R.E. Ewing ..... For example, if Raviart-Thomas pairs of nite ... For example, if the lowest order Raviart-Thomas spaces are used, then one pressure and four.
COMPUTATIONAL TECHNIQUES IN MULTIPHASE FLOW AND TRANSPORT IN POROUS MEDIA
R.E. Ewing Institute for Scienti c Computation Texas A&M University College Station, Texas 77843{3404 (409) 863{2716
R.D. Lazarov Department of Mathematics Texas A&M University College Station, Texas 77843{3368 (409) 845{7578
J.E. Pasciak Department of Applied Science Brookhaven National Laboratory Upton, New York 11973 (516) 282{4139
A.T. Vassilev Institute for Scienti c Computation Texas A&M University College Station, Texas 77843{3404 (409) 847{9086
I. INTRODUCTION In general the objectives of computer simulation of multiphase, multicomponent ow and transport in porous media are to understand better the complex physical and chemical processes. An example of such complex phenomenon is the transport of radionuclides in multiphase groundwater ow in combination with sorption, desorption and radioactive decay [1,2]. This knowledge can be used to successfully deal with urgent environmental problems. Using computer simulations ecient clean-up or remediation procedures can be developed. Flow and transport in porous media are described by a non-linear system of partial di erential equations of convection-di usion-reaction type. The formulation of the di erential model is usually based on the mass conservation principle enhanced with constitutive relations such as the Darcy's and Henry's laws. Analytic solutions of such di erential models are not possible. Accordingly numerical techniques provide the only feasible approach to solving these dicult problems. There has been an intense e ort focused on building such models for these equations during the last few decades. This work has resulted in the development of a number of simulators in the petroleum industry for ecient reservoir modeling. These techniques are also being applied to environmental problems. A typical environmental application involves a pollutant leaking from a source either in the atmosphere or underground. The accurate prediction of the dominant direction and speed of movement as well as concentration levels are among the primary tasks in such settings. The aim of this paper is to discuss one approach for building a numerical two phase uid ow and transport model for groundwater ow simulations. We discuss related discretization, solution methods and computer implementation. Numerical experiments involving a model groundwater application are provided. II. TWO PHASE FLUID FLOW AND TRANSPORT MODEL In this section we discuss a two phase uid ow and contaminant transport model. We use the well known fractional ow formulation coupled with a transport equation. This approach results in a mathematical formulation which is well behaved when solved numerically. Below we brie y describe this formulation. Let the porous medium be represented by a domain in the i-dimensional Eucledian space, i = 2; 3. The fractional ow formulation involves a global pressure p and total velocity u. This provides a two-phase water (w) and air (a) ow model which is described by the following equations [3]:
Sa ca
dp +rv = dt
v
? @ @t(p) + q(x; Sw );
x 2 ; t > 0;
= ?K(rp ? G);
x 2 ; t > 0; @ (p) @S + qw ; x 2 ; t > 0: w + r (fw v ? Ka qw g ? D(Sw ) rSw ) = ?Sw @t @t
(1) (2)
The global pressure and total velocity are de ned by
1 Z S a ? w dpc d and v = v + v : 1 w a 2 2 Sc d Here, dtd @t@ + vSaa r, Sw and Sa are the saturation for the water and air phases, (p) is the porosity, = w + a is the total mobility, K is the absolute permeability tensor, and = kr , = w; a, is the mobility for water and air, where kr is the relative permeability. The capillary pressure pc is given by pc = pa ? pw . The gravity forces G and capillary di usion term D(S ) are expressed as G = w w + aa g and D(S ) = ?K f dpc p = (pw + pa ) +
and the compressibility ca is de ned by
a w
ca =
dS
1 da : dp a
a
The phase velocities for water and air, which are needed in transport calculations, are given by:
vw = fw v + Kafw rpc ? Kafw g; va = fav ? Kw farpc + Kw fag;
(3)
where f = =, = w; a, and = a ? w . To complete the model, we assume constitutive relations betwen capilary pressure and saturation and also between relative permeabilities and saturation, i.e. pc = pc (Sw ) and kr = kr (Sw ):
Within the groundwater literature, the pressure normally is scaled by the gravity potential function. Equation (1) would then be given in terms of the pressure head. We should also note that if the Richards approximation [3], pa = 0, is valid, (2) can be replaced by: pc (Sw ) = ?pw . The phase velocity for air is given by (3) even if the Richards approximation is used. In conjunction with the above ow model weconsider transport equations based on a kinetic phase transfer. The rate of contaminant transfer from one phase to the other is determined by the degree of disequilibrium, a kinetic phase transfer coecient and the water content. The concentration form of the transport equations is as follows: @ (w cw ) + r (q c ) ? r ( D rc ) + c = k (c ? Hc ); @t
w w
w w
w
w w w
a a
w
(4)
@ (aca ) + r (qaca ) ? r (a Darca ) + a a ca = ?ka(a!a ? Hw !w ) : @t Here c is the concentration of phase , k is the coecient of mass transfer, H is the Henry's constant, D is the dispersion tensor for phase , is the reaction rate for phase , and q is the volumetric ux of phase .
