constitutive laws for geomaterials 1999 06 papamichos v54n6

8/10/2019 Constitutive Laws for Geomaterials 1999 06 Papamichos v54n6

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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 contraintes,

conditions aux limites et divers chemins de chargement, pressions de pore, tempratures, etc.Llaboration de lois constitutives suit deux tendances concurrentes : la tendance laborer une loiunifie dcrivant le comportement du matriau sous les conditions les plus gnrales, et le besoin de loisjustifi par une utilisation efficace dans la pratique. La dernire tendance inclut la fois la possibilitdune calibration par des essais sur des chantillons disponibles, et la convivialit de la loi. des finsdingnierie, la seconde tendance domine, lingnieur doit valuer le problme et slectionner le modlele plus appropri pour dcrire le phnomne dominant observ. Dans cet article, un examen des diverseslois constitutives pour les gomatriaux, et en particulier pour leurs applications, est prsent. Diversexemples de problmes concernant lingnierie et les gomatriaux appliqus au ptrole sont utiliss afinde 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 two

competing 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 incorporates

both the possibility of calibration from available specimens and the user-friendliness of the law. For

engineering 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 review

of various constitutive laws for geomaterials is presented with an emphasis on their application. Various

examples 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 whoformulate continuum theories believe matter really is

continuous, denying the existence of molecules. This is not

so. Continuum physics presumes nothing regarding the

structure of the matter. It confines itself to relations among

gross phenomena, neglecting the structure of a material on

a smaller scale. Whether the continuum approach is

justified, in any particular case, is a matter, not for the

philosophy or methodology of science, but for the

experimental 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 ofthe 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 constitutivemodels have obvious advantages. Examples of such problemswith emphasis on petroleum applications are given todemonstrate how to select or develop a constitutive modelbased on experimental and/or field data and observations.Section 1 describes a sandstone model for open hole stabilityand sand production problems, Section 2 a chalk model for

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

1 SANDSTONE MODEL FOR OPEN HOLE STABILITYAND SAND PRODUCTION

Open hole stability and sand production problems are oftenencountered in sandstone reservoirs during hydrocarbonproduction. 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 meanstresses by high deviatoric stresses due to the stressconcentration around the hole.

A nonlinear elastic-plastic model has been developed todescribe the near well behavior of these sandstones (Sulem etal., 1999). In brief, the model has an elasticity, which isnonlinearly stress-dependent, and a plasticity based on theflow theory. The plasticity is related to a linear Mohr-Coulomb yield function F with friction hardening andcohesion softening. The plastic dilatancy is described by alinear Mohr-Coulomb plastic potential function, differentfrom the yield surface, which makes the flow rule not

associative. This allows for the description of bothcompactive and dilatant behaviors under shear straining. Onthe other hand, the appropriate plastic behavior underhydrostatic straining has not been incorporated in the presentmodel as plastic hydrostatic straining is not usuallysignificant around wellbores and perforations. Throughoutthis section compression is taken negative, as it is moreconvenient for calculations. Tests results and modelpredictions are given with compression positive as moreoften used in rock and soil mechanics.

1.1 Model Description

The development of the model is based on behavior of soft,weak sandstones as observed in triaxial compression andextension experiments (Papamichos et al., 1999), which arerelative experiments for obtaining the deviatoric behavior ofa material. For nonlinear materials, the constitutive relationsare formulated in the incremental form which, for time-independent behavior, is equivalent to the rate formemployed in the following. A dot over a quantity denotes therate (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 modulus G must be a function of the stress, thatis G = G(

ij). The reversibility and path independency of the

elastic strain energy requires that the stresses are obtained

from the strain energy function Was (Chen and Han, 1988):

(3)

With these assumptions and requirements, the elasticshear modulus G must be a function of the equivalent stress

e, defined in terms of the isotropic (mean) stress p and the

shear-stress intensity as:

(4)

The stress invariantsp and are defined as:

(5)

where I1 is the first stress invariant and J2, the seconddeviatoric stress invariant. The dependency of G on theequivalent stress

e, is described by the material function:

(6)

where the constants G0 to G3, Gpeak, and for the variousmaterials are obtained by fitting this function to theexperimental data. According to Equation (6), the elasticshear modulus G increases from the initial value G = G0 atzero stress to G = Gpeak at and remains constantthereafter. The elastic tangent stiffness tensor may thenbe written as:

