A Model for Flow and Deformation in Unsaturated Swelling Porous Media

A thermomechanical theory for multiphase transport in unsaturated swelling porous media is developed on the basis of Hybrid Mixture Theory (saturated systems can also be modeled as a special case of this general theory). The aim is to comprehensively and non-empirically describe the effect of viscoelastic deformation on fluid transport (and vice versa) for swelling porous materials. Three phases are considered in the system: the swelling solid matrix s, liquid l, and air a. The Coleman–Noll procedure is used to obtain the restrictions on the form of the constitutive equations. The form of Darcy’s law for the fluid phase, which takes into account both Fickian and non-Fickian transport, is slightly different from the forms obtained by other researchers though all the terms have been included. When the fluid phases interact with the swelling solid porous matrix, deformation occurs. Viscoelastic large deformation of the solid matrix is investigated. A simple form of differential-integral equation is obtained for the fluid transport under isothermal conditions, which can be coupled with the deformation of the solid matrix to solve for transport in an unsaturated system. The modeling theory thus developed, which involves two-way coupling of the viscoelastic solid deformation and fluid transport, can be applied to study the processing of biopolymers, for example, soaking of foodstuffs and stress-crack predictions. Moreover, extension and modification of this modeling theory can be applied to study a vast variety of problems, such as drying of gels, consolidation of clays, drug delivery, and absorption of liquids in diapers.     

In-Situ Capillary Trapping of CO<Subscript>2</Subscript> by Co-Injection

Co-injection of water with CO₂ is an effective scheme to control initial gas saturation in porous media. A fractional flow rate of water of approximately 5–10% is sufficient to reduce initial gas saturations. After water injection following the co-injection, most of the gas injected in the porous media is trapped by capillarity with a low fractional volume of migrating gas. In this study, we first derive an analytical model to predict the gas saturation levels for co-injection with water. The initial gas saturation is controlled by the fractional flow ratio in the co-injection process. Next, we experimentally investigate the effect of initial gas saturation on residual gas saturation at capillary trapping by co-injecting gas and water followed by pure water injection, using a water and nitrogen system at room temperature. Depending on relative permeability, initial gas saturation is reduced by co-injection of water. If the initial saturation in the Berea sandstone core is controlled at 20–40%, most of the gas is trapped by capillarity, and less than 20% of the gas with respect to the injected gas volume is migrated by water injection. In the packed bed of Toyoura standard sand, the initial gas saturation is approximately 20% for a wide range of gas with a fractional flow rate from 0.50 to 0.95. The residual gas saturation for these conditions is approximately 15%. Less than approximately 25% of the gas migrates by water injection. The amount of water required for co-injection systems is estimated on the basis of the analytical model and experimental results.     

Deposition Behavior of Mud in Sand Beds Under the Effects of Organic Properties

Mud transport in sand fractures has yielded a large body of information which is used to understand the sludge-ization of tidal flats, to improve the tidal flat environment, and to design artificial tidal flats. Over the past decades, studies of particle transport in saturated and unsaturated porous media provide a very clear understanding of the processes that govern particle transport. It has been reported that the particle transport and capture in porous media are significantly governed by hydraulic pressure gradient, porosity, and permeability distribution. The objective of this research is to present a study of mud transport in saturated sand beds, aiming to delineate the effects of the organic properties of mud on the general behavior of mud transport and deposition condition in pores. Laboratory experiments were undertaken to look into the effects of the organic properties. The experiments were conducted at a flow rate of 1.3 cm³/s. Many types of muds that have differences in the organic properties were injected into different sand beds at a concentration of 120 mg/L. It was observed that the deposition condition of mud in the beds was mostly due to the organic properties of mud. Mud containing high amounts of organic matter easily remained in sand beds. Furthermore, it was found that mud deposited in the pores as clusters rather than being separately adsorbed onto the surface of soil particles.     

