Miscible Displacement In Porous Media Research Articles

High-order bound-preserving discontinuous Galerkin methods for compressible miscible displacements in porous media on triangular meshes

In this paper, we develop high-order bound-preserving (BP) discontinuous Galerkin (DG) methods for the coupled system of compressible miscible displacements on triangular meshes. We consider the problem with multi-component fluid mixture and the (volumetric) concentration of the jth component, cj, should be between 0 and 1. There are three main difficulties. Firstly, cj does not satisfy a maximum-principle. Therefore, the numerical techniques introduced in Zhang and Shu (2010) [44] cannot be applied directly. The main idea is to apply the positivity-preserving techniques to all cj′s and enforce ∑jcj=1 simultaneously to obtain physically relevant approximations. By doing so, we have to treat the time derivative of the pressure dp/dt as a source in the concentration equation and choose suitable fluxes in the pressure and concentration equations. Secondly, it is not easy to construct first-order numerical fluxes for interior penalty DG methods on triangular meshes. One of the key points in the high-order BP technique applied in this paper is the combination of high-order and lower-order numerical fluxes. We will construct second-order BP schemes and use the second-order numerical fluxes as the lower-order one. Finally, the classical slope limiter cannot be applied to cj. To construct the BP technique, we will not approximate cj directly. Therefore, a new limiter will be introduced. Numerical experiments will be given to demonstrate the high-order accuracy and good performance of the numerical technique.

A hybrid mixed finite element method for miscible displacement problem with MCC procedure

A new combined method is developed for solving incompressible miscible displacement in porous media. In this method, a hybrid mixed element method is constructed for the pressure and velocity equation, while a mass-conservative characteristic (MCC) finite element method is presented for concentration equation. The introduction of these two numerical algorithms not only makes the coefficient matrixes symmetric positive definite, but also keeps the mass balance. The stability and consistence of this method are analyzed and the optimal error estimates in L2-norm for velocity, concentration and pressure are derived. Finally, some numerical results are presented.

Convergence analysis of a family of ELLAM schemes for a fully coupled model of miscible displacement in porous media

We analyse the convergence of numerical schemes in the GDM–ELLAM (Gradient Discretisation Method–Eulerian Lagrangian Localised Adjoint Method) framework for a strongly coupled elliptic-parabolic PDE which models miscible displacement in porous media. These schemes include, but are not limited to, Mixed Finite Element–ELLAM and Hybrid Mimetic Mixed–ELLAM schemes. A complete convergence analysis is presented on the coupled model, using only weak regularity assumptions on the solution (which are satisfied in practical applications), and not relying on $$L^\infty $$ bounds (which are impossible to ensure at the discrete level given the anisotropic diffusion tensors and the general grids used in applications).

A new combined characteristic mixed finite element method for compressible miscible displacement problem

A new combined characteristic mixed finite element method is developed for solving compressible miscible displacement in porous media. In this algorithm, the splitting mixed finite element (SMFE) method is applied for solving the parabolic-type pressure equation, and the mass-conservative characteristic (MCC) finite element method is applied for solving the convection-diffusion type concentration equation. The application of the splitting mixed element method results in a symmetric positive definite coefficient matrix of the separated mixed element system, while the mass-conservative characteristic finite element method not only does well in handling convection-dominant diffusion problem but also keeps mass balance. The convergence of this method is considered and the optimal L2-norm error estimate is also derived. Finally, some numerical examples are provided to confirm our theoretical analysis.

Longitudinal dispersion in heterogeneous layered porous media during stable and unstable pore-scale miscible displacements

Abstract Dispersion and mixing of fluids during miscible displacement in porous media are known to be strongly affected by heterogeneity and viscous fingering. In this study, direct pore-level numerical simulations are employed to model flow and solute transport during both stable and unstable miscible displacements in two-dimensional packings of circular grains. Longitudinal dispersion coefficients are then computed by fitting the numerically obtained concentration profiles to the analytical solution of the convection-dispersion equation. We observed that diffusive, mixed diffusive-advective, and mechanical dispersion regimes are present during stable displacements whereas the middle diffusive-advective regime is absent in unstable displacements due to the evolution of viscous fingers. Viscous fingering also increases the longitudinal dispersion coefficient several orders of magnitude compared to the stable displacement. The results reveal that the scale-dependency of longitudinal dispersion is a function of layering direction and viscosity contrast. During stable displacement in porous domains with no layering or with the layering perpendicular to the displacement direction, the longitudinal dispersion coefficient increases along the length of flow until it reaches an asymptotic value. When layering is in the direction of displacement, however, the scale-dependency behavior is a function of the layering pattern. In unstable displacements, however, longitudinal dispersion always exhibits an asymptotic behavior. Finally, it is observed that the effect of heterogeneity and layering on the magnitude of the longitudinal dispersion coefficient diminishes as viscosity ratio increases to higher adverse values.

