macro-darcy and nano-darcy, but in the Gulf we fractured the formation that is can be 10, 20, 50, middle of the Darcy formation, but under special condition that we are not going to be discussing right now. But those formation have been fractured and we've been analyzing the data. We'll get, when we get in the analysis I will show you an example of, of fracturing hyper formation. of fracturing hyper-information. So, basically, when you do that, it's originally was, was looked at as sound production control, but is actually a fracturing process. Fracturing implementation is very complex. You need knowledge, not only fluid flow, many of petroleum engineers, they end up getting trained only in fluid flow, but this is more than just a fluid flow. It is fluid flow, it is fracture mechanics, it is rock mechanics, it's some operation you need to look at prop and... So, there is a lot of


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flow. It is fluid flow, it is fracture mechanics, it is rock mechanics, it's some operation you need to look at prop and... So, there is a lot of things one has to, has to learn in order to be an expert in, in that area. And, it is fairly expensive there fairly expensive for a company to apply. This is the one reason you have specialized service companies that do the fracturing. Let's talk about this a little bit because this is also related to something that I've given lecture on to... to students some time ago, and you are going to see it also on YouTube. The Hydraulic Fraction, different between Hydraulic Fraction and the other Fracturing techniques. Because, one can say, Hey, I can fracture using something a little different, I can use explosive or propellant, I'll talk about both of that. And actually in... I did not talk about, I did not mention in history of hydraulic or fracturing. About... so at one time people used the... or it was an experiment, to use nuclear bomb. And


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I did not mention in history of hydraulic or fracturing. About... so at one time people used the... or it was an experiment, to use nuclear bomb. And it's actually was applied, and as you can see is a really bad idea but people have tried explosives and explosives is a bomb basically and the pressure goes up very fast so in a microsecond you reach many thousands, maybe a million PSI. And so, it is very damaging to the formation, it's basically solid rocket fuel. It burns, it burns very quickly, it reaches very high pressure, but it's uncontrolled. In milliseconds, you can reach very high pressure, so indicating multiple fractures. Explosives will do even, even more of that, it creates cracks all over. And then we have the plasma stimulation, that's what I discussed in waterless fracturing, sometime ago you'll find it in YouTube, you may, maybe you want to look at it sometime if you're interested. We also reach a very, in a few microsecond, we reach high pressure, but not as high as


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maybe you want to look at it sometime if you're interested. We also reach a very, in a few microsecond, we reach high pressure, but not as high as what you see in propellant or explosives. You end up having multiple fraction, but it is more controlled process than you see in propellant, as well as repeatable. In hydraulic fracturing, now you reach that maximum pressure which is fairly low relative to the, to this other three approaches, but it takes many seconds or even minutes. So it is, although everybody will talk about it as a dynamic process, but when you look at what properties we use in Hydraulic Fracturing, we use the static properties of, the static mechanical properties of the rock, because it is fairly, relatively slow, you're not vibrating the grains, you are cracking the rock, relatively slow, relatively slow compared to explosive propellant or plasma. Those take microseconds, hydraulic fracturing takes many seconds or even minutes. So this is basically a representation


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propellant or plasma. Those take microseconds, hydraulic fracturing takes many seconds or even minutes. So this is basically a representation of what happens. You can see explosives create cracks in every direction. And it may not be as effective when we get into the fluid flow. Actually, long hydraulic fracturing is definitely more effective than having an explosive that crushes the rock and it can make the rock somehow glassy. ... Multiple propellant, the rise time is fairly fast and it can create multiple fracts. There's other applications along that people have used, propellant as well as propellant as well as having high pressure Nitrogen to follow up, to follow in the, that fracturing of, of the well. And so there was a lot of applications on, in this approach back in, in the mid 80's and early 90s. It's not used as much as we look at. Then hydraulic fraction has been always very, very successful. When you look at the fluid flow, we'll get a, we'll show you a comparison between


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at. Then hydraulic fraction has been always very, very successful. When you look at the fluid flow, we'll get a, we'll show you a comparison between hydraulic fractions, how it changes production. And you can quickly will see a relative comparison with other applications. So let me look at the fluid flow. So hydraulic fracturing is a tensile failure. Just to remind you, if you have not studied that, ... if you, if you look at a rock, any rock, like sandstone, is very strong in compression. If you get a rock and you test it under compression, it takes very high pressure, even without surrounding the rock with stress. So, if you have unconfined compressive strength of the rock, unconfined meaning that, the sample you are, you're putting under compression, does not have any stresses on it sideways. The rock, the unconfined compressive strength of sandstone is in neighborhood of 4,500 to 5,000 PSI. So it's fairly strong and now if you apply confining stress, it increases very quickly. We


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of sandstone is in neighborhood of 4,500 to 5,000 PSI. So it's fairly strong and now if you apply confining stress, it increases very quickly. We are not going to talk much more about it than that, at least right now, but in tension rocks are very weak. The tensile strength of sandstone for example would be between 100 to 200 PSI. You see, you can see that comparison between 5,000 PSI and a couple of hundred PSI. So, ... so fractures ... in, in rocks will, is a, is a tensile failure, not a compressive failure, although sometimes people think it is, you're compressing the rock, it's, it's compressive failure, but actually you need to look at the tip of the fracture, it is breaking under tension. This is one reason many times, sometimes in equations, people may ignore that, the tensile strength of the rock. Okay. How does it work? You know, if I ask you why fracture works and you tell me that because you change probability of deformation, you would flunk this course. The fractures, it


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if I ask you why fracture works and you tell me that because you change probability of deformation, you would flunk this course. The fractures, it may change the permeability just around the fracture phase. But, fractures work not because they change permeability, but because they modify the flow pattern inside the reservoir. So, if I'm looking at a droplet of oil, instead of this droplet moving to the wheel bore taking a long path, I don't know whether you see my... So if I have droplet of oil here, rather than, than moving to, sorry, rather than moving to the wheel bore, it moves first to the hydraulic fraction and then this hydraulic fraction has very high permeability. The formation may be micro-Darcy or a fraction of mainly Darcy, but the permeability inside that fraction may be in Darcy's. Actually, depending on the type of fraction, it may be going from 10 Darcy to 100 Darcy. So, the pattern of fluid flow changes, is no longer, no longer radial flow. You know, you study these


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it may be going from 10 Darcy to 100 Darcy. So, the pattern of fluid flow changes, is no longer, no longer radial flow. You know, you study these equations, Darcy equation, most people will set Darcy equation and talk about radial flow. Here is not radial flow, it's a lot more complex than, than radial flow. We'll, we'll talk about the different type of radial flow, of flow inside reservoir in, in a second. So when I look at the hydraulic fractured weld, I would say, well, one wing of the fracture is, length is LF, and most of the time people will assume that the hydraulic fracture has symmetry. So both lengths are the same. Are they exactly the same? Probably not, but it's unless the difference is huge, which usually is not the case, you can't tell the difference. You can approximate it with symmetric fractures. So you're saying I have length of LF, I have fracture width of W, and W, this fracture width may, after the fracture closes, it may be 0.2 inch. 0.2 inch is about five


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have length of LF, I have fracture width of W, and W, this fracture width may, after the fracture closes, it may be 0.2 inch. 0.2 inch is about five millimeters. I tended to use English units, that's what we use in the field. So it's about 0.2 inch and when a fracture hyperventilates formation we do special things and we end up having maybe a wider frac, maybe going to half an inch or even up to three quarters of an inch. So that's a very wide frac. So, what would be the probability of a smooth surface open crack? If it's the width on the, it's 0.01 inch, not even 0.2 inch. And the probability if it is open and the surface is smooth, so I have two, basically like a wedge here, the probability would be 54 times 10 to the power 9 w squared. So you can see the probability would be extremely, extremely high. However, realistically speaking, because the fracture is not as smooth, it has propent inside, there may be some damage around the fracture, you can realistic fracture probability is


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because the fracture is not as smooth, it has propent inside, there may be some damage around the fracture, you can realistic fracture probability is can be go from anywhere from 1 to 200 Darcy. You get this equation just for a few, wanna find out how did I get this equation? This equation you get by equating Poiseuille's equation for open crack to Darcy equation. And so in most of our work when we talk about from now on, I'm going to assume is simple FRAC. We'll work with simple FRAC where one wing is LF. So when I say fracture length, actually I'm talking about one wing of the fracture. So when people talk about LF, they talk about one wing of the fracture. So when people talk about LAF, they talk about one wing of the fracture. And I'm going to say my original formation probability is K, formation probability is, fracture probability is KF, and which of the fracture is W, or WF. Okay. W, or WF. Okay? And I'm going to assume I'm penetrating the whole formation, so it have height H,


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is KF, and which of the fracture is W, or WF. Okay. W, or WF. Okay? And I'm going to assume I'm penetrating the whole formation, so it have height H, the P zone is have the same height. Obviously you can deviate from those conditions but that's not for this class here, we're looking at basic behavior. If you've studied petroleum engineering, you know some of those parameters, so let me say, let me look at the changes in those dimensionless parameters. So I'm assuming you already know what dimensionless pressure is for case of oil, liquid, case of gas, or in case of liquid here it is, in case of gas here it is. And if this one, this would be dimensionless rate, 1 over dimensionless rate. In case of when I deal with hydraulic fracture, now dimensionless time, when I have radial flow, we have this terms everything the same except I have Rw squared. In case of fracture, we replace the Rw with length of one wing of the fracture, so LF square. Now there is one parameter you need to remember


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In case of fracture, we replace the Rw with length of one wing of the fracture, so LF square. Now there is one parameter you need to remember always, which is dimensionless fracture conductivity, that's an important parameter, it's actually, it relates the fracture effectiveness basically to formation. So you have Kfw, this is your fracture width, fracture width and fracture permeability, divided by formation permeability and one length of the fracture, one wing of the fracture. So safety is very important parameters to remember. And some of the literature, they are going to call it F-sub-CD. Some people still use that F-sub-CD, it is the same thing. So FCD or CFD, SPE likes to use CFD right now. Kfw divided by KlF, you have to remember that. Some of the dimensions parameters, you know, if you are my student taking my class, I would expect you to know those definitions by heart. You can't always Google things. So let us talk about what kind of hydraulic fracture we have. I'm going to


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you to know those definitions by heart. You can't always Google things. So let us talk about what kind of hydraulic fracture we have. I'm going to talk about only two types. One is infinitely conductive fracture, so infinite conductivity fractures versus finite conductivity fractures. First is there anything as infinite conductivity fractures? The answer to this is no. There is nothing like infinite conductivity. However, it behaves, the system will behave as if it has such a high conductivity that for all practical purposes it is infinite. Let us say this in a different way. Basically, in infinite conductivity fracture, what we are saying is that the pressure drop inside the fracture is so small relative to the pressure drop inside the reservoir that I can ignore it, ignore the pressure drop inside the fracture. So if my pressure drop inside the fracture is one psi, and the pressure drop inside the formation is 500 psi, or a thousand psi, and then you're talking about way smaller


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inside the fracture is one psi, and the pressure drop inside the formation is 500 psi, or a thousand psi, and then you're talking about way smaller than 1%. So I can assume that as if there is no pressure drop inside the fracture. So many and many high conductivity fractures that you will see that behavior. Finite conductivity fracture basically what I'm saying is the conductivity, the pressure drop inside the fracture is significant relative to the pressure drop inside the formation. It's not higher, it's still a lot smaller than the pressure drop inside the formation, but the value cannot be ignored. Okay, let's look at CFD. If CFD is higher than 300, then you can assume that the fracture is infinitely conductive. That value, you know, if you read all the literature, it may be 500, some recent literature, they may even look down to 100, and say this is an infinite conductivity fracture. However, many of those fractures would behave as a finite conductivity fracture. So anything less


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and say this is an infinite conductivity fracture. However, many of those fractures would behave as a finite conductivity fracture. So anything less than 300, at least from theoretical point of view, will behave like finite conductivity fractures. So let's look at, this graph was published, this is the behavior of infinitely conductive fracture was published by Remy and some of his students. There's more than one student who worked with him on that. Remy was, Professor Remy was professor in Stanford University. And I had the privilege actually of taking classes with him. Was a great professor in all aspects, as a researcher, as a person, great man. He passed away many years ago. he passed away many years ago. And so that work is showing dimensionless pressure versus dimensionless time based on, for a fracture. So it has LF squared, and it shows something very interesting. It shows that the early time is a straight line with slope of one half, then you have curvature, and that


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and it shows something very interesting. It shows that the early time is a straight line with slope of one half, then you have curvature, and that curvature continue like this. Then it eventually, at about time of five, it behaves very much like radial flow. So the early time is slope of one half then the curvature that we can't really easily analyze and then about dimensionless time of five or if you if you're not very strict you can say three it the fracture behaves like radial flow as if there is no fracture and then if it sees a boundary later on, then the boundary effect will appear slow one to one. Okay. So let us look at this behavior. So the early time, slope of one half, basically if you take the logarithm of this equation, it gives you, it says that log of pressure, dimensionless pressure, or delta P, versus log of time, it gives you a slope of one half. And this equation basically is equation indicating linear flow. Anytime you see a square root of time behavior like this,


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you a slope of one half. And this equation basically is equation indicating linear flow. Anytime you see a square root of time behavior like this, usually this is linear flow. Anytime you see a square root of time behavior like this, usually this is linear flow. So early on, you have linear flow. Later on, after dimensionless time of five or three, if you're not too strict, if you're not too strict, you have behavior of dimensionless pressure is function of natural log of time. Does this equation remind you of anything? This, the first part at least, is what you see in radial flow. If you look at radial flow equation without any fracture, it is PD equal to one half of natural log of TD plus skin factor. So very early on is linear flow. After long time is it behaves like radial flow. long time is it behaves like radial flow. So that's assumption here is you can reach that radial flow. Dimensionless time of five since it has LF square in it can take long long time. In classes that I


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assumption here is you can reach that radial flow. Dimensionless time of five since it has LF square in it can take long long time. In classes that I teach, I get the students to calculate the time, say for a tide gas formation to reach that dimensionless of time of five or even three and you find out that it may be many many many years. You know, it's not uncommon to calculate 10, 15, 20 years, depending on the properties of the formation and on fracture length. So you, so that is, you don't always get to that, the radial flow, unless you have high permeability formation and shorter fractures and liquid in tight gas formation forget it you can't reach that radial flow. If you reach that radial flow and you see I'm saying if you reach that radial flow then that equation relating fraction length to skin factor would apply. So if you have tight gas formation, forget about this equation you cannot use. If you have high probability formation and it's liquid and short the fracture, then


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tight gas formation, forget about this equation you cannot use. If you have high probability formation and it's liquid and short the fracture, then yes, you could calculate the equivalent skin factor. I think we are going to go a little beyond one hour here. Maybe we'll go to about another 20 minutes or so. For finite conductivity fracture, it gets more complicated because early on, you are going to see linear flow inside the fracture, which takes place in only the first few seconds, maybe up to a minute or so usually. So although theoretically is interesting, practically is not of great importance, unless you run extremely precise tests. And then you get to buy linear flow. And that is very interesting behavior. Because now you have linear flow inside the formation. So fluid coming from close around the fracture, not very far. And so it's coming from the area around the fracture moving linearly into the fracture phase and inside the fracture is moving linearly to the wood board. And


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coming from the area around the fracture moving linearly into the fracture phase and inside the fracture is moving linearly to the wood board. And we're going to look at this a little bit more then you get to linear flow inside the formation. Once the pressure drop inside the formation is a lot higher than inside the fracture, then it behaves as if there is no pressure drop inside the fracture and you see linear flow inside the formation, then you end up seeing that elliptical fluid flow. That's very difficult to analyze, you can only match data, you cannot, there is no specific graph to it. It is not like square root of time or semi-log graph that you use in radial flow. Eventually you get to the pseudo-radial flow. If you reach that pseudo-radial flow, which is not common at all to reach it in fractured cases, unless you produce for a very long time, and you have a short fracture, you have fluid, you have liquid, and then you see that behavior too. So this is the early linear flow.


