ALBERT

Description: 〈h3〉Abstract〈/h3〉 〈p〉To assess salt damage risks in building materials and geomaterials, the key components to identify are the accumulation of salts and the damage propagation. Experimental data combining both are scarce but offer an additional richness for understanding the coupling between transport and mechanics in the context of salt crystallization in porous media. Here, we quantify the drying of sodium sulfate and sodium chloride solutions from Savonnières limestone together with the damaging character of anhydrous sodium sulfate and halite precipitation, respectively. Repeated wetting–drying cycles are performed, by using salt solutions in the rewetting phase and by drying at an elevated temperature of 45 °C. The drying and deformation dynamics are characterized by means of high-resolution neutron radiography, with a moisture content resolution of 0.04 kg/m〈sup〉3〈/sup〉 and a spatial resolution of 13.5 μm/pixel. Precipitation occurs inside the specimen by treating the upper volume of the specimen hydrophobically. High Peclet numbers are found, representing a long first drying stage leading to salt accumulation in a localized zone, increasing the damage risk. In-pore crystallization of halite during drying of 5.8 molal sodium chloride solutions is particularly damaging for our type of samples. Large deformations are observed already during the first cycle, indicative of crack formation. With 1.4 molal sodium sulfate solutions, no damage is observed upon precipitating the anhydrous sodium sulfate crystal, but the drying rate decreases with every cycle due to augmented pore clogging.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉The stress dependency of the porosity and permeability of porous rocks is described theoretically by representing the preferential flow paths in heterogeneous porous rocks by a bundle of tortuous cylindrical elastic tubes. A Lamé-type equation is applied to relate the radial displacement of the internal wall of the cylindrical elastic tubes and the porosity to the variation of the pore fluid pressure. The variation of the permeability of porous rocks by effective stress is determined by incorporating the radial displacement of the internal wall of the cylindrical elastic tubes into the Kozeny–Carman relationship. The fully analytical solutions of the mechanistic elastic pore-shell model developed by combining the Lamé and Kozeny–Carman equations are shown to lead to very accurate correlations of the stress dependency of both the porosity and the permeability of porous rocks.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Gas bubbles can be generated naturally or introduced artificially in porous media. Gas bubble migration through porous media governs the rate of gas emission to the atmosphere as well as the hydraulic and mechanical properties of sediments. The migration of air bubbles through water-wet porous media of uniform geometry was studied using a glass micromodel. Experiments were conducted to measure the velocity of bubbles of various lengths rising in a glass micromodel saturated with different test liquids and varying elevation angles. The results showed a linear dependency of the average bubble velocity on the bubble length and the sine of inclination angle of the micromodel. Comparisons were made using experimental data for air bubbles rising in kerosene, Soltrol 170 and dyed white oil. The effective permeability of the micromodel for the gas bubble, 〈em〉K〈/em〉〈sub〉g〈/sub〉, was calculated for different systems at different inclination angles, assuming that the effective length for viscous dissipation is equal to the initial static bubble length. It was found that the calculated permeability of the medium for gas bubbles had an increasing trend with increasing the bubble length. To visualize the periodic nature of the flow of rising bubbles in a porous medium, the motion of the air bubbles in white oil was video recorded by a digital camera, reviewed and analyzed using PowerDVDTM11 software. The bubble shape, exact positions of the bubble front and bubble tail during motion and, hence, the dynamic bubble length were determined through image analysis. Numerical simulation was performed by modifying an existing simulation code for the rise velocity of a gas bubble and the induced pressure field while it migrates through the pore network. The results showed that the rise velocity of a gas bubble is affected by the grid size of the pore network in the direction perpendicular to the bubble migration. The findings of this study can have important implications for studies on the migration of injected gas bubbles in geoenvironmental applications, as well as fundamental studies on bubble transport and behavior in porous media.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉To celebrate the tenth anniversary of InterPore, we present an interdisciplinary review of colloid transport through porous media. This review aims to explore both classical colloid transport and topics that fall outside that purview and thus offer transformative insights into the physics governing transport behavior. First, we discuss the unique colloid characteristics relative to molecules and larger particles. Then, the classical advection–dispersion–filtration models (both conceptual and mathematical) of colloid transport are introduced as well as anomalous transport behaviors. Next, the forces of interaction between colloids and porous media surfaces are discussed. Fourth, applications that are interested in maximizing the transport of colloids through porous media are considered. Then the concept of motile, active biocolloids is introduced, and finally, colloid swarming as a newly recognized mode of transport is summarized.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Natural convection in a porous enclosure in the presence of thermal dispersion is investigated. The Fourier–Galerkin (FG) spectral element method is adapted to solve the coupled equations of Darcy’s flow and heat transfer with a full velocity-dependent dispersion tensor, employing the stream function formulation. A sound implementation of the FG method is developed to obtain accurate solutions within affordable computational costs. In the spectral space, the stream function is expressed analytically in terms of temperature, and the spectral system is solved using temperature as the primary unknown. The FG method is compared to finite element solutions obtained using an in-house code (TRACES), OpenGeoSys and COMSOL Multiphysics〈sup〉®〈/sup〉. These comparisons show the high accuracy of the FG solution which avoids numerical artifacts related to time and spatial discretization. Several examples having different dispersion coefficients and Rayleigh numbers are tested to analyze the solution behavior and to gain physical insight into the thermal dispersion processes. The effect of thermal dispersion coefficients on heat transfer and convective flow in a porous square cavity has not been investigated previously. Here, taking advantage of the developed FG solution, a detailed parameter sensitivity analysis is carried out to address this gap. In the presence of thermal dispersion, the Rayleigh number mainly affects the convective velocity and the heat flux to the domain. At high Rayleigh numbers, the temperature distribution is mainly controlled by the longitudinal dispersion coefficient. Longitudinal dispersion flux is important along the adiabatic walls while transverse dispersion dominates the heat flux toward the isothermal walls. Correlations between the average Nusselt number and dispersion coefficients are derived for three Rayleigh number regimes.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Estimating the fluid imbibition flow in natural system composed of nanopores is challenging due to the strong fluid/rock molecular scale interaction and the invalidation of the macroscopic thermodynamics treatment. We develop an analytical model for Lennard-Jones fluid imbibition into an organic nanopore considering the phase transition and fluid/rock intermolecular interactions. In addition, we apply the proposed model on octane molecules imbibition into 1–10 nm slit-shape graphite nanopores under the standard and shale reservoir condition. Predicted velocity and density profiles of 2 nm model at the standard condition show that octane molecules first imbibe as vapor phase at around 200–300 m/s and form adsorbed layers near the pore wall. Velocity and density profiles are compared with the molecular dynamic simulation results. Calculated mean velocities of the analytical model and simulation are around 10〈sup〉3〈/sup〉–10〈sup〉4〈/sup〉 of those predicted by classical models, which are similar with previous experimental results. Reservoir condition results show octane can fast flow only when the driving pressure is greater than 0.12 MPa when the initial reservoir pressure is 5.72 MPa. Particularly, the impact of the fluid phase transition on the imbibition rate is significant in organic nanopores.