Solute Transport in Heterogeneous Porous Media

Solute mass transport in porous media is strongly correlated with pore fluid flow. The analysis of solute transport is an effective means for studying medium heterogeneities. In this study, we discuss the effects of heterogeneity on the tracer transport. Assuming steady fluid flow, we have simulated tracer transport in various permeability heterogeneities. The results show that the tracer distribution is very closely correlated with the medium heterogeneity, and anisotropy in tracer transport exists when there is permeability lineation and large permeability contrast between lowand high-permeability regions. An important feature by which the tracer transport differs from the fluid flow field is that the tracer transport tends to smear the effects of a thin non-permeable layer (or small permeability barriers) through diffusion into the low-permeability layer, while the fluid flow cannot penetrate the low-permeability layer. In addition, the modeling results also show that the tracer transport strongly depends on the tracer source dimension, as well as the flow source dimension. INTRODUCTION The problem of solute transport associated with fluid flow in porous media has become increasingly important in geophysical applications. In petroleum reservoir production, tracer transport tests are a common technique to study the interwell connectivity of a reservoir and anisotropy of reservoir permeability. These tests are also very useful in studying connectivity of a borehole fracture network (Raven and Novakowski, 1984). In environmental studies, the knowledge of contaminant transport and its spatial distributions is essential for designing field survey and remediations. The heterogeneity variations in a porous medium can have significant effects on the transport process. Numerous works have been performed to study the relationship between heterogeneity and the transport process. Atal et al. (1988) have studied how the macroscopic and microscopic fluid dispersion varies with reservoir heterogeneities. Tsang et al. (1988) and Moreno et al. (1988) used the existing flow field to predict the movement of solute concentration through a rough surface fracture using partical tracking techniques. Thompson (1991) has investigated the fluid transport problem in natural fractures with rough surfaces. Thompson (1991) showed that the surface roughness creates spatially 314 Zhao and Toksoz varying hydraulic conductivity along the fracture; it therefore causes fluid flow and tracer transport to be restricted to channels that occupy only a small percentage of the fracture volumes, resulting in significant channeling of the transport. Because of the heterogeneous nature of a geological medium, the solute transport in heterogeneous porous media can provide knowledge about the effects of heterogeneities on the solute transport and transport parameters that control these effects. As a result, such effects as permeability heterogeneity, permeability anisotropy, etc., can be estimated from measuring the solute mass transport behaviors in a reservoir. The primary goal of this study is to investigate the effects of formation permea:bility heterogeneity and anisotropy on the solute transport characteristics measured downhole. The governing equation for the solute mass transport problem is the advectiondispersion equation. For heterogeneous media, numerical models can easily deal with variability in the flow and transport parameters (for example, permeability, porosity, and dispersivity etc.). Thus they can be conveniently used to model geological structures with complex geometry. This study adopts an Alternating Direction Implicit (ADI, Ferziger, 1980; Zhao and Toksiiz, 1992) finite difference scheme to solve the timedependent tracer transport problem. By substituting the domain of fluid flow into the finite difference grid, varying parameter values are assigned to the numerical grid to account for medium heterogeneities, and the ADI finite difference technique is used to calculate the solute distribution at each given time step. In this way, we can simulate the complex solute plume shapes that develop in natural geological systems. Complicating the solute transport problem is the fluid velocity field that is very important in determining the advection of the transport process. Continuity conditions in the numerical solutions of the transport equation require an accurate representation of the velocity field. The fluid flow velocity field is obtained from simulation of flow in the heterogeneous porous medium, in which the velocity field is calculated using Darcy's law with given permeability distribution and known parameters and boundary conditions. This problem has been solved in Zhao and Toksiiz, (1991). The fluid flow field is assumed to be independent of the solute transport. In other words, the solute concentration does not influence the flow. In this situation, the flow and transport equations can be solved separately. THE ADVECTION-DIFFUSION-DISPERSION EQUATION FOR SOLUTE TRANSPORT For fluid flow in a porous medium, solute transports are due to three important mechanisms: diffusion, dispersion, and advection. Here we briefly describe the derivation of the advection-diffusion-dispersion equation, in order to introduce the flow and transport parameters that control the transport process. The solute mass transport equation is based on the mass conservation equation: (1) where J is the solute mass flux, r/> is porosity, and C is solute concentration. The product r/>C is mass per volume. Equation (1) states that the net mass output per unit volume Transport in Porous Media equals the time rate of change of mass within the volume. The flux J contains both diffusion flux and advection flux: J = -Do\7C + v(C). 