Showing papers in "Transport in Porous Media in 1988"

On the reliability of unsaturated hydraulic conductivity calculated from the moisture retention curve

Abstract: In comparison with direct measurements of unsaturated hydraulic conductivity, the methods of calculations from the moisture retention curve are attractive for their fast and simple use and low cost. These are the main reasons for their increasing use, mainly in spatial variability studies. On the other hand, it is known that their applicability is limited. The possibility of the use of the retention curve to indirectly determine hydraulic conductivities is analyzed as follows. The theoretical derivation of the relationK(h) − θ(h) is briefly discussed with regards to potential sources of inaccuracy. The sensitivity of the algorithm forK(h) calculation is studied as a response to possible inaccuracies in the retention curve determination. Conclusions about the usability of calculated hydraulic conductivities are drawn.

Two-phase flow in heterogeneous porous media: The method of large-scale averaging

Abstract: The analysis of two-phase flow in porous media begins with the Stokes equations and an appropriate set of boundary conditions. Local volume averaging can then be used to produce the well known extension of Darcy's law for two-phase flow. In addition, a method of closure exists that can be used to predict the individual permeability tensors for each phase. For a heterogeneous porous medium, the local volume average closure problem becomes exceedingly complex and an alternate theoretical resolution of the problem is necessary. This is provided by the method of large-scale averaging which is used to average the Darcy-scale equations over a region that is large compared to the length scale of the heterogeneities. In this paper we present the derivation of the large-scale averaged continuity and momentum equations, and we develop a method of closure that can be used to predict the large-scale permeability tensors and the large-scale capillary pressure. The closure problem is limited by the principle of local mechanical equilibrium. This means that the local fluid distribution is determined by capillary pressure-saturation relations and is not constrained by the solution of an evolutionary transport equation. Special attention is given to the fact that both fluids can be trapped in regions where the saturation is equal to the irreducible saturation, in addition to being trapped in regions where the saturation is greater than the irreducible saturation. Theoretical results are given for stratified porous media and a two-dimensional model for a heterogeneous porous medium.

Mathematical Modelling of Flow Through Consolidated Isotropic Porous Media

Abstract: A new mathematical model is proposed for time-independent laminar flow through a rigid isotropic and consolidated porous medium of spatially varying porosity. The model is based upon volumetric averaging concepts. Explicit assumptions regarding the mean geometric properties of the porous microstructure lead to a relationship between tortuosity and porosity. Microscopic inertial effects are introduced through consideration of flow development within the pores. A momentum transport equation is derived in terms of the fluid properties, the porous medium porosity and a characteristic length of the microstructure. In the limiting cases of porosity unity and zero, the model yields the required Navier-Stokes equation for free flow and no flow in a solid, respectively.

Condensation and isothermal water transfer in cement mortar Part I — Pore size distribution, equilibrium water condensation and imbibition

Abstract: The pore size distribution of cement mortar is studied in relation to water sorption experiments with the help of mercury intrusion and nitrogen sorption. The importance of adsorbed water is pointed out. Isothermal imbibition experiments at four temperatures are presented. The temperature-dependence of the mass transfer coefficients is compared to the one predicted by the classical model. Significant discrepancies are noticed.

Fluid topology, pore size and aspect ratio during imbibition

Abstract: Imbibition in glass micromodels for air-mercury and water-oil systems occurs by wetting phase (wp) cluster growth and frontal drive processes. Lower capillary number and higher wetting phase (wp) saturation at the start of imbibition favour cluster growth.

Measuring transport coefficients necessary for the description of coupled two-phase flow of immiscible fluids in porous media

Abstract: In the simplist cases of coupled two-phase flow of immiscible fluids in porous media, the governing equations usually are written to show that there are four independent transport coefficients that implicitly have to be separately measured. The analysis presented here accordingly indicates that two types of known experiments involving two measurements apiece are needed at each fluid saturation condition in order to provide the necessary and sufficient information by which the unsteady as well as the steady states of ensuing transport processes can be established and characterized. Apparently, however, the fact that methodologies are already available for the required laboratory work either is not widely appreciated or it is being overlooked. For this reason and others, mention is made of the surprising fact that the experimental difficulties to be confronted in the actual study of coupled transport processes are no greater than those that have already been dealt with by the advocates of classical relative permeability theory (i.e. the traditionalists who simplistically model two-phase flow as though no coupling effects are involved).

