Anisotropic Modelings of the Hydrothermal Convection in Layered Porous Media

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1 J. Phys. Earth, 40, , 1992 Anisotropic Modelings of the Hydrothermal Convection in Layered Porous Media Hidenori Masamoto* and Satoru Honda** Department of Earth and Planetary Systems Science, Faculty of Science, University of Hiroshima, Higashi-Hiroshima 724, Japan Equivalence of anisotropic modelings of the hydrothermal convection to the layered permeability distribution is examined by 2-D numerical simulations. Permeability varies sinusoidally in horizontal or vertical directions. For the horizontally layered cases, flow fields approach monotonically to the corresponding anisotropic models, as the wavelength of permeability variation decreases. We find that the anisotropic modeling for this case may become valid, when the wavelength of permeability variation is equal to or less than the thickness of thermal boundary layers. For the vertically layered cases, since the vertical motion of the fluid is sensitive to the horizontal variation of permeability, the cell structure is different from the corresponding anisotropic models. 1. Introduction Hydrothermal convection is considered to be an important process for the heat transport within the oceanic crust. Many researchers have conducted the numerical simulations of the hydrothermal convection in a variety of geological settings (e.g., Ribando et al., 1976; Schubert and Straus, 1977; Fehn and Cathles, 1979; Fehn et al., 1983; Rosenberg and Spera, 1990). Among many factors taken into account in those model calculations, a spatial variation of permeability may be the most important because of the ubiquitous existence of heterogeneities in rocks and their structures. Recently, Rosenberg and Spera (1990) pointed out that the layering and/or fractures with preferred orientations in the oceanic crust may produce apparent anisotropy in permeability and found that the flow in such a medium is strongly affected by the effective anisotropic permeability. However, they did not discuss the feasibility of such an anisotropic modeling of layered or cracked rocks in detail. In this study, we investigate such feasibility using a simple model in which the permeability changes periodically in space. The equivalence of the anisotropic modeling to the layered permeability distribution has been studied by several workers (e.g., McKibbin and Tyvand, 1982). However, these studies were generally focused on the onset of convection or slightly Received June 4, 1992; Accepted July 8, 1992 * Now at Tokushima Prefectural Office, Tokushima, Japan ** To whom correspondence should be addressed. 555

2 J 556 H. Masamoto and S. Honda super-critical state. Here, numerical calculations with Ra (Rayleigh number; see below) around 100, which is an appropriate order of magnitude for the hydrothermal convection in the oceanic crust, will be reported. We consider the cases in which the layering of permeability is horizontal and vertical in 2-D models. 2. Theory The basic 2-D equations describing the hydrothermal convection in porous media may be written as follows (e.g., Turcotte and Schubert, 1982): (1) (2a) (2b) where the meaning of each symbol is given in Table 1. Equation (1) gives the (3) (4) Table 1. Meaning of the symbols used in this study.. Phys. Earth

3 Anisotropic Modelings of the Hydrothermal Convection 557 (a) Fig. 1. Schematic diagrams showing the vertically layered case. The shaded and white areas indicate where the permeability is large and small, respectively. If the flow direction is horizontal (a), the horizontal flow rate u is independent of x. On the contrary, if the flow direction is vertical (b), the pressure is independent of x. (b) conservation of mass for the fluid. Equations (2) are the well-known Darcy's law with the effect of gravity taken into account. Equation (3) means the conservation of energy and Eq. (4) shows the equation of state of the fluid. The equivalence of the anisotropic media to the layered homogeneous media may be understood as follows, assuming that only the permeability k is variable. Suppose that the permeability is a function of only x (denoted by case V, hereafter; Fig. 1) and the fluid flows uniformly in the horizontal direction (i.e., x direction: Fig. 1(a)). Then, the horizontal flow rate u is independent of x. The total pressure drop p from x=0 to x=l may be estimated using (2a) as or (5) where Compared with Eqs. (2a), (5) implies that the effective (or bulk) permeability for the horizontal flow is given by ke.h. For the vertical flow (i.e., flow in the z direction: Fig. 1(b)), the pressure p is independent of x, otherwise horizontal flow will be invoked. Using (2b), the horizontally averaged flow rate v- may be calculated as, Vol. 40, No. 4, 1992

4 558 H. Masamoto and S. Honda or (6) where Equation (6) implies that the effective (or bulk) permeability for the vertical flow is given by ke.v. Thus, we obtain the ratio of the effective horizontal permeability to the vertical as (7) Similar arguments are given for the case in which the permeability varies as a function of z (denoted by case H, hereafter). The ratio of the effective horizontal permeability to the vertical permeability for this case may be given by (8) In this study, we assume a sinusoidal variation of permeability in space given as (9a) and (9b) From Eqs. (7), (8), and (9), we obtain (10a) and (10b) J. Phys. Earth

