Analytical Solution of Unsteady-state Forchheimer Flow Problem in an Infinite Reservoir: The Boltzmann Transform Approach

Abstract

For several decades, attempts have been made by several authors to develop models suitable for predicting the effects of Forchheimer flow on pressure transients in porous media. However, due to the complexity of the problem, they employed numerical and/or semi-analytical approaches, which greatly affected the accuracy and range of applicability of their results. Therefore, in order to increase accuracy and range of applicability, a purely analytical approach to solving this problem has been introduced and applied. Therefore, the objective of this paper is to develop a mathematical model suitable for quantifying the effects of turbulence on pressure transients in porous media by employing a purely analytical approach. The partial differential equation (PDE) that governs the unsteady-state flow in porous media under turbulent conditions is obtained by combining the Forchheimer equation with the continuity equation and equations of state. The obtained partial differential equation (PDE) is then presented in dimensionless form (by defining appropriate dimensionless variables) in order to enhance more generalization in application, and the method of Boltzmann Transform is employed to obtain an exact analytical solution of the dimensionless equation. Finally, the logarithmic approximation (for larger times) of the analytical solution is derived. Moreover, after rigorous mathematical modeling and analysis, a novel mathematical relationship between dimensionless time, dimensionless pressure, and dimensionless radius was obtained for an infinite reservoir dominated by turbulent flow. It was observed that this mathematical relationship bears some similarities with that of unsteady-state flow under laminar conditions. Their logarithmic approximations also share some similarities. In addition, the results obtained show the efficiency and accuracy of the Boltzmann Transform approach in solving this kind of complex problem.