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The Pennsylvania State University
The Graduate School
Department of Energy and Mineral Engineering
ANALYSIS OF INSTANTANEOUS SHUT-IN PRESSURE
IN SHALE OIL AND GAS RESERVOIRS
A Thesis in
Energy and Mineral Engineering
by
Ahsen Ozesen
2017 Ahsen Ozesen
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Master of Science
August 2017
ii
The thesis of Ahsen Ozesen was reviewed and approved* by the following:
John Yilin Wang
Associate Professor of Petroleum and Natural Gas Engineering
Thesis Advisor
Hamid Emami-Meybodi
Assistant Professor of Petroleum and Natural Gas Engineering
Shimin Liu
Assistant Professor of Energy and Mineral Engineering
Luis F. Ayala H.
William A. Fustos Family Professor
Professor of Petroleum and Natural Gas Engineering
Associate Department Head for Graduate Education
*Signatures are on file in the Graduate School
iii
ABSTRACT
Over the past six decades, hydraulic fracturing stimulations have been used to boost
hydrocarbon production from shale oil and gas formations. Thus, the development of hydraulic
fracturing treatment methods to improve production has attracted attention. There are several key
parameters and critical considerations that are required in the design of hydraulic fracture
treatments. Since these changes impact on hydraulic fracture initiation and propagation, alterations
in pressure are one of the pertinent factors in shale reservoirs. A hydraulic fracture propagates
perpendicular to the minimum horizontal stress, after producing fractures in the formation and the
minimum horizontal stress is usually assumed equal to the instantaneous shut-in pressure.
This study aims to analyze instantaneous shut-in pressure in shale oil and gas reservoirs
considering all the pertinent factors as an attempt to express why instantaneous shut-in pressure
might be higher than the minimum horizontal stress and to understand how to obtain minimum
horizontal stress from the instantaneous shut-in pressure. These calculations were determined by
leakoff, wellbore storage, pore pressure, and thermal effects. The effects of closure time in different
permeability values, the comparison of pressure difference in different total wellbore volume,
changes in pore pressure and temperature have indicated that instantaneous shut-in pressure is not
equal to minimum-in situ stress. Analyses in this study help engineers to gain a better understanding
of the relationship between instantaneous shut-in pressure and minimum horizontal stress.
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TABLE OF CONTENTS
List of Figures ......................................................................................................................... vi
List of Tables ........................................................................................................................... ix
Acknowledgements .................................................................................................................. x
Chapter 1 Introduction ............................................................................................................. 1
Chapter 2 Literature Review .................................................................................................... 3
2.1 Unconventional Reservoirs ....................................................................................... 3
2.2 Hydraulic Fracturing ................................................................................................. 6
2.3 Identifying Instantaneous Shut-in Pressure (ISIP) .................................................... 15
2.4 The Effect of Various Factors on the Determination of In-situ Stress from ISIP ..... 24
Chapter 3 Statement of the Problem ....................................................................................... 29
Chapter 4 Analysis of Factors Affecting Instantaneous Shut-in Pressure (ISIP) ..................... 30
4.1 Effect of Leakoff on ISIP……………………………………………………………30
4.1.1 Effect of Fluid Loss Coefficients in Crosslinked Gel Fracture Treatments .... 33
4.1.2 Effect of Fluid Loss Coefficients in Linear Gel Fracture Treatments ............. 46
4.1.3 Effect of Fluid Loss Coefficients in Water Fracture Treatments .................... 49
4.2 Effect of Wellbore Storage on ISIP……………………………………………….....52
4.2.1 Effect of Wellbore Storage on End of Wellbore Storage ................................ 53
4.2.2 Effect of Wellbore Storage on Pressure Difference ....................................... 56
4.3 Effect of Pore Pressure on ISIP…...……………………………………………….....59
4.4 Effect of Temperature on ISIP…………………………………………………….....62
Chapter 5 Results and Discussion ............................................................................................ 65
Chapter 6 Conclusions and Recommendations ........................................................................ 70
Nomenclature ........................................................................................................................... 72
v
References ................................................................................................................................ 75
vi
LIST OF FIGURES
Figure 1. World Primary Energy Demand……………………………………………...….3
Figure 2. Life Span; Conventional vs Unconventional Shale Gas Wells………………..…4
Figure 3. U.S. Dry Natural Gas Production ………………………………………………..5
Figure 4. A Schematic Sketch of Hydraulic Fracturing for Shale Gas………………..…...7
Figure 5. Stress Element and Preferred Plane of Fracture………………………………….8
Figure 6. A Schematic of the Wellbore and the Fracture with Pressures………………….10
Figure 7. Interpretation of Fracturing Plots……………………………………………….13
Figure 8. Total Test Overview Plot……………………………………………………….16
Figure 9. Inflection Point Method..……………………………………………………….17
Figure 10. Pw versus log (t+∆t)/ ∆t Method……………………………………………….18
Figure 11. Pw versus log ∆t Method………………………………………………………19
Figure 12. log (Pw-Pa) versus log ∆t Method……………………………………………...20
Figure 13. log Pw versus log ∆t Method…………………………………………………..21
Figure 14. dPw / dt versus Pw Method……………………………………………………..22
Figure 15. Pw versus √∆𝑡………………………………………………………………..23
Figure 16. Maximum Curvature Method………………………………………………....23
Figure 17. 2 A Schematic View of Crossing and the Result of Experimental Study……...26
Figure 18. 3 A Schematic View of Crossing and the Result of Experimental Study……...26
Figure 19. The Effect of Pore Pressure on Stress………………………………………....27
Figure 20. Schematic of Fracture Propagation and Development of Zones……………....32
Figure 21. Determination of Cw and Spurt from Cumulative Leak-off Volume………….32
Figure 22. Average Compressibility of Distilled Water…………………………………..35
Figure 23. Rock Compressibility……………………………………………………...….36
vii
Figure 24. Pseudocritical Properties of Natural Gases……………………………………37
Figure 25. crTr Values at (1.05=Tr=1.4; 0.2=Pr=15)……………………………………..38
Figure 26. crTr Values at (1.4=Tr=3; 0.2=Pr=15)………………………………………...39
Figure 27. Viscosity of HC Gases as a Function of Molecular Weight and Temperature.40
Figure 28. Water Viscosity at Reservoir Temperatures…………………………………..41
Figure 29. Gelling Agent Concentration versus Cw for Complexed HPG Fluids at 125ºF.42
Figure 30. Temperature Correction for Cw……………………………………………….43
Figure 31. Permeability versus Overall Coefficient, Ct for 40 lbm/1000gal for Complexed
HPG Fluids…………………………………………………………………..……..44
Figure 32. Permeability versus Closure Time for 40 lbm/1000gal for Complexed HPG
Fluids………………………………………………………………………………..45
Figure 33. Guar Polymer Concentration vs. the Wall-building Coefficient………………46
Figure 34. Effect of Formation Temperature on Cw……………………………………...47
Figure 35. Permeability versus Closure Time for 10 lbm/1000gal for Linear Gel………...48
Figure 36. Permeability versus an Overall Coefficient, Ct for 10 lbm/1000gal for Linear
Gel…………………………………………………………………………………..49
Figure 37. Permeability vs Leakoff Coefficient, Cvc…………………………………….50
Figure 38. Permeability vs Closure Time………………………………………………...51
Figure 39. Variations in Compressibility and Viscosity…………………………………55
Figure 40. Total Wellbore Volume vs End of Wellbore Storage………………………….56
Figure 41. Total Wellbore Volume vs Pressure Difference for Injection of Water……….57
Figure 42. Total Wellbore Volume vs Pressure Difference………………………………58
Figure 43. Pore Pressure vs Minimum Horizontal Stress…………………………………60
Figure 44. Change in Pore Pressure vs Difference in Minimum Horizontal Stress…….…61
Figure 45. Changes in Temperature vs Changes in Thermal Expansion Stress…………...64
viii
Figure 46. The Comparison of Crosslinked Gel and Linear Gel Treatments……………...66
Figure 47. Permeability versus Cvc for Water Fracture Treatments……………………...67
Figure 48. Permeability versus Closure Time for Crosslinked Gel, Linear Gel, and Water
Treatments ………………………………………………………………………….68
ix
LIST OF TABLES
Table 1. Nolte-Smith Analysis Pressure Response Modes ...................................................... 14
Table 2. The Reservoir Conditions .......................................................................................... 34
x
ACKNOWLEDGEMENTS
First of all, I would like to express my deepest thanks to my advisor Dr. John Yilin Wang,
not only for being such a great mentor but also for his unlimited support and encouragement
throughout my studies here at Pennsylvania State University. I would like to thank Dr. Turgay
Ertekin for supporting me in all conditions. Without his support and guidance, I would not have
had the opportunity to have my graduate education at Pennsylvania State University. Additionally,
I would like to appreciate Dr. Hamid Emami-Meybodi who helped me at every step of graduate
education and Dr. Shimin Liu for their interests in serving as committee members.
