A unified analysis of fully mixed virtual element method for wormhole propagation arising in the petroleum engineering

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Abstract

Wormhole propagation, arising in petroleum engineering, is used to describe the distribution of acid and the increase of porosity in carbonate reservoir under the dissolution of injected acid and plays a very important role in the product enhancement of oil and gas reservoirs. In this paper, a fully mixed virtual element method (VEM) is employed to discretize this problem, in which mixed VEM is used not only for the Darcy flow equations but also for approximation the concentration equation by introducing an auxiliary flux variable to guarantee full mass conservation. The stability, existence and uniqueness of solution of the associated mixed VEM are proved by fixed point theory. Also, we obtain unconditionally optimal error estimate for concentration and auxiliary flux variable of convection-diffusion equation, as well as for the velocity and pressure of Darcy equations in the L2 norm. Finally, several numerical experiments are presented to support the theoretical analysis of convergence and to illustrate the applicability for solving actual problems.

Section snippets

Scope

Due to the huge oil and gas reserves in some countries and the economic and extraordinary importance of these materials, which are the source of income in some countries, the importance of issues related to oil engineering is determined and in this regard, doing things to increase the efficiency of oil wells in this industry seems very important. In order to exploit a newly excavated oil well, first to control the pressure and fluid flow, the walls of the well are latticed in the different ways

Analysis of the continuous problem

Now, we stablish the main aspects of the continuous problem, namely, existence, uniqueness and stability. Let us now discuss the stability properties of the forms in (1.15).

Virtual element approximation

The chief target of this section is to present the VE spaces and discrete bilinear (and trilinear) forms that are required for creating a VEM scheme. For simplicity of the presentation we restrict the construction to the 2D case.

Convergence analysis

We split the error analysis in two steps. First one estimates the velocity and pressure discretization errors, ‖un−uhn‖0 and ‖pn−phn‖0, respectively; and the second stage corresponds to establishing bounds for the concentration error ‖cn−chn‖0 and its flux, i.e., ‖σn−σhn‖0.

Lemma 4.1 Discrete Gronwall's inequality

Let τ,B and ak,bk,ck,γk, for integers k≥0, be non-negative numbers such thatan+τ∑k=0nbk≤τ∑k=0nγkak+τ∑k=0nck+B,forn≥0, suppose that τγk<1, for all k, and set σk=(1−τγk)−1. Thenan+τ∑k=0nbk≤exp⁡(τ∑k=0nγkσk)(τ∑k=0nck+B),forn≥0.

Numerical results

In this section, we provide numerical experiments to show the performance of the fully mixed virtual element technique to solve incompressible wormhole propagation. In all examples, we use pair space (X1h,Y1h) for Dracy and the concentrations equation in mixed formulation, unless otherwise stated.

Conclusion

This paper presents the solvability and convergence analysis of a four-field formulation of the wormhole propagation model where the unknown variables are the pressure and velocity (for Darcy equation), concentration, and the flux depending on the diffusive and velocity (concentration equation). Our approach is based on the mixed virtual element method (VEM) to discretize both Darcy and concentration equations in space, leading to a fully mixed formulation and an efficient scheme compared to

Acknowledgements

The authors are very grateful to the anonymous reviewers for carefully reading this paper and for their comments and suggestions, which have improved the paper.

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