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Copyright 2002, Society of Petroleum Engineers Inc.

This paper was prepared for presentation at the SPE International Symposium and Exhibitionon Formation Damage Control held in Lafayette, Louisiana, 2021 February 2002.

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, aspresented, have not been reviewed by the Society of Petroleum Engineers and are subject tocorrection by the author(s). The material, as presented, does not necessarily reflect anyposition of the Society of Petroleum Engineers, its officers, or members. Papers presented at

SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers isprohibited. Permission to reproduce in print is restricted to an abstract of not more than 300words; illustrations may not be copied. The abstract must contain conspicuousacknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O.Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435.

AbstractIn oil/gas reservoirs, the state of stress changes as fluid

production/injection from/into the reservoir takes place. Also, petrophysical and geomechanical properties may change dueto the variation of the effective stress. However, thisconsideration is seldom taken into account in welltest analysis.

This paper presents a numerical, fully coupled, fluid-flow/geomechanical model to perform well test analysis instress-sensitive reservoirs. The governing equations aredeveloped in cylindrical coordinates honoring the geometry of the flow lines characterizing the drainage area in most welltests. The model is a 3D, point distributed, finite-differencesimulator which applies a fully implicit discretization schemeto ensure maximum stability.

The model assumes isothermal, single-phase fluid-flow(slightly compressible fluid). Infinite and finite acting

behavior is allowed during the well test. Reservoir propertiesare allowed to change from one layer to another. Both casesisotropic and anisotropic rock property behaviors areconsidered in the model. The rock behaves as elastic system

whose deformation is described by nonlinear theory(the mechanical properties are function of the meaneffective stress).

The results show that in stress-sensitive reservoirs the permeability decreases with production time reaching itsminimum value near the wellbore and moving more and moreinto the reservoir as production time increases. After a certain

production time, the permeability distribution reaches aconstant value. An important finding from this study is that thedamage caused by rock deformation is irreversible; therefore,

its early detection and treatment is essential for optimumreservoir management.

IntroductionConventionally, well test analysis models are based on thefollowing implicit assumptions: (i) constant fluid-flow and

geomechanical rock properties, and (ii) constant stress state.However, published laboratory studies show that in stresssensitive reservoirs rock properties may change significantlywith variation of the pore pressure and the stress state. 1-7 Also,the variation in the well pressure produces a variation in thestress state. In fact, the largest deviations from the initial stressstate are found at the borehole wall and its neighborhoodwhere the pore pressure variation is always maximum.

To study the impact that the above mentioned assumptionshave on well testing, it is necessary to solve the governingequations describing the deformation of the solid part of therock coupled with the governing equations describing thechanges in the pore pressure. Due to their strong nonlinear

behavior, the solution of this set of differential equations must be performed numerically.This paper presents a 3D, point-distributed, finite-

difference model discribing the formation damage caused byrock deformation in stress-sensitive reservoirs. The physicalsystem is represented in cilyndrical coordinates anddiscretized by means of a point-distributed grid ( Fig. 1 ). Thegoverning equations are based mainly on the followingassumptions: (i) isothermal, single-phase fluid flow, (ii) thedeformation of the solid part of the rock behaves as anonlinear elastic medium with small strains, and (iii) themechanical and fluid-flow properties are assumed to befunctions of the mean effective stress. The primary variablesin the resulting system of governing equations are theincremental displacements and the pore pressure.

The nonlinear set of equations is discretized using second-order approximations in space. A fully-implicit procedure isapplied to achieve maximum numerical stability. The resultingset of algebraic equations are arranged as a 4 x 4 block matrixsystem corresponding to each of the four primary unknowns(i.e., the incremental displacements in the r -, - and z directions and the pore pressure). The numerical procedureused to solve this system of equations involves an iterative

SPE 73742

A Numerical Model to Study the Formation Damage by Rock Deformation from WellTest AnalysisJose Gildardo Osorio, SPE, Universidad Nacional de Colombia, Alejandro Wills, SPE, New Mexico Institute of Mining andTechnology, Osmar Rene Alcalde, SPE, New Mexico Institute of Mining and Technology

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2 J.G. OSORIO, A. WILLS, O.R. ALCALDE SPE 73742

sequence that includes evaluation of nonlinear properties asfunction of pore pressure and stress state.

Mathematical FormulationThe model used in this study assumes an isothermal single-

phase fluid flowing through a deformable rock skeleton. Thedeformation of the solid part of the rock behaves as a

nonlinear elastic medium with small strains. The fluid-flowand geomechanical properties are functions of the averageeffective stress and may vary from layer to layer. The modelmay be applied to both infinite and finite acting reservoirs.

The physical model is represented by a successive set of concentric cylinders of variable height and thickness.Horizontally, the system is divided into arcs of differentangular size. The governing equations are represented incylindrical coordinates.

The mathematical formulation of the model presented inthis paper accounts for the multicomponent nature of thereservoir rock (one fluid component and one solidcomponent). Therefore, the formulation comes from thecoupling of two different models: a fluid-flow modeldescribing the motion of the pore fluid and a stress-deformation model describing the deformation of the rock solid skeleton.