Finally, I would like to express my unlimited thanks to my family, my father Mehmet
Celal Ozesen and my mother Nazmiye Ozesen. It would not have been possible to successfully
complete my studies without their support, love, and encouragement.
Ahsen Ozesen
University Park, Pennsylvania
August, 2017
Chapter 1
Introduction
Unconventional reservoirs such as shale gas and oil resources are one of the most challenging
reservoirs to produce due to their complexity. Yet technological advances have played a vital role
in making oil and natural gas trapped in shale formations accessible. A very common key technique
to extract natural gas and oil trapped in shale formation is hydraulic fracturing. To produce at
economical rates, this treatment technology has been used over the 50 past years. For efficient
extraction of gas and oil from shale formations, understanding of hydraulic fracturing pressures
and rock properties have high importance.
The minimum principle stress has a considerable significance in the design and evaluation
of hydraulic fracturing. This is due to the fact that stresses are controlling the direction of
propagation, the fracture width, the height growth, and well performances. Moreover, it is
important to note that instantaneous shut-in pressure, closure pressure, fracture propagation
pressure are directly connected to minimum horizontal stress. The minimum horizontal stress is
usually assumed to be equal to the instantaneous shut-in pressure in hydraulic fracturing treatments.
Nevertheless, the instantaneous shut-in pressure (ISIP) is often not equal to in-situ stress. Therefore,
various models have been developed to interpret shut-in pressure. Inflection point method, Pw
versus log (t+∆t)/ ∆t method, Pw versus log ∆t method, log (Pw-Pa) versus log ∆t method, log Pw
versus log ∆t method, dPw / dt versus Pw method, Pw versus √∆𝑡 method, and maximum curvature
method were developed. Pw is defined as a bottomhole pressure, t is the injection time, ∆t is the
time since shut-in, and Pa is a trial value at the asymptotic pressure.
2
The goal of this study is to analyze the instantaneous shut-in pressure (ISIP) in shale oil/gas
formations considering all the pertinent factors as an attempt to explain why instantaneous shut-in
pressure (ISIP) is ‘too’ high in the field and to understand how to obtain in-situ stress from ISIP.
The procedure of my research is outlined below:
Chapter 2: The literature review of hydraulic fracturing on shale formations are
reviewed.
Chapter 3: The problem is stated.
Chapter 4: The factors, including leakoff coefficients, wellbore storage, pore pressure,
and temperature are analyzed.
Chapter 5: The results, conclusions, and recommendations are provided.
3
Chapter 2
Literature Review
2.1 Unconventional Reservoirs
Natural gas is an important energy source, and there are several reasons for the increased
importance of natural gas. As stated in Mohan (2008), natural gas now has better transportation
infrastructure, burns more efficiently and environmental friendly for power generation, and new
technologies have been developed such as gas-to-liquid technology. As a cleaner fuel, natural gas
is playing an increasingly important role in satisfying the energy demand. World primary energy
demand is shown in Figure 1, where the global demands for natural gas will rise dramatically in
the next 20 years.
Figure 1. World Primary Energy Demand (IEA World Energy Outlook, 2011)
Unconventional resources such as shale gas have been gaining attention in the past decade.
In Figure 2, it is clear that unconventional reservoirs have a longer lifespan due to its low
permeability than conventional reservoirs. Reserves of unconventional reservoirs increases as
4
technology develops and economic changes. In addition, the technological advancements, long-
term potential, environmental benefits and attractive gas prices bring unconventional gas resources
more rather than oil into the forefront of our energy future (Zahid, 2007).
Figure 2. Life Span; Conventional vs Unconventional Shale Gas Wells (Encana Website)
Unconventional gas resources, which includes shale gas, coal bed methane, and tight gas reservoirs,
have great potential. According to EIA report in 2015, the report estimated there was 622.5 TCF of
recoverable shale gas in the U.S., enough to provide the U.S. with about 27 years’ worth of natural
gas at current usage rates. Due to leading to a new great amount of natural gas supply with the
improved technology, shale gas is the fastest growing source.
5
2.1.1 Overview of Shale Oil / Gas Reservoirs
Unconventional sources (e.g., shale gas, coal bed methane, tight gas) have played a prominent role
because a large portion of natural gas comes from these sources. For instance, in the United States,
shale gas production accounted for more than half of U.S. natural gas production in 2015 and is
projected to more than double from 37 Bcf/d in 2015 to 79 Bcf/d by 2040, which is 7-% of total
U.S. natural gas production in the AEO2016 Reference case by 2040 (EIA, 2016). Figure 3 shows
the United States natural gas production by a source in the reference case 1990-2040. As it can be
seen, shale gas will be the largest contributor to the production growth.
Figure 3. U.S. Dry Natural Gas Production (EIA, 2012)
Shales are fine-grained sedimentary rocks and shale gas reservoirs have very low matrix
permeability and low matrix porosity. A matrix permeability is about 1 to 100 nd and a porosity is
less than 10%. Due to the low permeability of the rock, the rock traps the gas and prevents it from
6
migrating towards to surface. Therefore, it is important to note that as unconventional reservoirs,
shale gas reservoirs does not produce economic volumes of gas without applying massive
stimulation treatments or special recovery technologies. To exploit these low permeability
reservoirs successfully, the advancement of drilling and completion technologies can be used.
Hydraulic fracturing (also called hydro fracking or fracking) became an efficient and effective
stimulation technique for the extraction of natural gas. Over the past decade, the combination of
hydraulic fracturing and horizontal drilling has allowed access to large volumes of shale gas.
2.2 Hydraulic Fracturing
Hydraulic fracturing is the major technique in the exploration of shale gas reservoir to enhance the
permeability by increasing the contact area between fracture and matrix. Used in over one million
wells in the United States for more than 60 years, fracking has been successfully used to retrieve
more than 7 billion barrels of oil and over 600 trillion cubic feet of natural gas (Loris, 2012).
Figure 4 illustrates a schematic sketch of hydraulic fracturing for shale gas. A hydraulic
fracture is formed by pumping the fracturing fluid into the wellbore at a rate sufficient to increase
pressure downhole to exceed the minimum in-situ stress. After the fracturing fluid enters the
formation, natural fractures are opened. In order to maintain the fracture width, proppants are added
to the injected fluid. Proppant such as sands are common to apply to prevent the fracture from
closing when the injection pressure is removed or reduced below the breakdown pressure. The
induced fracture with proppant is permeable enough to allow the formation fluid to flow from
formation to wellbore. Upon completion of fracturing, in-situ stress or geologic pressure of the
formation will drive the fracturing fluid to rise to the surface, which is referred as flowback period.
The recovered liquid can either be recycled or injected into the disposal well below water zone. To
sum up, fracture stimulation has been crucial in the development of shale gas industry.
7
Figure 4. A Schematic Sketch of Hydraulic Fracturing for Shale Gas (Farris et. al., 1947)
2.2.1 Rock Mechanics and In-situ Stresses
The design of fracturing operations involves the noticeable amount of engineering and rock
mechanics. Engineering tools and methods to estimate hydraulic fracture propagation and
geometry, fracture conductivity and hydrocarbon productivity of the reservoir are necessary to
design the optimal fracture treatment in unconventional reservoirs (Meyer et al., 2013).
Fracture propagation is an important part of the fundamental principles of hydraulic
fracturing design. Fractures propagate along the path of least resistance, and a fracture avoids the
greatest stress in a three-dimensional stress regime. It is a fundamental principle to understand
fracture orientation and the stress regime that a fracture will propagate parallel to the greatest
principal stress and perpendicular to the plane of the least principle stress.
8
Figure 5. Stress Element and Preferred Plane of Fracture (Hubbert et. al, 1957)
A hydraulic fracture’s propagation perpendicular to the minimum principle stress made
the in-situ stress recognized as a basic parameter in the design of hydraulic fracturing. In-situ stress
is also referred to as minimum horizontal principle stress or closure pressure. This pressure is
required to keep the fracture open. Minimum stress can be calculated from the three methods:
Method 1 by Ben Eaton,
𝑉
1−𝑉 (σob –Pr) + Pr + σtectonics
(Eq. 1)
σtectonics = Tectonic stress, psi
Pr = Reservoir fluid pressure, psi
σob = Overburden stress, psi
𝑉= Poisson’s Ratio
9
Method 2 by Hubbert and Willis:
σmin = 1
3 (1+2Pr)
(Eq. 2)
σmin = 1
2 (1+Pr)
(Eq. 3)
Method 3 by Matthews and Kelly:
σmin = σv Ki+Pr
(Eq. 4)
2.2.2 Definitions of Pressure Differences
During the hydraulic fracturing process, it is crucial to interpret pressures because they play a key
role for the sources of energy gain and energy loss. There are eight critical pressures, including
wellhead pressure, hydrostatic pressure, fluid friction pressure, bottomhole pressure, perforation
friction pressure, tortuosity pressure, fracturing fluid pressure, and net pressure. Figure 6
demonstrates a schematic of significant pressure during fractures.
