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.     

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.     

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.     

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.     

Analytical solutions to the transient,unsaturated transport of water and contaminants through horizontal porous media

We present a range of analytical solutions to the combined transient water and solute transport for horizontal flow. We adopt the concept of a scale and time dependent dispersivity used for contaminant transport in aquifers and apply it to transient, unsaturated horizontal flow to develop similarity solutions for both constant solute concentration and solute flux boundary conditions. Through the use of a specific form of the water profile as used by Brutsaert [Water Resour Res 1968:4;785], the solute profiles can be reduced to a simple quadrature. We also derive a solution for the instantaneous injection of water and solute into a horizontal media for an arbitrary dispersivity. It is found that the solute concentration remains constant in both space and time as the water redistributes, suggesting that the solute does not disperse relative to the water.     

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.     

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.     

油水两相渗流多孔介质弹性波传播动力学模型研究

弹性波在储层渗流场中的传播与衰减规律是研究波场强化采油动力学机理的重要基础.基于等效流体理论和饱和静态流体弹性波传播Biot理论,建立油水两非混相流体渗流条件下储层多孔介质中弹性波传播的动力学模型,通过算例求解与分析,发现含油水两相渗流储层多孔介质中同时存在着3种纵波P1、P2、P3和1种横波S;受频率和含水饱和度的影响,各波波速和品质因子呈现出不同变化规律,4种体波波速与频率、饱和度正相关,P1、P2波品质因子与饱和度正相关,P3和S波品质因子与饱和度负相关;最后,通过与传统静态弹性波模型结果对比,进一步分析了宏观渗流场对弹性波传播特征的影响规律,为揭示低频人工地震波辅助强化采油技术的动力学机理和工艺参数优化提供了重要理论依据.

     

Pore-scale characteristics of multiphase flow in porous media: A comparison of air–water and oil–water experiments

Studies of NAPL dissolution in porous media have demonstrated that measurement of saturation alone is insufficient to describe the rate of dissolution. Quantification of the NAPL–water interfacial area provides a measure of the expected area available for mass transfer and will likely be a primary determinant of NAPL removal efficiency. To measure the interfacial area, we have used a synchrotron-based CMT technique to obtain high-resolution 3D images of flow in a Soltrol–water–glass bead system. The interfacial area is found to increase as the wetting phase saturation decreases, reach a maximum, and then decrease as the wetting phase saturation goes to zero. These results are compared to previous findings for an air–water–glass bead study; The Soltrol–water interfacial areas were found to peak at similar saturations as those measured for the air–water system (20–35% saturation range), however, the peak values were in some cases almost twice as high for the oil-water system. We believe that the observed differences between the air–water and oil–water systems to a large degree can be explained by the differences in interfacial tensions for the two systems.     

Asymptotic behaviour of solutions to the problem of wetting fronts in one-dimensional,horizontal and infinite porous media

The asymptotic behavior of solutions to the problem of wetting fronts is studied in one-dimensional, horizontal and infinite porous media with the soil-water diffusivity proportional to some power of the water content. The uniqueness of the similarity solution for the problem is studied and the properties of this solution are presented. It is shown that the similarity solution is an asymptotic solution of a wide class of initial value problems of wetting fronts in the media. The use of the similarity solution is discussed for the experimental determination of the soil-water diffusivity.     

Wave damping by a vertical porous structure placed near and away from a rigid vertical wall