(7)

With respect to plasticity, the standard flow theory is usedwhere the plastic tangent stiffness tensor is expressed as:

(8)

with F and Q being the yield and plastic potential functionsrespectively. The hardening modulus h is given by:

(9)h F C Q Q F g Fq qgijijkle

kl

s p p= +

dd

dd

Ch

C Q F

Cijklp

ijmn

e

mn st

stkl

e= 1

Cijklp

GG G

il jk ij kl

e

e

e e

+ + +( )

2 1 2

d d

d d

C Gijkle

ij kl ik jl il jk =

+ +

2

1 2

Cijkle

e e= peak

epeak

G for

for

= ( )=+

+( )+

>

GG

G G

G

G

e

e e

e

e e

e e

0

2 3

11

peak

peak

peak

p I J= =1 23,

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

p

ij

ij

=

W

C C Cijklep

ijkl

e

ijkl

p=

Cijklep

ijij

ij ijklep

kl= C

761

0

5

10

15

20

25

30

35

40

0 5 10 15 20 25

Axial strain (x 10-3)

Ax

ialstress(MPa)

Red Wildmoor

Confining stress 6 MPa

Confining stress 2.5 MPa


a

-8

-6

-4

-2

0

2

4

Volumetricstrain

(x10-3)

Confining stress 6 MPa



Axial strain (x 10-3)

0 5 10 15 20 25

b

Figure 1

Triaxial compression test results on humid Red Wildmoor sandstone. a: axial stress, and b: volumetric strain versus axial strain at threeconfining stresses (Papamichos et 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 (Papamichos et 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 intensity gp, and after the

peak stress is reached, the material softens. The yield Fandplastic potential Q surfaces are then functions of the stress

Figure 2

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

ij

and the accumulated plastic shear-strain intensity gp. Theuse of nonassociativity where the yield and plastic potentialfunctions do not coincide is necessary in soft weaksandstones. The experimental results in Figure 1b show thatthe material initially compacts and then dilates as the peakstress is approached. With associative behavior, dilation isalways predicted. For the Mohr-Coulomb criterion, Fand Q

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

(11)

where the stress invariant is called the Lode angle anddescribes the relative magnitude of the intermediate principalstress. It is defined through the second J2 and third J3deviatoric stress invariants as:

(12)

The parameter q = q(gp) is the tension cut-off, and = (gp) and = (gp) are the mobilized friction anddilatancy coefficients, respectively. These are related to themobilized friction angle

m, dilation angle

m, and cohesion

c through the expressions:

(13)

The parameter Qs

in Equation (9) is related to the plasticpotential surface Q and it is given as a function of the stressinvariantsJ2,J3 as:

(14)

where:

(15)

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

(16)

ij

p

ij ij

ijkl

e

kl

Q FC= , =

1

h

ijp

BQ

J

BQ

J

Q

Q

12

23

2

1 3 3

3

3

4 3 6

3

4 3

3

0

= =

+ + ( )

+

= =+

cos tan tan tan tan

sin coscos

2

if =

if =6

32

if =6

QJ B J B B J B

s

Q Q Q Q

= + +

23

32 1 3 1 2 2

22

2 2

= = =sin , sin , cosm m mc q

=

1

3

3 3

2

3

23 2a

J

Jsin

F F g q p

Q Q g p

ij

p

ij

p

= ( )=

( )=

= ( )=

+

, cossin

, cossin

30

3

0

10

20

30

40

50

60

0 2 4 6 8 10 12

Confining stress (MPa)

Axialstress

(MPa)

Red Wildmoor sandstone

1

1 0 0

0 0 0 0

=

= >

< =

, for and

for or and

FF

C

F FF

C

ij

ijkl

e

kl

ij

ijkl

e

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 materialfunctions q = q(gp), = (gp) and = (gp). 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-off q, which starts to decrease fromits initial value q0 after the friction coefficient has reached itspeak value, and thus after the plastic shear strain gp hasreached the value . In particular, the following functionswere calibrated for each material:

(19)

where the constants q0, q1, ,f0 tof3, fpeak, and d1 to d3,dpeak are obtained by fitting the functions (19) on experi-mental data.