10
Figure 6. A Schematic of the Wellbore and the Fracture with Definition of Pressure Differences
Wellhead Pressure, Ps
It is the pressure at the top of the well and measured by pressure gauges of the wellhead fittings.
This is also known as surface treating pressure or injection pressure.
Hydrostatic Pressure, Ph
The pressure exerted by the fracture fluid at a given point as a result of the total vertical depth and
density changes. The amount of hydrostatic pressure increases with depth and density.
Pℎ= 0.052 ∗ 𝜌 ∗ ℎ (Eq. 5)
h = The total vertical depth, ft
𝜌 = The slurry density, lb/gal
11
Fluid Friction Pressure, Pfric
Due to the friction effects in the wellbore, while fluids are injected, the friction pressure (also called
tubing or wellbore friction pressure) is a pressure loss. The actions of the friction pressure are like
the opposite direction of fluid flow.
Bottomhole Treating Pressure, (BHTP), Pwb
It is the pressure at the bottom of the well. It is also referred to as wellbore pressure in the center
of the interval being treated. It can be calculated as follow:
Pwb = Ps+ Ph - Pfric
(Eq. 6)
Perforation Friction Pressure, △Pperf
This is the pressure drop experienced by the fracturing fluid passes through narrow restrictions of
the perforations. The equation below shows the calculation of the perforation friction pressure.
ΔPperf = 0.2369 *q2∗ρ
Np2 Dp4 Cd2
(Eq. 7)
𝜌 = The slurry density, lb/gal
q = The total flow rate, bpm
Np = The number of perforations
Dp = The perforation’s diameter, inches
Cd = The discharge coefficient
Tortuosity Pressure, Ptort
This is the pressure drop experienced by the fracturing fluid passes through to restrained flow’s
region between the main part of the fracture and the perforation.
12
Fracturing Fluid Pressure, Pfrac
Fracturing fluid pressure measures inside the main body of the fracture. It is calculated by:
Pfrac = Pwb - △Pperf
(Eq. 8)
Net Pressure, Pfrac
It is defined as the pressure in the fracture minus the minimum in-situ stress (also referred to closure
pressure). Net pressure can be calculated as follows:
Pnet = Pfrac – 𝜎min
(Eq. 9)
Net pressure is very important to interpret fracture propagation during the hydraulic fracturing
treatment. It directly affects the geometry of the fracture. Therefore, to interpret net pressure, The
Nolte-Smith analysis was introduced in 1981. Nolte and Smith analyzed the fracturing pressure
response with using PKN, KGD, and radial models. Then fracture behaviors were predicted by
analyzing these pressure responses. Figure 7 explains the interpretation of fracturing pressures.
13
Figure 7. Interpretation of Fracturing Plots (Nolte and Smith, 1981)
For Mode 1, the fracture is propagating normally with confined height (PKN fracture geometry).
For Mode 2, because of stable height growth or increasing in fluid loss, the constant pressure region
occurs.
For Mode 3, net pressure is directly proportional to time. This positive slope shows that there is a
flow restriction in the fracture. The distance from the wellbore is a difference between Mode 3A
and 3B. When the distance is small, the screenout might happen near the wellbore. On the contrary,
if the distance is large, a screenout occurs near the tip.
For Mode 4, this negative slope represents rapid height growth (KGD or radial fracture geometry).
Table 1 summarizes the characterization of each mode below.
14
Table 1. Nolte-Smith Analysis Pressure Response Modes
2.2.3 MiniFrac Analysis in Unconventional Reservoirs
Before the main fracture treatment, in order to determine the reservoir and fracture properties for
the main stimulation design, a mini-frac test is an injection falloff diagnostic test performed.
Without using any proppant, a goal of the minifrac test is to create a small fracture. Determination
of the reservoir pressure and formation permeability and estimation of fracture design parameters
such as fracture closure pressure, net pressure, fracturing fluid efficiency, fracture gradient, flow
regimes, and leak-off coefficients are the main reasons to perform a mini-frac test. The fracture
fluid is injected into the well and then pressurized to create a fracture in the reservoir. To initiate
the crack in the reservoir, the downhole pressure must overcome the breakdown pressure. After the
crack is created, the downhole pressure decreases while fracturing continues to propagate into the
reservoir. The fracture closure pressure can be evaluated after injection is stopped (Holditch et al.,
2016).
The types of minifrac analysis have been divided into two specific categories. These are
Before Closure Analysis and After Closure Analysis. They are separated than each other by fracture
closure pressure. McLennan et al. (1982) states that fracture closure pressure is the pressure
Mode Slope of the line Interpretation
I 1/8 - 1/4 Unrestricted linear extension of the fracture; restricted height
II 0 (straight line) Moderate height growth; fracture extension continues
III 1 in 1 (45 degrees) or Restricted extension: two wings
2 in 1 (63.4 degrees) Restricted extension: one wing
IV Negative gradient Unstable height growth
15
required to hold the fracture open after initiation and at the same time it is a pressure required to
keep the fracture from just closing. During the pre-closure period, while the open fracture is closing,
closure pressure is identified and the early pressure falloff period is analyzed. The following
parameters are obtained from the Before Closure Analysis.
Instantaneous Shut-In Pressure (ISIP)
Fracture Closure Pressure
Fracture Gradient
Net Fracture Pressure
Fluid Efficiency
After Closure Analysis gives information for the estimation of reservoir pressure and
permeability. In order to interpret the after closure period of minifrac tests to define flow regimes
and estimate reservoir parameters, Nolte (1997), Nolte et al. (1997), and Talley et al. (1999) have
developed several techniques.
2.3 Identifying Instantaneous Shut-in Pressure (ISIP)
After the end of the pumping, the shut-in pressure is the pressure as a function of time. After shut-
in, Instantaneous Shut-in Pressure occurs immediately. Figure 8 illustrates the pressure test versus
time. As seen in Figure 9, identifying the Instantaneous Shut-in Pressure is sometimes difficult.
16
Figure 8. Total Test Overview Plot (Fekete, 2011)
To deal with the indistinct shut-in pressure, various methods have developed. In 1993, Guo,
Morgenstern, and Scott summarized the shut-in pressure identification techniques. There are eight
methods to determine the shut-in pressure. They are discussed in the following paragraphs.
2.3.1 Inflection Point Method
Gronseth and Kry obtained the shut-in pressure with a simple graphical technique in 1981. The
pressure at which the pressure-time record departs from the tangent line is defined as the shut-in
pressure (Guo et al, 1993). Figure 9 demonstrates the inflection point method.
17
Figure 9. Inflection Point Method (Gronseth and Kry, 1981)
2.3.2 Pw versus log (t+∆t)/ ∆t Method
Mclennan and Roegiers proposed this technique in 1981. It shows the determination of shut-in
pressure with the inflection point of Pw versus log (t+∆t)/ ∆t. Figure 10 shows the application of
this method. Pw is the bottomhole pressure, ∆t is the time since shut-in, and t is the injection time.
18
Figure 10. Pw versus log (t+∆t)/ ∆t Method (Mclennan and Roegiers, 1981)
2.3.3 Pw versus log ∆t Method
In order to obtain the shut-in pressure, Doe and Hustrulid (1981) used the plot of Pw versus log ∆t
after the first breakdown. This plot was advisable for interpreting hydraulic fracturing under slow
pumping cycles. This method is shown in Figure 11.
19
Figure 11. Pw versus log ∆t Method (Doe and Hustrulid, 1981)
2.3.4 log (Pw-Pa) versus log ∆t Method
This method is also known as Muskat Method. Aamodt and Kuriyagawa (1981) state that the
pressure after shut-in approaches some value asymptotically and the extrapolation of shut-in
pressure is obtained from the straight line. Furthermore, this acquired pressure plus Pa is taken as
the shut-in pressure. Figure 12 shows how to obtain the shut-in pressure from this plot.
20
Figure 12. log (Pw-Pa) versus log ∆t Method (Aamodt and Kuriyagawa, 1981)
2.3.5 log Pw versus log ∆t Method
This method has made by Zoback and Haimson (1982). As can be seen in Figure 13, the pressure
versus time curve is bilinear. These lines’ intersection gives the shut-in pressure.
21
Figure 13. log Pw versus log ∆t Method (Zoback and Haimson, 1982)
2.3.6 dPw / dt versus Pw Method
Tunbridge also considered that the shut-in curve is bilinear in this plot. The shut-in pressure obtains
the intersection of the two lines. This plot is illustrated in Figure 14.