Linear stability analysis and nonlinear simulation of the channeling effect on viscous fingering instability in miscible displacement

In this paper, the effect of channeling on viscous fingering instability of miscible displacement in porous media is studied. In fact, channeling is introduced as a solution to stabilize the viscous fingering instability. In this solution, narrow channels were placed next to the walls, and by considering an exponential function to model the channeling effect, a heterogeneous media is assumed. In linear stability analysis, the governing equations are transferred to Fourier space, and by introducing a novel numerical method, the transferred equations are analyzed. The growth rate based on the wave number diagram has been drawn up in three sections of the medium. It is found that the flow becomes more stable at the center and unstable along the walls when the permeability ratio is increased. Also when the permeability ratio is approximately equal to one, the channeling has no significant effect. In nonlinear simulations, by using stream function and vortices, new equations have been rewritten and it is shown that channeling has a profound effect on the growth of the fingers and mechanisms. In addition to the superposition of velocity vectors and concentration contours, the development of instability is investigated using the mixing length and sweep efficiency diagram. The results show that although channeling reduces instability, it increases the displacement process time.

Two-grid method for compressible miscible displacement problem by CFEM–MFEM

This paper is concerned about the error analysis of two-grid method for compressible miscible displacement in porous medium. A characteristics finite element method (CFEM) is presented for the concentration equation to handle the convection part, and standard mixed finite element method (MFEM) is used for the pressure equation. Moreover, we linearize the equations based on the Newton iteration method, then, two-grid method is considered in this full discrete scheme problem. We prove the Lp error estimates for the pressure, Darcy velocity, concentration variables in the two-grid method. It is shown that coarse space can be extremely coarse and we achieve asymptotically optimal approximation. Finally, numerical experiment indicates that two-grid method is a very effective method for solving miscible displacement problem.

An Arbitrary-Order Scheme on Generic Meshes for Miscible Displacements in Porous Media

We design, analyse and implement an arbitrary order scheme applicable to generic meshes for a coupled elliptic-parabolic PDE system describing miscible displacement in porous media. The discretisation is based on several adaptations of the Hybrid-High-Order (HHO) method due to Di Pietro et al. [Computational Methods in Applied Mathematics, 14(4), (2014)]. The equation governing the pressure is discretised using an adaptation of the HHO method for variable diffusion, while the discrete concentration equation is based on the HHO method for advection-diffusion-reaction problems combined with numerically stable flux reconstructions for the advective velocity that we have derived using the results of Cockburn et al. [ESAIM: Mathematical Modelling and Numerical Analysis, 50(3), (2016)]. We perform some rigorous analysis of the method to demonstrate its $L^2$ stability under the irregular data often presented by reservoir engineering problems and present several numerical tests to demonstrate the quality of the results that are produced by the proposed scheme.

Conservative Local Discontinuous Galerkin Method for Compressible Miscible Displacements in Porous Media

In Guo et al. (Appl Math Comput 259:88---105, 2015), a nonconservative local discontinuous Galerkin (LDG) method for both flow and transport equations was introduced for the one-dimensional coupled system of compressible miscible displacement problem. In this paper, we will continue our effort and develop a conservative LDG method for the problem in two space dimensions. Optimal error estimates in $$L^{\infty }(0, T; L^{2})$$Lź(0,TźL2) norm for not only the solution itself but also the auxiliary variables will be derived. The main difficulty is how to treat the inter-element discontinuities of two independent solution variables (one from the flow equation and the other from the transport equation) at cell interfaces. Numerical experiments will be given to confirm the accuracy and efficiency of the scheme.

Characteristic block-centered finite difference method for compressible miscible displacement in porous media

In this paper, the characteristic block-centered finite difference method is introduced and analyzed to solve compressible miscible displacement in porous media. Error estimates for the pressure, velocity, concentration and its flux in different discrete norms are established rigorously and carefully on non-uniform grids. Finally, some numerical experiments are presented to show that the convergence rates are in agreement with the theoretical analysis.

