Numerical algorithm to estimate formation factor of digital rocks

Abstract

The paper presents a numerical algorithm of estimating the formation factor for digital rock samples constructed using microtomographic images. The algorithm is based on the numerical solution of the three-dimensional Poisson equation with rapidly changing high-contrast coefficients. The arising system of linear algebraic equations turns out to be ill-conditioned, so to speed up the convergence it is necessary to use a preconditioner. The conjugate gradient method is applied to solve the system and the preconditioner is constructed as the inverse Laplace operator corresponding to the homogeneous model. In its turn, the spectral decomposition of two tridiagonal matrices appearing in the approximation of one-dimensional derivatives is used for inversion. The resulting series of one dimensional problems is solved by the Thomas algorithm. The used preconditioner can be effectively applied both to the original problem with rapidly changing high-contrast coefficients and to the problem, in which the solution is calculated only in the pore space. In the latter case, the method provides the same accuracy of results as the original one for inhomogeneous models, but it converges almost twice as fast. Implementation using graphics processors allows solving problems up to 109 unknowns with a single GPU.