THM Processes in a Fluid-Saturated Poroelastic Geomaterial: Comparison of Analytical Results and Computational Estimates

Transcription

1 ROCKENG09: Proceedings of the rd CANUS Rock Mechanics Syposiu, Toronto, May 009 (Ed: M.Diederichs and G.Grasselli) THM Processes in a Fluid-Saturated Poroelastic Geoaterial: Coparison of Analytical Results and Coputational Estiates A. P. Suvorov & A.P.S. Selvadurai McGill University, Montreal, Canada ABSTRACT: The theory of poroelasticity, extended to include theral effects, provides a useful odel for the study of the quasi-static response of fluid-saturated geoaterials subjected to heating. The basic odel has been used quite extensively for odeling thero-hydroechanical responses of rock encountered in nuclear waste anageent endeavours and geotheral energy extraction endeavours. The practical application of these theories invariably requires recourse to coputational approaches where the governing partial differential equations are solved using Galerkin finite eleent techniques along with a suitable tie-integration technique that assures stability of the solution. In recent years a nuber of ulti-physics codes and general purpose finite eleent codes have been advocated for the study of such THM responses. The unconditional accuracy of these coputational approaches can be assessed only by recourse to coparisons with known analytical solutions. This paper exaines the capabilities of two coputational codes in predicting the thero-poroelastic response in a colun of heated geoaterial subjected to heat diffusion and pore pressure dissipation through the upper surface, and to surface tractions. 1 INTRODUCTION This paper deals with the theroelastic effects in saturated geoaterials. Teperature changes can cause deforations of the pore water and the solid phase, which can lead to changes in pore pressure and effective stress. An increase in the pore pressure, in particular, can cause daage to the solid skeleton. The isotheral theory of three-diensional consolidation proposed by Biot (1941, 1956) was extended by Rice and Cleary (1976) to include the effects of copressibility of the water and solid phase. Booker and Savvidou (1985, 1989) specifically solved the proble of consolidation of a porous edia in the presence of heat sources: point, disk, cylindrical and spherical sources. The investigations of Selvadurai & Nguyen (1995, 1996) and Nguyen and Selvadurai (1995) deal with the thero-hydro-echanical behavior of fractured and intact geoaterials proposed for storing heat-eitting nuclear fuel waste. Rutqvist et al. (001) presented a coparative study of four theories and coputational ipleentations for both fully and partially saturated geoaterials which are subject to theral loadings. Thero-hydroechanical probles for soils with an elastoplastic skeletal aterial were considered by Lewis et al. (1986) and Hueckel et al. (1987). Reviews of recent developents in the theory of linear poroelasticity with applications are given by Selvadurai (1996), Wang (000), Auriault et al. (00), Coussy (004) and Abousleian et al. (005). This paper deonstrates the use of the coputational ulti-physics code COMSOL for solving a thero-hydro-echanical proble in geoechanics. COMSOL is a finite eleent code that allows the user to enter, as an input, the governing partial differential equations. For the purpose of validating the COMSOL software, we use an analytical solution for a onediensional proble of a fluid-saturated geoaterial, initially at a unifor teperature and flu- PAPER 410 1

