## Mathematical modeling, analysis and simulation of the productivity index for non-linear flow in porous media, with applications in reservoir engineering

##### Zusammenfassung

Fluid flow in porous media has been often modeled using Darcy’s Law. Because it has been observed that Darcy’s Law is valid only for low velocities, there have been several attempts to solve this problem by using modifications of Darcy’s equation. Forchheimer modified Darcy’s equation by adding a new term to account for inertia caused by high velocity flow initially thought to occur only in gas reservoir but later confirmed that some oil reservoirs may also exhibit non-Darcy flow behavior. This paper takes a look into high rate oil wells to determine if they may experience nonlinearity due to non-Darcy flow and to examine the discrepancy between the differential equation derived from the traditional Darcy’s Law and a new approach using a non-linear Darcy-Forchheimer equation.
Mathematical modeling of Darcy and Darcy-Forchheimer diffusivity equations will be performed to simulate high velocity flow. Finite differences methods will be used to approximate the solution of the partial differential equations. In this paper, both explicit and implicit methods will be used to evaluate the differential equations. Implicit methods are more commonly used for domain discretization because they are unconditionally stable. Stability analysis will be conducted to determine the condition for stability of the explicit method used. The results will then be compared to results from finite element method software.
The productivity index from the Darcy-Forchheimer diffusivity equation will be calculated with different boundary conditions to evaluate the effect of nonlinear flow on well potential, pressure gradient, and velocity.