Numerical computation of hydrothermal uid circulation in fractured Earth structures

Transcription

1 Geophys. J. Int. (1998) 135, 627^649 Numerical computation of hydrothermal uid circulation in fractured Earth structures Jianwen Yang, K. atychev and R. N. Edwards Geophysics Division, Department of Physics, University of Toronto, Toronto, Ontario, M5S 1A7, Canada Accepted 1998 June 25. Received 1998 May 8; in original form 1997 December 1 SUMMARY Hydrothermal uid circulation through porous Earth materials is an important physical phenomenon occurring in both submarine and continental environments. Irregularly interconnected discrete fractures are pervasive in nearly all Earth materials, providing preferential paths for uid ow and controlling the circulating uid patterns. Most mathematical algorithms addressing hydrothermal convection problems treat rocks as piecewise continuous media. The representation of local, large changes in permeability requires a high level of discretization for accurate results and a corresponding large number of unknowns. The alternative is to incorporate fractures discretely through special adaptation of the numerical code. We adopt this approach to solve the coupled, time-dependent heat and uid transport di erential equations using the nite element method. The nal algorithm is validated against both an analytical solution and numerical solutions from a complementary but less general nite di erence scheme. Case studies of some simpli ed fractured models indicate that fractures can induce and maintain hydrothermal uid circulation in media which would otherwise be passive. Fracture location can control both convection pattern and vigour in a closed system. Discrete fractures can also signi cantly change an established convection pattern. Multiply fractured porous media are comparable with the homogeneously anisotropic media in the numerical solutions if the e ective average horizontal and vertical permeabilities are kept the same. Key words: anisotropic medium, discrete fracture, nite di erence method, nite element method, fractured porous medium, hydrothermal circulation. 1 INTRODUCTION Hydrothermal uid circulation is an important physical phenomenon characteristic of the subsurface of the Earth. For example, in the submarine environment, it is an extremely e cient mechanism for the exchange of heat and matter between sea water and oceanic crust (Williams & Von Herzen 1974; Sclater, Jaupart & Galson 1980; Stein & Stein 1994; Scott 1985; owell & Rona1985; Rona et al. 1986; Brimhall 1991). In the continental environment, circulating groundwater driven by the natural geothermal gradient or local heat sources, such as heat generated by buried nuclear fuel waste, may carry toxic contaminants that are a hazard to the health of humans and other living organisms. Earth structures universally contain fractures along which uid can be highly mobile. Fractures are pervasive in both oceanic crust and continental geologic strata (MacDonald 1982; Morgan 1991; Choukroune, Francheteau & Hekinian 1984; Herzog et al. 1989; Cherry 1989; Sudicky & Mcaren 1992). Fractured zones control the discharge of hydrothermal uid in both continental and submarine systems. Obvious examples are warm and hot springs on land, and the recently discovered black and white smoker plumes on the sea oor at the mid-ocean ridge (Germanovich & owell 1992; owell, Rona & Von Herzen 1995). Numerical computation of hydrothermal uid ow involves the solution of the pertinent coupled time-dependent mass and energy conservation di erential equations subject to conditions on the temperature and uid ow on boundaries (Fehn & Cathles 1979; Fehn, Green & Von Herzen 1983; Fisher et al. 1990, 1994; Fisher & Becker 1995; Rosenberg & Spera 1990; Rosenberg, Spera & Haymon 1993; Bessler, Smith & Davis 1994; Travis, Janecky & Rosenberg 1991; owell et al. 1995). Steady-state solutions may also be derived. The previous algorithms have represented the permeability parameter as a smoothly varying variable, requiring a high sampling density to represent properly the rapid variation in permeability across a fracture. An alternative is to write special governing and boundary conditions for the fracture and thereby include it implicitly within the software. Our group has developed a ß1998RAS 627 GJI000 16/10/98 08:59:07 3B2 version 5.20

2 628 J. Yang, K. atychev and R. N. Edwards nite element computational algorithm to simulate hydrothermal uid circulation in discretely fractured porous media, based on the latter concept. We have used the algorithm to verify a theory to explain the origin of small-scale sea oor heat- ow variations with a characteristic wavelength of about 1000 m over a sediment-sealed ridge ank (Yang et al. 1996a) and also applied it in the investigation of the hydrothermal system within the Trans-Atlantic Geotraverse (TAG) active sulphide mound. We obtained the internal temperature and uid velocity elds (Yang et al. 1996b). We are also applying the method to a broad range of other problems (e.g. Ryan, Yang & Edwards 1998a,b). This paper introduces for the rst time details of the computational method, both the principles and the schemes for their implemention into nite element software. The numerical algorithm is validated against an analytical solution of temperature distribution within a 2-D model containing a single fracture situated in an impervious host rock. Further, a complementary, limited-application, nite di erence algorithm is introduced to cross-check results for a single fracture in a permeable host. A number of unpublished case studies is added to illustrate the applicability of the code. 2 GOVERNING EQUATIONS The governing equations for the hydrothermal uid transport problem in porous media can be derived by considering the continuity of uid and heat in conjuction with Darcy's law. Detailed derivation of these equations can be found in standard textbooks (e.g. Bear 1972). We list the results brie y here. 2.1 Darcy equation The Darcy equation for uid ow is q~{ k p k w n xz p y n yz p zo wg n z, (1) where q is the Darcy ux vector, n x, n y and n z are three unit vectors in the x, y and z directions, respectively, and k, p, g, T are the permeability, the pressure, the gravitational acceleration, andthetemperature. The temperature-dependent variables k w and o w arethedynamic uidviscosityandthe uid density. By employing the `equivalent freshwater head' h de ned as (Frind 1982) h~ p zz, (2) o 0 g where o 0 is a reference freshwater density of 1000 kg m {3 and z is the elevation head, the Darcy equation (1) can be written in a simpli ed form as q~{k h n xz h y n yz h zo r n z, (3) where K is the hydraulic conductivity de ned by K~ ko 0g (4) k w and o r is the relative density of uid de ned by o r ~ o w o 0 {1. (5) 2.2 Continuity equations for uid and heat transport If the porous medium is non-deformable and there exists a thermal equilibrium between the uid and the solid matrix, the uid mass and energy conservation equations can be written as K h z y K h z K h y zko r ~0 (6) and j m T z y j m T y z T j m { (c wo w q x T) { y (c wo w q y T){ (c T wo w q z T){c m o m ~0, (7) t where q x, q y and q z are the Darcy ux components in the x, y and z directions, j m is the thermal conductivity of the rock matrix, c w is the uid speci c heat, t is the time and c m and o m are the speci c heat and density of the rock matrix, respectively. 2.3 Case of a discrete fracture Consider a planar fracture of width 2b which is bounded by two smooth and parallel walls. If the uid owing along the fracture is assumed to be in the laminar range and principally along the plane of the fracture, then the hydraulic conductivity in the fracture K f can be de ned as K f ~ o 0g(2b) 2. (8) 12k w This expression has been derived by many authors for the ow in fractures with parallel walls (e.g., amb 1945; Bear 1972; Bear, Marsily & Tsang 1993). It is also valid even if the walls undulate, a phenomenon associated with the application of stress (e.g. Witherspoon et al. 1980). The uid transport equations along the fracture can be obtained by integrating eq. (6) over the fracture width 2b. They are (2b) K h f z y K h f zq n j y I z{q n j I {~0 (9) and (2b) K h f z h K f zk fo r zq n j I z{q n j I {~0, (10) where eq. (9) corresponds to a horizontal fracture in the xoy plane, and eq. (10) to a vertical fracture in the xoy plane. The last two terms in these equations represent normal components of the uid leakage ux across the interface between the fracture and the porous medium. Similarly, if we assume that the uid is always instantaneously and thoroughly mixed across the fracture, so that thermal equilibrium exists between the uid and the solid matrix, and further that the variation in temperature across the fracture aperture is negligible and the temperature in the host rock immediately outside the fracture is approximately equal to the temperature of the fracture, then the heat transport equation can be also obtained by integrating eq. (7) over the GJI000 16/10/98 08:59:28 3B2 version 5.20