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very long time, and you have a short fracture, you have fluid, you have liquid, and then you see that behavior too. So this is the early linear flow. It is, as I said, it happens only for short time. I'm not going to go through the equations and details, but this is the only behavior where you do not see the fracture permeability, the Kf and fracture width together. In all other behavior, you are going to see KfW together as a product. And if you see KfW together as a product, that tells you, you can't analyze those pressure behavior to separate Kf from W. And this is one reason in analysis people talk about KfW or you talk about CfD, you don't talk about fraction probability. However in the design part you always talk about Kf and W as two separate components. Bilinear flow will not show if the conductivity is very low and if the conductivity is very high. It's very high, you don't have bilinear flow, it goes from linear flow directly to electrical flow. If the conductivity of the


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is very high. It's very high, you don't have bilinear flow, it goes from linear flow directly to electrical flow. If the conductivity of the fracture is really, really low, and then you don't see that bilinear flow, you actually, the system will behave as if it is radial flow. So if your conductivity of the fracture is too low, then basically you don't have an effective fracture. There is a lot more stuff that one can discuss in this that it's beyond the scope of this lecture. Then I, for finite conductivity fracture, you can see, there's something interesting that appears here, is in early time, you can see that the slope is about quarter when I put log of dimensionless pressure, or log of delta P versus log of time, I see quarter slope, I see straight line that all parallel and see a quarter slope, I see straight line that all parallel and a half quarter slope. That makes the analysis of data like this very difficult. I'm, when there's implication to it, we're not going to discuss


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and a half quarter slope. That makes the analysis of data like this very difficult. I'm, when there's implication to it, we're not going to discuss today. Today is just an introduction. However, it is something interesting because I look at the equation, dimensionless pressures versus dimensionless time. And I look at equation that looks like this. So where dimensionless pressure is a function of dimensionless time raised to the power of 0.25. And the equation has CFD, square root of CFD. If I go and, this is one of my favorite questions to students, I will let you think about it, and maybe Dr. Elgarhi will give it to you as a problem. If I substitute the parameters of dimensionless pressure, dimensionless time and dimensionless CFD, dimensionless fracture conductivity, and I start the cancelling parameters, I find out that fracture length cancels out. And the question would be, isn't it strange that the dimensionless pressure in a fractured system is independent of fraction length


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cancels out. And the question would be, isn't it strange that the dimensionless pressure in a fractured system is independent of fraction length during the bilinear flow? So that sounds kind of strange, actually sounds very strange. And the question would be what is the implication of that? This is fairly important. It actually leads to very interesting conclusion. I will let you reach that conclusion yourself like what I do to my student when I teach. that I do to my student when I teach. Here, if I look at the linear flow inside the formation, you can see if CFD is very small, I don't have linear flow inside the formation at all. That's what basically, basically I'm getting to radial flow behavior to begin with. So CFD of 0.1 is too small. If I get to very high CFD, then I see that linear flow for some time. And as the conductivity gets, fraction conductivity gets smaller, the linear flow will start later in time and ends later in time. So it gets smaller and smaller. So you have a


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fraction conductivity gets smaller, the linear flow will start later in time and ends later in time. So it gets smaller and smaller. So you have a profile, something like this, like what we saw in here, in between, between the quarter slope and this area. Okay. And again for finite conductivity fracture, we get an equation similar to what we see in infinitely conductive fracture, except now there is another term in the set of 1.1, the term here is function of dimensionless conductivity. Equivalent skin fracture can be calculated under specific conditions, as I mentioned, only happens when you, when, when the fracture length is very short, you have liquid, so that does not happen very often. We saw the equation in infinitely conductive fracture. For finite conductivity fracture, you can have this term N in it. There is a couple of approaches, that approach using N, it's the one I have been using, Sankule and others have another approach, we'll talk about it maybe later on. But the


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that approach using N, it's the one I have been using, Sankule and others have another approach, we'll talk about it maybe later on. But the behavior N is very straightforward approach. I'm showing to you basically what it says that this term N is changes with, the equation would be one over, you have an N here, and you can see, N is, if you have very high conductivity, N is 2, is actually should be 1 over N, and if you're, as the conductivity gets smaller and smaller, you see going down this way, and it gets larger and larger. So, here Lf, sometimes you write it Lf over N, so Lf over N is, if you have infinite conductivity fraction like what we saw in the previous equation, L is equal to 2, as, as conductivity gets smaller and smaller, and it gets larger and larger, meaning that, basically the effective fraction length, it gets smaller. So as if your, your finite conductivity fraction, as if it can be represented by shorter infinite conductivity fracture. This happens only, you have


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So as if your, your finite conductivity fraction, as if it can be represented by shorter infinite conductivity fracture. This happens only, you have to remember that, that's, that this is not just a general equation, this happens only after you reach the radial flow or the pseudo-radial flow behavior. Many people take it wrong and they think that you can always represent fractured well with radial flow system. That is incorrect. This is one reason when I predict production I don't want to use equivalent skin factor, I want to use fractional dimensions. Planning for fractional treatment, you can look at steady state or pseudo-steady state, people have done that before, or you can use transient behavior. use transient behavior. For, for steady state, there was a very interesting correlation, that it was originally published by, by Tensley and the other several people. Unfortunately, it had... had some problems, it was... I hate to say it, but there was some mistakes in... in... in... in


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the other several people. Unfortunately, it had... had some problems, it was... I hate to say it, but there was some mistakes in... in... in... in the equation, I... I ended taking it and I... I redid the correlation and I published this back in, I think 1980 or 81, I can't remember, it was a long time ago, 40 years ago. But it is very interesting, because, let me, let me go to the equation where showing one is, is clearer. It, it tells you, basically, you, when you design a fracture, you don't want to be on that sharp curve. Because that sharp slope, basically, if you lose conductivity, you're sliding... Well, let me go back and say, here this is basically productivity and indices relationship. The difference between hydraulic fractured well and unfractured well. So you wanna be as high as possible. Here, mostly it is function of CFD. So, you don't want to design a fracture where you're sitting here. Because if you lose conductivity, and you're bound to lose conductivity with time,


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So, you don't want to design a fracture where you're sitting here. Because if you lose conductivity, and you're bound to lose conductivity with time, you... the conductivity, the productivity of the weld declines very fast. You want to be on the flat line, but you don't want to go so far in the flat line, because, once you get to a certain point here, you can increase conductivity very much, but you don't increase productivity that much. So that graph was actually very interesting, and that is the reason, you know, you always hear about dimensions, fractional conductivity of 30. Actually we published that many, years ago and that is this curve is the reason for that the mesh's conductivity of 30. At the time we published it based on another term that will be 10 but it had pi in it so it was 31 and so on. So that gives you the historical reason, because when you get to CFD of 30, you're actually on the flat line, a little deep in the flat line. If you go to 50, it does not increase


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reason, because when you get to CFD of 30, you're actually on the flat line, a little deep in the flat line. If you go to 50, it does not increase productivity very much. It increases a little bit, but not very much. It increases a little bit, but not very much. So you, but you want to avoid being on that sharp curve. The reading suggested reading material would be, would be SVE monograph number 23, chapter 11. That's something that I wrote, if several, about 20 years ago. And it is very interesting looking at analysis and it talks also about fracturing. There is more stuff is written in chapter 15 in book, Will Construction. Unfortunately, the book Will Construction Construction, is out of print. It's very expensive to get. And so I do not know what to tell you about it. S.P. Monograph, number 23, that can be ordered from S.P. It is a very good book. It's about well testing. So if you are in well testing, I would recommend this book and the chapter 11 talks about well testing of


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a very good book. It's about well testing. So if you are in well testing, I would recommend this book and the chapter 11 talks about well testing of hydraulic fractured well, but I've added a lot of stuff on hydraulic fracture design, some of the stuff we're talking about here. So these two chapters I wrote some time ago. That is, that is takes care of my presentation. Thank you for attending and we'll take just a couple of questions, other questions, if you have so many we can always answer them in email. If you want to access my email, I am with the University, it is very easy to get my email address, it is mysolomon, like what you see here, M-Y Solomon at uh.edu. That email does, that's an alias, but also you can find it as msolomon at central.uh.edu. So I have two of them. I will, you can reach either one. As I said, the presentation, PDF file of the presentation will be posted on research again in the next couple of days. So you can download it and look at it at your leisure.


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PDF file of the presentation will be posted on research again in the next couple of days. So you can download it and look at it at your leisure. Thank you, Dr. Solomon for this enlightening session. Now we'll move on to answering some questions by our honorable audience. Someone says, what kind of potential dangers questions. I'll move on to answering some questions by our honorable audience. Someone says what kind of potential, um, dangers oil based mud, um. Or sorry, oil based fracturing fluid can cause to the formation. We are going to when we talk about fractured design about fracture design, we'll be addressing issues like this, but simply said, you know, I work for a service company for some time, we always brought, if it's a new reservoir, the new reservoir that we're fracturing, you bring a sample of the rock, and you test it in different ways, mechanical testing to get Young's modulus for Sanz ratio and so on and so forth, like what we are going to discuss when we get into in


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ways, mechanical testing to get Young's modulus for Sanz ratio and so on and so forth, like what we are going to discuss when we get into in situ stresses, so we can design the fracture properly from propagation point of view. You also test it against the different fluid that you want to recommend and you add additives you are going to see when we get in fracture designs there's so many additives you add to make sure that you don't damage the formation. So that is that is the brief answer when we talk about fracture design there will be more discussion on the subject. Okay, thank you. Now we have another question that says, what kind of benefits can we get from fracturing high permeability reservoirs? That's a good question. I don't have a presentation for this course on high permeability reservoirs, but let me tell you, in high permeability reservoirs like in the Gulf of Mexico, those reservoirs are not only high permeability, they are usually weak, meaning that as they deplete the


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reservoirs like in the Gulf of Mexico, those reservoirs are not only high permeability, they are usually weak, meaning that as they deplete the formation, they collapse and the end losing permeability around the wood bore. It goes from 50 millidarcy down to maybe 1 millidarcy or even less and it goes into depth of several feet. So, acidizing or perforation does not help. So you fracture this formation and usually you design a shorter fracture. So those fractures, very high permeability fractures, we call them frac-pac. At least that was the commercial name for Haliburton, but other companies will call them frac and pac, for example, or hyper fracturing. So you end up having wide fractures, have high problem concentration, that is doing two things, is doing sand control, and spreading the productivity or the fluid flow over a long distance. So it can produce the same amount of fluid or more with a lot less pressure drop. This is the reason people call it frac pack because you're


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So it can produce the same amount of fluid or more with a lot less pressure drop. This is the reason people call it frac pack because you're packing the fracture. When we get into testing, I am going to show maybe not exactly the design, but some of the testing. But it is an application that is very, very important. We have seen it a lot in the Gulf. You can't produce those worlds on the radial flow condition for too long without losing that productivity. Did I answer your question? I may go back and look at the fracture design and we add a little bit more on infraction and hyper. Okay, thank you. Someone asks, are there any problems in fracture orientation and inclined, not vertical? That's a very good question. I was not going to get into that, but absolutely. We are going to get into the extreme cases horizontal well. We are going to look at fractional horizontal well a little bit. But inclined well is even more complicated than horizontal wells. We've done some paper on the


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to look at fractional horizontal well a little bit. But inclined well is even more complicated than horizontal wells. We've done some paper on the subject that we presented years ago. I do not know that they can see me or not. Can audience see me? Because I want to do something. Yes, they can see you, but I believe the camera is showing the top half of your head. Yes, now it's way better. You know, when you have a world that is inclined, inclination now becomes more complicated because you have to look at inclination in space. So it depends on the three stresses. Let us leave this question until I get to in situ stress. I may add a slide or two of things that we have done in the past. Most people don't look at it. However, we did fracture inclined wells. And you need to, when you say inclined wells, there are two things. There is deviation and there is inclination. They are two different things. Deviation is deviation from the vertical. So as, so deviation of zero would be a vertical


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and there is inclination. They are two different things. Deviation is deviation from the vertical. So as, so deviation of zero would be a vertical world, deviation of 90 would be a horizontal world. And then there is inclination in space. So, so this is more a problem than it appears. One thing we did years ago, where fracturing deviated well, that's also inclined in space. And the idea was, since obviously we can't control the stresses, the stresses are the way, whatever they are, what we try to do is to make it not inclined from stresses, but deviated. So if you do deviated well, and you perforate in the top and bottom, the top side of the well and the bottom side of the well, you end up having a vertical fracture that is propagating from the well. If you, if it's inclined with respect to stresses, if you, and you do that, then you still, if you perforate throughout the whole pay zone, you still get a vertical fracture that get inclined in space. I have done some more work in the


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if you perforate throughout the whole pay zone, you still get a vertical fracture that get inclined in space. I have done some more work in the area, and it is really, really interesting. It is more complex than even horizontal ones. Thank you Dr. Solomon. That's about it for the questions today. It was a pleasure having you on Paiapetro. Thanks for joining us, and thank you everyone for tuning in with us today. Today's session will be uploaded on Payopetro YouTube channel. You might want to check that out for your quiz. Thanks again. Have a great day. Thank you very much ladies and gentlemen. you


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Good evening, good morning and good afternoon. Hope you are all doing well and staying safe. I'm Maryam Shraif, a senior petroleum engineering student at the Lebanese American University, and I'll be moderating today's session. So it's my pleasure on the behalf of Payo Petro to welcome you all to the second lecture of the online course Hydraulic Fracturing, which has been presented by a special distinguished speaker, Dr. Mohamed Salomon. I'm sure his participation with us today the Department of Petroleum Engineering and Manufacturing. Which has been presented by a special distinguished speaker, Dr Mohammed Solomon. I'm sure his participation with us today will benefit all participants. With that I advise you to give your full attention to his informative session today. So let me begin by reintroducing our guest speaker for today. Professor Solomon. So Professor Solomon, the chairman of the Petroleum Engineering Department at University of Houston. Dr. Solomon has a long industrial


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Solomon. So Professor Solomon, the chairman of the Petroleum Engineering Department at University of Houston. Dr. Solomon has a long industrial and academic experience. He holds 35 patents on fracturing operations and analysis, testing and conformance applications. He is the author and co-author of over 250 technical papers and articles in various areas of petroleum engineering. Dr. Solomon is a registered professional engineer in the state of Texas. He is also a distinguished member of the Society of Petroleum Engineers and Fellow of the National Academy of the Inventors. To highlight, he has authored several book chapters on stimulation and testing. Additionally, he is the primary editor of Fracturing Horizontal Wells, published by Macro Hill in 2016. He is also the co-author of Optimization of Hydraulic Fracture Stages and Sequences in Unconventional Formations, published in 2018 by CRC. Finally, Dr. Solomon holds a Bachelor of Science with top honors from Cairo University and a