22
Figure 14. dPw / dt versus Pw Method (Tunbridge, 1989)
2.3.7 Pw versus √∆𝒕 Method
The fracture closes after the plot departs from the straight line. The corresponding bottomhole
pressure is the fracture closure pressure of the shut-in pressure (Guo et al, 1993). Figure 15 is a plot
of Pw versus √∆𝑡.
23
Figure 15. Pw versus √∆𝑡 (Sookprasong, 1986)
2.3.8 Maximum Curvature Method
In the shut-in curve, the bottomhole pressure at the maximum point assumed as the shut-in pressure.
Figure 16 illustrates this method.
Figure 16. Maximum Curvature Method (Hayashi and Sakurai, 1989)
24
2.4 The Effect of Various Factors on the Determination of Minimum In-situ Stress from
Instantaneous Shut-in Pressure (ISIP)
Previous studies (Kehle, 1964; von Schoenfeldt, 1970; Haimson, 1972; Roegiers, 1974; Bredehoeft
et al. 1976; Zoback et al., 1977; Mclennan, 1980) stated that instantaneous shut-in pressures are
regularly used as indicators of the minimum in-situ principal stresses, and the minimum principle
stress is usually assumed to be equal to the shut-in pressure in hydraulic fracturing stress
measurements. However, there are significant factors that affect the accuracy of in-situ stress
determination from ISIP.
Leakoff
Leakoff is a factor, which influences the determination of the minimum principle stress from shut-
in pressure. For hydraulic fracturing stimulation designs, a knowledge of the leakoff characteristics
is critical, since the loss of fracturing fluid restricts the efficiency of the fracturing. To investigate
the leakoff effects, several studies have been conducted. According to the analyses of previous lab
studies, the leakoff during hydraulic fracturing has a dominant effect on the geometry of fracture
and fracturing pressure. Carter (1957) developed a leakoff model. Based on this fracturing fluid
loss model, hydraulic head is constant in time and the fluid loss is one-dimensional. Williams
(1970) improved Carter’s equation and pointed out that the rate of fluid loss would be influenced
by fluid pressure. In this theory, the wall-building mechanism is defined as a dominant fluid loss
coefficient. Yi and Peden (1994) investigated the pressure effects, filter cake erosion, and boundary
conditions on the fluid loss.
In order to achieve the desired geometry or desired fracture length, leakoff coefficients are
leading parameters of fracturing fluids to be tuned. In other words, the accurate understanding of
25
fracturing-fluid leakoff coefficient is a crucial guidance of controlling the rate of fluid leakoff.
Three leakoff parameters express the viscosity, compressibility, and wall-building effects.
The Presence of Natural Fractures
It is substantial that the opening of natural fractures create excessive leakoff and affect the hydraulic
fracture propagation significantly. Because of these major concerns, it is worthy to determine the
criteria for the opening of natural fractures and the rate of fluid loss. Stress is the main factor for
natural fracture opening. Ahn et al. (2014) presented a hydraulic fracture model in order to simulate
propagation of complex fractures under the presence of natural fractures. The influential factors
such as pressure differences in the fracture and changes in minimum stresses were analyzed. This
parametric study was shown how to quantify the effect of differential stress on fracture geometry
after the treatment. It was concluded that the alterations in the induced fracture network are
controlled by a decrease in horizontal differential stress. It changes from elliptical to radial shape.
In addition, it was observed that higher volume of fluid loss caused by smaller differential stress.
Change in Fracture Orientation
Due to the complexity of unconventional oil or gas reservoirs, it is difficult to estimate the
propagation of hydraulic fractures. For example, the profile of hydraulic fractures would be
influenced by the geological properties and the stress related or mechanical properties, and the
presence of perforations or natural fractures. For this reason, it is crucial to understand how
hydraulic fracture propagates in complex geological settings. Based on the development of the
geologic control, it might be more dominant than stress control. Figure 17 illustrates that
propagation of hydraulic fractures crosses the natural fracture and maintain without any significant
alterations in its path, and Figure 18 shows that how hydraulic fracture turns into the natural
fractures and propagates along as below. The created fracture reorients itself depends on the
26
approach angle and stress conditions and propagates adequately. As a consequence, the shut-in
pressure becomes inconsistent and is not equal to minimum in-situ stress.
Figure 17. 2 A Schematic View of Crossing and the Result of Experimental Study (Keshavarzi,
2013)
Figure 18. 3 A Schematic View of Crossing and the Result of Experimental Study (Keshavarzi,
2013)
27
Pore Pressure
As has been emphasized in the past, in-situ stresses control the propagation of hydraulic fractures
The accuracy of stress magnitude depends on the pore pressure. The effect of pore pressure on
stress can be explained using a Mohr diagram. Higher pore pressure has the effect of translating
the stress circles to the left and gets closer to the failure. Figure 19 illustrates the failure criterion.
Figure 19. The Effect of Pore Pressure on Stress (Hillis, 2000)
In other words, the secondary principal stress calculated from the instantaneous shut-in pressure
might not be an indicator of the correct principal stress due to pore pressure. According to case
studies of Salz (1977) and Smith (1981), in deep, over-pressured wells, the minimum principal
stress has been usually found to decrease with the production of the wells and consequent
drawdown causing some changes of the stress field. During the stress measurement operations, the
alterations of pore pressures are a considerable factor while total stress levels are being determined
on the instantaneous shut-in pressure in permeable formations.
In conclusion, due to fracture fluid loss during the hydraulic fracturing treatment, pore
pressure increases and it causes a stress increase in the rock. With the pore pressure coefficient, the
relationship between stress and pore pressure can be obtained.
28
Wellbore Storage (Afterflow)
Numerous techniques are used to estimate the magnitude of the minimum in-situ stress. Yet there
are various factors that complicate the determination of the appropriate minimum principal in-situ
stress from pressure records. Wellbore storage has a prominent effect on pressure behavior. To
illustrate, when there is a large amount of wellbore storage, some errors may occur. This means
that the instantaneous shut-in pressure is not taken as a good approximation of the minimum in-situ
stress.
After the well is shut-in at the surface, wellbore storage is a flow of fluids into the
formation. The fluid is stored in the wellbore and this storage causes a continued of fluid into
formation although the surface rate is zero. The duration of wellbore storage is controlled by the
formation permeability, the fluid compressibility, and the wellbore volume. Low formation
permeability, the large compressibility of gas wells and large volume increase the period of
wellbore storage.
29
Chapter 3
Statement of the Problem
The minimum horizontal stress is an important parameter in the design of hydraulic fracturing. The
minimum principle stress is usually assumed to be equal to the instantaneous shut-in pressure
(ISIP), but they are not the same or even close in shale based on our observations of field data. The
objectives of my study are to analyze the pertinent factors to show that the minimum in-situ stress
is not assumed to be equal to the instantaneous shut-in pressure (ISIP) and evaluate the
identification of minimum horizontal stress from the instantaneous shut-in pressure (ISIP). These
assumptions and calculations are determined by:
Fluid loss coefficients
Wellbore storage effects
Changes in pore pressure
Temperature changes
To use the plot of bottomhole pressure versus time appropriately for the prediction of minimum
horizontal stress, it is important to control these factors. Analyses for the accurate interpretation of
the minimum in situ-stress from the instantaneous shut-in pressure (ISIP) will be explained in the
next chapter.
30
Chapter 4
Analysis of Factors Affecting Instantaneous Shut-in Pressure (ISIP)
This chapter attempted to analyze how factors affect the instantaneous shut-in pressure (ISIP) and
how to interpret minimum in-situ stress. To have a better understanding, the mathematical
equations were used in this chapter. This chapter is divided into four parts:
Effect of Leakoff on ISIP
Effect of Wellbore Storage on ISIP
Effect of Pore Pressure on ISIP
Effect of Temperature on ISIP
4.1 Effect of Leakoff on ISIP
Understanding of fluid loss is the key to more successful stimulation. The rate of fluid leakoff
affects fracture closure time as well as impacts the fluid efficiency, proppant scheduling, and
treatment size. Therefore, during the pumping operation, the minimization of fluid loss is one of
the important purposes. With the use of fracturing fluids, the control of fluid loss has developed. In
this method, each leakoff coefficient is obtained for three different fracturing fluids.
There are three types of fluid loss coefficients, including Cv, Cc, and Cw. Cv, the effects of
effluent viscosity and relative permeability, is obtained from reservoir data. In the invaded zone,
the filtrate’s viscosity is responsible for controlling the fluid loss.
Cv = 0.0469 [(kL *∆p*⏀) / (µa)] ½
(Eq. 10)
kL = Nonreactive liquid permeability, md
31
∆p = Difference between bottomhole treating pressure and bottomhole pressure, psi
⏀ = Fractional porosity of formation, % :
(⏀)*(1-Sor-Swr)
(Eq. 11)
µa = Effluent’s viscosity, cp
Cc, reservoir-fluid viscosity/compressibility effects, can be calculated by using reservoir data
and fracturing-fluid viscosity.