2 ROCKENG09: Proceedings of the rd CANUS Rock Mechanics Syposiu, Toronto, May 009 (Ed: M.Diederichs and G.Grasselli) id pressure, which undergoes heat diffusion and pore pressure dissipation due to the reduction of the surface teperature and pressure to zero. In addition, the geoaterial colun is subjected to a noral traction at the surface. A further aspect of the research is to provide an inter-code validation of the accuracy of COMSOL through a coparison with results obtained for the onediensional proble, using the general purpose finite eleent code ABAQUS. The paper is organized as follows: The governing equations and the three-diensional theory are presented in Section 1. An analytical solution for the one-diensional proble is presented in Section. In Section we describe the inforation the COMSOL user should provide to solve a thero-hydro-echanical proble via the finite eleent approach. Finally, in Section 4 we copare the results of the analytical solution fro COMSOL and ABAQUS. GOVERNING EQUATIONS Consider a fully saturated poroelastic ediu subjected to external echanical loading and heating. The poroelastic ediu consists of two phases - the porous solid and the liquid occupying the pore space. The porous solid is assued to be isotropic, linearly elastic, and locally non-deforable (i.e. rigid grain aterial). It is convenient to use concise notation in which spatial coordinates x, y, z are replaced by x ( x1, x, x) In the absence of body forces, the total stresses in the poroelastic ediu σij ust satisfy equilibriu equations, σ ij, j = 0 (1) The Duhael-Neuann for of the constitutive relationship that accounts for theral effects due tot and pore fluid pressure effects due to p takes the for, σ GD ij = GDε ij + ( K D ) εv δ ij K D α δ δ () st ij p where p is the excess pore pressure, T is the teperature change, ε ij are the strain coponents, ε V = ε kk is the voluetric strain, α s is the linear theral expansion coefficient of the solid phase, K D, GD are the bulk and shear odulus of the geoaterial under drained conditions. Note that absence of the theral expansion coefficient of the fluid α f in () does not iply that it has to be zero. The theral expansion of the fluid, along with the theral expansion of the solid phase, affects the value of the pressure. It is worth noting that when drainage is allowed, the fluid pressure in the geoaterial will dissipate with tie. Thus, the echanical properties K D, G D and theral expansion coefficient of the drained geoaterial will be equivalent to those for a porous geoaterial skeleton with epty pores. However, even when pressure is zero, the liquid is technically present in the porous fully saturated geoaterial. Therefore, the teperature field in () ust be obtained as a solution of the heat transfer (conduction) equation for a porous ediu with liquid filled pores. The infinitesial strain coponents are related to the displaceent coponents u i by the relationship 1 ε ij = ( u i, j + u j, i ) () Substituting () and () into the equilibriu equations (1) leads to, GD ( K D + ) uk, ki + GD ui p, i K D α st, i = 0 (4) Another governing equation can be derived fro the void occupancy equation and Darcy s law. The void occupancy equation states that the volue outflow fro a geoaterial eleent ust atch the decrease in volue of the eleent plus any increase in volue of the constituents due to an increase in teperature: t p n v dt+ ε + n = n( α ) T + (1 n)( α ) T (5) 0 ll, V f s K f ij PAPER 410

3 ROCKENG09: Proceedings of the rd CANUS Rock Mechanics Syposiu, Toronto, May 009 (Ed: M.Diederichs and G.Grasselli) where v l, l is the divergence of fluid velocity, n is porosity, In turn, Darcy s law can be written as, kl n vl p, K f is the bulk odulus of the fluid. = (6) γ where kl are coponents of the pereability tensor, and γ is the unit weight of the fluid. Substitution of (6) into (5) and differentiation of (5) with respect to tie results in another governing equation k du l n dp dt dt p, + + = n( ) + (1 n)( ) (7) γ dt K dt dt dt ll, l α f αs f The teperature field ust satisfy the heat conduction equation, T ceq keq T = 0 (8) t where c eq is the effective specific heat of the porous ediu and k eq is the effective theral conductivity of the porous ediu. To find these coefficients, it is possible to use, for exaple, volue averaged estiates. Thus, the specific heat and conductivity can be estiated as, ceq = nρ f c f + ( 1 n) ρscs k eq nk f + ( 1 n) ks = (9) where ρ f and ρs are the densities of the liquid and solid phase, c f and cs are their specific heat capacities, and k f, ks are the theral conductivity of the liquid and solid aterial. The governing equations (4), (7) and (8) ust be accopanied by initial and boundary conditions. Initial conditions are prescribed for the teperature and the pressure within the doain Ω, p ( x, t = 0) = p0 in Ω The boundary conditions on the boundary Γ = Ω are, T ( x, t = 0) = T 0 ( x ) in Ω (10) T ( x, = TΓ on Γ T1; keq T, i ( x, = qi on Γ T ; p( x, = pγ on Γ P1 kl p, = hl on Γ P ; ui ( x, = uiγ on Γ U1 (11) γ where σ ij ( x, n j = ti on Γ U ΓT 1 ΓT = Γ, ΓP 1 ΓP = Γ, ΓU 1 ΓU = Γ. SOLUTION OF ONE - DIMENSIONAL PROBLEM Consider an idealized proble of a one-diensional colun of fluid-saturated geoaterial occupying the region 0 x L being initially at an elevated teperaturet 0. During this heating, fluid drainage across the boundary x = 0 is prevented, which results in an initial fluid pressure build-up. Subsequently, the teperature and pressure at the surface of the region are reduced to zero and a copressive load σ 0 is applied at the upper surface of the colun. The heat flux, fluid velocity and displaceent are zero at the botto surface of the colun. Assuing that there are no lateral displaceents u x, = u ( x, 0 ; ε V = u, (1) 1 ( = The boundary conditions for the teperature, pressure, displaceents and stresses are: p p ( x = 0, = 0 ; ( x = L, = 0 x PAPER 410

