constitutive laws for geomaterials

8/11/2019 Constitutive Laws for Geomaterials

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Oil & Gas Science and Technology Rev. IFP, Vol. 54 (1999), No. 6, pp. 759-771Copyright 1999, ditions Technip

Constitutive Laws for GeomaterialsE. Papamichos1

1 SINTEF Petroleum Research, N-7465 Trondheim - Norway e-mail: [email protected]

Rsum Lois constitutives pour les gomatriaux Il existe un nombre important de loisconstitutives permettant d'approcher le comportement des gomatriaux sous diverses contrainteconditions aux limites et divers chemins de chargement, pressions de pore, tempratures, etcLlaboration de lois constitutives suit deux tendances concurrentes : la tendance laborer une lounifie dcrivant le comportement du matriau sous les conditions les plus gnrales, et le besoin de lo justifi par une utilisation efficace dans la pratique. La dernire tendance inclut la fois la possibilidune calibration par des essais sur des chantillons disponibles, et la convivialit de la loi. des findingnierie, la seconde tendance domine, lingnieur doit valuer le problme et slectionner le modle plus appropri pour dcrire le phnomne dominant observ. Dans cet article, un examen des diverslois constitutives pour les gomatriaux, et en particulier pour leurs applications, est prsent. Diveexemples de problmes concernant lingnierie et les gomatriaux appliqus au ptrole sont utiliss ade dmontrer que la loi doit tre aussi simple que possible, mais pas trop non plus.Mots-cls : loi constitutive, grs, craie, injection deau, endommagement.

Abstract Constitutive Laws for Geomaterials An abundance of constitutive laws exists toapproximate the behavior of geomaterials under various stresses, boundary conditions and loading

paths, pore pressures, temperatures and so on. The construction of constitutive laws is driven by twocompeting trends: the tendency for a unifying law describing the material behavior under the most general conditions, and the need for laws that can be used efficiently in practice. The latter incorporatesboth the possibility of calibration from available specimens and the user-friendliness of the law. Forengineering purposes, the second tendency dominates and the engineer scientist has to evaluate the

problem and select the most appropriate model to describe the dominant phenomena on hand. A reviewof various constitutive laws for geomaterials is presented with an emphasis on their application. Variousexamples of engineering problems and geomaterials with emphasis on petroleum applications are used to demonstrate that the law must be as simple as possible but not simpler.

Keywords: constitutive law, sandstone, chalk, water injection, core damage.

INTRODUCTION

The constitutive laws that are presented describe the behaviorof a geomaterial as a continuum. The perspective of thecontinuum material description can be appreciated in thefollowing quote by Truesdell and Noll (1965):

Widespread is the misconception that those who formulate continuum theories believe matter really is

continuous, denying the existence of molecules. This is not so. Continuum physics presumes nothing regarding thestructure of the matter. It confines itself to relations amonggross phenomena, neglecting the structure of a material ona smaller scale. Whether the continuum approach is

justif ied, in any particular case, is a matter, not for the philosophy or methodology of science, but for theexperimental test


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Oil & Gas Science and Technology Rev. IFP, Vol. 54 (1999), No. 6

Continuum models are indeed widely used today for bothresearch and engineering applications and provide usefulsolutions to technical problems. This of course is not to saythat micromechanical and discontinuum models should bedisregarded as purely scientific in nature. Without a doubtsuch models have shown a significant potential and in somecases they constitute the preferred way of modeling.

The literature on constitutive laws for geomaterials is richin models. The reason is that a wide variety of geomaterialswith often very dissimilar behaviors has to be appropriatelydescribed, such as the behavior of a sandstone, chalk, sand,limestone, shale, concrete, etc. In addition to the variousgeomaterials, the behavior of a particular material maydepend significantly on the application at hand, that is on theboundary or loading conditions, the pore pressures, the porefluids, the scale of the problem, the time periods involved,the temperature, etc.

There are two trends in constitutive modeling. The first isto make the model as general as possible in the expense of complexity and the other to make it as problem-specific aspossible with the advantage of simplicity. The characteristicsof a general constitutive model are: Various material types are included, such as elastic, plastic,

creep, pore pressure, temperature, water saturation, etc. Various problems can be solved. The model can be used in

various different problems such as wellbore stability, sandproduction, hydraulic fracturing, compaction, reservoirmodeling, etc.

The role of the engineer is limited. Extensive calibration is needed. The model requires the

input of many material parameters, which are oftendifficult to obtain, as in the case of core material whereenough test specimens are not available, or expensive toobtain, as large test series are required.

Low efficiency. The model is cumbersome to use and oftenthe results are difficult to interpret, as the importantphysical mechanisms are not always obvious.On the other hand the characteristics of a problem-specific

constitutive model are: Limited number of material types are included, such as

elastic or elastic-plastic or creep or pore pressure ortemperature or water saturation, etc.

Limited number of problems can be solved, as for examplewellbore stability or sand production or hydraulicfracturing or compaction or reservoir modeling, etc.