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in Unconventional Formations, published in 2018 by CRC. Finally, Dr. Solomon holds a Bachelor of Science with top honors from Cairo University and a Master's and a PhD degree from Stanford University. Thank you everyone for attending, thank you Maryam for the introduction, thank you Ahmad for arranging this teaching, giving me more work to do than I... as if I don't need any more work. Anyway, this is the second lecture. We talked last week about the history of hydraulic fracturing. I should have also mentioned last week that Mr. Buddy McDaniel is an old friend of mine and a colleague that we did a lot of work together, was the biggest contributor to the writing on fracturing, history of fracturing, fracturing was both vertical and horizontal. So if you, when you read this presentation, please remember that I should have acknowledged him as a very knowledgeable person that we did a lot of work together. During, in last week I also talked about fluid flow, and I mentioned the pseudo


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as a very knowledgeable person that we did a lot of work together. During, in last week I also talked about fluid flow, and I mentioned the pseudo radial flow and elliptical flow. So what I did, I'm having another four slides here to show you some numerical simulation of how the fluid flow would look. Just a simple case where I have a vertical hydraulic fracture in a formation and make it penetrating the whole formation, to show you how the fluid flow would look. This, I'm showing a couple of illustrations, it will be two slides on infinite conductivity fractures and the two full slides on finite conductivity fractures. So here I'm saying CFD and if you remember the definition of CFD which is permeability of the fracture times the width of the fracture divided by formation permeability, and also divided by fracture half-length, one wing of the fracture. And here I have very tight gas formation, so it takes, so the conductivity basically infinite, infinite meaning that the pressure


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of the fracture. And here I have very tight gas formation, so it takes, so the conductivity basically infinite, infinite meaning that the pressure drop inside the fracture is negligible relative to the pressure drop inside the formation. And you can see here, the behavior looks very much as if I have just a line inside, inside the formation. That's what will give me linear flow. After a long time you can see the pressure drop around the formation is demonstrated here. If I look at the outer boundary where the pressure has been significantly affected, you know, the rest, the red one area, the red area, basically saying we have not affected this area at all. So nothing coming out of, or very, very little coming out of this red area. Most of the fluid is coming out from this affected area. And you look at the affected area, it's elliptical shape, but it looks almost like a circle. So it is almost radial flow, and this is the reason you call it pseudo-radial flow. If I have finite


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shape, but it looks almost like a circle. So it is almost radial flow, and this is the reason you call it pseudo-radial flow. If I have finite conductivity fracture, you can see the effect is not as strong and you don't see the end of the fractures right away. So it takes some time to see the end of the fractures right away. So it takes some time to see the end of the fracture. That is the bilinear flow. If you remember, we talked about that a little bit. And you can see with time, you end up seeing further and further into the formation, but the elliptical behavior is more dominant here till eventually you reach the pseudo radial flow. Pseudo radial flow which is almost radial flow, it is still kind of elliptical, however the major and minor axis of the ellipse are almost equal. If they are almost equal so they are almost having radial flow. So that's the reason you would call it pseudo-radial flow. Fracturing part of it is fluid flow. So we discussed that. We need today we want to


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flow. So that's the reason you would call it pseudo-radial flow. Fracturing part of it is fluid flow. So we discussed that. We need today we want to talk a little bit about rack mechanics and in-situ stresses. Next week we'll talk about fraction mechanics and last week we'll talk about testing. And please remember, please remember that this is very introductory, you know, we don't go in things in huge details. We teach graduate class in, I taught it in Texas Tech and we'll teach it here at UH, where we go into all those areas in a lot more details and theory. So it is over 16 week, three hours per week and homework and midterms and all kind of things. So let us talk a little bit about Rack Mechanics because that we need to understand just before we get into the stresses. So what is R mechanics? Basically, it is a discipline that you, and that you use to describe the behavior of underground formation of rock under in-situ stresses or sometimes externally applied loading conditions.


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and that you use to describe the behavior of underground formation of rock under in-situ stresses or sometimes externally applied loading conditions. That's what you do in hydraulic fracturing. And when you couple the fluid and the rock mechanics together, that's when you talk about geomechanics. And we'll be talking about both in the rest of the hour. Is it important to study rock mechanics and geomechanics in petroleum industry? Absolutely. Absolutely, unfortunately it has been ignored for a long time in teaching undergraduate and graduate classes and that's a mistake because it's important in drilling, borehole stability, what happens as you drill with it. We'll mention a few little things as we talk here. Formation breakdown, if you're down a whole pressure or the wheelbarrow pressure, if it's too high, you could fracture the formation. If it is too low, you could collapse the wheelbarrow. So it's very important to understand all this relationship with the rock mechanics and the


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If it is too low, you could collapse the wheelbarrow. So it's very important to understand all this relationship with the rock mechanics and the fluid flow. Stimulation, obviously, is very important to understand all this relationship with the drug mechanics and the fluid flow. Stimulation obviously is very important. Fracture initiation pressure, we are going to look at one equation here. We spend a little bit of time on it for fracture initiation pressure because it's very important to understand. Then next week we'll talk more about the fracture propagation pressure. To initiate the fracture requires certain pressure. It's going to be a little high pressure. People observe that once you fracture the formation, then your propagation will be at a lower pressure, but you can pressure that can be increasing with time. And one has to be able to analyze that propagation pressure. So we'll be talking about that both next week as well as in the testing part. Then we, all of that related to


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to analyze that propagation pressure. So we'll be talking about that both next week as well as in the testing part. Then we, all of that related to fracture geometry and designing of a fracture, that's all important. Then you get into production, and then ... then there is a lot of aspects, so in rock mechanics, failure of the formation, sanding. We're not going to get into sanding at all in this ... in this course, because it's more hydraulic fracturing, but sanding is very important and ... hydraulic fraction. But sanding is very important and in a graduate class we end talking about sanding, the onset of sanding and the calculation of how much sand you produce. You want to talk about some of the assumptions we work with. we can say well formation, many times, especially when you study well testing, they tell you homogeneous, the formation is homogeneous and isotropic. And there are two different things. Homogeneous formation means that if you take one element in the formation, it's


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is homogeneous and isotropic. And there are two different things. Homogeneous formation means that if you take one element in the formation, it's basically represented the whole body as the whole formation. That's a little different for being, so many of the formation we look at, many of the assumptions we make is the formation is homogeneous. Really if you look at the formation at a very small element, they are really not homogeneous, but it is distributed so much that many of the formations, they behave as if they are homogeneous formations. So, we'll look at some of this, hopefully we can spend more time on it, because some of heterogeneities can be very important. When you get into heterogeneous formation, such as dual prostheses, shale, carbonate formation, this have very strong effect on fracture propagation. We deal with it a lot in our work, especially when you get into shell formation and when you analyze data. Isotropy, whether you call formation isotropic or anisotropic,


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in our work, especially when you get into shell formation and when you analyze data. Isotropy, whether you call formation isotropic or anisotropic, that means a dependence on direction. So a formation may be isotropic and homogeneous, it may be anisotropic and homogeneous. So if I have a formation that is say permeability in one direction is different from the other, so it can be homogeneous but it's still anisotropic. Isotropic would mean that properties in various directions are the same. Now when you talk about rock mechanics, many times we make the assumption that it is, the formation is linearly elastic. Basically linearly elastic is as if you're looking at spring. So you push the spring down, you compress it, you relax, it goes back to original shape. So, and we'll describe it a little bit more detail mathematically in a minute. But basically all we're saying here is the formation, it will be, will disappear once you remove the external forces. So how realistic they are?


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But basically all we're saying here is the formation, it will be, will disappear once you remove the external forces. So how realistic they are? Essentially I talked about that. Homogeneity many times is a good assumption unless you get into a dual prostate or natural fracture formation or it becomes important. We are going to see that the importance of heterogeneities when you talk about fracture design and especially when you talk about analysis. When I get into analysis I will show you something that we have been doing, and especially for shell formation, that heterogeneities have to be considered, the presence of natural fractures will have to be considered, which makes the formation heterogeneous. Isotropy, usually in a horizontal plane, it may not, probably not very, very much, but we many times we assume isotropy, but you are going to see in testing again, and even in design, it can have an effect. Linearly elastic material, if you have, if we have hard rock and we'll talk more


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to see in testing again, and even in design, it can have an effect. Linearly elastic material, if you have, if we have hard rock and we'll talk more definition of what hard rock is, if you have, if we have hard rock like sandstone, then the behavior would look very much like linearly elastic, but if you get a soft rock, such as when we fracture in Gulf of Mexico and I'm sure in areas where you live and you work. So in Gulf of Mexico they are soft rock, so the behavior is different. It cannot be exactly considered linearly elastic. So there is other ways to handle that behavior. So if I look at it, when I say, and there will be more of this in a minute, when I say linearly elastic, if I plot stress versus strain, and we'll talk about this in more in a few minutes. If I increase the stress, I get more strain. So basically it is like if I have a core and I compress it very much as if I'm compressing, you look here on the left hand side as if I'm compressing a spring. So if I compress it,


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a core and I compress it very much as if I'm compressing, you look here on the left hand side as if I'm compressing a spring. So if I compress it, it gets shorter. And then I release it, it goes back to original value. And if you look at here, stress versus strain, so it's going in a straight line. If I release it, it goes back on that straight line. So not only this is linearly elastic, but there is no hysteresis. So it goes back on the same line. Does it always work this way? No, but this is a simplification that many people use. In reality, it may look like the curve we have here beside it and that early part of behavior which is an elastic strain we use in some special testing. Unfortunately, I'm not going to get into that even in the testing part. It is too specialized, but we have used it many times. And when we get to testing, we may be able to say a word or two about it. And so eventually in any of these cases, if you have, if you've studied material science, you will, they


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may be able to say a word or two about it. And so eventually in any of these cases, if you have, if you've studied material science, you will, they will tell you, you know, after some time, if your stress gets too high the formation may fail. So you can have failure of the formation that's a plastic behavior so it cannot be reversed. Poor elasticity is very important concept to understand with it affects the fracture, affects the drug behavior, and we'll talk more about it in a few minutes. For example, here, this would having two straight lines showing whether it is drained, same like the one above, stress versus strain. I probably should have defined what stress is. Stress is and have the units like pressure is psi, force per unit area. However it is not the same as pressure. Pressure usually is is used to describe fluids. So pressure is a scalar quantity. Stress is a vectorial quantity, meaning that it has orientation. We'll see that in a minute. So I can, here I'm saying whether


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is a scalar quantity. Stress is a vectorial quantity, meaning that it has orientation. We'll see that in a minute. So I can, here I'm saying whether the rock does not have any fluid in it, it will follow different curve or different line than if it has fluid on it. And then we'll talk about stiffness versus frequency. You know, we'll get into that in a little bit, because we can sometimes measure the behavior of the rock using a sonic lab or seismic or what one would call a static behavior basically like what you are seeing here. So what are the important parameters that you want to look at? The most important parameters for elastic properties you want to look at are Young's modulus, Hossain's ratio, and Biot's constant. And we'll talk a little bit about each of those. For stresses, I want to look at the total and effective stress. So, keep that in mind, I will describe each one of them in a minute. So, when I look at stress, if I have a piece of rock, like what I have on the left,


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keep that in mind, I will describe each one of them in a minute. So, when I look at stress, if I have a piece of rock, like what I have on the left, and I'm having stress, and you can see, I have a force, the force has an orientation. That can be, I can break it into, it's two main components, very much like what you study in math. So, I can break this two compo... this force in two components, one parallel, and one perpendicular to the rock. So the parallel one is my shear force and here is the perpendicular one is normal force. So if I take that normal force divided by the area, that gives me the stress, that would be my normal stress. The shear stress is again the same thing, take the parallel force divided by the area. So if I, if I look at a piece of rock, you look at that nice looking graph, if I have a piece of rock downhole many hundreds or thousands of feet deep, it's exposed to stresses in all directions. It's not only in one direction, it's exposed to stress in all


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many hundreds or thousands of feet deep, it's exposed to stresses in all directions. It's not only in one direction, it's exposed to stress in all directions, but you can bring those stresses into three principal stresses. If you study electrical engineering, you remember the right hand rule and left hand rule. So they have three, you break those stresses into three perpendicular stresses and you call them three principal stresses. So they are all perpendicular to one another. Usually, so usually one is vertical and two are horizontal. In some cases there may be some orientation to it. You know, if you are beside a mountain, you know, like near the Himalayas, near the Andes mountain here in United States, near mountains these things will change, we'll talk about that in a minute. But you break the stresses into three perpendicular stresses, a vertical and two horizontal. So, I will call that SV, that's my total stress. So, that's vertical stress. The two horizontal stress, one usually


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a vertical and two horizontal. So, I will call that SV, that's my total stress. So, that's vertical stress. The two horizontal stress, one usually is bigger than the other. So, I'm going to call them SH-Max and SH-Min. And then your fluid will have, and that will have a pressure, so there is a pore pressure inside the pores, because we don't have completely solid, so there's pores inside, and this will have pore pressure, when it's important to think about that. And then you have also, the compressive strength of the rock, which is, if I, the un, the unconfined compressive strength is the, the stress that will cause the fra... the, for me, the rock to feel under compression, when it is completely unconfined, and you call that, unconfined compressive stress, and you have the tensile strength, stress, strength of the rock. We talked about that last week. The one thing you have to remember, like anything in nature, you move from one point to another, you want to use or you want to follow


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last week. The one thing you have to remember, like anything in nature, you move from one point to another, you want to use or you want to follow the path of least resistance. So if you are sitting at home, you want to go out, the path of least resistance, you get out of the door, you go downstairs and so on and so forth. The same thing with fracture, it has to follow the path of least resistance. What is the path of least resistance for the formation for a fracture to go? You can see if I have three stresses, the fracture will have to become perpendicular to the smallest stress. Because that is, that is the path of least resistance that we, we have. So, so a hydraulic fracture will be perpendicular to the smallest principle stress, the least principle stress. So that is my path of least resistance. Don't tell me, you know if I ask you this question, you have to tell me it is perpendicular to the minimum stress. It's perpendicular to minimum stress, not parallel to the highest stress.


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question, you have to tell me it is perpendicular to the minimum stress. It's perpendicular to minimum stress, not parallel to the highest stress. They are, mathematically they are different. You know, the answer would be when I ask what's the orientation, it is always perpendicular to the minimum principle stress. So let us look at, let's look at stresses and basic theory. This is very famous, grab been published by so many different people. So I just took it from, I think the first company that published it was GMI, although when I was in Halliburton, we use something very, very similar to that. And so, so here I'm going to look at different types of falls. And that will be an indication of the, of the regime, the stress regime that we've got. So if, if I have the stress regime that we've got. So if I have a stress regime in what I'm looking at, the formation I'm looking at, where the vertical stress is the highest, and then you have the maximum horizontal stress, and then you have


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at, the formation I'm looking at, where the vertical stress is the highest, and then you have the maximum horizontal stress, and then you have the minimum horizontal stress. If you have a fault, then the fault will go, I always like to show it with the illustration, but anyway this side of the fault will go down. It goes down because you have very high stress pushing into it. So if I So if I fracture this formation, if I have this situation which I'm going to call it normal stress regime. If I fracture this situation, what kind of fracture do you get? Well the fracture has to be perpendicular to the smallest stress. And here the smallest stress is the horizontal stress, SHMN. So that tells me that the fracture will be vertical and it is perpendicular to SHMN. So I'm going to jump to the reverse stress regime, the last one. Here, this is not as common. Usually the normal stress regime is the common regime. You see in most situation for formations that are deeper than 1500 feet or so,


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not as common. Usually the normal stress regime is the common regime. You see in most situation for formations that are deeper than 1500 feet or so, which is the cases that we usually fracture. Some are shallower than that, but usually you, it's deeper formation and what you see is normal stress. And when I say usually, there is always exceptions to the rules. So let's go back to the, no go down now to reverse stress regime. What reverse stress regime, when your vertical stress is the smallest. So your maximum horizontal stress is the highest, then followed by minimum horizontal stress, followed by the vertical stress. So if I fracture a formation like this, what do I see? Well, I'm going to see a horizontal fracture. And do you see it sometimes? Yes, it could happen. As we are going to look at what causes the stress in underground formation. I've seen it, not very often, but I've seen it sometime, especially near mountains. I've seen a case like this in South America and I,


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formation. I've seen it, not very often, but I've seen it sometime, especially near mountains. I've seen a case like this in South America and I, unfortunately, we never got the data. I wanted to get the data very badly, because it was, it was a very interesting, a different situation than what you see. And it has its signature, when you run testing and stuff, when logging, you see a different signature of the horizontal fracture. So, you don't see that very often. Now, in the middle, I have a strike-slip situation. A strike-slip situation and when the vertical stress is in the middle. So the highest stress is HS max, the smallest stress is SHMN. The minimum horizontal, minimum horizontal stress is the smallest. So, if I fracture it here, what happens? Well, what happens is, we already know that, that a fracture is from a perpendicular to minimum horizontal stress or minimum stress, so the minimum stress is horizontal, so I have a vertical fracture perpendicular to my minimum stress.


