Incompressible Miscible Flow Research Articles

A new optimal error analysis of a mixed finite element method for advection–diffusion–reaction Brinkman flow

AbstractThis article deals with the error analysis of a Galerkin‐mixed finite element methods for the advection–reaction–diffusion Brinkman flow in porous media. Numerical methods for incompressible miscible flow in porous media have been studied extensively in the last several decades. In practical applications, the lowest‐order Galerkin‐mixed method is the most popular one, where the linear Lagrange element is used for the concentration and the lowest‐order Raviart–Thomas element, the lowest‐order Nédélec edge element and piece‐wise constant discontinuous Galerkin element are used for the velocity, vorticity and pressure, respectively. The existing error estimate of this lowest‐order finite element method is only for all variables in spatial direction, which is not optimal for the concentration variable. This paper focuses on a new and optimal error estimate of a linearized backward Euler Galerkin‐mixed FEMs, where the second‐order accuracy for the concentration in spatial directions is established unconditionally. The key to our optimal error analysis is a new negative norm estimate for Nédélec edge element. Moreover, based on the computed numerical concentration, we propose a simple one‐step recovery technique to obtain a new numerical velocity, vorticity and pressure with second‐order accuracy. Numerical experiments are provided to confirm our theoretical analysis.

Analysis of Lowest-Order Characteristics-Mixed FEMs for Incompressible Miscible Flow in Porous Media

The time discrete scheme of characteristics type is especially effective for convection-dominated diffusion problems. The scheme has been used in various engineering areas with different approximations in spatial direction. The lowest-order mixed method is the most popular one for miscible flow in porous media. The method is based on a linear Lagrange approximation to the concentration and the zero-order Raviart-Thomas approximation to the pressure/velocity. However, the optimal error estimate for the lowest-order characteristics-mixed FEM has not been presented although numerous effort has been made in last several decades. In all previous works, only first-order accuracy in spatial direction was proved under certain time-step and mesh size restrictions. The main purpose of this paper is to establish optimal error estimates, $i.e.$, the second-order in $L^2$-norm for the concentration and the first-order for the pressure/velocity, while the concentration is more important physical component for the underlying model. For this purpose, an elliptic quasi-projection is introduced in our analysis to clean up the pollution of the numerical velocity through the nonlinear dispersion-diffusion tensor and the concentration-dependent viscosity. Moreover, the numerical pressure/velocity of the second-order accuracy can be obtained by re-solving the (elliptic) pressure equation at a given time level with a higher-order approximation. Numerical results are presented to confirm our theoretical analysis.

New analysis of Galerkin-mixed FEMs for incompressible miscible flow in porous media

Analysis of Galerkin-mixed FEMs for incompressible miscible flow in porous media has been investigated extensively in the last several decades. Of particular interest in practical applications is the lowest-order Galerkin-mixed method, in which a linear Lagrange FE approximation is used for the concentration and the lowest-order Raviart-Thomas FE approximation is used for the velocity/pressure. The previous works only showed the first-order accuracy of the method in L 2 L^2 -norm in spatial direction, which however is not optimal and valid only under certain extra restrictions on both time step and spatial mesh. In this paper, we provide new and optimal L 2 L^2 -norm error estimates of Galerkin-mixed FEMs for all three components in a general case. In particular, for the lowest-order Galerkin-mixed FEM, we show unconditionally the second-order accuracy in L 2 L^2 -norm for the concentration. Numerical results for both two- and three-dimensional models are presented to confirm our theoretical analysis. More important is that our approach can be extended to the analysis of mixed FEMs for many strongly coupled systems to obtain optimal error estimates for all components.

