Free-Boundary Problems for Nonlinear Models of Fluid Filtration in Inhomogeneous Porous Media

Existence theorems are proved for solutions of problems of nonlinear gravity fluid filtration in regions with specified boundaries of complex geometry. The theory developed can be used to design the underground flow net of a hydraulic structure with specified filtration characteristics.     

A Resistance Model for Newtonian and Power-Law Non-Newtonian Fluid Transport in Porous Media

A theoretical model based on the force balance between pressure, viscous force, and inertia force is proposed to predict the flow resistance of Newtonian and power-law non-Newtonian fluids through porous packed beds. The present model takes inertia effect into consideration, and the flow regime can be extended from Darcy flow to non-Darcy flow. It is demonstrated that the present model can predict most available experimental data well. The present results are also compared to the Ergun equation and other drag correlations.     

The Alternative Model of Water Vapour Sorption in Porous Building Materials

Mathematical modelling of water sorption in porous building materials is considered. The explanations of inadequacies of both Brunauer?CDeming?CDeming?CTeller model for the high relative humidities range and Frenkel?CHalsey?CHill model for the low relative humidities range reported by Pavlík et?al. (Transp. Porous Med. 91:939?C954, 2012) are proposed. The generalized D??Arcy and Watt (GDW) model is proposed as a simpler alternative for a procedure of experimental isotherms fitting proposed by Pavlík et?al. The suitability of the GDW equation to describe the water sorption isotherms in the building materials for the whole range of relative humidities is confirmed.     

幂律流体在多孔介质中径向渗流的分形模型

基于分形理论及毛细管模型,本文研究了非牛顿幂律流体在各向同性多孔介质中径向流动问题,推导了幂律流体径向有效渗透率的分形解析表达式．研究结果表明,幂律流体径向有效无量纲渗透率模型和Chang and Yortsos’s模型吻合很好;同时还得出幂律流体径向有效渗透率随孔隙率、幂指数的增加而增加,随迂曲度分形维数的增加而减少．     

An Orthorhombic Lattice Boltzmann Model for Pore-Scale Simulation of Fluid Flow in Porous Media

One application of the lattice Boltzmann equation (LBE) models is in combination with tomography to simulate pore-scale flow and transport processes in porous media. Most LBE models in the literature are based on cubic lattice, and if the voxels in a tomography image are not cubic or cannot be divided into cubes due to computational limitations, these models will lose most of their advantages. How to deal with such images is, hence, an interest in use of the LBE model to simulate pore-scale processes. In this paper, we present an orthorhombic LBE model based on the single-relaxation time approach with the relaxation parameter varying with lattice directions. The equilibrium distribution functions in the standard LBE model were modified to correct the anisotropy induced by the non-cubic lattice, and the calculations of the fluid density and momentum were also redefined in order to maintain the conservation of mass and momentum during the collision. We tested the model against analytical solution for fluid flow in a tube, and against the standard cubic-based LBE model for fluid flow in a duct with an island inside. The model was then applied to simulate fluid flow in a 3D image in attempts to analyse the errors if the voxels in the image are not cubic but are assumed to be cubic.     

A Numerical Approach for Non-Linear Moisture Flow in Porous Materials with Account to Sorption Hysteresis

A numerical approach for moisture transport in porous materials like concrete is presented. The model considers mass balance equations for the vapour phase and the water phase in the material together with constitutive equations for the mass flows and for the exchange of mass between the two phases. History-dependent sorption behaviour is introduced by considering scanning curves between the bounding desorption and absorption curves. The method, therefore, makes it possible to calculate equilibrium water contents for arbitrary relative humidity variations at every material point considered. The scanning curves for different wetting and drying conditions are constructed by using third degree polynomial expressions. The three coefficients describing the scanning curves is determined for each wetting and drying case by assuming a relation between the slope of boundary sorption curve and the scanning curve at the point where the moisture response enters the scanning domain. Furthermore, assuming that the slope of the scanning curve is the same as the boundary curve at the junction point, that is, at the point where the scanning curve hits the boundary curve once leaving the scanning domain, a complete cyclic behaviour can be considered. A finite element approach is described, which is capable of solving the non-linear coupled equation system. The numerical calculation is based on a Taylor expansion of the residual of the stated problem together with the establishment of a Newton–Raphson equilibrium iteration scheme within the time steps. Examples are presented illustrating the performance and potential of the model. Two different types of measurements on moisture content profiles in concrete are used to verify the relevance of the novel proposed model for moisture transport and sorption. It is shown that a good match between experimental results and model predictions can be obtained by fitting the included material constants and parameters.     

