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Impact of Stress Sensitive Permeability on Production Data Analysis

Impact of Stress Sensitive Permeability on Production Data Analysis
Impact of Stress Sensitive Permeability on Production Data Analysis

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

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 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

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

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