In the case of radionuclide transport, cw and ca are vectors of the concentrations of various radionuclides in the wetting and non-wetting phases. Then one should add a set of ordinary di erential equations to the above model that describes the change of the corresponding concentrations in the rock as a consequence of the processes
of sorption and desorption. The boundary conditions are an important element of the above model. Standard types of boundary conditions, namely Dirichlet, Neumann and Robin, are assumed for the transport equations (4). The speci c pressuresaturation formulation of the ow model (1)-(2) deserves special attention. If ? is the boundary of , general boundary conditions for (1)-(2) can be given by a combination of the following expressions: p = p? (x; t);
x 2 ?1; t > 0;
v + b(x; t; Sw)p = G(x; t; Sw);
Z ?3
v = g(t);
x 2 ?2; t > 0;
and p = p? (x; t) + d(t); Sw = S? (x; t);
x 2 ?3; t > 0;
x 2 ?4 ;
(fw v + Ka fw (rpc ? g)) + bw (x; t; Sw )p = Gw (x; t; Sw ); x 2 ?5; t > 0; where ?i, i = 1; 2 : : : 5 are given partitions of ?. Since the total presure is not a physical quantity, its boundary conditions will be non-linear functions of the boundary values of the physical variables in the original two-pressure formulation [4]. This means that we have to iterate on the boundary conditions as a part of the solution process. III. DISCRETIZATION In this section we consider discretizations of the models discussed in the previous section. Three important factors in selecting a discretization are the mathematical properties of the di erential model, the geometry of the domain and the linearization of the non-linear model. Obviously, the coupled ow and transport model (1)-(4) is non-linear. There are non-linearities on multiple levels, within each of the equations, between the pressure and saturation equations and between the pressure and transport equations. One way of solving this system is to rst linearize each equation by lagging in time the setup of the coecients in order to get an initial guess. After that a Picard or Newton iteration can be applied in order to resolve the non-linearities. Such approach is used successfully in [4] in the case of unsaturated ows. Another approach to linearization is discussed in [5]. We will not further consider the linearization aspect in this paper. Since we are interested in groundwater applications which involve domains with complex geometry and geology, we have considered only nite element discretizations. It is well known that attempts for nite di erence discretizations of problems on domains with general geometry lead to unsurmountable diculties. The mathematical nature of the transport (4), saturation (2) and pressure (1) equations are di erent and speci c methods for their approximation should be considered. Typically, the transport and the saturation equations are convection dominated and thus special care should be taken in their discretization. Also, the di usion terms there are small but important and cannot be neglected. The pressure equation, however, is not convection dominated and this fact should be taken into account when the discretization method is selected. This equation has a strong elliptic part. Based on these observations, we have selected two types of nite element discretizations|the standard conforming formulation and the mixed method. Advantages of the former are its simplicity, smaller number of unknowns, isoparametric techniques for arbitrary shaped domains, and availability of a vast variety of ecient methods for solving the resulting system of linear equations. These features are particularly important for three
dimensional problems. Combined with upstream weighting, such discretizations can be used for the transport and the saturation equations. Enhanced performace results from the use of logically rectangular grids. Among the disadvantages of the conforming discretizations are the lack of local mass conservation of the numerical model and some diculties in computing the phase velocities needed in the transport and saturation equations. The straightforward numerical di erentiation is far from being justi able in problems formulated in highly heterogeneous medium with complex geometry. On the other hand, the mixed nite element method [6] o ers an attractive alternative. In fact, this method conserves mass cell by cell and produces a direct approximation of the two variables of interest { pressure and velocity. Moreover, it naturally takes into account constraints that appear in the variational formulation of a given di erential model, e.g., r u = f . Such methods can be used for the discretization of the pressure equation. Typically the resulting system of linear equations has the form of a saddle point problem de ned on a pair of nite dimensional Hilbert spaces H1 and H2 :