Optimal error analysis of Crank–Nicolson lowest‐order Galerkin‐mixed finite element method for incompressible miscible flow in porous media

AbstractNumerical methods for incompressible miscible flow in porous media have been studied extensively in the last several decades. In practical applications, the lowest‐order Galerkin‐mixed method is the most popular one, where the linear Lagrange element is used for the concentration and the lowest order Raviart–Thomas mixed element pair is used for the Darcy velocity and pressure. The existing error estimate of the method in L2‐norm is in the order in spatial direction, which however is not optimal and valid only under certain extra restrictions on both time step and spatial meshes, excluding the most commonly used mesh h = hp = hc. This paper focuses on new and optimal error estimates of a linearized Crank–Nicolson lowest‐order Galerkin‐mixed finite element method (FEM), where the second‐order accuracy for the concentration in both time and spatial directions is established unconditionally. The key to our optimal error analysis is an elliptic quasi‐projection. Moreover, we propose a simple one‐step recovery technique to obtain a new numerical Darcy velocity and pressure of second‐order accuracy. Numerical results for both two and three‐dimensional models are provided to confirm our theoretical analysis.

Convergence Analysis of Crank–Nicolson Galerkin–Galerkin FEMs for Miscible Displacement in Porous Media

We propose a fully discrete linearized Crank–Nicolson Galerkin–Galerkin finite element method for solving the partial differential equations which govern incompressible miscible flow in porous media. We prove optimal-order convergence of the fully discrete finite element solutions without any restrictions on the step size of time discretization. Numerical examples are provided to illustrate the theoretical results.

Multiple-relaxation-time lattice Boltzmann model for incompressible miscible flow with large viscosity ratio and high Péclet number.

A lattice Boltzmann model with a multiple-relaxation-time (MRT) collision operator is proposed for incompressible miscible flow with a large viscosity ratio as well as a high Péclet number in this paper. The equilibria in the present model are motivated by the lattice kinetic scheme previously developed by Inamuro et al. [Philos. Trans. R. Soc. London, Ser. A 360, 477 (2002)]. The fluid viscosity and diffusion coefficient depend on both the corresponding relaxation times and additional adjustable parameters in this model. As a result, the corresponding relaxation times can be adjusted in proper ranges to enhance the performance of the model. Numerical validations of the Poiseuille flow and a diffusion-reaction problem demonstrate that the proposed model has second-order accuracy in space. Thereafter, the model is used to simulate flow through a porous medium, and the results show that the proposed model has the advantage to obtain a viscosity-independent permeability, which makes it a robust method for simulating flow in porous media. Finally, a set of simulations are conducted on the viscous miscible displacement between two parallel plates. The results reveal that the present model can be used to simulate, to a high level of accuracy, flows with large viscosity ratios and/or high Péclet numbers. Moreover, the present model is shown to provide superior stability in the limit of high kinematic viscosity. In summary, the numerical results indicate that the present lattice Boltzmann model is an ideal numerical tool for simulating flow with a large viscosity ratio and/or a high Péclet number.

Regularity of the Diffusion-Dispersion Tensor and Error Analysis of Galerkin FEMs for a Porous Medium Flow

We study Galerkin finite element methods for an incompressible miscible flow in porous media with the commonly used Bear--Scheidegger diffusion-dispersion tensor $D({\bf u}) = \Phi d_m I + |{\bf u}| ( \alpha_T I + (\alpha_L - \alpha_T) \frac{{\bf u} \otimes {\bf u}}{|{\bf u}|^2} )$. The traditional approach to optimal $L^\infty((0,T);L^2)$ error estimates is based on an elliptic Ritz projection, which usually requires the regularity of $\nabla_x\partial_tD({\bf u}(x,t)) \in L^p(\Omega_T)$. However, the Bear--Scheidegger diffusion-dispersion tensor may not satisfy the regularity condition even for a smooth velocity field ${\bf u}$. A new approach is presented in this paper, in terms of a parabolic projection, which only requires the Lipschitz continuity of $D({\bf u})$. With the new approach, we establish optimal $L^p$ error estimates and an almost optimal $L^\infty$ error estimate.