Fluid-Flow Model . Four basic relations constitute the fluid-flow model: fluid mass conservation, solid mass conservation,Darcys law, and the equation of state. Mathematically, theserelations can be expressed as follows

Fluid mass conservation:

=

z

uu

r

1

r

r u

r

1 fz f f f fr f

) f

t f

t

q~t

V

V 1 +

................................................. (1)

Solid mass conservation:

( )[ ] ( )

f s sr s u1

r 1

r r u1

r 1

( ) ( )[ ] s

t f

t

fz sq~

t

1V

V 1

z

u1+

=

.................. (2)

Darcys law:

( ) pk uu f

si fi = ........................................ (3)

Equation of state (isothermal fluid compressibility):

p1

c f

f f

=

................................................... (4)

In Eq. 1 through 4, is the density, is the effective

porosity, u is the velocity, t V is the bulk volume, ~q is thesource/sink term expressed as mass rate per unit of bulk volume (we adopt the convention that the minus sign representa source and the plus sign a sink), k is the permeabilitytensor, is the fluid viscosity, p is the fluid pressure, t is

time, and c is compressibility. The subscript f refers to fluid.The subscripts r , , and z refer to the r , , and z directions, respectively. The subscripts s and l refer to theliquid and solid phases, respectively. The symbol denotesgradient.

For a slightly compressible fluid. Eq. 4 yields:

) p p( c fo f

0 f e= ............................................... (5)

In Eq. 4, the subcript o refers to a reference state.Combining Eqs. 1, 2, 3 and 5 gives the following fluid

flow equation:

+

+

pk r 1

r pk

r r r

1

f f

f

r f

f f P

p f

f

z f q

~t

pc

t V

V 1

z pk

z +

+

=

........ (6)

In Eq. 6, pV is the pore volume. Eq. 3 assumes that the

solid source/sink term is zero.Following Zimmerman 8, the change of the pore volume

term in Eq. 6 can be expressed as

( )( ) ( )

+=

bc

vr bcr bc

p

p cdt d

ccdt dp

1cc1

dt

dV

V 1

(7)

In Eq. 7, v is the volumetric strain (in this study the stress

is positive if compressive), c s is the compressibility of therock-matrix material measured from an unjacketedcompressibility test, and bcc is the bulk compressibilityexpressing the effect of the mean stress variation on the bulk volume at constant pore pressure 8,

p

t

t bc

V

V 1

c

=

.............................................. (8)

In Eq. 8, is the mean stress.Substitution of Eq. 7 into Eq. 6 gives,

( ) f bc

vr bct f

f f qcdt

ccdt dp

c p ~+

=

K

...................................................................................... (9)

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SPE 73742 A NUMERICAL MODEL TO STUDY THE FORMATION DAMAGE BY ROCK DEFORMATION FROM WELL TEST ANALYSIS 3

In Eq. 9, denotes divergence; t c is a compressibilityterm given by:

( ) f r bct cccc ++= 1 ...............................................(10)

The porosity is a pressure and stress-dependent property

which can be expressed as

( )( )( )dpd ccd r bc += 1 ..................................(11)

Stress-Deformation Model. The fundamental assumption of the stress-deformation model is that the relationship betweenincrements in stress and increments in strain from time t totime t dt + is described by the theory of nonlinear elasticdeformation with small strains.

The stress-deformation model is based on three basicrelations: stress-equilibrium, strain-displacement and stress-strain-pressure equations

Stress-equilibrium equations. To preserve equilibrium of

forces after a time increment t , the stress-equilibriumequations must satisfy that 9

++

+

+

+

r r r rz r r r r r z r r 11

00000

0=+

+r z

r zr .........................................(12)

r r r z r r r r z r

+

++

+

+

1210000

02 =+

+r z

r r ................................................(13)

r z r r r z rz z rz z rz z

+

++

+

+

0000 1

01 =+

+

r r rz z

................................................(14)

In Eqs. 12 through 14, i , i = r, , z , is the incrementaltotal stresses in the i direction; rz , r and z areincremental shear stresses.

Strain-displacement equations. In incremental form, theincremental displacements and incremental strains are related

by 10, 11

r u r

r = .................................................................(15)

+=

uu

r r 1

...............................................(16)

z u z

z = .................................................................(17)

r u

uu

r r

r +

=

21

................................(18)

+

=

z u

r u r z

rz 21

...........................................(19)

+

= z

uu

r

z z

1

2

1.......................................(20)

In Eqs. 15 through 20, i and iu , i = r, , z , are theincremental normal strains and the displacements,respectively, in the i direction; r , rz and z areincremental shear stresses.

Stress-strain-pressure equations. The stress-strain- pressure equations in incremental form are given by 4

ij ij v ij ijG p= + +2 .............................(21)

In Eq. 21, p is the incremental pore pressure; G , and

are the shear modulus, the Lames constant and the Biots poroelastic constant, respectively; ij is Kroneckers delta

( ij =1 for i=j , ij = 0 for i j); v is the incrementalvolumetric strain defined as

z uu

ur r

u z r

r v

+

++

=

1.....................(22)

The effective stress is a measure of the actual stress carried by the solid skeleton of the rock. The incremental effectivestress law defines the relation combining the incremental totalstress and the incremental pore pressure to obtain theincremental effective stress. Following the usual definition for the effective stress, the incremental effective stress is given by

ijijij p = ...................................................(23)

The Biots poroelastic parameter in Eq. 23 is a function of the stress state (effective stress) at which the increment ijtakes place. Notice that Eq. 21 can be written as

ijvijij G2 += ............................................(24)

Governing Equations of the Stress-Deformation Model inTerms of Displacements and Pore Pressure. The stress-deformation model can be written in terms of the incrementaldisplacements and incremental pore pressure by introducingEqs. 15 through 20 into the set of equations represented by Eq.24, and substituting the resulting equations into Eqs. 12through 14. This results in

