Compressible Flow In Porous Media Research Articles

Efficient implementation of mixed finite element methods for parabolic problems

It is well-known that conventional mixed finite element method (FEM) results in a large saddle point system or non-symmetric system. A splitting mixed FEM is introduced by Yang (Numer. Methods Partial Differential Equations 17: 229–249, 2001), which computes the flux variable by solving a symmetric positive definite system. Then, the original unknown is solved based on the computed flux. In this paper, we prove that the splitting mixed FEM proposed by Yang is equivalent to the conventional mixed FEM provided that the initial data is carefully taken. Moreover, we show that the equivalence also holds for more complicated models, such as splitting mixed FEM for the compressible flow in porous media. Therefore, all available theoretical results of conventional mixed FEMs hold for the splitting ones. Numerical comparisons are provided for the conventional mixed FEM and the splitting mixed FEM. Efficiency of the splitting mixed method is clearly demonstrated. Corresponding results can be extended to many other mixed FEMs, such as Brezzi–Douglas–Marini mixed FEM.

Convergence of a finite volume scheme for immiscible compressible two-phase flow in porous media by the concept of the global pressure

This paper deals with development and analysis of a finite volume (FV) method for the coupled system describing immiscible compressible two-phase flow, such as water-gas, in porous media, capillary and gravity effects being taken into account. We investigate a fully coupled fully implicit cell-centered “phase-by-phase” FV scheme for the discretization of such system. The main goal is to incorporate some of the most recent improvements in the scheme and the convergence of the numerical approximation to the weak solution of such models. The spatial discretization uses a TPFA scheme and a new strategy for handling the upwinding. Based on a priori estimates and compactness arguments, we prove the convergence of the numerical approximation to the weak solution. The particular feature in this convergence analysis of the classical engineering scheme based on the “phase-by-phase” upwinding on an orthogonal mesh relies on the global pressure–saturation fractional flow formulation as was defined relatively recently for immiscible compressible flow in porous media. We have developed and implemented this scheme in a new module in the context of the open source platform DuMuX. Two numerical experiments are presented to demonstrate the efficiency of this scheme. The first test addresses the evolution in 2D of gas migration through engineered and geological barriers for a deep repository for radioactive waste. The second test case is chosen to test the ability of the method to approximate solutions for 3D problems modeling scenarios of CO2 injection in a fully water-saturated domain.

An efficient multigrid-DEIM semi-reduced-order model for simulation of single-phase compressible flow in porous media

In this paper, an efficient multigrid-DEIM semi-reduced-order model is developed to accelerate the simulation of unsteady single-phase compressible flow in porous media. The cornerstone of the proposed model is that the full approximate storage multigrid method is used to accelerate the solution of flow equation in original full-order space, and the discrete empirical interpolation method (DEIM) is applied to speed up the solution of Peng–Robinson equation of state in reduced-order subspace. The multigrid-DEIM semi-reduced-order model combines the computation both in full-order space and in reduced-order subspace, which not only preserves good prediction accuracy of full-order model, but also gains dramatic computational acceleration by multigrid and DEIM. Numerical performances including accuracy and acceleration of the proposed model are carefully evaluated by comparing with that of the standard semi-implicit method. In addition, the selection of interpolation points for constructing the low-dimensional subspace for solving the Peng–Robinson equation of state is demonstrated and carried out in detail. Comparison results indicate that the multigrid-DEIM semi-reduced-order model can speed up the simulation substantially at the same time preserve good computational accuracy with negligible errors. The general acceleration is up to 50–60 times faster than that of standard semi-implicit method in two-dimensional simulations, but the average relative errors of numerical results between these two methods only have the order of magnitude 10−4–10−6%.

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We present continuum-scale modeling of multiphase compressible flow in porous media with applications to hydrocarbon reservoir engineering. A new black-oil model is developed and compared with a fully compositional simulator to model the thermodynamic phase behavior. In the context of black-oil modeling, where components are lumped into a gas and liquid pseudocomponent with only the gas transferring between liquid and gas phases, we allow for a variable bubble point pressure (e.g., when gas enters an undersaturated zone). Traditionally, a primary variable switching strategy has been used, which is known to be prone to convergence and phase identification issues. Instead, we adopt an overall molar composition-based framework that can robustly model phase appearance or disappearance. Phase properties across a broad range of pressures for different black-oil compositions are constructed from compositional phase split calculations to correctly model the phase transition. Mass transport is updated explicitly by a locally mass conserving multilinear discontinuous Galerkin method. Globally continuous pressure and velocity fields are obtained through an implicit mixed hybrid FE scheme. The robustness and accuracy of our FE simulator are demonstrated in several problems, where we have attained considerable speed-up and maintained the accuracy with the new black-oil model.