3 Hydrothermal uid circulation 629 fracture width. We obtain (2b) j T w z y j T w y T {c w o w q fy y {c T wo w t {c w o w q fx T z n j I z{ n j I {~0, (11) where q fx and q fy are the Darcy ux components over the fracture in the xoy plane. The last two terms represent the loss or gain of heat ux across the interface between the fracture and the host medium. 2.4 Supplemental equations The Darcy, mass and energy conservation equations are not alone su cient to describe a hydrothermal uid circulation system. We have to de ne also the dependence of uid density and viscosity on temperature. Empirical formulation can be obtained using polynomial interpolation functions to t the published database (Molson, Frind & Palmer 1992) and can be easily implemented in computer code (Yang et al. 1996a,b). In this paper, however, we assume that the uid viscosity is a constant and that the uid density is a linear function of temperature in order to introduce a dimensionless parameter, the Rayleigh number. The uid density is assumed to have the form o w ~o 0 (1{a v T), (12) where a v is the uid volume thermal expansion coe cient. When the temperature is small, the above assumptions are reasonable. 3 FINITE EEMENT (FE) METHOD 3.1 The Galerkin nite element technique The mass and energy conservation equations (6) and (7) above form a transient, non-linear system, coupled through Darcy's equation (3). The Galerkin nite element method and the Picard iterative technique are used to solve these equations (Huyakorn & Pinder 1983). Speci cally, we rst update the uid density and viscosity using the latest temperature T l, and determine the head distribution h lz1 on the basis of the updated uid properties by solving eq. (6), where l is the level of iteration. We then determine the Darcy ux distribution q lz1 on the basis of Darcy's law (eq. 3) using h lz1, and evaluate the temperature distribution T lz1 by solving the matrix eq. (7). Finally, we need to investigate whether the convergence criteria Max jh lz1 j {h l j j dh, Max jtj lz1 {Tj l j dt are satis ed. If not, h and T must be updated and the iterative process is repeated. A priori speci cation of the small quantities dh and dt de nes the convergence criteria for the ow and heat transport equations, respectively. In designing the nite element grid, the primary considerations are the grid Peclet and Courant criteria, de ned as (Daus, Frind & Sudicky 1985): P x ~ o x*x 2, P y ~ o y*y 2, P z ~ o z*z 2 (13) k m k m k m and C x ~ o x*t *x 1, C y~ o y*t *y 1, C z~ o z*t 1, (14) *z where k m is the thermal di usivity of the porous medium, *x, *y and *z are the grid spacings in three directions and *t is time step. 3.2 Numerical algorithm for a discretely fractured porous medium The numerical technique presented below was suggested by the work of Sudicky & Mcaren (1992), who applied a similar method to model groundwater movement and contaminant transport in fractured porous media. We use similar coupled equations to model groundwater movement and heat transport. The fracture network is included explicitly by superimposing a number of planar elements representing individual fracture segments onto the sides of the background volume elements. By arranging the planar and volume elements in this manner, the continuity of temperature and head at the fracture^matrix interface is automatically satis ed because each element type shares common nodes along the interface. Moreover, upon element assembly, the uid and heat exchange uxes across the interface are accounted for naturally, such that explicit calculation of the normal components of the uid leakage ux and the in ux heat ow and e ux heat ow is unnecessary. We will prove this point formally later. We shall use a 2-D model to explain the coding in the vicinity of a fracture. Planar fractures in 2-D are in nitely long in the strike direction and appear as lines in a cross-sectional view. Fig. 1(a) shows a 2-D fractured porous medium, in which the bold line represents a vertical discrete fracture with an aperture of 2b. The host porous medium is discretized as a rectangular grid, with 10 elements in the x direction and ve elements in the z direction, giving a total of 66 nodes, as shown in Fig. 1(b). The vertical fracture is incorporated onto the sides of the rectangular elements, and it shares the nodes 26, 27 and 28 with them. The model may be separated into the porousmedium part (Fig. 1c) and the vertical-fracture part (Fig. 1d), maintaining the same node numbering. To demonstrate the fundamental principle of the numerical scheme, let us consider the uid ow equation in two dimensions. If the head h is continuous across the interface between the fracture and the host medium, then the uid ow equations can be obtained for both the porous medium and the fracture. For the host porous medium, the governing di erential equation is K h z K h zko r ( 0 for the regions outside the fracture ~ (15) c gain for the regions containing the fracture and, for the particular case of a vertical fracture, it is h K f zk fo r ~{c loss, (16) where c gain, a uid ux per unit area, represents the uid gain of the host medium from the fracture, and {c loss represents the uid loss of the fracture to the host medium. To approximate the ow equations by the Galerkin nite element method, we discretize the head h using a trial function GJI000 16/10/98 08:59:39 3B2 version 5.20