Damping of water waves by a vertical porous structure placed at some distance from a vertical wall is investigated within the framework of linear water wave theory. The rectangular porous structure is placed on a small rectangular elevation. An incident wave of small amplitude propagates through the structure – some portion gets reflected back while some portion gets transmitted to a third region bounded vertically by a rigid wall which is considered to be at a distance near the porous structure, and also away from the wall at a large distance as a separate case. Boundary value problems are set up in all three regions and, by using the matching conditions along the vertical boundaries, a system of linear equations is deduced. The roots of the relevant dispersion relation are used in setting up the system of equations. The overall scattering phenomenon is studied with respect to different relevant parameters. The dependence of the coefficients on the thickness (width) of the porous structure is investigated for different numbers of modes and porosity. It is observed that, except for the case when the porous structure is thin, the reflection and transmission coefficients give rise to values as expected. In the case, when the rigid wall is nearer to the structure, the reflection coefficient decreases rapidly for a thin structure and converges for all numbers of evanescent modes afterwards. The transmission coefficient also decreases as the width increases, ultimately converging and vanishing for a wide structure. When the wall is at a large distance away from the structure, the behavior of both the reflection and transmission coefficients remain the same. For both cases of the wall being nearer and away from the structure, higher porosity gives rise to lower reflection coefficients and higher transmission coefficients. However, the transmission coefficients converge and vanish when the porous structure is very wide. We also discuss the energy loss against the width of the porous structure for different values of number of modes and porosity. Irrespective of the positioning of the rigid wall, we observe that for higher values of porosity, energy loss is more pronounced when the structure is not thin, whereas energy loss is same for all numbers of modes. All our observations are supported by graphs. Good agreement of our result with earlier results justifies our model.     

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.     

Real gas flow through heterogeneous porous media: theoretical aspects of upscaling

We consider the problem of upscaling transient real gas flow through heterogeneous bounded reservoirs. One of the commonly used methods for deriving effective permeabilities is based on stochastic averaging of nonlinear flow equations. Such an approach, however, would require rather restrictive assumptions about pressure-dependent coefficients. Instead, we use Kirchhoff transformation to linearize the governing stochastic equations prior to their averaging. The linearized problem is similar to that used in stochastic analysis of groundwater flow. We discuss the effects of temporal localization of the nonlocal averaged Darcy's law, as well as boundary effects, on the upscaled gas permeability. Extension of the results obtained by means of small perturbation analysis to highly heterogeneous porous formations is also discussed.     

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.     

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.     

Influence of heterogeneous air entry pressure on large scale unsaturated flow in porous media

The paper presents numerical simulations of water infiltration in unsaturated porous media containing coarse-textured inclusions embedded in fine-textured background material. The calculations are performed using the two-phase model for water and air flow and a simplified model known as the Richards equation. It is shown that the Richards equation cannot correctly describe flow in the presence of heterogeneities. However, its performance can be improved by introducing appropriately defined effective capillary and permeability functions, representing largescale behaviour of the heterogeneous medium.     

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.     

Some exact solutions for the flow of fluid through tubes with uniformly porous walls

Summary This paper considers an incompressible fluid flowing through a straight, circular tube whose walls are uniformly porous. The flow is steady and one dimensional. The loss of fluid through the wall is proportional to the mean static pressure in the tube. Several formulations of the wall shear stress are considered; these formulations were motivated by the results from Hamel's radial flow problem, boundary layer flows/and boundary layer suction profiles. For each of these formulations exact solutions for the mean axial velocity and the mean static pressure of the fluid are obtained. Sample results are plotted on graphs. For the constant wall shear stress problem, the theoretical solutions compare favorably with some experimental results.Notations A, B, D, E constant parameters - a, b constant parameters - Ai(z), Bi(z) Airy functions - Ai, Bi derivatives of Airy functions - k constant of proportionality betweenV andp - L length of pores - p,p mean static pressure - p ₀ static pressure outside the tube - p ₀ value ofp atx=0 - Q constant exponent - R inside radius of the tube - T wall shear stress - T ₀ shear parameter - t wall thickness - U free stream velocity - ,u mean axial velocity - u ₀ value ofu atx=0 - V,V mean seepage velocity through the wall - v ₀ mean seepage velocity - x,x axial distance along the tube - z transformed axial distance - z ₀ value ofz atx=0 - mean outflow angle through the wall - cos - density of the fluid - wall shear stress - dynamic viscosity of the fluid - over-bar dimensional terms - no bar nondimensional terms The National Center for Atmospheric Research is sponsored by the National Science Foundation