Simulation Of Viscous Fingering Research Articles

Numerical Simulations of Viscous Fingering in Fractured Porous Media

AbstractThe effect of heterogeneity induced by highly permeable fracture networks on viscous miscible fingering in porous media is examined using high-resolution numerical simulations. We consider the planar injection of a less viscous fluid into a two-dimensional fractured porous medium that is saturated with a more viscous fluid. This problem contains two sets of fundamentally different preferential flow regimes; the first is caused by the viscous fingering, and the second is due to the permeability contrasts between the fractures and the rock matrix. We study the transition from the regime where the flow is dominated by the viscous instabilities, to the regime where the heterogeneity induced by the fractures define the flow paths. Our findings reveal that even minor permeability differences between the rock matrix and fractures significantly influence the behavior of viscous fingering. The interplay between the viscosity contrast and permeability contrast leads to the preferential channeling of the less viscous fluid through the fractures. Consequently, this channeling process stabilizes the displacement front within the rock matrix, ultimately suppressing the occurrence of viscous fingering, particularly for higher permeability contrasts. We explore three fracture geometries: two structured and one random configuration and identify a complex interaction between these geometries and the development of unstable flow. While we find that the most important factor determining the effect of the fracture network is the ratio of fluid volume flowing through the fractures and the rock matrix, the exact point for the cross-over regime is dependent on the geometry of the fracture network.

High resolution viscous fingering simulation in miscible displacement using a p-adaptive discontinuous Galerkin method with algebraic multigrid preconditioner

High resolution simulation of viscous fingering can offer an accurate and detailed prediction for subsurface engineering processes involving fingering phenomena. The fully implicit discontinuous Galerkin (DG) method has been shown to be an accurate and stable method to model viscous fingering with high Peclet number and mobility ratio. In this paper, we present two techniques to speedup large scale simulations of this kind. The first technique relies on a simple p-adaptive scheme in which high order basis functions are employed only in elements near the finger fronts where the concentration has a sharp change. As a result, the number of degrees of freedom is significantly reduced and the simulation yields almost identical results to the more expensive simulation with uniform high order elements throughout the mesh. The second technique for speedup involves improving the solver efficiency. We present an algebraic multigrid (AMG) preconditioner which allows the DG matrix to leverage the robust AMG preconditioner designed for the continuous Galerkin (CG) finite element method. The resulting preconditioner works effectively for fixed order DG as well as p-adaptive DG problems. With the improvements provided by the p-adaptivity and AMG preconditioning, we can perform high resolution three-dimensional viscous fingering simulations required for miscible displacement with high Peclet number and mobility ratio in greater detail than before for well injection problems.

Towards Smoothed Particle Hydrodynamics Simulation of Viscous Fingering in Porous Media

Towards Smoothed Particle Hydrodynamics Simulation of Viscous Fingering in Porous Media

Numerical Simulation of Viscous Fingering and Flow Channeling Phenomena in Perturbed Stochastic Fields: Finite Volume Approach with Tracer Injection Tests

Modeling viscous fingering perturbations in stochastic fields have always been a formidable task for geoscientists. Although fingering and channeling are often used synonymously to describe uneven front displacements, in fact, both phenomena represent different bypassing effects on hydrocarbon ultimate recovery. However, fingering perturbations in stochastic fields to invoke them during simulation often lead to severe channeling. Hence, the focus of this paper is to develop a finite volume numerical simulator with a level of certainty where fingering perturbations in stochastic fields do not result into severe channeling. The simulator development involves discretization of convection–diffusion and pressure equations with total variation diminishing and central differencing schemes, respectively. Correlated stochastic fields for simulation runs were created using convolution with a Gaussian filter achieved through 2D fast and inverse Fourier transform, Dykstra–Parsons coefficient, random porosity generator, and autocorrelation lengths. The obtained pressure and tracer concentration profiles were solved by iterative implicit pressure and explicit concentration approach, respectively. Numerical results of concentration profiles indicated that, unlike flow channeling, viscous fingering occurred in both homogeneous and heterogeneous fields. The obtained flow pattern map captures these flow regimes, namely fluid mobility fingering, channeling, and heterogeneous fingering. Sensitivity analyses of diffusion coefficient contributions to channeling and fingering phenomena are of extreme importance. The simulator shows a good validation with analytical solution while insensitive to grid refinements and spurious oscillations. The simulator is an improvement on first-order scheme simulators with the capability of computationally tracking fingering perturbations in stochastic fields which often result into channeling anomalies.