4 630 J. Yang, K. atychev and R. N. Edwards Figure 1. (a) A 2-D fractured porous medium including a vertical fracture; (b) discretization of the model by a 2-D equal-spaced grid; (c) the host porous medium; (d) the discrete fracture. The host medium and fracture share the nodes numbered 26, 27 and 28. of the form h ª (x, z)~ X66 and h ª (z)~ X28 j~26 j~1 h j Nj 2 (x, z) for the porous medium (17) h j Nj 1 (z) for the vertical fracture, (18) where Nj 1(z) andn2 j (x, z) are, respectively, 1-D and 2-D basis functions, and h j are nodal values of h. The uid gain and loss variable c on the right-hand side of eqs (15) and (16) is a function of position and time. If we assume it is constant over a rectangular cell or a linear fracture element, we may write the integrated total uid,, lost by the fracture element gained by the cell as 2b z c loss and x z c gain, respectively, where x and z are the cell size in the x and z directions. We start by substituting eq. (17) into eq. (15), and eq. (18) into eq. (16). If we apply the Galerkin residual theory, we obtain " # K hª z K hª zko r Ni 2 (x, z)dxdz A ª (i~1, 2,..., 66) ~ Ni 2 (x, z)dxdz A x z (i~26, 27, 28) (19) and " # h K f A zk fo r Ni 1 (z)dz ~{ Ni 1 (z)dz A 2b z (i~26, 27, 28), (20) where A is the total area of the porous medium, A is the total length of the vertical fracture and the range (i~26, 27, 28) indicates that the vertical fracture shares nodes 26, 27 and 28 with the host porous medium. Eq. (20) can be rewritten as (2b) A " h K ª # f zk fo r Ni 1 (z)dz ~{ Ni 1 (z)dz (i~26, 27, 28). (21) A z Adding eq. (19) to eq. (21) yields a set of 66 equations, indexed by i, A " # K hª z K hª zko r z(2b) A ~ A x z { A z " K h ª # f zk fo r N 2 i N 1 i N 1 i N 2 i (x, z)dxdz (i~26, 27, 28) (x, z)dxdz (i~1, 2,...,66) (z)dz (i~26, 27, 28) (z)dz (i~26, 27, 28). (22) Eq. (22) is not the nal result. It simpli es signi cantly because the right-hand side actually vanishes. The proof is quite lengthy but is given here because it can be a source of debate. For the 2-D rectangular element as shown in Fig. 1(c), i and j~1, 2, 3, 4. The basis functions can be expressed as GJI000 16/10/98 08:59:49 3B2 version 5.20

5 Hydrothermal uid circulation 631 Figure 2. Diagram illustrating the FD discretization near the fracture. (Huyakorn & Pinder 1983) N 2 1 (x, z)~1{x/ x{z/ z zxz/ x z, (23) N 2 2 (x, z)~x/ x{xz/ x z, (24) N 2 3 (x, z)~xz/ x z, (25) N 2 4 (x, z)~z/ z{xz/ x z. (26) For the 1-D linear element as shown in Fig. 1(d), i~1, 2. The basis functions are N1 1 (z)~1{z/ z and N2 1 (z)~z/ z. (27) By direct integration of the right-hand side of eq. (22), on the basis of eqs (23)^(27), we have 2 3 I /2 Ni 2 (x, z)dxdz (i~26, 27, 28)~ 6 I /2z II /2 7 A x z 4 5 II /2 where I and II denote the uid gain or loss, respectively, over two linear fracture elements. Note that the fracture consists of two elements, as shown in Fig. 1(d). Substituting eqs (28) and (29) into eq. (22), we obtain " # K hª z K hª zko r Ni 2 (x, z)dxdz A z(2b) A " K h ª # f zk fo r N 1 i (i~1, 2,...,66) (z)dz (i~26, 27, 28)~0. (30) (28) and { Ni 1 A z 2 3 { I /2 (z)dz (i~26, 27, 28)~ 6 { I /2{ II / , { II /2 (29) Figure 3. A simple fracture^matrix system. GJI000 16/10/98 08:59:59 3B2 version 5.20

6 632 J. Yang, K. atychev and R. N. Edwards Eq. (30) indicates that, upon the element assembly shown in Fig. 1(b), the uid exchange uxes across the fracture interface are accounted for naturally and that explicit inclusion into the software of the normal components of the uid leakage ux in eqs (9) and (10) is unnecessary. Solving this equation gives rise to the nal solution of h for the fractured porous medium as a whole. A similar procedure can be also applied to solve the temperature distribution for the discretely fractured porous medium The details are available from the authors. 4 FINITE DIFFERENCE (FD) METHOD We have mentioned that the number of analytical solutions available to check the FE code is rather limited. We felt it necessary to introduce a second numerical method to solve a limited set of special cases to verify correct operation of the FE software. Our logic was to nd a simple scheme that could incorporate the main routines unmodi ed from a proven, widely distributed algorithm. The selected technique may be employed when the problem of interest contains, for example, Figure 4. Normalized temperature contours at a uid velocity of 5 10 {3 ms {1 : (a), (b) and (c) are the analytical solutions; (d), (e) and (f) are the numerical solutions, corresponding to the time levels of 1, 2 and 3 days. The contour intervals are from 0.1 to 1.0. GJI000 16/10/98 09:00:09 3B2 version 5.20

