SPE 114166
Impact of Stress Sensitive Permeability on Production Data Analysis
Rosalind Archer, University of Auckland
Copyright 2008, Society of Petroleum Engineers
This paper was prepared for presentation at the 2008 SPE Unconventional Reservoirs Conference held in Keystone, Colorado, U.S.A., 10–12 February 2008.
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
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Abstract
The dependence of reservoir permeability and porosity on pressure can have a significant impact on well performance. This
paper addresses this issue theoretically, via reservoir simulation, and via production data analysis type curves. This paper
presents the single phase fluid flow equation for a slightly compressible fluid in a stress sensitive reservoir in a form which
can be solved in Laplace space to generate profiles of well flow rate versus time (for constant bottomhole pressure
production). This is achieved by writing the flow equation in terms of the derivatives of a function involving the sum of
pressure and pressure-squared terms. Reservoir simulation results concur with the theoretical prediction that depletion in a
stress sensitive reservoir will be slower than in a non-stress sensitive reservoir, however ultimate recovery in the simulation
cases was not affected by stress sensitivity. p/z plots for stress sensitive reservoirs show volumetrics are unaffected by stress
sensitivity. Analysis of simulated production data from stress sensitive reservoirs shows that drainage area is likely to be
underestimated by this approach. Permeabilities predicted from analysis using standard type curves are lower than the
permeability at initial reservoir pressure. The magnitude of the reduction in the interpreted permeability naturally depends on
the magnitude of the stress sensitivity.
Introduction
Stress sensitivity is a particularly important phenomenon in tight gas reservoirs. Rushing and Newsham (2001) noted this
and recommended measuring rock properties at a range of stress conditions. McKinney et al. (2002) attribute the hyperbolic
decline often observed in tight gas reservoirs to stress sensitivity. Cox et al. (2007) considered the impact of stress sensitivity
on a couple of simulated cases in a wider study but did not comprehensively address the impact of stress sensitivity of
production data analysis. The topic is of considerable practical interest to reservoir engineers e.g. Amar et al. (1995), Lei et
al. (2007) and Hedong et al. (2007).
Much of theoretical analysis of flow in stress sensitive reservoirs has focused on pressure transient analysis (e.g. Pedrosa,
1986, Samaniego and Cinco-Ley, 1989). This study considers rate transient analysis which has received less attention. The
approach presented in this paper avoids the use of a pseudopressure to handle stress sensitivity, however the approach
presented in this paper is limited to a linear variation of permeability and porosity with pressure change.
Theory
Fluid Flow Equations
To assess the impact of stress sensitive permeability on well performance the nature of the differential equation governing
single phase fluid flow was considered. Note that this analysis uses a form of the fluid flow equation which is applicable to a
slightly compressible (i.e. liquid flow) case. This section aims to provide some theoretical insight to compliment
observations based on reservoir simulation results presented in later sections.
First consider the case of fluid flow in a non-stress sensitive reservoir. This is governed by:
D
D D D D t p r p r r r ??=????????????1 (1)
This equation is typically solved in Laplace space (Raghavan, 2003) to give a solution of the form:
2 SPE 114166
)()(s r BK s r AI p D o D o D +=
(2)
where A and B are constants which can be determined by ensuring that the solution satisfies suitable boundary conditions
(e.g. constant pressure or constant rate at the wellbore, no flow or constant pressure at the outer boundary).