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minimum horizontal stress or minimum stress, so the minimum stress is horizontal, so I have a vertical fracture perpendicular to my minimum stress. However, situations like this you could have a problem. One is many times it becomes a T-shaped fracture. It hits the boundaries between layers and you end up having a T-shape. So you have a horizontal part of the fracture and that when this happens, it will cause all kind of issues during the hydraulic fracturing. Another big issue that we are not going to get into is during drilling. Because now if you, especially if you plan to eventually drill a horizontal well, as the well is deviated you need, because as we say, is deviation, deviation from vertical and inclination from north, from your stresses. So you could have a lot of problem during drilling. We are not going to get into that, but this is a very important issue. So if you suspected that you have a slack slip situation you have to test the formation, you have to test it, because


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this is a very important issue. So if you suspected that you have a slack slip situation you have to test the formation, you have to test it, because it's important to know which regime you are in. That gets you designed a drilling pad and a fracturing later on. Years ago we actually, a company, spent a lot of money, we worked with them in order to figure out whether it was a strike, slip, or normal stress. Luckily for them, eventually after we ran all the testing, it was normal stress. So let us look at a couple of cases here kind of quickly. Here I have a fault. What is this normal stress? Usually when I teach, I ask a student to tell me, here I cannot see you. And so I'm going to tell me here I cannot see you and so I'm going to tell you here you can see this party here went lower you see this is a little higher so that is that is normal stress regime. This is a similar situation. You see the fault here and you see that little man is working, right? You can see that this side, the


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stress regime. This is a similar situation. You see the fault here and you see that little man is working, right? You can see that this side, the right-hand side, part of the formation dipped down. So this, at the time time it was normal stress regime. How about this one? If you look at it, and you look at it from like from far away, you can see as if the formation just slipped sideways. That's a strike-slip situation. How about this? You see the fault here? And rock actually bent as if it is, as if it's a soft thing, but under high pressure and the high temperature, even rock will turn around. That's the reason we see an antic line and all this geologic formation. So this is, you can see this right-hand side dropped below this right side, that left hand side part of the rock, that is reverse faulting. So this must have been, at the time it happened, must have been extremely deep. And not only deep, that's high pressure and high temperature, but also was exposed to some external, very


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it happened, must have been extremely deep. And not only deep, that's high pressure and high temperature, but also was exposed to some external, very high external forces, that, that caused the horizontal stresses to be higher than the vertical stresses. We'll see why. So what, what causes all those stresses? We'll go through some of them, not all of them. The biggest one is the Tectonic plates. of them, not all of them. The biggest one is the tectonic plates. Let us look at it, then we'll talk about topography, the mountains, lateral variation of the thickness of Earth, sometimes localized load, and it may be just the weight of, weight of the rock. Thermal stresses are, can be important also. We, we sometimes take advantage of this in the fracturing process. I will talk about it a little bit. So, tectonic plates, if I, if I look at this map and this, when this map was created, as you can see at the bottom, here, by Karlsruhe University. You know, I, when I hear that, it's like this,


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map and this, when this map was created, as you can see at the bottom, here, by Karlsruhe University. You know, I, when I hear that, it's like this, because I, I spent one summer actually, as a graduate student at Karlsruhe University, University of Karstu. It's a interesting place. A very strong, technically. ... You, you look and you find the Earth is broken into pieces. ... You look at this, for example you look at ... Africa is sitting on a tectonic plate, that looks very much like Africa, only part of it is above water, the rest is under water. South America is similar, North America is similar, ... Asia, Euro Asia it is one tectonic plate. Actually it can break the tectonic plates into, there is major ones and, but you can break it into sub-major ones. ... I can tell you, here in, in United States, you have this tectonic plates here and it can cause earthquakes, and that's the reason for behind the earthquake in San Francisco, back in 1905. Your tectonic plates here, you can see


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and it can cause earthquakes, and that's the reason for behind the earthquake in San Francisco, back in 1905. Your tectonic plates here, you can see breaking in ... through Iceland. We'll look at this in ... in a second. So, Earth is alive. Actually, people did not realize that that there is such a thing, ... as tectonic plates, until early 20th century. ... Someone, someone wrote a paper on the subject, and people did not believe him. They, because, they thought it's just a coincidence, that Africa and, and North and South America look the way they look. Till after Second World War, and the exploration of bottom of the ocean, revealed that there's actually this like a volcano type line here, and it was in this area, if you see around Indonesia, this is what caused the tsunami back in 2000 something, 2005 maybe, maybe around that time. But it was a huge tsunami that killed a quarter of a million people. And, it was caused because of the tectonic plates, because tectonic plates they


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But it was a huge tsunami that killed a quarter of a million people. And, it was caused because of the tectonic plates, because tectonic plates they keep moving. ... For example, the movement here across Iceland, it ... basically the Euro-Asian plate and the North American plate, they go away from one another about an inch per year. It sounds like it's, it's very small amount, it's a very small number, but over millions of years that's a, becomes a huge, huge distance. ... So, so here, what, what they did in Karst Ruins, they were trying to find the stresses. So, for example, an, an area like here, South Africa, this is, this is thick arrows, this is an indication of a strike-slip situation, when you have arrows going into one another like that. If they are going away from one another like this red arrows, that tells me I have normal stresses. I usually give students a homework to go download another map of your area. homework to go download another map of your area. In US, I asked


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I usually give students a homework to go download another map of your area. homework to go download another map of your area. In US, I asked them to download the map for United States and explain what the stresses that we have. Those are regional stresses, does not mean that local stresses will be exactly the same. That is, we already found that a long time ago that this is not the case in South Texas, although it looks like you have some strike-slip here, where many, many areas were actually normal stress, and the stress orientation changed across, across a fault. Okay. ... Let's look, this is just showing part of the Alps is actually, ... is from Africa, because you have the African plate and, and ... European North, Euro-Asian plate are colliding, and the African plate is basically going on top of the Euro-Asian plate. That causes a lot of stresses. If you go back to this map and you see, it's huge stresses near the Himalayas, near anytime you have a mountain. In Iceland, I


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a lot of stresses. If you go back to this map and you see, it's huge stresses near the Himalayas, near anytime you have a mountain. In Iceland, I downloaded this map from Google, you can see in one side this is Europe, on the other side this is North America. So there is a bridge that you go from one side to another, you are crossing from the Euro-Asian plate to North American plate. And this distance keeps increasing. So, in lifetime of a person, it goes away by 6 feet. And the way it happens, and that's the reason behind the Tsunami, is one tectonic plate may be going under the other or getting separated. And what happened in the tsunami, is you know, although we say it moved about one inch per year, does not mean it is going very smoothly. In the tsunami there was no movement for a long time, and then it jumped, it built up a very high stress, and basically one plate jumped over the other, it moved 20, the estimation is 25 feet, it moved the water, although it may not, if you are


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stress, and basically one plate jumped over the other, it moved 20, the estimation is 25 feet, it moved the water, although it may not, if you are sitting in the top of the water here in a boat, you don't really feel it very much, but as it comes closer to the, to the shore, it goes from, from inches into many, many feet, and so it has catastrophic effect. And that's what it causes a lot of stress. Another, the thinning of Earth, basically the lithosphere. And you look at, I have to give you example, the Gulf of Mexico is getting thinner and thinner, so the stresses change. So you find that your minimum stress, if you are here near New Orleans, your minimum stress is like this, and your fracture is perpendicular to it. If you are in Mexico, the minimum stress in different direction, and the fracture is perpendicular to it. So, does not mean that in the Gulf the orientation is the same everywhere. So, that is one reason people will have to test the formation to, to tell the orientation


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that in the Gulf the orientation is the same everywhere. So, that is one reason people will have to test the formation to, to tell the orientation of stresses and magnitude of stress. We'll talk about ways to do that next week. Localized loads like, like salt domes, so you, you have salt domes, a salt, salt domes is ... under the high pressure and temperature it, it flows. Salt will ... although it may be hard, when you take it to the surface, but then down hold it, it flows. The same thing with chalk, you know the... we have fractured the chalk formation, and if you look at chalk, in the surface, when you take a sample, the surface is very hard, but downhole is flows like, not exactly like fluid, but it, it creeps very much. And so, soft dome, it moves and it can cause stresses, so, I don't have a graph for it, but, if you go down hole, if you take a cross section, you can see for, it's pushing the formation up and the oil will be trapped, will be trapped between the solid dome and


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if you take a cross section, you can see for, it's pushing the formation up and the oil will be trapped, will be trapped between the solid dome and the formation, because solid dome has basically zero permeability. So you drill on the side of the solid dome, so as you move around, the orientation of fractures will change, because orientation of stresses will change. Gravitational stress is ... is interesting, ... you, basically it is weight of the rock. So your vertical stress in most cases it is nothing but the weight of the rock. So if I have various formation between surface and down hole, all you are doing is you are integrating in that way. You know, so it is an easy equation to do this integration, if it is, you can break it into various layers and add the stresses. If you do not know the value, most people will say, okay, what is my specific gravity of sandstone? If you remember, if you multiply the specific gravity by 0.433, you get your gradient. And so that gradient for


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is my specific gravity of sandstone? If you remember, if you multiply the specific gravity by 0.433, you get your gradient. And so that gradient for sandstone is about 1 to 1.1 psi per foot. So if I do not know anything about the formation, I'm just making a very quick estimate, I would say okay my formation is 5,000 feet deep multiplied by 1 psi or 1.1 PSI per foot, then my vertical stress should be between 5,000 to 5,500 PSI. So that is an easy way to do it. If I give you a formation and said, okay we have all this formation, here is the specific gravity of each rock, some of them maybe 3, 3.6, maybe less, multiplied by 0.433, so if it is a sandstone is 2.65, then approximately. So, 2.65 times 0.433 is about 1.1 feet, type of foot or so. But, if I give you information, you need to do it correctly, you, you information you need to do it correctly. You can do it easily. And that's what exactly what log would do when they calculate stress for you. All they are doing is calculating the


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correctly. You can do it easily. And that's what exactly what log would do when they calculate stress for you. All they are doing is calculating the vertical stress. We'll talk in a minute about calculating the horizontal stress. So you basically are integrating. If I'm in water, then part of it is water, you can see the equation, part of it is in water, part of it in, in, in the rock. So, in water, the specific gravity is 1 times 0.433, so, I have 0.433 psi per foot. So if I'm drilling in, in offshore well and water depth is, let's say, 2,500 feet, which is not, not really that deep in what we do now. And so you, that will be 2,500 times 0.433, so it's almost a thousand PSI, a little higher than a thousand PSI, and plus the weight of the rug. So let's look at the mechanical properties, the Young's modulus and Poisson's ratio, and Barth's constant. So we do it experimentally like what we have done that personally, we've done that in the lab many times, and we do it also from logs. So


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constant. So we do it experimentally like what we have done that personally, we've done that in the lab many times, and we do it also from logs. So if I do it in the lab, and there is a difference between the two, the difference is kind of important. If I do it in the lab, you know, I get a plug, or a core, and apply stress, I keep varying that stress, the force, I increase the force, so I'm increasing the stress, and I build, then I take the change in height of that plug, and divide it by the total length. That give me the stress. So delta L over L is the strain. The force divided by the area is my stress and draw this straight line. And the slope of the straight line is my Young's modulus. Okay, Young's modulus for a hard sandstone may be, may be 5, 6, 7 million psi. This is, the stress is in psi, the strain is unitless, is dimensionless parameter, because it's length divided by length. So, ... so if I have hard sandstone, maybe 5, 6 million, we'll give you a table later on, about


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parameter, because it's length divided by length. So, ... so if I have hard sandstone, maybe 5, 6 million, we'll give you a table later on, about the values. So 5, 6 million PSI. If you work in industry, most people will say 5 or 6, meaning 5 or 6 million, not just the five or six, but... So, five, six million PSI, and I'm talking in, in English units. If you work with metric units, then you need to translate that. ... And the stiffer the rock is, the higher the slope. So, if I go into softer rock, you get smallerness, the slope is smaller, so is smaller Young's modulus. Okay? Now, Poisson's ratio, let's talk about Poisson's ratio, then we'll give you a table. ... Poisson's ratio, let's talk about Poisson's ratio, then we'll give you a table. If I repeat the same, if I'm running this experiment again, and I'm applying this force, not only I'm measuring the change in, in strain, I'm getting, this rock is getting wider, so there is, there is axial strain, which is Delta L over L, but


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I'm measuring the change in, in strain, I'm getting, this rock is getting wider, so there is, there is axial strain, which is Delta L over L, but there's also lateral strain which is the change in radius divided by radius, or you can call it change of diameter divided by diameter. And if I take the ratio of the lateral strain divided by the axial strain, that gives me what we call Poisson's ratio. Poisson was famous mathematician, French mathematician back in many years ago, in 1700s or so, you can see this. That ratio has to be less than 0.5 for a rock. If you run, you're running these experiments on a rock, that ratio, that Poisson's ratio is definitely less than 0.5. If you, if I give you a problem, and I've done that before, trick the students, if I give you a problem and you end up calculating Poisson's ratio 0.5 or larger, you, unless I wrote the problem wrong, which is very unlikely, or you made a mistake, which is probably the case. It happened in an exam that I have given.