7 Hydrothermal uid circulation 633 a single fracture or an equivalent highly permeable vertical column. Consider the simple case of stationary convection in a 2-D isotropic domain attached to a vertical fracture of constant aperture 2b extending from the base to the top of the domain along its left lateral boundary, as shown in Fig. 2. The right boundary is assumed adiabatic and impermeable. The upper and lower boundaries are isothermal and impermeable. The solution sought is assumed symmetric about the plane containing the fracture. Physics requires that the normal uxes and the tangential elds at the fracture^medium interface be continuous. The iterative procedure is to compute the normal ux into the fracture from the domain, update the elds in the fracture and give this update back to the medium as boundary conditions. The numerical process follows this path iteratively until all the boundary conditions are satis- ed simultaneously. Theoretically, there is clearly a second option. The elds in the domain can be transferred to the fracture and updates of the normal uxes returned Figure 5. Normalized temperature contours at a uid velocity of 10 {2 ms {1 : (a), (b) and (c) are the analytical solutions; (d), (e) and (f) are the numerical solutions, corresponding to the time levels of 0.5, 1.0 and 1.5 days. The contour intervals are from 0.1 to 1.0. GJI000 16/10/98 09:00:10 3B2 version 5.20

8 634 J. Yang, K. atychev and R. N. Edwards to the domain. This option proved less stable; in fact, it is often divergent. To simplify the algebra, it will be convenient to introduce a stream function ( connected to q by q x ~{ 1 o w (, q z~ 1 o w (. (31) It is clear from eq. (31) that on the three impermeable boundaries, ( assumes a constant value, which we have the option of setting to zero. In terms of ( and T, the governing equations for the medium, eqs (6) and (7), become j m c w 1 o w K ( z 1 o w K ( 2 T 2 z 2 T 2 z ( T { ( ~{ o r, (32) T ~0, (33) where j m, c w and k w are constants, and both /y and T/t are equal to zero. This system is solved iteratively, incrementing the iteration index. At a given iteration step, eqs (32) and (33) are solved on a uniform rectangular grid, shown in Fig. 2, utilizing the reputable, user-friendly MUDPACK collection of nitedi erence multigrid solvers for elliptic PDEs (Adams 1989). The solvers operate in a black-box mode. The user is required to supply the coe cients of the equation and the boundary conditions. We select the option of default multigrid cycling, always starting it at the coarsest grid level, and Gauss^Seidel line relaxation in both the x and z directions, the latter chosen for its robustness for the type of problem at hand. The termination of the iterations is controlled by the maximum norm of the relative di erence between the last two computed approximations at the solution grid. Normally, it is set to 0.1^1 per cent for both eqs (32) and (33). et us derive the fracture boundary conditions starting from the stream function. Consider the portion of the fracture stretching from 0 to z, width2b, andleto x and o z be the x and z components of the uid velocity inside the fracture. Integration of eq. (16) over the fracture aperture and the vertical coordinate z results in o w o z 2b~{2 z 0 o x j x~b o w dz. (34) Figure 6. Normalized temperature distribution along the fracture. The solid lines denote the analytical solution and the open boxes the numerical solution: (a) Fluid velocity 5 10 {3 ms {1 ; for times of 1.0, 2.0 and 3.0 days, the averaged relative error between analytical and numerical solutions is 15:7, 13:4 and 10 per cent, respectively. (b) Fluid velocity 1 10 {2 ms {1 ; for times of 0.5, 1.0 and 1.5 days, the averaged relative error between analytical and numerical solutions is 10:4, 11:4 and 7:1 per cent, respectively. where Q~bK f /K. If Q is small, eq. (36) tends to the impermeable limit, (j x~b ~0. Note that the stream function boundary condition (eq. 36) can be routinely handled by MUDPACK as a mixed-derivative boundary condition. The fracture temperature boundary condition is obtained next. First, the stationary energy conservation equation, eq. (7), for the fracture element, shaded in Fig. 2, is cast in The factor of 2 in front of the integral on the right-hand side of eq. (34) occurs because of symmetry. Note that because the normal component of the mass ux across the interface must be continuous, o x ~q x at x~b. The continuity of both temperature and pressure at x~b and Darcy's law (eq. 1) yield Ko z j x~b ~K f q z j x~b. (35) Then, from eqs (31), (34) and (35), and using (j z~0 ~0, one obtains the stream function boundary condition ( { 1 Q ( ~0, (36) x~b,z~[0,h] Figure 7. A 2-D porous medium containing a single vertical fracture. GJI000 16/10/98 09:00:12 3B2 version 5.20

9 Hydrothermal uid circulation 635 the form T T +. To w c w o x {j w n x z To w c w o z {j w n z ~0, (37) with all quantities taken inside the control volume. In the gure, the horizontal size of the fracture is exaggerated for clarity. In practice, the relation 2b%jz z {z { j must hold. We integrate eq. (37) over the element area making use of the 2-D Gauss theorem, the symmetry about x~0and the continuityof the normal uxes, o x j x~b ~{ 1 ( o w x~b and j w T fracture T medium ~j m. x~b x~b The result is ( T z { zz dz{ bj w T x~0 c w x~0 z z ~ j m z { c w zz z { T medium x~b dz. (38) The second term in eq. (38), representing the heat conduction along the fracture, can be neglected in most practical cases as Figure 8. Numerical simulation results for the single-fractured model shown in Fig. 7 at the steady state: (a), (b) and (c), temperature, uid velocity eld in the porous medium (the maximum value is 1:21 10 {5 ms {1 ), and uid velocity eld in the fracture (the maximum value is 1:52 10 {3 ms {1 ) derived from the nite element method; and (d), (e) and (f), corresponding quantities derived from the nite di erence method. GJI000 16/10/98 09:00:26 3B2 version 5.20