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( )r r rz r r uG

r z r r +

+

+

+

00000 1

( ) ( ) ( )

+

+

+ uGr r

pr r

G2

1u

u

022

22=

u

r

Gu

r

Gr ...........................................(25)

( ) ( ) ( )

+

+

+ r uG

r u

r

G pr r 22

211u

( )0

31 =+

+

r u

r G

r uG

r r uG

r .............(26)

++

+

+

r r r z rz z rz z 0000 1

( ) ( ) ( ) 0 z

pu

z z u

GuG z =+

+

+

.......................................................................................(27)

In Eqs. 25 through 27, u is the incremental displacementvector given by

( )T z r uuuu ++= ......................................(28)

Initial Boundary Conditions. The solution of the governingequations requires the definition of fluid-flow andgeomechanical initial conditions.

Fluid-Flow Initial Condition. The initial pore pressure is afunction of the depth. Mathematically,

( ) z p ) z , y , x( p o= , 0t = ..............................................(29)

Geomechanical Initial Conditions. The simulation isinitialized with zero incremental displacements.Mathematically,

0u i = , ; z , ,r i = 0=t ..........................................(30)

Outer Boundary Conditions. Outer boundary conditions arerequired for both the fluid-flow and the stress-deformationequations.

Fluid-Flow Boundary Conditions. The model assumesconstant flow rate at the wellbore and no fluid-flow at theouter boundaries of the reservoir. Mathematically,

( ) t tanConskAqr p t r r w == = ................................(31)

( ) 0= = er r r p .............................................................(32)

( ) 0 z p topand bottom z = = .............................................(33)

Stress-Deformation Boundary Conditions . Zeroincremental displacements are considered at the outer

boundaries, except at the reservoir top which is treated as aconstant stress boundary and equal to the overburden.Mathematically,

0 )uew r ,r r i

== , for z , ,r i = ...................................(34)

0 )u bottomi = , for z , ,r i = ....................................(35)

At the reservoir top 9, 11, 12 ,

( ) ++

+= r r r r r

1r 1

1

0 z zr = ...............................................................(36)

0211 =+

++= z z r r r r

...................................................................................... (37)

0r 1

r 1

1 z r rz z z z =+

++=

...................................................................................... (38)

In Eqs. 36 through 38, i , z , , r i = , are thecomponents of the incremental total stress tensor applied onthe top boundary, and i are the components of a unit vector

normal to the surface and pointing outward..Periodic boundary conditions are considered in the

direction (the dependent variables and reservoir physical properties at 0= and 2= are identical 9).

Numerical Analysis ApproachThe governing equations (Eqs. 9 through 11and Eqs. 25through 38) are strongly nonlinear and, therefore, their solution requires the application of a numerical approach.

In this study, the governing equations are discretized usingthe finite difference method. The physical system isrepresented in cylindrical coordinates and is discretized bymeans of a point-distributed grid. All equations are solved by

using second-order approximations in the r -, - and z directions (the grid notations in these directions are given by i j and k , respectively). A fully implicit time marching procedure is adopted to assure maximum numerical stability.

The finite difference approximation of Eq. 9 and Eqs. 25through 27 can be expressed as seven-point stencils of theform

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1n1k , j ,ik , j ,i X

000

0 B0

000+

+1nk , j ,i

k , j ,i

k , j ,ik , j ,ik , j ,i

k , j ,i

X

0S 0

E C W

0 N 0+

+1n

1k , j ,ik , j ,i X 0000T 0

000+

+ = F i,j,k ..................................(39)

In Eq. 39, the stencil elements Bi j k , , , N i j k , , , W i j k , , ,

C i j k , , , E i j k , , , S i j k , , and T i j k , , represent the matrix

coefficients of the difference equation for , X 1n 1k , j ,i+

, X 1n

k ,1 j ,i++

, X 1n k , j ,1i+

, X 1nk , j ,i

+ , X 1n k , j ,1i+

+ , X 1n

k ,1 j ,i+ and , X

1n1k , j ,i

++

respectively. The variables X n+1 represent the unknown valuesof one of the dependent variables (pore pressure or one of theincremental displacements) in the discretized equation at node( )k , j ,i . The stencil coefficients and the values F i j k , , arefunctions of the unknown variables, X n+1 .

Discretization of the Pore Pressure Equation. Following thenotation in Eq. 39, the finite difference approximation of Eq. 9can be written as

1,,,,

1,,1,,

1,1,,,

11,,,,

++

+

+ +++ n k jik jin k jik jin k jik jin k jik ji P C P W P S P B

k jin

k jik jin

k jik jin

k jik ji F P T P N P E ..1

1,,,,1

,1,,,1

,,1,, =+++ + +++++ ....(40)

The stencil coefficients k ji B ,, , k jiS ,, , etc. in Eq. 40 are

defined in Appendix A.1

,,

+nk ji P represents the discrete value of

the pore pressure at node ( )k , j ,i and time level n+1.

Discretization of the Stress-Deformation Model Equations.T he finite difference approximations of Eqs. 25 through 27can be written as

++ + + 1n k ,1 j ,i ,mk , j ,i ,m1n 1k , j ,i ,mk , j ,i ,m U S U B ++ ++ 1n k , j ,i ,mk , j ,i ,m1n k , j ,1i ,mk , j ,i ,m U C U W

1nk ,1 j ,i ,mk , j ,i ,m

1nk , j ,1i ,mk , j ,i ,m U N U E

++

++ +

k jimn

k jimk jim F U T ,,,1

1,,,,,, =+ + + .....................................(41)

In Eq. 41, m = r, , z , depending if it refers to Eq. 25, Eq.

