Convergence analysis of a DDFV scheme for a system describing miscible fluid flows in porous media

In this article, we prove the convergence of a discrete duality finite volume scheme for a system of partial differential equations describing miscible displacement in porous media. This system is made of two coupled equations: an anisotropic diffusion equation on the pressure and a convection‐diffusion‐dispersion equation on the concentration. We first establish some a priori estimates satisfied by the sequences of approximate solutions. Then, it yields the compactness of these sequences. Passing to the limit in the numerical scheme, we finally obtain that the limit of the sequence of approximate solutions is a weak solution to the problem under study. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 723–760, 2015     

垂直裂缝井试井分析模型和方法

根据压裂井的流动机制,综合考虑井筒储存、裂缝壁面污染和各种边界条件,通过组合线性流模型与有效井径模型,建立了垂直裂缝井试井分析的新模型,提出了确定垂直裂缝井有效井径的方法,给出了有效井径随裂缝长度、裂缝导流能力和裂缝表皮系数的变化关系．该模型形式简明,曲线完整,计算速度快,可满足实时计算和快速响应的试井解释要求．以实测的压力或产量为拟合目标函数,建立了识别地层和水力裂缝参数的最优化模型,提出了综合应用逐步线性最小二乘法和约束变尺度法的垂直裂缝井试井分析自动拟合方法,并通过算例说明了该方法的准确性和可靠性．应用情况表明,该技术可科学合理地分析评估压裂施工质量,指导并改进压裂设计,提高压裂设计水平和施工效果．     

Calculation of an unsteady moving-boundary flow in a porous medium by the generalized method of integral relations

Dorodnicyn’s generalized method of integral relations is used to compute a Verigin-type single-phase unsteady flow in a porous medium. This problem describes the pumping of a gas through a gallery in a bounded horizontal aquifer and is associated with underground gas storage in aquifers. The case of an isothermal process and an ideal gas are considered. The viscosity of the gas is neglected. Sines are used as smoothing functions. The results obtained in the first and third approximations are presented and analyzed. The solution is compared with a finite-difference solution and that produced by the method of integral relations. The results are given in a table.     

Numerical analysis of a finite volume scheme for a seawater intrusion model with cross‐diffusion in an unconfined aquifer

We consider a degenerate parabolic system modeling the flow of fresh and saltwater in a porous medium in the context of seawater intrusion. We propose and analyze a finite volume scheme based on two‐point flux approximation with upwind mobilities. The scheme preserves at the discrete level the main features of the continuous problem, namely the nonnegativity of the solutions, the decay of the energy and the control of the entropy and its dissipation. Based on these nonlinear stability results, we show that the scheme converges toward a weak solution to the problem. Numerical results are provided to illustrate the behavior of the model and of the scheme.     

Combined effect of magnetic field and heat absorption on unsteady free convection and heat transfer flow in a micropolar fluid past a semi-infinite moving plate with viscous dissipation using element free Galerkin method

The fully developed electrically conducting micropolar fluid flow and heat transfer along a semi-infinite vertical porous moving plate is studied including the effect of viscous heating and in the presence of a magnetic field applied transversely to the direction of the flow. The Darcy-Brinkman-Forchheimer model which includes the effects of boundary and inertia forces is employed. The differential equations governing the problem have been transformed by a similarity transformation into a system of non-dimensional differential equations which are solved numerically by element free Galerkin method. Profiles for velocity, microrotation and temperature are presented for a wide range of plate velocity, viscosity ratio, Darcy number, Forchhimer number, magnetic field parameter, heat absorption parameter and the micropolar parameter. The skin friction and Nusselt numbers at the plates are also shown graphically. The present problem has significant applications in chemical engineering, materials processing, solar porous wafer absorber systems and metallurgy.     

Group method analysis of combined heat and mass transfer by MHD non-Darcy non-Newtonian natural convection adjacent to horizontal cylinder in a saturated porous medium

The group theoretic method is applied for solving problem of combined magneto-hydrodynamic heat and mass transfer of non-Darcy natural convection about an impermeable horizontal cylinder in a non-Newtonian power law fluid embedded in porous medium under coupled thermal and mass diffusion, inertia resistance, magnetic field, thermal radiation effects. The application of one-parameter groups reduces the number of independent variables by one and consequently, the system of governing partial differential equations with the boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions. The ordinary differential equations are solved numerically for the velocity using shooting method. The effects of magnetic parameter M, Ergun number Er, power law (viscosity) index n, buoyancy ratio N, radiation parameter R_d, Prandtl number Pr and Lewis number Le on the velocity, temperature fields within the boundary layer, heat and mass transfer are presented graphically and discussed.     

Nonsimilar boundary layer analysis of double-diffusive convection from a vertical truncated cone in a porous medium with variable viscosity

This work presents a boundary layer analysis about variable viscosity effects on the double-diffusive convection near a vertical truncated cone in a fluid-saturated porous medium with constant wall temperature and concentration. The viscosity of the fluid is assumed to be an inverse linear function of the temperature. A boundary layer analysis is employed to derive the nondimensional nonsimilar governing equations, and the transformed boundary layer governing equations are solved by the cubic spline collocation method to yield computationally efficient numerical solutions. The obtained results are found to be in good agreement with previous papers on special cases of the problem. Results for local Nusselt and Sherwood numbers are presented as functions of viscosity-variation parameter, buoyancy ratio, and Lewis number. For a porous medium saturated with a Newtonian fluid with viscosity proportional to an inverse linear function of temperature, higher value of viscosity-variation parameter leads to the decrease of the viscosity in fluid flow, thus increasing the fluid velocity as well as the local Nusselt number and the local Sherwood number.     

Convergence analysis of a vertex‐centered finite volume scheme for a copper heap leaching model

In this paper a two‐dimensional solute transport model is considered to simulate the leaching of copper ore tailing using sulfuric acid as the leaching agent. The mathematical model consists in a system of differential equations: two diffusion–convection‐reaction equations with Neumann boundary conditions, and one ordinary differential equation. The numerical scheme consists in a combination of finite volume and finite element methods. A Godunov scheme is used for the convection term and an P1‐FEM for the diffusion term. The convergence analysis is based on standard compactness results in L². Some numerical examples illustrate the effectiveness of the scheme. Copyright © 2009 John Wiley & Sons, Ltd.