Convergence Analysis of a Numerical Method for a Fractional Model of Fluid Flow in Fractured Porous Media

Abstract

The present paper is devoted to the construction and study of numerical methods for solving an initial boundary value problem for a differential equation containing several terms with fractional time derivatives in the sense of Caputo. This equation is suitable for describing the process of fluid flow in fractured porous media under some physical assumptions, and has an important applied significance in petroleum engineering. Two different approaches to constructing numerical schemes depending on orders of the fractional derivatives are proposed. The semi-discrete and fully discrete numerical schemes for solving the problem are analyzed. The construction of a fully discrete scheme is based on applying the finite difference approximation to time derivatives and the finite element method in the spatial direction. The approximation of the fractional derivatives in the sense of Caputo is carried out using the L1-method. The convergence of both numerical schemes is rigorously proved. The results of numerical tests conducted for model problems are provided to confirm the theoretical analysis. In addition, the proposed computational method is applied to study the flow of oil in a fractured porous medium within the framework of the considered model. Based on the results of the numerical tests, it was concluded that the model reproduces the characteristic features of the fluid flow process in the medium under consideration.

An Analytical Solution for Variable Viscosity Flow in Fractured Media: Development and Comparative Analysis With Numerical Simulations

“…The uniform fluid viscosity assumption is commonly applied in modeling flow through porous and fractured media, as it simplifies the mathematical formulation and solution of the problem (Baigereyev et al., 2021; Berre et al., 2019). However, fluid viscosity can actually vary due to different factors (Reddy & Reddy, 2013).…”

Application of Haar wavelet to shear-wave equation and corresponding fractional differential equation

“…Hence, one can increase the flexibility of controlling a system from point to a space [10]. There are many more phenomena in science and engineering that are modelled using the concept of fractional calculus [11][12][13][14][15] Besides modelling, fractional calculus facilitates finding the solution techniques' reliabilities. There is a need to develop numerical methods for finding the solution for fractional differential equations.…”

Error estimates of the numerical method for the filtration problem with Caputo-Fabrizio fractional derivatives

“…In [23,24], the authors considered the issues of the numerical solution of fractional differential equations, to which the filtration equations are reduced, using the methods of the theory of difference schemes, and they carried out a rigorous theoretical study of the convergence order of the proposed schemes. In the previous work [25], two finite element schemes of the convergence order O (τ 2−ν ), ν = max {α, β, γ}, α, β, γ ∈ (0, 1) were constructed for solving the initial boundary value problem for an equation of the form (1) with a fractional Caputo derivative. In this paper, we continue this endeavor, but unlike [25], we use a fractional derivative in the sense of Caputo-Fabrizio and assume that its use provides a more realistic description of the fluid flow process and helps to better capture the dynamic behavior of real phenomena as discussed in works [26,27].…”

“…In the previous work [25], two finite element schemes of the convergence order O (τ 2−ν ), ν = max {α, β, γ}, α, β, γ ∈ (0, 1) were constructed for solving the initial boundary value problem for an equation of the form (1) with a fractional Caputo derivative. In this paper, we continue this endeavor, but unlike [25], we use a fractional derivative in the sense of Caputo-Fabrizio and assume that its use provides a more realistic description of the fluid flow process and helps to better capture the dynamic behavior of real phenomena as discussed in works [26,27]. In addition, the use of the Caputo-Fabrizio derivative eliminates the difficulty of a degenerate singular kernel, which makes it difficult to apply approximate methods of its discretization.…”