26 or Eq. 27, respectively. 1 ,,,+ n k jimU represents the discrete

value of the incremental displacements at node ( )k , j ,i andtime level n+1. The stencil coefficients k , j ,im B , k , j ,imS , etc.

in Eq. 41 are defined in Appendix B.

Grid Generation in the r -Direction. To have a better gridresolution near the wellbore, the node positions in the r direction are located according to the following geometric

progression 13, 14 (Fig. 2 ):

( ) ( ) 11

1=+

N

weii r r r r .....................................................(42)

In Eq. 42, ir and 1+ir are the radial positions of the nodes

( )k , j ,i and ( )k , j ,1i + , respectively; N is the number of nodes in the r -direction; er and wr are the outer and wellboreradius, respectively.

A volume control is assigned to each node. The lower andupper radial limits of the volume controls are calculated byapplication of a logarithmic average as described by thefollowing equations 13, 14 (Fig . 2):

( )iiii

i r r r r

r 1

12/1 ln +

++

= ........................................................ (43)

( )11

2/1 ln

=

ii

iii r r

r r r .......................................................(44)

Discretization of the Porosity Equation. The discretizationof Eq. 11 is given by

( )( ) ( ( ) ( )[ ]n , j ,in k , j ,i1n , j ,i1n k , j ,i1n k , j ,bci

n , j ,i

nk , j ,i

1n , j ,i

1nk , j ,ir

1nk , j ,bci

nk , j ,i1n

k , j ,i p pc1

p pcc

= ++++++

+

...................................................................................... (45)

Numerical Solution Procedure. Given the nonlinear behavior of the governing equations and their discretized forms, their solution must be found iteratively. The numerical procedureused in this study involves a Picard-like, block Gauss-Seideliteration where the nonlinear terms are updated as soon as newvalues for one of the dependent variables are computed. Theiterative sequence that includes evaluation of nonlinearities isas follows:

1. An initial guess is assigned to the pore pressure andincremental displacements at time level 1n + .

2. Eq. 41 is applied to the incremental displacement in

the radial direction ( )1n k , j ,ir 1n k , j ,i ,m U U ++ = . Thus, the nonlinear coefficients and vector F i j k , , are updated and Eq. 41 is solved

for 1n k , j ,ir U + .

3. The strains and effective stresses are calculated byapplying discretized forms of Eqs. 15 through 20 and Eq. 24.

In this calculations, the latest estimated values for 1n k , j ,ir U +

are used.

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4. Steps 2 and 3 are repeated for 1n k , j ,iU +

instead of

1nk , j ,ir U

+ .

5. Steps 2 and 3 are repeated using the latest estimated

values of 1n k , j ,iU +

and1nk , j ,ir U

+ to obtain an estimate for

1n

k , j ,i z U

+ .

6. Steps 2 and 3 are repeated using the latest estimated

values of 1n k , j ,iU +

,1nk , j ,ir U

+ and 1n k , j ,i z U + to obtain an

estimate for 1n k , j ,i P + . Notice that 1n k , j ,i P

+ is estimated by solving

Eq. 40.7. Steps 2 through 6 are repeated until convergence is

found (the incremental displacements and pore pressuredistributions are the same, within certain tolerance, after twoconsecutive iterations).

Once covergence is achieved, all dependent variables attime level n replaced by their latest calculated values. Theabove procedure is performed for calculation at a new time

level 1n + .

Example of ApplicationThis paper includes only an example of application of themodel developed in this study and presents a limiteddiscussion of results obtained from this example. There aretwo reasons for limiting the discussion of results. First, thelength of this paper would not allow an extensive discussionof several of important aspects withdrawn from the applicationof the model. Second, given the importance of the topic, theauthors are preparing a second paper devoted exclusively tothe formation damage behavior of stress-sensitive reservoirs.

Fig. 3 shows the permeability curve and Table 1 the

parameters used as input data. The reservoir consists of eightlayers. The two upper and lower layers exhibit low permeablity and porosity values to be consistent with whatreally takes place in the reservoir.

Production rate is constant during the first 21.6 hours of the well test. Then, it is decreased gradually becoming zeroafter a productin time of 25 hours.

Effect of Stress-Sensitivity on a Horner Plot . Fig. 4 shows aHorner plot for two different drawdown tests. The first testrefers to a stress-sensitive reservoir whose permeabilitychanges as described by the curve with base permeabilityequal to 45 md in Fig. 3 (in this study, the base permeability is

defined as the value obtained from the permeability curve at amean effective stress equal to zero). The conditions for thesecond test are the same as those used for the first test, exceptthat in the second test the permeability is kept constant andequal to the base permeability (45 md).

As shown in Fig. 4, the pressure drop at the wellbore of astress-senstive reservoir is different from the case in which the

permeability is considered to remain constant durinf the welltest. This difference is due to the fact that, in stress-sensitivereservoirs, the rock undergoes some deformation as the local

stress state varies with pressure drawdown. This deformationcauses formation damage in a region under compressive stresslocated around the wellbore where the pore pressure is alwaysminimum.

Fig. 5 shows the Horner plot for two build-up tests carriedout under the same conditions as the drawdown curves

presented in Fig. 4. For the build-up case, it is observed that

both curves tend to the same reservoir static pressure at highshut-in times. However, at early shut-in times both curves tendto separate, what confirms the presence of a formation damagecomponent due to rock deformation as observed in Fig. 4.

