Physical splitting of nonlinear effects in high-velocity stable flow through porous media

For a high-velocity stable flow through a periodic corrugated channel representing an element of porous medium, we suggest splitting the overall nonlinear macroscopic effects into two kinds of different physical origin: a pure inertia effect produced by the convective term of Navier–Stokes equations and an inertia–viscous cross effect representing a variation of the viscous dissipation due to a streamline deformation by inertia forces. We will show that the inertia–viscous cross effects may be revealed by simulating a periodic flow, whilst the pure inertia effects are produced by the microscale flow nonperiodicity. We will reveal the individual flow law for each nonlinear component and analyze the relative role of both components numerically by using the finite element method applied to the Navier–Stokes equations. Both the pure inertia and the inertia–viscous cross effects are revealed to be exponential prior to quadratic or cubic ones. The influence of the dead volume is analyzed. The inertia–viscous cross phenomena are shown to be negligible when the flow structure is clearly nonperiodic.     

Experimental analysis of pore-scale flow and transport in porous media

A novel, non-intrusive fluorescence imaging technique has been used to quantitatively measure the pore geometry, fluid velocity, and solute concentration within a saturated, three-dimensional porous medium. Discrete numerical averages of these quantities have been made over a representative volume of the medium and used to estimate macroscopic quantities that appear in conventional continuum models of flow and transport. The approach is meant to illustrate how microscopic information can be measured, averaged, and used to characterize medium-scale processes that are typically approximated constitutively. The experimental system consisted of a clear, cylindrical column packed with clear spherical beads and a refractive index-matched fluid seeded with fluorescent tracer particles and solute dye. By illuminating the fluid within the column with a scanning planar laser beam, details of flow and concentration within the pore spaces can be quantitatively observed, allowing for three-dimensional, dimensional, time dependent information to be obtained at good resolution. In time dependent information to be obtained at good resolution. In the current experiment, volumetrically averaged velocities and void-to-volume ratios are first compared with bulk measurements of fluid flux and medium porosity. Microscopic measurements of concentration are then used to construct cross-sectionally averaged profiles, mean breakthrough curves, and direct measurements of the dispersive flux, velocity variance, and concentration variance. In turn, the dispersive flux measurements are compared with mean concentration gradients to provide a basis for confirming the Fickian dispersion model and estimating dispersion coefficients for the medium. Coefficients determined in this manner are compared with others based upon traditional length-scale arguments, mean breakthrough analyses, and curve fits with numerical simulations.     

Simulation of density-driven flow in fractured porous media

We study density-driven flow in a fractured porous medium in which the fractures are represented as manifolds of reduced dimensionality. Fractures are assumed to be thin regions of space filled with a porous material whose properties differ from those of the porous medium enclosing them. The interfaces separating the fractures from the embedding medium are assumed to be ideal. We consider two approaches: (i) the fractures have the same dimension, d, as the embedding medium and are said to be d-dimensional; (ii) the fractures are considered as (d − 1)-dimensional manifolds, and the equations of density-driven flow are found by averaging the d-dimensional laws over the fracture width. We show that the second approach is a valid alternative to the first one. For this purpose, we perform numerical experiments using finite-volume discretization for both approaches. The results obtained by the two methods are in good agreement with each other.     

Similarity solution of axisymmetric flow in porous media

Applications of the axisymmetric Boussinesq equation to groundwater hydrology and reservoir engineering have long been recognised. An archetypal example is invasion by drilling fluid into a permeable bed where there is initially no such fluid present, a circumstance of some importance in the oil industry. It is well known that the governing Boussinesq model can be reduced to a nonlinear ordinary differential equation using a similarity variable, a transformation that is valid for a certain time-dependent flux at the origin. Here, a new analytical approximation is obtained for this case. The new solution,, which has a simple form, is demonstrated to be highly accurate.     

Nonequilibrium effects in models of three-phase flow in porous media

In this paper we extend to three-phase flow the nonequilibrium formalism proposed by Barenblatt and co-workers for two-phase porous media flow. The underlying idea is to include nonequilibrium effects by introducing a pair of effective water and gas saturations, which are linked to the actual saturations by a local evolution equation. We illustrate and analyze how nonequilibrium effects lead to qualitative and quantitative differences in the solution of the three-phase flow equations.     

Effective parameters for flow in saturated heterogeneous porous media

Effective parameters for flow in saturated porous media are obtained via Taylor-Aris-Brenner moment analysis considering both periodic as well as stationary porous medium properties. It is assumed that a slug is instantaneously introduced into an unbounded, anisotropic porous medium having a compressible matrix, and that the correlation length of the local hydraulic conductivity and specific storage fluctuations is smaller than the correlation length of hydraulic head fluctuations (gradually varying flow). It is shown that the effective specific storage is equal to its volume average. The effective hydraulic conductivity is derived by a small-perturbation analysis and it is shown to consist of its volume average and of a second term which accounts for the ‘small’ local conductivity fluctuations.     