5 Anisotropic Modelings of the Hydrothermal Convection Numerical Simulations and Results In this report, we compare the results obtained for a variety of n, which is the cycle of the permeability variation (see (9a) and (9b)), with those for corresponding anisotropic models. To solve the basic Eqs. (1) to (4), we use a stream-function formulation of the flow field. They are non-dimensionalized (see Table 2) and the final non-dimensional equations to be solved are (11a) (11b) where Ra is the Rayleigh number defined by (12) (right-hand side of (12) is dimensional values; see Table 1) and ě is the stream defined by For the case of anisotropic permeability, Eq. (11a) is replaced by (13) (11a') In the following discussions, all the symbols are non-dimensional values, unless stated. Physical properties except for permeability are assumed to be constant. In deriving (11b), we assume ĕmcm=ĕfcf (dimensional) which is approximately valid for the hydrothermal convection within the oceanic crust. A finite difference analogue of (11) is constructed with the central difference in space and the explicit forward difference in time. Equation (11a) or (11a') is solved using the iterative SOR method and the advection terms in (11b) are integrated using Arakawa's scheme (e.g., Roache, 1972). Geometry and boundary conditions of our models are the same as those used by Rosenberg and Table 2. Non-dimensional scheme used in this study. Here, the symbols with a dash imply non-dimensional values. Vol. 40, No. 4, 1992

6 560 H. Masamoto and S. Honda (a) Fig. 2. (a) Relation between n and the Nusselt number normalized by those calculated for the corresponding anisotropic permeability models (horizontally layered case). Nua is the Nusselt number obtained for the corresponding anisotropic model. (b) Same as Fig. 1(a) except for vertically layered cases. (b) Spera (1990): aspect ratio of one (i.e., l=1), fixed temperature at bottom (z=0; T=1) and top (z=1; T=0), symmetric temperature boundary conditions at both sides (x=0, 1; ÝT/ Ýx=0) and impermeable at all the walls (i.e., ƒõ=0). An equal distance mesh with 51 ~51 and with 101 ~101 points are used. For some cases and the cases with high n values (e.g., Ra=150; n=10; Ra=300; n=8, n=10, n=11), the mesh of 201 ~201 is used to check or assure the accuracy of solutions. Parameters TH and rv are set as rh=rv= ã3/2, which gives RV=1/2 and RH = 2. (Note that the actual variation of permeability is (1+r(H or V)/(1-r(H or V))=(1+( ã3/2))/(1-( ã3/2)) 14.) We calculate both the flow field (ƒõ,t) and the volume averaged Nusselt number defined by (14) which is a good measure of the total flow field variation. In general, the results with the 51 ~51 mesh agrees well with those of 101 ~101 at n 4, while the results of 201 ~201 agrees with those of 101 ~101 with a discrepancy less than 2 percents for all the cases in which a comparison is made. Based on these comparisons and those with the results presented by Rosenberg and Spera (1990), we think that the overall error of the results in this study is within a few percents. The conductive temperature profile (i.e., T=1-z) is used for the initial conditions and we add a temperature perturbation with the magnitude of 0.1 at the top half of the right side. All the results shown in the following discussions are the steady-state results. In Fig. 2, we show the volume averaged Nusselt numbers (Nu) normalized by those obtained by the corresponding anisotropic permeability models (Nua) for different Ra and n. Figure 2(a) and (b) are for cases H and V, respectively. Both cases show that the Nu is greater than the corresponding Nua, which suggests that the flow J. Ph yes. Earth

7 Anisotropic Modelings of the Hydrothermal Convection 561 (a) Fig. 3. (a) Examples of the stream function and the temperature calculated for horizontally layered cases (Ra = 300). The lowermost figures of the right column are those of the corresponding anisotropic case. Contours are drawn at the interval of a tenth of the maximum absolute values which are shown in the top of each figure. "Time" is the non-dimensional time elapsed since each calculation starts. Arrows show the places where the permeability becomes maximum. (b) Same as Fig. 2(a) except for vertically layered cases (Ra=300). is confined to the high permeability layers. Figure 3 shows how the flow field changes (Ra=300), as n increases. Generally speaking, case H shows a well-defined behavior, that is, the flow and temperature fields approach monotonically with an increase of n to the corresponding anisotropic permeability models. However, as the Rayleigh number increases, this tendency degrades. This degradation might be understood as a result of the decrease in the thickness of thermal boundary layers. The flow with high Rayleigh numbers is controlled by the thermal boundary layers rather than by the whole flow field. We infer that the number of the layers covered by the thermal Vol. 40, No. 4, 1992

8 562 H. Masamoto and S. Honda Fig. 3 (b) boundary layers is the key factor in determining whether the flow may be treated as "anisotropic" or not. The thickness of the thermal boundary layers may be roughly estimated as (Note that there are two boundary layers in the region.) Using the Nusselt numbers obtained for anisotropic models, we get (15) Wavelengths of the permeability variation (Ď=1/n), at which Nu becomes approximately Nua, are J. Phys. Earth