Unconditional Convergence and Optimal Error Estimates of a Galerkin-Mixed FEM for Incompressible Miscible Flow in Porous Media

In this paper, we study the unconditional convergence and error estimates of a Galerkin-mixed FEM with the linearized semi-implicit Euler scheme for the equations of incompressible miscible flow in porous media. We prove that the optimal $L^2$ error estimates hold without any time-step (convergence) conditions, while all previous works require certain time-step restrictions. Our theoretical results provide a new understanding on commonly used linearized schemes. The proof is based on a splitting of the error into two parts: the error from the time discretization of the PDEs and the error from the finite element discretization of corresponding time-discrete PDEs. The approach used in this paper can be applied to more general nonlinear parabolic systems and many other linearized (semi)-implicit time discretizations.

Characteristics difference method for incompressible miscible flow in porous media

Two-phase, incompressible miscible flow in prous media is governed by a system of nonlinear partial differential equations. The pressure equation, which is elliptic in appearance, is discretized by a standard five-points difference method. The concentration equation is treated by an implicit finite difference method that applies a form of the method of characteristics to the transport terms. A class of biquadratic interpolation is introduced for the method of characteristics. Convergene rate is proved to be O(Δt+h 2).

A generalized streamline method to predict reservoir flow

We present a generalized streamline method to model flow in porous media, including the effects of gravity and dispersion. We first describe the theory and discuss the approximations of the method, and then compare the predictions using the streamline technique against two-dimensional numerical simulations of incompressible miscible flow. In the cases we studied, the streamline method predicts recovery with an average error of at most 5%, where the principal flow directions are governed by the pattern of permeability, and for gravity numbers less than or equal to 1 and mobility ratios of 10 or less. The streamline method does not suffer from numerical dispersion and is more than 100 times faster than conventional simulation.

A Numerical Solution with Second-Order Accuracy for Multicomponent Compressible Stable Miscible Flow

Abstract This paper presents an extension of an earlier one, which presented an improved numerical technique for solving multicomponent stable miscible flow equations. Diffusion-convection equations for each component together with a pressure equation are solved. Second-order correct approximations for individual component flows, resulting in numerical dispersion correction terms, are employed in solving the diffusion-convection equations. Backward-in-time implicit and Crank-Nicolson approximation techniques are used to solve the pressure equation. Pressure and saturation distributions using first- and second-order correct finite-difference approximations for individual component flows are compared. One- and two-dimensional systems are considered for these comparisons. The results show that in a compressible miscible displacement process, first-order correct finite-difference scheme process, first-order correct finite-difference scheme for individual component flows results in considerable numerical smearing in the saturation distribution. The pressure distribution, which depends upon the saturation distribution, is very sensitive to numerical smearing. Therefore, high-order correct approximations for individual component flours are necessary. The results also demonstrate That the earlier approach employed to reduce numerical smearing in the numerical solution of incompressible miscible flow processes can be extended to compressible miscible flow systems. Introduction Many investigators have tried to solve in some fashion the system of equations governing the miscible displacement process. In most cases, even though the system was compressible, an incompressible system was considered. One may state several reasons for this. Since the system of equations governing compressible miscible flow is nonlinear, analytical solution of such a system, even in a one-dimensional case, is not known In a multidimensional case it may be a practical impossibility. The only methods known to solve the above problem are the numerical methods. One of such methods is presented here. Basic flow equations for the individual components present in the compressible miscible flow system present in the compressible miscible flow system are assumed. These flow equations together with the individual component continuity equations are used to arrive at the pressure equation. Crank-Nicolson and backward-in-time approximation schemes are used in an iterative procedure to solve the pressure equation. Flows of individual components are calculated with first- and second-order correct finite-difference approximations. It is also shown that a semirigorous technique, which eliminates most of the numerical smearing from the numerical solution of an incompressible miscible system, can also be used for a compressible miscible flow system. It has been observed that pressure distributions in a one-dimensional case are pressure distributions in a one-dimensional case are significantly different when the individual component flow terms are approximated with first- and second-order correct finite-difference approximations. Also, the order of approximation of the flow equations has more effect on the pressure distribution than the order of approximation (time approximation) used to solve the pressure equation.