315 (2) The first term on the right hand side is the diffusion flux, Do being the molecular diffusion coefficient. The second term is the advection flux, which is caused by V, the velocity of the pore fluid flow. During mass transport, mechanical dispersion is also an important mechanism (Tang et al., 1981; Sudicky and Frind, 1982; Grisak and Pickens, 1980). The effect of this dispersion is mathematically treated by changing the Do in Equation (2) to (Domenico and Schwartz, 1990; Thompson, 1991) D = Do + aU (3) where a is known as the dynamic dispersivity which is an important property of the porous medium. The coefficient D now is called the hydrodynamic dispersion coefficient. Depending on the value of a, the dispersive process due to the fluid velocity U contributes to the mechanical mixing of solute. Here U is the magnitude of the velocity. In the case of multi-dimensional flow, U is defined as U = Ilvll = Jv~ + v~ + v;. (4) This approach was used by Mur~y and Scott (1977), who assumed that D is proportional to the full magnitude of fluid flow velocity. For a heterogeneous porous medium, we also allow v to vary spatially if the velocity field varies because of permeability variation of the medium. In this case, U = U(x, y, z) is the local magnitude of the velocity field. Letting U vary spatially is important for modeling solute transport in a heterogeneous porous medium, because in such a medium velocity values may differ greatly at different locations of the medium, especially if the driving pressure is a localized (or point) source. By substituting Equations (2) and (3) into Equation (1), we get \7. [D\7C] \7. [v(C)] = a(:~) . The second term on the left hand side of Equation (5) can be written as \7. [V(C)] = (\7. V)C + v' \7(C) . (5) (6) If we assume that the solute transport process does not change the density of the fluid, then the solute transport does not affect the flow velocity field v, and vcan be calculated independent of the solute concentration field. We further assume that solute transport takes place in a steady fluid flow field, which is governed by Darcy's law v= -1<,/fJ-\7P, where I<, is permeability, fJis fluid viscosity and \7P is the pressure gradient. In this case, the divergence of the fluid velocity field vanishes, as in the following equation, (7) 316 Zhao and Toksoz Equation (5) is the governing equation for the fluid pressure in a porous medium with a heterogeneous permeability distribution, which can be solved to compute the fluid velocity field v for the given permeability distribution and boundary conditions (see Zhao and Toksiiz, 1991). Under the steady state flow assumption, Equation (5) is simplified to become V'. [¢DV'C] v' V'(¢C) = fJ(¢C) . (8) fJt Equation (8) is the governing equation for solute transport in a steady fluid flow field. This equation includes the effects of diffusion, dispersion, and advection transport in heterogeneous media. This equation is the basis for the numerical modeling of solute transport in heterogeneous media. From the governing equation, it can be seen that the solute transport is a complicated process affected by a number of factors. Because of the molecular diffusion process, over a certain period of. time diffusion can cause solute mass to move considerable distances, even in media with very low permeability. The pore fluid velocity field v plays an important role in the solute transport process. Therefore diffusion and advection processes compete with each other in the transport process. When diffusion dominates, the solute tends to be homogeneously dispersed in the medium. On the other hand, according to Darcy's law, the velocity field v is controlled by medium permeability and the pressure gradient that drives the flow. Therefore, when advection dominates, the transport process will mainly reflect the effect of medium permeability. In this later case, effects of permeability and porosity heterogeneities become an important issue. Because of this mechanism, tracer tests are used as diagnostics of formation permeability. The primary goal of this study is to investigate the effects of medium heterogeneous properties, especially permeability and porosity, on the solute transport process. Aimed at geophysical applications, we will model solute transport or tracer experiments made in boreholes, because the majority of such measurements are made downhole. The transport process, will be simulated in a Cartesian coordinate system to study the vertical variation of solute concentration in the crosshole experiment. By modeling the transport process for a point source, we can also simulate the azimuthal variation of solute transport due to tracer injection from a borehole. FORMULATION IN CARTESIAN COORDINATES Theoretical formulation In this study, we model solute transport in a two-dimensional domain. For the 2-D case, Equation (8) can be written as !-. (DfJC) !-. ("'DfJC) _ fJ(¢C) _ fJ(¢C) = fJ(¢C) fJx ¢ fJx + fJz 'P fJz V x fJx Vz fJz fJt' The pore fluid volumetric flow velocities V x and V z are computed as follows. solve the following equation for the pressure field :x [a(x,z)~:] + ~ [a(x,z)~~] =0 , (9) We first