Dynamics of a Highly Viscous Circular Blob in Homogeneous Porous Media

Viscous fingering is ubiquitous in miscible displacements in porous media, in particular, oil recovery, contaminant transport in aquifers, chromatography separation, and geological CO2 sequestration. The viscosity contrasts between heavy oil and water is several orders of magnitude larger than typical viscosity contrasts considered in the majority of the literature. We use the finite element method (FEM)-based COMSOL Multiphysics simulator to simulate miscible displacements in homogeneous porous media with very large viscosity contrasts. Our numerical model is suitable for a wide range of viscosity contrasts covering chromatographic separation as well as heavy oil recovery. We have successfully captured some interesting and previously unexplored dynamics of miscible blobs with very large viscosity contrasts in homogeneous porous media. We study the effect of viscosity contrast on the spreading and the degree of mixing of the blob. Spreading (variance of transversely averaged concentration) follows the power law t 3 . 34 for the blobs with viscosity ∼ O ( 10 2 ) and higher, while degree of mixing is found to vary non-monotonically with log-mobility ratio. Moreover, in the limit of very large viscosity contrast, the circular blob behaves like an erodible solid body and the degree of mixing approaches the viscosity-matched case.

The effect of numerical integration in Galerkin methods for compressible miscible displacement in porous media

The effect of numerical integration in Galerkin methods for compressible miscible displacement in porous media

Bound-Preserving Discontinuous Galerkin Method for Compressible Miscible Displacement in Porous Media

In this paper, we develop bound-preserving discontinuous Galerkin (DG) methods for the coupled system of compressible miscible displacement problems. We consider the problem with two components and the (volumetric) concentration of the $i$th component of the fluid mixture, $c_i$, should be between $0$ and $1$. However, $c_i$ does not satisfy the maximum principle. Therefore, the numerical techniques introduced in (X. Zhang and C.-W. Shu, Journal of Computational Physics, 229 (2010), 3091-3120) cannot be applied directly. The main idea is to apply the positivity-preserving techniques to both $c_1$ and $c_2$, respectively and enforce $c_1+c_2=1$ simultaneously to obtain physically relevant approximations. By doing so, we have to treat the time derivative of the pressure $dp/dt$ as a source in the concentration equation. Moreover, $c_i's$ are not the conservative variables, as a result, the classical bound-preserving limiter in (X. Zhang and C.-W. Shu, Journal of Computational Physics, 229 (2010), 3091-3120) cannot be applied. Therefore, another limiter will be introduced. Numerical experiments will be given to demonstrate the accuracy in $L^\infty$-norm and good performance of the numerical technique.

Coupled effect of viscosity and density gradients on fingering instabilities of a miscible slice in porous media

Miscible displacements in porous media exhibit interesting spatio-temporal patterns. A deeper understanding of the physical mechanisms of these emergent patterns is relevant in a number of physicochemical processes. Here, we have numerically investigated the instabilities in a miscible slice in vertical porous media. Depending on the viscosity and density gradients at the two interfaces, four distinct flow configurations are obtained, which are partitioned into two different groups, each containing a pair of equivalent flows until the interaction between the two interfaces. An analysis of the pressure drop around the respective unstable interface(s) supports numerical results. We classify the stabilizing and destabilizing scenarios in a parameter space spanned by the log-mobility ratio (R) and the displacement velocity (U). When the viscosity and density gradients are unstably stratified at the opposite interfaces, the stability characteristics are very complex. The most notable findings of this paper are the existence of a stable region between two unstable regions in the R–U plane and occurrence of secondary instabilities. We further show that the stability regions in the R–U plane depend strongly on the slice width, and beyond a threshold value of it the stable zone remains almost unaltered. For thin sample, the stable region expands and the secondary instabilities disappear.

A combined discontinuous Galerkin finite element method for miscible displacement problem

A new combined method is constructed for solving incompressible miscible displacement in porous media. In this procedure, a hybrid mixed element method is constructed for the pressure and velocity equation, while a symmetric discontinuous Galerkin finite element method is proposed for the concentration equation. The introduction of these two numerical methods not only makes the coefficient matrixes symmetric positive definite, but also conserves the local mass balance. The stability and consistency of the method are analyzed and the optimal error estimate in L∞(L2) for velocity and concentration and the super convergence in L∞(H1) for pressure are derived. Finally, some numerical results are presented.