The role of the engineer is significant since evaluation of the geomaterial and problem at hand and selection of theappropriate model are required.

Limited calibration is needed, as only the absolutelynecessary material parameters are required.

Optimum efficiency is attained as non-important effectsare excluded.

For engineering problems the problem-specific constitutivmodels have obvious advantages. Examples of such problemwith emphasis on petroleum applications are given tdemonstrate how to select or develop a constitutive modbased on experimental and/or field data and observationSection 1 describes a sandstone model for open hole stabiliand sand production problems, Section 2 a chalk model fo

reservoir compaction with water saturation effects, anSection 3 a sandstone model for core damage effects. Finalwe present the concluding remarks.

1 SANDSTONE MODEL FOR OPEN HOLE STABILI AND SAND PRODUCTION

Open hole stability and sand production problems are ofteencountered in sandstone reservoirs during hydrocarboproduction. The reservoir rocks are usually soft, weaksandstones of considerable porosity in the range of 10-30%In this problem, the rock is exposed to relatively low meastresses by high deviatoric stresses due to the stresconcentration around the hole.

A nonlinear elastic-plastic model has been developed tdescribe the near well behavior of these sandstones (Sulemet al., 1999). In brief, the model has an elasticity, which inonlinearly stress-dependent, and a plasticity based on thflow theory. The plasticity is related to a linear MohrCoulomb yield functionF with friction hardening andcohesion softening. The plastic dilatancy is described by linear Mohr-Coulomb plastic potential function, differenfrom the yield surface, which makes the flow rule no

associative. This allows for the description of bothcompactive and dilatant behaviors under shear straining. Othe other hand, the appropriate plastic behavior undehydrostatic straining has not been incorporated in the presemodel as plastic hydrostatic straining is not usuallysignificant around wellbores and perforations. Throughouthis section compression is taken negative, as it is morconvenient for calculations. Tests results and modepredictions are given with compression positive as moroften used in rock and soil mechanics.

1.1 Model Description

The development of the model is based on behavior of sofweak sandstones as observed in triaxial compression anextension experiments (Papamichoset al., 1999), which arerelative experiments for obtaining the deviatoric behavior a material. For nonlinear materials, the constitutive relationare formulated in the incremental form which, for timeindependent behavior, is equivalent to the rate formemployed in the following. A dot over a quantity denotes thrate (or increment) of the quantity.

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E Papamichos / Constitutive Laws for Geomaterials

The rate form of the constitutive stress-strain relationshipscan be written as:

(1)

where and are the stress and strain rates, respec-tively, and the elastic-plastic tangent stiffness tensor,

which can be written as:(2)

Figure 1 shows triaxial-compression test results on humidRed Wildmoor sandstone at three confining stresses. Startingfrom the elasticity of the material, the load-unload cyclesshow that the elasticity is nonlinear and depends on thestress. In order to keep the model as simple as possible, it isassumed that the Poissons ratio is constant and thus theelastic shear modulusG must be a function of the stress, thatis G = G(ij ). The reversibility and path independency of theelastic strain energy requires that the stresses are obtainedfrom the strain energy functionW as (Chen and Han, 1988):

(3)

With these assumptions and requirements, the elasticshear modulusG must be a function of the equivalent stresse, defined in terms of the isotropic (mean) stress p and theshear-stress intensity as:

(4)

The stress invariants p and are defined as:(5)

where I 1 is the first stress invariant and J 2, the secondeviatoric stress invariant. The dependency ofG on thequivalent stresse, is described by the material function

(6

where the constantsG0 to G3, Gpeak, and for the variomaterials are obtained by fitting this function texperimental data. According to Equation (6), the shear modulusG increases from the initial valueG = G0 azero stress toG = Gpeak at and remains constthereafter. The elastic tangent stiffness tensor mabe written as:

(7

With respect to plasticity, the standard flow theory iwhere the plastic tangent stiffness tensor is expres

(8

withF andQ being the yield and plastic potential funcrespectively. The hardening modulush is given by:

(9h F C Q Q F g F q qgij ijkle

kls p p= + dd dd

C h

C Q F

C ijkl p

ijmne

mn st stkle= 1

C ijkl p

GG Gil jk ij kl

e

e

e e+ + +( ) 2 1 2

d dd d

C Gijkle

ij kl ik jl il jk =

+ +21 2

C ijkle

e e= peak

epeak

G for

for

= ( ) =+

+( )+

>G

GG G

G

Ge

e e

ee e

e e

02 3

11peak

peakpeak

p I J = =1 23,

e = ( ) + +( )3 1 2 2 12 2 p

ij ij

=

W

C C C ijklep

ijkle

ijkl p=

C ijklep

ij ij

ij ijklep kl= C

761

0

5

10

15

20

25

30

35

40

0 5 10 15 20 25Axial strain (x 10 -3)

A x i a l s t r e s s

( M P a

)

Red Wildmoor

Confining stress 6 MPa

Confining stress 2.5 MPa


a

-8

-6

-4

-2

0

2

4

V o

l u m e

t r i c s t r a

i n ( x 1 0 - 3 )

Confining stress 6 MPa



Axial strain (x 10 -3)

0 5 10 15 20 25

b

Figure 1Triaxial compression test results on humid Red Wildmoor sandstone. a: axial stress, and b: volumetric strain versus axial strain confining stresses (Papamichoset al., 1999).


