Numerical Schemes and Stability Analysis for Fractional Models of Transient Uni- and Bidimensional flow in Petroleum Reservoirs

Abstract

Objective: To propose a modeling for the flow in oil reservoirs using Caputo’s definition of fractional derivative applied to the spatial coordinate of the problem. Furthermore, delimit a region of numerical stability for the explicit method of solving the model equations.   Methodology: Starting from the material balance in a differential control volume and the generalized Darcy's law, and coupling it with an equation of state, models describing flow in porous media were derived. A numerical solution scheme using an explicit discretization was proposed. For this purpose, the L1 method for approximating spatial fractional derivatives and finite difference approximation for temporal derivatives were employed..   Results and Discussion: Through the Von Neumann stability analysis, it was observed that the explicit solution schemes for the models are conditionally stable and dependent on the order of the fractional derivative. Additionally, numerical stability regions were constructed for one and two-dimensional transient schemes, revealing that the one-dimensional scheme is stable for , and the two-dimensional scheme is stable for  for all derivative orders between 0 and 1.   Implications of the research: Historically, flow models in petroleum reservoirs have been based on the application of Darcy's law; however, its use is limited due to some constraints. Recently, fractional calculus has played a significant role in generalizing and obtaining more accurate models in various application areas. In this context, Darcy's law has been generalized, allowing for the derivation of fractional models that are more suitable for describing flow in porous media. These models, in general, are solved using a numerical method, and it is crucial to understand the stability region of the method for their resolution.