The results presented in Figs. 4 and 5 show that theapplication of conventional well test techniques to analyzewell test data from stress-sensitive reservoirs may yield tounder or over estimate the well formation damage.

Effect of Production Time on Formation Damage. Fig. 6shows the reduced permeability at different drawdown testtimes. The qualitative behavior of the curves presented inFig.6 shows that the permeability reduction is maximum near the wellbore. As the radial distance from the wellvoreincreases, the permeability tends to a higher constant value.Again, this behavior is due to the fact that the stress

perturbation inside the reservoir is less than in the wellboreneighborhood. As production time increases, the radius of investigation travels deeper inside the reservoir, what causes a

propagation of the formation damage region inside thereservoir. This indicates that for long production times, theformation damage caused by rock deformation affects not onlythe wellbore neighborhood, but also reservoir regions far fromthe wellbore. The impact of this type of formation damage onreservoir productivity, under different production scenarios, iscurrently under investigation.

Figs. 7 , 8 and 9 show the permeability reduction asfunction of the radial distance in the producing layer as well asin the upper and lower layers adjacent to the producing layer.The curve refer to three different flow times: 0.2 (Fig. 7), 11(Fig. 8) and 21 hours (Fig. 9).

After 0.2 hours, the permeability reduction in the wellboreneighborhood is greater in the producing layer than in theadjacent layers. However, at a radial distance of 2.5 feetapproximately, the permeability reduction is greater in theupper layer adjacent to producing layer than in the producinglayer itself. The permeability reduction is always less in thelower adjacent layer than in the producing layer. This behavior is caused by the superposition of two effects: (i) thecompaction caused by the overburden, which is stronger in the

upper layers due to the constant stress boundary condition, and(ii) the rock deformation cuased by fluid production, which isstronger in the producing layer and in the region close to thewellbore.

Given the pressure reduction, rock deformation is the predominant effect in the wellbore neighborhood of the producing layer. This is the reason why the permeabilityreduction close to the wellbore is greater in the producinglayer than in the upper and lower adjacent layers. As the radialdistance from the wellbore increases, the effect of rock

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deformation on formation damage decreases, because of thelower pressure drop, and compaction becomes the

predominant effect. This is the reason why the permeabilityreduction, in the region far from the wellbore, is greater in theupper layer than in the producing and lower layers.

Conclusions

This paper develops a numerical, fully coupled, fluid-flow/geomechanical model to perform well test analysis instress-sensitive reservoirs. The application of the model isillustrated through a simple example. On the basis of theresults from this study, the following general conclusions arederived:

1. The application of conventional well test techniquesto analyze well test data from stress-sensitive reservoirs mayyield to a wrong estimation of the well formation damage.

2. The permeability of a stress-senstive reservoir decreases with producing time. This permeability reduction iscaused by variations in the stress state due to a decrease in the

pore pressure. The permeability reduction is greater in theregion near the wellbore than inside the reservoir. However, as

production time increases, the damage region penetratesdeeper inside the reservoir.

3. Producing layers undergo a strong permeabilityreduction near the wellbore due to: (i) the rock deformationcaused by the pore pressure decline, and (ii) the reservoir compaction caused by the overburden. Far from the wellbore,the permeability reduction in the upper layers is greater than inthe producing layers. In this latter case, the predominantmechanism for permeablity reduction is the rock compactioncaused by the overburden. As production time increases, thiseffect penetrates deeper and deeper into the reservoir.

Nomenclature B = formation volume factor; also stencil element

representing the dependent variable coefficient of the cell below the cell in reference

c = compressibility, Lt 2/mC = stencil element representing the dependent variable

coefficient of the cell in reference E = s tencil element representing the dependent variable

coefficient of the cell east to the cell in referenceW = stencil element representing the dependent variable

coefficient of the cell west to the cell in reference F = stencil element representing the right-hand side

termG = shear modulus, m/Lt 2

i, j = integer denoting cell location in the r-, -direction,respectively

k = permeability, L 2; also, integer indicating celllocation in the z-direction

N = stencil element representing the dependent variablecoefficient of the cell north to the cell inreference

P = pressure (discrete approximation), m/Lt 2 q = voumetric flow rate, L 3/t

~q = mass rate per unit of bulk volum, m/L 3tr = radius, LS = stencil element representing the dependent variable

coefficient of the cell south to the cell inreference

t = time, tT = total stress acting on the top boundary, m/Lt 2; also ,

stencil element representing the dependentvariable coefficient of the cell above the cell inreference

u = displacement (continuous function), L; also,velocity, L/t

U = displacement (discrete approximation), LV = volume, L 3

W = stencil element representing the dependent variablecoefficient of the cell west to the cell in reference

r, z = distance, L X = discrete approximation of a dependent variable = Biots poroelastic constant, dimensionless

ij = Kroneckers delta

= increment = strain = Lames constant, m/Lt 2 = outward normal vector = porosity, fraction = angle, radians = viscosity, m/Lt; also component of the unit vector = density, m/L 3 = stress, m/Lt 2

= mean stress, m/Lt 2

Subscriptsb = bulk

bc = bulk volume, with mean stress changingi, j, k = cell location in the r-, -, z -direction, respectively

f = fluide = outer radius o = reference state

p = porousr = rock

s = solid v = volumetric t = totalw = wellbore

r, , z = r-, -, z -direction, repectively

Superscripts N = number of nodes in the r- direction = direction of the unit vector n = integer indicating time level 0 = initial state = effective

References1. Vairogs, et al .: Effect of Rock Stress on Gas Production From

Low-Permeability Reservoirs, JPT (Sept. 1971) 1161-67.