10 636 J. Yang, K. atychev and R. N. Edwards being much smaller than the right-hand side. Moreover, the temperature variation across the fracture aperture is negligible, as it was asssumed in integrating eq. (7) over the fracture width, so approximately we have T fracture x~0 ~ T fracture x~b ~ T and nally arrive at c w ( T T ~ j m x~b,z~[0,h] medium x~b,, (39) x~b,z~[0,h] where both temperature derivatives are evaluated in the medium. Eqs (36) and (39) are the required fracture boundary conditions. The steps in obtaining the full solution are as follows. We start from the solution for the unfractured case, setting the boundary conditions on both vertical boundaries to be adiabatic and impermeable. Then, about 5^6 iterations over the governing eqs (32) and (33) are performed with the mixedderivative boundary condition (eq. 36) for the stream function and a Dirichlet boundary condition for the temperature along the fracture. The computation scheme is repeated following an update of the Dirichlet temperature. To update T (l) known at iteration step l, we evaluate the right-hand side of eq. (39), i.e. the heat ux into the fracture, and then treat it as an ODE in z for the boundary temperature. To integrate this ODE, we use the fourth-order Bode quadrature rules. If we denote the result of this integration as T (lz1=2), then the boundary temperature is updated according to the following over-relaxation scheme: 5 VAIDATION OF THE FE NUMERICA AGORITHM FOR THE FRACTURED POROUS MEDIUM 5.1 Comparison with the analytical solution It would be most satisfactory if we could compare results from the FE algorithm with a large range of analytical solutions in one, two and three dimensions. Such solutions do not exist. We can consider the case of a thin, rigid fracture situated in an impermeable host (Fig. 3). The water velocity in the fracture o is assumed constant, and a heat source of the constant temperature T 0 exists at the origin of the fracture. We assume that the width of the fracture 2b is much smaller than its length so that the variation in temperature across the fracture's aperture is negligible, and the uid is always instantaneously and thoroughly mixed across the fracture, thus thermal equilibrium exists between uid and solid matrix. Heat transport in the matrix is mainly by molecular di usion (i.e. conduction), heat transport along the fracture is much faster than transport within the matrix, and the temperature variation is so small that the uid density o w is constant throughout the system. These assumptions provide the basis for a 1-D representation of heat transport along the fracture itself and for taking the direction of heat ux in the porous matrix to be perpendicular to the fracture. This results in the simpli cation of the 2-D system to two orthogonal, coupled 1-D systems. The detailed derivation of analytical solutions of temperature distributions is given in Appendix A. T (lz1) ~T (l) zm(t (lz1=2) {T (l) ), (40) where m is called the over-relaxation parameter (Iserles 1996). If is the required relative error tolerance, the process is terminated when DT (lz1) {T (l) D < 0:5mDT (l) zt (lz1) D (41) for all boundary points. Numerical experiments have revealed that for the convergence to take place, m should not exceed 0.1^0.2. Usually, 10^30 boundary iterations with this m are su cient. It should be mentioned that in solving eq. (39) as an ODE, it is important to work with the correct value of T/ at the base of the fracture. This value cannot be obtained from the righthand side of eq. (39) because (j z~0 ~0, so we have to resort to the governing equations. If T is subscripted at the fracture boundary as in Fig. 2, then to the second order in * z, 2 T 2 0,1 ~ {3 T/ 0,1z4 T/ 0,2 { T/ 0,3 zo(* 2 z 2* ). z (42) On the other hand, it follows from the temperature governing eq. (33) that 2 T 2 ~{ 2 T 2 (43) 0,1 0,1 because q z j 0,1 ~0and T/ 0,1 ~0. Then T/ 0,1 can be determined from eqs (42 ) and (43). Figure 9. Temperature and uid velocity distributions along the fracture at the steady state: (a) normalized temperature distribution; (b) normalized uid velocity eld. GJI000 16/10/98 09:00:41 3B2 version 5.20

11 Hydrothermal uid circulation 637 An equivalent FE model consists of a water-saturated 2-D system of 10 m 10 m with a 10 m long fracture included in its centre. The fracture is given an aperture of 1 mm. The permeability and porosity of the host medium surrounding the fracture are assigned such small values that the host rock can be considered as impermeable and the uid velocity along the fracture is constant. The top boundary has a xed temperature of 0 0 C. All other boundaries are assumed to be adiabatic except for the bottom of the fracture, at which temperature is assigned a constant value of T 0 ~1 0 C. Both the lower and upper boundaries are assumed to be permeable for uid ow, whereas the side boundaries are assumed impermeable. On the upper and lower boundaries, the constant head is chosen in such a way that the magnitude of the uid velocity in the fracture is equal to 5 10 {3 ms {1 and 10 {2 ms {1, respectively. The uid velocities in the host matrix can, for practical purposes, be considered to be zero. Both the initial temperature and the uid velocity are assumed to be equal to zero all over the solution domain. The simulation domain is discretized by a 2-D equal-spaced grid, and the number of elements in each of the x and z directions is 20. The fracture^matrix system is given the following thermal transport parameters: c m ~800 J kg {1 0 C {1, o m ~2650 kg m {3, c w ~4174 J kg {1 0 C {1, o w ~1000 kg m {3, j m ~2:0 Jm {1 s {1 0 C {1, j w ~0:5 Jm {1 s {1 0 C {1, and porosity h~0:001. When the uid velocity along the fracture is equal to 5 10 {3 ms {1, the normalized temperature distributions within the solution domain at di erent time levels of 1.0, 2.0 and 3.0 days are illustrated in Fig. 4. Analytical solutions are shown in Figs 4(a)^(c), whereas numercial solutions are shown in Figs 4(d)^(f). It can be seen that the numerical simulation results are fairly close to the analytical solutions at all times. When the uid velocity is equal to 10 {2 ms {1,thenormalized temperature contours at di erent time levels of 0.5, 1.0 and 1.5 days are shown in Fig. 5. Clearly, the numerical solutions (Figs 5d^f) are again in a good agreement with the analytical solutions (Figs 5a^c). To quantify the di erence between the numerical solution and the analytical solution, we plot the normalized temperature pro le along the vertical fracture, as shown in Fig. 6, where the solid lines denote the analytical solution and the little open boxes represent the numerical solution. We calculated the relative error between analytical and numerical solutions at each node point of the fracture, and printed out the averaged value for each time level, as shown in the caption of Fig. 6. It can be seen that the averaged relative error varies from 7:1 per cent to 15:7 per cent. Although the numerical solutions are quite close to the analytical solutions in an overall sense, the di erence between them is still visible. We believe that the di erent boundary conditions applied in the numerical simulation may be the major reason. In the simulation, the lower boundary is assumed adiabatic except at the bottom of the fracture, at which point the temperature is assigned a constant value. The lateral boundaries are also assumed adiabatic. Thus heat cannot be lost through the lower and lateral boundaries. However, in deriving the analytical solution, we have just xed the temperature at the bottom of the fracture and at in nity. In particular, heat can ow downwards from Figure 10. Numerical simulation results for a 2-D water-saturated unfractured porous medium: (a) and (b), initial temperature and uid velocity perturbations (maximum initial velocity is 3:35 10 {7 ms {1 ); (c) and (d), steady-state temperature contours and uid velocity eld (maximum value is 1:53 10 {11 ms {1, i.e. at the numerical `noise' level). GJI000 16/10/98 09:00:55 3B2 version 5.20

