Incompressible Flow In Porous Media Research Articles

Stationary Stokes solver for single-phase flow in porous media: A blastingly fast solution based on Algebraic Multigrid Method using GPU

The paper is focused on high efficiency Stokes solver that is applied to the incompressible flow in porous media. Computational domains are represented by binarized 3D computed tomography voxel models. The solution procedure is constructed around a fast algebraic multigrid solver that utilizes the power of graphics processing units (GPUs). In order to minimize memory footprint and accelerate the solver, a simple MAC-type staggered finite difference discretization is used and a coupled Stokes saddle-point type system is solved directly on a GPU. The MAC discretization is discussed, taking particular care of internal solid boundaries around pores. The method includes topological domain analysis on a GPU, which removes isolated volumes with no flow in the domain (based on the connected component labeling), depending on the boundary conditions. We consider various types of boundary conditions and efficient parallel strategies for the GPUs, including fast matrix assembly and residual regularization. Our method is extensively benchmarked both against analytical solutions and applied problems from digital rock physics in terms of computational wall time, precision, approximation order, and convergence. We demonstrate that it takes up to 5–23 s on a modern GPU card to obtain a solution with 1⋅10−6 error residuals for 3D geometries with 300–4503 voxels and a porosity range of 5–37%.

Linear stability analysis and nonlinear simulations of convective dissolution in an inclined porous layer between impermeable surfaces.

Convective dissolution can occur in porous media when a given solute dissolves in a host layer from above and increases the density of the host solution. Buoyancy-driven fingering can then develop, which increases the transfer flux of the solute. We investigate here numerically the properties of this convective dissolution when the porous host layer is inclined by an angle θ relative to the horizontal direction. We consider an incompressible flow in porous media governed by Darcy's law, driven by density gradients associated with the concentration of the dissolving solute. The model problem focuses on the case of a very long (infinite) tilted porous layer limited by two parallel impermeable surfaces. A linear stability analysis and nonlinear simulations are performed using the Boussinesq approximation. A vorticity-stream function formulation is adopted to solve the two-dimensional hydrodynamic field through the finite element method. We find that the inclination of the interface decreases the growth rate of the instability and the range of unstable wavenumbers, delaying or even suppressing the onset of the fingering instability. Moreover, it introduces a drift velocity on the perturbations, which is characterized here in both the linear stability analysis and the nonlinear simulations.

Temporary velocity correction-based immersed boundary–lattice Boltzmann method for incompressible flows in porous media at representative elementary volume scale

The immersed boundary (IB)–lattice Boltzmann (LB) method is an effective strategy for complex boundary condition treatment. By adding an extra body force term in the LB equation properly, the specific velocity boundary condition can be enforced in this method. However, when it comes to incompressible flows through porous media at the representative elementary volume (REV) scale, the conventional IB–LB method fails because the velocity and the force term induced by porous media are coupled. In order to solve this problem, a temporary velocity is used to construct the IB-induced force term with the enforcement of the velocity boundary condition. The temporary velocity is decomposed into the intermediate temporary velocity and the corresponding correction. By this treatment, the temporary velocity correction is the linear function of the IB-induced force term. Furthermore, to obtain the force term accounting for the IB, the velocity boundary condition is transformed to the temporary velocity. Consequently, a temporary velocity correction-based IB–LB method is established for the incompressible flows at the REV scale. To avoid the error of explicitly calculating the IB-induced force term, the multi-direct-forcing scheme is employed in which iteration is carried out in terms of the specific boundary condition. The proposed IB–LB method for REV-scale incompressible flows is applied for the Couette flow in a porous annulus and lid driven flow in a square cavity filled with porous matrix. Numerical results show the computational accuracy of the developed IB–LB method.

An energy stable linear numerical method for thermodynamically consistent modeling of two-phase incompressible flow in porous media

In this paper, we consider numerical approximation of a thermodynamically consistent model of two-phase flow in porous media, which obeys an intrinsic energy dissipation law. The model under consideration is newly-developed, so there is no energy stable numerical scheme proposed for it at present. This model consists of two nonlinear degenerate parabolic equations and a saturation constraint, but lacking an independent equation for the pressure, so for the purpose of designing efficient numerical scheme, we reformulate the model forms as well as the free energy function, and further prove the corresponding energy dissipation inequality. Based on the alternative reformulations, using the invariant energy quadratization approach and subtle semi-implicit treatments for the pressure and saturation, we for the first time propose a linear and energy stable numerical method for this model. The fully discrete scheme is devised combining the upwind approach for the phase mobilities and the cell-centered finite difference method. The unique solvability of numerical solutions and unconditional energy stability are rigorously proved for both the semi-discrete time scheme and the fully discrete scheme. Moreover, the scheme can guarantee the local mass conservation for both phases. We also show that the upwind mobility approach plays an essential role in preserving energy stability of the fully discrete scheme. Numerical results are presented to demonstrate the performance of the proposed scheme.