Figure 3

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

The model can be calibrated through a minimum of twotriaxial compression tests with load-unload cycles at variousstress levels. Table 1 lists values of the various elasticand plastic material constants for humid Red Wildmoorsandstone.

TABLE 1

Humid Red Wildmoor sandstone: elastic and plastic material constants

[ ] 0.2 f0 [ ] 0.067148 q0 [MPa] 3.82

G0 [MPa] 349.17 f1 [ ] 109.88 q1 [MPa] 25.0

G1 [MPa-1] 0.042907 f2 [ ] 273.75

G2 [ - ] 250.32 f3 [ ] 8946.6 d1 [ ] 5.834

G3 [MPa-1] 0.0 fpeak [ ] 0.94523 d2 [ ] 0.78636

Gpeak [MPa] 6183.2 [ ] 0.0099069 d3 [ ] 90.453

[MPa] dpeak [ - ] 0.94523

1.2 Calculation Examples

The back analysis of the triaxial compression and reducedextension test results is shown in Figure 4. The comparisonwith the experimental results for both test types is verysatisfactory despite the complexity of the soft weaksandstone behavior. The model was also used to predict theresponse of Red Wildmoor sandstone hollow cylindersloaded isotropically (Papamichos et al., 1996). In these testsa thick-wall hollow cylinder was loaded externally, while theinternal hole remained unsupported. The external radial stress

was equal to the axial stress. The internal hole and externaldeformations of the cylinder were measured. In Figure 5experimental results for the external radial stress versus theinternal and external tangential strains are compared with themodel predictions. The nonlinear elastic-plastic modelpredicts very well the hollow cylinder response up to thepoint of failure of the inner hole. This is expected since afterthis point the continuum description of the material ceases.

The point of initial failure is predicted through thebifurcation theory and continua with microstructure. Inaddition to the nonlinear elastic-plastic model, predictionswere made using a simpler model where the nonlinear

elasticity was suppressed and an average value of elasticshear modulus was used. It can be seen in Figure 5 that thelinear elastic-plastic model predictions are rather poorwhich illustrates the need for a nonlinear elasticity modelin this case.

For weakly cemented, unconsolidated sandstones theconstitutive law may need to be enriched with additionalfeatures to model adequately the near well behavior ofthe rock. In the presented model, a linear Mohr-Coulombmodel was adopted for relatively low mean stresses. In

epeak

gpeakp

p

T

q q0

c0

cp

c(0)

(1)

(2)1

f0

fp1

gp

peak

q q g

q q g g

f

ff

ff 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

3

2 31

3

2 3

2

for g

g for g

,for

for

, d

p

p p

peak

peak peak

peak

peak peak

11

11

g

d gg g

d g g

p

p

p p

p p

( )+( )

>

for

for

peak

peak peak

gp

peak

g gp p= peak

ije

ij ij

p=

ijp

g Qp s=

gp

763


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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 isimportant in many aspects of oil production such as reservoircompaction and solids production. Among several factorsaffecting the mechanical behavior of such rocks is the relativefluid saturation (gas/oil/water), which has an impact on thecapillary forces between the grains. Capillarity arises from afluid property known as surface tension, a phenomenon

764

-0.015 0.000 0.015 0.030

Radial strain Axial strain

0

20

40

60

10.3 MPa

6.9 MPa3.5 MPa

1.4 MPa

0.4 MPa

Simulation of triaxial compression tests(complete model)

Data Computed

Axialst

ress(MPa)

a

-0.015 0.000 0.0150

20

40

60

Radial strain Axial strain

Simulation of triaxialextension tests

(complete model)

Data

Computed

Axial

stress(MPa)

60 MPa52.5 MPa

45 MPa

b

Figure 4

Back analysis of a: triaxial compression, and b: reduced triaxial extension test results (Sulem et al., 1999).