12 638 J. Yang, K. atychev and R. N. Edwards the source, explaining why, near the lower boundary, the numerical solutions are globally slightly larger than the analytical solutions. Similarly, on the upper boundary we have xed the numerical solution at 0 0 C; however, only at in nity does the analytical solution approach 0 0 C, explaining the globally slightly smaller numerical than analytical solutions there. 5.2 Comparison with the nite di erence method Consider a 2-D water-saturated porous medium with dimensions of 10 m 20 m, as illustrated in Fig. 7. All four outer boundaries are assumed impermeable. The temperatures of the upper and lower boundaries are xed at 0 and 10 0 C, respectively. The side boundaries are assumed adiabatic. Permeability, thermal conductivity, porosity and uid volume thermal expansion coe cient are assigned values of 10 {10 m 2, 5.0 J s {1 0 C {1, 10 per cent and 8 10 {4 l 0 C {1, respectively. The calculated Rayleigh number corresponding to these physical parameters is equal to 36, smaller than the critical value of 4n 2 for convection to occur (e.g. Bear 1972). There is a vertical fracture having an aperture of 3 mm and rising from the bottom to the top of the cell. The initial temperature distribution varies linearly with depth, and the initial uid velocity is zero over the whole solution domain. The FE solution is obtained on a 2-D equal-spaced grid. There are 40 and20elementsinthex and z directions, respectively. The FD equations are solved on half of the domain, on a grid of 21 by 21 nodes, which corresponds to 20 by 20 elements. As mentioned above, the nite element algorithm produces time-dependent solutions. For this simple model, at each time step the nite element method takes only 0.8 s of CPU time on our IBM RISC workstation to complete the calculation. When the intrinsic time step is 0.01 day, it takes about 6 hr to cover time steps in order to reach the steady state. In contrast, the nite di erence algorithm deals only with the steady-state problem, which derives it directly. On the same workstation, it takes only 15 s of CPU time to converge to the nal solution with a relative error tolerance of 1 per cent The FE equations can be forced to nal solution in signi cantly less time at the expense of losing accurate intermediate solutions. Fig. 8 illustrates the comparison between the nite element solutions and the nite di erence solutions. Clearly, two di erent numerical techniques produce almost the same Figure 11. Numerical simulation results at the steady state when the vertical fracture is located in the middle of the model and its aperture is 1.5 mm: (a) temperature contours; (b) uid velocity eld in porous medium (the maximum value is 1:9 10 {6 ms {1 ); (c) uid velocity eld in the vertical fracture (the maximum value is 4:67 10 {4 ms {1 ). Figure 12. E ect of the fracture's aperture on temperature contours: (a) 2.0 mm; (b) 2.5 mm. The fracture is located in the centre of the model. GJI000 16/10/98 09:01:00 3B2 version 5.20

13 Hydrothermal uid circulation 639 results. To quantify the di erence between the numerical schemes, we plot the normalized temperature and uid velocity pro le along the vertical fracture as shown in Fig 9, where the solid line denotes the FE solution and the open boxes represent the FD solution. The uid velocity eld is normalized by the maximum value over the fracture. We calculate the relative error between the FD and FE solutions at each node point of the fracture. The average value is 5:4 per cent for the temperature and 4:3 per cent for the uid velocity eld, respectively. 6 CASE STUDIES OF FOW IN FRACTURED POROUS MEDIA We focus our attention on the kinds of problem that may be studied with the software, and on hydrothermal convection in media which, on average, are subcritical but which locally have permeability anomalies due to the presence of fractures. In the following models, the choice of physical parameters is speci ed in such a way that the calculated Rayleigh number is very close to the critical value. Figure 13. E ect of the fracture's lateral location on numerical simulation results at the steady state: (a) temperature contours; (b) uid velocity eld in porous medium (the maximum value is 2:23 10 {6 ms {1 ); (c) uid velocity eld in the vertical fracture (the maximum value is 3:69 10 {4 ms {1 ). The fracture's aperture is 1.5 mm. Figure 14. E ect of the fracture's lateral location on numerical simulation results at the steady state: (a) temperature contours; (b) uid velocity eld in porous medium (the maximum value is 5:8 10 {8 ms {1 ); (c) uid velocity eld in the vertical fracture (the maximum value is 1:26 10 {5 ms {1 ). The fracture's aperture is 1.5 mm. GJI000 16/10/98 09:01:06 3B2 version 5.20

