CMWR'94, XTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL METHODS IN WATER RESOURCES, HEIDELBERG, GERMANY, J ULY 19-22, 1994

CONTINUUM MODELING OF COUPLED THERMO-HYDRO-MECHANICAL PROCESSES IN FRACTURED ROCK R. Ababou*, A. Millard*, E. Treille*, M. Durin*, F. Plas** *Commissariat à l'Energie Atomique, Centre d'Etudes de Saclay, DRN/DMT/SEMT, 91191 Gif-sur-Yvette Cedex, France.

**Agence Nationale pour la Gestion des Déchets Radioactifs, 92260 Fontenay-aux-Roses, France.

REFERENCE OF THIS PAPER : R. Ababou, A. Millard, E. Treille, M. Durin, F. Plas, 1994 : "Continuum Modeling of Coupled Thermo-Hydro-Mechanical Processes in Fractured Rock". Computational Methods in Water Resources X. A.Peters et al (eds.), Kluwer Academic Publishers, Netherlands, pp.651-658.

DOCUMENT FORMAT, ADDENDUM, ERRATUM : This document reproduces the contents of the paper by Ababou et al 1994 (referenced just above). Only the format of the original paper has been changed, resulting in a larger number of pages. The contents (text and figures) are identical to the original paper, except for one erratum (in equation 1) and one addendum (in the list of references) as indicated in the footnotes.

CONTINUUM MODELING OF COUPLED THERMO-HYDRO-MECHANICAL PROCESSES IN FRACTURED ROCK R. ABABOU*, A. MILLARD*, E. TREILLE*, M. DURIN*, F. PLAS** *Commissariat à l'Energie Atomique, Centre d'Etudes de Saclay, DRN/DMT/SEMT, 91191 Gif-sur-Yvette Cedex, France. **Agence Nationale pour la Gestion des Déchets Radioactifs, 92260 Fontenay-aux-Roses, France.

ABSTRACT An equivalent continuum approach for modeling coupled Thermo-Hydro-Mechanical (THM) processes in 2-D and 3-D fractured rock is presented. The equivalent, homogenized, model can be used for studying the stability of nuclear waste geologic repositories, e.g., the response of water-saturated fractured rock, to disturbances caused by excavation and heat production. The continuum approach leads to anisotropic hydromechanical equations, equivalent to Darcy's law and to Biot's poro-elastic equations, with smoothly variable tensorial coefficients. A hypothetical 2-D test problem, involving a complex network of thousands of fluid-filled joints, is used for illustration of the approach. The effect of homogenization scale on hydraulic conductivity is examined.

INTRODUCTION Brittle rocks like granite are considered as a potential host rock for the underground storage of high level nuclear wastes. The overall response of the rock mass and of groundwater, due to excavation and to subsequent heat production by wastes, generally involves coupled Thermo-Hydro-Mechanical (THM) phenomena. It is important to model such phenomena in order to appraise the environmental safety of a mined geologic repository. In this paper, we present an effective approach for modeling coupled THM processes in water-filled fractured rock. The processes are modeled as if they were taking place in a continuum, without explicitly simulating individual joints. It is assumed that natural fractures can be classified in two categories: (i) major fractures or faults extending through a large part of the domain; and (ii) fractures or joints of a lesser extent. The model presented in this paper concerns the second category, where the joints may have irregular or 'random' spatial distribution, and a broad spectrum of sizes and orientations. Major fractures or faults can be taken into account separately using a discrete approach. The advantage of the continuum approach is that it can be used for modeling coupled THM processes in the presence of many, variously oriented joints, while a discrete joints approach would become rapidly untractable as the number of joints and their geometrical complexity increases.

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Starting from a known rock matrix with a given distribution of joints, the constitutive laws of the equivalent continuum are obtained by a linear superposition approach, based on the methods developed by several authors, in particular Snow [1] for hydraulics, and Oda [2] for hydro-mechanics. The continuum equations are formulated for a 2-D or 3-D fractured rock made up of an elastic impervious matrix, and irregularly distributed, water-filled elastic joints. A hypothetical 2-D test problem involving thousands of joints is used for illustration. Here, we focus on certain essential features of the coupled continuum model, rather than on detailed simulation results.