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and the following definition of the switch functionexpressing the plastic loading criterion:

(10)

Figure 2 shows the peak axial stress during triaxialcompression at various confining stresses, which gives theshape of yield surface at peak stress. The yield surface isapproximated with a straight line in order not to includeadditional nonlinearities in the model, such as for example apressure dependency of the strength parameters. At higherconfining stresses the curvature of the yield surface may belarge enough, such that the exclusion of pressure-dependenteffects on the strength parameters may not be possible. Forthe selection of the yield surface, reduced triaxial extensiontest data suggest that the material follows an intermediatebehavior between the Mohr-Coulomb and the Drucker-

Prager yield surfaces (Papamichoset al., 1999). Otheranalyses and comparisons between model predictions andtest results suggest that the Mohr-Coulomb model is moreappropriate for geomaterials. The Mohr-Coulomb is a modelwidely used in geomechanics and a lot of experience andstrength parameters, such as cohesion and friction angle, areavailable.

Figure 1a shows that the material behaves plasticallyalmost from the start of straining as demonstrated by theresidual plastic strains during the unload-reload cycles.During the stress increase, the material hardens with theaccumulated plastic shear-strain intensityg p, and after the

peak stress is reached, the material softens. The yieldF andplastic potentialQ surfaces are then functions of the stress

Figure 2Triaxial compression test results on humid Red Wildmoorsandstone. Peak axial stress versus confining stress andstraight line fit to the data (Papamichoset al., 1999).

ij and the accumulated plastic shear-strain intensityg p. Theuse of nonassociativity where the yield and plastic potentifunctions do not coincide is necessary in soft weaksandstones. The experimental results in Figure 1b show ththe material initially compacts and then dilates as the peastress is approached. With associative behavior, dilation always predicted. For the Mohr-Coulomb criterion,F andQ

can be written in terms of the stress invariants p, , and, as:

(11)

where the stress invariant is called the Lode angle anddescribes the relative magnitude of the intermediate principstress. It is defined through the second J 2 and third J 3deviatoric stress invariants as:

(12)

The parameterq = q(g p) is the tension cut-off, and = (g p) and = (g p) are the mobilized friction anddilatancy coefficients, respectively. These are related to thmobilized friction anglem, dilation angle m, and cohesionc through the expressions:

(13)

The parameterQs in Equation (9) is related to the plasticpotential surfaceQ and it is given as a function of the stressinvariants J 2, J 3 as:

(14)

where:

(15)

Finally, the plastic strain-rates are given by the flowrules:

(16)

ij p

ij ij ijkle

klQ F

C = , =1h

ij p

BQ

J

BQ

J

Q

Q

12

23

2

1 3 33

34 3 634 3

3

0

= =

+ + ( )

+

= =+

cos tan tan tan tan

sin coscos

2

if =

if =6

32

if =6

Q J B J B B J B

s

Q Q Q Q

= + +2 33

2 1 3 1 2 22

22 2

= = =sin , sin , cosm m mc q

= 1

3

3 3

23

23 2a

J

J sin

F F g q p

Q Q g p

ij p

ij p

= ( ) = ( ) =

= ( ) = +

, cos sin

, cos sin3

0

3

0

10

20

30

40

50

60

0 2 4 6 8 10 12Confining stress (MPa)

A x i a

l s t r e s s

( M P a

)

Red Wildmoor sandstone

1

1 0 0

0 0 0 0=

= >

< =

, for and

for or and

F F

C

F F F

C

ij ijkle

kl

ij ijkle

kl

,

1

762


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which result into the following expression for the plasticshear-strain intensity rate :

(17)For the elastic strain rates it holds that:

(18)

The model requires the calibration of the materialfunctionsq = q(g p), = (g p) and = (g p). For thesesandstones, a model with both hardening and softening withplastic shear strain was used, as illustrated in Figure 3.Hardening was modeled by an increasing friction coefficient,which reaches its maximum value at plastic shear strain

and remains constant thereafter. Softening wasmodeled by a tension cut-offq, which starts to decrease fromits initial valueq0 after the friction coefficient has reached itspeak value, and thus after the plastic shear straing p hasreached the value . In particular, the following functionswere calibrated for each material:

(19)

where the constantsq0, q1, , f 0 to f 3, f peak, andd 1 to d 3,d peak are obtained by fitting the functions (19) on experi-mental data.

Figure 3Motion of the Mohr-Coulomb yield surface in stress-space:(0-1) friction-hardening phase, (1-2) tension-softening phase(Sulemet al., 1999).

The model can be calibrated through a minimum triaxial compression tests with load-unload cycles at vstress levels. Table 1 lists values of the various eand plastic material constants for humid Red Wildsandstone.

