A NUMERICAL FORMULATION FOR THE SIMULATION OF HIGHLY HETEROGENEOUS POROUS MEDIA

Abstract

Although it is a tradition to investigate porous media flow with the aid of Darcy's law, there are several applications in which heterogeneity makes this approach inadequate. Carbonate reservoirs found in the pre-salt layer in Brazil are examples of highly heterogeneous naturally fractured formations, with severe variations in their petrophysical properties. Rocks submitted to acidifying treatments are another example of highly heterogeneous porous media, where, by the injection of an acid system in the rock matrix, wormholes (highly conductive channels) are created. In this work, we numerically compare the employment of Darcy's equation with a more general formulation based on the average conservation equations for highly heterogeneous porous media. The coupled continuity and momentum equations are solved employing the open source software OpenFOAM. We apply the new formulation to three cases. The first is more academic, followed by two more applied situations associated with 2D and 3D flows. Different values of the Reynolds number (Re) and different permeability ratios were tested. Since the pressure drop was imposed, an error measure based on the flow rate was computed. We show that higher values of Re and permeability ratios lead to more discrepant results between the two approaches. Analyzing the Brinkman model for one of the cases, which takes into account diffusive effects, we found that the error with respect to the complete model, which in addition considers inertial effects, decreases but is still significant. Hence, the classical extension of the Darcy model, namely Brinkman and Forchheimer, cannot handle alone the situations of high Re and/or high heterogeneity, since both effects neglected in the Darcy model are important. As a consequence, a formulation that generalizes Darcy's law is required for more accurate results in these cases.