Filtration in a Porous Granular Medium: 2. Application of Bubble Model to 1-D Column Experiments

This paper describes the formulation of a quasi-1-D network model, referred to as the ‘bubble model’, and its application for simulating particle transport and filtration through a granular filter bed. The model comprises a series of homogeneous sites linked through bundles of cylindrical bonds that represent flow pathways through distributions of pores and pore throats. This model incorporates pore scale processes of particle sieving and infiltration are based on numerical simulations described in a companion paper. The modeling of infiltration is further refined based on detailed experimental observations and measurements of the filtration of a dilute suspension of acrylic particles through a column of glass beads reported by Yoon et al. (2005 Water Resour. Res., to appear). Their data distinguish (a) between the collection of particles on grain surfaces and at grain-to-grain contact points, and (b) between particles that are fully entrapped and those that are hindered (temporarily collected) and can later become detached. These effects are represented by two parameters that characterize the probability of attachment and are linked to the surface roughness of the grains; one that describes the minimum particle size that can be fully entrapped, and one that describes the detachment rate. These parameters can be readily calibrated from conventional measurements of effluent concentration and effluent particle size distribution. Detailed comparisons with the data reported by Yoon et al. show that the proposed bubble model is able to achieve reliable predictions of the spatial distribution of particles within the filter bed following phases of particle injection and washing.     

A Spatially Non-Local Model for Flow in Porous Media

A general mathematical model of steady-state transport driven by spatially non-local driving potential differences is proposed. The porous medium is considered to be a network of short-, medium-, and long-range interstitial channels with impermeable walls and at a continuum of length scales, and the flow rate in each channel is assumed to be linear with respect to the pressure difference between its ends. The flow rate in the model is thus a functional of the non-local driving pressure differences. As special cases, the model reduces to familiar forms of transport equations that are commonly used. An important situation arises when the phenomenon is almost, but not quite, locally dependent. The one-dimensional form of the model discussed here can be extended to multiple dimensions, temporal non-locality, and to heat, mass, and momentum transfer.     

A Subgrid-Scale Model for Turbulent Flow in Porous Media

Transport in Porous Media - Given the analogy between the filtered equations of large eddy simulation and volume-averaged Navier–Stokes equations in porous media, a subgrid-scale model is...     

A New Model for Immiscible Displacement in Porous Media

Immiscible displacement is regarded as the superposition of forward flows of both water and oil, due to injection of water into the medium, and of additional forward flow of water coupled with reverse flow of oil, caused by the existence of capillary pressure gradients. The model has been evaluated numerically for the prediction of the evolution of saturation profiles in waterfloods covering a wide range of water injection rates. In agreement with experimentation, saturation profiles ranging from a completely flat shape to piston-shape, depending on the injection rate, have been obtained. Also in agreement with experimentation, numerical evaluation of the model for the case of a closed system with an initial step-function saturation profile has predicted a gradual spreading of the piston front into S-shaped profiles with an increasing variance. The final profile corresponds to uniform saturation everywhere in the medium.     

Reactive Solute Transport in a Nonstationary,Fractured Porous Medium: A Dual-Porosity Approach

A numerical method of moment is developed for solute flux through a nonstationary, fractured porous medium. Solute flux is described as a space-time process where time refers to the solute flux breakthrough and space refers to the transverse displacement distribution at a control plane. A first-order mass diffusion model is applied to describe interregional mass diffusion between fracture (advection) and matrix (nonadvection) regions. The chemical is under linear equilibrium sorption in both fracture and matrix regions. Hydraulic conductivity in the fracture region is assumed to be a spatial random variable. In this study, the general framework of Zhang et al.(2000) is adopted for solute flux in a nonstationary flow field. A time retention function related to physical and chemical sorption in the dual-porosity medium is developed and coupled with solute advection along random trajectories. The mean and variance of total solute flux are expressed in terms of the probability density function of the parcel travel time and transverse displacement. The influences of various factors on solute transport are investigated. These factors include the interregional mass diffusion rate between fracture and matrix regions, chemical sorption coefficients in both regions, water contents in both regions, and location of the solute source. In comparison with solute transport in a one-region medium, breakthrough curves of the mean and variance of the total solute flux in a two-region medium have lower peaks and longer tails. As compared with the classical stochastic studies on solute transport in fractured media, the numerical method of moment provides an approach for applying the stochastic method to study solute transport in more complicated fractured media.     

Correction to: A Two-Parametric Model for Gas Flow in Low-Permeable Porous Materials

Transport in Porous Media - The original publication of the article includes the error in Fig. 9 legend.     