Analytical solution for one-dimensional three-phase incompressible flow in porous media for concave relative permeability curves

Three-phase flow in porous media may appear in different scenarios during the production life of a hydrocarbon reservoir. The simultaneous flow of different phases is modeled by relative permeability curves which are fundamental to petroleum production analysis and forecast. Laboratory experiments are the main source of data for relative permeability curves. Mathematical solutions for multiphase flow in porous medium are key for determining relative permeability curves from laboratory data, to check numerical reservoir simulation results and for screening an enhanced oil recovery technique. The complexity of reservoir modeling and the use of numerical optimization to history match the laboratory data have shown the importance of concave relative permeability curves. In this paper we present the analytical solution for one-dimensional incompressible immiscible three-phase flow in porous media, where the relative permeability functions are described by concave curves. The hyperbolic system of equations that results from mass conservation is solved by the method of characteristics. The results show close agreement to numerical solutions.

Energy stable modeling of two-phase flow in porous media with fluid–fluid friction force using a Maxwell–Stefan–Darcy approach

Two-phase incompressible flow in porous media plays an important role in various fields including subsurface flow and oil reservoir engineering. Due to the interaction between two phases flowing through the pores, the fluid–fluid friction force may have a significant effect on each phase velocity. In this paper, we propose an energy stable (thermodynamically consistent) Maxwell–Stefan–Darcy model for two-phase flow in porous media, which accounts for the fluid–fluid friction. Different from the classical models of two-phase flow in porous media, the proposed model uses the free energy to characterize the capillarity effect. This allows us to employ the Maxwell–Stefan model to describe the relationships between the driving forces and the friction forces. The driving forces include the pressure gradient and chemical potential gradients, while both fluid–solid and fluid–fluid friction forces are taken into consideration. Thermodynamical consistency is the other interesting merit of the proposed model; that is, it satisfies an energy dissipation law and also obeys the famous Onsager's reciprocal principle. A linear semi-implicit numerical method is also developed to simulate the model. Numerical simulation results are provided to show that the fluid–fluid friction force can improve the oil recovery substantially during the oil displacement process.

Preconditioned multiple-relaxation-time lattice Boltzmann equation model for incompressible flow in porous media

An improved preconditioned multiple-relaxation-time lattice Boltzmann equation model for incompressible flow (IPMRT-LBE) in porous media is proposed. Motivated by previous LBE models [Guo et al., Phys. Rev. E 70, 066706 (2004); Premnath et al., J. Comput. Phys. 228, 746 (2009); Guo et al., J. Comput. Phys. 165, 288 (2000)], the current model is demonstrated to have the advantages of accurate implementation of the no-slip boundary condition, reducing the compressible effect as well as fast convergence rate compared with standard LBE models. To validate the IPMRT-LBE model, flows in two- and three-dimensional synthetic porous media (square array of cylinders and body-centered cubic array of spheres) are simulated. The results show that the current model can predict the macroscopic property (such as permeability) accurately with significantly accelerated convergence rate. Furthermore, simulations of flow through a three-dimensional sandpack confirm the applicability and advantages of the IPMRT-LBE model.

Numerical Simulation of Free Surface Fluid Flows Through Porous Media by Using the Explicit MPS Method

An extension of the explicit moving particle semi-implicit (MPS) method is proposed for simulating free surface flow through a porous structure object. The model formulation is based on the local volume averaging equations, and the porous drag is given by a general closure function. It can compute the macroscopic behaviors of incompressible flows in porous media. Specifically, the present study addresses numerical difficulties of existing MPS models in two respects, namely the conservation of macroscopic fluid volume and the well-balanced hydrostatic equilibrium where the fluid and porous media interpenetrate. Numerical tests are performed to examine the validity and accuracy of the proposed model, from which good agreements are found between the simulation results and validation data. In comparison with existing models, it is also demonstrated that the numerical techniques developed in this study are important to obtain consistent predictions of flow behaviors. Therefore, the present MPS method is shown to be suitable for modeling free surface flows interacting with porous structures.

An HMM–ELLAM scheme on generic polygonal meshes for miscible incompressible flows in porous media

We design a numerical approximation of a system of partial differential equations modelling the miscible displacement of a fluid by another in a porous medium. The advective part of the system is discretised using a characteristic method, and the diffusive parts by a finite volume method. The scheme is applicable on generic (possibly non-conforming) meshes as encountered in applications. The main features of our work are the reconstruction of a Darcy velocity, from the discrete pressure fluxes, that enjoys a local consistency property, an analysis of implementation issues faced when tracking, via the characteristic method, distorted cells, and a new treatment of cells near the injection well that accounts better for the conservativity of the injected fluid.