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉The first experimental evidence of polymer adsorption in natural carbonates obtained from streaming potential measurements is presented. The surface probing of the Estaillades core was realized in a highly saline environment with 〈em〉I〈/em〉 ~ 3 M similar to formation water naturally found in some oil reservoirs, which are typical target for enhanced oil recovery (EOR) by Smart Water Injection Methods (SWIM) and Hybrid SWIM. A highly sensitive setup with an exceptional signal-to-noise ratio even at high salinity, allowing to discriminate voltage fluctuations below 0.1 mV, was used to probe the streaming potential coupling coefficient. The injection of partially hydrolyzed polyacrylamide Flopaam 3130S was followed by chase flooding with formation water for several pore volumes, to assure that any changes measured after the polymer flooding are due to the retained polymer. We show that the streaming potential coupling coefficient changes sign after polymer adsorption. This result not only is an unambiguous evidence for polymer adsorption but has implications on chemical EOR scenarios that need to consider wettability changes due to inversion of surface charge.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Understanding mechanism of transmitted and stored heat in porous materials was extremely important for improving thermal protective performance of clothing. A coupled heat and moisture transfer model in a three-layer fabric system while exposing to a low-level thermal radiation was developed in this study. The model simulated the transmitted and stored heat in porous materials, and considered the effect of moisture transport on the transmitted and stored heat. The predicted results from the coupled model were validated with the experimental results, and compared with the predicted results from the previous model without considering the moisture effect. It was found that the prediction accuracies in skin burn and skin temperature through the coupled model were further improved. The coupled model was used to examine the moisture effect on heat transport and storage in porous materials. The results demonstrated that the moisture within porous materials increased the heat storage and discharge, but decreased the heat transport. The increases in initial moisture content and fiber moisture regain, while increasing the thermal hazardous effect, greatly enhanced the thermal protective performance of clothing. Therefore, it suggested that the moisture management in porous materials was a key consideration for thermal functional design of fabric.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉A high-fidelity physics-based model of mixed-gas transport coupled with kinetic and equilibrium adsorption is derived, and experiments were performed in order to calibrate and exercise the model. In the literature, a continuum-scale model that couples Fickian diffusion with Henry’s law absorption, and kinetic Langmuir adsorption was previously developed to describe the diffusion and sorption of moisture in porous materials. Here, we expand the model to gases, rather than moisture, derive, and implement a competitive adsorption mechanism into the model to enable mixed-gas sorption. This model facilitates a mechanistic-based understanding of the sorption and diffusion processes of mixed gases in polymeric materials. Diffusion and sorption experiments were conducted for a range of partial pressures; model validation and calibration were carried out by comparing modeled mass uptake and experimental data considering the uncertainties of conceptualized (or assumed) physical processes and system parameters. Uncertainty quantification and sensitivity analysis methods are described and exercised here to demonstrate the capability of this predictive model.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉The RANS modelling of turbulence across fluid-porous interface regions within ribbed channels has been investigated by applying double (both volume and Reynolds) averaging to the Navier–Stokes equations. In this study, turbulence is represented by using the Launder and Sharma (Lett Heat Mass Transf 1:131–137, 1974) low-Reynolds-number 〈span〉 〈span〉\(k-\varepsilon \)〈/span〉 〈/span〉 turbulence model, modified via proposals by either Nakayama and Kuwahara (J Fluids Eng 130:101205, 2008) or Pedras and de Lemos (Int Commun Heat Mass Transf 27:211–220, 2000), for extra source terms in turbulent transport equations to account for the porous structure. One important region of the flow, for modelling purposes, is the interface region between the porous medium and clear fluid regions. Here, corrections have been proposed to the above porous drag/source terms in the 〈em〉k〈/em〉 and 〈span〉 〈span〉\(\varepsilon \)〈/span〉 〈/span〉 transport equations that are designed to account for the effective increase in porosity across a thin near-interface region of the porous medium, and which bring about significant improvements in predictive accuracy. These terms are based on proposals put forward by Kuwata and Suga (Int J Heat Fluid Flow 43:35–51, 2013), for second-moment closures. Two types of porous channel flows have been considered. The first case is a fully developed turbulent porous channel flow, where the results are compared with DNS predictions obtained by Breugem et al. (J Fluid Mech 562:35–72, 2006) and experimental data produced by Suga et al. (Int J Heat Fluid Flow 31:974–984, 2010). The second case is a turbulent solid/porous rib channel flow to examine the behaviour of flow through and around the solid/porous rib, which is validated against experimental work carried out by Suga et al. (Flow Turbul Combust 91:19–40, 2013). Cases are simulated covering a range of porous properties, such as permeability and porosity. Through the comparisons with the available data, it is demonstrated that the extended model proposed here shows generally satisfactory accuracy, except for some predictive weaknesses in regions of either impingement or adverse pressure gradients, associated with the underlying eddy-viscosity turbulence model formulation. 〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Various processes such as heterogeneous reactions or biofilm growth alter a porous medium’s underlying geometric structure. This significantly affects its hydrodynamic parameters, in particular the medium’s effective permeability. An accurate, quantitative description of the permeability is, however, essential for predictive flow and transport modeling. Well-established relations such as the Kozeny–Carman equation or power law approaches including fitting parameters relate the porous medium’s porosity to a scalar permeability coefficient. Opposed to this, upscaling methods directly enable calculating the full, potentially anisotropic, permeability tensor. As input, only the geometric information in terms of a representative elementary volume is needed. To compute the porosity–permeability relations, supplementary cell problems must be solved numerically on this volume and their solutions must be integrated. We apply this approach to provide easy-to-use quantitative porosity–permeability relations that are based on representative single grain, platy, blocky, prismatic soil structures, porous networks, and real geometries obtained from CT-data. As a discretization method, we use discontinuous Galerkin method on structured grids. To make the relations explicit, interpolation of the obtained data is used. We compare the outcome with the well-established relations and investigate the ranges of the validity. From our investigations, we conclude whether Kozeny–Carman-type or power law-type porosity–permeability relations are more reasonable for various prototypic representative elementary volumes. Finally, we investigate the impact of a microporous solid matrix onto the permeability.