TABLE 1

Humid Red Wildmoor sandstone: elastic and plastic material con

[ ] 0.2 f 0 [ ] 0.067148 q0 [MPa] 3.8

G0 [MPa] 349.17 f 1 [ ] 109.88 q1 [MPa] 25.

G1 [MPa-1] 0.042907 f 2 [ ] 273.75

G2 [ - ] 250.32 f 3 [ ] 8946.6 d 1 [ ] 5.83

G3 [MPa-1] 0.0 f peak[ ] 0.94523 d 2 [ ] 0.7863

Gpeak[MPa] 6183.2 [ ] 0.0099069d 3 [ ] 90.45

[MPa] d peak[ - ] 0.9452

1.2 Calculation ExamplesThe back analysis of the triaxial compression and reextension test results is shown in Figure 4. The compwith the experimental results for both test types isatisfactory despite the complexity of the soft sandstone behavior. The model was also used to predresponse of Red Wildmoor sandstone hollow cylloaded isotropically (Papamichoset al., 1996). In these tea thick-wall hollow cylinder was loaded externally, whinternal hole remained unsupported. The external radiawas equal to the axial stress. The internal hole and edeformations of the cylinder were measured. In Fiexperimental results for the external radial stress versinternal and external tangential strains are compared wmodel predictions. The nonlinear elastic-plastic predicts very well the hollow cylinder response up point of failure of the inner hole. This is expected sincthis point the continuum description of the material cea

The point of initial failure is predicted througbifurcation theory and continua with microstructuaddition to the nonlinear elastic-plastic model, predwere made using a simpler model where the non

elasticity was suppressed and an average value of shear modulus was used. It can be seen in Figure 5 tlinear elastic-plastic model predictions are ratherwhich illustrates the need for a nonlinear elasticity in this case.

For weakly cemented, unconsolidated sandstonconstitutive law may need to be enriched with addfeatures to model adequately the near well behavthe rock. In the presented model, a linear Mohr-Comodel was adopted for relatively low mean stress

epeak

gpeak p

p

T

q q 0

c 0

c p

c (0)

(1)

(2)1f 0

f p 1

g ppeak

q q gq q g g

f f

f f

f f g g

f gg g

f g g

d d

d d g d

p

p p

p p

p p p

p p

p

= ( ) >

=+

=+

+( )+

>

=+

=

+ +

0

0 1

02 3

1

2 3

32 3

1

32 3

2

for gg for g

, for

for

, d

p

p p

peak

peak peak

peak

peak peak

11

11g

d gg g

d g g

p

p p p

p p

( )+( )

>

for

for

peak

peak peak

g ppeak

g g p p= peak

ij e ij ij p=

ij p g Q p s=

g p

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unconsolidated sandstones, however, a curved yield surfacemay be required even at low mean stresses. This may bemodeled with a pressure-sensitive friction angle. Theaddition of a pressure cap may also be necessary. Anotherobservation in these sandstones is their volumetric responseupon shearing. Figure 6 shows the volumetric response of anunconsolidated reservoir sandstone tested in triaxialcompression at three confining stresses. The experimentalresults clearly indicate a pressure-sensitive dilatancy, whichmay be necessary to be included in the constitutive law.

2 CHALK MODEL FOR RESERVOIR COMPACTIONDUE TO WATER INJECTION

The mechanical behavior of weak, high-porosity chalks important in many aspects of oil production such as reservocompaction and solids production. Among several factoaffecting the mechanical behavior of such rocks is the relativfluid saturation (gas/oil/water), which has an impact on thcapillary forces between the grains. Capillarity arises from fluid property known as surface tension, a phenomeno

764

-0.015 0.000 0.015 0.030Radial strain Axial strain

0

20

40

60

10.3 MPa

6.9 MPa3.5 MPa

1.4 MPa0.4 MPa

Simulation of triaxial compression tests(complete model)

Data Computed

A x

i a l s t r e s s

( M P a

)

a-0.015 0.000 0.015

0

20

40

60

Radial strain Axial strain

Simulation of triaxialextension tests

(complete model)DataComputed

A x

i a l s t r e s s

( M P a

)

60 MPa52.5 MPa

45 MPa

b

Figure 4Back analysis of a: triaxial compression, and b: reduced triaxial extension test results (Sulemet al., 1999).

Figure 5Hollow cylinder stability. Nonlinear elastic-plastic versus linear elastic-plastic model simulations and comparison with experimental data.