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I wrote the problem wrong, which is very unlikely, or you made a mistake, which is probably the case. It happened in an exam that I have given. And so I had to tell the students, Hey, you calculate a value of 0.5 or higher for percentage ratio, you are making a mistake. You need to look back into it. And why is that? And, why is that? Point five means that the rock cannot carry any load. So it's almost like fluid, like liquid. So obviously that is not the case. And I used to give a problem to, for students to understand that better. Okay? Then, then you have shear modulus, shear modulus basically deformation, if you apply force, and then you, shear force, then, then it will deform the rock by an angle, Theta. And, shear stress, shear modulus, it would be shear stress divided by this angle, in radians, not in degrees. And, you know the, Pi is basically 180 degrees. Okay, but you can actually, you can calculate shear modulus, it is related to Young's modulus and Poisson's ratio. So if


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Pi is basically 180 degrees. Okay, but you can actually, you can calculate shear modulus, it is related to Young's modulus and Poisson's ratio. So if you know Young's modulus and Poisson's ratio, you can calculate shear, shear modulus. It's interesting that they are all things that are related to one another. Young's modulus and Poisson's ratio are independent variables. I'm falling behind very much, but, so it's probably another half an hour or so. Bulk modulus is actually the inverse of compressibility. So if I apply stress around the rock, the volume gets smaller. So it is essentially one over compressibility, except that we call it, in rocks we call it bulk modulus, and again its bulk modulus is related to Young's modulus and Poisson's ratio. So here is the relationship between one and the other in this table. That's assuming isotropic material. The values you have, you can see, Young's modulus for granite, it can be 10 million, and this all in PSI. In shale it may be, shale,


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material. The values you have, you can see, Young's modulus for granite, it can be 10 million, and this all in PSI. In shale it may be, shale, shale can, you can have soft shale and hard shale. Most shales are not as hard. This is here, we're talking about hard shale that is almost as high as, as rock. But steel would be 20 to 30 million. Coal seam is 0.3 million. Let's look at sandstone. Sandstone can get higher than 4 million. With percent ratio, it's usually higher than zero, but percent ratio has to be less than 0.5. Usually, in, in what we deal with, the confining shale usually is softer than sandstone. And this is a very interesting parameter, we are going to look at it in a minute. But let me first talk about the parameters we looked at so far, Young's modulus and Poisson's ratio, shear modulus, this we call them static measurements. Dynamic measurements is where you are vibrating the grains. So you want to vibrate the grains, what you do is, you apply a sonic wave. Sonic wave


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Dynamic measurements is where you are vibrating the grains. So you want to vibrate the grains, what you do is, you apply a sonic wave. Sonic wave will vibrate the grains. And, your vibration happens either, either longitudinal, as you can see in this picture, or transverse. So, if you are in one place, and you can see the, the reference of, of this slide, and I find it like an interesting slide. So, you can see there is vibration going longitudinal or transverse. So if I apply a sonic wave, the wave either will go compressional wave, so it moves from one place to another. So if I have a source in one, say at the bottom of the well, and I have a receiver a little higher, I can hear the compressional wave coming in, and then I hear the shear wave coming after it. Okay, so, I hear the compressional, then I hear the shear wave. I hear the compressional, then I hear the shear wave. The difference, the difference in time or in velocity of the two, is an indication of Poisson's ratio. So, so


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then I hear the shear wave. The difference, the difference in time or in velocity of the two, is an indication of Poisson's ratio. So, so here if I'm listening to it, I think that, So here, if I'm listening to it, I think that, DTS was the shear divided by compression. This is taking longer time to arrive than the compression wave. The shear wave look taking longer time to arrive. I divided one another, square it, take one half minus one and use this equation, I can calculate for Sons ratio. And that is the way logs actually calculate for Sons ratio and then in a minute we'll see how you calculate the stress out of that. Because you run sonic log, you're getting density of the rock, so you can calculate the vertical stress, if you integrate the whole thing, you get the vertical stress. If I'm also doing Sonic-Log and I have to have enough distance between the transmission and the receiver, in order to be able to see able to see the compressional and shear wave distinctly from one


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distance between the transmission and the receiver, in order to be able to see able to see the compressional and shear wave distinctly from one another. Once I get the Poisson's ratio, then I can get shear modulus. Here is a simple equation where A is a constant here, and from there I can calculate Young's modulus and you will be in business. So you can do it, you can do it with locks, okay? Or if you have velocity, here is the shear wave and the compression wave, the V sub V. So before we leave this area, okay, let's say I calculated Young's modulus and Poisson's ratio here from a log and I also calculated from any lab experiments, which is a static. So which one would I use? Are they equal? The first question, should we have an agreement? Should these two parameters, whether static or dynamic, be equal? Well, probably not. And there are two different types of measurements, one is static and one dynamic. And they will be different. Which one is larger depends on a lot of parameters.


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two different types of measurements, one is static and one dynamic. And they will be different. Which one is larger depends on a lot of parameters. So we're not going to get into that, but which one would you use? Mostly we're going to say dynamic, and the answer is no, it is use static values in designing the fractures. Why is that? Well, the dynamic values is actually you are vibrating the grain, but what you, when you look at your fracture, although the fracture is obviously a dynamic process, but it is not depending on, on vibrating the grain, it's actually pushing the rock very much like what you do in experiment. And so, what most, what we would do is, you develop a relationship between the dynamic and the static properties in your formation. It's easier to, to run logs, you say, run the logs, you take the values and boom, through your correlation between static and dynamic you get the static and use that in designing the fracture. Okay? I may have said it a little fast, so you


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your correlation between static and dynamic you get the static and use that in designing the fracture. Okay? I may have said it a little fast, so you may want to listen to the game. So here is the stresses, now, this area is a little dry I have to admit, but it's important to understand. So, now what stress would you use? Well, the stresses that you have, the total stress is carried by both the grains, sandy grains and by the fluid. grains, sandy grains and by the fluid. Okay, so the, the... and what is affecting the propagation or sanding or collapse of the rock, is the effective stress, not the total stress. Effective, because the stress is being carried by both the grains and by poor pressure. The first attempt to do that, to look at it, was Terzaki's effective stress. He said, hey, what is being carried by the grains is you just take the total stress and subtract out of it your poor pressure. However, people found out later on that poor pressure may not be 100% effective in


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take the total stress and subtract out of it your poor pressure. However, people found out later on that poor pressure may not be 100% effective in carrying that stress. Because it moves out as you press it, it moves out. So, and you say, well, there is a factor here that is usually less than one one which will buy it coefficient. So rather than using assuming it's 100% efficient, we're totally efficient. So that's about constant anywhere between 0.7 to 1. It depends, most people will probably use about 0.9 or so. But, it is important to measure that parameters, especially if you're going to look at sanding. That is very important to do. So, you can see in some situation, your biotrch constant for Ottawa sand, prostate if it is dry is 0.33. You can see that's in megapascal. You wanna relate with mega, this is megapascal is 10 Atmosphere, so you multiply any of those numbers by about 145. One Atmosphere is 0.1 Mega Pascal. So, you can see there is, depending on the rock, it may be, it


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so you multiply any of those numbers by about 145. One Atmosphere is 0.1 Mega Pascal. So, you can see there is, depending on the rock, it may be, it may be up to 0.9 and in some rocks it may be down less than 0.5. So it's important to, to test the rock, you can only find it by testing. You test the rock and you find what, what batch constant, batch coefficient you... you have, because that's going to affect your sanding issue. So, we... we've done that, we... this is just a table to give you specific graph of different material, in case I... important, in case you get a problem on calculating that stress. So, fresh water is one, this is your specific gravity, is dimensionless. So, to get the... the stress down hold, is the depth, the integration of the depth time, times ... specific gravity time, point 433, which is ... the gradient of fresh water. So, in case of Quartz is about 2.65, 2.65 and so 0.433 times 2.65 it gives you whatever value is a little bit more than one psi per foot.


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water. So, in case of Quartz is about 2.65, 2.65 and so 0.433 times 2.65 it gives you whatever value is a little bit more than one psi per foot. Now without going into, hmm, I, let me look at the first equation. I usually, we go through the derivation of this equation, which is very easy to do. And I would, my advice is you go, if you Google it you'll find the derivation. Derivation is very simple. Basically all, all you're saying is that when we calculate Poisson's ratio, we are saying it's unconfined and we calculate how much lateral strain you have. But if that rock, where you are applying stress, is part of a huge formation, then it doesn't have room to expand. Then it's applying stress. So Poisson's ratio also is saying when I apply stress vertically, part of it is going to transfer horizontally. And so if I apply this concept, then I can say when I apply effective stress vertically, which is sigma V dash, there will be sigma H dash, there will be a horizontal stress, effective


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I can say when I apply effective stress vertically, which is sigma V dash, there will be sigma H dash, there will be a horizontal stress, effective horizontal stress moving sideways. The relationship between the two is based on Poisson's ratio. Okay, so you can see Poisson's ratio has to be less than 0.5 because if Poisson's ratio is 0.5 that's saying the stress is transferred horizontally 100% that the rock cannot carry any stress which obviously that is false. So then the rock cannot carry any stress, which obviously that is false. So, if the rock can carry some stress, then Poisson's ratio has to be less than 0.5. In sandstone it may be 0.35 or so. So you can look at it as a 0.35 divided by 0.65. So it's a little bit more than half the effective vertical stress. Okay, in case of shale, if it's softer shale, it may be 0.4, a little higher than 0.4, not much higher. Sandstone, maybe even less than 0.35, maybe 0.3. So you can see that this is the reason that softer shale will have


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higher than 0.4, not much higher. Sandstone, maybe even less than 0.35, maybe 0.3. So you can see that this is the reason that softer shale will have higher horizontal stress than hard rock, sandstone. And this, as we are going to talk about next week, this provide us with some confinement of hydraulic fracturing. So nature was nice to fracturing, to the oil industry, to create fractions that are somewhat confined. We'll talk more about that next week. And so here I'm just giving you this equation that I usually ask students to derive. You can see in the first equation is the relationship that very easy to derive. You may wanna Google it. I may add the derivation to the presentation that will put it in research gate. And we send you a link of the last week presentation. You're going to receive another link for this week's presentation. Then if I substitute the definition of the effective stress, I end getting here an equation relating the total horizontal stress to the total vertical


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if I substitute the definition of the effective stress, I end getting here an equation relating the total horizontal stress to the total vertical stress, and obviously, pore pressure will appear, and also, Alpha will appear, if we're in class, I would have asked you if you remember what Alpha is, it is Biot's constant. In situ stress orientation is very important, there is multiple techniques to, to calculate it, that will... Let us wait till when we talk about testing next week. We... there's a lot of ways to, to do it. Next week we'll talk about the, the MicroFrac testing. I may, may add something, but I was not planning on it, but an Elastic Strain Relaxation, this is something, it's very interesting that we, we have done previously, we'll, we'll talk about it next week. ... we'll talk about it next week. ... I'll probably will not talk about tiltimeters, it's been used before, it's very interesting technique, but we may be able to... Then people will use well testing and doing it.


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about tiltimeters, it's been used before, it's very interesting technique, but we may be able to... Then people will use well testing and doing it. Geological rules of thumb, ... we... I have personally used that, we work with a company and the company was not, they really did not want to run any testing, say well can you guess what it would be from our geologic map, I looked at it and they said, yes, yeah we can do that. You, you look at it, it was an anticline, there was already long natural fractures, there was faults in the formation, so it would be parallel to the hydraulic fraction, most probably would be parallel to this fault, so you follow the anticline. And so that is one way of doing it. Let me look at the effect of drilling a well bore, because this is something I want to cover before we... I mean, it is fairly important to understand the breakdown of a well bore to create a fracture. So when you drill a well, to create a fracture. So when you drill a well, Oh, I, I could


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the breakdown of a well bore to create a fracture. So when you drill a well, to create a fracture. So when you drill a well, Oh, I, I could have had a graph here. When I, drill a well bore, I, I change all the stresses around that well bore. Okay. Many years ago, this equation was developed, when I say many, I'm talking about more than a hundred years ago. You know, if you, if you know mathematics is, is more advanced than any other, other technology. People develop stuff in math that takes many years before it gets applied. So, if I have a rock, and let me show you, basically I did that by hand, couple of days in October, the weekend. If I, if I have, drill a wheelbarrow like this, and the minimum stress is this way, the, the minimum horizontal stress, the minimum, maximum horizontal stress is in the other way, and I'm looking at the top of the wheelbarrow. So, the vertical stress is going vertically into, into the picture. And, I'm going to look at an angle of Theta here. So, I, I


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of the wheelbarrow. So, the vertical stress is going vertically into, into the picture. And, I'm going to look at an angle of Theta here. So, I, I can look at any point in the reservoir, not inside the wall bore. Any point inside this reservoir, this is my, the white area is my reservoir, and if I draw a straight line to the center of the wheelbarrow, I have this angle theta. So the stress at any point inside this reservoir is defined by these two equations. So Sigma RR is whatever stress that is going perpendicular going towards the center of the whirlpool. Sigma theta theta is the stress adjacent or tangential tangential to the wall bore or parallel to that wall bore. So you you can see it is moving like a circle around around this this wall bore. So this equation would describe so now you know the definition of Sigma H, Sigma H, Sigma H max, and the Theta. Rw is the Wilbur radius, and the R is whatever distance you are away from, from that Wilbur, from the center of the Wilbur. So


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H max, and the Theta. Rw is the Wilbur radius, and the R is whatever distance you are away from, from that Wilbur, from the center of the Wilbur. So I'm going to do it just mentally here, rather than, usually it would be, I want you to, it may be a good exercise for you to, to eventually sit down with a piece of paper, I'm going to do it very quickly here. So the first question here is, what, what, what is Sigma RR at the wheelbore? Without looking at the equation. Well, Sigma rr at the wheelbore, if the wheelbore is stable, should be zero. So let us substitute here. So at the wheelbore, RW is equal R. So this is 1. 1 minus 1 is 0, so the first term disappears. And here was 1 minus 4 times 1 plus 3 times 1. So that is again 1 minus 4 plus 3. That is 0, so the second term is zero. So sigma RR at the well bore is zero regardless of theta. So that's now I have a stable well. If it is not a stable well, we'll find out later on, maybe next week we'll talk about testing, we'll see other


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So that's now I have a stable well. If it is not a stable well, we'll find out later on, maybe next week we'll talk about testing, we'll see other situations. So Sigma R is regardless of Theta at the Wilbur it is zero. That sounds very good. So now let us look at Sigma Theta Theta which is tangential to the, to the wool board. So add, add the wool board, Rw squared divided by R, well R is equal to Rw, then this is, this is one, one plus one is two, two here cancels out, so I end up with sigma H plus sigma H max, minus this second term. I'm going to do it now for theta equals zero and theta equals 90 degrees. Okay, so everyone get a piece of paper and follow what I'm saying. So at R equal rw, so I have sigma h plus sigma h max minus, this is one plus three is four, then it becomes a minus, I'm going to open parentheses two sigma H minus two sigma H max. Time cosine, if I'm at zero, cosine zero is one. So when I put this all together, it ends being three sigma H max minus Sigma H min.


