Non-Darcy Porous Media Flow According to the Barree and Conway Model: Laboratory and Numerical Modeling Studies

Abstract

Abstract This paper presents supplementary laboratory data to show that a non-Darcy flow model, proposed by Barree and Conway in 2004, is capable of overcoming the limitation with the Forchheimer non-Darcy equation in high flow rates while describing the entire range of relationships between rate and potential gradient from low- to high-flow rates through proppant packs using a single equation or model. To supplement these laboratory findings, a numerical model is developed that incorporates the Barree and Conway model into a general-purpose reservoir simulator for modeling single-phase non-Darcy flow in porous and fractured media. In the numerical approach, flow through fractured rock is handled using a general multi-continuum approach, applicable to both continuum and discrete fracture conceptual models. The numerical formulation is based on a discretization using an unstructured grid of regular or irregular meshes, followed by time discretization carried out with a backward, first-order, finite-difference method. The final discrete nonlinear equations are handled fully implicitly, using Newton iteration. Additionally, an analytical solution under steady-state linear flow condition is derived and used to verify numerical simulation results for the steady-state linear flow case. The numerical model is applied to evaluate the transient flow behavior at an injection well for non-Darcy flow according to the Barree and Conway model. Results show that the parameter of characteristic length, τ, is more sensitive than other parameters; while the impact of the minimum permeability plateau is shown only at extremely large flow rates or pressure gradients. The proposed numerical modeling approach is suitable for modeling various types of multi-dimensional non-Darcy flow through porous and fractured heterogeneous reservoirs.