A Numerical Solution to Groundwater Flow in Full Tensorial Permeability Fields

The description of flow through heterogeneous porous media is of primary interest to many fields such as reservoir engineering or hydrology. Because of porosity and permeability heterogeneity, flow equations are usually solved on the basis of numerical techniques. The discretization technique most commonly used is the finite-difference method. Thus, a conventional discretization scheme involves a 5 point stencil for two-dimensional media and a 7 point stencil for three-dimensional media. Continuity of flux and pressure is readily incorporated into the discretization by deriving the coefficients at the interface between adjacent gridblocks from the harmonic average of the permeabilities of the two contiguous gridblocks. This approximation holds as far as permeability tensor is diagonal. <br><br>Today, very detailed geological models are built on grids containing millions of gridblocks. Although computers are growing ever more powerful, fluid flow simulations on such grids are tremendously CPU-time consuming. To make them tractable, the number of gridblocks must be reduced to about 105. This motivated the development of techniques for upscaling the geological model to a manageable level of detail. Upscaling consists in determining the equivalent permeabilities of the coarse gridblocks. These equivalent permeabilities are usually full tensors. Discretization techniques in which interface fluxes are determined from more than two points have been developed to solve full tensorial flow equations. In this case, the discretization stencil requires 9 points for two dimensional media and even more for three-dimensional media. These techniques, known as multipoint flux approximations (MPFA) have recently enjoyed increased popularity. To date, to our knowledge, there is no straightforward generalization of the harmonic average to estimate fluxes at interfaces with MPFAs. <br><br>We develop an alternative to MPFAs based upon numerical spectral methods so that the full tensorial pressure equation is solved on regular grids. We also consider source or sink terms as well as gravity. Contrary to MPFAs, the proposed spectral technique does not call for any kind of approximation to estimate fluxes at the interface between adjacent gridblocks. Specific applications are presented and discussed.<br>