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Growing plants under microgravity conditions in a space ship is essential for future long-term missions to supply needs for food and oxygen. Although plant growth modules for microgravity have been developed and tested for more than 40 years, creating optimal saturation conditions for plant growth in the absence of gravity still remains a challenge. In this study, we present results from a series of spontaneous imbibition experiments designed to approximate microgravity conditions by using density-matched fluid pairs. Porous media with patterned wettability characteristics are used to manipulate macroscopic fluid saturation and microscopic fluid interfacial configurations. These are compared to an additional experiment under Earth gravity, wherein we observe spontaneous imbibition of water into common potting soil. Patterning grains of different wettabilities under microgravity conditions proves to be an effective method to manipulate spatial distributions and saturations of fluids. These wettability patterns can be optimised to fine-tune residual fluid characteristics, e.g. non-wetting phase saturation, connectivity and interfacial area. Furthermore, we present tomographic evidence supporting previous work which was suggesting enhanced snap-off and disconnection of the gas phase in porous media under microgravity. 〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Suspension flow through porous medium was studied using the Stokesian dynamics simulation method. Stokesian dynamics is an efficient tool to carry out numerical simulations for suspension of rigid particles interacting through hydrodynamic and non-hydrodynamic forces. After validating the simulation method for a single particle flowing through an array of fixed grain particles, we have analysed the suspension transport through porous medium. It was observed that the hydrodynamic interactions and the inter-particle non-hydrodynamic forces between the moving and fixed grain particles have a strong influence on the particle trajectories. This was apparent from the change in particle flux with the fractional channel width in the presence of non-hydrodynamic forces. Hydrodynamic interaction between the suspension and grain particles was also studied for large-scale porous system that was generated by a random arrangement of the particles in a periodic cell. It was found that the change in porosity leads to change in the average fluctuation velocity of the suspension. The fluctuation velocity was observed to vary linearly with the particle concentration and average suspension velocity. Finally, a comparative study was performed with suspension flow in a straight channel and it was observed that the shear-induced particle migration in porous medium is altered by the presence of grain particles.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉The onset of convection in a porous layer saturated by a power-law fluid is here investigated. The walls are considered to be isothermal, isobaric and permeable in such a way that a vertical throughflow is described. The threshold for a buoyancy-driven cellular flow is investigated by means of a linear stability analysis. This study consists in introducing disturbances with small amplitude. The disturbances are plane waves, i.e. a normal modes stability analysis of the basic stationary solution is performed. The resulting problem is an ordinary differential equation eigenvalue problem which is solved numerically by coupling the Runge–Kutta method with the shooting method. Results are presented in the form of marginal stability curves and their critical points representing the values of the control parameters such that the growth rate of the disturbances is zero. It is found that, among roll disturbances, the most unstable modes are stationary and uniform with infinite wavelength. For this reason, an asymptotic analysis for vanishing wave numbers is carried out. The results of this asymptotic analysis are obtained analytically displaying a very good agreement with the numerical solution. It is found that vertical throughflow plays a destabilising role for pseudoplastic fluids and a stabilising role for dilatant fluids.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉We review fundamental aspects of linear poro-elasticity. In contrast to most available textbooks and review articles, our treatment of poro-elastic media is based on the continuum Mixture Theory. Kinematic state variables and dynamic variables are introduced and formally linearized before the fundamental constitutive relations, between pairs of these, are extensively discussed. The role of porosity in linear poro-elasticity is highlighted, and it is shown that porosity is one of the possible choices for one of the two kinematic state variables, and therefore, relations to alternative pairs of kinematic variables can be formulated. The treatment is concluded by the formulation of the governing set of partial differential equations that constitute the basis for analytical or numerical investigations of boundary value problems.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Capillary pressure saturation (〈em〉P〈/em〉〈sub〉c〈/sub〉–〈em〉S〈/em〉〈sub〉w〈/sub〉) relationship plays a central role in the description of fluid flow in porous media. In this research, the light transmission visualization (LTV) technique was applied to characterize the 〈em〉P〈/em〉〈sub〉c〈/sub〉–〈em〉S〈/em〉〈sub〉w〈/sub〉 relationship in a double-porosity medium. Four experiments were conducted in two-dimensional (2-D) flow chambers packed with a double-porosity medium composed of a mixture of silica sand and sintered kaolin clay spheres. In each experiment, a different volumetric fraction of macropores and micropores was used. The experiment was also repeated by compacting the flow chamber with silica sand only to represent single-porosity medium. Variable saturations of water across the height of the system were applied by controlling the capillary pressure. Images of the 2-D model were collected using a digital camera and analyzed pixel by pixel to determine water saturation in the double-porosity medium. Results from the LTV technique showed that the 〈em〉P〈/em〉〈sub〉c〈/sub〉–〈em〉S〈/em〉〈sub〉w〈/sub〉 relationships for all experiments in double-porosity soil medium were similar in shape but varied depending on the porous media composition. Comparison with the pressure cell test results showed that the 〈em〉P〈/em〉〈sub〉c〈/sub〉–〈em〉S〈/em〉〈sub〉w〈/sub〉 curves for all experiments consistent comparable to those obtained by the LTV technique. The 〈em〉P〈/em〉〈sub〉c〈/sub〉–〈em〉S〈/em〉〈sub〉w〈/sub〉 curves were also fit to van Genuchten model for comparison and validation. For double-porosity media, the best-fit parameters were consistent with published data for sandy clay. Moreover, little variability was observed in the best-fit 〈em〉α〈/em〉 and 〈em〉n〈/em〉 values for the different double porosity. Overall, this study proves that the LTV technique is a noninvasive laboratory tool that can provide high-resolution spatial data for water saturation distribution in different types of porous media and is capable of producing highly resolved 〈em〉P〈/em〉〈sub〉c〈/sub〉–〈em〉S〈/em〉〈sub〉w〈/sub〉 relationships. 〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉 In this attractive paper (Battiato et al. in Transp Porous Media, 〈span〉2019〈/span〉), the authors review different methods of upscaling heterogeneous media descriptions to continuous macroscopic equivalent descriptions. I would like to introduce some comments to complete the presentation of the homogenisation theory in Section 7. 〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉The two key parameters of the Brinkman’s model for fluid flow in porous media—permeability and effective viscosity—are determined theoretically. The analytical solution of a 1D problem for the Brinkman equation for the fluid flow in a porous medium between two solid walls is compared with results of the numerical Stokes fully resolved flow model in a two-dimensional periodic cells containing solid inclusions. The boundary value problem for the Stokes flow is solved using the boundary element method. The dependencies of the permeability and effective viscosity on porosity for various types of configurations and shapes of solid inclusions are numerically studied. The product of permeability and effective viscosity is introduced as a new parameter. It is shown that this parameter is almost constant for a wide range of values of porosity excluding the small interval close to the unity value. The analytical estimation confirms the numerically obtained values of the parameter. The obtained dependencies are verified by the solution of the 2D Brinkman flow problem. It is shown that the permeability and effective viscosity determined give good agreement between the pressure fields and flow streamlines of the exact solution of the two-dimensional Brinkman equation and the numerical solution of the Stokes equations. The approximations for the dependencies of permeability and effective viscosity on the porosity are constructed on the basis of the numerical data obtained. The applicability of the Brinkman and Darcy models is discussed.