A Fractal Model for Oil Transport in Tight Porous Media

Tight porous media are mainly composed of micro/nano-pores and throats, which leads to obvious microscale effect and nonlinear seepage characteristics. Based on the capillary bundle model and the fractal theory, a new nonlinear seepage equation was deduced, and a further fractal permeability model was obtained for oil transport in tight porous media by considering the effect of the boundary layer. The predictions of the model were then compared with experimental data to demonstrate that the model is valid. This model clarifies the oil transport mechanisms in tight porous media: the effective permeability is no longer a constant value and is governed by properties of tight porous media and oil. Furthermore, parameters influencing effective permeability were analyzed. The model can accurately present the seepage characteristics of the oil in tight porous media and provide a reliable basis for the development of unconventional reservoirs.     

A Two-Parametric Model for Gas Flow in Low-Permeable Porous Materials

Transport in Porous Media - Capillary flow at low pressure gradients is determined by the pore structure of the matrix and the viscosity of the fluid. Traditionally, the conductivity of porous...     

A Stochastic Model for Particulate Suspension Flow in Porous Media

A population balance model for a particulate suspension transport with size exclusion capture of particles by porous rock is derived. The model accounts for particle flux reduction and pore space accessibility due to restriction for large particles to move through smaller pores – a particle is captured by a smaller pore and passes through a larger pore. Analytical solutions are obtained for a uniform pore size medium, and also for a medium with small pore size variation. For both cases, the equations for averaged concentrations significantly differ from the classical deep bed filtration model.     

A Dynamic Network Model for Two-Phase Flow in Porous Media

We present a dynamic model of immiscible two-phase flow in a network representation of a porous medium. The model is based on the governing equations describing two-phase flow in porous media, and can handle both drainage, imbibition, and steady-state displacement. Dynamic wetting layers in corners of the pore space are incorporated, with focus on modeling resistivity measurements on saturated rocks at different capillary numbers. The flow simulations are performed on a realistic network of a sandpack which is perfectly water-wet. Our numerical results show saturation profiles for imbibition in agreement with experiments. For free spontaneous imbibition we find that the imbibition rate follows the Washburn relation, i.e., the water saturation increases proportionally to the square root of time. We also reproduce rate effects in the resistivity index for drainage and imbibition.     

A Mathematical Model for Hysteretic Two-Phase Flow in Porous Media

We develop a mathematical model for hysteretic two-phase flow (of oil and water) in waterwet porous media. To account for relative permeability hysteresis, an irreversible trapping-coalescence process is described. According to this process, oil ganglia are created (during imbibition) and released (during drainage) at different rates, leading to history-dependent saturations of trapped and connected oil. As a result, the relative permeability to oil, modelled as a unique function of the connected oil saturation, is subject to saturation history. A saturation history is reflected by history parameters, that is by both the saturation state (of connected and trapped oil) at the most recent flow reversal and the most recent water saturation at which the flow was a primary drainage. Disregarding capillary diffusion, the flow is described by a hyperbolic equation with the connected oil saturation as unknown. This equation contains functional relationships which depend on the flow mode (drainage or imbibition) and the history parameters. The solution consists of continuous waves (expansion waves and constant states), shock waves (possibly connecting different modes) and stationary discontinuities (connecting different saturation histories). The entropy condition for travelling waves is generalized to include admissible shock waves which coincide with flow reversals. It turns out that saturation history generally has a strong influence on both the type and the speed of the waves from which the solution is constructed.     

Generalized Macroscopic Models for Fluid Flow in Deformable Porous Media I: Theories

This study developed generalized mathematical models to describe the motion of fluids in porous media, and applied these models to harmonic excitation applications. The problem of fluid flow in small channels of a periodic elastic solid matrix was studied at the pore scale, and the homogenization technique was applied to predict the macroscopic behavior of reservoirs. Based on the homogenization study, five separate characteristic macroscopic models were identified according to the relation between a length scale parameter and a property contrast number. These five models can be used to interpret the corresponding responses of a saturated porous medium. The relation to existing theories, such as Darcy's law, the Telegrapher's equation and Biot's theory, was investigated. The numerical results and applications are presented in Part II of the study.     

Effects of Cilia Movement on Fluid Velocity: I Model of Fluid Flow due to a Moving Solid in a Porous Media Framework

Transport in Porous Media - Cilia, hair-like, organelles that are found in the respiratory tract (nasal cavity, pharynx, trachea, and bronchi) rhythmically beat to clear mucus from the airways....     

A Two-Equation Model for Heat Conduction in Porous Media (I: Theory)

A two-equation model is presented which describes the conservation of heat in each phase of a porous medium in which diffusion is the predominant means of heat transfer, and of which the phases are not in thermal equilibrium with each other. The model is derived using the method of local volume averaging. This formulation, together with the introduction of characteristic temperature distributions, yields the definition of an effective and a coupled thermal conductivity tensor.