4 ROCKENG09: Proceedings of the rd CANUS Rock Mechanics Syposiu, Toronto, May 009 (Ed: M.Diederichs and G.Grasselli) T T ( x = 0, = 0 ; ( x = L, = 0 x σ x = ; 0 ( = 0, σ 0 σ 1 ( x = 0, = u ( x = L, = 0 The initial conditions for the teperature and pressure are: (1) T ( x, t 0) = T0 p ( x, t 0) = p0 = (14a) = (14b) The teperature field satisfying the partial differential equation (8) governing heat conduction and the boundary/initial conditions (1), (14a) is given by, (15) k T( x, T sin( x )exp( ; L c 4 π = 0 κ κ = π 4L Note that the total noral stress σ inside the colun is equal to the applied stress, eq eq Fro the constitutive equation () (, σ 0 σ x = (16) and, therefore, the noral strain is 4 ( K + GD ) ε K D α s T p = σ = σ 0 D (17) 1 ε = εv = ( σ 0 + K D α s T + p) (18) 4 K D + GD Equation (18) can be used in (7) to obtain a differential equation for the fluid pressure, i.e. k n 1 dp dt dt K D dt p, + [ + ] = nα f + (1 n)α s α s (19) γ K f 4 dt dt dt 4 dt K D + GD K D + GD Solution of equation (19) is obtained as the su of the solution of the hoogeneous equation H * p, and a particular solution of the inhoogeneous equation p, i.e., * p = H p + p (0) The hoogenous equation is obtained fro the equation above by setting T = 0, and the general solution is given by, H 4 k ( K + G ) π D D = ω ω = (1) L 4L n 4 γ (1 + ( KD + GD)) K f p ( x, C sin( x )exp( ; where C are unknown coefficients. Now we need to find a particular solution of the given equation by considering application of teperaturet. The particular solution of (19) is taken in the sae for as the teperature field (15), * π eq (, ) = sin( )exp( κ ) ; κ = () L 4L c eq p x t A x t To find the unknown coefficients A, () is substituted into (19). This gives k PAPER 410 4

5 ROCKENG09: Proceedings of the rd CANUS Rock Mechanics Syposiu, Toronto, May 009 (Ed: M.Diederichs and G.Grasselli) A K D 4T0 ( α s nα f (1 n)α s ) K D + 4GD / k π 1 n ( + ) κ γ 4L K D + 4GD / K f π = () Since the total fluid pressure ust satisfy the initial condition, 4 p( x, t = 0) = p0 = p0 sin L 1,,5 Fro (0) and (4) we can establish the following connection between the coefficients, 4 C = p A κ (4) 0 (5) The initial fluid pressure can be found fro the void occupancy equation (5) at tie t = 0 and constitutive equation (17), σ 0 + [ nα f + (1 n)α s ] T0 ( K D + 4GD / ) K D α st0 p0 = n 1+ ( K D + 4GD / ) K f To suarize, the total solution for the fluid pressure is given by, π p( x, = C sin( x ) exp( ω + A sin( x ) exp( κ L L where A is found fro (), C fro (5) and the initial pressure p0 fro (6). This copletes the solution for the pressure. Now the strain is derived using constitutive equation (18), 1 ε = ε V = ( σ 0 + K D α s T p) K D + 4GD / + The displaceent field can be found by integrating the strain, π (6) (7) 1 u = ( σ 0 + K D α s T + p) dx + U (8) K D + 4GD / where U is the constant of integration. This gives 1 8L D αs 0 κ KD + 4 GD / π L L cos( C x )exp( ωt ) L u = [ K T cos( x )exp( L cos( A x )exp( κt ) + σ0 ( y L )] L The constant of integration is found by requiring that the displaceent be zero at the lower surface, i.e., u x = L, 0. ( = (9) 4 RESULTS OBTAINED USING COMSOL AND ABAQUS The solution for the one-diensional proble described in the previous section is used to validate perforance of the ulti-physics code COMSOL. The solution was also verified using the ABAQUS finite eleent progra. We can define a sequence of auxiliary probles. In each auxiliary proble, the teperature is applied gradually within a short period of tie, starting PAPER 410 5

