Bridging Effective Stress and Soil Water Retention Equations in Deforming Unsaturated Porous Media: A Thermodynamic Approach

The finite deformation of an unsaturated porous medium is analysed from first principles of mixture theory. An expression for Bishop’s effective stress is derived from (1) the deformation-dependent Brooks and Corey’s water retention curve and (2) the restrictions on the constitutive relationships of an unsaturated medium subject to finite deformation. The resulting expression for the effective stress parameter \(\chi \) is reasonably consistent with experimental data from the literature. Hence, it is shown that Bishop’s equation is constitutively linked to water retention curves in deforming media.     

A New Method for Generating Pore-Network Models of Porous Media

In this study, we have developed a new method to generate a multi-directional pore network for representing a porous medium. The method is based on a regular cubic lattice network, which has two elements: pore bodies located at the regular lattice points and pore throats connecting the pore bodies. One of the main features of our network is that pore throats can be oriented in 13 different directions, allowing a maximum coordination number of 26 that is possible in a regular lattice in 3D space. The coordination number of pore bodies ranges from 0 to 26, with a pre-specified average value for the whole network. We have applied this method to reconstruct real sandstone and granular sand samples through utilizing information on their coordination number distributions. Good agreement was found between simulation results and observation data on coordination number distribution and other network properties, such as number of pore bodies and pore throats and average coordination number. Our method can be especially useful in studying the effect of structure and coordination number distribution of pore networks on transport and multiphase flow in porous media systems.     

Uniqueness of Specific Interfacial Area–Capillary Pressure–Saturation Relationship Under Non-Equilibrium Conditions in Two-Phase Porous Media Flow

The capillary pressure?Csaturation (P ^c?CS ^w) relationship is one of the central constitutive relationships used in two-phase flow simulations. There are two major concerns regarding this relation. These concerns are partially studied in a hypothetical porous medium using a dynamic pore-network model called DYPOSIT, which has been employed and extended for this study: (a) P ^c?CS ^w relationship is measured empirically under equilibrium conditions. It is then used in Darcy-based simulations for all dynamic conditions. This is only valid if there is a guarantee that this relationship is unique for a given flow process (drainage or imbibition) independent of dynamic conditions; (b) It is also known that P ^c?CS ^w relationship is flow process dependent. Depending on drainage and imbibition, different curves can be achieved, which are referred to as ??hysteresis??. A thermodynamically derived theory (Hassanizadeh and Gray, Water Resour Res 29: 3389?C3904, 1993a) suggests that, by introducing a new state variable, called the specific interfacial area (a ^nw, defined as the ratio of fluid?Cfluid interfacial area to the total volume of the domain), it is possible to define a unique relation between capillary pressure, saturation, and interfacial area. This study investigates these two aspects of capillary pressure?Csaturation relationship using a dynamic pore-network model. The simulation results imply that P ^c?CS ^w relation not only depends on flow process (drainage and imbibition) but also on dynamic conditions for a given flow process. Moreover, this study attempts to obtain the first preliminary insights into the global functionality of capillary pressure?Csaturation?Cinterfacial area relationship under equilibrium and non-equilibrium conditions and the uniqueness of P ^c?CS ^w?Ca ^nw relationship.     

Modeling species transport by concentrated brine in aggregated porous media

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.     

A Theoretical Model of Hysteresis and Dynamic Effects in the Capillary Relation for Two-phase Flow in Porous Media

It is well known that the relationship between capillary pressure and saturation, in two-phase flow problems demonstrates memory effects and, in particular, hysteresis. Explicit representation of full hysteresis with a myriad of scanning curves in models of multiphase flow has been a difficult problem. A second complication relates to the fact that P ^c–S relationships, determined under static conditions, are not necessarily valid in dynamics. There exist P ^c–S relationships which take into account dynamic effects. But the combination of hysteretic and dynamic effects in the capillary relationship has not been considered yet. In this paper, we have developed new models of capillary hysteresis which also include dynamic effects. In doing so, thermodynamic considerations are employed to ensure the admissibility of the new relationships. The simplest model is constructed around main imbibition and drainage curves and assumes that all scanning curves are vertical lines. The dynamic effect is taken into account by introducing a damping coefficient in P ^c–S equation. A second-order model of hysteresis with inclined scanning curves is also developed. The simplest version of proposed models is applied to two-phase incompressible flow and an example problem is solved.     

What is the Correct Definition of Average Pressure?

Use of a correct definition of average pressure is important in numerical modeling of oil reservoirs and aquifers, where the simulated domain can be very large. Also, the average pressure needs to be defined in the application of pore-network modeling of (two-phase) flow in porous media, as well as in the (theoretical) upscaling of flow equations. Almost always the so-called intrinsic phase-volume average operator, which weighs point pressure values with point saturation values, is employed. Here, we introduce and investigate four other potentially plausible averaging operators. Among them is the centroid-corrected phase-average pressure, which corrects the intrinsic phase-volume average pressure for the distance between the centroid of the averaging volume and the phase. We consider static equilibrium of two immiscible fluids in a homogeneous, one-dimensional, vertical porous medium domain under a series of (static) drainage conditions. An important feature of static equilibrium is that the total potential (i.e., the sum of pressure and gravity potentials) is constant for each phase over the whole domain. Therefore, its average will be equal to the same constant. It is argued that the correct average pressure must preserve the fact that fluid potentials are constant. We have found that the intrinsic phase-volume average pressure results in a gradient in the total phase potential, i.e., the above criterion is violated. In fact, only the centroid-corrected operator satisfies this criterion. However, at high saturations, use of the centroid-corrected average can give rise to negative values of the difference between the average nonwetting and wetting phase pressures. For main drainage, differences among various averaging operators are significantly less because both phases are present initially, such that the difference between the centroids of phases, and the middle of the domain are relatively small.     

Study of Hydraulic Properties of Uncoated Paper: Image Analysis and Pore-Scale Modeling

In this study, uncoated paper was characterized. Three-dimensional structure of the layer was reconstructed using imaging results of micro-CT scanning with a relatively high resolution \((0.9~\upmu \hbox {m})\). Image analysis provided the pore space of the layer, which was used to determine its porosity and pore size distribution. Representative elementary volume (REV) size was determined by calculating values of porosity and permeability values for varying domain sizes. We found that those values remained unchanged for domain sizes of \(400\times 400\times 150\,\upmu \hbox {m}^{3}\) and larger; this was chosen as the REV size. The determined REV size was verified by determining capillary pressure–saturation Open image in new window imbibition curves for various domain sizes. We studied the directional dependence of Open image in new window curves by simulating water penetration into the layer from various directions. We did not find any significant difference between Open image in new window curves in different directions. We studied the effect of compression of paper on Open image in new window curves. We found that up to 30% compression of the paper layer had very small effect on the Open image in new window curve. Relative permeability as a function of saturation was also calculated. Water penetration into paper was visualized using confocal laser scanning microscopy. Dynamic visualization of water flow in the paper showed that water moves along the fibers first and then fills the pores between them.