BASIC ASSUMPTIONS Dimensionality and Geometry In 3-D space, the fractured rock is assumed to be made up of intact rock and planar joints or fractures, with known shapes, lengths, orientations, and apertures. The 2-D case arises when the fracture planes are all parallel to a given direction, e.g. horizontal. In this case, fractures may be represented by line segments in the plane orthogonal to all fractures, e.g. vertical. The assumption of plane strains can be used to model THM processes within the plane. This is the case for the 2-D model problem to be described further below (application section).

Thermal Processes In general, THM processes are fully coupled. However, assuming that fluid velocities in the joints are sufficiently small, heat convection effects can be neglected. Moreover, heat conduction can be approximated based solely on intact matrix properties. With these assumptions, heat transport is only 'one-way' coupled to hydro-mechanical processes. The temperature field history can therefore be calculated independently, then injected into the hydro-mechanical model. Full thermal coupling, including heat convection and the effect of joints, is postponed to a later stage.

Hydro-Mechanical Processes The mechanical behaviour of joints is assumed to be linear-elastic in compression, tension, shear. Nonlinear joint behaviour, e.g. through a Mohr-Coulomb elastoplastic model, may be considered later. The hydraulic behavior of joints is assumed to be governed by Poiseuille's law, which can be viewed as an approximation to the full NavierStokes equations (neglecting inertial terms, transient effects, and non-planar flow components within each joint). The intact rock matrix is a homogeneous, isotropic, elastic medium, satisfying Hooke's law. Finally, the matrix is assumed impervious given the low permeability of intact granite.

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Local Constitutive Laws and Equations Ø For the rock matrix, we use Hooke's law

σij = λekkδij+2µeij, where λ and µ are the matrix Lamé coefficients (λ ↔ compression, µ ↔ shear). Ø Within each water-saturated fracture, we use Terzhagi's "effective" stress concept, namely :

σ'ij = σij+pδij, where the convention of negative compressive stress is used. That is, (σ'ij) is equal to total stress (σ ij) minus negative fluid stress (-pδij). Ø We then assume that each joint behaves elastically. Thus, starting from an equilibrium state, crack aperture 'a' varies linearly with normal effective stress (σ'n) :

δσ'n = Knδa. Similarly, crack length l varies linearly with shear effective stress (σ's) :

δσ's = Ksδ l . The normal and shear crack stiffness coefficients, Kn and Ks , are taken constant, independent of stress, same for all fractures (Kn ≈ 1011 Pa/m, Ks ≈ 1010 Pa/m). Ø Finally, fracture flow obeys Poiseuille's law :

V = a2(g/12ν) J , where V is the average velocity vector in the joint, 'g' the acceleration of gravity, 'ν' the kinematic viscosity, and J=-∇ ∇ (p+ρgz)/ρg the hydraulic gradient.

EQUIVALENT CONTINUUM PROPERTIES We now consider an arbitrary set composed of N fractures having various apertures, lengths, and orientations. Equivalent homogenized properties are determined based on a linear superposition approximation, which may be applied either to the whole domain, or more generally, to a subdomain. The size of the homogenization subdomain, or homogenization scale l H , may be larger or smaller than the Representative Elementary Volume (REV). The choice of l H , and its relation to REV size, may have an important effect on the interpretation of results obtained with the equivalent continuum model. This point will be discussed later.

Equivalent Darcy Law (Hydraulics) Following Snow's superposition approach [1,2,3], a homogenized Darcy law is developed for quasi-steady flow in fractured rock. First, we use Poiseuille's cubic law to compute

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the discharge rate q of a single joint as a function of the hydraulic gradient J imposed at the scale of the domain. This yields

q = (a3g/12ν) J' , where J' is the projection of J on the fracture plane. If n is the unit vector normal to the fracture, then

J' = J - (J.n) n . Secondly, we superimpose the contributions of all fractures to the global flow, by arithmetic summation of the local q's, with J constant for all fractures. This yields a linear relation between the global or mean discharge rate Q (output), and the global hydraulic gradient J (input). The equivalent permeability tensor Kij , such that

Qi = Kij Jj , is expressed in terms of fracture apertures and lengths, or alternatively, fracture porosities and specific areas : N

(1)

Kij =

∑ K f (δ ij

−ni , f n j,f

)

f =1

with (special case of cartesian fracture geometry): (1b)

3 g af Kf = 12 í L f

or (for more general geometry) : (1c)

4 g Φ 3f Kf = , 12 í σ 2f

where Kf is the directional conductivity of fracture number 'f'. In these formulas, Ø ni,f is the ith component of the unit vector normal to fracture 'f', Ø af is fracture aperture, and Ø Lf is fracture spacing (2-D case) 1.