Analytical and numerical solutions for carbonated waterflooding

We develop a Riemann solver for transport problems including geochemistry related to oil recovery. The example considered here concerns one-dimensional incompressible flow in porous media and the transport for several chemical components, namely H2O, H+, OH−, CO2, $\text {CO}_{3}^{2-}$ , $\text {HCO}_{3}^{-}$ , and decane; they are in chemical equilibrium in the aqueous and oleic phases, leading to mass transfer of CO2 between the oleic and aqueous phases. In our ionic model, we employ equations with zero diffusion coefficients. We do so because it is well known that for upscaled equations, the convection terms dominate the diffusion terms. The Riemann solution for this model can therefore be applied for upscaled transport processes in enhanced oil recovery involving geochemical aspects. In our example, we formulate the conservation equations of hydrogen, oxygen, hydrogen, and decane, in which we substitute regression expressions that are obtained by geochemical software. This can be readily done because Gibbs phase rule together with charge balance shows that all compositions can be rewritten in terms of a single composition, which we choose to be the hydrogen ion concentration (p H). In our example, we use the initial and boundary conditions for the carbonated aqueous phase injection in an oil reservoir containing connate water with some carbon dioxide. We compare the Riemann solution with a numerical solution, which includes capillary and diffusion effects. The significant new contribution is the effective Riemann solver we developed to obtain solutions for oil recovery problems including geochemistry and a variable total Darcy velocity, a situation in which fractional flow theory does not readily apply. We thus obtain an accurate solution for a carbonated waterflood, which elucidates some mechanisms of low salinity carbonated waterflooding.

On the well-posedness of the incompressible flow in porous media

On the well-posedness of the incompressible flow in porous media

A theoretical model for the porosity–permeability relationship

An analytical permeability model is proposed for incompressible flow in porous media. Based on a general non-equilibrium thermodynamic stability theory, the model is derived from maximizing the entropy production rate to achieve a stable system. The resulting formula is in a form similar to previous empirical equations, but with calculable coefficients. Both three-dimensional lid-driven cavity and cubicle natural convection problems have been computed, via a modern computational fluid dynamics (CFD) solver, for numerical investigations of its performance.

Numerical Investigation of the “Poor Man’s Navier–Stokes Equations” with Darcy and Forchheimer Terms

In this paper, a discrete dynamical system (DDS) is derived from the generalized Navier–Stokes equations for incompressible flow in porous media via a Galerkin procedure. The main difference from the previously studied poor man’s Navier–Stokes equations is the addition of forcing terms accounting for linear and nonlinear drag forces of the medium — Darcy and Forchheimer terms. A detailed numerical investigation focusing on the bifurcation parameters due to these additional terms is provided in the form of regime maps, time series, power spectra, phase portraits and basins of attraction, which indicate system behaviors in agreement with expected physical fluid flow through porous media. As concluded from the previous studies, this DDS can be employed in subgrid-scale models of synthetic-velocity form for large-eddy simulation of turbulent flow through porous media.

An efficient discontinuous Galerkin method for two-phase flow modeling by conservative velocity projection

Purpose – The purpose of this paper is to present a novel sequential implicit discontinuous Galerkin (DG) method for two-phase incompressible flow in porous media. It is based on the wetting phase pressure-saturation formulation with Robin boundary condition (Klieber and Riviere, 2006) using H(div) velocity projection. Design/methodology/approach – The local mass conservation and continuity of normal component of velocity across elements interfaces are enforced by a simple H(div) velocity projection in lowest order Raviart-Thomas (RT0) space. As further improvements, the authors use the weighted averages and the scaled penalties in spatial DG discretization. Moreover, the Chavent-Jaffre slope limiter, as a consistent non-oscillatory limiter, is used for saturation values to avoid the spurious oscillations. Findings – The proposed model is verified by a pseudo 1D Buckley-Leverett problem in homogeneous media. Two homogeneous and heterogeneous quarter five-spot benchmark problems and a random permeable medium are used to show the accuracy of the method at capturing the sharp front and illustrate the impact of proposed improvements. Research limitations/implications – The work illustrates incompressible two-phase flow behavior and the capillary pressure heterogeneity between different geological layers is assumed to be negligible. Practical implications – The proposed model can efficiently be used for modeling of two-phase flow in secondary recovery of petroleum reservoirs and tracing the immiscible contamination in porous media. Originality/value – The authors present an efficient sequential DG method for immiscible incompressible two-phase flow in porous media with improved performance for detection of sharp frontal interfaces and discontinuities.