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two sigma H max. Time cosine, if I'm at zero, cosine zero is one. So when I put this all together, it ends being three sigma H max minus Sigma H min. Right? Okay now I'm going to do the same thing again except I'm going to put Theta of theta of pi over 2, 90 degrees. So pi over 2 times 2, that is, that's pi, that's 180 degrees. What is cosine 180 degrees? I hope you remember, it is minus one. So if I put, now multiply that second quantity by minus one, I will find out at 90 degrees, the equation is fairly similar, it is sigma, 3 sigma H min minus sigma H max. Okay? Remember that, because this equation, I will be asking you, if you know anything about hydraulic fraction, you remember the CFD definition and you remember this equation too. As well as the equation for Poisson's ratio, the relationship between the vertical and minimum horizontal stress. There's three equations you need to remember. So if I look in here, I find out the sigma RR at this theta equal to zero is three sigma H


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stress. There's three equations you need to remember. So if I look in here, I find out the sigma RR at this theta equal to zero is three sigma H max minus sigma H min. And at 90 degrees, which is, would be vertical here, the sigma theta theta would be vertical. It is three sigma H Min minus Sigma H Max. Now if I'm ballooning, if I'm injecting fluid and I'm getting this well to expand, to get a little bigger, where would it break? It would break at this point where it has the smallest stress, because what I find out, that this would be the smallest stress here, and the high, talking about tangential stress, and this is the highest tangential stress there. So, fracture is going to break here and propagate perpendicular to minimum stress. Which it, that's the way it should happen. and propagate perpendicular to minimum stress. Which is, that's the way it should happen. I look at the bring down pressure and it looks something like that. So sigma RR at the whirlpool, regardless of where


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the way it should happen. I look at the bring down pressure and it looks something like that. So sigma RR at the whirlpool, regardless of where you are, this is sigma RR, is zero. And when you go away, it goes back to the far field stress. Sigma theta theta depends on your orientation. So if I have a graph like this, which sometimes people call it hoop stress. So hoop stress, here is the largest, here is the smallest, so it depends, so if I'm ballooning the fracture it will break at the smallest value. There are a lot more to it because as you fracture the weld, you change the stress regime. So here I'm saying this is weld board that has not been fractured yet. So I apply stress, it should look something like this. If I increase the stress, eventually it will break at theta equal 90 degrees. It's in here, like what we show it's. So the fracture would propagate this way, perpendicular to minimum stress. How about breakdown? Have you ever looked at a formation and you notice it's


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the fracture would propagate this way, perpendicular to minimum stress. How about breakdown? Have you ever looked at a formation and you notice it's breaking down? Well, when breakdown is under compression, so actually breakdown will happen in this orientation, because now I have very high maximum stress, it's causing the formation and lowering the pressure inside the wheel bore very much that you have a breakdown here and here. Then if I look at a breakdown I know immediately that the minimum stress is in the other direction. So, how to take care of it, is you increase the weight of the mud. And, here is the equation, considering the, the pore pressure, and the depletion of, of the formation. and the depletion of the formation. So here I have Wilbur pressure, and I have pore pressure, and obviously here you are be considering Biot's constant. So what equation you would remember, you want to remember, let's go back to this equation. You want to remember, not this whole equation, but


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So what equation you would remember, you want to remember, let's go back to this equation. You want to remember, not this whole equation, but you wanna remember the three sigma max minus sigma h, because that appears very much. You wanna make it more accurate, you add the tensile stress to it, you wanna remember CFD, and you want to remember CFD, and you want to remember the whole sense ratio. So let us entertain a couple of questions. Again I'm going to put this presentation on Researchigate, we'll send you the link to Dr. Elgarhi, who will communicate with you, but you can, you don't actually even need the link, if you know what research you're getting, you can get on it and you can download anything you want, assuming it isn't there. I've sent also, Dr. Algarhia will send you a link to some write-up I did, it's a long time ago, in mid-90s, early to mid-90s, however it is still a very good writer. It's mostly for fractional, horizontal ones and I would suggest that you read it for


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early to mid-90s, however it is still a very good writer. It's mostly for fractional, horizontal ones and I would suggest that you read it for your own enjoyment. Anyway, let's entertain a couple of questions. Okay, so thank you Dr. Solomon for this informative session. Now we will begin by addressing the questions. So the first question is that should we drill the horizontal wells in the direction of minimum horizontal stress direction and why? That's a good question. Actually drilling a well in the direction of minimum stress, we found out if you have normal stress regime, actually this is the most stable situation you've got. So second, we don't have, I don't have a presentation on that, well I do have in my computer but not in here. People always talk about what is the optimum or the best way to deplete a formation. You wanna drill horizontal well and fracture it usually in very tight formation. And the best way to do that in a tie formation is to create multiple fractures in that


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well and fracture it usually in very tight formation. And the best way to do that in a tie formation is to create multiple fractures in that horizontal wells. If we look at history of what we have done, we all were the first who started, and the first fracture of the horizontal well was in late 80s. Actually, I do have a paper on the subject you can download from SPE that we, I wrote back and presented it in 88 and it was published in 1990. Well, I can't send you a link of it because SPE owns the copyright. So you have to download it from SBE. And so we used to talk about four to six wells. Now we fracture not even 10 or 20, sometimes a hundred fractures in one horizontal well because the formation is extremely tight. You can't, you can do that only with horizontal well, where you create multiple, multiple fractures. If we have time at the end, I will show you. Actually, I do have a nice presentation on, Actually I do have a nice presentation on, that I, we did eight, ten years ago. I


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end, I will show you. Actually, I do have a nice presentation on, Actually I do have a nice presentation on, that I, we did eight, ten years ago. I may put it also in Researchgate, because it is my presentation. I cannot put papers, but I would put it in Researchgate, look at it, it's a very interesting presentation and it deals only with fracturing horizontal ones. And that usually, I used to cover it, it takes, it's only mostly about fluid flow and it used to take me about three hours to present but we put the presentation you may wanna you may wanna look at it. Okay great. Sometimes in the future we may present it to the group in the distant future. Go ahead. Okay, great, thank you. And, the second question is that, is it possible to see hydraulic fractures in real life? And, if yes, is there any real example for that? You see it real life? You see it in real life? Yes I think he means if it's visible, like to see it in the naked eye. No not visible. Horizontal. Let me say


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it real life? You see it in real life? Yes I think he means if it's visible, like to see it in the naked eye. No not visible. Horizontal. Let me say something about that. We have, many years ago, when I was in the Halliburton company, we actually, Halliburton developed a tool. I was involved in it. There was a lot of people were involved in that specific tool. And it was a joint project between Halliburton and TUTAL. And, it was called the tool, T-H-E, for TUTAL-Halliburton extensometers. Basically, we worked on, myself and my group, we worked on the rock mechanics side of it, and those other groups worked on the electric, the electrical part, another group worked on the mechanical part. Where you go in, and you inject the fluid, and you balloon the well, so you, you don't observe it by eye, but you observe the behavior of ballooning of the fraction of the well bore and eventually you observe or measure the behavior of the fracturing process and the widening of the fracture. Dr.


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of the fraction of the well bore and eventually you observe or measure the behavior of the fracturing process and the widening of the fracture. Dr. Solomon, I believe that question mean instead of having vertical fracture, he's من التحطيق. د. سليمان ، أعتقد أن هذا السؤال يعني أنه بدلاً من الحصول على تحطيق الوضع الوضعي ، يتحدث عن تحطيق الوضع الوضعي. هل هذا حال ، حال حقيقي ، أن نرى تحطيق الوضع الوضعي في الحياة الحقيقية؟ هل قال أنه تحطيق الوضع الوضعي؟ نعم ، تحطيق الوضع الوضعي ، نعم. حسناً ، لقد رأيت ، لقد رأيت حالات Yes, horizontal fractures, yes. Well, I have seen cases, at least there was one very clear case where it was a horizontal fracture. I have another case that you don't, let's not say see, but observe the behavior of horizontal fracture. We've seen that. It's not very common, but observe the behavior of horizontal fractures. We've seen that. It's not very common, but we have seen it. Yes, it can be done in a vertical well, you see horizontal fractures. Okay, great. Thank you.


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seen that. It's not very common, but we have seen it. Yes, it can be done in a vertical well, you see horizontal fractures. Okay, great. Thank you. And the last question for today, can we use the dynamic Young's modulus to design a hydraulic factoring operation or job without converting it to static Young's modulus? The answer is no. You need to convert it to the static values. There have been a lot of correlations. You have to build the correlation based on your reservoir. So many people have done that. You can't just, there is no, and also there is no general relationship between the two. It has to be developed for a reservoir. So thank you Dr. Salomon for joining us this afternoon and we are pleasure to have you today and I just want to highlight that this session was been recorded and it will be uploaded soon on Pio Petra's YouTube channel. And also for those who are considering to join this course, they already missed a quiz. However, they can still join, yet they should pass the


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And also for those who are considering to join this course, they already missed a quiz. However, they can still join, yet they should pass the three remaining quizzes and do the final exam. And for those who already joined, and please continue doing the quizzes and do the final exam. Now, best of luck. And thank you again again Dr. Solomon and have a nice day and see you in future webinars. Thank you. Alright. Bye everybody.


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Hello everyone, I hope all of you are doing great and staying safe. On behalf of Biopetrol, Arab Oil and Gas Academy and SPE Egypt section, I'd like to welcome you to our third interesting webinar about hydraulic fracturing. This course consists of four webinars, four quizzes and the final exam. First, let me introduce myself. My name is Selma Rashidi. I'm a petroleum engineer. I owned my bachelor's degree from the American University of Ras al-Khaimah. And my name is Shahad Bajet. I'm a third-year petroleum engineering student at KNU in Kurdistan, Iraq. And we will be your moderators for today. Our lecture today will be delivered by a very special guest from the University of Houston. Professor Mohamed Salman is the chairman of the Petroleum Engineering Department. He holds 35 patents on hydraulic fracturing operations, analysis, testing, and confirmance applications. He is the author or co-author of over 250 technical papers and articles in petroleum engineering. Professor Solomon


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and confirmance applications. He is the author or co-author of over 250 technical papers and articles in petroleum engineering. Professor Solomon is a registered professional engineer in the state of Texas. He is also a well-known member of the Society of Petroleum Engineers and the National Academy of Inventors. He has authored several book chapters on stimulation testing. Also, Professor Solomon is the primary editor of Fracturing Horizontal Wells, published by McRaw Hill in 2016. He's a co-author of Optimization of Hydraulic Fracture Stages and Sequences in Unconventional Formations, published by CRC in 2018. Professor Solomon holds a bachelor's degree with honors from Cairo University, the mic is yours. Thank you very much. Welcome everyone. Hope you enjoy the lectures. Uh Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Thank you very much. Welcome everyone. I hope you enjoyed the lecture. This lecture will talk a little bit about fracture design. We'll address several


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you very much. Welcome everyone. I hope you enjoyed the lecture. This lecture will talk a little bit about fracture design. We'll address several issues that we have in fracturing. I hope you enjoy it. that we, we have in, in fracturing. I hope you enjoy it. And if you have questions at the end, we'll try to address a few questions, but you can always send other questions to me on my email, which it would be mysolomonatuh.edu. If we run a fracturing treatment, this is, was basically a test, which we ran a very small fracturing treatment, but it shows what, what usually happens. In this specific case that we ran many, we ran it many years ago, where ... where we had downhole gauge, and we monitored the pressure rate, and we also monitored ... propent if we, if we have some. ... So you, you can see here, you, like what we talked yesterday, and I'm going to show another slide that even shows that even more clearly. ... As you inject, as you inject the fluid, you can see if I have


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and I'm going to show another slide that even shows that even more clearly. ... As you inject, as you inject the fluid, you can see if I have injection rate, I have down hole pressure. ... As I inject, the pressure will increase at one point, I'll ... break down the formation. If you, if you attended last week, and we talked about it, there is a breakdown, that we, if you remember, we looked at the equation, I ask you to, maybe go through the derivation a little bit. It is ... 3 Sigma H H min minus sigma H max. And the fracture is perpendicular to minimum stress. And you can see it, once you bring down the wellbore, the formation, pressure will decline, then it should stay constant or increase a little bit. We'll talk about that a little bit more in this lecture. Most of the models we see some increase in bottom hole pressure that has... people use it in analysis, we'll discuss that. ... but, that would be indication of fracture propagation, you can see I have a slurry rate here, I


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people use it in analysis, we'll discuss that. ... but, that would be indication of fracture propagation, you can see I have a slurry rate here, I have... ... there is not much of problem to concentration, this is mostly like a test. And, you can see at one point I'm going to shut down, stop injection, ... pressure would decline, the sharp decline in pressure, we'll, we'll see more exaggerated case in a minute. ... That will be an indication of near wall bore ... friction ... inside. With the, with down hole gauge, that's what I'm going to see, the near wall bore friction. If I'm having surface gauge, then I see sharper decline, which will indicate the, the friction inside the, the tubing. Or ... or, or whatever ... inside the tubing, or ... or whatever inside the wall board, going from surface down to down hole. And once I... we stop, there is some pressure decline, we'll... Next ... next week we'll talk about analysis, very much analysis of this area here, what happens when you stop


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is some pressure decline, we'll... Next ... next week we'll talk about analysis, very much analysis of this area here, what happens when you stop injection and the pressure will decline, that will cover it in fracture testing. This is another, another case that we, we looked at, and you can see sometimes people either they monitor like this orange line, the tubing pressure, this is the surface pressure, or you monitor ... down hole pressure. So there is a lot of things that one will monitor during fractioning treatment, and it is not monitored for no reason. We are going to see, as we talk about today and the next week, a lot of this data need to be analyzed, people analyze it. I'm not sure, some of it is real time, some of it is after the fact. We'll see how we do that. Okay, this is, was a test and which run many, many years ago. Actually I was on that job. It is, it's interesting. It's what we call micro-FRAC. I'm going to discuss it more in detail what micro-FRAC is later on. But


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I was on that job. It is, it's interesting. It's what we call micro-FRAC. I'm going to discuss it more in detail what micro-FRAC is later on. But the reason I'm showing you this slide is to show a very controlled test. You can look here, I have injection rate showing here, I'm injecting, I'm shutting in, injecting, shutting and so on. So I actually have fractured this wellbore three times. The very first time, you can see pressure increased very much, and at one point, the... here we see very sharp drop in pressure. Actually, this point here, this sharp point, is where you broke down the formation. That is the breakdown. Once you break it down, this sharp resistance, the hoop stress we talked about last week, you can see here once I overcome that hoop stress, pressure will decline very much. So there is a drop here. One make, you know, some people have taken this value and say, well, if I know minimum stress, I can also calculate maximum horizontal stress from here. Sometimes it


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some people have taken this value and say, well, if I know minimum stress, I can also calculate maximum horizontal stress from here. Sometimes it works. So here I'm injecting, so there is some propagation, and then at one point I'm shutting down. So you go from here to here, very sharp with decline. Eventually you lose pressure inside the fracture and the fracture will close. And so, one will ... can use data here from the shut-in down to fracture closure, and from fracture closure down you can do some other analysis. So that will be before the closure analysis, this is the after closure analysis, we'll discuss it a little bit next week. The fracture closure pressure is very important value, and the reason for that, if, if, if we're running a test, and the amount of fluid that we're injecting is fairly small, that closure pressure will be very close to the minimum in situ stress. You know, the far field distress we talked about last week. So what happens now if I come back, if I come


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very close to the minimum in situ stress. You know, the far field distress we talked about last week. So what happens now if I come back, if I come back and I, after some time, I start injecting again, would I see that breakdown? Of course not, because you can break down the formation. And this is a vertical one. You can break it down only once. So when I start injecting again, at one point I'm going to reopen that fracture again. So I'm reopening the fracture. Opening the fracture is a lot smaller than the reopening pressure is a lot smaller of course than the breakdown pressure. Because here I'm just overcoming the minimum stress in the breakdown pressure, overcoming the hoop stress which would be 3 sigma min minus sigma max. So here I'm doing, we did the same thing so we repeated the test a couple of more times. And we wanted to see whether I have consistency in fracture closure pressure, what we did, we had consistency of ... ice IP, that's near, near fracture, fracture pressure.