6 ROCKENG09: Proceedings of the rd CANUS Rock Mechanics Syposiu, Toronto, May 009 (Ed: M.Diederichs and G.Grasselli) fro a zero value and reaching a axiu value T 0. The sequence is fored by setting the duration over which the teperature rise takes place to zero. Note that the initial conditions for the original proble, in which the teperature is raised instantaneously tot 0, can be considered as a liit of the sequence of the solutions of these auxiliary probles. When the user solves the auxiliary probles with the help of ABAQUS, the step GEOSTAT- IC is not needed, since the initial teperature and all initial fields are indeed zero. (The step GEOSTATIC is used to find equilibriu state for the geoaterial subject to initial pressure and initial stresses.) One can confir that the solutions of the auxiliary probles converge, at tie t = 0, to the current solution, when the teperature is applied instantaneously. Therefore, the step GEOSTATIC ust not be used if the user solves the original proble with ABAQUS. In fact, this step will give a wrong solution, in which for exaple, the displaceent is zero at tie t = 0 for the case of incopressible fluid. Also, note that ABAQUS requires the input of the void ratio e instead of porosity. They are related as, n e = 1 n In the input file the initial pressure can be set equal to zero; the user does not have to evaluate the value of the initial pressure (6). When solving the proble with COMSOL the user has to odify certain dialog boxes. First of all, in the dialog box of the elasticity proble, in the section Load the user has to check the checkbox Include theral expansion and in the field Tep put in the nae of the variable denoting the teperaturet. Then, in the sae dialog box, the user has to add the body forces that are caused by the pressure gradient. Thus, according to (4), the paraeter body load F x ust be assigned the value p, x, the paraeter F y takes the value p, y. Note that the ters containing derivatives of teperature T, i need not be added to the fields of this dialog box since, for the thero-elastic proble, the teperature gradient is already included as a body force. Next, the user has to odify the ass conservation equation. COMSOL solves the Darcy s equation in the for of a piezo-conduction equation (Barenblatt et al., 1990; Selvadurai et al., 005; Selvadurai, 009) p ks S [ ( p + ρ f gd)] = Qs t η where S is the storage ter, Qs is the liquid source, g is the gravitational acceleration, ρ f is the fluid density, ks is the saturated pereability coefficient, η is the viscosity of the fluid. To atch this for of ass conservation law with the given equation (7), in the dialog box of the Darcy s equation, in the section Coefficients, the user has to add source ters according to (7). For the right-hand side of the Darcy s equation Q s, the user has to input vyt + n * * α f * ( Tt + T 0 / t0 * ( t < t0)) + (1 n) * * α s * ( Tt + T 0 / t0 * ( t < t0)) Here the vertical displaceent in COMSOL is denoted by v, the coordinate y = x, vyt is the derivative of v with respect to y and tie t, T 0 is the initial (applied) teperature, t0 is an arbitrarily chosen but sall value of tie. The expression ( t < t0) is 1 if the condition inside the brackets is satisfied, and it is 0 otherwise. The ter containing T 0 / t0hshould not be oitted since it represents an approxiate derivative of the Heaviside function T 0 () t with respect to tie. To eliinate the effect of self weight, the user can set gravitational acceleration equal to zero, i.e., g = 0. The storage paraeter S ust be set to n / K f. Thus, in the case of an incopressible fluid S = 0. Also, the user has to ake sure that k s / η = k / γ as in (7). As in ABAQUS, there is no need for the user to know the initial pressure (6), it can be set to zero. In the dialog box Conduction in the section Init, the user has to specify the initial teperature T 0. Also, in the section Theral Properties, the user can specify effective heat capacity and effective theral conductivity for the porous edia. PAPER 410 6

