Mechanical Behaviour of a Porous Chalk and Water / Chalk Interaction Part II : Numerical Modelling

Mechanical Behaviour of a Porous Chalk and Water/Chalk Interaction, Part II: Numerical Modelling* — This paper is the second part of a general work on the mechanical behaviour of a porous chalk and the effect of saturating fluid. It presents the development of an elastoplastic model with two yield surfaces to describe the chalk behaviour. The water induced plastic deformation observed experimentally is described by using an additional plastic potential, based on the behaviour jump between two material states of chalk saturated with two different fluids. A good correlation is obtained between numerical simulations and experimental data obtained in the experimental study presented in the first part.


INTRODUCTION
Some valuable theoretical and numerical developments have been formulated for the description of water induced plastic deformation.Piau andMaury (1994, 1995) and Piau et al. (1998) proposed a conceptual model by introducing an additional stress increment which is seen as responsible to the plastic deformation induced by water injection.This model provides a simple conceptual framework.However, the physical meaning of the additional stress and the theoretical background of the formulation remain confuse.Papamichos et al. (1997) extended the concepts of unsaturated soils to develop an elastoplastic model.However, in the case of chalk, the validity of suction concept is questionable.Furthermore, continuous variations of chalk properties as functions of water saturation are not clearly shown by experimental data.
On the basis of test data by Schroeder, the authors (Homand et al., 1998) noticed that there is a critical value of water saturation at which a sudden transition from "oil-like" behaviour to "water-like" behaviour takes place.Then a preliminary version of an elastoplastic model has been proposed by introducing the mechanism of yield surface jump when chalk is flooded by water.A similar approach has been used by Piau et al. (1998) in Cam-Clay model.However, the validity of the model was not tested against laboratory tests even for basic loading conditions.Furthermore, Schroeder (1995) performed hydrostatic tests on samples saturated with fluids with different viscosity and he has concluded that there is no clear correlation between chalk behaviour and fluid viscosity.
However, two groups of behaviour can clearly be identified when the chalk is saturated by "water-like" fluid and "oil-like" fluid.Therefore, a viscoplastic modelling is not suitable for the description of chalk/water interaction.Nevertheless, chalk usually exhibits time-dependent plastic deformation (Shao et al., 1995) which could be enhanced by the water flooding.But this feature is not investigated in this work.
The purpose of this paper is to develop a new constitutive model for the description of basic mechanical properties of chalk and water induced deformation.The preliminary version of the elastoplastic model proposed by the authors (Homand et al., 1998) is completed and improved.The parameters have been determined from test data presented in the paper concerning the experimental results (Homand and Shao, 2000).A new way is proposed to describe additional plastic deformation induced by water injection.
Comparisons between numerical simulations and experimental data have been done, both for conventional triaxial tests, lateral extension tests, proportional loading tests and specific water injection tests.

Background
The experimental results suggest that the chalk behaviour can be modelled in terms of elastoplastic model with two plastic deformation mechanisms (Halleux et al., 1990;Shao and Henry, 1991;Monjoie et al., 1995;Homand and Shao, 2000).Drucker et al. (1957) were the first to introduce a spherical end-cap to the Drucker-Prager model in order to control the plastic volumetric change of soil.Within the framework of small strain elastoplasticity, it is assumed that the total strain increment dε ij is obtained from the superposition of an elastic and a plastic parts, dε e ij and dε p ij , respectively.The plastic part is then divided into plastic collapse and plastic deviatoric parts, dε c ij and dε d ij , respectively: The elastic strains dε e ij due to an applied stress increment dσ ij are obtained from an elastic stress-strain relation of the form: (2) where E 0 and v 0 are the elastic parameters, respectively the Young modulus and the Poisson's ratio in drained conditions.