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I have consistency in fracture closure pressure, what we did, we had consistency of ... ice IP, that's near, near fracture, fracture pressure. fracture ... fraction pressure. And another issue I want you to... we're not going to... we're going to... it's going to affect what we do in testing, but you need to know a little bit about it, because this is a very famous equation that was developed many years ago, people call it Carter's equation. Basically showing ... as you're injecting fluid inside the fracture, you expect a couple of things, you expect the fracture is propagating, so there is some fluid staying inside the fracture otherwise it would not be opening, so I have fracture volume, but I have some fluid that is leaking off. And that, that fluid that's leaking off, I'm going to call it leak off volume. ... In many of the experiments we do in the lab, to determine how much fluid would leak off, we do in the lab to determine how much fluid would leak off, would eventually it


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we do in the lab, to determine how much fluid would leak off, we do in the lab to determine how much fluid would leak off, would eventually it follows a shape like this curve. Basically linear fluid, there is some early fluid, fluid that will leak off very fast. I call that a spurt loss. So that as soon as the version of formation gets exposed to a high pressure fraction fluid, I'm going to see, I'm going to see fluid leaking off very quickly, more or less instantaneously, I call that a spurt loss, and then it follows a straight line, that I'm plotting versus square root of time, and the slope of that straight line is my Lekhov coefficient. If you remember in the first week we talked, two weeks ago, we talked about linear flow being, always shows itself as a square root of time. You may not remember that, but let us repeat it here. So usually when you have linear flow, it is linear function of the square root of time. If you remember when we talked about infinitely conductive


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So usually when you have linear flow, it is linear function of the square root of time. If you remember when we talked about infinitely conductive fracture. So however in many of development of analysis like in testing, since the spirit loss is small, they approximate the whole thing with a straight line going through zero. So there is some more extra work that one may do. We actually published some work in the area where rather than assuming the spirit loss is too small, we actually consider the value to it. It's kind of an interesting solution. But we have developed in the industry so many new solutions. Now, where we, rather than using leak-off coefficient, we depend on fracture permeability and say, well, this is dependent also on fracture permeability, and one can develop solutions for that. We'll talk about some of those solutions next week. With a fracture you... I'm going to give you just a little bit of ... what is ... equation that people will use in simulating fracture and


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next week. With a fracture you... I'm going to give you just a little bit of ... what is ... equation that people will use in simulating fracture and solving those equations. Basically, most people will assume that the pressure inside the fracture is constant. So there is not much of a pressure blow inside the fracture during compression. So that's except near the tip. Near the tip of the fracture, the pressure gets very narrow. And so many of the solution we'll talk about today will have this assumption here, saying the width of the fracture is function of E prime. E prime, I defined it here, is the plane of the modulus, which would be equal to Young's modulus divided by one minus the square of Poisson's ratio. So if Poisson's ratio is 0.3, so square is 0.09, so it is Young's modulus divided by 0.91, if it's 0.3. One, one important parameter that you, people look at, is stress intensity factor, which is nothing but the pressure inside the fracture, times the square root of length,


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that you, people look at, is stress intensity factor, which is nothing but the pressure inside the fracture, times the square root of length, the fracture length. And you, usually this term, people look at it as, Psi per square root of inch, not foot. So you can see when the fraction is very, very short, just starting, this value may be fairly, fairly small. Once the fraction length gets a little long, this value gets really large. Why I'm talking about it? Because this value needs to be greater than, than KIC. KIC is a critical in stress intensity factor or you call it fracture toughness, toughness which is function of the formation. So let us look at what values we have for formation for this KIC. It is sandstone is maybe 1200 to 1500. So you know if your pressure inside your fracture is of any significance, once it becomes a couple of feet, the fracture length is couple of feet, it will be, the KIC doesn't come into play anymore. It's not that important. Which it, when you think


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of feet, the fracture length is couple of feet, it will be, the KIC doesn't come into play anymore. It's not that important. Which it, when you think about it, if once you, you can see if you hold a piece of say wood and it has a little crack and you try to extend that crack, if it's very short it is very difficult to extend it. Once it gets along, you know, your the pressure time length becomes so high that you can overcome the resistance, the fracture toughness of, of the formation. You can see here, sandstone to 1,400 to 1,500, shale maybe a little smaller, granite is, is fairly high. One would, one would say, why do you worry about granite? Actually, there is a lot of new work being done on Granite, including here at University of Houston, but that is a research subject we can talk about some other time. One thing, when we talk about plane strain, many of your... is 2D and pseudo-3 models, we'll talk about plane strain. We're assuming that whatever happens, if I have fracture,


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plane strain, many of your... is 2D and pseudo-3 models, we'll talk about plane strain. We're assuming that whatever happens, if I have fracture, whatever happens, all the strain remains in this plane. When I deal with pseudo 3D models, I have a couple of slides at the end comparing the two. When I deal with full 3D model, now I'm going to say every place, everywhere inside the formation will have an effect on the fracture everywhere. So that it becomes a lot more sophisticated and more difficult to solve. At UH as well as in other places, everyone that developed his own models basically for research and we did the same. So if we deal with plane strain, which is very good in explaining a lot of things and we use it in data analysis, simplifies things and what it also gives a very good results. So, we're saying all the stress and the resulting strain remain on a plane which can be repeated infinite times. So every plane is by itself. And you can look at it either in vertically or


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resulting strain remain on a plane which can be repeated infinite times. So every plane is by itself. And you can look at it either in vertically or horizontally. If you look at it in a vertical, vertical plane strain, and then the model that was developed years ago for that 2D model, which we are going to use in analysis of data is Perkins and the Kernel model, the PKM. And the horizontal plane is KGT, KGD is after, is, we used to call it Krasnovich and Zoltov model, the original name. That's saying I'll do it in the horizontal plane. There is evidence that PKN works better, and I'm going to explain why I, that's my own personal opinion. Of course, other people may disagree with it, but I'll tell you why I think this is the case. So if I, if I look at the basic PKN, in a regular class we'll go through modeling and writing equations and maybe software for it. For PKN, I'm going to assume that the fracture cross-section is elliptical and as it is constant, the height more or less,


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and maybe software for it. For PKN, I'm going to assume that the fracture cross-section is elliptical and as it is constant, the height more or less, although we allow that to change, and is elliptical in the horizontal direction. And from there I can write an equation for width and I can write equation for pressure. And equation for pressure is going to show that, that was PKM model. Pressure, if I don't have sharp increase in heights or changes, serious changes, in most models, when you see some increase in pressure was time. A KGD assumes that I have a rectangular shaped cross-section, essentially saying we have a slippage on the boundary and that will reflect into constant pressure or even declining pressure. In most cases, in field cases, we have seen that the pressure does, when you have a successful job, pressure does increase with time a little bit, in some slope. We'll get into that slope in just a few minutes. I will tell you it's between, no, I need to wait till, because I


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time a little bit, in some slope. We'll get into that slope in just a few minutes. I will tell you it's between, no, I need to wait till, because I need to talk more about it and see what we plot there to see that pressure increase. In those cases, we're looking at fracture height and fracture height prediction. Is containment important? Yes, it is important. I'm going to show you some experiments that we have done in the lab before. Is it critical? Absolutely. You can, if you don't watch for it, you don't want to fracture into water. If you end fractures, say propagating downward and the fracture into, into water like some of the shale formation in North, in Northern US, United States. You know, you watch very much for fracture height because you have water zones below. And if you fracture into water, you essentially have a water well. So you want to control it. And it's not only that you get too much water, but you waste the prop and the fluid. Can it be controlled passively? Was


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So you want to control it. And it's not only that you get too much water, but you waste the prop and the fluid. Can it be controlled passively? Was that has been usually successful by looking at the injection pressure and monitoring what the pressure you're injecting. You may be monitoring the job. There is some ways to do that. We'll talk about some of them. But, but that's in a passive way, so I don't want to exceed certain pressure, so I don't break into, into another zone. ... Some people have tried it actively, ... using say, if, if you have downhole, you have a zone below your pay zone, that is, that, ... that is ... maybe carrying water and you don't wanna get into it, for water or whatever other reason. You could, some people have used the heavier propant. So they say, well, the heavier propant will settle and will prevent the fracture from, you end up having very high friction pressure in order to propagate that fracture. So you, it prevents a further propagation into the


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fracture from, you end up having very high friction pressure in order to propagate that fracture. So you, it prevents a further propagation into the lower zone. People have done it by creating a fracture with propant and then going back, perforating above into the pay zone and doing the, the fracturing, so it, it may work okay. There is ... the success of those active using Propant is, is not completely proven. Passively, yes it's proven by, especially when you monitor what is going on, using temperature, maybe pressure to know where, whether you're propagating down or not. You go, propagating downward or not. The containment happens due to one of, the natural containment happens because if you have a stress contrast, if you have a change in lithology. Let us show some slides, some experiments that we did several years ago on that subject. First, to stresses, it's very important to, in in advance to calculate the stress, both in, both in various layers and ... to calculate or measure


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First, to stresses, it's very important to, in in advance to calculate the stress, both in, both in various layers and ... to calculate or measure the, the stress in your pay zone, above the pay zone and below the pay zone. ... If, many time the stress variation is just due to variation in Poisson's ratio. If you don't have lithology change, but it can be due to just change in Poisson's ratio. If you remember, we talked about this upper equation here, and this is the effective stress in horizontal direction is Poisson's ratio or one minus Poisson's ratio times the effective vertical stress. If I substitute the definition of effective stress with bar its ratio, I decided to, I did not write it as a slide here, but I, I did a little bit of calculation, saying okay, and maybe you should give you this as a homework assignment. If, if you have sandstone and Poisson's ratio is 0.25, if I have 6,000 foot formation, I, I could end calculating my, my horizontal stress if I assume one piece I


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sandstone and Poisson's ratio is 0.25, if I have 6,000 foot formation, I, I could end calculating my, my horizontal stress if I assume one piece I for, for, for vertical stress, for total vertical stress and you substitute all the, but constant and pull pressure, we say pull, pull, pull pressure is gradient, say of ... here I said of 0.4, I end calculating ... vertical stress of 6,000 PSI, horizontal stress of 3,000 PSI, horizontal stress of 3,600 PSI. But, if I... if this ... Phase Zone is surrounded by... by shale, that's softer, and ... let's say for size ratio, in this case is 0.4, then my horizontal stress would be calculated at 4,800 psi. That is 1,100 psi stress contrast between the P zone and the shale zone. And if you have something like this, that will be sufficient to confine the fracture. So let's look at, oh, here is the problem. You will have it, you can solve it and see whether I made a mistake in my calculations or not. But here, here what I'm trying to show, if I have


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You will have it, you can solve it and see whether I made a mistake in my calculations or not. But here, here what I'm trying to show, if I have soft shale around the pay zone, sandstone reservoir, it may be confined. Well let us see if we, if it's really confined or not, because we have to talk about theory. Here is an experiment that we ran several years ago. This was a piece of rock that was 18 inch in length. It was 12 inch in height here, and the thickness was six inch. And we created a well in the middle, we had a perforation like you see here, I hope you see my mouse, my around the mouse, you can see here, I have, I perforated or have this an open area, and the rest is closed, I have a I perforated or have this an open area, and the rest is closed. I have a pipe here, and I started injecting, and you can see I created, very much, the fracture that I have here. The reason for that is I've got, basically I did not stress in the middle here, and I put 1200 PSI above and below. So


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the fracture that I have here. The reason for that is I've got, basically I did not stress in the middle here, and I put 1200 PSI above and below. So I have a stress contrast of 1200 PSI, that was sufficient to confine the fracture, although the formation is homogeneous, and there's absolutely no change in lithology between the middle part and the upper, upper and lower parts of the back. This is with the same piece of rug, the only difference is the stress. So we applied vertical stress of 1400 psi, I applied side stress here of 2000 psi, of 2000 PSI and then the stress perpendicular to the, to the screen is zero in the middle 1200 above 1200 below. So those are the lowest stress, so I'm expecting a fraction that will propagate along the screen. And you can see here, if you look carefully in it, and hopefully you can see it, you can see that it's, it looks like, almost like lines in here, like streamlines, where the fluid is going in and opening up this fracture. Experiment looks


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see that it's, it looks like, almost like lines in here, like streamlines, where the fluid is going in and opening up this fracture. Experiment looks really impressive. Well, in the other case, if I have negative stress contrast, if I have negative stress contrast, if I have negative stress contrast, if I, if I say okay maybe I have sandstone here, I have shale, maybe I have water, although I may have a barrier here, but below that shale I've got loose, loose stress formation and, and I don't want to get into it, but we did not realize that, so we fractured it anyway. So, this is a similar case, a similar size rock, and you can see here, the smallest stress, or smallest stresses are the one, perpendicular to the, to, to the screen. And you can, you can look here. This is just a highlighter. And you can see it basically to compare the size. This is 18 inches again, and 18 by 12 by six inches in thickness. And here we're doing it in a fraction in the middle. So I have 600, 1000 PSI,


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size. This is 18 inches again, and 18 by 12 by six inches in thickness. And here we're doing it in a fraction in the middle. So I have 600, 1000 PSI, 1000 PSI and I have here zero stress. So what happens here, when I, when I fracture in the 600 PSI, initially if you, if you see that same piece of rock, fracture propagated very much like what you would expect, but once it's so the low stress, you can see the propagation went down hole. So, so went downwards, and boom, all you can see much of the propagation went to the lower stress. So one has to be very careful. So knowledge of stresses is, is not just something that theoretical thing you need to do. It is really practical thing to do. This is the reason people have been, have been measuring and calculating stresses. We talked about calculating stresses from logs, like what we, the equation we saw. We're going to talk about measuring stress. Well, part of it, that fracture closure pressure, if I don't inject very much of fluid and I


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we saw. We're going to talk about measuring stress. Well, part of it, that fracture closure pressure, if I don't inject very much of fluid and I monitor the pressure carefully, I will see the fracture closure pressure. It's a very important parameter. We are doing a lot of research in the area right now. Believe it or not, it's still not, not a, is still open for interpretation. So we have our own technique, we may mention it a little bit, this is more of a research area, but we have companies really believe in what we're doing. So there is still a lot of work to be done in the area, nothing is final, science and technology never stop. But the important thing about this slide, you can see here, it went down all your much of or most of your, or your propagation is happening in the lower stress zone, and it ended breaking even into that lower, the stress, this lower, higher stress zone. stress, this lower, higher stress zone. So what is the problem with that? The problem with that is


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lower, the stress, this lower, higher stress zone. stress, this lower, higher stress zone. So what is the problem with that? The problem with that is your perforation are up here, your problem to may be sitting all down here. So you may not have communication even if this zone is is productive, there is no communication with the perforation. It is worse if this formation is having water or having issues, or so you're wasting a lot of energy, a lot of money, your province, your fluid, to get a fracture that is not effective. So you fracture it and say, oh, I am not producing nearly as much as I expected. Well, maybe because the fracture went in the wrong direction. I saw the opposite too. This was not in the slide, but I saw the opposite too, where we've been involved in a case for, can't tell you the oil company, but here in a smaller oil company in United States, several years ago, where they fractured and most of the fluid it came from a layer that they did not know was productive.


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company in United States, several years ago, where they fractured and most of the fluid it came from a layer that they did not know was productive. And the reason for that is the fracture propagated upward and intersected that formation that they were not intending to fracture at the end. Luckily for them, they ended getting a lot more production than they expected. But, still that's not right, because you... one has to know in advance, you need to make at least ... some knowledge of what is going on, so you design the best diffraction treatment possible. Well, if you, if you have some lithology change, you could have problems too. So a situation like this, where, where there is a fracture that's propagated upward and it ended big since I have a barrier here and that's not very strong and the fracture although there is some breakage through it but most of the propagation went sideways so that's again another another issue. This is more of discontinuity. We have seen that of


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through it but most of the propagation went sideways so that's again another another issue. This is more of discontinuity. We have seen that of discontinuity. We have seen that, actually, I have a case that we wrote a paper on several years ago. If you're interested in it, we had the logs, we had everything, and we ran very nice design, and we said, okay, we'll go apply it in the field. And after we applied it, we found out the fracture height, actually the fracture was better than expected. So we go back and measure it and the fracture height was smaller and eventually we realized that the logs did not show some break in lithology. So once you have a break in lithology, that can be a barrier. So you end to have confined fracture. So this is something again, we did not know. There is, people have seen, especially in shallower formation, this from paper by Warpansky. It was a fairly shallow formation that they went back and mined back the formation. You see there is multiple fractures.