Cc = 0.0374*∆p [(ki*ct*⏀) / (µf)] ½
(Eq. 12)
ki = Formation permeability to mobile reservoir fluid, md
ct = Total formation compressibility, psi-1
µf = Viscosity of mobile formation, cp
Cw that is wall-building effects (filter cake zone) can be computed by using reservoir data
and fracturing-fluid viscosity. Because of the deposition of polymer and fluid loss additives, a filter
cake begins to build up while the fracturing fluid leaks off into the formation. Figure 20
demonstrates the fracture propagation and zones. On the fracture face, the filter cake has a critical
barrier. Thus, the wall-building effects are characterized by the wall-building coefficient and spurt
loss.
Cw = (0.0164*m) / A
(Eq. 13)
m= Slope of cumulative filtrate volume versus time1/2
plot (Fig 21)
32
A= Area of core that is used in laboratory
Figure 20. Schematic of Fracture Propagation and Development of Zones (Gadiyar, 1997)
Figure 21. Determination of Cw and Spurt from Cumulative Leak-off Volume (Gadiyar, 1997)
33
There are numerous methods for integrating the three separate leakoff coefficients. In many
fluid loss calculations, the combination of Cv and Cc is used as a Cvc.
Cvc = (2* Cv* Cc) / [Cv+ (Cv2+4Cc2)1/2]
(Eq. 14)
Another method is to calculate Cv, Cc, and Cw separately and combine these values to get
an overall coefficient, Ct.
Ct = (2* Cv* Cc* Cw) / [Cv Cw + (Cw 2Cv2+4Cc2 (Cv2+ Cw 2)) 1/2]
(Eq. 15)
In addition, Cvc controls the leakoff rate of the spurt volume, Vs. This combined fluid loss
coefficient is used for situations without filter cake deposition. The spurt time, ts is,
ts= [Vs/2Cvc]2
(Eq. 16)
Total leakoff volume per unit area;
V= Vs+2*(Cw or Ct)*t1/2
(Eq. 17)
4.1.1 Effect of Fluid Loss Coefficients in Crosslinked Gel Fracture Treatments
This method was applied to 40lbm/1000gal crosslinked hydroxypropyl guar (HPG) for different
permeability values and shows how the fracturing-fluid leakoff coefficients impact the fracturing
closure time. The reservoir parameters are given in Table 2.
34
Table 2. The Reservoir Parameters
The leakoff coefficient is a function of the permeability. Therefore, different permeability
values were selected for various cases. It was within the range of 0.0001 md to 1000 md.
It is assumed that the shale reservoir was occupied by water and gas. Thus, the total
compressibility varies with the compressibility and saturation of gas and water as well as
the rock. The total compressibility is defined as:
ct = Sw *cw + Sg*cg + cr
(Eq. 18)
Considering reservoir conditions such as reservoir temperature and bottomhole treating
pressure, the compressibility of water were selected from Figure 22.
Bottomhole Pressure, psi 3600
Bottomhole Treating Pressure, psi 4600
Reservoir Temperature, R 660
Porosity, % 11
Oil Saturation, % 0
Residual Oil Saturation, % 0
Water Saturation, % 32
Residual Water Saturation, % 32
Gas Saturation, % 68
Specific Gas Gravity 0.65
35
Figure 22. Average Compressibility of Water (Long and Chierici, 1961)
The initial porosity value with Hall’s correlation in Figure 23 is a good approximation for
the rock compressibility that was used in the calculation.
Since this is a shale gas reservoir, the gas compressibility has a dominant effect on the
total compressibility. Instead of using correlations, the pseudocritical pressure and
temperature were estimated from Figure 24.
As a function of reduced temperature and pressure, the reduced compressibility and
temperature product (cr Tr ) were obtained with Figure 25 or Figure 26. Then the
compressibility of gas calculated.
Gas is the only mobile reservoir fluid, so the gas viscosity was found from Figure 27.
36
Figure 23. Rock Compressibility (Newman, 1973)
37
Figure 24. Pseudocritical Properties of Natural Gases (Carr et al., 1954)
38
Figure 25. crTr Values at (1.05=Tr=1.4; 0.2=Pr=15) (Mattar et al., 1975)
39
Figure 26. crTr Values at (1.4=Tr=3; 0.2=Pr=15) (Mattar et al., 1975)
40
Figure 27. Viscosity of Hydrocarbon Gases as a Function of Molecular Weight and Temperature
(Carr et al., 1954)
In addition, because of using a crosslinked gel in shale reservoir conditions, assuming
that effluent is a water and µa was calculated from Figure 28.
41
Figure 28. Water Viscosity at Reservoir Temperatures (Matthews and Russel, 1967)
The viscosity and compressibility controlled coefficients were calculated under given reservoir
conditions. Due to using crosslinked HPG gel, a filter cake is formed on the fracture. These gelling
agents do not enter the formation easily. Thus, the fluid loss through filter cake was found by the
wall-building coefficient. In order to reach the rate of flow of fluid through filter cake, Cw was
read for 40lbm/ 1000 gal crosslinked HPG gel from Figure 29. For temperature correction of Cw,
Figure 30 can be used.
42
Figure 29. Gelling Agent Concentration versus Cw for Complexed HPG Fluids at 125ºF (Penny
and Conway, 1989)
43
Figure 30. Temperature Correction for Cw (Penny and Conway, 1989)
During fracturing treatment, all three fluid losses occur simultaneously. In this case, the reservoir
leakoff coefficient is the combination of these three leakoff mechanisms which is the overall
coefficient, Ct. Since the leakoff rate is dominated by Ct, closure time is calculated by t = [V/ (2Ct)]
1/2. The leakoff rate of spurt volume (Vs) is essentially assumed to be zero for all fluids if the
permeability is less than 0.5 md. For higher permeabilities, the volume of spurt is proportional to
permeability which is controlled by Ct. Total frac volume per unit area was obtained to 0.01 with
the 0.24-inch fracture width. These calculations were refined by calculating Cw, Cc, Ct, Cv, Cvc and
44
closure time for permeabilities ranging from 0.0001 to 1000 md. As the result of calculations,
Figure 31 below displays the relationship between the permeabilities and overall leakoff
coefficient, Ct and Figure 32 compares the closure time in different permeability ranges.
Figure 31. Permeability versus Overall Coefficient, Ct for 40 lbm/1000gal for Complexed HPG
fluids
It can be clearly seen that there are sharp increases from 0.0001 md to 1 md, while the overall
leakoff coefficient remains relatively steady between 10 md to 1000 md. For 0.001 md, the
overall fluid loss coefficient is 0.00048 ft/ min0.5 and it is 0.0036 ft/ min0.5 for 100 md. This
implies that in high permeable reservoirs, the volume of fluid leakoff into the formation is higher
than the low permeable formation.
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
0.0001 0.001 0.01 0.1 1 10 100 1000
Ct,
ft/m
in0
.5
Permeability, md
Permeability vs Overall Leakoff coefficient, Ct
45
Figure 32. Permeability versus Closure Time for 40 lbm/1000gal for Complexed HPG fluids
The closure time is inversely proportional to the overall coefficient. The closure time of 0.0001 md
is roughly 1260 min while the closure time is 1.95 min for 10 md. As can be seen that for 0.0001
md case, the closure time of fracture takes hours. Since fracture does not close instantly after
pumping is stopped, ISIP is always higher bound on the minimum horizontal stress. Therefore, the
instantaneous shut-in pressure (ISIP) is not equal to closure stress.
0
200
400
600
800
1000
1200
1400
0.0001 0.001 0.01 0.1 1 10 100 1000
Clo
sure
Tim
e, m
in
Permeability, md
Permeability vs Closure Time
46
4.1.2 Effect of Fluid Loss Coefficients in Linear Gel Fracture Treatments
This method was applied to 10lbm/1000gal linear HPG gel for different permeability values
because the use of either a crosslinked gel or a linear gel is dependent on the formation permeability.
Additionally, same reservoir parameters were applied.
Previous mathematical calculations and correlations were used to calculate fluid loss
coefficients. The measured gas permeability (kg) values range between 0.0001 to 1000
md again.
The leakoff coefficient of a filter cake, Cw was acquired from the Figure 33 for linear gel
guar. It is important to note that gelling agent concentrations and temperature affect the
value of Cw. Temperature correction factor to be applied are illustrated in Figure 34.
Figure 33. Guar Polymer Concentration vs. the Wall-building Coefficient (Smith &
Montgomery, 2015)
47
Figure 34. Effect of Formation Temperature on Cw (Smith & Montgomery, 2015)
Figure 35 represents the total fluid loss coefficient in the different range of permeabilities and
Figure 36 demonstrates the impacts of permeability on closure time.