Free convection from a vertical plate with nonuniform surface heat flux and embedded in a porous medium

Abstract: An analysis is presented for the calculation of heat transfer due to free convective flow along a vertical plate embedded in a porous medium with an arbitrarily varying surface heat flux By applying the appropriate coordinate transformations and the Merk series, the governing energy equation is expressed as a set of ordinary differential equations Numerical solutions are presented for these equations which represent universal functions and several computational examples are provided

Advances in Modeling of Water in the Unsaturated Zone

Abstract: This paper reviews recent advances in modeling of water flow in the unsaturated zone. The Richards model remains the most widely accepted and fertile framework for water flow analyses. More general formulations are reserved for the analysis of problems involving macroporosity, thermal effects, and air pressure effects. Many exact and approximate solutions have been derived for particular boundary value problems of homogeneous soils using methods such as quasi-linear analysis, Green-Ampt analysis, perturbation, and the kinematic wave approximation. Numerical simulators have become bigger and more accurate due to improvements in the areas of nonlinear solution procedures, mass conservation, computational efficiency, and computer hardware. Problems of natural heterogeneity have been addressed primarily through application of various stochastic methods to the Richards model. The stochastic formulations generally refute the concept of simple “equivalent” homogeneous properties, but do themselves offer a certain limited potential for a predictive capability.

Fractal porous media III: Transversal Stokes flow through random and Sierpinski carpets

Abstract: The transversal Stokes flow of a Newtonian fluid through random and Sierpinski carpets is numerically calculated and the transversal permeability derived. In random carpets derived from site percolation, the average macroscopic permeability varies as (e- ɛc)3/2, close to the critical porosityɛc. This exponent is found to be slightly different from the conductivity exponent. Results for Sierpinski carpets are presented up to the fourth generation. The Carman equation is not verified in these two model porous media.

Numerical study of the linear stability of immiscible displacement in porous media

Abstract: In this paper the linear stability of immiscible displacement in porous media is examined by numerical methods. The method of matched initial value problems is used to solve the eigenvalue problem for displacement processes pertaining to initially mobile phases. Both non capillary and capillary displacement in rectilinear flow geometries is studied. The results obtained are in agreement with recent asymptotic studies. A sensitivity analysis with respect to process parameters is carried out. Similarities and differences with the stability of Hele-Shaw flows are delineated.

Advances in modeling of water of the unsaturated zone

Abstract: Discussion de methodes recentes pour la solution du probleme de l'ecoulement en milieu poreux, dans sols rigides, en zone non saturee. Au-dela du modele de Richards, formulations plus generales pour problemes de macroporosite, effets thermiques, effets de la pression de l'air; simulations numeriques; applications de methodes stochastiques au modele de Richards, pour milieux heterogenes

The Comparison Model Method - A New Arithmetic Approach to the Discrete Inverse Problem of Groundwater Hydrology .1. One-Dimensional Flow

Abstract: Let the steady-state pressure z(·) of a fluid in a one-dimensional domain be governed by the equation d x (a d x z) = f subject to Dirichlet boundary conditions. We consider the identification of the transmissivity a (·), given f(·), and measured pressure z(·) by the comparison model method, a direct method which has been known and applied for some time but lacked theoretical background. With reference to a domain in one spatial dimension, we examine both the infinite-(‘continuous’) and finite-(discrete) dimensional cases. In the former, the method is based on the solution p(·) of an auxiliary flow equation, where f(·) and the two-point boundary conditions are unchanged with respect to the original or z(·) equation, whereas a tentative constant value b is assigned to the auxiliary transmissivity. The ratio of the first derivatives of p(·) and z(·) multiplied by b yields a solution a(·) to the inverse problem. We examine in detail the nonuniqueness of a(·) as a function of b, some cases where existence implies uniqueness, the role of positivity constraints, and a special feature: self-identifiability. We then translate all available results into the discrete case, where the good unknowns for the inverse problem are the internode coefficients. Several algebraic and numerical examples are presented.