The acoustic signature of fluid flow in complex porous media

Effective medium approximations for the frequency-dependent and complex-valued effective stiffness tensors of cracked/porous rocks with multiple solid constituents are developed on the basis of the T-matrix approach (based on integral equation methods for quasi-static composites), the elastic–viscoelastic correspondence principle, and a unified treatment of the local and global flow mechanisms, which is consistent with the principle of fluid mass conservation. The main advantage of using the T-matrix approach, rather than the first-order approach of Eshelby or the second-order approach of Hudson, is that it produces physically plausible results even when the volume concentrations of inclusions or cavities are no longer small. The new formulae, which operates with an arbitrary homogeneous (anisotropic) reference medium and contains terms of all order in the volume concentrations of solid particles and communicating cavities, take explicitly account of inclusion shape and spatial distribution independently. We show analytically that an expansion of the T-matrix formulae to first order in the volume concentration of cavities (in agreement with the dilute estimate of Eshelby) has the correct dependence on the properties of the saturating fluid, in the sense that it is consistent with the Brown–Korringa relation, when the frequency is sufficiently low. We present numerical results for the (anisotropic) effective viscoelastic properties of a cracked permeable medium with finite storage porosity, indicating that the complete T-matrix formulae (including the higher-order terms) are generally consistent with the Brown–Korringa relation, at least if we assume the spatial distribution of cavities to be the same for all cavity pairs. We have found an efficient way to treat statistical correlations in the shapes and orientations of the communicating cavities, and also obtained a reasonable match between theoretical predictions (based on a dual porosity model for quartz–clay mixtures, involving relatively flat clay-related pores and more rounded quartz-related pores) and laboratory results for the ultrasonic velocity and attenuation spectra of a suite of typical reservoir rocks.     

Adaptive finite element simulation of Stokes flow in porous media

The Stokes problem describes flow of an incompressible constant-viscosity fluid when the Reynolds number is small so that inertial and transient-time effects are negligible. The numerical solution of the Stokes problem requires special care, since classical finite element discretization schemes, such as piecewise linear interpolation for both the velocity and the pressure, fail to perform. Even when an appropriate scheme is adopted, the grid must be selected so that the error is as small as possible. Much of the challenge in solving Stokes problems is how to account for complex geometry and to capture important features such as flow separation. This paper applies adaptive mesh techniques, using a posteriori error estimates, in the finite element solution of the Stokes equations that model flow at pore scales. Different selected numerical test cases associated with various porous geometrics are presented and discussed to demonstrate the accuracy and efficiency of our methodology.     

A master equation for reactive solute transport in porous media

The mean value of a density of a cloud of points described by a generalized Liouville equation associated with a convection dispersion equation governing adsorbing solute transport yields a joint concentration probability density. The general technique can be applied for either linear or nonlinear adsorption; here the application is restricted to linear adsorption in one-dimensional transport. The equation generated for the joint concentration probability density is in the general form of a Fokker-Planck equation, but with a suitable coordinate transformation, it is possible to represent it as a diffusion equation with variable coefficients.     

Multiscale flow and transport model in three-dimensional fractal porous media

This paper proposes a multiscale flow and transport model which can be used in three-dimensional fractal random fields. The fractal random field effectively describes a field with a high degree of variability to satisfy the one-point statistics of Levy-stable distribution and the two-point statistics of fractional Levy motion (fLm). To overcome the difficulty of using infinite variance of Levy-stable distribution and to provide the physical meaning of a finite domain in real space, truncated power variograms are utilized for the fLm fields. The fLm model is general in the sense that both stationary and commonly used fractional Brownian motion (fBm) models are its special cases. When the upper cutoff of the truncated power variogram is close to the lower cutoff, the stationary model is well approximated. The commonly used fBm model is recovered when the Levy index of fLm is 2. Flow and solute transport were analyzed using the first-order perturbation method. Mean velocity, velocity covariance, and effective hydraulic conductivity in a three-dimensional fractal random field were derived. Analytical results for particle displacement covariance and macrodispersion coefficients are also presented. The results show that the plume in an fLm field moves slower at early time and has more significant long-tailing behavior at late time than in fBm or stationary exponential fields. The proposed fractal transport model has broader applications than those of stationary and fBm models. Flow and solute transport can be simulated for various scenarios by adjusting the Levy index and cutoffs of fLm to yield more accurate modeling results.     

