Numerical Simulation of the Nonlinear Difusion-Type Equation in a Two Phase Media with Functional Porosity and Permeability Models

This work presents a mathematical model for a radial flow of a two phase fluid in a porous media (petroleum reservoir). Because the porosity and permeability were pressure dependent, the resulting diffusivity equation was a nonlinear pressure diffusion type equation. The rock porosity and permeability are mathematically modelled as functions of pressure. The resulting nonlinear pressure diffusion type equation is solved numerically retaining its nonlinearity with an explicit backward-forward finite difference method using MATLAB software. A constant porosity and permeability model was first simulated to test the code and to serve as a baseline as it has well known analytic results. The different models for the porosity and permeability were simulated and results obtained and discussed. The results showed that the pressure distribution in a reservoir is dependent on the porosity and permeability models. The slopes from the pressure distribution curves were used to calculate the Darcy flux towards the wellbore. The inverse pressure porosity and permeability model was observed to be the model with the highest fluid yield towards the wellbore as it recorded the least pressure at the well bore at all times, that is, 1000 Pascal at t=10 hours, 500 Pascal at T=25 hours, 400 Pascal at T=50 hours and 250 Pascal at t=100 hours. It had a Darcy flux at T=20 hours, r=1000 as 13.33m/hr and at r=6000 meters as 9.11m/hr. Also at T=50 hours, r=1000 meters as 6.33m/hr and r=6000 meters as 6.94m/hr.