Pore Collapse Mechanism
From the experimental data (Homand and Shao, 2000), the plastic yield function for the pore collapse mechanism is expressed by: (4) with: (5 where k is the hardening function for the pore collapse mechanism and h a material parameter defining the ratio of the two axes of the elliptic yield surface. p and q respectively designate the mean stress and the deviator of the stress tensor.As the pore collapse mechanism is physically related to irreversible modification of pore volume, it is assumed that the material hardening for this mechanism depends on the plastic volumetric strain ε p c .Therefore, the following plastic hardening law is proposed: where k 0 and a are respectively the initial pore collapse threshold and a plastic hardening parameter.
According to experimental data from porous chalks, it has been found that a non-associated flow rule should be used for this type of rock for the pore collapse mechanism.In the present model an elliptical flow rule has been chosen in accordance with the form of the yield surface: where parameter h c defines some deviation of the plastic flow from the normal to the yield surface.It makes possible to better describe the path dependency of collapse plastic strains.

Deviatoric Mechanism
During deviatoric loading, the plastic deformation results essentially from sliding and from microrotations of grains: material failure occurs by shearing.From experiment data, the plastic yield function for the matrix shearing is proposed by extending the classical Drucker-Prager criterion: where the effect of Lode angle is included: In order to account for chalk responses in complex loading paths, the yield function of the deviatoric mechanism depends on the three basic stress invariants p, q and J 3 (or Lode angle noted θ).This way to take into account the Lode angle dependency of material behaviour was used by several authors, for instance Papamichos et al. (1995).α is the hardening function representing the current value of material friction and p 0 the hydrostatic tensile strength assumed to be constant.The strain-hardening rule of the deviatoric mechanism is described by the evolution of the friction α from its initial value α 0 to its ultimate value α m .The following equation is proposed: and: (10 where e d i j is the deviatoric plastic strain and b a material hardening parameter.
Like most rocks, the plastic flow in deviatoric mechanism is generally non-associated and the following plastic potential is proposed: where β is the plastic dilatancy coefficient.In order to consider the progressive transition from plastic compressibility to dilatancy in most rocks it is proposed that β decreases with the generalised deviatoric plastic strain as follows: where β 0 defines the initial value of β and b (the same as that in Eq. ( 10)) is a material hardening parameter.

Determination of the Model Parameters
The parameters of this model may be determined from conventional laboratory tests (triaxial and hydrostatic tests).
The method used for their determination will be given below.

The Pore Collapse Mechanism Parameters
The plastic yield function and the plastic potential concerning the pore collapse mechanism involve four parameters called h, k 0 , a and h c .The value of h can be determined by drawing the elliptic surface from initial yield stresses obtained from hydrostatic and triaxial tests with "high" confining pressure (Fig. 1a).k 0 can be directly evaluated from the hydrostatic test: its value corresponds to the mean pressure after which irreversible deformations occur in a material during this test.a is obtained from a hydrostatic compression test, it represents the slope of the curve plotting the plastic volume deformation as a function of the logarithm of the mean stress divided by the pore collapse threshold ratio (Fig. 1b): (13) Finally, the parameter h c involved in the plastic potential can be identified by plotting the increment of plastic strains at a given stress state during a triaxial test with high confining pressure where only the collapse mechanism is active (Homand, 2000).

The Deviatoric Mechanism Parameters
The plastic yield function and the plastic potential concerning the deviatoric mechanism involve five parameters called p 0 , α m , α 0 , β 0 and b.The values of p 0 , α m and α 0 can be determined from a minimum of three triaxial tests at low confining pressure conducted until failure of sample.The plot in the p-q plane of the initial yield stresses and of the peak stresses provides two straight lines with respective slopes of α* 0 and α* m .Hydrostatic tensile strength p* 0 is given by the intersection of these two straight lines with the p (mean stress) axis (Fig. 1c).For triaxial compression, by putting the value of the Lode angle θ = -π/6 respectively into the initial yield condition (α = α 0 ) and failure condition (α = α m ) in Equation ( 8), we obtain easily the values of α 0 and α m from α* 0 and α* m as well as p 0 = p* 0 .
To determine the hardening parameter b, we use the plastic loading phase of a triaxial test.As the parameters p 0 , α m and α 0 are previously determined, the use of the consistence conditions f d = 0 allows to calculate the current value of friction coefficient α: The strain hardening function (Eq.( 10)) can be rewritten in logarithmic form: (15) As the generalised deviatoric plastic strain γ d can be calculated at each stress state during plastic phase, the plot of Equation ( 15) allows to determine b from a linear regression (Fig. 1d).However, as can be expected from the different shapes obtained for the different triaxial tests, this parameter may be sensible to the confining pressure.The solution used the start of the plastic loading (Fig. 1e).
For the chalk studied here, two sets of model parameters corresponding to the two different saturating fluids are determined for the two material states of chalk (Table 1), by using the procedure presented here above.