A Fully Discretized Finite Element Approximation for an Incompressible Flow in Porous Media

In this paper, we will study the fully discretized finite element approximation for an incompressible flow in porous media. The model consists of the heat equation, the equation for the concentration and the equations of motion under the Darcy law. The model is rewritten using the stream function-vorticity formulation. The Stability of the fully discrete problem is established. Optimal a priori error estimates are given.

Multiple-relaxation-time lattice Boltzmann modeling of incompressible flows in porous media

In this paper, a two-dimensional eight-velocity multiple-relaxation-time (MRT) lattice Boltzmann (LB) model is proposed for incompressible porous flows at the representative elementary volume scale based on the Brinkman–Forchheimer-extended Darcy model. In the model, the porosity is included into the pressure-based equilibrium moments, and the linear and nonlinear drag forces of the porous matrix are incorporated into the model by adding a forcing term to the MRT-LB equation in the moment space. Through the Chapman–Enskog analysis, the incompressible generalized Navier–Stokes equations can be recovered. Numerical simulations of several typical porous flows are carried out to validate the present MRT-LB model. It is found that the present numerical results agree well with the analytical solutions and/or other numerical results reported in the literature.

On the Stream Function-Vorticity Finite Element Formulation for Incompressible Flow in Porous Media

Stream function-vorticity finite element formulation for incompressible flow in porous media is presented. The model consists of the heat equation, the equation for the concentration, and the equations of motion under the Darcy law. The existence of solution for the discrete problem is established. Optimal a priori error estimates are given.

A control volume finite element method for adaptive mesh simulation of flow in heap leaching

Accurate determination of fluid flow within heap leaching is crucial for understanding and improving performance of the process. Numerical methods have the potential to assist by modelling the process and studying the transport phenomena within the porous medium. This paper presents an adaptive mesh numerical scheme to solve for unsaturated incompressible flow in porous media with applications to heap leaching. The governing equations are Darcy’s law and the conservation of mass. An implicit pressure explicit saturation method is used to decouple the pressure and saturation equations. Pressure is discretized using a control volume (CV) finite element method, while for saturation a node-centred CV method is employed. The scheme is equipped with dynamic anisotropic mesh adaptivity to update the mesh resolution as the leaching solution infiltrates through the heap. This allows for high-fidelity modelling of multiscale features within the flow. The method is verified against the Buckley–Leverett problem where a quasi-analytical solution is available. It is applied for two-phase flow of air and leaching solution in a simplified two-dimensional heap geometry. We compare the accuracy and CPU efficiency of an adaptive mesh against a static mesh and demonstrate the potential to achieve high spatial accuracy at low computational cost through the use of anisotropic mesh adaptivity.

NUMERICAL STUDY OF BÉNARD CONVECTION WITH TEMPERATURE-DEPENDENT VISCOSITY IN A POROUS CAVITY VIA LATTICE BOLTZMANN METHOD

In this paper, the heat transfer characteristics of a two-dimensional steady Bénard convection flow with a temperature-dependent viscosity are studied numerically by the lattice Boltzmann method (LBM). The double-distribution model for LBM is proposed, one is to simulate incompressible flow in porous media and the other is to solve the volume averaged energy equation. The method is validated by comparing the numerical results with those existing literature. The effect of viscosity dependent on temperature is investigated. The average Nusselt numbers for the cases of exponential form of viscosity-temperature and effective Rayleigh number based on average temperature (T ref = 0.5 (Th +Tc)) are compared. A new formula of reference temperature (T ref = Tc +f (b) (Th -Tc)) is proposed and the numerical results show that the average Nusselt numbers predicted by this method have higher precision than those obtained by average temperature.

A parallel second-order adaptive mesh algorithm for incompressible flow in porous media

In this paper, we present a second-order accurate adaptive algorithm for solving multi-phase, incompressible flow in porous media. We assume a multi-phase form of Darcy's law with relative permeabilities given as a function of the phase saturation. The remaining equations express conservation of mass for the fluid constituents. In this setting, the total velocity, defined to be the sum of the phase velocities, is divergence free. The basic integration method is based on a total-velocity splitting approach in which we solve a second-order elliptic pressure equation to obtain a total velocity. This total velocity is then used to recast component conservation equations as nonlinear hyperbolic equations. Our approach to adaptive refinement uses a nested hierarchy of logically rectangular grids with simultaneous refinement of the grids in both space and time. The integration algorithm on the grid hierarchy is a recursive procedure in which coarse grids are advanced in time, fine grids are advanced multiple steps to reach the same time as the coarse grids and the data at different levels are then synchronized. The single-grid algorithm is described briefly, but the emphasis here is on the time-stepping procedure for the adaptive hierarchy. Numerical examples are presented to demonstrate the algorithm's accuracy and convergence properties and to illustrate the behaviour of the method.