Figure 5

Hollow 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

xternalradialstress

(MPa)

SimulationLinear elastic-plasticsimulation

Test 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

Externalradialstress

(MPa)

Simulation

Linear elastic-plasticsimulation

Test 21Cav04

b


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Figure 6

Triaxial 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 ofsaturation, 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 a

dry 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 ofcohesion. 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 ofmarine animals. In some chalks, it is found that capillarymenisci formed by connate water are the only cohesionalforce that holds together the coccoliths. Upon elimination ofthe 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

(Papamichos et al., 1997). The model is based on experi-mental evidence from tests on Pietra Leccese chalk andincorporates Bishops effective stress principle for thecapillary pressure, a water-saturation-dependent elasticstiffness, and a Mohr-Coulomb yield surface with a pressurecap. The cohesion and pressure cap parameters in the yieldsurface are water-saturation-dependent to model the capillary

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

2.1 Experimental Evidence

The model is based on experimental evidence from triaxialand hydrostatic compression tests on Pietra Leccese chalk atvarious degrees of water saturation (Papamichos et al., 1997)The Pietra Leccese chalk is a high-porosity outcrop bio-calcarenite from the south of Italy. Figure 7 shows uniaxialcompression test results for specimens at various watersaturations. The results show that the uniaxial compressivestrength (UCS) decreases substantially with increasing watersaturation. Load-unload cycles in these tests showed that forthis material, the elasticity does not change significantly withthe applied axial or confining stress. However, a significantdecrease in the Youngs modulus with increasing watersaturation was observed. The Poissons ratio on the otherhand did not appear to be influenced significantly by thesaturation. From the test data, an average elastic Youngsmodulus E for each saturation, and a constant averagePoissons ratio for all saturations were calculated.

In Figure 8, the peak axial stress obtained at uniaxial andtriaxial compression tests is plotted versus the confiningstress and saturation. From this plot and assuming a linearMohr-Coulomb yield criterion for the peak axial stress, the

Figure 7

Uniaxial compression test results on Pietra Leccese chalk.Axial stress versus axial strain for various water saturations S.

-50

-45

-25

-20

-40

-15

-35

-10

-30

-5

0

-0.004-0.003-0.002-0.0010

Axial strain

Axialstress

(MPa)

Plt05 S= 0.016Plt33 S= 0.025

Plt09 S= 0.047Plt30 S= 0.055Plt10 S= 1

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04Axial strain

Volumetricstrain

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

765


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peak friction angle and cohesion c can be calculated for thetwo saturations where triaxial test data are available. Thecalculated values of peak friction angle and cohesion aretabulated in Table 2. From these limited test results, itappears that changes in the water saturation affect primarilythe cohesion of the material, which reduces substantially infully water-saturated specimens. This is attributed to thedestruction, with increasing water saturation, of the capillarymenisci, which act as cohesional bonds between the grains.On the other hand, the internal friction angle, which isprimarily influenced by such factors as porosity, particleshape and size, and grain size distribution, appears not to be

affected significantly.

TABLE 2

Pietra Leccese chalk: peak friction angle and cohesion

at various water saturations calculated from triaxial compression test data

Saturation Peak friction angle () Peak cohesion (MPa)

0.047 29.1 10.34

1 27.5 6.46

Figure 9 shows the results from four drained hydrostaticcompression tests performed on specimens at different water

saturations, ranging from 0.013 to 1. During plastic yieldunder hydrostatic compression, the specimens undergo acharacteristic structural change for collapsible materialswhich is referred to as pore collapse. This gradual structuralchange is characterized with a significant increase incompressibility. The results show that the elastic bulkmodulus and the plastic yield stress decrease significantlywith increasing water saturation. The tangent bulk modulusdoes not change substantially until plastic yield and thus theelasticity of Pietra Leccese can be considered at a first

approximation as stress-independent. The post-pore-collapsetangent bulk modulus is approximately the same in all testedspecimens, and thus it can be considered as saturation-independent.

Three water-injection tests were also performed. Thewater-injection tests comprise of three phases. In the firstphase, a specimen with initial saturation 0.013 or 0.019 ishydrostatically compressed under drained conditions until aprescribed hydrostatic stress level is reached. In the secondphase, the hydrostatic stress is maintained constant, and waterinjection is performed by flowing few pore volumes ofequilibrium water through the specimen at a constant flow