Now consider flow of a slightly compressible fluid in a stress sensitive reservoir. For the purposes of illustration a linear variation in permeability and porosity is considered. The flow is governed by:
t
p c p p r p r p p r r k t i i ????=????????????μβφα))(1())(1( (3)
This can be expanded in terms of p and p 2 to give: t
p c t p c p r p r r r k r p r r r p k t t i i ??+???=????????????+???????????2
22)1(2)1(φβμμβφαα (4)
t
p p k c t p p k c p r p r r r p r p r r r i t i t i i ???+????=?????????????+??????????22)1(2)1()1(1)1(21αφβμαμβφαα (5)
Solution Strategy
To make further progress analytically it is necessary to assume that α = β. This is a limiting assumption but is required to
ensure a solution can be found. In many formations the impact of stress sensitivity would be more pronounced on
permeability than on porosity. Under this assumption we have:
k c p k c p t i t i αφβμαμβφ=??)1()1(
(6) . Equation (5) can now be written in the form:
),()
1(2),(,12t r p p t r p f where t f r f r r r i ααγ?+=??=?????????? (7)
Equation 7 can be solved in Laplace space using the same procedures used to solve equation 1 to produce equation 2. For
production data analysis the boundary conditions of most interest are a constant initial reservoir pressure, a constant pressure
at the wellbore (r = r w ) and a closed outer boundary (r = r e ), i.e.:
(8)
00)1()1(2)1(2220=??=???+??=???+=?+====r p by satisfied be would which r p p p r p r f p p p f p p p f i r r w i w r r i i i t e w αααααα
SPE 114166 3 For production data analysis the most useful quantity to solve for is w
r r r f
=??, which can be readily done in Laplace space (and then inverted to real space via the Stehfest algorithm, 1970). This leads directly to w
r r r p
=??(note that p = p w at this point so it is not necessary to solve a nonlinear equation to determine
r
p ??).
This derivation highlights the impact of stress sensitivity on well and reservoir performance. For a case in which α = β =
0.0001 (i.e. the reservoir permeability and porosity would be reduced by 50% if the pressure was decreased from 5000 psi to 0 psi), the group
)1(2i p αα? is on the order 10-8, so the second term in equation 5 is much smaller than the first term and could be neglected. The terms of the right hand side of equation 5 control the rate at which reservoir pressure is depleted. It is
clear that the right hand side of equation 5 would be always be greater than it would be in the case that α = β = 0. One way to
view this is that the reservoir will initially behave like a conventional reservoir with porosity larger than the true porosity,
hence depletion of a stress sensitive reservoir will take longer than that of a reservoir which is not stress sensitive.
Results
Reservoir Simulation
To assess the impact of stress sensitivity on well performance a radial reservoir simulation was used. A single unfractured
production well was located at the centre of this model. The reservoir contained single phase gas (γg = 0.71). This well was
operated at a bottomhole pressure of 350 psi and was constrained to produce at a maximum rate of 10,000 Mscf/d. The initial
reservoir pressure was 5000 psi. The reservoir porosity was 10% and cases of 0.1, 0.5, 1 and 10md permeability (at initial
reservoir pressure) were considered. Drainage areas of 10, 15 and 20 acres were considered. Very small timesteps
(increasing logarithmically) were used initially to simulate rate-transient behaviour. Ten years of well performance was then
simulated. This reservoir simulation model is deliberately simple. A homogenous reservoir with a radial drainage area was
chosen to make sure that any departures from conventional reservoir behaviour could be clearly attributable to the effects of
stress-sensitive reservoir permeability.
Four cases of stress sensitivity were considered. In the reservoir simulation package used in this study stress sensitivity was
implemented as a drawdown dependent transmissibility multiplier. In the first two cases the transmissibility multiplier
varied exponentially with the drawdown, in the second two cases the transmissibility multiplier varied linearly with
drawdown. These transmissibility multipliers are plotted in Figure 1. No multiplier was applied to porosity in the
simulation model runs.
Figures 2 to 4 summarize some the reservoir simulation results which were obtained. Figure 2 shows gas rate versus time for
a non-stress sensitive case and four stress sensitive cases. This data supports the observation from the previous section
(though for a slightly compressible liquid case) that stress sensitive reservoirs would in theory take longer to deplete than
non-stress sensitive reservoirs. Figure 3 shows gas rate versus cumulative gas production profiles for a stress sensitive case
(exponential SS#1). This shows the curve scale with permeability (note production in the 10md is capped at a plateau rate of
10,000 Mscf/D). Figure 4 compares p/z curves for several cases of stress sensitivity and shows that stress sensitivity has
essentially no effect on the overall volumetric behaviour of the reservoir.