0

5

10

15

20

25

30

35

0 0.005 0.01 0.015 0.02 0.025 0.03

Internal tangential strain

E x t e r n a

l r a

d i a l s t r e s s

( M P a

)

SimulationLinear elastic-plasticsimulationTest 21Cav04

Initial failure

a

0

5

10

15

20

25

30

0 0.0005 0.001 0.0015 0.002 0.0025

External tangential strain

E x t e r n a

l r a

d i a l s t r e s s

( M P a

)SimulationLinear elastic-plasticsimulationTest 21Cav04

b


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Figure 6Triaxial compression test results on an unconsolidatedreservoir sandstone. Volumetric strain versus axial strain atthree confining stresses, showing pressure-sensitivedilatancy.

that occurs at the interface between different materials. Forrocks and soils it occurs between surfaces of water (normallythe wetting phase), mineral grains, and air, gas or oil(normally the nonwetting phase). At low degrees of saturation, water is bound in small pores and in the narrowspaces between grains forming capillary menisci. Lowdegrees of water saturation are often representative of theinitial conditions in oil or gas reservoirs. It is noted that forsome rocks, bridges of connate water are known to be stablefor temperatures up to 300C, posing problems in defining adry reference state. The surface tension at the boundarybetween the water and the air in the adjoining voids pulls thegrains together with a force known as contact pressure. Thefrictional resistance produced by the contact pressure has thesame effect as if the grains were held with a certain amount of cohesion. At higher degrees of saturation, the voids betweenthe grains are filled with water, the surface tension stronglydecreases and the contact pressure vanishes. The magnitudeof the capillary suction can reach the order of several MPa,depending on the radius of the capillary menisci, which is afunction of the pore size. Another characteristic of chalks istheir collapsible behavior. Chalks have a high porosity

because they comprise of coccoliths, that is, skeletons of marine animals. In some chalks, it is found that capillarymenisci formed by connate water are the only cohesionalforce that holds together the coccoliths. Upon elimination of the cohesional forces between the coccoliths, the chalkstructure collapses with a significant reduction in porosity.Similar collapsible materials are the bulking structuresobserved in unconsolidated sands and sandstones.

An elastic-plastic model with water effects was developedto model the behavior of partially water-saturated chalks

(Papamichoset al. , 1997). The model is based on expmental evidence from tests on Pietra Leccese chalincorporates Bishops effective stress principle focapillary pressure, a water-saturation-dependent estiffness, and a Mohr-Coulomb yield surface with a prcap. The cohesion and pressure cap parameters in thsurface are water-saturation-dependent to model the ca

effect on the strength parameters. Compression is negative in this section.

2.1 Experimental Evidence

The model is based on experimental evidence from tand hydrostatic compression tests on Pietra Leccese cvarious degrees of water saturation (Papamichoset al., 1997)The Pietra Leccese chalk is a high-porosity outcrocalcarenite from the south of Italy. Figure 7 shows ucompression test results for specimens at various saturations. The results show that the uniaxial compstrength (UCS) decreases substantially with increasingsaturation. Load-unload cycles in these tests showed tthis material, the elasticity does not change significantthe applied axial or confining stress. However, a signdecrease in the Youngs modulus with increasing saturation was observed. The Poissons ratio on thehand did not appear to be influenced significantly saturation. From the test data, an average elastic Yomodulus E for each saturation, and a constant avePoissons ratio for all saturations were calculated.

In Figure 8, the peak axial stress obtained at uniaxtriaxial compression tests is plotted versus the constress and saturation. From this plot and assuming aMohr-Coulomb yield criterion for the peak axial stre

Figure 7Uniaxial compression test results on Pietra Leccese chalkAxial stress versus axial strain for various water saturationsS .

-50

-45

-25

-20

-40

-15

-35

-10

-30

-5

0-0.004-0.003-0.002-0.0010

Axial strain

A x

i a l s

t r e s s

( M P a

)

Plt05 S = 0.016Plt33 S = 0.025Plt09 S = 0.047Plt30 S = 0.055Plt10 S = 1

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04Axial strain

V o l u m e

t r i c s t r a

i n

Atlas 04 Confining stress 10 MPa

Atlas 01 Confining stress 2 MPa

Atlas 05 Confining stress 0.5 MPa

0.060 0.02 0.04 0.08 0.1 0.12

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Figure 10

Water-injection tests on Pietra Leccese chalk at hydrostaticstress 10, 20 and 40 MPa. The results from the hydrostaticcompression tests without water injection are included forcomparison.

2.2 Model Description

In partially saturated media, the generalized effective stressis given as:

(20)

whereij is the total stress,S is the water saturation, pgasand pw the pore gas and pore water pressures, respectively, and

ij the Kronecker delta. By referring all pressures to theatmospheric pressure, pgas is zero when equal to theatmospheric, and Equation (20) can be written as:

(21)

where the pore-water capillary suction p cp, defined as p cp = pgas pw , is a function of the saturationS , that is pcp = pcp (S ). Plastic yield is associated with an isotropic yieldsurfaceF , the size of which depends on the accumulatedplastic strain parameter and the saturationS , that is

. The total strain rate is decomposed intotwo parts:

(22)

The elastic strain is related to the effective stressby a linear elastic law as:

(23)

where the elastic stiffness tensor is assumed to be afunction of the saturationS , that is . Time

differentiation of Equation (23) yields a relation betweffective stress rate and the elastic strain rate :

(24

The plastic strain rate is obtained by a nonassoflow rule as:

(25

where is the plastic potential functionis a scalar function. From the consistency condition