rate. The flow rate is kept low to ensure no significant porepressure built-up in the specimen. In the third phase, fluidflow is halted and the specimen is hydrostatically unloadedunder drained conditions to zero stress. Figure 10 shows plotsof the experimental results of water-injection tests at stressesabove (test Plw17) and below (tests Plw19 and Plw19a),respectively, the hydrostatic plastic yield stress of the fullysaturated material. The load-unload cycles for tests Plw19and Plw19a show indeed an elastic behavior both beforeand after water injection. In the figure, the results fromthe hydrostatic tests without water injection on specimenswith saturation 0.013, 0.019 and 1 are superimposed for

comparison. During water injection the specimens experienceinitially a small dilation, followed by compaction. The totalfinal compaction approaches the compaction obtained in thefully saturated specimen at the corresponding stress level. Inthe specimen water-injected at a stress level above the plasticyield stress of the fully saturated material, the additionalwater-injection-related compaction is significant, due to porecollapse during water injection. The initial dilation may berelated to the elimination of the capillary suction duringwater injection.

766

Figure 8

Uniaxial and triaxial compression test results on PietraLeccese chalk. Peak axial stress versus confining stress andsaturation S.

Figure 9

Hydrostatic compression tests on Pietra Leccese chalk showingthe influence of water saturation on the pore collapse stress.

-70

-60

-50

-40

-30

-20

-10

0-12-10-8-6-4-20

Confining stress (MPa)

Axia

lstress

(MPa)

S= 1

S= 0.055S= 0.047S= 0.025S= 0.016

-180

-160

-140

-120

-100

-80

-60

-40

-20

0-0.2-0.15-0.1-0.050

Volumetric strain

Hydrostaticstress

(MPa)

Plw10 S= 0.013Plw22 S= 0.019Plw23 S= 0.047Plw31 S= 1

Pore collapse


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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)

where ij

is the total stress, Sis the water saturation,pgas andp

wthe 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 pcp

, defined asp

cp=pgaspw , is a function of the saturation S, that is

pcp

=pcp

(S). Plastic yield is associated with an isotropic yieldsurface F, the size of which depends on the accumulatedplastic strain parameter and the saturation S, 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 saturation S, that is . Time

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

(24)

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

(25)

where is the plastic potential function, and

is a scalar function. From the consistency condition:

(26)

and using Equations (22), (24) and (25), can be solved as:

(27)

For plastic loading, is a nonnegative factor and h isalways positive. It then follows from Equation (27) that theplastic loading criterion can be expressed in terms of theswitch function defined as:

(28)

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

(29)

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

(30) = + ij ijklep

kl ij

epC D S

ij

ij

p

ij st

stkl

e

kl

ij kl

kl

e

h

Q F

C

h

Q F

S

FD S

=

+

+

1

1

1

1 0

0

0

0

=

+ +

>

+ +

for F = and

for F < 0 or F = 0 and

FC

F

S

FD S

FC

F

S

FD S

ij

ijkl

e

kl

ij

ij

e

ij

ijkl

e

ij

ij

e

1

=

+ +

=

1 1h

FC

h

F

S

FD S

hF

CQ F

ij

ijkl

e

kl

ij

ij

e

ij

ijkl

e

kl

FF F F

SS

ij

ij=

+ + =

0

Q Q Sij= ( ) , ,

ij

p

ij

Q=

ijp

= + = ij ijkle

kl

e

ij

e

ij

e ijkl

e

kl

eC D S D

C

S,

d

d

ije ij

C C Sijkle

ijkl

e= ( )

Cijkle

= ij ijkle

kl

eC

ij ij

ij ij e

ij

p= +

ijF Sij = , , 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

Hydrosta

ticstress(MPa)

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

767


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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 time

differentiation 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)

where G 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 saturation Sisreflected only in the shear modulus G, 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 the

expression:

(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 I1 the first invariant of , and J2, J3 the second andthird invariants of the deviatoric stress S

ij, respectively. In

Equation (34) the hardening/softening parameters q,pcand

represent the size of the yield surface. The parameters q andp

care identified as the intercepts of the yield surface Fwith

the 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 asquare root term for the mean pressure p and a newparameterp

crelated to the pressure cap. Figure 11 compares

the Mohr-Coulomb and the modified Mohr-Coulomb yieldsurfaces in the-p plane for = 0.6, q = 5MPa,p

c= 10 MPa,

and conditions of conventional triaxial compression where

= /6.