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2. Thomas, R.D. and Ward, D.C.: Effect of Overburden Pressureand Water Saturation on Gas Permeability of Tight SandstoneCores, JPT (Feb. 1972) 120-24.

3. Jones, F.O. and Owens, W.W.: A Laboratory Study of Low-Permeability Gas Sands, JPT (Sept. 1980) 1631-40.

4. Warpinski, N.R. and Teufel, L.W.: Determination of theEffective-Stress Law for Permeability and Deformation in Low-Permeability Rocks, paper SPE 20572 presented at the 1990

SPE Annual Technical Conference and Exhibition, NewOrleans, Sept. 23-26.

5. Holt, R.M.: Permeability Reduction Induced by a NonhydrostaticStress Field, SPEFE (Dec. 1990) 444-48.

6. Rhett, D.W. and Teufel, L.W.: Effect of Reservoir Stress Path onCompressibility and Permeability of Sandstones, paper SPE24756 presented at the 1992 SPE Annual Technical Conferenceand Exhibition, Washington, DC, October 4-7.

7. Morita, N. et. al .: Rock-Property Changes During Reservoir Compaction, SPEFE (Sept. 1992) 197-205.

8. Zimmerman, R.W.: Compressibility of Porous Rocks , J.Geophys. Res . (1986) 91 , 12765-77.

9. Alcalde, O.R. y Wills, A.: Anlisis de Pruebas de Presin enYacimientos Sensitivos a Esfuerzos y Deformaciones,Universidad Nacional de Colombia, Sede Medelln, Facultad deMinas. Tesis, 2001.

10. Fjaer, E., Holt, R.M., Horsrud, P., Raaen, A.M.I., and Risnes, R.:Petroleum Related Rocks Mechanics, Elsevier science

publishing company inc. 1992.11. Chou, Pei C., and Pagano, N.J.: Elasticity: Tensor, Dyadic and

Engineering Approaches, Dover publications, Inc., New York,1992.

12. Osorio, J.G., Chen, H.G., and Teufel, L.W.: NumericalSimulation of the Impact of Flow Induced GeomechanicalResponses on the Productivity of Stress Sensitive Reservoirs,

paper SPE 51929, presentated at the 1999 SPE Reservoir Simulation Symposium, Houston, Texas, february 14-17.

13. Settari, A., and Aziz, K.: Petroleum Reservoir Simulation,Elsevier applied science publishers, Londres, 1979.

14. Gmez, C.O.A., y Vanegas, R.: Simulacin de Pruebas de Flujo yRestauracin de Presin, Universidad Nacional de Colombia,Sede Medelln, Facultad de Minas. Tesis, 1992.

Appendix A- Stencil Coeficientes in Eq. 40.Finite-difference, point-distributed, discretization of Eq. 9yields the definition of the stencil coefficients in Eq. 40. Thesecoefficientes are as follows:

12/1,,,,

++=

nk jik ji T B .......................................................(A-1)

1,2/1,,,

+= n k jik ji T S .......................................................(A-2)

1,,2/1,,

+= n k jik ji T W ........................................................(A-3)

1,,2/1,,

++= n k jik ji T E ........................................................(A-4)

1,2/1,,,

++=

nk jik ji T N .....................................................(A-5)

12/1,,,,

+=

nk jik ji T T ...................................................... (A-6)

( 1 ,,2/11 ,,2/11 ,2/1,1 2/1,,,, +++++ +++= n k jin k jin k jin k jik ji T T T T C

+++ + +

++ k jik ji t f k ji

nk ji

nk ji cT T ,,,,,,

12/1,,

1,2/1, ............. (A-7)

k jik ji f f n

k jik jtik j fik jik ji Q pc F ,,,,,,,,,,,,,, +=

( ) ( ) ( ++ z z r r t k ji f k jbi uur ur V k ji 1

,,,, ,,

................................................................................ (A-8)

In Eqs. A-1 though A-8, the T terms are defines as:

( )k jik , j ,i

2k , j ,2 / 1i

2k , j ,2 / 1i jk ,2 / 1ik , j ,2 / 1ii

k , j ,2 / 1i F F r r 4

r r r F T

++++

=

................................................................................ (A-9)

( )k j1ik , j ,i

2k , j ,2 / 1i

2k , j ,2 / 1i jk ,2 / 1ik , j ,2 / 1ii

k , j ,2 / 1i F F r r 4

r r r F T

+

=

................................................................................ (A-10)

( )k j

2k , j ,i

2k , j ,2 / 1i

2k , j ,2 / 1ik ,2 / 1 j ,i

k ,2 / 1 j ,i F r 4

r r T

= +++

................................................................................ (A-11)

( )k 1 j

2k , j ,i

2

k , j ,2 / 1i

2

k , j ,2 / 1ik ,2 / 1 j ,ik ,2 / 1 j ,i F r 4

r r T

+ =

....... (A-12)

( ) jk

2k , j ,2 / 1i

2k , j ,2 / 1i2 / 1k , j ,i

2 / 1k , j ,i F z 4

r r T

= +++

....... (A-13)

( ) j1k

2k , j ,2 / 1i

2k , j ,2 / 1i2 / 1k , j ,i

2 / 1k , j ,i F z 4

r r T

+

=

....... (A-14)

The following definitions hold for Eqs. A-1 through A-14:

t V k jibk ji = ,,,,,

( ) ( ) t nnt = + 1

( )nn

mmm mm +

=

+

1

11 , m = r, , z y n = i, j, k .