14 640 J. Yang, K. atychev and R. N. Edwards 6.1 An unfractured porous medium The model is exactly same as that shown in Fig. 7 except that there is no fracture. The Rayleigh number is 36, smaller than the critical value of 4n 2. If we introduce an initial temperature perturbation and uid velocity eld with a maximum value of 3:35 10 {7 ms {1, as illustrated in Figs 10(a) and (b), then the initial temperature/ uid perturbation is damped as time progresses, and the strength of thermal convection becomes weaker and weaker towards the nal steady state. The nal temperature contours are a set of horizontal lines, demonstrating no signi cant convective e ect, and the maximum uid velocity now has a value of 1:53 10 {11 ms {1,asshown in Figs 10(c) and (d), respectively. The uid velocity eld shown in Fig. 10(d) is at the numerical `noise' level, which cannot produce a signi cant e ect on the temperature distribution shown in Fig. 10(c). 6.2 A single fractured porous medium A vertical fracture is introduced into the previous model, rising from the bottom to the top of the cell, as shown in Fig. 7. The initial temperature distribution varies linearly with depth, and the initial uid velocity is zero over the whole solution domain. When the fracture aperture is 1.5 mm, the averaged vertical and horizontal permeabilities (McKibbin & Tyvand 1984) are Figure 15. E ect of the fracture's lateral location on numerical simulation results at the steady state: (a) temperature contours; (b) uid velocity eld in porous medium (the maximum value is 4:46 10 {7 ms {1 ); (c) uid velocity eld in the vertical fracture (the maximum value is 4:6 10 {5 ms {1 ). The fracture's aperture is 1.5 mm. Figure 16. E ect of the fracture's lateral location on numerical simulation results at the steady state: (a) temperature contours; (b) uid velocity eld in porous medium (the maximum value is 4:7 10 {6 ms {1 ); (c) uid velocity eld in the vertical fracture (the maximum value is 5:55 10 {4 ms {1 ). The fracture's aperture is 1.5 mm. GJI000 16/10/98 09:01:11 3B2 version 5.20

15 Hydrothermal uid circulation {10 m 2 and 10 {10 m 2, respectively. The steady-state numerical simulation results are shown in Fig. 11. It can be seen that hydrothermal convection has developed in the host porous medium. Along the vertical fracture, uid moves upwards because of the buoyancy force; the corresponding ow in the host porous medium near the vertical fracture is an upwelling zone. The isotherms are deformed from parallel lines into lines which are convex to the upper boundary. The presence of even a single fracture can therefore initiate and maintain free convection. The Rayleigh number for the host is 36. If we consider the host medium and the fracture as a whole, the Rayleigh number on average is 39, still smaller than the critical value. ocally, some hitherto unde ned Rayleigh number is large enough for convection to be sustained. The maximum values of the uid velocity elds in the host medium and the fracture are equal to 1:9 10 {6 ms {1 and 4:67 10 {4 ms {1, respectively. 6.3 E ect of the fracture aperture We now vary the fracture aperture from 2 mm to 2.5 mm when it is located at the centre. The computed steadystate temperature contours are shown in Fig. 12. Comparing Fig. 11(a) with Figs 12(a) and (b), it can be seen that the curvature of isothermal lines increases with increasing fracture aperture. The larger the fracture aperture, the greater the e ect of the fracture on the convection system. The patterns of uid velocity elds in both the vertical fracture and the host porous medium are similar to those shown in Figs 11(b) and (c). However, corresponding to apertures of 2.0 and 2.5 mm, the maximum values of uid velocities are increased to 6:7 10 {6 ms {1 and 9:87 10 {6 ms {1, respectively, in the host porous medium, and 1:19 10 {3 ms {1 and 1:46 10 {3 ms {1 along the vertical fracture. 6.4 E ect of the fracture location We here x the fracture aperture at 1.5 mm, but vary its horizontal location with o -centre distances of 2.5 m, 5.0 m, 7.5 m and 9.5 m. The numerical simulation results at the steady state are shown in Figs 13, 14, 15 and 16, respectively. Again, hydrothermal convection has developed in the host porous medium because of the uid movement along the fracture. No matter where the vertical fracture is located, a common feature is that uid always moves upwards along the fracture and the upwelling zone is always near it. When the fracture is located in the middle (see Figs 11 and 13), the upwelling zone is in the central part, with the downwelling zone close to the lateral boundaries. When the fracture is close to the right wall (see Figs 15 and 16), the upwelling zone is close to the right boundary, with the downwelling zone located in the middle. The isotherms in the central region are deformed from convex lines to the upper boundary (see Figs 11 and 13) into concave lines (see Figs 15 and 16). The convection vigour Figure 17. Numerical simulation results at steady state: (a) and (c), temperature contours; (b) and (d), uid velocity elds, where (a) and (b) correspond to the orthogonally multiply fractured model, and (c) and (d) correspond to the homogeneously isotropic model. The averaged relative error between (a) and (c) is 1.55 per cent. GJI000 16/10/98 09:01:16 3B2 version 5.20

16 642 J. Yang, K. atychev and R. N. Edwards seems also to depend on the location of the fracture. When the fracture is located in the very centre or close to the side boundary, convection is strongest, whereas when the fracture is located in other positions, convection strength becomes relatively weak (see Figs 14 and 15). A particularly interesting phenomenon can be seen in Fig. 14, where the fracture has an o -centre distance of 5 m. Now three convection cells have been developed, but they are so weak that the curvature of temperature contours is almost invisible. Another common feature is that no matter where the fracture is located, the cell containing the fracture is dominant, and the other is relatively weak. Thus, both the convection pattern and the vigour produced in a closed system are very strongly dependent on the location of the fracture. 7 EQUIVAENCE BETWEEN HOMOGENEOUSY ANISOTROPIC MEDIA AND MUTIPY FRACTURED MEDIA As stated above, open fracture permeability is mainly dependent on its aperture, and it is usually much larger than the permeability of the host medium. Thus, a multiply fractured porous medium, as a whole, is made up of alternate layers with very di erent permeabilities. For these types of material, the e ective average horizontal and vertical permeabilities can be estimated (McKibbin & Tyvand 1984). When the alternate layers are in the vertical direction, they are 1/k x ~ 1 d X N i~1 d i k i, (44) k z ~ 1 d X N i~1 d i k i, (45) where d i and k i are the thickness and permeability of the layer i (i~1, 2,..., N), with N being the total number of layers making up the total thickness d. In the following studies, the thermal conductivity is decreased to 3.6 J s {1 0 C {1,andall other parameters and conditions are kept the same. 7.1 Isotropic porous medium et us rst consider an orthogonally multiply fractured model. Assume the host medium is impermeable and there are mm-wide vertical fractures rising from the bottom to Figure 18. Numerical simulation results for the vertically multiply fractured model at steady state (the fracture aperture is 1 mm): (a) temperature contours; (b) uid velocity eld in the host medium; (c) uid velocity eld in the discrete fractures. Figure 19. Numerical simulation results for the homogeneously anisotropic medium (k x ~10 {10 m 2 and k z ~2:63 10 {10 m 2 ): (a) temperature contours; (b) uid velocity eld. GJI000 16/10/98 09:01:22 3B2 version 5.20