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉The problem of natural convection in an inclined square cavity filled with a fluid-saturated porous medium with sinusoidal temperature distribution on the side walls is presented, and the numerical results are obtained. Dimensionless equations governing the mathematical model together with the boundary conditions are obtained, and the problem is solved by finite differential method of second-order accuracy. The numerical results of streamlines and isotherms are investigated, and the effect of different inclination angles of the cavity, amplitude ratio of the sinusoidal temperature, and phase deviation are discussed. The cavity inclination angle and the periodic thermal boundary conditions are proposed to be control parameters for heat and fluid flow inside the cavity. Interesting results for these new parameters have been obtained, such as, the behaviour of the convective cells is completely changed compared with no inclination of the cavity and the increase in inclination angle determines an increase in the average Nusselt number and fluid flow rate. The results obtained for no inclination of the cavity are compared and successfully validated with previous reported results of the literature.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉The dispersion process in particulate porous media at low saturation levels takes place over the surface elements of constituent particles and, as we have found previously by comparison with experiments, can be accurately described by superfast nonlinear diffusion partial differential equations. To enhance the predictive power of the mathematical model in practical applications, one requires the knowledge of the effective surface permeability of the particle-in-contact ensemble, which can be directly related with the macroscopic permeability of the particulate media. We have shown previously that permeability of a single particulate element can be accurately determined through the solution of the Laplace–Beltrami Dirichlet boundary value problem. Here, we demonstrate how that methodology can be applied to study permeability of a randomly packed ensemble of interconnected particles. Using surface finite element techniques, we examine numerical solutions to the Laplace–Beltrami problem set in the multiply-connected domains of interconnected particles. We are able to directly estimate tortuosity effects of the surface flows in the particle ensemble setting.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉The fourth-order Darcy–Bénard eigenvalue problem for onset of thermal convection in a 2D rectangular porous box is investigated. The conventional type of solution has normal-mode dependency in at least one of the two spatial directions. The present eigenfunctions are of non-normal-mode type in both the horizontal and the vertical direction. A numerical solution is found by the finite element method, since no analytical method is known for this non-degenerate fourth-order eigenvalue problem. All four boundaries of the rectangle are impermeable. The thermal conditions are handpicked to be incompatible with normal modes: The lower boundary and the right-hand wall are heat conductors. The upper boundary has given heat flux. The left-hand wall is thermally insulating. The computed eigenfunctions have novel types of complicated cell structures, with intricate internal cell walls.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Based on microfocus X-ray computed tomography analyses, the relationship between the characteristics of pore structure and fluid flow behavior was investigated through the single-phase flow experiment. The macroscopic bulk fluid seepage behavior was fundamentally explained by the microscale flow mechanism considering all the porous microstructure analysis of artificial cores. The results indicate that the pore size and connectivity have significant effects on the initial permeability of the rock cores. The permeabilities obtained from different methods have a linear law relationship with porosities in tested artificial cores. The results also suggest that the permeability of core decreases exponentially with the increase in effective stress. The inner different pore structures have an important influence on the stress-dependent fluid flow in synthetic porous rocks. The polynomial equation yields well fittings of the artificial core which has the poor pore sorting characteristic. The permeabilities of the artificial core are more significantly affected by changes in the low effective stress range. The larger the pore channel of the artificial core is, the greater influence on permeability the pressure will have. The stress sensitivity of artificial core increases as the grain diameter decreases. The heterogeneous coefficient and the stress sensitivity of permeability are positively correlated.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉There have been several foam field applications in recent years. Foam treatments targeting gas mobility control in injectors as well as gas blocking in production wells have been performed without causing operational problems. The most widely used injection strategy of foam has been injecting alternating slugs of surfactant in brine with gas injection. This procedure seems to be beneficial as injection is easy to perform and control below fracturing pressure. Simultaneous injection of surfactant solution and gas may give difficulties, especially with interpretation of the tests, if fracturing pressure are exceeded during the injection period. This paper reviews critical aspects of foam for reservoir applications and intends to motivate for further field trials. Key parameters for qualification of foam are: foam generation, propagation in porous medium, foam strength and stability of foam. Stability is discussed, especially in the presence of oil at reservoir conditions. Data on each of these topics are included, as well as extracted summary of relevant literature. Experimental studies have shown that foam is generated at low surfactant concentration even below the CMC (critical micelle concentration). Results indicate that in situ foam generation in porous medium may depend on available nucleation sites. In situ generation of foam is complex and has been found to be especially difficult in oil wet carbonate rocks. Foam propagation in porous medium has been summarized, and propagation rate for a given experiment seems to be constant with time and distance. Laboratory studies confirm a propagation rate of 1–3 m/day. Field tests performed have not given reliable information of foam propagation rate, and future field pilots are encouraged to include observation wells in order to gain information of field-scale foam propagation. Foam strength is generally high with all gases. The exception is CO〈sub〉2〈/sub〉 at high pressure where CO〈sub〉2〈/sub〉 becomes supercritical. Stability of foam has been studied in laboratory and field tests, and has confirmed long-term stability of foam.〈/p〉