Figure 11

Illustration 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 shearstrain gp, that is = (gp), = (gp). The parameter q isexpressed as a function of the saturation S, while the

parameter pc as a function of the saturation and theaccumulated plastic volumetric strain vp, that is q = q (S),p

c=p

c(S, vp). The plastic shear gp and volumetric vp strains

are defined as:

(37)

where is the plastic deviatoric strain.eijp

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 pp p

q p

ij c

c

c

( )=

( ) +

+

, , , , cos sin

3

-12 -8 -4 0 4

p (MPa)

(MPa)

-pc q

Mohr-Coulomb

Mohr-Coulombw/pressure cap

0

8

12

4

ij

p I J aJ

J= = =

1

3

1

3

3 3

21 23

23 2

, , sin

ij

F q p S

q pp p

q p

ij c

c

c

( )=

( ) +

+ =

, , , , cos sin

3

0

C G S

DG

G

S

ijkl

e

ij kl ik jl il jk

ij

e

ij

= ( )

+ +

=

2

1 2

1

d

d

Dije

Cijklep

,d

ij ijklep

kl ij ij

cp

ij ij

epC P S P

Sp

SD= + =

( )+

d

ij

ij

C Ch

CQ F

C

D Dh

CQ F

S

FD

ijkl

ep

ijkl

e

ijmn

e

mn st

stkl

e

ij

ep

ij

e

ijkl

e

kl st

st

e

=

=

+

1

1

Dijep

Cijklep

768


11/13


2.3 Calculation Examples

The calibration procedure and the model parameters for thePietra Leccese chalk can be found in Papamichos et al.(1997). The hydrostatic compression tests on the specimenswith saturation S= 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 initial

loading 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 12

Theoretical simulations and experimental results ofhydrostatic compression tests on specimens with variouswater saturations S.

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 propertiesas a result of drill-out and retrieval of the core from the in situenvironment to the surface. For reservoir rocks, stress releaseduring coring is thought to be a major core damagemechanism. Holt et al. (1998) have investigated core damagein reservoir sandstones using a synthetic sandstoneanalogous to the reservoir rock with respect to rockmechanical and petrophysical properties and texture. The

synthetic sandstone is created under in situ stress conditionsto simulate the diagenesis process. Comparative tests are thenperformed to simulate virgin compaction behavior, and thebehavior of an unloaded and reloaded simulated core.

The stress paths followed in the virgin compaction testsand the simulated core compaction tests are shown inFigure 14. In the virgin compaction tests, sand was loaded tothe in situ stress conditions where cementation took placeAfter cementation, the specimen was compacted underoedometric K0 conditions. In the simulated core compactiontests, after cementation at in situ stress conditions, thespecimen was unloaded to simulate the coring process

Subsequently the specimen was loaded to the in situ stressconditions and an oedometric K0 compaction test wasperformed to simulate a test on a cored specimen. Figure 15shows a comparison between the virgin compaction and thesimulated core compaction for this sandstone. Figure 15ashows only the K0 part of the tests. The virgin compactioncurves are nonlinear with a higher initial stiffness. Thestiffness during virgin compaction matches the stiffness ofthe core compaction above 45 MPa. Thus, if core data areused directly for reservoir rock compaction estimation, the

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

-25-20-15-10-50

Hydrostaticstress(MPa)

0 -0.01 -0.02 -0.03 -0.04 -0.05

Volumetric strain

Test Plw10: S = 0.013

Test Plw31: S = 1

Test Plw17: water injection

Model

-180

-160

-140

-120

-100

-80

-60

-40-20

0-0.2-0.15-0.1-0.050

Volumetric strain

Hy

drostaticstress

(MPa)

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 at in situ stresses isloaded and thus, during coring, it simply unloads. The yieldsurface for sand is centered at zero stresses and extends such

that the in situ stress conditions lie on it. During virgin K0loading, the sand matrix is further loaded and the yieldsurface extends further such that the current stress conditionslie always on the yield surface. This is illustrated schematicallyin Figure16.