Production Data Analysis
Production data analysis of the simulated well performance data from the previous section was performed using techniques
proposed by Doublet et al. (1994). This analysis approach compares well data to type curves describing dimensionless rate,
rate integral and rate integral derivative versus material balance time. By construction these type curves align all late time
data on the b=1 harmonic decline curve stem. The type curves presented by Doublet et al. are for a conventional reservoir
case in which permeability is not stress sensitive. Therefore it would be surprising for the simulated well performance data
to completely match the type curves. However given that a reservoir engineer may not have a priori knowledge that a
reservoir is stress sensitive this section explores what impact stress sensitivity has on production data analysis results.
Figures 5 to 8 present the production data analysis of simulated production data for a well producing from a 15 acre drainage
area, with a permeability of 1md. Four different stress sensitivity scenarios (see Figure 1) were used.
114166 4 SPE
In the type curve matching process the rate the simulated production data was match to the b=1 harmonic decline curve stem,
while achieving the best possible match to the rest of the type curves. In all cases the stress-sensitive behaviour of the
reservoir was not evident from the rate or rate-integral curves. The only significant departure from standard type curves
observed in the results from the stress-sensitive cases was a change in the "hump" in the rate-integral-derivative curve. When
field data is used to compute the rate-integral-derivative this data may be noisy so the change in the shape of this hump may
not always be detected.
The production data analysis results are summarized in Table 1. In every case the drainage area interpreted from production
data analysis was less than the actual 15 acre drainage area. As expected the interpreted permeability values were less than
the 1md reservoir permeability at initial conditions. It should be noted though that every production data interpretation also
resulted in skin factors between 1.3 and 1.6, when the underlying reservoir simulation model used a zero skin factor.
Conclusions
This paper has presented an analysis of the single phase flow equation for a slightly compressible fluid in a stress-sensitive
reservoir. This analysis showed this differential equation can be viewed as one governing the sum of pressure and pressure-
squared terms. A solution strategy to calculate rate transients in stress sensitive reservoirs is proposed. This solution can be
calculated via a simple manipulation (in Laplace space) of the solution for transient flow rates in a non-stress sensitive
reservoir.
Reservoir simulation results for stress sensitive gas reservoirs show that well rates are less that what would be observed
without stress sensitivity. Ultimate recovery however is comparable. These slower depletion rates are in keeping with the
findings from the theoretical section of this study.
Production data analysis using "Blasingame" type curves shows that the dimensionless rate and rate-integral curves can be
reasonably well matched for the cases considered in this study. Stress sensitivity causes a change in the "hump" in the rate-
integral-derivative curve. Production data analysis of well performance data from the reservoir simulation model inferred
smaller drainage areas than those in the simulation model and always predicted lower permeabilities than the permeability at
initial reservoir pressure. This decrease in permeability depended strongly on the degree of stress sensitivity.
All results presented are for single porosity systems. Further exploration of production analysis behaviour of stress sensitive
dual porosity systems would be of interest in the future.
Acknowledgments
The author would like to acknowledge Schlumberger for providing access to reservoir simulation software. The author
would also like to thank Tom Blasingame for many fruitful discussions about production data analysis over the years.
Nomenclature
A, B Constants in Bessel function solution of diffusivity equation in Laplace space
c t Total compressibility, psi-1
f Function satisfyin
g the diffusion equation
I0, K0 Bessel
functions
k Permeability, md (at initial reservoir pressure)
p Pressure,
psi
p i Initial reservoir pressure, psi
p D Dimensionless pressure
r Radial distance, ft
r D Dimensionless radial distance
r Wellbore radius, ft
s Laplace transform variable
hr
t Time,
t D Dimensionless time
αConstant defining linear decrease in permeability
βConstant defining linear decrease in porosity
fraction
φ Porosity,
cp
μ Fluid
viscosity,
SPE 114166 5 References
Amar, Z. Altunbay, M., and Barr D.:"Stress Sensitivity in the Dulang Field – How it is Related to Productivity", SPE 30092 presented at the 1995 SPE European Formation Damage Conference, The Hague, Netherlands, 15 – 16 May.