9 Anisotropic Modelings of the Hydrothermal Convection 563 These values are similar to the thickness of the thermal boundary layers estimated before, which implies that the anisotropic modeling of the horizontally layered case is appropriate, when the wavelength of the permeability variation is equal to or less than the thickness of the thermal boundary layers. For case V, the relation between the normalized Nusselt number and n shows more irregular nature, which may reflect the sensitivity of the cell structure to the horizontal variation of permeability. We find that the number of the cells changes from two (at n=2; not shown in Fig. 3(b)) to three as n increases (Fig. 3(b)). This is in contrast with the corresponding anisotropic case (the last panel of Fig. 3(b)) in which two-cell configuration appears. The vertical motion of the fluid tends to concentrate to the places where the permeability is high. Since such places change their positions with n, the cell structure adjusts its position irregularly as n varies. This is the cause of the fluctuation of the Nusselt number. 4. Discussion and Conclusion In our modelings, we choose fairly restricted values of the permeability variations (i.e., RH or RV) and aspect ratio (i.e., l). The reason for these choices depends mainly on the numerical restrictions. Changing both parameters may require more computer time because of more need to increase the mesh sizes. In such studies, we feel that we have to use more efficient methods such as FEM instead of the finite difference method used in this study. However, we think that the main results, which are given below, are fairly consistent with our physical intuition, so that they may be valid for a wide range of parameters. Even if this is not the case, our results become good indicators to be checked for the future studies of this kind. At this stage, the most unpredictable factor is the time-dependent behavior of layered and the corresponding anisotropic cases, since it is impossible to apply the arguments which are valid for the steady-state cases. It is definitely necessary to conduct numerical simulations for unsteady problems. To summarize, although we agree with the suggestion of Rosenberg and Spera (1990), that is, a possible anisotropic effect of permeability is important, the anisotropic modeling of the layered porous media should be treated with care. Especially, for the vertically layered cases, the flow structure may be fairly different from corresponding anisotropic models as we have shown, since the cell structure is sensitive to the horizontal variation of the permeability. We also note that 3-D effects may be more important for this case than for the horizontally layered case. There are three effective permeabilities in 3-D modeling of the layered media, i.e., the effective permeabilities for z-direction (denoted, hereafter, kz), x-direction (kx), and y-direction (ky,) which is perpendicular to the x-z plane. For the horizontally layered case, kx is equal to ky because of the symmetry of the problem. However, for the vertically layered case, ky (=(1/1) çl0kdx) is greater than kx (=l/( çl0(1/k)dx)). In such a case, we expect that the flow perpendicular to the x-z plane (i.e., the flow confined to the high permeability layers), which we cannot deal with in this study, may occur easily. Thus, it is necessary to conduct 3-D calculations for the vertically layered case. However, such studies are beyond the scope of present paper and they should be treated in the future. For the horizontally layered case, we found that the anisotropic modeling may Vol. 40, No. 4, 1992

10 564 H. Masamoto and S. Honda be appropriate, when the wavelength of the layering is comparable to the thickness of the thermal boundary layers. We thank Yasuyuki. Iwase, Kayoko Tsuruga, Takaharu K ono, and Masahiko Masuda for their valuable discussions and comments. Kiyoshi Yomogida read the manuscript carefully and gave us constructive criticisms. We also thank two anonymous referees for their quick reviewing and the comments to improve the manuscript. Some of the present calculations were done at Ocean Research Institute, the University of Tokyo. REFERENCES Fehn, U. and L. M. Cathles, Hydrothermal convection at slow-spreading mid-ocean ridges, Tectonophysics, 55, , Fehn, U., K. E. Green, R. P. Von Herzen, and L. M. Cathles, Numerical models for the hydrothermal field at the Galapagos spreading center, J. Geophys. Res., 88, , McKibbin, R. and P. A. Tyvand, Anisotropic modeling of thermal convection in multilayered porous media, J. Fluid Mech., 118, , Ribando, R. J., K. E. Torrance, and D. L. Turcotte, Numerical models for hydrothermal circulation in the oceanic crust, J. Geophys. Res., 81, , Roache, P. J., Computational Fluid Dynamics, Hermosa, Albuquerque, N. M., Rosenberg, N. D. and F. J. Spera, Role of anisotropic and/or layered permeability in hydrothermal convection, Geophys. Res. Lett., 17, , Schubert, G. and J. M. Straus, Two-phase convection in a porous medium, J. Geophys. Res., 82, , Turcotte, D. L. and G. Schubert, Geodynamics Applications of Continuum Physics to Geological Problems, John Wiley & Sons, New York, J. Phys. Earth