Plastic Flow Rules
The flow rules establish the direction of the plastic strain increments by taking the gradient to the plastic potential.Therefore, for the pore collapse and deviatoric mechanism the plastic strain increments are respectively: (17) and: In the case of multisurface model, the plastic multipliers are determined by solving a linear system expressing the consistency conditions for each surface (Simo et al., 1988;Pramono and Williams, 1989): (18)

Effect of Saturating Fluid on Chalk Behaviour
The laboratory results presented in the first part of the study (Homand and Shao, 2000) have clearly shown that the mechanical behaviour of chalk is strongly sensitive to the nature of saturating fluid.In this study, the validation of the new approach for numerical modelling of this phenomenon proposed by Homand et al. (1998) is made possible by using the laboratory investigations mentioned above.This approach assumed that there are two chalk material states, dried or oil saturated and water saturated, superposed on the same material point.When the water saturation is less than the critical value S c w , the "oil" chalk behaviour is active.On the other hand, the "water" chalk should be considered when the water saturation is higher than the critical value.Moreover, the two chalk materials can be described by the same formal constitutive model, but associated with two distinct sets of material parameters.Consequently, water injection causes a jump of yield surfaces from the "oil state" to "water state" and the plastic equilibrium is disturbed.The way to re-establish this plastic equilibrium is to produce an instantaneous plastic deformation.Therefore, the water induced plastic strains can be expressed by: w is the critical water saturation degree.g w is a potential which determines the direction of water induced plastic strain and is determined from laboratory testing.In this paper, a linear potential is proposed: The value of the plastic multiplier ∆λ w is determined from the differences in material parameters between the two chalk material states.In a simplified scheme, we assume that the water flooding effect is represented by a jump of the elliptic yield surface only through a reduction of the initial pore collapse threshold k 0 .Therefore, the plastic multiplier ∆λ w can be explicitly determined from the plastic equilibrium condition between two material states (under constant stresses) as follows: (22) where k 0 oil and k 0 water are respectively the pore collapse threshold for "oil" and "water" states of chalk.