7 ROCKENG09: Proceedings of the rd CANUS Rock Mechanics Syposiu, Toronto, May 009 (Ed: M.Diederichs and G.Grasselli) 5 NUMERICAL RESULTS Consider the THM proble for a 1D colun of height L = 10. The colun occupies the doain 0 x L. The colun is initially at the elevated teperature equal tot0 = 100C and is acted upon by the non-zero pore pressure that builds up initially due to the absence of fluid drainage across the boundaries. The pressure and the teperature at the upper surface x = 0 are reduced to zero and the constant copressive stress σ 0 = 10 MPa is applied at the upper surface. The following properties are used: Table 1. Properties of soil Property Value Porosity n 0.5 Young s odulus E = 60*10 9 Pa Poisson s ratio ν = 0. Unit weight of water 9800 N/ Fluid pereability k =.94e-1 /s Effective theral conductivity k eq = 4 (W/ºC) Effective specific heat c eq = (J/kg/ºC) Linear theral expansion of solid phase α s = 8. *10-6 (1/ ºC) Linear theral expansion of liquid neglected Fluid bulk odulus K f = or.*10 9 Pa (two values are used to exaine effect of fluid copressibility) In the figures presented here the analytical solution is shown with a solid line and the COM- SOL solution is shown in dotted lines. Figure 1 shows the teperature distribution through the depth of the geoaterial colun at 1, 100, and 65 days after initiation of heat diffusion. At 0 tie t = 0 the teperature change was equal tot0 = 100 C. For t > 0 on the upper surface x = 0 the teperature change is reduced to zero. The coputational results obtained using the ABAQUS code are very close to the analytical and COMSOL results, and, thus, are not shown here. Figure shows the distribution of pore fluid pressure with depth within the colun subjected to the axial stressσ 0, aforeentioned teperature change, and zero pressure at the upper surface. In Figure, the results obtained using ABAQUS are shown as circles. As expected, with tie, the pressure dissipates inside the geoaterial. The axiu pressure is attained here at tie t = 0, for the case of an incopressible fluid. Note that the initial fluid pressure in the absence of the teperature change is equal to 10 MPa (which is consistent with undrained response of a saturated geoaterial) but with the teperature increase included, the pressure reaches the value 6. MPa. This value can be obtained fro the result (6). The positive value of the initial pressure caused by teperature change alone ( =6. MPa) can be explained in the following way: Theral expansion of the solid phase and zero theral expansion of the fluid would lead to pore fluid tension, and thus negative pressures, if the pores were free to defor in the lateral direction. However, due to constraint on lateral displaceents, u 1 = u = 0, the resulting fluid pressure is positive, i.e., the fluid is still in copression. Figure shows the absolute value of the surface displaceent. The displaceent is a axiu at tie t = 0 and at that tie it is caused entirely by heating. Then the displaceent is reduced due to the heat dissipation and the peranently applied copressive stress. Note that the heating (positive teperature change) causes the upward displaceent, while the applied copressive stress leads to the downward displaceent. Figure 4 shows the pressure distribution versus tie at depth 10, where the pressure takes the axiu value. The pressure is shown for two values of copressibility: zero (incopressible fluid) and 4.54e-10 1/Pa. Note that the pressure gets saller for larger values of copressibility, which is the expected result. Siilarly, Figure 5 shows the surface displaceent versus tie for the porous ediu saturated with an incopressible fluid and a fluid with copressibility 4.54e-10 1/Pa. PAPER 410 7

8 ROCKENG09: Proceedings of the rd CANUS Rock Mechanics Syposiu, Toronto, May 009 (Ed: M.Diederichs and G.Grasselli) Figure 1. Teperature distribution within a one-diensional eleent subjected to initial teperature change 100 o C and subsequent heat dissipation due to reduction of teperature at the upper surface to zero. Figure. Pore fluid pressure distribution within a one-diensional eleent subjected to the teperature change in Figure 1, and a copressive load 10 MPa at the upper surface. Pore fluid pressure at the upper surface is zero, and the fluid velocity is zero at the botto surface. Figure. Displaceent distribution within the one-diensional eleent subjected to the teperature change shown in Figure 1, and copressive load 10 MPa at the upper surface. Displaceent is zero at the base. PAPER 410 8