Modeling species transport by concentrated brine in aggregated porous media

Abstract: Basic equations governing the transport of species by concentrated brine flowing through an aggregated porous medium are developed. Some simple examples are solved numerically. The medium is considered to be composed of porous rock aggregates separated by ‘macropores’ through which the brine flows and transport of salt and low-concentration species takes place. The aggregates contain dead-end pores, cracks, and stationary pockets collectively called ‘micropores’. The micropore space does not contribute to the flow, but it serves as a storage for salt and species. Adsorption of fluid species takes place at internal surface of aggregates where it is assumed that a linear equilibrium isotherm describes the process. The effects of high salt concentrations are accounted for in the brine density relation, the viscosity relation, Darcy's and Fick's laws, and the rate of mass transfer between macropores and micropores. Mass balance equations, supplemented by extended forms of Darcy's and Fick's laws, are employed to arrive at two sets of equations. One set consists of seven coupled equations for the salt mass fraction and fluid density in macropores, salt mass fraction in micropores, fluid velocity vector, and the fluid pressure. The other set consists of two coupled equations to be solved for the mass fractions of low-concentration species in micropores and macropores. Based on these equations, a mathematical model called TORISM is developed. Using this model, the potential significance of modifications to Darcy's Law are demonstrated.

Matrix and Fissure Water Movement through Unsaturated Calcareous Sandstone

Abstract: Evaluation of pollution endangering groundwater resources beneath fractured sediments may be achieved by estimating the transport rates and recharge amounts of both the matrix and the fissure components This study examines the transport of water by matrix and fissure flow in the unsaturated zone using environmental tritium as a natural tracer A 35-year record of tritium concentration along 40 m calcareous sandstone column was reconstructed It was found that on the average, 40 mm yr1 (8% of the yearly rain) percolated downward through the matrix pores at a velocity of 11 m yr1 An additional amount of more than 20 mm yr1 (more 4% of the rains) percolated rapidly through fissure network These field data fit and support the model proposed by Wang and Narashimham (1985) that the bulk of the water movement under unsaturated conditions occurs through interconnected pores in the matrix

Theoretical studies of mercury intrusion in some networks: testing the applicability of mercury intrusion in the size characterisation of the lacunar pore space of soil samples

Abstract: At the textural level, it is possible to consider the organization of soil components as a characteristic feature of those materials. This organization confers a pore space specific to the clay phase and a lacunar pore space which is present between clay masses. The lacunar pore space may be regarded as a generally continuous network, whose dimensions are much larger than those of the clay pore space.

Effect of capillary pressure on the approach to residual saturation

Abstract: The approach to residual oil saturation during the immiscible displacement of oil as predicted by the multiphase Darcy equations is studied. It is well known that when the capillary pressure term is neglected, one arrives at the Buckley-Leverett formulation according to which the inlet face attains residual oil saturation instantaneously. This result may, however, be strongly influenced by the inclusion of the capillary pressure term. In this paper it is shown that when the relative permeability and capillary pressure functions have power law dependencies on the saturation deviation from residual oil condition, the long time solution exhibits a power law decay toward residual saturation. Moreover, the power law decay solution is found to be unique and independent of the initial condition. The relationship of this solution to the classical Buckley-Leverett result is shown. Finally, generalization to the time varying flow rate case is addressed. As a verification of the theoretical conjectures, the power law solution is compared with direct numerical simulation of the two phase flow equations.

Coupling between liquid flow and heat flow in porous media: A connection between two classical approaches.

Abstract: The presented work addresses exclusively to the transport in the liquid (sub)phases occurring in porous media. By analysing the thermodynamics of the solid-liquid and liquid-gas interfaces present within a porous solid-liquid-gas system, it is shown that the forces acting on two distinct subphases of the liquid, due to the presence of a macroscopic temperature gradient, tend to balance each other. Exact counterbalance of the resulting fluxes implies that liquid flow in porous media under nonisothermal conditions is adequately described by the product of isothermal liquid diffusivity and the gradient of volumetric liquid content.