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from paper by Warpansky. It was a fairly shallow formation that they went back and mined back the formation. You see there is multiple fractures. Part of it, this can be misleading a little bit because it was formation was a little bit too shallow to have the stresses that basically overcome heterogeneities in the formation. So the fracture followed the heterogeneities. You see things like this when you fracture shears because of the natural fractures you end up opening those natural fractures and they have very significant effect on what is going on. opening those natural fractures, and they have very significant effect on, on ... on what is going on. We are going to see some cases, real cases, actually with deep formation here in United States, Shield formation that we, we analyzed using some technique that we have developed. This is ... multiple fractures, they were actually, this also from Morpensky's paper, but there was a recent paper, couple of years ago, by ConocoPhillips,


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... multiple fractures, they were actually, this also from Morpensky's paper, but there was a recent paper, couple of years ago, by ConocoPhillips, where they did something similar, but except they did not, they did not mine back the... the well, because that was too deep to do that. Here we're talking about a few couple of hundred feet. But in ConocoPhillips' situation it was a very deep formation and they did not find problem in where they thought you would have problems. So, the fracture propagated in multiple... we have multiple fractures in multiple areas. It was a horizontal one, this one here, it was a vertical one in this example. ... You can see here also, you can see the fracture propagation, it looks like jumped, obviously it went around here, and then there's another crack. So there is discontinuity in the fracture. And, any time, any time you have turning of fracture, that's usually not good news, because it have very high friction pressure, and the turning around of


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any time, any time you have turning of fracture, that's usually not good news, because it have very high friction pressure, and the turning around of fracture happens on steps. the turning around of fracture happens on steps. So it does not happen very smoothly and that you would have a narrow, very narrow effective fracture width, which usually can, can cause sand out. The sand out meaning that the sand cannot continue to move, it's friction pressure is so high, that basically, eventually you have to shut down the treatment. So, here I'm going to give you some equations that were published in literature. There's so many different equations looking at things like that. And so, here if I have stresses, you can, you can, you can see, if fraction is propagating through higher stress, it may give you a pinch point, it may go up and, if I get higher stress here and we have, there'll be an upper tip and lower tip and with a pinch point. So let's look at, what would it would take to


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if I get higher stress here and we have, there'll be an upper tip and lower tip and with a pinch point. So let's look at, what would it would take to propagate a fracture. Usually we do it numerically. There is a lot of simulators in commercial as well as research simulators for fraction propagation. But sometimes it is good to look at cases where you have an analytical solution and you can look at it yourself in case you wanna do something very quick. So in a situation where this is symmetric case, I have a zone like this, a P zone like this, and surrounded by the higher stress. And here, since it's symmetric, I'm going to assume Sigma 2 and Sigma 2 here, they are same value, same rock. And I, I have this P-Zone in the middle. So the, if this is the case, if I ignore the, the effect of height, there is a little difference in height, so nothing is really, truly, truly symmetric. Because even if, even if I, if the stresses are the same, there is Hydrostatic head that is different. But


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nothing is really, truly, truly symmetric. Because even if, even if I, if the stresses are the same, there is Hydrostatic head that is different. But here I'm going to ignore this effect, and say fracture propagates upward and downward, as similarly. So I end having a single, the, the pressure, I can calculate the pressure that I need to, in order to fracture this formation. This pressure is related to the stresses that I, I have above and below and in the P-zone. So, since it is symmetric, Sigma 2, Sigma 2 here, pressure Sigma 1, and you can write this equation. I usually give a problem in this, for people to see what happens when the fraction height is very large. Once it gets very large, the effect of KIC becomes very small. And so, this confirms what, what we talked about earlier in this lecture. If it's a height, as when height is very small and then you, and then KIC becomes important. So the important of the second term, it gets smaller and smaller with time. There was, there


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very small and then you, and then KIC becomes important. So the important of the second term, it gets smaller and smaller with time. There was, there was fractional propagation and height, eventually it, it becomes essentially negligible. If you use this equation, please remember, when you write something like this, you know, it is not in degrees, it's in radium. How many, I've seen students that put this, get the angle and write it in degrees and say well the answer doesn't make any sense, of course it doesn't make sense because this should be in radium not degrees. If I have a case that's not symmetric it gets a little bit more complicated because the propagation in height above and below is different. Basically the center of the fracture moves in this case upward because the upper layer stress is smaller so I'm expecting more fraction propagation in upper zone than the lower zone. So you end this equation and you I'm just writing this equation for your knowledge, that's something


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propagation in upper zone than the lower zone. So you end this equation and you I'm just writing this equation for your knowledge, that's something to choose but you are going to find many equations like this in ... in literature. So, here if I have non-symmetric case, even if KIC is all the same, 1000 psi per square inch, you can see that the center of the fracture is moving upward. It can move actually upward above even the ... the P zone, depending on stresses, depending on your treating pressure, and those time. So, let's look at the material balance variable. This is the variable here, I'm injecting, you have fluid leak off, it's widening, the fracture, and I want to look at this, in general, and say, say well what happens, how we can analyze data. There was a paper that was written, a bunch of papers were written, but there is one were written by Nolte Smith that we are going to depend very much on the first part on this analysis, then we'll talk about a paper that ... myself


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were written by Nolte Smith that we are going to depend very much on the first part on this analysis, then we'll talk about a paper that ... myself and my students, actually multiple papers that we wrote on, on the subject. Most people don't realize that many of the models will depend, and the analysis techniques will, during fractional propagation, will depend on what is called the power law assumption. Basically it is saying, fraction length and the hidden area, we're writing in terms of area, but if you assume that the height does not change very much, then, then the length is, is fraction of time raised to some power. is function of time raised to some power. So, if I, if I take this concept, and going into treating pressure analysis, and I need to refer you to paper by Nolte, by Nolte and Smith, that was published several years ago. I, I should have written the paper number. I thought I did. I'm sorry, I, I let me see. I did not write it. I, I apologize for that, but I think we


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I, I should have written the paper number. I thought I did. I'm sorry, I, I let me see. I did not write it. I, I apologize for that, but I think we need to... I'm going to modify the presentation before putting it in, into Researchgate to have that SPA number, because this is, this is, it's not just to, to make sure that I'm referring to the right place, but also it is, it interesting for you if you wanna, wanna pursue the subject to read that paper. It's very fundamental paper by Nolte-Smith and it was published long time ago, but it is really an excellent paper. Basically, they, they said, okay if I plot the net pressure, net pressure is the pressure you're observing, the down hold pressure minus, minus closure pressure, or minimum in-situ stress. If I plot that net pressure, and you, you please notice when I say net pressure, that's a pressure difference, from minimum in-situ stress, not poor pressure. People that are doing well testing the Delta P is from poor pressure. So here,


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a pressure difference, from minimum in-situ stress, not poor pressure. People that are doing well testing the Delta P is from poor pressure. So here, my down-hole treating pressure minus minimum in-situ stress, if I take the log of this value and plot it log of time, they said, hey we can get one of various modes, and they called it one, two, three, four. Okay. Let's look at this modes. Mode one, which I see pressure going, maybe between 0.2 to 0.35, although I'm writing what they wrote, but I think it's usually a little higher than, than that. So, if I get slope like that, that tells me that the fraction is propagating normally, because that's what I expect to see from that power law model. to see from that power law model. If everything is okay, I'm going to see about 0.2, 0.25, 0.3, 0.35, you know, in that range, that will tell me that the fracture is basically a fairly confined, so I have some containment. There may be obviously some fractional height growth, but it cannot be


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that the fracture is basically a fairly confined, so I have some containment. There may be obviously some fractional height growth, but it cannot be excessive at all, and so, so that what you would hope to see throughout a fractional treatment, obviously, because now you see a nice fraction is propagating, you're getting a lot of length. Okay. Then they looked at model two, and they said uncontrolled fractional height growth. That's what I'm going to differ from them later on and what you're going to see in a minute. And they said, now you have an issue, and many times it may be natural fractures and you may, may it end turning into model, mode four, where you say, okay, my mode four is, let me go to mode 4, pressure is declining, I added this intersecting or opening natural fracture, breaking out of zone, lower extension pressure. Actually in their paper, if you read their paper, model 2, they had an issue of explaining model two, we have a different way of doing it right now. Then


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in their paper, if you read their paper, model 2, they had an issue of explaining model two, we have a different way of doing it right now. Then model three, nobody disagrees with that, is saying, if I see a very sharp pressure increase where it goes to slow one to one, that is usually a screen out. So one-to-one, that is usually screen out. And so you better be ready to stop injection. So that's tell me that the fracture is ballooning, is getting wider rather than getting longer in length. Sometimes you see cases where it is even going higher, the slope is higher than one to one. Okay, so we look at that and we, there was some recent decline in use of this model because there is a model, mode one and model three, there's really no disagreement in them. Two and four were some problems. So we developed our own new technique. So here I'm referring Maltesmith technique. The one issue that we had with it is the reference point is all based on zero time, which is a start of injection. We


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referring Maltesmith technique. The one issue that we had with it is the reference point is all based on zero time, which is a start of injection. We look at some experiment that we have done in the lab and some observation in the field and we think that fracture does not continue propagating. And once it stops propagating everything as if it's the time is reset. So we say the process that's in the technique we developed when the the process is this discontinuous. So we build another approach. I'm going to go through it a little quick because it's mostly research area or very recent area, but I encourage you to read the papers. We wrote a few papers on the subject. So very interesting. papers, we wrote a few papers on the subject, very interesting. So here I took a case and we applied Nolte-Smith analysis and this is taken from this paper here and this paper will give you a reference to Nolte-mith. I should have done it anyway. Sorry for that. And it looked at something like this. And


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here and this paper will give you a reference to Nolte-mith. I should have done it anyway. Sorry for that. And it looked at something like this. And when you look at it, the early part looks okay, but then you can see now it's going, the net pressure, this is log-log graph. The net pressure is flat, but if you are analyzing the data, you say, well, I'm not going to do anything. Well, I do not know what is going on. And then you get in here, you have to wait till this is 10, 20, 30 minutes. This is what people call frack pack. This is fracturing high permeability formation in the Gulf of Mexico. And we got it from a company that did that work. And so, but we can, all I can tell you is in Gulf of Mexico. So basically they waited, this is 30, 40, after 40 minutes of injection, usually those runs are usually fairly small. You stop. So we analyze it using our technique. Our technique is saying, every time there is some changes, I reset everything. I reset the reference point. So we call it


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it using our technique. Our technique is saying, every time there is some changes, I reset everything. I reset the reference point. So we call it moving reference point technique. So the reference point is not always zero. And I don't use logarithmic, I'm changing this and I'm plotting my exponent versus time. So here, this area here, the green is saying I have fracture propagation, and the red is saying I have a send out. So this is telling me that fracture was propagating okay here there was some issues after about seven, about seven, eight minutes in fracture propagation. Actually, I don't have as much propagation as I thought I have. And then it went back into normal propagation, did the same thing. Then I have normal propagation here, not opening natural fracture or anything. I have normal propagation. And then at this point I started seeing movement towards sand out, and here I know we are in trouble. So, many times in, in frac pack, you don't stop as soon as you see evidence of


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seeing movement towards sand out, and here I know we are in trouble. So, many times in, in frac pack, you don't stop as soon as you see evidence of sand out. You actually, you continue to inject. The reason for that is because the formation is very high probability, you want to create very wide fracture. So, the area, the early area you call it, you're fracturing, the later area here, you are still injecting to widen the fracture or to pack that fracture. And this is the reason we call it frac pack. Frac pack is a commercial name for Halliburton. Use it because I worked for Halliburton for 30 plus years before moving to academia 10 years ago. So, but with some others will call it frack and pack. So there is various names to it. But here you are widening that fracture. So many times people continue to inject, they are aware of it, that is it in sand out mode, but you continue to inject anyway, because you don't wanna get narrow fracture, you wanna get a wide fracture, because you're


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is it in sand out mode, but you continue to inject anyway, because you don't wanna get narrow fracture, you wanna get a wide fracture, because you're dealing with the hyper-ability formation. And this is another example, and when I saw decline in net pressure here, and I see in this area, there was, there's actually, it seems like we propagated in height or intersected natural fracture significantly before we moved into fracture propagation. And so this was very different situation from what we expected before. And we went into the packing of fracture for most of the fracture treatment. Then we applied it to shale formation. We applied it to two cases, one Marcellus shale, that's both in the United States, and one was the Shale, that's both in the United States, and one was the Eagle Ford Shale, also in the United States. Eagle Ford Shale has produced a huge amount of oil and gas. And so, and that was an interesting situation, not going to go into much details on how we did it, but


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has produced a huge amount of oil and gas. And so, and that was an interesting situation, not going to go into much details on how we did it, but the interesting thing, it was more or less like a blind test. And we can see here, this, every time you intersect a natural fracture, it dips. So we say we have natural fractures here, here, here. So in those areas, I can see very clear natural fracture. You pass the natural fracture and then I end getting into the fracture propagation. If you propagate the fracture some distance and boom, you intersect swarms of natural fracture and you can see it right away. So that, to me, that was, if you look at it, let me see, I don't think I applied the Nolte-Smith technique to it here. For the Egelford Shale, I saw similar things. And you can see here again, I have the intersection of natural fracture after some time, most of my fluid is leaking into natural fractures, so after, this is 190 minutes, you know that you're no fracture propagation


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after some time, most of my fluid is leaking into natural fractures, so after, this is 190 minutes, you know that you're no fracture propagation anymore, you're just leaking off, the fluid is leaking off into, into natural fractures. So I, if I'm running the job, I say, well, at this point, I need to stop the job. Let me look at, usually going back to regular fracturing treatment, what do you do? In a typical treatment, that's what you do. And the little things have been changing a little bit these days. You have a pre-pad basically to initiate the fracture. Pre-pad usually is a lighter fluid and you're initiating the fracture, and you follow with the pad, which is basically, fracturing fluid, except it does not have propent, so you wanna, you want to widen the fracture to get it ready to, to, to bring propent in. You want the problem, you want the fracture to be a little wide when you problem going in, otherwise it will sand out immediately. I've seen a case where, where, where,


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you want the fracture to be a little wide when you problem going in, otherwise it will sand out immediately. I've seen a case where, where, where, where using a hundred mesh sand and since there was not enough pad or pre-pad, the fracture sand it out immediately. You don't want this to happen. So, you inject the fluid, you open up the fracture, you widen the fracture, and then you end, putting propant carrying fluid. You may end with just, again, fluid without propant, in order to, to displace the treatment deeper into fracturing, you clean the whole board, and ... sometimes this works, sometimes it doesn't, this will... ... that would take a lot of time to discuss, so I'm going to limit at that. So, what's your fraction material? You a propane selection, you have fraction fluid, you have additives, and you have fluid testing. So we'll touch in each of those a little bit, but you will have the presentation, you can look at it, and if you... I have actually a lot longer presentation


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touch in each of those a little bit, but you will have the presentation, you can look at it, and if you... I have actually a lot longer presentation that is... goes into all those aspects in more detail. And I'll probably, what I will do, I'll put the more, the longer presentation on, send you the PDF file. May not cover it here, but will give you the presentation anyway. So let us ... for Provence, Provence is important to provide the necessary conductivity to produce the formation. If you remember, I put this equation again, if you remember, I told you almost to remember this equation if you work with fracturing. You... it's important to have fairly decent fracture conductivity. So you wanna get to, to good dimensions fracture conductivity, and this dimensions fracture conductivity is Kf which is coming from the problem, the width of the fracture which is what you injected for how long and what the pressure is, formation probability and the fracture length. So you want to get the