Simulation of Viscous Fingering in Rectangular Porous Media with Lateral Injection and Two- and Three-Phase Flows

Understanding multiphase flow in porous media is of tremendous importance for many industrial and environmental applications at various spatial and temporal scales. The present study consequently focuses on modeling multiphase flows by the volume-of-fluid method (sharp interface) in porous media with a simplified Darcy-scale approach and shows simulations of Saffman–Taylor fingering. The simplification of the Darcy-scale approach is performed by assuming sharp interfaces between pure phases. The volume-of-fluid method with octree mesh refinement is used. It is implemented in the Gerris (Popinet in J Comput Phys 190(2):572–600, 2003. doi: 10.1016/S0021-9991(03)00298-5 ; J Comput Phys 228(16):5838–5866, 2009. doi: 10.1016/j.jcp.2009.04.042 ) code which allows efficient parallel computations. We measure the scaling properties of the fractal viscous fingering patterns that appear in the numerical simulations. One of these properties is the fractal or Hausdorff dimension $$D_\mathrm{F}$$ . The other is the variation of the area A of the viscous fingering cluster with the length L of its perimeter, which varies as a simple power law $$A \sim L^{\alpha }$$ . The injection of an intermediate-viscosity Newtonian fluid as a second step is also simulated. We are thus able to observe an increase of recovery of the high-viscosity fluid behind the fingering front, due to the reduction of the viscosity contrast. Some of these results are compared to waterflooding experiments of extra-heavy oils in quasi-2D square slab geometries of Bentheimer sandstone.

Numerical simulation of viscous fingering in porous media

The formation and growth of three-dimensional branching patterns that occur when a less viscous fluid penetrates into a porous media in more viscous surroundings are studied numerically on the basis of a simplified filtration model that corresponds to the zero dispersion and high capillary number limit. The balance equation for the saturation of a porous medium with a less viscous fluid is approximated on the basis of the high-order accurate weighted non-oscillatory (WENO) scheme. Tip-splitting is found to be the dominant way of pattern formation in the numerical solutions. A qualitative illustration of the influence of the numerical scheme is presented.

Phase-field simulations of viscous fingering in shear-thinning fluids

A phase-field model for the Hele-Shaw flow of non-Newtonian fluids is developed. It extends a previous model for Newtonian fluids to a wide range of fluids with a shear-dependent viscosity. The model is applied to simulate viscous fingering in shear-thinning fluids and found to capture the complete crossover from the Newtonian regime at low-shear rate to the strongly shear-thinning regime. The width selection of a single steady-state finger is studied in detail for a two-plateau shear-thinning law (Carreau’s law) in both its weakly and strongly shear-thinning limits, and the results are related to the previous analyses. For power-law (Ostwald–de Waele) fluids in the strongly shear-thinning regime, good agreement with experimental data from the literature is obtained.

Finite element simulation of viscous fingering in miscible displacements at high mobility-ratios

Numerical simulations of viscous fingering instabilities in miscible displacements at high mobility-ratios are presented. Anisotropic dispersion and monotonic viscosity profiles are considered. The coupled set of partial differential equations is approximated by the semi-discrete SUPG stabilized finite element formulation plus a discontinuity capturing technique to improve stability around the moving sharp fronts. The pressure equation is discretized by the standard Galerkin method, and a post-processing scheme is used to improve the numerical evaluation of Darcy’s velocity. In the resulting scheme all variables (concentration, pressure and velocity) are approximated by equal order linear triangular elements. A homogeneous channel and a radial system were studied. Complex nonlinear viscous fingering mechanisms for high mobility-ratio miscible displacements were observed.

Computer simulation of viscous fingering in a lifting Hele-Shaw cell with grooved plates

We simulate viscous fingering generated by separating two plates with a constant force, in a lifting Hele-Shaw cell. Variation in the patterns for different fluid viscosity and lifting force is studied. Viscous fingering is strongly affected by anisotropy. We report a computer simulation study of fingering patterns, where circular or square grooves are etched on to the lower plate. Results are compared with experiments.