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f f k =

( )11 += nnn mm F

For n = i, j, k and m = r, , z, respectively.

Appendix B- Stencil Coeficientes in Eq. 41.Finite-difference, point-distributed, discretization of Eq. 25yields the definition of the stencil coefficients in Eq. 41. Thesecoefficientes are as follows:

2/1,,1,,, 2 = k jik k jir GC B .........................................(B-1)

( ) k jik ji jk jir Gr C S ,2/1.2 ,,1,,, 2 = .............................(B-2)

k , j ,2 / 1i1ik , j ,2 / 1ik , j ,2 / 1ik , j ,i1ik , j ,i ,r C 2r Gr C 4W +=

k jik jii r F ,,2/1,,2/1 ...........................................(B-3)

=k , j ,i ,r C

+

+ ++ k , j ,2 / 1ik , j ,2 / 1i

k , j ,i

1ik , j ,2 / 1ik , j ,2 / 1i

k , j ,i

i r Gr C 4

r Gr

C 4

( ) ++ + k ,2 / 1 j ,i2 k , j ,i1 jk ,2 / 1 j.i2 k , j ,i j Gr C 2Gr C 2 +++ + 2 / 1k , j ,i1k 2 / 1k , j ,ik GC 2GC 2

k , j ,2 / 1i1ik , j ,2 / 1ii C 2C 2 + +

( ) ( ) )2 k , j ,ik , j ,ik , j ,2 / 1iik , j ,2 / 1ik , j ,2 / 1ii r G2r F r F ++ ++ ....................................................................................(B-4)

) ++= +++ k , j ,2 / 1iik , j ,2 / 1ik , j ,2 / 11k , j ,iik , j ,i ,r C 2r Gr C 4 E ) k , j ,2 / 1ik , j ,2 / 1ii r F ++ ..............................................(B-5)

( ) k jik ji jk jir Gr C N ,2/1.2 ,,,,, 2 += ...............................(B-5)

2/1,,,,, 2 += k jik k jir GC T .............................................(B-6)

F k , j ,i ,r =

+

+ + ++++ U r

F F G

r

F F 1nk ,1 j ,1i ,k , j ,2 / 1i

k , j ,i

jik ,2 / 1 j ,i

k , j ,i

ji

+

+ + ++ 1n k ,1 j ,1i ,k , j ,2 / 1i

k , j ,i

jik ,2 / 1 j ,i

k , j ,i

jiU

r

F F G

r

F F

+ + ++

+

1nk ,1 j ,1i ,k , j ,2 / 1i

k , j ,2 / 1i

jik ,2 / 1 j ,i

k , j ,i

jiU

r

F F G

r

F F

+ +

1nk ,1 j ,1i ,k , j ,2 / 1i

k , j ,2 / 1i

jik ,2 / 1 j ,i

k , j ,i

jiU

r

F F G

r

F F

+++ 1n k , j ,1i ,k ,2 / 1 j ,i

k , j ,i

jik ,2 / 1 j ,i

k , j ,i

jiU G

r

F F G

r

F F

++ k , j ,2 / 1ik , j ,2 / 1i

jik , j ,2 / 1i

k , j ,2 / 1i

ji

r

F F

r

F F

+

+ ++ 1n k ,1 j ,i ,k , j.i2

k , j ,i

jk ,2 / 1 j.i2

k , j ,i

jU G

r

F 2G

r

F

+

++ 1n k , j ,1i ,k ,2 / 1 j ,i

k , j ,i

jik ,2 / 1 j ,i

k , j ,i

jiU G

r

F F G

r

F F

+

+k , j ,2 / 1i

k , j ,2 / 1i

jik , j ,2 / 1i

k , j ,2 / 1i

ji

r

F F

r

F F

+

+ 1n k ,1 j ,i ,k , j.i

2 k , j ,i

jk ,2 / 1 j.i

2 k , j ,i

jU G

r

F 2G

r

F

++ 1n k , j ,i ,k ,2 / 1 j ,i2

k , j ,i

jk ,2 / 1 j.i2

k , j ,i

jU G

r

F G

r

F

( ) ++ + ++++ 1n 1k , j ,1i , z k , j ,2 / 1ik i2 / 1k , j ,ik i U F F G F F ( ) ++ + ++ 1n 1k , j ,1i , z k , j ,2 / 1ik i2 / 1k , j ,ik i U F F G F F ( ) + + ++ 1n 1k , j ,1i , z k , j ,2 / 1ik i2 / 1k , j ,ik i U F F G F F ( ) ++ + 1n 1k , j ,1i , z k , j ,2 / 1ik i2 / 1k , j ,ik i U F F G F F ( ) ++ +++ 1n k , j ,1i , z 2 / 1k , j ,ik i2 / 1k , j ,ik i U G F F G F F

( ) ++ 1n k , j ,1i , z 2 / 1k , j ,ik i2 / 1k , j ,ik i U G F F G F F ( ) + + ++ 1n 1k , j ,i , z k , j ,2 / 1ik ik , j ,2 / 1ik i U F F F F ( ) + + + 1n 1k , j ,i , z k , j ,2 / 1ik ik , j ,2 / 1ik i U F F F F

+ ++++ 1n k , j ,1ik , j ,2 / 1ii1n k , j ,1ik , j ,2 / 1ii P F P F

( ) + ++ 1n k , j ,ik , j ,2 / 1iik , j ,2 / 1ii P F F

( ) ( )++ ++ n k ,1 j ,ir n k ,1 j ,ir k , j ,i

jnk , j ,1ir

nk , j ,1ir i r

F F

( ) ( )n k , j ,in k , j ,ir k , j ,i

n1k , j ,irz

n1k , j ,irz k r

1 F + + ....(B-7)

The following operators hold in Eqs. B-1 through B-7:

( )[ ]11 += nnnn mmmC

( )[ ]111 1 += nnnn mmmC

For n = i, j, k and m = r, , z, respectively.