Nonequilibrium Effects During Spontaneous Imbibition

Accurate models of multiphase flow in porous media and predictions of oil recovery require a thorough understanding of the physics of fluid flow. Current simulators assume, generally, that local capillary equilibrium is reached instantaneously during any flow mode. Consequently, capillary pressure and relative permeability curves are functions solely of water saturation. In the case of imbibition, the assumption of instantaneous local capillary equilibrium allows the balance equations to be cast in the form of a self-similar, diffusion-like problem. Li et al. [J. Petrol. Sci. Eng. 39(3) (2003), 309–326] analyzed oil production data from spontaneous countercurrent imbibition experiments and inferred that they observed the self-similar behavior expected from the mathematical equations. Others (Barenblatt et al. [Soc. Petrol. Eng. J. 8(4) (2002), 409–416]; Silin and Patzek [Transport in Porous Media 54 (2004), 297–322]) assert that local equilibirum is not reached in porous media during spontaneous imbibition and nonequilibirium effects should be taken into account. Simulations and definitive experiments are conducted at core scale in this work to reveal unequivocally nonequilbirium effects. Experimental in-situ saturation data obtained with a computerized tomography scanner illustrate significant deviation from the numerical local-equilibrium based results. The data indicates: (i) capillary imbibition is an inherently nonequilibrium process and (ii) the traditional, multi-phase, reservoir simulation equations may not well represent the true physics of the process.     

Measurement of Capillary Pressure–Saturation Relationships Under Defined Compression Levels for Gas Diffusion Media of PEM Fuel Cells

Measurements of capillary pressure–saturation relationships under defined levels of compression for gas diffusion layers (GDL) of polymer electrolyte membrane fuel cells as thin, mixed-wettable porous media have been carried out in a newly constructed device. This article lines out the construction principle of the cell and the preconditioning procedure of the sample to measure the capillary pressure–saturation relationships under well-defined conditions and loads of compression. Three commercial GDLs (Freudenberg H2315T10A, H2315T10AC1, and SGL Carbon BA24) have been examined and a compression-depending hysteresis of the capillary pressure–saturation relationship has been measured and identified.     

Heat and mass transfer in unsaturated porous media with solid–liquid change

 A model on heat and mass transfer in unsaturated porous media with solid/liquid phase change was developed with extending the three-variable model previously proposed. The movement of air phase and its effect on the motion of water is considered. The model has been checked with comparison of the experimental results of the temperature distribution for two dimensional freezing process. The evolution of air pressure, water and ice saturation were predicted by solving the governing equations. The ice segregation and moisture movement toward the front of freezing were numerically simulated. Received on 8 December 1999     

Experimental Investigation and Modelling of Colloidal Release in Porous Media

Gas production from underground storage reservoirs is sometimes associated with solid particles eroded from the rock matrix. This phenomenon often called sand production can cause damage to the storage equipments, leading the operator to choke the wells and prevent them from producing at full capacity. Colloid release is often associated as a precursor of larger solid production. Indeed, in sandstone storage sites, clay release induced by the presence of condensed water associated with the gas production in the near-wellbore region can be a forecast of intergranular cement erosion. The objective of this work is twofold: firstly to experimentally investigate colloidal particle detachment through ionic strength reduction (absence of salinity of the condensed water) in porous media and secondly to determine its evolution with time and to model it. Laboratory experiments with model systems are developed to reproduce the particle generation and their transport in porous media. The model porous medium is a packed column of two powders: silicon carbide particles of 50 μm and silica particles of 0.5 μm (3% by weight) initially mixed together. Brine flows at different concentrations are imposed through the porous sample and, at very low salt concentration, colloid silica particles are massively released from the medium. Experimental evolutions of the particle concentration with time are compared to solutions of the advection–dispersion equation including first-order source terms for colloid release. The dispersion coefficients of the porous medium have been determined with tracer tests. The experimental results exhibit a different behaviour at short- and long-time intervals and a model has been built to predict the colloid production evolution with the introduction of two different time scales for the eroded rate. The model can be used in a core test to evaluate the amount of detachable fines and the rate of erosion.     