(26

and using Equations (22), (24) and (25), can be solv

(27

For plastic loading, is a nonnegative factor andh isalways positive. It then follows from Equation (27) tplastic loading criterion can be expressed in terms switch function defined as:

(28

The plastic strain rate can be calculated from the floEquation (25), and Equation (27) as:

(29

The effective stress rate can be determined Equation (24), using Equations (22) and (29) as:

(30 = + ij ijklep kl ij epC D S

ij

ij p

ij st stkle

kl

ij klkle

hQ F

C

hQ F

S F

D S

=

+

+

1

1

1

1 00

0

0

=

+ +

>

+ +

for F = and

for F < 0 or F = 0 and

F C

F S

F D S

F C

F S

F D S

ij ijkle

klij

ij e

ij ijkle

ij ij e

1

=

+ +

=

1 1h

F C

hF S

F D S

hF

C Q F

ij ijkle

klij

ij e

ij ijkle

kl

F F F F S

S ij

ij = + + =

0

Q Q S ij = ( ) , ,

ij

p

ij

Q=

ij p

= + = ij ijkle kle ij e ij eijkle

kleC D S D

C

S ,

dd

ij e ij

C C S ijkle

ijkle= ( )

C ijkle

= ij ijkle

kleC

ij ij

ij ij e ij p= +

ij F S ij = , , 0

= ij ij cp ij Sp

= + ( ) ij ij ij w ij p S p pgas gas

ij

-70

-60

-50

-40

-30

-20

-10

0-0.03-0.025-0.02-0.015-0.01-0.0050

Volumetric strain

H y

d r o s t a

t i c s

t r e s s

( M P a

)Plw10 S = 0.013Plw22 S = 0.019Plw31 S = 1Plw17 wat. inj.Plw19a wat. inj.Plw19 wat. inj.

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where the tangent elasto-plastic stiffness and saturationtensors are given as:

(31)

Finally, the total stress rate can be obtained by timedifferentiation of Equation (21) and substitution of fromEquation (30), as:

(32)

The employed elasticity is isotropic and linear with stress,in which case the elastic stiffness and saturationtensors are given by the expressions:

(33)

whereG is the elastic shear modulus and the Poisson'sratio. Based on the experimental data for the Pietra Leccesechalk, the dependency of the elasticity on the saturationS isreflected only in the shear modulusG, which is assumed as afunction of the saturation.

The employed yield surface is a modified Mohr-Coulombyield function to include a pressure cap. It is described by theexpression:

(34)

where three invariants, termed the isotropic (mean) effectivestress p, the shear stress intensity , and the Lode angle,have been introduced for the effective stress :

(35)

with I 1 the first invariant of , and J 2, J 3 the second andthird invariants of the deviatoric stressS ij , respectively. InEquation (34) the hardening/softening parametersq, pc andrepresent the size of the yield surface. The parametersq and

pc are identified as the intercepts of the yield surfaceF withthe positive (tension cut-off) and negative (pressure cap)

p-axis, respectively. A comparison of Equation (34) with theMohr-Coulomb yield function shows that the parameter

conceptually replaces the sinus of the friction angle, andthat the modified yield surface has been augmented with square root term for the mean pressure p and a newparameter pc related to the pressure cap. Figure 11 comparesthe Mohr-Coulomb and the modified Mohr-Coulomb yiesurfaces in the - p plane for = 0.6,q = 5MPa, pc = 10 MPa,and conditions of conventional triaxial compression whe

= /6.

Figure 11Illustration in the - p plane of the original Mohr-Coulomband the Mohr-Coulomb with a pressure cap yield surfaces.

The corresponding plastic potential function to Equa

tion (34) is:

(36)

where the parameter replaces the sinus of the dilatancy anglein the Mohr-Coulomb model. The hardening parameters, are expressed as functions of the accumulated plastic sheastraing p, that is = (g p), = (g p). The parameterq isexpressed as a function of the saturationS , while the

parameter p c as a function of the saturation and theaccumulated plastic volumetric strainv p, that isq = q (S ), pc = pc (S , v p). The plastic shearg p and volumetricv p strainsare defined as:

(37)

where is the plastic deviatoric strain.eij p

g g t g

v v t

p p p

p p p

=

=

,

,

d

d

= 2e e

v =

ij p

ij p

kk p

Q q p S

q p p pq p

ij c

c

c

( ) =

( ) ++

, , , , cos sin

3

-12 -8 -4 0 4

p (MPa)

(

M P a

)

-p c q

Mohr-Coulomb

Mohr-Coulombw /pressure cap

0

8

12

4

ij

p I J a J

J = = =

1

3

1

3

3 3

21 23

23 2, , sin

ij

F q p S

q p p pq p

ij c

c

c

( ) =

( ) ++

=

, , , , cos sin

3

0

C G S

DG

GS

ijkle

ij kl ik jl il jk

ij e

ij

= ( )