A rescaling scheme with application to the long-time simulation of viscous fingering in a Hele–Shaw cell

In this paper, we present a time and space rescaling scheme for the computation of moving interface problems. The idea is to map time–space such that the interfaces can evolve exponentially fast in the new time scale while the area/volume enclosed by the interface remains unchanged. The rescaling scheme significantly reduces the computation time (especially for slow growth), and enables one to accurately simulate the very long-time dynamics of moving interfaces. We then implement this scheme in a Hele–Shaw problem, examine the dynamics for a number of different injection fluxes, and present the largest and most pronounced viscous fingering simulations to date.

Numerical Simulation of Viscous Fingering in Non-Newtonian Fluids in a Hele-Shaw Cell.

Viscous fingering in non-Newtonian fluids in a rectangular Hele-Shaw cell was numerically simulated using the finite element method together with the volume of fluid (VOF) method for the representation of an interface between two viscous fluids. Interfacial tension was treated as a volumeric force using the continuum surface force (CSF) model. The Carreau model was applied for describing the shear-thinning viscosity. In addition, the viscous fingering in a Newtonian fluid was also calculated for comparison. The computation was performed under various conditions of initial finger shape, interfacial tension, and mean velocity. The typical patterns such as `shielding', `spreading', and `splitting' were simulated and the numerical results predicted that the shear-thinning viscosity increased the shielding effect. The present results indicated the numerical approach was effective.

Numerical simulation of viscous fingering of shear‐thinning fluids

AbstractThe viscous fingering instability of miscible shear‐thinning fluids has been examined using a pseudo‐spectral numerical technique based on the Hartley transform. The instability was studied for a flow in a rectilinear Hele‐Shaw cell, and the shear‐thinning character of the fluids has been modelied using the Carreau equation. New mechanisms of viscous fingering not previously observed in the case of similar Newtonian flow displacements have been identified. These mechanisms, which are reminiscent of the fractal patterns observed in experimental studies, were interpreted in terms of the velocity‐dependent mobility of the flow.

Effect of No-Flow Boundaries on Viscous Fingering in Porous Media of Large Aspect Ratio

Summary We report on a boundary effect associated with the preferential propagation of viscous fingers along no-flow boundaries in miscible displacements in two-dimensional (2D) rectilinear geometries near conditions of transverse equilibrium (TE) (large values of the generalized aspect ratio RL). It is shown that this effect intensifies with an increase in the mobility ratio, and diminishes with increased permeability disorder, and as the system departs from conditions of TE. The effect disappears altogether if periodic boundary conditions are used. We show that this effect results from the violation of the no-slip condition at no-flow boundaries, associated with Darcy's law. It must be considered in simulations of viscous fingering in relatively homogeneous media, near conditions of TE, particularly when gravity segregation occurs.

Computer simulation of viscous fingering in Sierpinski carpet

A new method?mapping dilation method is proposed in this paper to construct Sierpinski carpet. Viscous fingering (VF) in Sierpinski carpet, based on the assumption that bond radii are beta distribution, is investigated by means of successive over-relaxation techniques. The topology and the geometry of the porous media have a strong effect on displacement processes. In the Sierpinski network, the VF pattern of porous media in the limit M???? is found to be similar to the diffusion-limited-aggregation pattern. The fractal dimension for VF in fractal space is calculated and the fractal dimension D can be reasonably regarded as a useful parameter to evaluate the sweep efficiencies and oil recoveries. We have also found that the geometry of the porous medium also has strong effects on the displacement processes and the structure of the VF. Moreover, we find that the sweep efficiency of the displacement processes mainly depends upon the length of the network system and also on the viscosity ratio M. This shows that the current method can be used to solve VF problems in complex structures if the structures are self-similar, or they can be reduced to a self-similar structure.