0 10 1 0 1 T A B @ A@ X A = @ F A; B ?D Y G
(5)
BU )2 c0kV k2 for all V 2 H2 ; sup ((V; U 2H1 AU; U )
(6)
where F 2 H1 and G 2 H2 are given and X 2 H1 and Y 2 H2 are the unknowns. Here A : H1 7! H1 is a linear, symmetric, and positive de nite operator. In addition, the linear map BT : H2 7! H1 is the adjoint of B : H1 7! H2 . D : H2 7! H2 is either 1 I with being a small parameter or 0, depending on the type of the problem being solved|parabolic or elliptic. The existence and uniqueness of a solution is guaranteed by an additional requirement for the pair of spaces H1 H2 . This is known as the inf-sup condition [6] given by for some positive constant c0 .
This discretization leads to inde nite systems with a considerably larger number of unknowns. These systems are more dicult to solve. However, the popularity of mixed methods has increased considerably as a consequence of the progress made in the recent years in developing ecient methods for solving these equations. The presense of lower order terms in the di erential model results in non-symmetric terms in the model which requires careful treatment. This can be e ectively resolved using the modi ed method of characteristics (MMOC) [7,8] or the related ELLAM method [9]. These methods can be viewed as a special time-stepping procedure that can be combined with any spatial discretization. MMOC gives rise to a symmetric discrete model which can be e ectively solved by iterative methods. However, this method requires accurate approximation of the velocity in order to accurately follow the characteristics. For this reason combinations of MMOC with mixed discretizations lead to better numerical models. IV. EFFICIENT ITERATIVE METHODS In this section we give an overview of some relevant iterative methods for solving the systems of linear equations arising from nite element discretizations of the di erential problems discussed above. Because of signi cant progress made in developiong iterative techniques for systems coming from Galerkin discretizations, shall not concentrate on this issue. We only mention that domain decomposition preconditioners are particularly useful when the models are implemented on distributed parallel computers [10]. The inde nite systems resulting from mixed nite elements are much more dicult to solve iteratively. In general, the properties of the discrete operators involved in the de nition of the system must be understood well in order to design an ecient iterative scheme for solving it. For example, if Raviart-Thomas pairs of nite dimensional spaces are used (see [6]) then the operator A is well conditioned, whereas the Schur complement operator BA?1BT exhibits a condition number growth like h?2, where h is the discretization parameter. It is
well known that in our setting BA?1BT behaves like a discretization of a second order operator. A classical approach [11] to the solution of (5) is to perform Gaussian block elimination and obtain the reduced system
BA?1BT Y = BA?1F ? G:
(7) The classical Uzawa algorithm is a linear iteration for solving (7). Alternatively, one can apply preconditioned conjugate gradient to (7). A disadvantage of these approaches is that they require the action of A?1 which substantially reduces their eciency. Recent convergence results for inexact variants of the Uzawa algorithm [12], where the evaluation of A?1 is avoided, make such algorithms still attractive. The approach here uses appropriately scaled preconditioners. It is worth mentioning that there are ecient preconditioners that are scaled appropriately by default. For example, multigrid is automatically scaled properly according to the theory of the inexact Uzawa algorithms whereas a scaling constant must be computed for the incomplete Cholesky factorization. The conjugate residual method applied to (5) is another approach to the solution of this problem. The two level iteration