Analytical and numerical studies on a moving boundary problem of non-Newtonian Bingham fluid flow in fractal porous media

Abstract

Non-Darcy flow with a threshold in fractal porous media has been widely used in the development of unconventional petroleum resources such as heavy oil and tight oil. Mathematical modeling of such challenging “threshold flow” problems with strong nonlinearity has great significance in improving petroleum science and technology. Based on a fractal theory, a new non-Darcy kinematic equation with a fractal threshold pressure gradient (TPG) is mathematically deduced in order to describe the non-Darcy flow of a non-Newtonian Bingham fluid with a threshold in fractal porous media. Then mathematical modeling of planar radial non-Darcy flow in a fractal heavy oil reservoir is performed as a nonlinear moving boundary problem. In addition, a steady analytical solution method and a transient numerical solution method are developed. The analytical solution of an ordinary differential equation system for a simple steady model is derived directly, and the transient numerical solution of a partial differential equation system for an unsteady flow model is obtained based on the finite element method with good convergence. These two model solutions are validated by cross-comparisons. It is found from the calculation results that for the steady state, the extremely disturbed moving boundary and its corresponding pressure distribution are affected only by a TPG, production pressure, and a transport exponent; by contrast, for the unsteady state, the moving boundary and its corresponding pressure distribution are affected by many more factors including the fractal dimension. Furthermore, neglect of the fractal TPG and the induced moving boundary can lead to high overestimation of well productivity.