9 ROCKENG09: Proceedings of the rd CANUS Rock Mechanics Syposiu, Toronto, May 009 (Ed: M.Diederichs and G.Grasselli) Figure 4. Pore fluid pressure distribution at a 10 depth in a one-diensional eleent subjected to the teperature change of Figure 1, and copressive load 10 MPa at the upper surface. Figure 5. Surface displaceent of a one-diensional eleent subjected to the teperature change shown in Figure 1, and copressive load 10 MPa at the upper surface. 6 CONCLUSIONS In the present paper we have exained the effect of heating on the deforation of a fluidsaturated porous ediu. The solution for the one-diensional proble of a geoaterial colun initially at a unifor teperature and pore fluid pressure subjected to subsequent heat dissipation, is obtained analytically in the for of a power series expansion. In addition, the authors deonstrate the applicability of two coercial finite eleent codes COMSOL and ABAQUS to solve thero-hydro-echanical probles. The coputational results for a onediensional proble are validated through coparison with an analytical solution and satisfactory agreeent is observed. Acknowledgeents The work described in this paper was supported in part through a Nuclear Waste Manageent Organization Research Contract and in part through a NSERC Discovery Grant awarded to A.P.S. Selvadurai. PAPER 410 9

10 ROCKENG09: Proceedings of the rd CANUS Rock Mechanics Syposiu, Toronto, May 009 (Ed: M.Diederichs and G.Grasselli) 7 REFERENCES Abousleian, Y., Cheng, A.H.-D.& Ul, J.-F. (Eds.) 005. Poroechanics III, Proceedings of the rd Biot Conference in Poroechanics, Noran, OK, USA, A.A. Balkea, The Netherlands. Auriault, J.-L., Geindreau, C., Royer, P. Bloch, J.-F., Boutin, C.& Lewandowska, J. (Eds.) 00. Poroechanics II, Proceedings of the nd Biot Conference in Poroechanics, Grenoble, France, A.A. Balkea, The Netherlands. Barenblatt, G.I., Entov, V.M. & Ryzhik, V.M Theory of Fluid Flows Through Natural Rocks, Kluwer Acadeic Publ., Dordrecht. Biot, M.A General theory of three-diensional consolidation, Journal of Applied Physics 1: Biot, M.A Theory of deforation of a porous viscoelastic anisotropic solid, Journal of Applied Physics 1: Booker J.R. & Savvidou C Consolidation around a point heat source, International Journal for Nuerical and Analytical Methods in Geoechanics 9: Booker J.R. & Savvidou C Consolidation around a heat source buried deep in a porous theroelastic ediu with anisotropic flow properties, International Journal for Nuerical and Analytical Methods in Geoechanics 1: Coussy, O Poroechanics, John Wiley and Sons, New York. Hueckel, T., Borsetto M. & Peano, A Modelling of coupled thero-elastoplastic-hydraulic response of clays subjected to nuclear waste heat. In Nuerical Methods in Transient and Coupled Probles: (R.W. Lewis, E. Hinton, P. Bettess and B.A. Schrefler, Eds.),Wiley, Chichester, pp Lewis, R.W., Majorana, C.E. & Schrefler, B.A A coupled finite eleent odel for consolidation of nonisotheral elastoplastic edia, Transport in Porous Media: 1: Nguyen, T.S. & Selvadurai, A.P.S Coupled theral-echanical hydrological behavior of sparsely fractured rock : Iplications for nuclear fuel waste disposal, International Journal of Rock Mechanics and Mining Sciences (5): Rice, J. R. & Cleary, M.P Soe basic stress-diffusion solutions for fluid saturated elastic porous edia with copressible constituents, Reviews of Geophysics and Space Physics 14: Rutqvist, J., Borgesson L., Chijiatsu M., Kobayashi A., Jing L., Nguyen T.S., Noorishad J., &Tsang, C.F Therodynaics of partially saturated geological edia: governing equations and forulation of four finite eleent odels, International Journal of Rock Mechanics and Mining Sciences 8: Selvadurai, A.P.S Influence of residual hydraulic gradients on decay curves for one-diensional pulse tests, Geophysical Journal International (in press) Selvadurai, A.P.S Mechanics of Poroelastic Media, Kluwer Acadeic Publ., The Netherlands. Selvadurai A.P.S. & Nguyen, T.S Coputational odeling of isotheral consolidation of fractured porous edia, Coputers and Geotechnics 17: 9-7. Selvadurai, A.P.S. & Nguyen, T.S Scoping analysis of the coupled theral-hydrologicalechanical behaviour of the rock ass around a nuclear fuel waste repository, Engineering Geology, 47: Selvadurai, A.P.S., Boulon, M.J. & Nguyen, T.S Pereability of intact granite, Pure and Applied Geophysics 16: Wang, H.F Theory of Linear Poroelasticity with applications to Geoechanics and Hydrogeology, Princeton Univ. Press, Princeton. PAPER