48
Figure 35. Permeability versus Closure Time for 10 lbm/1000gal for Linear Gel
The increase of permeability contributes to the increase of leakoff. However, 0.0001 md has
0.00014 ft/min0.5 leakoff coefficient, it is 0.0071 ft/min0.5 for 1000 md. These results point out
that fracturing fluid leaks high permeable channels more. Therefore, high-efficiency fluid is
required for high permeable reservoirs. In addition, it can be important that the opening of natural
fractures in shale reservoirs results in more leakoff.
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.0001 0.001 0.01 0.1 1 10 100 1000
Ct,
ft/m
in0
.5
Permeability, md
Permeability vs Overall Leakoff coefficient, Ct
49
Figure 36. Permeability versus an Overall Coefficient, Ct for 10 lbm/1000gal for linear gel
These results show that the fracture does not close rapidly if less fluid leaks off. While closure
time is 1258 min for 0.0001 md, it is 1.5 min for 0.1 md. This is due to the fact that for 0.1 md the
overall leakoff coefficient is 0.004 ft/min0.5 which means that there is too much fluid leaks off.
Thus, the control of fluid loss is important in hydraulic fracturing treatments.
4.1.3 Effect of Fluid Loss Coefficients on Water Fracture Treatments
Same reservoir conditions and gas permeability (kg) values were applied in this previous section.
Water fracture treatments are more different than crosslinked gel and linear gel treatments. The
filter cake is not formed during this treatment because water does not contain any gelling agents
0
200
400
600
800
1000
1200
1400
0.0001 0.001 0.01 0.1 1 10 100 1000
Clo
sure
Tim
e, m
in
Permeability, md
Permeability vs Closure Time
50
and enters the formation easily. Therefore, the fluid loss of the fracture surface without filter cake
deposition can be modeled by a different equation.
V= 2* Cvc* √𝑡
(Eq. 19)
The leakoff rate is dominated by Cvc which is assumed to approximately close to Ct.
Figure 37 depicts the variation of Cvc in different ranges of permeabilities.
Figure 37. Permeability vs Leakoff Coefficient, Cvc
Due to the effects of viscosity, permeability and compressibility effects, the fluid loss coefficient
slightly increases up to 0.53 ft/min0.5. It can be observed that high permeable reservoirs are
influenced more by fluid loss coefficient.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.0001 0.001 0.01 0.1 1 10 100 1000
Cvc
, ft
/min
0.5
Permability, md
Permeability vs Cvc
51
Figure 38. Permeability vs Closure Time
The relationship of permeability and closure time is illustrated in Figure 38. Closure time is 103
min at 0.001 md, while it 1.9 min at 100 md. This implies that reduction of permeability makes
the fracture closure time longer. In 0.0002 md shale formation, the closure time of fracture is
1250 min after the end of treatment, which means that this dramatic result can have a significant
effect in the design of hydraulic fracturing treatments due to the lower overall leakoff coefficient.
0
200
400
600
800
1000
1200
1400
0.0001 0.001 0.01 0.1 1 10 100 1000
Clo
sure
Tim
e, m
in
Permeability, md
Permeability vs Closure Time
52
4.2 Effect of Wellbore Storage on ISIP
The wellbore storage effect plays an important role because of the changes in pressure behavior
and end of wellbore storage time. These changes in pressure cause two different types of wellbore
storage. The first type is defined as the wellbore storage due to fluid expansion. The effect of fluid
expansion is a function of the total wellbore volume and the compressibility of wellbore fluid. The
second wellbore storage caused by changing liquid level in the annulus. In terms of wellbore
storage constant, C, the change in fluid level and effect of fluid expansion can be computed
separately. In this study, the change in fluid level was assumed to be zero.
For a well filled a single phase gas or liquid, the wellbore storage constant:
Cs = cwb Vwb
(Eq. 20)
Vwb = The total wellbore fluid volume, bbl
cwb = The average compressibility of fluid in the wellbore, psi-1
Cs= (ΔVwb) / (ΔP)
(Eq. 21)
ΔVwb = Changing in the volume of fluid in the wellbore, (Vi- V(t)), bbl
ΔP = The pressure difference (Pi-Pwf), psi
To determine well storage constant, selected point on the log-log unit-slope straight line
can be used. (t, ΔP)
Cs= (q*t) / (24*ΔP)
ΔP= (Pi-Pwf) = [(q) / (24* Cs)]*t
ΔP= (Pi-Pwf) = [(Q*B) / (24* Cs)]*t
(Eq. 22)
53
t = Time, hrs
ΔP = The pressure difference, psi
Pi = Initial wellbore pressure before unloading, psi
Pwf = Wellbore pressure during unloading, psi
q= Flow rate, bbl/ day
Q= Flow rate, STB/day
B=Formation volume factor, bbl/STB
Approximate the time required for wellbore storage influence to end from: (total time that
marks the end of wellbore storage effect and the beginning of the semi-log straight line).
4.2.1 Effect of Wellbore Storage on End of Wellbore Storage
Three case studies were conducted to understand how wellbore volume impacts the
duration of the end of wellbore storage. Case studies are the injection of water, oil, and gas with
various total wellbore volume which is 50, 200, 300, 600, and 1000 bbl. To find the duration of
wellbore storage impact, the first step is to calculate wellbore storage constant by fluid expansion
for different wellbore volumes because the period of wellbore storage is dependent on three crucial
factors: the permeability of the formation, the fluid compressibility, and the total wellbore volume.
Therefore, the wellbore storage constant formula for fluid expansion, Eq. 20 was used.
Reservoir properties such as permeability, thickness, injection rate and skin factor are
same for all cases. The reservoir pressure is 4000 psi and temperature is 200 F.
54
The compressibility of water and viscosity were calculated by Osif correlation. This
correlation is,
Cw = -1
𝐵𝑤 (
𝜕𝐵𝑤
𝜕𝑃)T
=1/ (7.033 P+541.5 Cs -537.0 T +403,300)
(Eq. 23)
Mw1 =A TB
(Eq. 24)
where
A = 109.574 -8.40564 S+ 0.313314 S2 +8.72213x10-3 S3
(Eq. 25)
B= -1.12166 + 2.63951x10-2 S -6.79461x10-4 S2 -5.47119x10-5 S3 +1.55586x10-6 S4
(Eq. 26)
The oil compressibility and viscosity were estimated by using Vasques and Beggs
correlation.
Co = (5Rsob + 17.2 T- 1180 yg +12.61 yAPI -1433)/ (P x 105)
(Eq. 27)
Mo =Mob (P/Pb)m
(Eq. 28)
The gas compressibility and viscosity were obtained from Figure 39.
55
Figure 39. Variations in Compressibility and Viscosity (Penny and Conway, 1989)
According to calculations, it is obvious that the fluid expansion is mainly small because of
the compressibility of water and oil. On the other hand, the gas expansion is significant due to gas
expansion. With the wellbore storage constant, the total time that shows the end of wellbore storage
effect was obtained by using Eq. 27.
td > 170,000 𝐶𝑠 𝑒^0.14𝑠
𝑘ℎ/𝑀
(Eq. 29)
Figure 40 illustrates how the volume of fluids stored in the wellbore controls the duration of
wellbore storage with different wellbore volumes. The duration of wellbore storage is higher due
to the injection of gas. On the contrary, wellbore storage effects are finished much more quickly
for water case. Based on the plot, it is noticeable that larger compressibilities (gas) and larger
volumes increase the period of wellbore storage. To illustrate, for 1000 bbl wellbore volumes,
56
whereas the end of wellbore storage effect is 11 min for water, this duration for gas is 116 min. A
lazy shape of early time region is a good indicator of the storage effects.
Figure 40. Total Wellbore Volume vs End of Wellbore Storage
4.2.2 Effect of Wellbore Volume on the Difference in Pressure
The estimation of the minimum in-situ stress from pressure records is less accurate when there is a
large amount of wellbore storage because pressure behaviors act differently with the storage of
fluids in the wellbore than the absence of the wellbore storage effects. During the early time, the
0
20
40
60
80
100
120
140
0 200 400 600 800 1000 1200
Tim
e, m
in
Total wellbore volume, bbl
Total wellbore volume vs End of wellbore storage
injection_gas injection_water injection_oil
57
existence of wellbore storage effects is identified by S-shaped. After the unit slope line, taking 1.5
log cycle represents the end of wellbore storage. This unit-slope straight line is very important to
estimate pressure difference before the end of wellbore storage time. Based on the calculations of
the duration of wellbore storage effects, the differences in pressure were found at the same exact
time for all cases by Eq. 22.
C=𝑞𝑡
24𝛥𝑃 =
𝑄𝐵𝑡
24𝛥𝑃
The pressure difference is the pressure before shut-in minus the pressure during shut-in.
Figure 41 represents the pressure difference after the injection of water. As can be seen that, when
all reservoir parameters are same and injection rate is constant, the sandface pressure in large
wellbore volume increases more than the small wellbore volume. Due to the pressure build-up at
the sandface and high wellbore storage constant, pressure difference decreases dramatically.