The cement at in situ stresses is not loaded and thus duringeither coring or virgin K0 loading, it loads, and after an elasticregion it yields. The yield surface for the cement is centeredat the in situ stress conditions and extends such that it givesan initial elastic region upon reduction of the stresses forcoring simulation or increase of the stresses for virgin K0loading. Yielding leads to the destruction of the cementcohesion, such that the cement yield surface reduces in sizeduring stress changes, which can be either due to coring orvirgin K0 loading. Upon reloading to in situ stresses,

770

Time

Axialan

dradialstresses

Forming

Virgin uniaxial compaction (K0)

Axial stress

Radial stress

aTime

Axialan

dradialstresses

Forming

Coring simulation

Uniaxial compaction (K0)of core

Axial stress

Radial stress

b

Figure 14

Stress paths in compaction tests. a: virgin K0 compaction, and b: coring simulation, reloading to in situ conditions and K0 compaction (Holt etal., 1998).

30

40

50

60

0 2 4 6 8

Axial strain (millistrain)

Axialstress(MPa)

Virgin compaction

Simulatedcore compaction

A Gron01

A Gron03

A Gron14

B1s Gron05

B1s Gron04

B1f Gron15

a

0

10

20

30

40

50

60

70

-0.02 -0.015 -0.01 -0.005 0 0.005 0.01

Axial strain

Axialstress(MPa)

Simulated core compaction

VirginK0compaction

b

Figure 15

Compaction behavior of virgin versus simulated core of a synthetic reservoir sandstone (Holt et al., 1998). a: K0 phase of the tests, and b: alltests.


13/13


Figure 16

Yield surfaces F for the sand matrix and the cement andstress paths during virgin and core compaction.

Figure 17

Simulation of virgin K0 compaction and coring K0 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 from in 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 the twomaterials, the sand matrix and the cement, act in oppositeways in the case of unloading, that is coring. Internallyviewed the compression in the sand matrix is counterbalanced by tension in the cement, resulting in zero externalstress.

Figure 17 shows a simulation of virgin K0 compaction and

coring K0 compaction. The model results compare favorablewith the experimental results in Figure 15a showing thepotential of the model to predict core damage in reservoirsandstones.

CONCLUSIONS

The presented modeling examples show that the behavior ofgeomaterials is often too complex to generalize the modelsEnough testing material is often not available for acalibration of a general constitutive model at a reasonablecost. General models lead also to cumbersome computationswhich often obscure the physics of the problem at hand. Therole of the engineer becomes thus important, as he is requiredto identify and model first-order effects.

REFERENCES

Chen, W.F. and Han, D.J. (1988) Plasticity for StructuraEngineers, Springer-Verlag, 173.

Holt, R.M., Brignoli, M., Kenter, C.J., Meij, R. and SchutjensP.M.T.M (1998) From Core Compaction to Reservoir CompactionCorrection for Core Damage Effects. SPE/ISRM47263. ProcEurock 98, 1, 311-320.

Papamichos, E., Brignoli, M. and Santarelli, F.J. (1997) AnExperimental and Theoretical Study of a Partially-SaturatedCollapsible Rock.Mech. Cohes.-Frict. Mater., 2, 3, 251-278.

Papamichos, E., Tronvoll, J., Skjrstein, A., Unander, T.E.Vardoulakis, I. and Sulem, J. (1996) The Effect of Plane-Strainand Isotropic Loading in Hollow-Cylinder Strength. ProcEurock 96, Torino, Barla, G. (ed.), 1, Balkema, Rotterdam, 197204.

Papamichos, E., Tronvoll, J., Vardoulakis, I., Labuz, J.F.Skjrstein, A., Unander, T.E. and Sulem, J. (1999) ConstitutiveTesting of Red Wildmoor Sandstone.Mech. Cohes.-Frict. Mater., 4.

Sulem, J., Vardoulakis, I., Papamichos, E., Oulahna, A. andTronvoll, J. (1999) Elasto-Plastic Behaviour of Red WildmoorSandstone.Mech. Cohes.-Frict. Mater., 4.

Truesdell, C. and Noll, W. (1965) The Non-Linear Field Theoriesof Mechanics. InEncyclopedia of Physics, Flgge, S. (ed.), III/3Sect. 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

Axialstress(MPa)

Coring + K0

Direct K0

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

Mean stress p(MPa)

Deviato

ricstress(MPa)

Ffor sand matrix

K0loading stress path

Coring stress pathF for cement

2

4

6

8

10

12

14

16

0

771

constitutive laws for geomaterials 1999 06 papamichos v54n6

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