A Semi-Analytical Model for Two Phase Immiscible Flow in Porous Media Honouring Capillary Pressure

This article describes a semi-analytical model for two-phase immiscible flow in porous media. The model incorporates the effect of capillary pressure gradient on fluid displacement. It also includes a correction to the capillarity-free Buckley–Leverett saturation profile for the stabilized-zone around the displacement front and the end-effects near the core outlet. The model is valid for both drainage and imbibition oil–water displacements in porous media with different wettability conditions. A stepwise procedure is presented to derive relative permeabilities from coreflood displacements using the proposed semi-analytical model. The procedure can be utilized for both before and after breakthrough data and hence is capable to generate a continuous relative permeability curve unlike other analytical/semi-analytical approaches. The model predictions are compared with numerical simulations and laboratory experiments. The comparison shows that the model predictions for drainage process agree well with the numerical simulations for different capillary numbers, whereas there is mismatch between the relative permeability derived using the Johnson–Bossler–Naumann (JBN) method and the simulations. The coreflood experiments carried out on a Berea sandstone core suggest that the proposed model works better than the JBN method for a drainage process in strongly wet rocks. Both methods give similar results for imbibition processes.     

The Blocking Ability and Flowing Characteristics of Steady Foams in Porous Media

Foam flow experiments were carried out to study the influence factors such as surfactant concentration, foam quality, injection rate of liquid and gas, permeability of porous media, temperature, and oil saturation on blocking ability and flowing characteristics of steady foams in porous media. Foam blocking mechanisms and flowing characteristics were summarized according to the experimental results and foam migration behavior. The results showed that the pressure distribution of flowing foams was linearly descending in porous media at steady state. The results further showed that the foam size and quality in pores along the sand pack were almost uniform, that is, foam generation and destruction gradually reached dynamic equilibrium at steady state. In porous media, the blocking ability of steady foams increased with the concentration of the foaming agent and the increase in the permeability of porous media, but the blocking ability decreased with the increase in the temperature, the shearing rate, and the oil saturation of the porous media. Foam resistance factor reached maximal value at the foam quality of 85% in porous media.     

Trapped Gas Fraction During Steady-State Foam Flow

Trapped or stationary gas contributes significantly to the extent of gas mobility reduction for aqueous foams. Simultaneous measurements of effluent bubble sizes and trapped gas saturation in sandstone are reported for the first time. Roughly 80% of the gas saturation in an aqueous foam is stationary at steady state in this permeable porous medium. The experiments show that as gas velocity increases, the trapped gas fraction decreases. Similarly, as injected gas–liquid ratio increases, the trapped gas fraction decreases. Hence, the absolute velocities of gas and aqueous surfactant solution are fundamental to foamed-gas mobility reduction for they help determine in situ foam texture. Effluent foam bubbles range in size from 60 to 120 μm in diameter. The smaller the effluent bubble, the smaller is the fraction of mobile gas. Scaling laws from network percolation theory are used to engender a mechanistic understanding of the various parameters identified as important in the experimental program. The closed form approimation predicts that the trapped gas fraction is a weak function of pressure gradient, foam-bubble size, and the permeability of the porous medium. Moreover, the theory reproduces well the newly obtained experimental data.     

Comparison of Three FE-FV Numerical Schemes for Single- and Two-Phase Flow Simulation of Fractured Porous Media

We benchmark a family of hybrid finite element–node-centered finite volume discretization methods (FEFV) for single- and two-phase flow/transport through porous media with discrete fracture representations. Special emphasis is placed on a new method we call DFEFVM in which the mesh is split along fracture–matrix interfaces so that discontinuities in concentration or saturation can evolve rather than being suppressed by nodal averaging of these variables. The main objective is to illustrate differences among three discretization schemes suitable for discrete fracture modeling: (a) FEFVM with volumetric finite elements for both fractures and porous rock matrix, (b) FEFVM with lower dimensional finite elements for fractures and volumetric finite elements for the matrix, and (c) DFEFVM with a mesh that is split along material discontinuities. Fracture discontinuities strongly influence single- and multi-phase fluid flow. Continuum methods, when used to model transport across such interfaces, smear out concentration/saturation. We show that the new DFEFVM addresses this problem producing significantly more accurate results. Sealed and open single fractures as well as a realistic fracture geometry are used to conduct tracer and water-flooding numerical experiments. The benchmarking results also reveal the limitations/mesh refinement requirements of FE node-centered FV hybrid methods. We show that the DFEFVM method produces more accurate results even for much coarser meshes.     