A new perturbation expansion for horizontal infiltration and sorptivity estimates

Abstract: A new asymptotic expansion is proposed for the nonlinear diffusion equation used to model the horizontal infiltration of water into soils. The expansion is based on a small parameter associated with the nonlinearity in the diffusivityD. The expansion leads to simpler integral formulae than previous series solutions and most previous iterative methods. To first order, it yields a formula for sorptivity identical with the ‘double-integration’ result, derived in a less rigorous manner by Parlange (1975). Rigorous conditions for the validity of the asymptotic expansion and the double-integration technique are discussed.

Fluid capacity distributions of random porous media

Abstract: As a quantitative measure of the microstructure in a statistically homogeneous porous material, we introduce the notion of thefluid capacity at a specified length scale λ. In two dimensions, fluid capacity is the void space per unit area for a square of side λ and in three dimensions it is the void space per unit volume for a cube of side λ. The most random distribution of fluid capacity, for a prescribed mean fluid capacity, corresponds to an exponential distribution. The distribution of fluid capacity is important during unstable fluid displacements in porous media where viscous fingering occurs. For a material with an exponential fluid capacity distribution, an unstable displacement process can be simulated by simple stochastic algorithms related to diffusion-limited aggregation. We measure the two-dimensional fluid capacity distributions of published cross-section photomicrographs of sandstone, salt, and packed beds of glass beads, for various length scales A. The form of the distribution depends upon the magnitude of the length scale λ. For the sandstone and salt packs, appropriate length scales are found on which the fluid capacity has, to a good approximation, an exponential distribution. An exponential distribution appears to be inappropriate for the packed bed of glass beads on all length scales.

Stability of vertical miscible displacements with developing density and viscosity gradients

Abstract: Viscous fingering and gravity tonguing are the consequences of an unstable miscible displacement. Chang and Slattery (1986) performed a linear stability analysis for a miscible displacement considering only the effect of viscosity. Here the effect of gravity is included as well for either a step change or a graduated change in concentration at the injection face during a downward, vertical displacement. If both the mobility ratio and the density ratio are favorable (the viscosity of the displacing fluid is greater than the viscosity of the displaced fluid and, for a downward vertical displacement, the density of the displacing fluid is less than the density of the displaced fluid), the displacement will be stable. If either the mobility ratio or the density ratio is unfavorable, instabilities can form at the injection boundary as the result of infinitesimal perturbations. But if the concentration is changed sufficiently slowly with time at the entrance to the system, the displacement can be stabilized, even if both the mobility ratio and the density ratio are unfavorable. A displacement is more likely to be stable as the aspect ratio (ratio of thickness to width, which is assumed to be less than one) is increased. Commonly the laboratory tests supporting a field trial use nearly the same fluids, porous media, and displacement rates as the field trial they are intended to support. For the laboratory test, the aspect ratio may be the order of one; for the field trial, it may be two orders of magnitude smaller. This means that a laboratory test could indicate that a displacement was stable, while an unstable displacement may be observed in the field.

Two-dimensional adaptive Eulerian-Lagrangian method for mass transport with spatial velocity distribution

Abstract: A Eulerian-Lagrangian scheme is used to solve the two-dimensional advection-dispersion equation. Concentration and its partial differential operator are decomposed into advection and dispersion terms. Thus, advection is formally decoupled from dispersion and solved by continuous forward particle tracking. Dispersion is handled by implicit finite elements on a fixed Eulerian grid. Translation of steep gradients of concentration in advection-dominated flow regimes, is done without numerical distortion. Continuous spatial distribution of velocities are evaluated by using Galerkin's approach in conjunction with Darcy's law based on hydraulic input data from each element. The method was implemented on coarse FE grid with linear shape functions, demonstrating no over/under shooting and practically no numerical dispersion. Simulations, covering a wide range of Peclet numbers, yield high agreement with analytic and practical results.

Estimation of three-phase relative permeabilities

Abstract: Starting from the statistical structural model of Alemanet al. (1988), we have developed an alternative to Stone's (1970, 1973; Aziz and Settari, 1979) methods for estimating steady-state, three-phase relative permeabilities from two sets of steady-state, two-phase relative permeabilities. Our result reduces to Stone's (1970; Aziz and Settari, 1979) first method, when the steady-state, two-phase relative permeability of the intermediate-wetting phase with respect to either the wetting phase or the nonwetting phase is a linear function of the saturation of the intermediate-wetting phase. As the curvature of either of these relative permeability functions increases, the deviation of our result from Stone's (1970; Aziz and Settari, 1979) first method increases. Currently, there are no data available that are sufficiently complete to form the basis of a comparison between our result and either of the methods of Stone (1970, 1973; Aziz and Settari, 1979).