Adjoint based state estimation of compressible flow in porous media

Abstract In this paper, we study the state estimation of compressible single phase flow in compressible porous media. The initial pressure distribution is estimated according to discrete adjoint approach based on the collected well pressure data. The first-order Tykhonov regularization method is used to obtain reasonable estimation. By analyzing the optimality condition of estimation problem, the discrete adjoint state equation and discrete adjoint gradient are derived based on the numerical scheme of the continuous equations. A quasi-Newton numerical optimization method related to adjoint gradient is proposed to solve the estimation problem. The estimation results with different regularization coefficients are compared and analyzed by numerical experiments. The deviation between the estimated pressure obtained without regularization and the real pressure is large. Estimation result with smaller deviation and higher smoothness can be obtained through appropriate regularization coefficient. When the observation error is large, the observed values generated by the estimated pressure fit well with the real pressure.

Parallel reservoir simulators for fully implicit complementarity formulation of multicomponent compressible flows

The numerical simulation of multicomponent compressible flow in porous media is an important research topic in reservoir modeling. Traditional semi-implicit methods for such problems are usually conditionally stable, suffer from large splitting errors, and may accompany with violations of the boundedness requirement of the numerical solution. In this study we reformulate the original multicomponent equations into a nonlinear complementarity problem and discretize it using a fully implicit finite element method. We solve the resultant nonsmooth nonlinear system of equations arising at each time step by a parallel, scalable, and nonlinearly preconditioned semismooth Newton algorithm, which is able to preserve the boundedness of the solution and meanwhile treats the possibly imbalanced nonlinearity of the system. Some numerical results are presented to demonstrate the robustness and efficiency of the proposed algorithm on the Tianhe-2 supercomputer for both standard benchmarks as well as realistic problems in highly heterogeneous media.

A fully mass-conservative iterative IMPEC method for multicomponent compressible flow in porous media

In this paper we consider efficient and fully mass-conservative numerical methods for the multicomponent compressible single-phase Darcy flow in porous media. Compared with the classical IMplicit Pressure Explicit Concentration (IMPEC) scheme by which one of the components may be not mass-conservative, the new scheme enjoys an appealing feature that the conservation of mass is retained for each of the components. The pressure–velocity system is obtained by the summation of the discrete conservation equation for each component multiplying an unknown parameter which is nonlinearly dependent of the molar concentrations. This approach is quite different from the conventional method which is used in the classical IMPEC scheme. We utilize a fully mass-conservative iterative IMPEC method to solve the nonlinear system for molar concentration, pressure and velocity fields. The upwind mixed finite element methods are used to solve the pressure–velocity system. Although the Peng–Robinson equation of state (EOS) is utilized to describe the pressure as a function of the molar concentrations, our method is suitable for any type of EOS. Under some reasonable conditions, the iterative scheme can be proved to be convergent, and the molar concentration of each component is positivity-preserving. Several interesting examples of multicomponent compressible flow in porous media are presented to demonstrate the robustness of the new algorithm.

Multipoint flux mixed finite element methods for slightly compressible flow in porous media

In this paper, we consider multipoint flux mixed finite element discretizations for slightly compressible Darcy flow in porous media. The methods are formulated on general meshes composed of triangles, quadrilaterals, tetrahedra or hexahedra. An inexact Newton method that allows for local velocity elimination is proposed for the solution of the nonlinear fully discrete scheme. We derive optimal error estimates for both the scalar and vector unknowns in the semidiscrete formulation. Numerical examples illustrate the convergence behavior of the methods, and their performance on test problems including permeability coefficients with increasing heterogeneity.

Higher-order black-oil and compositional modeling of multiphase compressible flow in porous media

We present continuum-scale modeling of multiphase compressible flow in porous media with applications to hydrocarbon reservoir engineering. A new black-oil model is developed and compared with a fully compositional simulator to model the thermodynamic phase behavior. In the context of black-oil modeling, where components are lumped into a gas and liquid pseudocomponent with only the gas transferring between liquid and gas phases, we allow for a variable bubble point pressure (e.g., when gas enters an undersaturated zone). Traditionally, a primary variable switching strategy has been used, which is known to be prone to convergence and phase identification issues. Instead, we adopt an overall molar composition-based framework that can robustly model phase appearance or disappearance. Phase properties across a broad range of pressures for different black-oil compositions are constructed from compositional phase split calculations to correctly model the phase transition. Mass transport is updated explicitly by a locally mass conserving multilinear discontinuous Galerkin method. Globally continuous pressure and velocity fields are obtained through an implicit mixed hybrid FE scheme. The robustness and accuracy of our FE simulator are demonstrated in several problems, where we have attained considerable speed-up and maintained the accuracy with the new black-oil model.

A Block-Centered Finite Difference Method for Slightly Compressible Darcy–Forchheimer Flow in Porous Media

A block-centered finite difference method is introduced to solve an initial and boundary value problem for a nonlinear parabolic equation to model the slightly compressible flow in porous media, in which the velocity–pressure relation is described by Darcy–Forchheimer’s Law. The method can be thought as the lowest order Raviart–Thomas mixed element method with proper quadrature formulation. By using the method the velocity and pressure can be approximated simultaneously. We established the second-order error estimates for pressure and velocity in proper discrete norms on non-uniform rectangular grid. No time-step restriction is needed for the error estimates. The numerical experiments using the scheme show that the convergence rates of the method are in agreement with the theoretical analysis.