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ERRATUM - We have corrected here the final part of the sentence of the original CMWR 1994 proceedings paper, which mistakenly stated "and l f is fracture length (2-D case)" instead of the correct statement "and Lf is fracture spacing (2-D case)". Eq.(1b) was also incorrect in the original paper, with fracture length l f in the denominator, instead of fracture spacing Lf as it should be. The two forms of Kf (special 1b and general 1c) are connected through the following relations in the case of simple cartesian fracture geometry: Φ f = af/Lf and σf = 2(af+Lf)/Lf2 ≈ 2/Lf.

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The first Kf formula (1b) can be found in [1,2,3]. More generally, the second formula (1c) gives Kf in terms fracture porosity Φ f and fracture specific area σ f (accounting for both fracture walls). The second formulation is applicable to 2-D as well as 3-D fracture sets [3], including the case of arbitrarily shaped plane fractures. Remarkably, it generalizes the classical isotropic Kozeny-Carman formula. Finally, the equivalent hydraulic porosity is given by : (2)

Φ=

N

∑ Φf

.

f =1

Equivalent Poro-Elastic Mechanics)

Stiffness

Coefficients

(Mechanics

&

Hydro-

A strain-based superposition approach was developed in [2] to obtain equivalent hydromechanical laws for an elastic rock containing many cracks. The individual cracks or joints were assumed to behave elastically or quasilinearly under compression and shear [2], and to satisfy Terzaghi's "effective stress" approximation [4]. Our implementation assumes linear elastic laws with constant coefficients. The mean strain, due to the imposed global stress tensor σ ij, is calculated by linear superposition of the local displacements occuring throughout the intact rock matrix and the discrete joints, keeping the global stress constant. This leads to linear hydro-mechanical laws coupling solid stress and fluid pressure to solid strain and fluid strain (or fluid production), similar to the poro-elastic laws developed earlier by Biot [5]. The results can be summarized as follows. A relation between global variables eij , σ ij , p , i.e. fractured rock strain, total stress, and fluid pressure in the joints, is obtained : (3)

eij = Tijk′ l σkl + B′ij p

where T' ijkl and B' ij are equivalent homogenized coefficients, namely the total compliance coefficient (4th rank tensor), and the strain-pressure coupling coefficient (2nd rank tensor). Another relation can be derived from the same superposition method, based on fluid/solid mass conservation. This relation couples linearly 'fluid production' (ξ) to "effective" stress σ'ij : (4)

ξ=

1 1 N Fij σ ij′ with Fij = l f σ f (ni ) f (n j ) f , 2 f =1 Knl

∑

Note that ξ represents the net variation of volume of fluid per unit volume of the deformable medium; it is related to the spherical strains of the solid (e = ekk/3) and fluid ( ε = εkk/3) by the simple conservation equation

ξ = -3Φ (ε-e) .

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The 2nd rank tensor Fij is purely geometrical. Note that the 'prime' sign in equation (3) serves to distinguish the strain vs. stress formulation. The tensorial relations (3)-(4) need to be inverted in order to obtain suitable forms of the continuum model (for numerical reasons and for comparison purposes). The new unprimed coefficients involved in the stress vs. strain formulation below are stiffness coefficients, 'inverse' in some sense of the primed compliance coefficients. Now, the 4th rank tensorial equation (3) is inverted by using appropriate tensor algebra, and exploiting the particular symmetry properties of the elastic tensors, for the general case of non-orthotropic media. The inverted relation is found to be of the form : (5)

σij = Tijkl ekl − Bij p

Comparing term by term equations (3) and (5), and taking also into account that T and B satisfy the required symmetry properties of elastic tensors, one obtains indeed the correct relation between a symmetric 4th rank tensor and its inverse : (6)

(

)

Tk′lmn Tmnij = δ ki δ lj + δ kj δ li / 2 .

