The dependence of rock permeability on direction, or permeability anisotropy, is confirmed by numerous field examples. Ability to measure permeability anisotropy is very important for correct placement of wells. There exist several methods of identifying the maximum and minimum permeability coefficients by interpreting specially designed well tests. Here we develop a new method of estimating the angle of orientation of permeability tensor away from the well. We assume that the principal permeabilities near a cored or otherwise logged well are already known. To accomplish this task we need pressure measurements in monitoring wells.
The orientation of permeability tensor is identified by minimizing the square of the differences between measured and computed data. The underlying boundary-value problem is discretized by the support-operators method, which imposes few restrictions on the grid structure; produces a conservative finite-difference scheme and can be applied to very heterogeneous systems. The gradient and second differential of the minimization criterion are obtained through two systems of adjoint equations, called the first and second adjoint systems. The gradient of the criterion is obtained both in continuous and discrete versions. As an illustration, we analyze a steady-state isothermal, single-phase pressure equation in an anisotropic porous medium. A few applications of the technique developed in this paper for rectangular and curvilinear grids confirm good performance of the algorithm.