Figure 41. Total Wellbore volume vs Pressure Difference for Injection of Water
0
200
400
600
800
1000
1200
0 200 400 600 800 1000 1200
Pre
ssu
re D
iffe
ren
ce, p
si
Total wellbore volume, bbl
Total wellbore volume vs Pressure Difference
58
To prove that the instantaneous shut-in pressure is not equal to closure pressure or minimum in-
situ stress, a case study for Marcellus oil shale formation was conducted. Before the end of
wellbore storage effects, pressure difference with different wellbore volume was calculated with
constant rate and wellbore storage factor. Figure 42 indicates the fall in pressure difference. The
pressure difference is 237 psi at 50 bbl wellbore volume. This means that at the exact time after
shut-in, pressure changes 237 psi. When the ISIP is assumed to be 2000 psi, and the pressure
difference is obtained 237 psi from the plot for 50 bbl wellbore volume, the minimum in-situ stress
was calculated by the subtraction of these two values. Therefore, the minimum in-situ stress is
2000-237=1763 psi. In conclusion, this result represents the effect of wellbore storage on pressure
response and proves that instantaneous shut-in pressure is not equal to the minimum in-situ stress.
Figure 42. Total Wellbore Volume vs Pressure Difference
0
50
100
150
200
250
0 200 400 600 800 1000 1200
Pre
ssu
re D
iffe
ren
ce, p
si
Total wellbore volume, bbl
Total wellbore volume vs Pressure difference
59
4.3 Effect of Pore Pressure on ISIP
The minimum horizontal stress changes with pore pressure are one of the primary considerations
in hydraulic fracture treatment design. The elastic uniaxial strain model was proposed to evaluate
the variation of horizontal stress magnitude from a knowledge of Poisson’s ratio, the total vertical
stress, and pore pressure (Hubbert and Willis, 1957). The equation of the model is expressed by the
following equation:
𝜎min= (V / (1-V) (𝜎ob
-Pp) +Pp +𝜎tectonics
(Eq. 30)
𝜎min = The minimum horizontal stress, psi
𝜎ob = Total vertical stress, psi
V= Poisson’s ratio
𝜎tectonics = Tectonics stress, psi
Pp = Pore pressure, psi
This model was applied to a specific case study based on the Marcellus shale gas play. For purposes
of this model, a depth of shale formation is 8000 ft. Considering P-waves and S-waves velocities,
an essential mechanical parameter, Poisson’s ratio is assumed to be 0.3. In addition, for shale
reservoirs, the overburden or vertical stress gradient is normally between 1.0 to 1.1 psi/ft. For this
case, overburden stress gradient selected is 1.05 psi/ft. In addition, the tectonics forces are
neglected because the stresses are controlled by the overburden load. In the basins, the boundaries
are assumed to be fixed, and the horizontal stresses are equal to each other in both directions. The
ratio of horizontal stress versus vertical stress is constant and elastic properties are isotropic and
homogeneous. Therefore, the effects of tectonic stresses are neglected due to the extensional and
compressional forces. The magnitude of the minimum horizontal stress was obtained from Eq. 30,
60
as summarized in Figure 43.
Figure 43. Pore Pressure vs Minimum Horizontal Stress
The results show that minimum principle stress is for 3000 psi is about 5314 psi which is
571 psi higher than 2000 psi. This is significant that a decline in pore pressure results in a decrease
in minimum horizontal stress. The difference for two cases is due to the reservoir depletion or
geologic factors at a given depth. On the contrary, larger pore pressure leads to increase principle
horizontal stress. This change occurs when the reservoir is overpressured.
0
2000
4000
6000
8000
10000
12000
14000
0 2000 4000 6000 8000 10000 12000 14000 16000 18000
Min
imu
m h
ori
zon
tal s
tres
s, p
si
Pore Pressure, psi
Minimum horizontal stress vs Pore Pressure
61
Figure 44. Change in Pore Pressure vs Difference in Minimum Horizontal Stress
Figure 44 displays the difference in minimum horizontal stress data estimated from the changes in
pore pressure. It is obvious that changes in pore pressure cause significant changes in the stress
field. While 3000 psi increase in pressure difference induces 1714 psi growth in stress changes,
2000 psi decrease in pressure causes to be 1942 psi reduction in minimum horizontal stress.
Finally, as seen in both figures minimum in-situ stresses are calculated taking into account
different pore pressure and for these different scenarios, it is concluded that minimum horizontal
stress is controlled by pore pressure. As a result of the influence of pore pressure on horizontal
stress in the shale formation, the instantaneous shut-in pressure (ISIP) is not equal to the
-6000
-4000
-2000
0
2000
4000
6000
8000
-8000 -6000 -4000 -2000 0 2000 4000 6000 8000 10000 12000
Dif
fen
ce in
Min
imu
m H
ori
zon
tal S
tres
s, p
si
Change in Pore Pressure, psi
Difference in Pore Pressure vs Difference in Minimum Horizontal Stress
62
minimum horizontal stress. The reason is that a pressure drop after shut-in causes a reduction in
the magnitude of minimum principle stress.
4.4 Effect of Temperature on ISIP
The temperature is important to be determined in order to analyze well responses because the
changes in stresses would be caused by temperature changes in the formation. Stress changes within
the reservoir are obtained with thermoelastic stresses and poroelastic parameters. Change of stress
due to thermal effects can be calculated by the following equation (Kaarstad, 1999):
ΔσT = 𝐸
1−𝑉 αT (TT-Tf)
(Eq. 31)
αT = Thermal expansion coefficient, ºC-1
E = Young’s modulus, psi
V = Poisson’s ratio
TT = Formation temperature after treatment, ºC
Tf = Original formation temperature, ºC
Shear Modulus, G and Young Modulus, E can be expressed as:
G= 1.34 x 1010 x𝑞𝑏
𝛥𝑡𝑠∗∆𝑡𝑠
(Eq. 32)
E=2G (1+V)
(Eq. 33)
This case study of Marcellus oil formation was conducted to demonstrate thermal effects on
minimum horizontal stress. In order to estimate stress changes from Eq. 31, P-wave velocity and
63
S-wave velocity were used as 13500 ft/sec and 7150 ft/sec respectively. In terms of P-wave velocity
and S-wave velocity, Poisson’s ratio was calculated. The formation’s bulk density is also assumed
to be 2.5 g/cc. Based on Fjaer et al. (1992) thermal expansion coefficient is assumed 10-5°C-1.
Figure 45 illustrates how the alterations in temperature have impacts on the thermal expansion
stress. As is presented in the graph, there has been a gradual increase. As a consequence of a
stimulation operation, temperature differences can occur in the formation. Due to the fact that the
temperature of injecting fluid is lower than the temperature of reservoir rock, the area around the
well becomes cold. This cooling effect causes reduction of stresses. As seen while the difference
in temperature is -50°C, thermal expansion stress is -2896 psi. Moreover, when the temperature
difference is -100° C which is cooler than -50°C, a decrease in thermal expansion stress occurs. On
the other hand, after the injection of fluid stops, the formation near the well slowly heats. For
example, it is clear that there is an increasing thermal expansion stress which is 579 psi in response
to 50°C changes in temperature. These results point out how minimum horizontal stress is more
different than the instantaneous shut-in pressure regarding the effect of thermal expansion.
64
Figure 45. Changes in Temperature vs Changes in Thermal Expansion Stress
-7000
-6000
-5000
-4000
-3000
-2000
-1000
0
1000
2000
3000
4000
-120 -100 -80 -60 -40 -20 0 20 40 60
Ch
ange
s in
Th
erm
al E
xpan
sio
n S
tre
ss, p
si
Changes in Temperature, ∘C
Changes in Temperature vs Changes in Thermal Expension Stress
65
Chapter 5
Results and Discussion
This chapter presents and discusses the results obtained from the mathematical approaches
to analyze the instantaneous shut-in pressure (ISIP) in shale oil and gas reservoir. A total of four
different factors were used for this study to express why the minimum horizontal stress is not the
same or even close in shale based on our observations of field data.
First of all, the leakoff coefficients and closure time accomplished for different fracturing
fluid system based on the Marcellus shale gas reservoir parameters. The fracturing fluids tend to
leakoff or filter into the formation and the fluid loss of the fracturing fluid is also dependent on the
formation permeability. Therefore, numerical calculations were performed for different ranges of
permeabilities to quantify the impact of fluid leak off coefficients on crosslinked gel fracture
treatments, linear gel treatments, and water fracture treatments. This comparison study allows us
to understand that the instantaneous shut-in pressure (ISIP) is not taken as a good approximation
of the minimum in-situ stress. Due to fluid leakoff, the stress estimation can be in error. Thus, an
accurate understanding of fracturing fluid loss behavior is crucial to design a successful treatment.