Comparing again (3) and (5) yields a relation between Biot's coefficient B and its reciprocal (B') : (7)

Bmn = Tmnij B′ij or Tk′lmn Bmn = B′kl

Finally, inserting σ'ij = σ ij+pδij in (4) and using previously established relations, gives also the equation coupling pressure to strains, or rather, to solid strain e and fluid production ξ : (8)

p = −G (Bkl ekl − ξ )

Remarkably, eqs.(5) and (8) are of the same form as those of Biot's theory [5], provided the latter are properly generalized for fully anisotropic poroelasticity. More precisely, it can be shown as a consequence of linear superposition and conservation equations, that the "Biot" coefficient Bij which appears in the stress/strain law (5), is identical to that appearing in the pressure/strain law (8). For fractured rock, the reciprocal Biot coefficient and the Biot modulus G are given by : (9)

B′ij = Fij /( Kn l) and G = {Bij′ (δ ij − Tijk l B′kl )}−1 = {Tijk′ l B kl (δ ij − Bij )}−1 ,

where l stands for mean fracture length. It will be interesting to specialize these relations to orthotropic and isotropic cases. In the isotropic case, let Kb be the volumetric or 'bulk' stiffness modulus of the drained medium, Ks that of the individual solid 'grains', and Kw that of water. Biot's coefficient is classically given as

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β = 1-Kb/Ks , Biot's modulus is given as

G = [(β-Φ)/Ks+Φ/Kw]-1 , and bulk stiffness is related to Lamé constants by

Kb = (λ+2µ)/3. Our tensorial relations for fractured rock constitute a generalization of such isotropic poroelasticity formulas.

2-D APPLICATION : VARIABILITY, ANISOTROPY, SCALE EFFECTS For specific applications, an appropriate homogenization scale l H must be selected. When l H is chosen less than domain size, the equivalent coefficients and constitutive laws are non-unique, being spatially variable, and scale-dependent. Also, the 'Representative Elementary Volume' is worth investigating. We focus here on the equivalent hydraulic conductivity tensor.

Fractured Rock Data (2-D) The 50 m × 50 m domain represents a 2-D vertical cross-section of a hypothetical rock formation. The rock mass contains 6580 fractures of various orientations, apertures, and lengths. Figure 1 depicts the entire fracture network. Each fracture is represented as a thin segment, so that the various apertures are not being displayed. The geometric data of the network are : (i) fracture counter; (ii) (x,y) location of fracture midpoint; (iii) angle θ of the unit normal vector n; (iv) fracture aperture 'a'; and (v) fracture length 'l '. Several statistical properties were analyzed, including probability distributions of a, l , θ: Ø Fracture orientations were only roughly uniform (isotropic). A "deficit" of fractures was observed for normal angles θ between 3π/4 and π (fractures inclined at angles between π/4 and π/2 over horizontal axis x). Ø Fracture lengths have positively skewed distribution, similar to lognormal, with a mean of 1.83 m and a coefficient of variation (CV) of 58%. Ø Apertures are also positively skewed, but closer to an exponential distribution, with a mean of 4.57 µ (microns) and a CV of 86%. Ø The equivalent porosity of the rock mass, by equation (2), is Φ ≈ 2.2 10-5. The mean fracture density is ρ ≈ 2.632 fractures/m2.

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Hydraulic Conductivity Tensor at the Largest Scale ( l H = 50m) The Kij tensor was computed at the scale of 50m x 50m, in the natural (x,y) system, using eq.(1). The principal components were then obtained by expressing Kij (m/s) in the principal system (x*,y*) :

K ij * =

+0.127E-08

0.000E+00

0.000E+00

0.000E+00

+0.767E-09

0.000E+00

0.000E+00

0.000E+00

+0.204E-08

with the convention that the first component is the largest, and the second is the smallest (the z component is irrelevant and therefore not concerned by this rule). The rotation angle from non-principal (x,y) to principal (x*,y*) system, is γ = -45.34 degrees. This gives the direction of largest conductivity, roughly orthogonal to the direction for which there is a deficit of fractures, as expected. The global anisotropy ratio is moderate, A = K11*/K22* ≈ 1.7.

Anisotropy and Spatial Variability The above calculations can be repeated at smaller scales, e.g. based on a partition of the 50m x 50m domain into subdomains. Figure 2 depicts conductivity tensors obtained on 6.25m x 6.25m subdomains, corresponding to a partition into 8x8 squares. The "local" Kij 's are represented by their anisotropy ellipse, or directional conductivity ellipse, of major axis (K11*)1/2 and minor axis (K22*)1/2. There are 64 such ellipses. Their orientations and aspect ratios indicate the principal directions and the square-root anisotropy ratio of the local Kij 's. The ellipses' spatial variability is quite apparent. Some of the local anisotropy ratios are significantly larger than the global ratio A≈1.7.