Pore-Network Simulation of Unstable Miscible Displacements in Porous Media

We develop a comprehensive pore-network model for simulation of miscible displacements in laboratory-scale porous media. The goal is to study the effect of pore-scale heterogeneity, and in particular pore-size distribution and pore connectivity, as well as mobility ratio M on miscible displacements, and compute the effective longitudinal dispersion coefficient as a function of M. At the pore level the phenomenon is described by the convective-diffusion equation with a Taylor–Aris dispersion coefficient. Extensive simulations indicate the strong effect of the connectivity and broadness of the pore-size distribution on the efficiency of the displacement process. Moreover, the model enables us to compute the effective dispersion coefficient as a function of the mobility ratio of the two fluids that, to our knowledge, has not been calculated before. Simple generalization of the model will make it possible to study more complex related phenomena, and in particular injection of $$\mathrm{CO}_2$$ into a pore space that contains saline water.

$$L^p$$ L p Error Estimates of Two-Grid Method for Miscible Displacement Problem

This paper is concerned about the $$L^p$$Lp error analysis of two-grid method for incompressible miscible displacement in porous medium. A characteristics finite element method is presented for the concentration equation to handle the convection part, and standard mixed finite element is used for the pressure equation. Mixed finite element method has an advantage of approximating the unknown variable and its diffusive flux across grid-cell interfaces simultaneously, which has been proven to be an effective numerical method for solving fluid problems. Moreover, we linearize the equations based on the Newton iteration method, then, two-grid algorithm is considered in this full discrete scheme problems. It shown that coarse space can be extremely coarse and we achieve asymptotically optimal approximation. Numerical experiments are presented finally to validate the theoretical analysis.

3-D visualization of diffusive and convective solvent transport processes in oil-saturated porous media using laser technology

A technique to visualize miscible displacement in porous media is introduced in this paper. After saturating the model made of different sized glass beads with oil, solvent was introduced to mix and displace it. The refractive indices of saturated and injected fluids were made the same by mixing them with lower and higher indices of refraction. Refractive index matching made the model transparent. Fluorescent dyes that were only visible with excitation of laser were dissolved in the solvent. A laser sheet scanned the model while synchronous pictures were taken by two high speed cameras from two sides of the model. Two groups of models representing diffusive (no injection) and convective (injection of solvent) interaction, respectively, were considered to test the visualization system: (1) solvent diffusion under purely static conditions, and (2) injection/production through a pair of horizontal wells. Visual data were analyzed and the limitations of the visualization technique introduced were presented.

A new MCC–MFE method for compressible miscible displacement in porous media

This paper presents a new mass-conservative finite element method for solving the compressible miscible displacement problem in porous media. This new method combines characteristic tracking with the mixed finite element method. Specifically, the mass-conservative characteristic finite element method is applied to the concentration equation whereas the mixed finite element method with continuous flux approximation is used for the flow equation. This procedure conserves mass globally. Convergence of the fully discrete scheme is analyzed and optimal L2-norm and H1-norm error estimates for the concentration are obtained. Numerical experiments are presented to illustrate the theoretical analysis.

Pore-Scale Study of Miscible Displacements in Porous Media Using Lattice Boltzmann Method

In the past few years, the miscible displacements in porous media were usually simulated by some semiempirical models based on the volume averaging at the representative elementary volume scale. To better understand the microscopic mechanism of the viscous fingering phenomenon in porous media for miscible fluids, in this paper the miscible displacements processes in porous media are studied using the lattice Boltzmann method (LBM) at the pore scale. First, the code of LBM is tested by simulating the displacement process of two miscible fluids with the same viscosity between two parallel plates which is the well-known Taylor–Aris dispersion problem, and comparing the results with the theoretical predictions. Then, the effects of the Peclet number Pe, the viscosity ratio M and the structure of the porous media on the displacement phenomenon are investigated, and the location and velocity of the finger tip, the displacement efficiency are also studied. In this paper, the displacement efficiency is calculated by $$1-m$$ , here the quantity m is defined as $$m=V_M/V_T$$ , where $$V_M$$ is the volume of more viscous fluids (the displaced fluid) left behind the finger tip, $$V_T$$ is the total pore volume behind the finger tip. It can be found that the “interface” of two fluids will become clearer with the increasing of the Peclet number. As Pe and M are large enough, the viscous fingering phenomenon will occur, and in the front of the finger, “mushroom-like” pattern can be observed. Besides, with the increasing of Pe or M the quantity m will be increased too, i.e., the displacement efficiency will be decreased. While Pe (or M) is greater than a certain value, the growth rate of the quantity m will slow down. The same trend was observed for the miscible displacement in capillary tubes or Hele–Shaw cells. Besides, changing the structure of the porous media makes the finger pattern different. The present simulation results provide a good understanding of the microscopic mechanism of the miscible displacement process in porous media, and also show that the LBM can be a useful tool for investigation miscible fluids behavior in porous media.