It is shown as a plot of the permeability versus an overall coefficient in Figure 46. The use
of both linear gel and crosslinked gel fracturing fluids have significant effects on fluid loss. For
reservoirs with permeability lower than 0.01 md, the overall leakoff coefficients are close to each
other. With an increase in permeability, the fluid loss coefficient is getting higher, so for both cases
using crosslinked gel and linear gel, the control of fluid loss is very important.
66
Figure 46. The Comparison of Crosslinked Gel and Linear Gel Treatments
There are usually two stages of the fluid loss. The first step is the initial invasion of the fracturing
fluid into the formation, and the second one is to generate a filter cake on the surface of the porous
medium. To illustrate, crosslinking rises gel strength and viscosity and create a wall building effects
which is a filter cake. On the other hand, some fluid leakoff without forming a filter cake. In this
case, the leakoff rate is governed by Cvc. Figure 47 demonstrates the results of the combination of
viscosity, relative permeability, and compressibility effects on different permeabilities for water
fracture treatments. As can be seen that it has a dramatic increase on Cvc after 1 md.
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.0001 0.001 0.01 0.1 1 10 100 1000
Ove
rall
Co
effi
cie
nt,
Ct,
ft/
min
0.5
Permeability, md
Permeability vs Ct
Crosslinked Gel Linear Gel
67
Figure 47. Permeability versus Cvc for Water Fracture Treatments
Figure 48 represents closure time for different permeability values. After the shut-down of
pumping, the shut-in pressure is the pressure as a function of time. Instantaneous shut-in pressure
is the pressure happens instantly after shut-in. As it can be seen in Figure 42, for Marcellus shale
gas formations, the fracture takes hours to close while especially using highly-viscous fracturing
fluids such as linear and crosslinked gels. For example, when the permeability is 0.0001 md, the
closure time is roughly 1259 min which is very high. The bottomhole pressure decreases below to
the instantaneous shut-in pressure because of longer closure time after pumps are turned off. It is
an evidence that ISIP is not equal to the closure pressure, and greater than the minimum horizontal
stress.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.0001 0.001 0.01 0.1 1 10 100 1000
Cvc
, ft/
min
0.5
Permability, md
Permeability vs Cvc
68
Figure 48. Permeability versus Closure time for Crosslinked Gel, Linear Gel, and Water
Treatments
Secondly, the effects of wellbore storage on pressure behaviors were investigated. To illustrate,
for Marcellus oil case study, the pressure difference is 60 psi at 300 bbl total wellbore volume. It
proves that pressure decreases 60 psi after the shut-in period while the instantaneous shut-in
pressure is 2000 psi. Therefore, the minimum horizontal stress is equal to 1940 psi. A decrease in
pressure results in changes in the minimum horizontal stress. Because of pressure drop, the
instantaneous shut-in pressure is greater bound on the closure stress. In conclusion, the results
have provided a better understanding of the relationship between the instantaneous shut-in
pressure and the minimum horizontal stress.
.
0.00001
0.0001
0.001
0.01
0.1
1
10
100
1000
10000
0.0001 0.001 0.01 0.1 1 10 100 1000
Clo
sure
tim
e, m
in
Permeability, md
Permeability vs Closure Time
Crosslinked Gel Linear Gel Water
69
Closure pressure has been influenced significantly by pore pressure. Results from Marcellus shale
gas formation outlined the minimum horizontal stress showing increased or decreased closure stress
with the pore pressure. Due to the reservoir depletion or overpressured wells, closure is not
considered constant. 2000 psi increase in pore pressure results in 1142 psi growth in minimum
horizontal stress while 3000 psi decrease in pore pressure causes to be 1714 psi reduction in closure
stress. Therefore if ISIP is assumed to be 4000 psi after shut-in, the minimum horizontal stress is
2286 psi due to the reduction in pore pressure. These results point out that the minimum horizontal
stress is not constant and these changes create pressure differences the instantaneous shut-in
pressure and the minimum horizontal stress.
To find how the surface and rock temperature affect the magnitude of horizontal stresses
which are generated thermally, the uniaxial strain model was used. A case study of Marcellus oil
formation was conducted. During the injecting fluid into the formation, the temperature around the
well decreases. While the changes in temperature is 10°C, the differences in thermal expension
stress declines to 579 psi. Assuming that ISIP is 2000 psi, minimum horizontal is 579 psi lower
than the instantaneous shut-in pressure. Regarding the effects of the temperature, the minimum
principal stress is lower bound on the instantaneous shut-in pressure which means that they are not
equal to each other.
70
Chapter 6
Conclusions and Recommendations
In this study, the purpose was to interpret the instantaneous shut-in pressure (ISIP) in shale oil and
gas reservoirs. To better understand the relationship between the instantaneous shut-in pressure and
minimum principal in-situ stress, the mathematical approaches were used. These calculations were
determined by leakoff coefficients, wellbore storage, pore pressure, and temperature. Main
important conclusions are:
Leakoff is the main factor affecting minimum in-situ stress and closure time. Use of
highly-viscous fracturing fluids such as linear and crosslinked gels sometimes make
significant alterations on fracture closure time. The fracture may take hours to close. This
is the reason why ISIP is higher bound on the minimum horizontal stress, and they are not
equal to each other.
Considering the presence of wellbore storage effects, the end of wellbore storage time for
the gas case is higher than oil and water case. This is due to the fact that large volumes and
large compressibilities increase the duration of wellbore storage time. Additionally, after
shut-in, the difference in pressure occurs and changes slightly because of the effects of
wellbore volume.
Closure pressure has been influenced significantly by pore pressure since pore pressure
causes stress changes in the formation. During depletion, pore pressure decreases and
results in a decline in minimum horizontal stress, whereas over pressured reservoir leads
to increase pore pressure and minimum principle stress.
71
Increasing the difference in temperature results in a linear growth of thermal expansion
stress. Injecting the lower fluid into the formation causes reduction of the stress. On the
other hand, after the injection of fluid stops, the formation near the well slowly heats.
There are more factors that I am going to study to improve an analysis of the instantaneous shut-in
pressure in the future. Factors such as the amount of clay content, the changes in fracture
orientation, stress levels, or the presence of natural fractures certainly affect the interpretation.
Thus, this work can be extended to investigate how dominant these factors are in the analysis of
the instantaneous shut-in pressure.
72
Nomenclature
B Formation Volume Factor [bbl/STB]
Cc Reservoir-Fluid Viscosity/Compressibility Effects [ft/min0.5]
Cd The Discharge Coefficient [ - ]
cg The Compressibility of Gas [psi-1]
ct The Total Compressibility [psi-1]
Ct The Overall Leakoff Coefficient [ft/min0.5]
cr The Rock Compressibility [psi-1]
Cs The Wellbore Storage Constant [ - ]
Cv The Effluent viscosity and relative permeability effects [ft/min0.5]
Cw Wall-Building Effects [ft/min0.5]
cw The Compressibility of Water [psi-1]
cwb The Average Compressibility of fluid in the wellbore [psi-1]
Dp The Perforation’s Diameter [inches]
E Young Modulus [psi]
G Shear Modulus [psi]
ℎ The Total Vertical Depth [ft]
kL Nonreactive Liquid Permeability [md]
73
m Slope of Cumulative Filtrate Volume vs Time1/2 Plot [ - ]
Np The number of perforations [ - ]
Pi Initial Wellbore Pressure [psi]
Pfrac Fracturing Fluid Pressure [psi]
Pfric Fluid Friction Pressure [psi]
Pℎ Hydrostatic Pressure [psi]
Pnet Net Pressure [psi]
Pp Pore Pressure [psi]
Pr Reservoir fluid pressure [psi]
Ps Wellhead Pressure [psi]
Ptort Tortuosity Pressure [psi]
Pwb Bottomhole treating pressure [psi]
∆P The Pressure Difference [psi]
ΔPperf Perforation Friction Pressure [psi]
Sg Gas Saturation [%]
Sor Residual Oil Saturation [%]
Sw Water Saturation [%]
Swr Residual Water Saturation [%]
td The End of Wellbore Storage Time [min]
ts The Spurt Time [min]
74
Tf Formation Temperature After Treatment [ºC]
TT Original Formation Temperature [ºC]
q The Total Flow Rate [bpm]
𝜌b The Bulk Density [g/cc]
Q Flow Rate [STB/day]
𝜌 The Slurry Density [lb/gal]
V Poisson’s Ratio [ - ]
Vwb The Total Wellbore Fluid Volume [bbl]
⏀ Porosity [%]
µa Effluent’s Viscosity [cp]
µf Viscosity of mobile formation [cp]
αT Thermal Expansion Coefficient [ºC-1]
𝜎min Minimum Horizontal Stress [psi]
σob Overburden Stress [psi]
σtectonics Tectonic Stress [psi]
ΔσT Thermal Expansion Stress [psi]
75
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