Investigation of the flow stability of liquids with incipient gas bubbles in porous media

The stability of the steady-state flow regimes of a liquid with dissolved gas in a porous medium is investigated in the region of the saturation pressure. It is shown that under certain conditions periodic and stochastic self-oscillations caused by the accumulation in the porous medium and subsequent entrainment of the very small gas bubbles formed as a result of pressure reduction may arise. Experimental data that confirmed the theoretical results are presented. Ufa. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 66–73, March–April, 1994.     

基于介观结构的饱和与非饱和多孔介质有效应力

基于描述含液颗粒材料介观结构的Voronoi 胞元模型和离散颗粒集合体与多孔连续体间的介-宏观均匀化过程, 定义饱和与非饱和多孔介质有效应力. 导出了计及孔隙液压引起之颗粒体积变形的饱和多孔介质广义有效应力. 用以定义广义有效应力的Biot 系数不仅依赖于颗粒材料的多孔连续体固体骨架及单个固体颗粒的体积模量(材料参数),同时与固体骨架当前平均广义有效应力及单个固体颗粒的体积应变(状态量) 有关. 提出了描述非饱和多孔介质中非混和固体颗粒、孔隙液体和气体等三相相互作用的具介观结构的Voronoi 胞元模型.具体考虑在低饱和度下双联(binary bond) 模式的摆动(pendular) 液桥系统介观结构. 导出了基于介观水力-力学模型的非饱和多孔介质的各向异性有效应力张量与有效压力张量. 考虑非饱和多孔介质Voronoi 胞元模型介观结构的各向同性情况,得到了与非饱和多孔连续体理论中唯象地假定的标量有效压力相同的有效压力形式.但本文定义的与确定非饱和多孔介质有效应力和有效压力相关联的Bishop 参数由基于三相介观水力-力学模型, 作为饱和度、孔隙度和介观结构参数的函数导出,而非唯象假定.      

Insights into the Relationships Among Capillary Pressure,Saturation, Interfacial Area and Relative Permeability Using Pore-Network Modeling

To gain insight in relationships among capillary pressure, interfacial area, saturation, and relative permeability in two-phase flow in porous media, we have developed two types of pore-network models. The first one, called tube model, has only one element type, namely pore throats. The second one is a sphere-and-tube model with both pore bodies and pore throats. We have shown that the two models produce distinctly different curves for capillary pressure and relative permeability. In particular, we find that the tube model cannot reproduce hysteresis. We have investigated some basic issues such as effect of network size, network dimension, and different trapping assumptions in the two networks. We have also obtained curves of fluid–fluid interfacial area versus saturation. We show that the trend of relationship between interfacial area and saturation is largely influenced by trapping assumptions. Through simulating primary and scanning drainage and imbibition cycles, we have generated two surfaces fitted to capillary pressure, saturation, and interfacial area (P _c –S _w –a _nw ) points as well as to relative permeability, saturation, and interfacial area (k _r –S _w –a _nw ) points. The two fitted three-dimensional surfaces show very good correlation with the data points. We have fitted two different surfaces to P _c –S _w –a _nw points for drainage and imbibition separately. The two surfaces do not completely coincide. But, their mean absolute difference decreases with increasing overlap in the statistical distributions of pore bodies and pore throats. We have shown that interfacial area can be considered as an essential variable for diminishing or eliminating the hysteresis observed in capillary pressure–saturation (P _c –S _w ) and the relative permeability–saturation (k _r –S _w ) curves.     

Experimental study on interfacial area transport of a vertical downward bubbly flow

In relation to the development of the interfacial area transport equation, local flow measurements of vertical downward air–water flows in a pipe with an inner diameter of 50.8 mm were performed at three axial locations of z/D=6.50, 34.0, and 66.5 as well as ten radial locations from r/R=0 to r/R=0.9 using a multi-sensor probe. In the experiment, the superficial liquid velocity and the void fraction ranged from –0.620 m/s to –2.49 m/s and from 0.21% to 8.4%, respectively. The dependence of the interfacial area transport on the liquid velocity, void fraction, and bubble size is discussed in detail.     

