Laminar hydromagnetic flow in an annulus with uniformly porous walls of different permeability

Summary The problem of steady laminar hydromagnetic flow in a porous annulus of different permeability in the presence of a magnetic field is considered and solved by a method of perturbation. The case when the injection or the withdrawal rate at the outer wall, depending on the sign of the normal velocity at this wall, is greater than or equal to such rate at the inner wall is given in details. Some numerical results regarding the friction coefficients at the walls and the axial pressure coefficient and velocity distribution have been appended.     

Steady hydromagnetic viscous flow in an annulus with porous walls

Summary An exact solution of electrically conducting viscous incompressible flow in an annulus with porous walls under an external radial magnetic field is obtained when the motion is due to both longitudinal motion of the inner boundary and a constant axial pressure gradient, and the fluid injection rate at one wall is equal to the fluid withdrawal rate at the other. The fluid may be injected at the outer wall and sucked at the inner or vice versa. The solution for the hydromagnetic flow between two flat plates has also been obtained as a limiting case of the annulus problem.     

Similarity solutions to the second boundary value problem of unsaturated flow through porous media

Similarity solutions to the second boundary value problem of unsaturated flow are studied in one-dimensional, semi-infinite porous media with the soil-water diffusivity proportional to some power of the water content. The existence and uniqueness of two types of similarity solutions to the problem are investigated and the properties of these solutions are presented. It is shown that these two types of similarity solutions exist and that they may not be unique for every parameter range studied. The use of the similarity solutions is discussed for the experimental determination of soil-water diffusivity.     

Some exact solutions of unsteady boundary layer equations-I

Summary Unsteady boundary layer flows generated in a homogeneous, non-rotating viscous fluid are considered. The method of Laplace transform is used to obtain exact solutions of the unsteady boundary layer equations in a more general situation. The structures of the unsteady velocity field and the associated boundary layers are determined. Several particular solutions are recovered as special cases of the present general theory. The physical implications of the mathematical results are investigated.     

Some exact solutions of unsteady boundary layer equations-II

Summary Some exact solutions are presented for the unsteady boundary layer flows of a homogeneous, viscous, incompressible fluid bounded by (i) an infinite rigid oscillating flat plate or (ii) two parallel rigid oscillating flat plates. An explicit representation of the velocity fields for both the configurations has been given. The structures of the associated periodic boundary layers are determined with physical interpretations. Several results of interest have been recovered as special cases of this general theory. The Heaviside operational calculus along with the theory of residues of analytic functions is adopted in finding the solutions.     

Flow of non-Newtonian fluid through circular tubes

Summary According to Newton's law of viscosity _y = Dv_y/dy. But experiments have shown that _y is indeed proportional to –dv _x/dy for all gases and for homogeneous nonpolymeric liquids. There are however, a few industrially important materials, e.g. plastics, asphalts, crystalline materials that are not described by the equation given by Newton's law of viscosity and they are referred to as non-Newtonian fluids. The steady state rheological behaviour of most fluids can be expressed by the generalised form, _y = –(dv_y/dy) where may be expressed as a function of eitherdv _x/dy or _y (where is independent of the rate of shear, the behaviour is Newtonian with =). Numerous empirical equations or models have been proposed to express the steady-state relation between _y anddv _x/dy. The flow of Newtonian fluids through circular tubes have been discussed before by many. Here we shall discuss the case of two such models of non-Newtonian fluids through circular tubes. The flow of fluids in circular tubes is encountered frequently in Physics, Chemistry, Biology and Engineering.     

Similarity solutions of the Cauchy problem of horizontal flow of water through porous media for experimental determination of diffusivity

An experimental method for determining diffusivity is studied by using similarity solutions of the Cauchy problem of horizontal flow of water through homogeneous porous media. The theoretical justification of the method is presented by applying a mathematical theorem recently derived by Van Duyn. Some important aspects of data analysis are discussed by using actual experimental data.     

Miscible flow through porous media with dispersion and adsorption

The partial differential nonlinear equation which describes the one-dimensional flow of miscible fluids through porous media with dispersion and Langmuir equilibrium adsorption is numerically solved by finite differences.Local truncation error is determined and von Neumann stability analysis is applied. In order to eliminate either numerical dispersion or unstability, weighting parameters and distance and time increments are conveniently adjusted.Finite differences results are verified with the exact solution for the linear adsorption case. They are obtained for different boundary conditions, whose influence is discussed.Numerical solutions are matched with experimental results from Szabo's¹ polymer flooding tests. Differences between numerical and experimental results are minimized applying optimization techniques to obtain the most suitable physical parameters.     

Random maze models of flow through porous media

Summary This paper discusses a class of stochastic models of flow through porous media in which the randomness is attached to the structure of the medium rather than to the flow path. These models are obtained by generalizing an earlier model available in the literature where a regular crystal was taken in which bonds (flow channels) were dammed in a random fashion, yielding a random maze. The hydraulic properties of general models of this type are calculated; in particular, it is shown that they exhibit the phenomenon of dispersion whereby the factor of dispersion turns out to be a linear function of the percolation velocity.     

Particular solutions to the problem of horizontal flow of water and air through porous media near a wetting front

Particular solutions to the problem of horizontal flow of water and air through homogeneous porous media are derived and regularity properties of the solutions are presented. It is found that a singularity occurs in the solutions at the wetting fronts. Effects of air flow on water flow are discussed.     

Particular solutions to the problem of vertical flow of water and air through porous media near a water table

Particular solutions to the problem of vertical flow of water and air through homogeneous porous media are derived and the regularity properties of the solutions are presented. It is found that under certain conditions a singularity occurs in the solutions at the water table, though it is not certain whether or not a singularity always occurs. Effects of air flow on water flow are discussed.     