A new description for dispersion

Abstract: Chang and Slattery (1986) introduced a simplified model for dispersion that contains only two empirical parameters, both of which can be determined in one-dimensional experiments. The traditional model for dispersion (Nikolaevskii, 1959; Scheidegger, 1961; de Josselin de Jong and Bossen, 1961; Bear, 1961a; Peaceman, 1966; Bear, 1972) has three empirical parameters, two of which can be measured in one-dimensional experiments while the third, the transverse dispersivity, must be measured in experiments in which a two-dimensional concentration profile develops. For the common one-dimensional experiment in which the signs of the concentration gradient and of the velocity field are different, the simplified model and the traditional model give identical results. For a one-dimensional experiment in which the signs of the concentration gradient and of the velocity field are at least sometimes the same and for two- and three-dimensional flows, the simplified model of Chang and Slattery (1986) gives results that can differ from those predicted using the traditional model. Only the experimental data of Bear (1961b) and of Yule and Gardner (1978) are sufficiently complete to permit a comparison of the two models. Considering the quality of the experimental data, we can not distinguish between the predicted profiles based upon the simplified model and those based upon the traditional model.

An analysis of the viability of polymer flooding as an enhanced oil recovery technology

Abstract: The concept of improving oil recovery through polymer flooding is analysed. It is shown that while the injection of a polymer solution improves reservoir conformance, this beneficial effect ceases as soon as one attempts to push the polymer solution with water. Once water injection begins, the water quickly passes through the polymer creating a path along which all future injected water flows. Thus, the volume of the polymer slug is important to the process and an efficient recovery would require that the vast majority of the reservoir be flooded by polymer. It is also shown that the concept of grading a polymer slug to match the mobilities of the fluids at the leading and trailing edges of a polymer slug does not work in a petroleum reservoir. While this process can supply some additional stability to the slug, it is shown that for the purposes of enhanced oil recovery this additional stability is not great enough to be of any practical use. It is found that in this case the instability has simply been hidden in the interior of the slug and causes the same sort of instability to occur as was the case for the uniform slug.

Stochastic analysis of groundwater flow in a bounded domain by spectral methods

Abstract: A saturated flow problem with spatially varying conductivity is studied in a rectangular domain. An expansion of the flow equation with respect to small perturbations of the conductivity is given. Discrete spectra are used to calculate the expected flux across the outflow boundary and its variability. The results obtained are compared with results based on Monte Carlo studies. Another way to deal with heterogeneous soils is to replace the actual conductivity by a smooth, so-called, effective conductivity. A comparison is made between results based on that approach and our results.

An adaptive domain decomposition method for simulation of transport in porous media

Abstract: A Galerkin finite element method is used along with a self-adaptive strategy of domain discretisation to model dispersion in an axisymmetric cylindrical porous medium A solution strategy is proposed based on the use of a Gear scheme for the time stepping and partial vectorisation of the code The domain is highly discretised in the area of the sharp transient front, while the remainder is coarsely discretised The area covered by the fine mesh is determined by the value of the local concentration gradients Numerical results are presented for the one and two dimensional cases

Saltwater transport in a heterogeneous formation: a case study

Abstract: Historical information of the hydraulic and salinity aspect, detailed geological information, and information on the physical characteristics of the different layers comprising the formation, are needed for simulating the saltwater transport process in aquifers. In most simulation studies of field situations, there is an inadequacy of data and the modeller has to make justifiable assumptions to analyze a particular situation in order to provide an insight into the problem.

A scale-dependent equation for solute transport in porous media

Abstract: The governing equation describing solute transport in porous media is reformulated using standard volume averaging techniques. The alternative formulation is based on a modified definition of the deviation, which allows for variation of macroscopic velocity across the REV. The new equation contains additional scale-dependent terms which are functions of the size of the averaging volume (REV). This result indicates that the scale-dependent nature of the dispersion phenomenon is inherent even at the scale of the REV.