A residual flow procedure and application for free surface flow in porous media

A finite element procedure based on the extension of the saturated flow domain into the partially saturated zone above the free surface is proposed. The finite element equations are derived by using a pseudo variational principle which results into a residual or correction load vector. The steady or transient free surface is corrected by using the residual load in an iterative scheme. The proposed procedure uses only one (initial) mesh and does not require modification of the mesh during iterations. It is compared (qualitatively) with other procedures such as variable mesh and variational inequalities. The procedure provides satisfactory comparisons with a number of closed-form solutions and laboratory test results; two applications involving the latter are described in the paper.     

Numerical investigation of the anisotropic hydraulic conductivity behavior in heterogeneous porous media

The behavior of the mean equivalent hydraulic conductivity normal and parallel to stratification (K₁, and K₂, respectively) is studied here through Monte Carlo simulations of three-dimensional, steady-state flow in statistically anisotropic, bounded, and heterogeneous media. For water flow normal to stratification in strongly heterogeneous porous media (²_Y=3) the value of K₁ is not unique; it ranges from an arithmetic to a geometric, and finally, to a harmonic mean behavior depending on field dimensions, and medium anisotropy. For a fixed anisotropy ratio and variance of Y = ln K, the larger the distance, in the direction perpendicular to stratification, over which water flow takes place, the faster the rate at which, K_H, behavior is approached. However, even for large anisotropy ratios, harmonic mean behavior appears to be a good approximation only for aquifer thickness L₁ that is large enough to allow stratified flow to occur. For small aquifer thickness (L₁/₁<8, where ₁ is the integral scale normal to stratification) the limiting behavior, for large anisotropy ratios, appears to be, instead, that of two-dimensional flow, i.e., water flows primarily parallel to the planes of stratification. When the aquifer thickness is very small compared to the horizontal dimensions (and with relative similar integral scales in the three directions) a behavior resembling arithmetic mean conditions is exhibited, i.e., water flow takes place through heterogeneous, vertical, soil volumes. The geostatistical expressions of Desbarats (1992a) for upscaling hydraulic conductivity values were utilized and closed form empirical relations were developed for the main components of the upscaled hydraulic conductivity tensor.     

Experimental investigation of turbulent processes in a complex terrain

In the present paper the first results of the international KOPEX-86 experiment are presented. The experiment took place at the Kopisty Atmospheric Observatory of the Institute of Physics of the Atmosphere in Prague as part of a special project of the Commission of the Academy of Sciences in Planetary Geophysics (KAPG) in June and July 1986. Using 4 ultrasonic anemometers at 4 levels up to 80 m, Doppler-SODAR and wind-, temperature- and radiation balance gradients from 2 to 80 m, a complex investigation of the atmospheric boundary layer in an industrial area was made. The authors present the first results of

Deterministic sensitivity analysis for a model for flow in porous media

A deterministic method for sensitivity analysis is developed and applied to a mathematical model for the simulation of flow in porous media. The method is based on the singular value decomposition (SVD) of the Jacobian matrix of the model. It is a local approach to sensitivity analysis providing a hierarchical classification of the directions in both the input space and of those in the output space reflecting the degree of sensitiveness of the latter to the former. Its low computational cost, in comparison with that of statistical approaches, allows the study of the variability of the results of the sensitivity analysis due to the variations of the input parameters of the model, and thus it can provide a quality criterion for the validity of more classical probabilistic global approaches. For the example treated here, however, this variability is weak, and deterministic and statistical methods yield similar sensitivity results.     

Efficient approximations for the simulation of density driven flow in porous media

Due to the non-linear coupling between flow and transport equations the simulation of real density driven flow problems requires a lot of computational time and/or heavy equipments. We suggest some approximations and numerical recipes to reduce the CPU costs for these strongly non-linear coupled equations without loss in accuracy.     

Comparison of Galerkin-type discretization techniques for two-phase flow in heterogeneous porous media

A comparison of Standard Galerkin, Petrov-Galerkin, and Fully-Upwind Galerkin methods for the simulation of two-phase flow in heterogeneous porous media is presented. On the basis of the coupled pressure-saturation equations, a generalized formulation for all three finite element methods is derived and analysed. For flow in homogeneous media, the Petrov-Galerkin method gives excellent results. But this method fails miserably for problems with heterogeneous media. This is because it is not able to capture correctly processes that take place at interfaces when, for instance, the capillary pressure-saturation relationship after Brooks and Corey is assumed. The Fully-Upwind Galerkin method is superior to the Petrov-Galerkin approach because it is able to give correct results for flow in homogeneous and heterogeneous media for the two models of van Genuchten and Brooks-Corey. The widely used formulation which is correct for the homogeneous case cannot be used for heterogeneous media. Instead the straightforward approach of gradp_c in combination with a chord-slope technique must be utilized.