Particular solutions to the problem of horizontal flow of water and air through porous media near a water table

Particular solutions to the problem of horizontal flow of water and air through porous media near a water table are derived and regularity properties of the solutions are presented. It is found that a singularity occurs in the solutions at the water table and the water table can be interpreted as an acceleration wave of the nth order in terms of either air or water flow where n is a positive integer. Effects of air flow on water flow are discussed.     

Steady Couette flow of a simple fluid between porous plates

Summary Steady Couette flow ofNoll's simple fluid between two porous plates has been considered. It is seen that such flow, though not strictly lineal, is viscometric and unique solution for the main flow exists, in general, under Lipschitz's condition on the shearing function. Special cases have been discussed.     

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.     

Estimates of effective permeability for non-Newtonian fluid flow in randomly heterogeneous porous media

An understanding of the interplay between non-Newtonian effects in porous media flow and field-scale domain heterogeneity is of great importance in several engineering and geological applications. Here we present a simplified approach to the derivation of an effective permeability for flow of a purely viscous power–law fluid with flow behavior index n in a randomly heterogeneous porous domain subject to a uniform pressure gradient. A standard form of the flow law generalizing the Darcy’s law to non-Newtonian fluids is adopted, with the permeability coefficient being the only source of randomness. The natural logarithm of the permeability is considered a spatially homogeneous and correlated Gaussian random field. Under the ergodic hypothesis, an effective permeability is first derived for two limit 1-D flow geometries: flow parallel to permeability variation (serial-type layers), and flow transverse to permeability variation (parallel-type layers). The effective permeability of a 2-D or 3-D isotropic domain is conjectured to be a power average of 1-D results, generalizing results valid for Newtonian fluids under the validity of Darcy’s law; the conjecture is validated comparing our results with previous literature findings. The conjecture is then extended, allowing the exponents of the power averaging to be functions of the flow behavior index. For Newtonian flow, novel expressions for the effective permeability reduce to those derived in the past. The effective permeability is shown to be a function of flow dimensionality, domain heterogeneity, and flow behavior index. The impact of heterogeneity is significant, especially for shear-thinning fluids with a low flow behavior index, which tend to exhibit channeling behavior.     

Necessary conditions for inverse modeling of flow through variably saturated porous media

Non-unique solutions of inverse problems arise from a lack of information that satisfies necessary conditions for the problem to be well defined. This paper investigates these conditions for inverse modeling of water flow through multi-dimensional variably saturated porous media. It shows that in order to obtain a unique estimate of hydraulic parameters, along each streamline of the flow field (1) spatial and temporal head observations must be given; (2) the number of spatial and temporal head observations required should be greater or equal to the number of unknown parameters; (3) the flux boundary condition or the pumping rate of a well must be specified for the homogeneous case and both boundary flux and pumping rate are a must for the heterogeneous case; (4) head observations must encompass both saturated and unsaturated conditions, and the functional relationships for unsaturated hydraulic conductivity/pressure head and for the moisture retention should be given, and (5) the residual water content value also need to be specified a priori or water content measurements are needed for the estimation of the saturated water content.For field problems, these necessary conditions can be collected or estimated but likely involve uncertainty. While the problems become well defined and have unique solutions, the solutions likely will be uncertain. Because of this uncertainty, stochastic approaches are deemed to be appropriate for inverse problems as they are for forward problems to address uncertainty. Nevertheless, knowledge of these necessary conditions is critical to reduce uncertainty in both characterization of the vadose zone and the aquifer, and prediction of water flow and solute migration in the subsurface.     

Analytical solutions for flow to a well through a fault

We present explicit analytical solutions to problems of steady groundwater flow to a pumping well in an aquifer divided by an infinite, linear fault. The transmissivity of the aquifer is allowed to jump from one side of the fault to the other to model the juxtaposition of host rocks with different hydrologic properties caused by faulting. The fault itself is represented as a thin anisotropic inhomogeneity; this allows the fault to act as a combined conduit–barrier to groundwater flow, as is commonly described in the literature. We show that the properties of the fault may be represented exactly by two lumped parameters—fault resistance and fault conductance—and that the effects of the fault on flow in the adjacent aquifer is independent of the fault width. We consider the limiting cases of a purely leaky and a purely conductive fault where the fault domain may be replaced exactly by internal boundary conditions, and we investigate the effects of fault properties on the flow behavior in the adjacent aquifers. We demonstrate that inferring fault properties based on field observations of head in the aquifer is inherently difficult, even when the fault may be described by one of the two limiting cases. In particular, the effects of a leaky fault and a conductive fault on heads and discharges in the aquifer opposite the fault from the well, are shown to be identical in some cases.     

A new finite element technique for the solution of two-phase flow through porous media

A new upstream weighting finite element technique is developed for improved solution of the two-phase immiscible flow equations. Unlike the upstream weighting technique used by previous investigators, the new technique does not employ finite difference concepts to achieve the required upstream weighting of relative permeabilities or mobilities. Instead, upstream weighting is achieved by (1) representing the relative permeabilities or mobilities as continuous functions expressed in terms of the shape functions and nodal values (2) using asymmetric weighting functions to weight the spatial terms in the flow equations. These weighting functions are constructed such that they are dependent on the flow direction along each side of an element.In conjunction with the proposed technique, two solution schemes for treating the resulting set of non-linear algebraic equations are presented. These are the fully-implicit chord slope incremental solution scheme and the Newton-Raphson solution scheme. Both schemes allow the use of large time steps without being unstable.The proposed numerical technique is applied to two problems (1) the one-dimensional Buckley-Leverett problem (2) the two-dimensional five-spot well flow problem. Results indicate that this technique is superior to not only earlier finite element schemes but also five-point upstream finite difference formulae.