Description: 〈p〉The original publication of the article includes the error in Fig. 9 legend.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Asymptotic multiple scale homogenisation allows to determine the effective behaviour of a porous medium by starting from the pore-scale description, when there is a large separation between the pore-scale and the macroscopic scale. When the scale ratio is “small but not too small”, the standard approach based on first-order homogenisation may break down since additional terms need to be taken into account in order to obtain an accurate picture of the overall response of the medium. The effect of low scale separation can be obtained by exploiting higher-order equations in the asymptotic homogenisation procedure. The aim of the present study is to investigate higher-order terms up to the third order of the advective–diffusive model to describe advection–diffusion in a macroscopically homogeneous porous medium at low scale separation. The main result of the study is that the low separation of scales induces dispersion effects. In particular, the second-order model is similar to the most currently used phenomenological model of dispersion: it is characterised by a dispersion tensor which can be decomposed into a purely diffusive component and a mechanical dispersion part, whilst this property is not verified in the homogenised dispersion model (obtained at higher Péclet number). The third-order description contains second and third concentration gradient terms, with a fourth-order tensor of diffusion and with a third-order and an additional second-order tensors of dispersion. The analysis of the macroscopic fluxes shows that the second- and the third-order macroscopic fluxes are distinct from the volume averages of the corresponding local fluxes and allows to determine expressions of the non-local effects.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Pore network models (PNMs) offer a computationally efficient way to analyse transport in porous media. Their effectiveness depends on how well they represent the topology and geometry of real pore systems, for example as imaged by X-ray CT. The performance of two popular algorithms, maximum ball and watershed, is evaluated for three porous systems: an idealised medium with known pore throat properties and two rocks with different morphogenesis—carbonate and sandstone. It is demonstrated that while the extracted PNM simulates simple flow (permeability) with acceptable accuracy, their topological and geometric properties are significantly different. This suggests that such PNM may not serve more complex studies, such as reactive/convective transport of contaminants or bacteria, and further research is necessary to improve the interpretation of real pore spaces with networks. Linear topology–geometry relations are derived and presented to stimulate development of more realistic PNM.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Understanding the fate of colloids in porous media plays a vital role in applications such as enhanced oil recovery (EOR), colloid-facilitated transport or groundwater remediation. Although considerable research has been dedicated toward the retention characteristics of colloids over the last couple of years, less attention has been paid to their transport characteristics in porous media, and hence little knowledge is available on how colloids flow through a medium of complex channel alignment. This lack of understanding becomes apparent for nanoparticle applications in EOR, where relatively concentrated nanofluids are injected into low-permeable sedimentary rocks. Here, two flow phenomena arise: an earlier breakthrough of nanoparticles compared to a conservative tracer and an apparent slip effect, by which the measured pressure drop is less than that calculated via Darcy’s law. The underlying mechanism that couples both phenomena is typically attributed to the depleted layer effect—a theory that presumes that the nanoparticles are excluded from the low-velocity region near a wall at a distance typically larger than the nanoparticles. This depletion causes the particles to move on average faster than the bulk fluid containing them and causes that fluid to exhibit a viscosity lower than when measured in a viscometer. However, the depleted layer theory is not predicated on any direct observation during flow in porous media. In a broader sense, this review paper not only aims at finding answers as to why particles drive away from a wall so that a depleted layer occurs. It also questions the aforementioned hypothesis that early breakthrough and reduced apparent viscosity can be interrelated, and assesses whether each phenomenon should be treated independently. In a strict sense, this review presents both potential mechanisms of particles flow enhancement and how its manifestation can be hindered, as well as opportunities for particle suspensions to experience enhanced flow as a whole. It is meant to be an overview for readers who are not familiar with the characteristic transport features of particles. Accordingly, this review highlights also outstanding challenges in allocating the underlying mechanisms behind the flow enhancement phenomena and shows thereby future research opportunities.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Permeability, one of the most significant parameters for CBM, is affected by many influencing factors. In this paper, an improved fully coupled permeability model was proposed to investigate the influence of the effective stress, adsorption, diffusion and variable Klinkenberg’s effect on the evolution of permeability. Then, the permeability model is validated by comparing with the experimental data in previous paper. Then, a series of cases were carried out to analyze the influence of various factors on the permeability evolution. Results showed that when the adsorption layer thickness is considered, the effective pore radius exhibited the remarkable difference. With the rising pore pressure, the adsorption layer thickness and sorption-induced swelling deformation increase, leading to the reduction in effective pore radius and thus the depression in gas permeability. Diffusion and Klinkenberg’s effect exhibit the remarkable enhancement effect on the permeability, especially at the low pore pressure conditions. It is noticed that compared with the previous constant Klinkenberg’s factor, this work constructed a variable Klinkenberg’s factor, which is affected by the effective pore radius. Finally, we constructed a governing equation for gas transport in coal seam to reveal the pressure evolution and the contribution of different influencing factors to the total permeability.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Reaction–diffusion processes in multiscale catalytic porous media are found in a wide range of scientific areas as, for example, electrochemical energy conversion and storage devices, geological systems and bioengineering. The dependency of effective transport properties on reaction rate has been long debated in the literature, and traditionally ignored in emerging fields, such as polymer electrolyte fuel cells (PEFCs). In this work, a 1D upscaling method is presented to evaluate the effective properties (effective diffusivity and catalyst utilization) of PEFC catalyst layers featuring first-order kinetics. Unlike Whitaker’s closure method, the present algorithm is easy to implement and well suited for porous media with arbitrarily complex 3D geometries. The numerical results show that the normalized effective diffusivity and catalyst utilization are not passive geometrical properties but are influenced by the reaction–diffusion coupling when the Thiele modulus is higher than 1. This effect can be important at high current densities in the cathode catalyst layer of state-of-the-art PEFCs.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉To characterise the sorption behaviour of wood or other hygroscopic materials with the use of water vapour sorption experiments, an accurate understanding of water vapour transport both to and through the sample material is necessary. Within the last decades, various modelling approaches on the sorption kinetics were developed, but there seems to be no general agreement on the relevance of water vapour transport. Using small amounts of grained wood, this study aims to estimate the influence of water vapour transport in sorption experiments on small sample sizes. A theoretical analysis based on a diffusion equation including a simplified sink/source term was carried out and compared with corresponding experiments. Nonlinear isotherm and temperature were shown to have a major impact on the effective diffusion of water vapour, whereas step size in humidity seems to mainly influence processes within the cell wall (e.g. relaxation and reorganisation of microstructure). Considering the paths of stagnant air above sample surface, the anomalous behaviour for a variation in sample thickness of grained wood can be explained. In conclusion, water vapour transport appreciably influences the sorption kinetics of small sample sizes and has to be considered in the modelling and interpretation of water vapour sorption experiments.