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The stencil coefficients for the discretized forms of Eqs. 26and 27 ( m = y z ) are similar to the coefficients writtes above(the detailed equations are provided in reference 9).

SI Metric Conversion Factorscp 1.0* E-03 = Pa.sft 3.048* E-01 = m

ft3 2.831 685 E-02 = m 3 md 9.869 233 E-04 = m2 psi 6.894 757 E+00 = kPa

R R/1.8 = K Conversion factor is exact.

Table 1-Parameters used as input data

Outer raidius, ft 1500Wellbore radius, ft 0.198Reservoir thickness, (feet) 20

API gravity 30

Gas gravity (air=1)0.8

Fluid compressibility, psi -1 1.0E-5

Initial pressure at the bottom layer, psi 4500

Initial stress at the top layer, psi 8000

Initial stress anisotropy state V H = Total compressibility, psi -1 3.0E-6

Rock compresibility, psi -1 1.0E-7

Poisson ratio 0.2

Initial porosity, fraction 0.150

0

20

40

60

80

10 0

12 0

0 1 00 0 2 00 0 30 0 0 4 00 0 50 00 60 00

Mean Effect ive s t ress (Ps i )

R e l a t

i v e

P e r m e a

b i l i t y

Base perm eabi l i ty= 3 ,86

Base perme abi l ity= 45,0 (md)

Base perme abi l ity= 632,0 (md)

Fig. 3 Permeability curves used as input data.

Fig. 4 Horner plot (drawdown test).

Fig. 5 Horner plot (build-up test).

l m-1/2 l + l m-1 l m l m+1

k n

k n-1/2 k n-1

k +

k +

m,n m+1,n m-1 n

+

m n-1k n-1

l m l m-1

Fig. 1 - Point-distributed grid.

Fig. 2 - Horizontal grid refinement.

r i-1/2

r i+1/2

r i

r e

Base Perm eability = 45.0 md

3 7 5 0

3 8 0 0

3 8 5 0

3 9 0 0

3 9 5 0

4 0 0 0

4 0 5 0

4 1 0 0

4 1 5 0

0.1 1 10 100Time (hours)

W e l

l b o r e

P r e s s u r e

( p s i

) Variable permeabilityConstant permeability

Base permeability = 45.0 md

4310

4315

4320

4325

4330

4335

110(Tp+DT) / DT

W e l

l b o r e

P r e s s u r e

( p s i

)

Variable permeability

Constant permeability

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Fig. 6 Permeability vs. radial distance (producing layer).

Fig. 7 Permeability vs. radial distance, t=0.2 hours (producingand adjacent layers).

B a s e p e r m e a b i l i ty = 4 5 , 0 m d( Ti m e = 11 h o u r s )

2 . 4 5 E + 0 1

2 . 4 6 E + 0 1

2 . 4 7 E + 0 1

2 . 4 8 E + 0 1

2 . 4 9 E + 0 1

0 1 2 3 4 5 6 7 8 9 1 0 11 12 13 14 15

Rad ia l d i s t ance ( f t )

P e r m e a

b i l i t y ( m d )

P e r m e a b i l i ty o f l a y e r = 4P e r m e a b i l i ty o f l a y e r = 5

P e r m e a b i l i ty o f l a y e r = 6

Fig. 8 Permeability vs. radial distance, t=11 hours (producingand adjacent layers).

Fig 9 Permeability vs. radial distance, t=21 hours (producingand adjacent layers).

Base Permeability = 45.0md

2.45E+01

2.46E+01

2.47E+01

2.48E+01

2.49E+01

0 10 20 30 40 50 60 70 80 90 100

Radial distance (ft)

P e r m e a

b i l i t y ( m

d )

Time = 0.2 hoursTime = 11 horas

Time = 21 hours

B a s e p e r m e a b i l it y = 4 5 m d( Ti m e = 0 , 2 h o u r s )

2 . 4 6 E + 0 1

2 . 4 7 E + 0 1

2 . 4 8 E + 0 1

2 . 4 9 E + 0 1

0 1 2 3 4 5 6 7 8 9 10

Radia l d is tance ( f t )

P e r m e a

b i l i t y

( m d )

Perm eab i l i ty o f l aye r = 4Perm eab i l i ty o f l aye r = 5Perm eab i l i ty o f l aye r = 6

Base permeabil i ty = 45,0 md(Time = 21 hours )

2.45E+01

2.46E+01

2.47E+01

2.48E+01

2.49E+01

0 1 2 3 4 5 6 7 8 9 1 0 11 12 13 14 15

Radial dis tance (f t )

P e r m e a

b i l i t y ( m

d )

Permeab i l i ty o f l aye r = 4Permeab i l i ty o f l aye r = 5Permeab i l i ty o f l aye r = 6

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