17 Hydrothermal uid circulation 643 the top, and mm-wide horizontal fracture spanning from the left wall to the right wall. These two sorts of fractures are spaced uniformly. Although the host medium itself is impermeable, the interconnected fractures provide the paths for hydrothermal uid circulation. It can be estimated that both k x and k z are equal to 10 {10 m 2. Thus, the system as a whole is isotropic in permeability. Now assume there are no discrete fractures but the host medium is permeable and has a permeability of 10 {10 m 2, keeping other parameters and conditions intact. The Rayleigh number for both systems is equal to 50, larger than the critical value. Hydrothermal convection can be initiated and maintained. The numerical simulations at steady state are shown in Fig. 17. Figs 17(a) and (c) show temperature contours of the orthogonally multiply fractured model and the homogeneously isotropic model, and Figs 17(b) and (d) show uid velocity elds in the orthogonal fractures and in the permeable host medium. Clearly, Figs 17(a) and (b) are comparable to Figs 17(c) and (d). The averaged relative error between Figs 17(a) and (c) is equal to 1:55 per cent. 7.2 Multiply fractured porous medium with many vertical fractures Now consider a vertically multiply fractured model. The host medium is assigned a permeability of 10 {10 m 2 and there are 39 uniformly spaced vertical fractures rising from the bottom to the top. Other conditions are kept the same. When the fracture aperture 2b is equal to 1, 1.5 and 2 mm, the estimated average vertical permeabilities are equal to 2:63 10 {10,6:48 10 {10 and {10 m 2, respectively, but the estimated average horizontal permeabilities are all equal to 10 {10 m 2. The numerical simulation results for these models at steady state are shown in Figs 18, 20 and 22, corresponding to 2b~1, 1:5 and 2 mm. Next let us consider a uniformly anisotropic porous model with a xed horizontal permeability of 10 {10 m 2. When the vertical permeability is assigned a value of 2:63 10 {10,6:48 10 {10 and {10 m 2, the steadystate numerical simulation results are shown in Figs 19, 21 and 23, respectively. It can be seen that when the fracture aperture 2b increases, and hence the e ective vertical permeabilities become larger, the temperature contours seem to be com- Figure 20. Numerical simulation results for the vertically multiply fractured model at steady state (the fracture aperture is 1.5 mm): (a) temperature contours; (b) uid velocity eld in the host medium; (c) uid velocity eld in the discrete fractures. Figure 21. Numerical simulation results for the homogeneously anisotropic medium (k x ~10 {10 m 2 and k z ~6:48 10 {10 m 2 ): (a) temperature contours; and (b) uid velocity eld. GJI000 16/10/98 09:01:35 3B2 version 5.20

18 644 J. Yang, K. atychev and R. N. Edwards Figure 23. Numerical simulation results for the homogeneously anisotropic medium (k x ~10 {10 m 2 and k z ~14 10 {10 m 2 ): (a) temperature contours; (b) uid velocity eld. Figs 20(a) and 21(a), and 22(a) and 23(a) are equal to 4.38, 5.83 and 7.16 per cent, respectively. Figure 22. Numerical simulation results for the vertically multiply fractured model at steady state (the fracture aperture is 2.0 mm): (a) temperature contours; (b) uid velocity eld in the host medium; (c) uid velocity eld in the discrete fractures. pressed more and more in the horizontal direction, and the convection cells in the host medium become more and more thin and narrow. This is not surprising, because the existence of the vertical fractures leads to less hydraulic resistance in the vertical direction, so uid ows more easily in the vertical direction than the horizontal direction. Observing the results in Figs 18, 20 and 22, it can be concluded that the vertical discrete fractures mainly contribute to the vertical transport of uid, whereas the host porous medium is most responsible for the horizontal transport of uid between the adjacent fractures. Also, the results of the vertically multiply fractured models are comparable to those of the homogeneously anisotropic models. The average relative errors between Figs 18(a) and 19(a), 7.3 Multiply fractured porous medium with many horizontal fractures Finally, let us consider a horizontally multiply fractured model. The host medium is assigned a permeability of 10 {10 m 2 and there are 19 uniformly spaced horizontal fractures across the model from the left to the right. These fractures have an aperture of 1.5 mm. Other conditions are kept the same. The estimated average horizontal and vertical permeabilities are equal to 6:34 10 {10 and 10 {10 m 2, respectively. The numerical simulation results are shown in Fig. 24. Next, let us consider a uniformly anisotropic porous model with xed horizontal and vertical permeabilities of 6:34 10 {10 m 2 and 10 {10 m 2, respectively. The sumulation results are shown in Fig. 25. Now the temperature contours seem to have been stretched along the horizontal direction. Only one convection cell has been formed in the host porous medium. The shape of the convection cell now becomes long and at with an aspect ratio of about 2. This is also reasonable because the horizontal fractures lead to less hydraulic resistance for uid motion. The results in Fig. 24 indicate that the horizontal fractures mainly contribute to the horizontal uid transport, whereas the host porous medium provides a surplus for the vertical uid transport between the adjacent fractures. Again the results of the horizontally multiply fractured model are comparable to those GJI000 16/10/98 09:01:41 3B2 version 5.20