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉TOUGH + Millstone has been developed for the analysis of coupled flow, thermal and geomechanical processes associated with the formation and/or dissociation of CH4-hydrates in geological media. It is composed of two constituent codes: (a) a significantly enhanced version of the TOUGH + HYDRATE simulator, V2.0, that accounts for all known flow, physical, thermodynamic and chemical processes associated with the behavior of hydrate-bearing systems undergoing changes and includes the most recent advances in the description of the system properties, coupled seamlessly with (b) Millstone V1.0, a new code that addresses the conceptual, computational and mathematical shortcomings of earlier codes used to describe the geomechanical response of these systems. The capabilities of TOUGH + Millstone are demonstrated in the simulation and analysis of the system flow, thermal and geomechanical behavior during gas production from a realistic complex offshore hydrate deposit. In the first paper of this series, we discuss the physics underlying the T + H hydrate simulator, the constitutive relationships describing the physical, chemical (equilibrium and kinetic) and thermal processes, the states of the 〈span〉 〈span〉\({\hbox {CH}}_4 + {\hbox {H}}_2 \hbox {O}\)〈/span〉 〈/span〉 system and the sources of critically important data, as well as the mathematical approaches used for the development of the of mass and energy balance equations and their solution. Additionally, we provide verification examples of the hydrate code against numerical results from the simulation of laboratory and field experiments.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉The viscosity exhibited by shear-thinning fluids within the interstices of a porous medium differs depending on pore dimensions and injection velocity. Therefore, predicting the macroscopic value of viscosity required as input to Darcy’s law is challenging and needs accurate identification of the characteristic microscopic dimensions dominating global pressure losses. The most common approach consists of defining an apparent “in situ” shear rate which can be used in the bulk constitutive equation of the fluid to predict viscosity during flow through the porous medium. The dependence of this apparent shear rate on Darcy velocity has traditionally been assumed to be linear, which is appropriate in the case of Newtonian fluids and power-law fluids. However, yield stress and plateau viscosities that can potentially affect such dependence are not captured by power-law model, so the linear assumption may lead to inaccurate viscosity predictions. For this reason, a set of two-dimensional (2D) flow problems were considered and solved numerically to assess the effects of the shear rheology model, the pore size distribution and the microstructural complexity on the value of the apparent shear rate. In order to facilitate the analysis, the microscopic features of all the investigated porous media were well characterized through pore network modelling. The present results prove the inability of traditional approaches to predict the macroscopic viscous pressure losses generated during the creeping flow of widely used Herschel–Bulkley and Carreau fluids. In particular, the existence of a yield stress or a plateau viscosity induces significant deviations from linearity in the relationship between apparent shear rate and Darcy velocity. The importance of these deviations is, in turn, shown to be highly affected by the dispersion of the pore size distribution and the degree of shear thinning. Moreover, the reasons for such observations are discussed using an analytical approach. 〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉We study the upscaling of advective pore-scale dispersion in terms of the Eulerian velocity distribution and advective tortuosity, both flow attributes, and of the average pore length, a medium attribute. The stochastic particle motion is modeled as a time-domain random walk, in which particles move along streamlines in equidistant spatial steps with random velocities and thus random transition times. Particle velocities describe stationary spatial Markov processes, which evolve along streamlines on the mean pore length. The streamwise motion is projected onto the mean flow direction using tortuosity. This upscaled stochastic particle model predicts accurately the (non-Fickian) transport dynamics obtained from direct numerical simulations of particle transport in a three-dimensional digitized Berea sandstone sample. It captures all aspects of transport and sheds light on the dependence of the upscaled transport behavior on the flow heterogeneity and the initial particle distribution, which are critical for the accurate modeling of dispersion from the pre-asymptotic to asymptotic regimes.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Estimating the values of dispersion and biochemical reaction rates of heterogeneous aquifers is critical to predicting the temporal evolution and fate of reactive solutes. While previous studies have investigated field-scale heterogeneity of transport and biochemical properties of porous media, effects of local dispersion have not been well understood. In this paper, longitudinal macro-dispersivity, effective decay rate, and effective solute velocity are derived for a stratified aquifer, and the effects of local dispersion, especially the local transverse dispersion, are studied. It is shown that the inclusion of local transverse dispersion leads to enlarged effective decay rate, and that ignoring it may significantly underestimate the effective rate. The Damkohler (〈em〉Da〈/em〉) number and the coefficient of variation (CV) of decay rate have slight influence to macro-coefficients under very small 〈em〉Pe〈/em〉 number (with large local transverse dispersion). However, 〈em〉Da〈/em〉 number has growing effect on the asymptotic effective decay rate with the decrease in 〈em〉Pe〈/em〉 number, and results in constant asymptotic values regardless of 〈em〉Da〈/em〉 number under the condition with very large 〈em〉Pe〈/em〉 number. Larger CV of decay rate leads to smaller effective decay rate and effective velocity, and longitudinal macro-coefficient. The longitudinal macro-dispersivity is found to depend on the correlation between the hydraulic conductivity and the decay rate if the local longitudinal dispersion is spatially variable.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉In order to study gas transport properties of fractured shale gas reservoirs for the accurate estimation of shale gas production, a new multiscale fractal transport model with an effective porosity model was proposed based on the fractal theory and the multilayer fractal Frenkel–Halsey–Hill (FHH) adsorption. In shale matrix, both fractal microstructures of pores (such as pore size distribution, flow path tortuosity, and pore surface roughness) and multiscale flow mechanisms (including slip flow and Knudsen diffusion) were coupled. In fracture network, fractal fracture length distribution, stress compaction, and gas pressure were introduced to formulate a new fracture permeability model. These permeability and effective porosity models were then incorporated into the governing equations of gas flow and the deformation equation of reservoirs to form a numerical model. This numerical model was solved within COMSOL Multiphysics for shale gas recovery. Both transport models in shale matrix and fracture network were validated by experimental data or compared with other models. Finally, sensitivity analysis was conducted to identify key parameters to gas recovery enhancement. It was found that the multilayer gas adsorption and fractal microstructures have great impacts on gas production in shale reservoirs. The cumulative gas production can be increased by 26% after 8000 days when the multilayer adsorbed gas is considered. Larger surface fractal dimension and larger tortuosity fractal dimension represent more roughness pore surface, higher flow resistance, and lower cumulative gas production. Bigger pore diameter fractal dimension means more pores, higher permeability, and higher cumulative gas production. Our model with fractal FHH adsorption was in better agreements with field data from Marcellus and Barnett shale reservoirs than other models.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉X-ray microtomography (〈span〉 〈span〉\(\upmu \hbox {CT}\)〈/span〉 〈/span〉) scanning provides high-resolution images in applications ranging from medical to material sciences and failure analysis. In general, CT scanning relies on X-ray absorption to produce a 3D computed image of the material. In Earth Sciences, 〈span〉 〈span〉\(\upmu \hbox {CT}\)〈/span〉 〈/span〉 scans are used to characterize porosity and pore size, shape and topology of rock samples. For sufficiently large pore systems, the resulting segmented images may be used for quantitative transport calculations. In this note, we infer the limitations of 〈span〉 〈span〉\(\upmu \hbox {CT}\)〈/span〉 〈/span〉 images of rock samples, caused by attainable resolution for a representative sample size. To this end, (1) we perform a systematic analysis with the aid of a resolution chart, (2) we present example scans of an Indiana limestone and a Berea sandstone mini-cores, and (3) we process and analyze the images to extract pore structures using different segmentation algorithms. Porosity estimates inferred from 〈span〉 〈span〉\(\upmu \hbox {CT}\)〈/span〉 〈/span〉 images tend to be lower than bulk measurements.