+ +

=

21 2

1

dd

Dij eC ijkl

ep

,d

ij ijklep

kl ij ij cp

ij ij epC P S P

Sp

S D= + =

( )+

d

ij

ij

C C h

C Q F

C

D Dh

C Q F

S F

D

ijklep

ijkle

ijmne

mn st stkle

ij ep

ij e

ijkle

kl st st e

=

=

+

1

1

Dij ep

C ijklep

768


11/13


2.3 Calculation Examples

The calibration procedure and the model parameters for thePietra Leccese chalk can be found in Papamichoset al.(1997). The hydrostatic compression tests on the specimenswith saturationS = 0.013, 0.019, 0.047, and 1 were back-analyzed with the developed model and the results are pres-ented and compared with the experimental data in Figure 12.The simulations compare well with the experimental data,showing that the model can simulate properly the saturation-dependent elastic stiffness during the initial part of the test,the saturation-dependent hydrostatic stress at plastic yield,and reasonably well the pore collapse behavior of the PietraLeccese chalk. The back analysis of the water-injection testat 40 MPa and the comparison with the experimental data arepresented in Figure 13. The specimen enters the plasticregime of deformation and experiences pore collapse duringthe water-injection phase. The simulation compares well withthe experimental data, and the model can simulate properlythe saturation-dependent elastic stiffness during the initialloading part of the test, the additional compaction during thewater-injection phase at both the elastic and plastic regimes,and the elastic stiffness during the unloading part of the test.In the model, the specimen straining during water injectioncomprises of three parts: a dilatant part due to the elimination of the capillary

suction, which essentially results in a reduction of theeffective compressive stress;

a compactive elastic part due to the reduction of the elasticstiffness with increasing water saturation;

a compactive plastic part due to plastification from the soft-ening of the pressure cap with increasing water saturation.

Figure 12Theoretical simulations and experimental results of hydrostatic compression tests on specimens with variouswater saturationsS .

Figure 13

Theoretical simulations and experimental results ofhydrostatic compression tests with water injection athydrostatic compressive stress 40 MPa.

3 SANDSTONE MODEL FOR CORE DAMAGE

Core damage is a permanent alteration of the rock proas a result of drill-out and retrieval of the core from thein situenvironment to the surface. For reservoir rocks, stress during coring is thought to be a major core damechanism. Holtet al. (1998) have investigated core damin reservoir sandstones using a synthetic sandanalogous to the reservoir rock with respect tomechanical and petrophysical properties and textur

synthetic sandstone is created underin situ stress conditioto simulate the diagenesis process. Comparative tests aperformed to simulate virgin compaction behavior, abehavior of an unloaded and reloaded simulated core.

The stress paths followed in the virgin compactioand the simulated core compaction tests are shoFigure 14. In the virgin compaction tests, sand was loathe in situ stress conditions where cementation took pAfter cementation, the specimen was compacted oedometricK 0 conditions. In the simulated core compatests, after cementation atin situ stress conditions, tspecimen was unloaded to simulate the coring pr

Subsequently the specimen was loaded to thein situ stresconditions and an oedometricK 0 compaction test wperformed to simulate a test on a cored specimen. Figshows a comparison between the virgin compaction asimulated core compaction for this sandstone. Figushows only theK 0 part of the tests. The virgin compaccurves are nonlinear with a higher initial stiffnessstiffness during virgin compaction matches the stiffnthe core compaction above 45 MPa. Thus, if core dused directly for reservoir rock compaction estimatio

-70-65-60-55-50-45-40-35-30

-25-20-15-10

-50

H y d r o s

t a t i c s t r e s s

( M P a

)

0 -0.01 -0.02 -0.03 -0.04 -0.05Volumetric strain

Test Plw10: S = 0.013Test Plw31: S = 1Test Plw17: water injectionModel

-180

-160

-140

-120

-100

-80

-60

-40-20

0-0.2-0.15-0.1-0.050

Volumetric strain

H y d r o s t a

t i c s t r e s s

( M P a

)

Data S = 0.013Data S = 0.019Data S = 0.047Data S = 1Model

769


12/13


initial compaction will be overestimated. In Figure 15b, theunloading during the coring simulation is also included. Thetwo specimens behave exactly the same above an axial stressof 45 MPa.

The test results show that during coring, unrecoverable

damage takes place with large plastic strains. This can bemodeled with an elastoplastic model with a pressure cap tomodel correctly the behavior under uniaxial compaction. Away to describe the core damage process is a model with twoyield surfaces. Both yield surfaces have a pressure cap. Suchmodels are referred to as overlay or mechanical sublayermodels. The one yield surface is for the sand matrix and theother for the cement. The sand matrix atin situ stresses isloaded and thus, during coring, it simply unloads. The yieldsurface for sand is centered at zero stresses and extends such

that thein situ stress conditions lie on it. During virginK 0loading, the sand matrix is further loaded and the yielsurface extends further such that the current stress conditionlie always on the yield surface. This is illustrated schematicalin Figure16.