Simulation of Laboratory Tests
In this section, numerical simulations of some representative laboratory tests are performed by using the proposed constitutive model.Comparisons between numerical results and test data make it possible to check the validity of the model.
In Figure 2, the simulations of two hydrostatic compression tests respectively performed on "oil saturated" and "water saturated" samples are shown.The good correlation between numerical and experimental data indicates that the weakening effect of water on plastic pore collapse of chalk is well described.In Figures 3 and 4, two series of triaxial compression tests with different confining pressures, respectively in water and oil saturated conditions, are simulated by using the proposed elastoplastic model and the sets of parameters given in Table 1.There is also an overall good agreement between numerical and experimental results.In Figure 5, three lateral extension tests are simulated.In this test, the sample is first submitted to a hydrostatic stress up to a prescribed value (noted P ci ).Then the lateral stress is decreased while the axial stress is kept constant.Axial and lateral strains are obtained as functions of the deviatoric stress.This kind of loading path is particularly interesting in rock mechanics field and used as a fundamental test to check the validity of constitutive models.Indeed, such a loading path is relevant to represent stress disturbances around a tunnel due to excavation.The comparison in Figure 5 shows a good correlation.In Figure 6, two proportional compression tests are simulated.In this test, the sample is submitted to a deviatoric stress state with a constant ratio between the lateral and the axial stresses.A good correlation is obtained both for high and low stress ratios.As a conclusion, the proposed model seems to be valid to describe the basic mechanical behaviour of porous rocks, both in "oil" and "water" saturation conditions.
In order to check the capacity of the proposed constitutive model to predict plastic deformation of chalk due to water flooding, the model is applied to simulate the water injection tests presented in the first part of this study.The simulations have been carried out by using a Finite Element Method code.However these tests have been performed on samples coming from a different block of chalk than those used for all the previous tests: pore collapse is higher for this new set of samples.Different model parameters should be determined.By putting the main aim here on the modelling of the water induced strain, some simplifications have been proposed.Only two representative plastic parameters depend on the saturating fluids (the pore collapse threshold k 0 and the tensile strength p 0 ), and the others have been considered as independent of the saturating fluid and an average value has been chosen for each one.The sets of parameters used for the following simulations are summarised in Table 2.
Numerical results in different water injection conditions are presented and compared with experimental data.In Figures 7 and 8, we present the evolution of axial and lateral strains at three height levels of sample (level locations are presented in the experimental study, see Homand and Shao (2000)).There is a good correlation between numerical and experimental results for the axial strain.Quantitative scatters are observed on the lateral strain.In Figures 9 and 10, we present the evolution of the axial and lateral strains at the middle point of sample, obtained in the different water injection tests under constant stress conditions listed in Table 3.   Once again, a good correlation is obtained.In each case, the qualitative response of chalk, for instance, the instantaneous and local character of the induced plastic deformation during the passage of water front, is fully reproduced.From a quantitative point of view, the magnitude of induced strains is also correctly recovered in most cases.
We should note that the induced plastic strains are not uniform in sample due to water flooding.The values measured by gauges represent an average strain in the small zone around the gauge location while the numerical values are obtained at the closest Gauss points to gauge centres.
In addition, a special surface treatment was necessary to glue strain gauges on porous chalk.Errors are certainly included in experimental data, in particular when strains are small like the case of lateral strain.Therefore, the quality of the numerical simulations seems to be satisfactory.Moreover, a linear water plastic potential g w has been used for this study and it seems that a more elaborated potential (elliptic potential for example) should be used.that the chalk presents a behaviour jump between two material states when it is flooded by water.The validity of the model is checked by simulating water injection tests under different stress conditions.A good quality of numerical predictions is obtained.

CONCLUSIONS
The proposed model will be applied to study oil reservoir compaction during water injection process.Simulation of the axial displacement induced by water injection in a soltrol saturated chalk sample under constant stress conditions.P c = 17 MPa, q = 3.5 MPa.

Figure 8
Simulation of the radial deformations induced by water injection in a soltrol saturated chalk sample under constant stress conditions.P c = 17 MPa, q = 3.5 MPa.
is to consider an average of the different values obtained for different confining pressures.β 0 is obtained by plotting the plastic volumetric strain ε d v as a function of the plastic deviatoric strain γ d : β 0 is the initial slope of this curve.This parameter actually governs the partition of the deformation tensor into its volumetric and deviatoric parts: (16) then: dε d v /dγ d = β with β = β 0 e -bγ d and: dε d v /dγ d | γ d =0 = β 0 at

Figure 2
Figure 2 Simulations of hydrostatic tests for different saturating fluids (water and soltrol).
Figure 7 Figure 9Simulation of the axial deformations induced by water injection in a soltrol saturated chalk sample under various constant stress conditions (gauges located on the middle of the sample).

Figure 10 Simulation
Figure 10Simulation of the radial deformations induced by water injection in a soltrol saturated chalk sample under various constant stress conditions (gauges located on the middle of the sample).

TABLE 1
Values of model parameters for two material states of chalk