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉A mathematical model is developed based on the empirical power law equation for post-laminar flow through porous media. Hydraulic conductivity and the critical Reynolds number are used as boundary conditions. The developed model can predict hydraulic gradients for specific velocities, irrespective of the media sizes or porosities, over the complete flow transition. Therefore, the model can be very useful to recognise the specific flow regime or to predict the velocity and hydraulic gradient for a given flow regime. A parametric study is carried out concerning the behaviour of binomial (Forchheimer) and power law (Izbash and Wilkins) coefficients subjected to different media sizes, porosities and flow regimes. The observed behaviour of Forchheimer and Izbash coefficients with different media sizes and porosities are similar to the experimental results reported in the literature. However, the values of these coefficients differ when subjected to different flow regimes for any specific packing. The ratios of non-Darcy and Darcy coefficients of the Forchheimer equation suggest an increasing influence of inertia towards the turbulent regime. The maximum and minimum values of 〈em〉β〈/em〉 are found to be 1.38 and 0.69 for laminar and turbulent regime, respectively. However, these values are found to be unaffected by the media size and porosity variation. The value of Wilkins coefficient 〈em〉w〈/em〉 in the laminar regime is found to be 4841.72 for all media sizes and porosities. However, the coefficient represents a decreasing variation trend towards the turbulent regime which is also dependent of the media size variation.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Charged porous media are pervasive, and modeling such systems is mathematically and computationally challenging due to the highly coupled hydrodynamic and electrochemical interactions caused by the presence of charged solid surfaces, ions in the fluid, and chemical reactions between the ions in the fluid and the solid surface. In addition to the microscopic physics, applied external potentials, such as hydrodynamic, electrical, and chemical potential gradients, control the macroscopic dynamics of the system. This paper aims to give fresh overview of modeling pore-scale and Darcy-scale coupled processes for different applications. At the microscale, fundamental microscopic concepts and corresponding mass and momentum balance equations for charged porous media are presented. Given the highly coupled nonlinear physiochemical processes in charged porous media as well as the huge discrepancy in length scales of these physiochemical phenomena versus the application, numerical simulation of these processes at the Darcy scale is even more challenging than the direct pore-scale simulation of multiphase flow in porous media. Thus, upscaling the microscopic processes up to the Darcy scale is essential and highly required for large-scale applications. Hence, we provide and discuss Darcy-scale porous medium theories obtained using the hybrid mixture theory and homogenization along with their corresponding assumptions. Then, application of these theoretical developments in clays, batteries, enhanced oil recovery, and biological systems is discussed.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉The data input for reservoir simulation of low-salinity waterflood in sandstones includes adsorption isotherms, or equilibrium constants for all cation-exchange reactions, and cation-exchange capacity (CEC). We develop a novel method for determination of those functions from the laboratory data of low-salinity coreflooding; the inverse problem is solved; rather, matching is applied. Changing independent time variable to Lagrangian coordinate transforms the governing system for two-phase multicomponent flow into a single-phase system of mass balance equations for each ion and the volumetric balance equation for water. It allows determining the adsorption isotherms for all ions, or equilibrium constants and CEC, from the breakthrough ion concentrations, using the exact solution of ion-exchange inverse problem. Further, the fractional flow and relative permeability can be determined from the effluent water cut and pressure drop histories, using the Welge’s and JBN methods; here each data point corresponds to the breakthrough values of ion concentrations. We show that for four-ion waterflooding, only two constants from two equilibrium constants and CEC can be determined. Treatment of five coreflood data sets shows a close agreement between the laboratory data and the results of inverse problem solution. 〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉The permeability of natural porous media, such as soils and rocks, usually possesses uncertainties due to the randomness and spatial variation of microscopic pore structures. It is of great importance to develop an effective methodology to obtain statistical properties of permeability for porous media. In this work, an efficient approach is developed by combining the sphere packing algorithm, lattice Boltzmann method (LBM), and probabilistic collocation method (PCM). The porous media are generated by sphere packings of a specified size distribution, and the isotropy and representative elementary volume are verified by statistical analyses. Fluid flow in the complex pore structures is numerically resolved by LBM, with the permeability calculated by Darcy’s law. The uncertainty of permeability can be quantified by PCM with only several porosity samplings required at predetermined collocation points. In addition, the porosity–permeability relationships can be acquired efficiently. Numerical results indicate that, with the proposed approach, the computational efforts are reduced by more than two orders of magnitude compared to the Monte Carlo simulations.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉A methodology has been developed to create a pore network model (PNM) from the geometrical/topological information extracted from the micro-tomographic images of a polyurethane (PU) foam sample. By solving fluid-flow equations on this model, we could estimate the permeability of PU foams and validate with experimental findings. Previous literature suggested that the permeability of open-cell PU foams was overestimated using the traditional Carman–Kozeny and Duplessis–Masliyah models. Inter-cell connectivity was deemed as the potential cause of this difference. Thus, taking into consideration the effects of spatial arrangement of pores of different sizes, the throat constriction between pores, and the percentage open-cell content, a PNM was employed where the entire void space is treated as a network of pores and throats of varying dimensions. To simulate flow through the network, the mass conservation equations were solved at each pore, the Darcy’s equation was applied globally, and thus, the overall permeability of the foam was estimated. Using our PNM, we were able to estimate the permeability with very good agreement with experimentally observed results. This indicates that pore connectivity has a significant effect on permeability. Parametric studies with PNM also revealed that the permeability depends on the square of pore size while is quite sensitive to throat constrictions with cubic dependence.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉This paper is devoted to the stability analysis of 3D convective flow in the horizontal channel filled by the porous medium. Solute precipitation (solute sorption) by the porous medium is taken into account within the linear MIM approach. The solute concentration difference across the channel and the external filtration flux are assumed as constant. As a result, conditions of the appearance of two-dimensional convection regimes with respect to possible three-dimensional perturbations were obtained, which made it possible to estimate the range of parameters in which the two-dimensional convection regimes can be observed. The dependencies of the parameters of two-dimensional regimes on the parameters of the problem and on the thickness of the channel are discussed.〈/p〉