Evaluation of compressible flow in spherical fueled reactors using the porous media model

In this study, using porous media approach, the compressible flow within the core of a Pebble Bed Modular Reactor (PBMR) is simulated. This reactor has been composed of coated fuel particles with compressible gas as a coolant and graphite as a moderator and reflector. Containing about 450,000 fuel complexes, the reactor core is considered a porous medium subject to high temperature and high pressure helium flow. The porosity and permeability parameters are calculated and utilized. The coolant compressibility has been introduced as an effective parameter in the thermal–hydraulic analysis. Accordingly, using the ANSYS CFX code, which is capable of simulating compressible flow in porous media, the reactor core is simulated and thermal–hydraulic parameters of the core are obtained through Computational Fluid Dynamic (CFD) approach. The heat flux in the core is first obtained in axial and radial coordinates by MCNP code and is then used in CFD simulation as a semi sine and an algebraic function. The major characteristics of the flow field have been calculated whereby the thermal–hydraulic parameters such as temperature and pressure profiles have been obtained and compared with other data. Comparing the results obtained with other codes and software, the outcomes show that the inclusion of compressibility is reasonable and will lead to a slight difference between the measured and actual temperature, pressure and velocity. In another stage, pressure drop, flow vortices and helium flow lines are explored for two fuel complexes. The empirical formula of pressure drop presented by Kugeler and Schulten is modified and gas density is considered as a function of the core length. Fuel complexes in the reactor core are randomly arranged. However, because Body Center Cubic (BCC) is the closest arrangement to the random distribution, flow parameters are obtained using the BCC arrangement and they are found to deviate very slightly when compared with predictions of other codes.

Two phase compressible flow in porous media

We study the mathematical model of two phase compressible flows through porous media. Under the condition that the compressibility of rock, oil, and water is small, we prove that the initial-boundary value problem of the nonlinear system of equations admits a weak solution.

Three-phase compressible flow in porous media: Total Differential Compatible interpolation of relative permeabilities

We describe the construction of Total Differential (TD) three-phase data for the implementation of the exact global pressure formulation for the modeling of three-phase compressible flow in porous media. This global formulation is preferred since it reduces the coupling between the pressure and saturation equations, compared to phase or weighted formulations. It simplifies the numerical analysis of the problem and boosts its computational efficiency. However, this global pressure approach exists only for three-phase data (relative permeabilities, capillary pressures) which satisfy a TD condition. Such TD three-phase data are determined by the choice of a global capillary pressure function and a global mobility function, which take both saturations and global pressure level as argument. Boundary conditions for global capillary pressure and global mobility are given such that the corresponding three-phase data are consistent with a given set of three two-phase data. The numerical construction of global capillary pressure and global mobility functions by C1 and C0 finite element is then performed using bi-Laplacian and Laplacian interpolation. Examples of the corresponding TD three-phase data are given for a compressible and an incompressible case.

On the effective density for compressible flow in porous media

This paper addresses the following question: Is there any kind of effective density for the description of momentum transport of a compressible fluid in porous media? Pioneering 1956 Biot’s results and subsequent experimental and theoretical work suggest the convenience of introducing an effective density to describe important physical phenomena, such as wave propagation in a saturated fluid porous medium. In this work we use a volume averaging method to obtain effective medium equations to describe the macroscopic phenomena in porous media. Our results indicate that an effective density can be theoretically determined and its computation can be obtained from the solution of a closure problem.

The upwind finite difference fractional steps methods for two‐phase compressible flow in porous media

AbstractThe upwind finite difference fractional steps methods are put forward for the two‐phase compressible displacement problem. Some techniques, such as calculus of variations, multiplicative commutation rule of difference operators, decomposition of high‐order difference operators, and prior estimates, are adopted. Optimal order estimates in L2 norm are derived to determine the error in the approximate solution. This method has already been applied to the numerical simulation of seawater intrusion and migration‐accumulation of oil resources. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 67–88, 2003

A splitting positive definite mixed element method for miscible displacement of compressible flow in porous media

AbstractA miscible displacement of one compressible fluid by another in a porous medium is governed by a nonlinear parabolic system. A new mixed finite element method, in which the mixed element system is symmetric positive definite and the flux equation is separated from pressure equation, is introduced to solve the pressure equation of parabolic type, and a standard Galerkin method is used to treat the convection‐diffusion equation of concentration of one of the fluids. The convergence of the approximate solution with an optimal accuracy in L2‐norm is proved. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 229–249, 2001

A characteristic mixed element method for displacement problems of compressible flow in porous media

A new characteristic mixed element scheme is formulated to solve numerically displacement problems of compressible fluids in porous media. A new mixed finite element method is introduced to solve the pressure equation of parabolic type, in which the mixed element system is symmetric positive definite and the pressure equation is separated from the flux equation. The modified method of characteristics is used to treat convection-dominated diffusion equations of the concentrations. The convergence with optimal accuracy is proved under the general condition.