Convective mass transfer across fluid interfaces in straight angular pores

Steady convective mass transfer to or from fluid interfaces in pores of angular cross-section is theoretically investigated. This situation is relevant to a variety of mass transport process in porous media, including the fate of residual non-aqueous phase liquid ganglia and gas bubbles. The model incorporates the essential physics of capillarity and solute mass transfer by convection and diffusion in corner fluid filaments. The geometry of the corner filaments, characterized by the fluid–fluid contact angle, the corner half-angle and the interface meniscus curvature, is accounted for. Boundary conditions of zero surface shear (‘perfect-slip’) and infinite surface shear (‘no-slip’) at the fluid–fluid interface are considered. The governing equations for laminar flow within the corner filament and convective diffusion to or from the fluid–fluid interface are solved using finite-element methods. Flow computations are verified by comparing the dimensionless resistance factor and hydraulic conductance of corner filaments against recent numerical solutions by Patzek and Kristensen (J. Colloid Interface Sci 236, 305–317 2001). Novel results are obtained for the average effluent concentration as a function of flow geometry and pore-scale Peclet number. These results are correlated to a characteristic corner length and local pore-scale Peclet number using empirical equations appropriate for implementation in pore network models. Finally, a previously published “2D-slit” approximation to the problem at hand is checked and found to be in considerable error.     

Three-Dimensional Analytical Models for Virus Transport in Saturated Porous Media

Analytical models for virus transport in saturated, homogeneous porous media are developed. The models account for three-dimensional dispersion in a uniform flow field, and first-order inactivation of suspended and deposited viruses with different inactivation rate coefficients. Virus deposition onto solid particles is described by two different processes: nonequilibrium adsorption which is applicable to viruses behaving as solutes; and colloid filtration which is applicable to viruses behaving as colloids. The governing virus transport equations are solved analytically by employing Laplace/Fourier transform techniques. Instantaneous and continuous/periodic virus loadings from a point source are examined.     

Dynamics of Viscous Entrapped Saturated Zones in Partially Wetted Porous Media

As a typical multiphase fluid flow process, drainage in porous media is of fundamental interest both in nature and in industrial applications. During drainage processes in unsaturated soils and porous media in general, saturated regions, or clusters, in which a liquid phase fully occupies the pore space between solid grains, affect the relative permeability and effective stress of the system. Here, we experimentally study drainage processes in unsaturated granular media as a model porous system. The distribution of saturated clusters is analysed by optical imaging under different drainage conditions, with pore-scale information from Voronoi and Delaunay tessellation used to characterise the topology of saturated cluster distributions. By employing statistical analyses, we describe the observed spatial and temporal evolution of multiphase flow and fluid entrapment in granular media. Results indicate that the distributions of both the crystallised cell size and pore size are positively correlated to the spatial and temporal distribution of saturated cluster sizes. The saturated cluster size is found to follow a lognormal distribution, in which the generalised Bond number (\( Bo^{*} \)) correlates negatively to the scale parameter (μ) and positively to the shape parameter (σ). With further consideration of the total surface energy obtained based on liquid–air interfaces, we were able to include additional grain-scale information in the constitutive modelling of unsaturated soils using both the degree of saturation and generalised Bond number. These findings can be used to connect pore-scale behaviour with overall hydro-mechanical characteristics in granular systems.     

Oil Displacement for One-Dimensional Three-Phase Flow with Phase Change in Fractured Media

In this article, the numerical simulations for one-dimensional three-phase flows in fractured porous media are implemented. The simulation results show that oil displacement in matrix is dominated by oil–water capillary pressure only under certain conditions. When conditions are changed to decrease the amount of water entering into the fractured media from the boundary of the flow field, water in fracture may be vaporized to superheated steam. In these cases, the appearance of superheated steam in fracture rather than in matrix will decrease the fracture pressure and generate the pressure difference between matrix and fracture, which results in oil flowing from matrix to fracture. Assuming that oil is wetting to steam, the matrix steam–oil capillary pressure will decrease the matrix oil-phase pressure as the matrix steam saturation increases. After the steam–oil capillary pressure finally exceeds the pressure difference due to the appearance of superheated steam in fracture, the oil displacement in matrix will stop. It is also shown that variations of the water relative permeability curve in matrix do not result in different mechanisms for oil displacement in matrix. The simulation results suggest that the amount of liquid water supply from the boundary of flow field fundamentally influence the mechanisms for oil displacement in matrix.