implementation considered in [13] suggests that due to the non-linearity of the outer iteration the accuracy of the inner solve should be increased considerably for stability. This, however, increases the computational cost of the algorithm. The positive de nite reformulation of (5) suggested in [14] is an attractive method for solving the inde nite problem. This approach utilizes preconditioners for A and BA?1BT and results in a positve de nite system which can be solved eciently by conjugate gradient. However, the preconditioner for A must be properly scaled. This scaling factor is either known a priory or can be obtained by employing computationally cheap procedures for estimating the largest eigenvalue, such as the power method. In general, the linear iteration corresponding to the inexact Uzawa methods converges slower than the conjugate gradient method just described. However, the Uzawa method may be advantageous when implementated on distributed memory parallel computers since inner products are not required. The hybrid mixed formulation [6] provides another way for computing the solution to (5). The idea behind this method is to impose continuity of the normal components of the velocity at the interelement interfaces by Lagrange multipliers. From a computational point of view, the main bene t in this formulation is in the fact that the resulting system of linear equation can be reduced to a system for the multipliers which is symmetric and positive de nite. The condition number of the latter system is like O(h?2) and should be preconditioned. Using equivalence arguments, a domain decomposition preconditioner for this problem is constructed in [15]. The preconditioner suggested in [16] is based on algebraic substructuring techniques and also uses equivalence arguments. Both approaches give rise to well conditioned linear systems. V. NUMERICAL SIMULATIONS In this section we discuss our experience in developing computer simulators of the models described above. Clearly, most interesting and dicult are three dimensional transient problems. For this reason we shall concentrate on issues related to such problems. The approach we have taken in de ning the triangulation of the computational domain is to have an underlying logically rectangular grid. Such grid o ers perhaps the most economical approach to building nite element approximations in terms of the number of unknowns. It essentially simpli es many coding issues and yet allows complex geometries to be handled. In fact, our computational grid can be as complex as any reasonable union of logically rectangular structures including toroidal or L-shaped domains. The basic logically rectangular grid is used for the de nition of Galerkin nite element method. To de ne the mixed method one has to split further each grid cell into tetrahedra. The most economical way is to split each cell into ve tetrahedra. For example, if the lowest order Raviart-Thomas spaces are used, then one pressure and four velocity unknowns are attached to every tetrahedron in the grid. Because of this it is clear that the numerical solution of such models requires extensive memory and CPU resources. In our opinion, only supercomputers are
capable of solving these numerical model in reasonable time. We have experimented primarily on distributed memory architectures such as the Intel's Paragon. We use a domain decomposition approach in order to utilize these machines. The original computational domain is decomposed into a set of logically rectangular structures each which is attached to a single node. Then a corresponding parallel algorithm for solving the problem is applied. We have used the system for remote procedure calls (IPX), developed at the Brookhaven National Laboratory [17], in the parallelization of our computer codes. This system provides the user with the ability to write parallel codes in a style that is very close the the common serial style of writing numerical codes and for this reason reduces considerably the complexity of the development software for distributed architectures. In addition, the resulting software is independent of the vendor supplied primitives for parallel processing which is important for the portability. Another interesting feature of IPX is that it can unite the exchange of data and a method of its processing among the computing