A semi-analytic porosity evolution scheme for simulating wormhole propagation with the Darcy–Brinkman–Forchheimer model

Abstract

In this paper, we consider numerical simulation of wormhole propagation with the Darcy-Brinkman-Forchheimer model. Matrix acidization in carbonate reservoirs is a widely practiced technique in the product enhancement of the oil and gas reservoir. A wormhole, i.e. a flow channel with high porosity, is generated during reactive dissolution of carbonates by the action of the injected acid. In the wormhole forming process, the porosity changes non-uniformly in space, and it even becomes close to unity in the central regions of a wormhole. The Darcy-Brinkman-Forchheimer model accounts for both the porous media and clear fluid area, so it can be used to model wormhole propagation perfectly. This model, however, strongly depends on porosity. Therefore, the time schemes for solving the evolutionary equation of porosity have a significant effect on accuracy and stability of the numerical simulation for wormhole propagation. We propose a semi-analytic time scheme, which solves the porosity equation analytically at each time step for given acid concentration. The proposed numerical method can improve the accuracy and stability of numerical simulation significantly. For theoretical analysis of the proposed time scheme for the wormhole simulation, we first reconstruct the analytical functions of porosity to analyze the time error of the porosity, and on the basis of error estimates of porosity, we employ a coupled analysis approach to achieve the estimates of pressure, velocity and solute concentration. The time error estimates for velocity, pressure, concentration and porosity are obtained in different norms. Finally, numerical results are provided to verify the effectiveness of the proposed scheme.

Solar-thermal energy conversion prediction of building envelope using thermochemical sorbent based on established reaction kinetics

“…(6) For the porous heat storage wall, the parameters such as porosity is invariable, the local thermal equilibrium between gas and solid is assumed, and the reaction enthalpy of the form-stable sorbent is constant. The Brinkman-Forchheimer extended Darcy's law is employed to model fluid flow in the porous wall [50].…”

“…The Darcy-Brinkman-Forchheimer equation is adopted to deal with the air flow within porous medium [50]:…”

A Newton’s Second Law Abided Darcy-Brinkman-Forchheimer Framework in Matrix Acidization Simulation

“…) , (22) in which 𝑘 𝑠0 is the surface reaction rate at temperature 𝑇 0 , 𝐸 𝑔 is the activation energy, and 𝑅 𝑔 is the molar gas constant [30]. As a result of that, the reaction rate also becomes a function of T, and its expression is rewritten as 𝑅(𝐶 𝑠 , 𝑇) in (21).…”

“…At the initial stage of the model, the momentum conservation equation in the Darcy scale is written from Darcy's law, so the two-scale model can also be called the Darcy framework in this work [15]- [18]. Later, due to the nature of matrix acidization, Wu et al [19]- [22] consider the Darcy framework is not accurate enough to describe matrix acidization, and provide the Darcy-Brinkman-Forchheimer (DBF) framework by adding Brinkman term and Forchheimer term to the momentum conservation equation. Li et al [23]- [25] further analyse the numerical stability and accuracy of the DBF framework.…”

A 6M digital twin for modeling and simulation in subsurface reservoirs

“…Reactive flow and transport in porous media: Reactive dissolution of carbonates by the action of the injected acid, also known as wormhole propagation, is a widely practiced technique in the product enhancement of petroleum industry. A semi-analytic scheme was proposed with a reconstruction of analytical porosity functions to analyze the time error of the porosity, and a coupled analysis approach was employed to achieve the estimates of pressure, velocity and solute concentration on the basis of porosity error estimation (Wu et al, 2015b;Kou et al, 2019). Meanwhile, various primal discontinuous Galerkin schemes, including NIPG, SIPG and IIPG have been investigated deeply for solving multi-component reactive transport and coupled with multiphase flow simulation in porous media (Sun andWheeler, 2005, 2006).…”