Simulation of viscous fingering in miscible displacements with nonmonotonic viscosity profiles

The nonlinear evolution of viscous fingering instabilities in miscible displacement flows in porous media with nonmonotonic viscosity profiles is investigated. The flow is accurately simulated using a Hartley transform based spectral method. A flow with nonmonotonic viscosity profile has an unstable region followed downstream by a stable region. Instabilities first begin in the unstable region and then grow and penetrate the stable region. The striking contrast between viscous fingering in flows with monotonic and nonmonotonic viscosity profiles is the direction of fluid penetration. The nonmonotonicity in the viscosity profile gives rise to a new phenomena of ‘‘reverse’’ fingering in which the displaced fluid fingers through the displacing fluid more readily than vice versa. A forward and a reverse mixing lengths are defined to characterize the growth of the mixing zone in the two directions. At large times, both the forward and reverse mixing lengths grow linearly in time. A model nonmonotonic viscosity profile is used to parametrically study the asymptotic mixing rates. The parametric study shows that for a given end point and maximum viscosities the growth rate of the mixing zone varies nonmonotonically with the length of the stable barrier in the viscosity profile. A physical mechanism is put forth to explain the observed phenomena of reverse fingering and its dependence on the parameters of the problem. Finally, a finger is isolated and its evolution is studied to understand the differences in the mechanisms that control the growth of a finger in flows with monotonic and nonmonotonic viscosity profiles.

Simulation of large‐scale transport of variable density and viscosity fluids using a stochastic mean model

Previous results of stochastic analysis of variable density and viscosity fluids in heterogeneous isotropic porous media show that longitudinal macrodispersivity is a function of mean concentration and concentration gradient. The dependence of macrodispersivity on mean concentration and concentration gradient gives rise to a nonlinear mean solute transport equation. To evaluate this behavior, a one‐dimensional finite difference formulation of the mean solute transport equation that incorporates the analytically derived expression for longitudinal macrodispersivity is used to simulate a step input problem. This approach allows effects of a spatially and temporally varying macrodispersivity on breakthrough curves to be observed and effects on dispersed‐zone width to be calculated. Simulation results show that, for solute concentrations of environmental significance, density effects give rise to macrodispersivities up to several orders of magnitude larger (for the unstable (fingering) case) or smaller (for the stable case) than would be predicted by analysis of an ideal tracer, and the smaller the velocity, the greater the effect. In miscible secondary petroleum recovery, where the displaced and displacing fluids differ in concentration by as much as 100%, viscosity effects have the predominant influence on macrodispersivity; in this case, simulations illustrate that viscous fingering effects give rise to macrodispersivities that are several orders of magnitude larger than predicted by classical analysis. Qualitatively these results agree with fine‐grid numerical simulations of viscous fingering and laboratory column experiments in that similar trends are observed for the effects of displacement orientation, velocity, and density and viscosity contrasts on dispersed‐zone width.

Modelling Viscous Fingering on a Parallel Computer

Abstract Diffusion limited aggregation (DLA) has proved very successful in modelling systems which display fractal characteristics, like viscous fingering. However, by nature, such simulations are very processor intensive, requiring large amounts of processor time even for relatively small models. We have performed simulations of viscous fingering on the NCUBE parallel computer which has hypercube architecture. We find that, as long as the number of processors used is much less than both the total number of walkers released and the overall dimensions of the model, the fractal dimensions obtained using serial and parallel algorithms give similar results whilst achieving a considerable speed-up in the parallel implementation. An average fractal dimension of 1.71 was obtained along with a speed-up of 106 (in the best case) and 83% efficiency using 128 processors.

A Proposed Technique for Simulation of Viscous Fingering in One-Dimensional Immiscible Flow

Summary One-dimensional (1D) oil/water displacement data for a very wide range of unfavorable mobility ratios obtained on three sandpacks show that for these sandpacks, the macroscopic effect of viscous fingering may be captured when its behavior is modeled with a reservoir simulator and one set of relative permeability values. This set may be calculated directly from the experimental data where viscous fingering occurred, or alternatively, it may be generated from relative permeability values obtained on systems with no frontal instabilities.

Fluctuations, viscous fingering, and diffusion-limited aggregation.

A physical interpretation of the diffusion-limited-aggregation (DLA) algorithm for simulating viscous fingering during fluid displacement in a porous medium is given in terms of the material properties of the porous medium. In one dimension, the flow patterns generated by a DLA simulation correspond to a simple exponential distribution of ``fluid capacity.'' Also, for any given distribution of fluid capacity, it is possible to establish a corresponding stochastic algorithm for the moving fluid interface. The discussion of this paper places the DLA simulation of viscous fingering on a physical basis and, more generally, establishes the mathematical equivalence of two models of stochastic interface evolution.