Cox, S.A., Sutton, R.P., Stoltz, R.P., Barree, R.D. and Conway, M.W.:"Evaluation of the Effect of Complex Reservoir Geometries and Completion Practices on Production Analysis", SPE 111285 presented at the 2007 Eastern Regional Meeting, Lexington, Kentucky, 17 – 19 October.
Doublet, L.E., Pande P.K., McCollum, T.J. and Blasingame, T.A.: "Decline Curve Analysis Using Type Curves – Analysis of Oil Well Production Data Using Material Balance Time: Application to Field Cases", SPE 28688 presented at the 1994 Petroleum Conference and Exhibition of Mexico, Veracruz, Mexico, 10-13 October.
Hedong, S., Xiangjiao, X., Jianping, Y. and Xhangfeng:"Study on Productivity Evaluation and Performance Prediction Method of Overpressured, Stress-Sensitive Gas Reservoirs", SPE 108451 presented at the Asia Pacific Oil and Gas Conference and Exhibition, Jakarta, Indonesia, 30 October – 1 November
McKinney, P.D., Rushing, J.A., and Sanders, L.A.:"Applied Reservoir Characterization for Maximizing Reserve Growth and Profitability in Tight Gas Sands: A Paradigm Shift in Development Strategies for Low-Permeability Gas Reservoirs", SPE 75708 presented at the 2002 SPE Gas Technology Symposium, Calgary, Canada, 30 April – 2 May.
Lei, Q., Xiong, W., Yuan J., Cui Y. And Wu, Y.:"Analysis of Stress Sensitivity and Its Influence on Oil Production from Tight Reservoirs", SPE 111148 presented at the 2007 Eastern Regional Meeting, Lexington, Kentucky, 17 – 19 October.
Pedrosa, O.:"Pressure Transient Response in Stress-Sensitive Formations", SPE 15115 presented at the 1986 California Regional Meeting, Oakland, California, 2 – 4 April.
Raghavan, R.:" Well Test Analysis", Prentice Hall, 1993.
Rushing, J.A. and Newsham, K.E.:"An Integrated Work-Flow Model to Characterize Unconventional Gas Resources: Part II – Formation Evaluation and Reservoir Modeling", SPE 71352 presented at the 2001 SPE Annual Technical Conference and Exhibition, New Orleans, Louisiana, 30 September – 3 October.
Samaniego, F. and Cinco-Ley, H.:"On the Determination of the Pressure-Dependent Characteristics of a Reservoir Through Transient Pressure Testing", SPE 19774 presented at the 1989 SPE Annual Technical Conference and Exhibition, San Antonio, Texas, 8 – 11 October.
Stehfest, H.:"Algorithm 368: Numerical inversion of Laplace transforms", Communications of the ACM, 13, 1, 47-49, Jan. 1970.
Tables
Case Drainage area, acres Permeability (at p i), md Skin
1 0
Base 15.0
14.8 0.0724 1.3789
1 - Strong exponential
decrease in permeability
2 - Exponential decrease in
10.9 0.1768 1.5334 permeability
3 - Strong linear decrease in
10.7 0.1556 1.5422 permeability
4 - Linear decrease in
11.0 0.7999 1.5293 permeability
Table 1: Summary of production data analysis results
6 SPE
114166 Figures
Figure 1: Transmissibility multipliers used in reservoir simulation study
Figure 2: Comparison of rate versus time profiles in 1md, 15acre drainage area (base case and four stress sensitive cases)
SPE 114166 7
Figure 3: Comparison of rate versus cumulative profiles for a 15 acre stress sensitive drainage area
Figure 4: Comparison of p/z curves for 1md reservoir, 15 acres drainage area
114166 8 SPE
Figure 5: Production data analysis, case 1
Figure 6: Production data analysis, case 2
SPE 114166 9
Figure 7: Production data analysis, case 3
Figure 8: Production data analysis, case 4