nodes which leads to more enhanced software environment for creating parallel programs. At the end of the section we give some computational results from a simulation of an groundwater problem. The application is the transport of a very dense contaminant that is leaking from a storage pit and is moving due to density through the unsaturated zone to the saturated zone and along a bedrock with highly variable geometry. A Richards equation formulation is used that eciently models this combination of unsaturated/saturated ow of miscible contaminants. The bedrock elevations were provided by eld personnel from seismic measurements. A simulation of a ow and transport problem in a domain with 136 128 10 grid cells was performed on a 56-node Intel Paragon. It is important to note that the properties of the medium are speci ed with respect to this grid. The computational results show very intersting phenomena of contaminant distribution with the time progressing, due to the combination of forces driving the process, the geometry of the domain and the properties of the medium. They are in good agreement with experimental results obtained by measuring concentrations in real sites with similar conditions. For example, eld tests conducted at the site indicated contaminant movement against the groundwater ow direction. Our simulation was essential in understanding why the pollutant owed upstream but \down" the bedrock slope. REFERENCES 1. K. Krauskopf, Radioactive Waste Disposal Geology, Chapman & Hill, London (1988). 2. M. Williams, \Stochastic Problems in the Transport of Radioactive Nuclides in Fractured Rock," Nuclear Sci. Eng. 112, 215{230 (1992). 3. G. Chavent and J. Ja re, Mathematical Models and Finite Elements for Reservoir Simulation: Single Phase, Multiphase and Multicomponent Flows Through Porous Media, North-Holland, Amsterdam (1986). 4. P. Binning, Modelling Unsaturated Zone Flow and Contaminant Transport in the Air and Water Phases, PhD thesis, Princeton University (1994). 5. M. Espedal and R. Ewing, \Characteristic Petrov-Galerkin Saturation Methods for Two-Phase Immiscible Flow," Comp. Meth. Appl. Mech. and Eng., 64, 113{135 (1987). 6. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York (1991). 7. R. E. Ewing, T. F. Russell, and M. F. Wheeler, \Convergence Analysis of an Approximation of Miscible Displacement in Porous Media by Mixed Finite Elements and a Modi ed Method of Characteristics," Computer Methods in Applied Mechanics and Engineering, 47, 73{92 (1984). 8. J. Douglas, Jr. and T. Russell, \Numerical Methods for Convection-Dominated Di usion Problems Based on Combining the Method of Characteristics with Finite Element or Finite Di erence Procedures," SIAM J. Numer. Anal., 19, 871{885 (1982).
9. M. Celia, T. F. Russell, I. Herrera, and R. Ewing, \An Eulerian-Lagrangian Localized Adjoint Method for Advection Di usion Equation," Advances in Water Resources, 13, 187{206 (1990). 10. J. Bramble, J. Pasciak, J. Wang, and J. Xu, \Convergence Estimates for Product Iterative Methods with Applications to Domain Decomposition," Math. Comp., 57, 1{21 (1991). 11. R. E. Ewing and M. F. Wheeler, \Computational Aspects of Mixed Finite Element Methods," Numerical Methods for Scienti c Computing (R. Steelman, ed.), pp. 163{172, North-Holland, Amsterdam, 1983. 12. J. Bramble, J. Pasciak, and A. Vassilev, Analysis of the Inexact Uzawa Algorithm for Saddle Point Problems, Technical Report ISC{94{09{MATH, Institute for Scienti c Computation, Texas A& M University, 1994. 13. T. Rusten and R. Winther, \A Preconditioned Iterative Method for Saddle Point Problems," SIAM J. Matrix Anal. Appl., 13, 887{904 (1992). 14. J. Bramble and J. Pasciak, \A Preconditioning Technique for Inde nite Systems Resulting from Mixed Approximations of Elliptic Problems," Math. Comp., 50, 1{18 (1988). 15. L. Cowsar, J. Mandel, and M. Wheeler, \Balancing Domain Decomposition for Mixed Finite Elements," Math. Comp., (to appear). 16. Z. Chen, R. Ewing, Y. Kuznetsov, R. Lazarov, and S. Maliassov, Multilevel Preconditioners for Mixed Methods for Second Order Elliptic Problems, Technical Report ISC-94-14-MATH, Institute for Scienti c Computation, Texas A& M University, 1994. 17. R. Marr, J. Pasciak, and R. Peierls, IPX { Preemptive Remote Procedure Execution for Concurrent Applications, Brookhaven National Laboratory, 1994.