Scale Effects Increasing homogenization scale l H, or coarsening the subdomains, should smooth out spatial variability. Let us define a 'spherical' conductivity, K* = (K11*2+K22*2)1/2, for each subdomain, and plot K*(x,y) using a standard plotting package. The results are shown in Figure 3, for l H = 5m and l H = 10m. The "tunnel" cavity in the middle of the domain gives the scale of inhomogeneity that must be resolved by the numerical mesh. Another analysis of the Kij 's was developed based on nested subdomains of increasing sizes, growing from the center of the domain. Each Kii* was plotted versus l H (not shown here for lack of space). The plots indicated an REV ≈ 25m. Plotting instead A and γ versus l H, suggests a smaller REV ≈ 15-20m. It should also be kept in mind that scaledependence and REV size are relative to the property being studied.

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SUMMARY AND OUTLOOK Equivalent hydromechanical laws were obtained by linear superpositions of local fluxes and local strains, and then identified with anisotropic (non-orthotropic) versions of Darcy's law and Biot's theory of poro-elastic mixtures. The equivalent permeability tensor, stiffness tensor, and pressure coupling coefficients, were expressed in terms of (i) elastic properties of rock matrix, (ii) hydraulic and elastic properties of joints, and (iii) geometric properties of joints. The anisotropic 2nd rank tensors were given explicitly (the non-orthotropic 4th rank stiffness coefficient Tijkl , not given here, is of similar form). The spatial variability and scale-dependence of Darcy's equivalent conductivity tensor was studied for a 2-D fractured rock containing thousands of joints. New criteria and guidelines need to be developed for the 'optimal' choice of homogenization scale. Our model is being applied to a hypothetical case involving excavation of a gallery, and heating, in a 2-D fractured rock mass. The coupled THM equations are solved numerically for the equivalent anisotropic continuum, using 'CASTEM 2000', a general-purpose finite element code of the Commissariat à l'Energie Atomique. The primal variables are temperatures, displacements with quadratic basis functions, and pressure with linear basis functions. The elements are 6 node triangles and 8 node quadrangles. Time integration is done with an implicit one-step scheme. While preliminary simulation tests were restricted to elastic behavior, the homogenization approach described in this paper has been extended to model the sensitivity of material properties with respect to the ambient fields of strains, stresses, etc. Such nonlinear feedback couplings will be taken into account in future. The possible effects of local nonlinear joint behavior should also be appraised. Comparisons of the continuum simulations with discrete joint or discrete element hydro-mechanical models, will help assess and validate key features of the continuum model.2

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ADDENDUM - Concerning latest developments of this work, i.e. results collected or analyzed soon after this paper was printed in 1994, see the subsequent collective paper by A.Stietel, A.Millard, E.Treille, E.Vuillod, A.Thoraval and R.Ababou 1996 (the reference of this paper has been added to the original list of references below).

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REFERENCES 3 [1] Snow D.T. (1969). Anisotropic Permeability of Fractured Media. Water Resour. Res., 5(6), 1273-1289. [2] Oda M. (1986). An Equivalent Continuum Model for Coupled Stress and Fluid Flow Analysis in Jointed Rock Masses. Water Resour. Res., 22(13), 1845-1856. [3] Ababou R. (1991). Approaches to Large Scale Unsaturated Flow in Heterogeneous and Fractured Geologic Media. Report NUREG/CR-5743, U.S.NRC, Washington D.C. [4] Terzaghi Von K. (1936). The Shearing Resistance of Saturated Soils and the Angle Between the Planes of Shear. First Int. Conf. Soil Mech., Vol.1, Harvard Univ., pp. 54-56. [5] Biot M.A. (1956.a & b). Theory of Deformation of a Porous Viscoelastic Anisotropic Solid. J. Appl. Phys. 27, 459-467 (1956.a). General Solutions of the Equations of Elasticity and Consolidation for a Porous Material. J. Appl. Mech. 23, 91-96 (1956.b).

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ADDITIONAL REFERENCE :

A.Stietel, A.Millard, E.Treille, E.Vuillod, A.Thoraval and R.Ababou 1996 : "Continuum Representation of Coupled Hydromechanic Processes of Fractured Media : Homogenisation and Parameter Identification", in Developments in Geotechnical Engineering, Vol. 79 : "Coupled Thermo-Hydro-Mechanical Processes of Fractured Media" (O.Stephansson, L.Jing and C-F.Tsang, eds.), Elsevier Science, 1996, pp.135-164.

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