The cement atin situ stresses is not loaded and thus duringeither coring or virginK 0 loading, it loads, and after an elasticregion it yields. The yield surface for the cement is centereat thein situ stress conditions and extends such that it givesan initial elastic region upon reduction of the stresses focoring simulation or increase of the stresses for virginK 0loading. Yielding leads to the destruction of the cemencohesion, such that the cement yield surface reduces in sizduring stress changes, which can be either due to coring ovirgin K 0 loading. Upon reloading toin situ stresses,

770

Time

A x

i a l a n

d r a

d i a l s t r e s s e s

Forming

Virgin uniaxial compaction ( K 0)

Axial stress

Radial stress

aTime

A x i a

l a n d r a

d i a l s t r e s s e s

Forming

Coring simulation

Uniaxial compaction ( K 0)of core

Axial stress

Radial stress

b

Figure 14Stress paths in compaction tests. a: virginK 0 compaction, and b: coring simulation, reloading toin situ conditions andK 0 compaction (Holtet al., 1998).

30

40

50

60

0 2 4 6 8Axial strain (millistrain)

A x i a

l s t r e s s

( M P a

)

Virgin compaction

Simulatedcore compaction

A Gron01A Gron03A Gron14B1s Gron05B1s Gron04B1f Gron15

a

0

10

20

30

40

50

60

70

-0.02 -0.015 -0.01 -0.005 0 0.005 0.01Axial strain

A x i a

l s t r e s s

( M P a

)

Simulated core compactionVirgin K 0 compaction

b

Figure 15Compaction behavior of virgin versus simulated core of a synthetic reservoir sandstone (Holtet al., 1998). a:K 0 phase of the tests, and b: alltests.


13/13


Figure 16Yield surfacesF for the sand matrix and the cement andstress paths during virgin and core compaction.

Figure 17Simulation of virginK 0 compaction and coringK 0 compaction.

nonlinear but elastic behavior is observed. After that thematerial continues to yield since the cement has beendestructed. Essentially destruction of the cement occurseither by loading or by unloading fromin situ conditions,such that after the cement destruction the two materialsbehave exactly the same.

The proposed two-material model can also be viewed as amodel with internal, often referred to also as residual,

stresses. The internal stresses result from the fact that tmaterials, the sand matrix and the cement, act in opways in the case of unloading, that is coring. Inteviewed the compression in the sand matrix is cobalanced by tension in the cement, resulting in zero estress.

Figure 17 shows a simulation of virginK 0 compaction an

coringK 0 compaction. The model results compare favowith the experimental results in Figure 15a showipotential of the model to predict core damage in resandstones.

CONCLUSIONS

The presented modeling examples show that the behageomaterials is often too complex to generalize the mEnough testing material is often not available calibration of a general constitutive model at a reascost. General models lead also to cumbersome compuwhich often obscure the physics of the problem at hanrole of the engineer becomes thus important, as he is reto identify and model first-order effects.

REFERENCESChen, W.F. and Han, D.J. (1988)Plasticity for Structural

Engineers , Springer-Verlag, 173.Holt, R.M., Brignoli, M., Kenter, C.J., Meij, R. and SchP.M.T.M (1998) From Core Compaction to Reservoir CompCorrection for Core Damage Effects.SPE/ISRM 47263.Proc.

Eurock 98 , 1, 311-320.

Papamichos, E., Brignoli, M. and Santarelli, F.J. (199Experimental and Theoretical Study of a Partially-SatCollapsible Rock. Mech. Cohes.-Frict. Mater., 2, 3, 251-278.Papamichos, E., Tronvoll, J., Skjrstein, A., Unander,Vardoulakis, I. and Sulem, J. (1996) The Effect of Plane-and Isotropic Loading in Hollow-Cylinder Strength.Proc.

Eurock 96 , Torino, Barla, G. (ed.), 1, Balkema, Rotterdam204.Papamichos, E., Tronvoll, J., Vardoulakis, I., Labuz,Skjrstein, A., Unander, T.E. and Sulem, J. (1999) ConstTesting of Red Wildmoor Sandstone. Mech. Cohes.-Frict. Mater., 4.Sulem, J., Vardoulakis, I., Papamichos, E., Oulahna, ATronvoll, J. (1999) Elasto-Plastic Behaviour of Red WilSandstone. Mech. Cohes.-Frict. Mater., 4.Truesdell, C. and Noll, W. (1965) The Non-Linear Field Thof Mechanics. In Encyclopedia of Physics , Flgge, S. (ed.), IIISect. 3, Springer-Verlag.

Final manuscript received in July 1999

-65

-60

-55

-50

-45

-40

-35

-30 -0.009-0.008-0.007-0.006-0.005-0.004-0.003-0.002-0.0010

Axial strain

A x i a

l s t r e s s

( M P a

)

Coring + K 0Direct K 0

-30 -25 -20 -15 -10 -5 0 5

Mean stress p (MPa)

D e v i a t o r i c s

t r e s s

( M P a

)

F for sand matrixK 0 loading stress pathCoring stress path

F for cement

2

46

8

10

12

14

16

0

771

constitutive laws for geomaterials

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