Description: 〈p〉The article “Quantitative In-situ Analysis of Water Transport in Concrete Completed Using X-ray Computed Tomography”, written by “Tyler Oesch, Frank Weise, Dietmar Meinel and Christian Gollwitzer”〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉This numerical analysis investigated the effects of heat and mass transfer characteristics on the mixed convection flow of non-Newtonian fluids over the vertical wedge in a saturated porous medium with Soret/Dufour effects and internal heat generation. The numerical modeling of this problem has attracted considerable attention from researchers because it has practical applications in biological sciences, electronic cooling, advanced nuclear systems, etc. The internal heat generation is assumed to be an exponential decaying form. The power-law model of Ostwald–de Waele for non-Newtonian fluids is considered. The surface of the vertical wedge is kept at variable wall temperature and concentration. In the analysis of mixed convection, which included free convection and forced convection, parameter varies from 0 (pure free convection) to 1 (pure forced convection). The transformed equations are obtained by using a suitable coordinate transformation, and then, Keller box method is utilized to solve the non-similar equations. Comparisons with data published previously showed good agreement. Both the local Nusselt number and the local Sherwood number increase with increasing the exponent of variable wall temperature/concentration. Increasing the internal heat generation coefficient decreases (increases) the local Nusselt (Sherwood) number. As the power-law index is increased, the local Nusselt and Sherwood numbers are decreased.〈/p〉

Description: 〈span〉 〈h3〉Abstract〈/h3〉 〈p〉Many processes in nature (e.g., physical and biogeochemical processes in hyporheic zones, and arterial mass transport) occur near the interface of free-porous media. A firm understanding of these processes needs an accurate prescription of flow dynamics near the interface which (in turn) hinges on an appropriate description of interface conditions along the interface of free-porous media. Although the conditions for the flow dynamics at the interface of free-porous media have received considerable attention, many of these studies were empirical and lacked a firm theoretical underpinning. In this paper, we derive a complete and self-consistent set of conditions for flow dynamics at the interface of free-porous media. We first propose a principle of virtual power by incorporating the virtual power expended at the interface of free-porous media. Then by appealing to the calculus of variations, we obtain a complete set of interface conditions for flows in coupled free-porous media. A noteworthy feature of our approach is that the derived interface conditions apply to a wide variety of porous media models. We also show that the two most popular interface conditions—the Beavers–Joseph condition and the Beavers–Joseph–Saffman condition—are special cases of the approach presented in this paper. The proposed principle of virtual power also provides a minimum power theorem for a class of flows in coupled free-porous media, which has a similar mathematical structure as the ones enjoyed by flows in uncoupled free and porous media.〈/p〉 〈/span〉 〈span〉 〈h3〉Graphic Abstract〈/h3〉 〈p〉The derived interface conditions are summarized along with a pictorial description of the problem, which pertains to the flow of an incompressible fluid in coupled free-porous media. 〈span〉 〈span〉\(\Psi \)〈/span〉 〈/span〉 is the power expended density along the interface. 〈span〉 〈span〉\(\mathbf {v}_{\text {free}}\)〈/span〉 〈/span〉 and 〈span〉 〈span〉\(\mathbf {v}_{\text {por}}\)〈/span〉 〈/span〉 are the velocities in the free and porous regions, respectively. A superposed asterisk on a (vectorial) quantity denotes its tangential component along the interface. 〈span〉 〈span〉\(v_n\)〈/span〉 〈/span〉 is the normal component of the velocity at the interface from the free region into the porous region. 〈span〉 〈span〉\(\mathbf {T}_{\text {free}}^{\text {extra}}\)〈/span〉 〈/span〉 and 〈span〉 〈span〉\(\mathbf {T}_{\text {por}}^{\text {extra}}\)〈/span〉 〈/span〉, respectively, denote the extra Cauchy stresses in the free and porous regions. 〈span〉 〈span〉\(\mathbf {t}_{\text {free}}\)〈/span〉 〈/span〉 and 〈span〉 〈span〉\(\mathbf {t}_{\text {por}}\)〈/span〉 〈/span〉, respectively, denote the tractions on the free and porous sides of the interface with outward normals 〈span〉 〈span〉\(\widehat{\mathbf {n}}_{\text {free}}\)〈/span〉 〈/span〉 and 〈span〉 〈span〉\(\widehat{\mathbf {n}}_{\text {por}}\)〈/span〉 〈/span〉. A unit tangential vector along the interface is denoted by 〈span〉 〈span〉\(\widehat{\mathbf {s}}\)〈/span〉 〈/span〉. 〈span〉 〈span〉 〈img alt="" src="https://static-content.springer.com/image/MediaObjects/11242_2019_1326_Figa_HTML.png"〉 〈/span〉 〈/span〉〈/p〉 〈/span〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉A reliable prediction of two-phase flow through porous media requires the development and validation of models for flow across multiple length scales. The generalized network model is a step towards efficient and accurate upscaling of flow from the pore to the core scale. This paper presents a validation of the generalized network model using micro-CT images of two-phase flow experiments on a pore-by-pore basis. Three experimental secondary imbibition datasets are studied for both sandstone and carbonate rock samples. We first present a quantification of uncertainties in the experimental measurements. Then, we show that the model can reproduce the experimental fluid occupancies and saturations with a good accuracy, which in some cases is comparable with the similarity between repeat experiments. However, high-resolution images need to be acquired to characterize the pore geometry for modelling, while the results are sensitive to the initial condition at the end of primary drainage. The results provide a methodology for improving our physical models using large experimental datasets which, at the pore scale, can be generated using micro-CT imaging of multiphase flow.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉The porosity and permeability of coal change with pore pressure, due to changes in effective stress and matrix swelling due to gas adsorption. Three analytical models to describe porosity and permeability change in this context have been presented in the literature, all of which are based on poroelastic theory and uniaxial strain conditions. However, each of the three models provides different results. Review articles have attributed these differences to the use of stress formulations or strain formulations. In this article, the three aforementioned porosity models are used to derive three associated expressions for the storage coefficient. A single mathematical equation for the storage coefficient in an aquifer under uniaxial strain conditions is well established. The storage coefficient represents the volume of fluid released per unit volume of a porous rock following a unit decline in pore pressure. It is shown that only one of the aforementioned three coal-bed methane porosity models leads to the correct equation for the uniaxial strain storage coefficient in the absence of gas sorption-induced strain.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉A novel finite volume method is presented that is applicable to discontinuous capillary pressure fields. The method is developed within the control-volume distributed multi-point flux approximation (CVD-MPFA) framework (Edwards and Rogers in Comput Geosci 02(04):259–290, 〈span〉1998〈/span〉; Friis et al. in SIAM J Sci Comput 31(02):1192–1220, 〈span〉2008〈/span〉). Results are computed on structured and unstructured grids that demonstrate the ability of the method to resolve flow in the presence of a discontinuous capillary pressure field for diagonal and full-tensor permeability fields. In addition to an upwind approximation for the saturation equation flux, the importance of upwinding on capillary pressure flux via a hybrid formulation is shown.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉The injection of seawater-like brines alters stiffness, strength and time-dependent deformation rates for water-saturated chalks. This study deals with the mechanical effects and oil production upon brine injection through wettability-altered samples. The results from two test programs are presented: (a) ‘Wettability determination program’ and (b) ‘triaxial test program’. Kansas chalk samples were saturated by a mixture of oil and water and aged over time at 90 °C. The wettability index of the altered samples was estimated using chromatographic separation tests by co-injecting sulphate ions that adsorb on the 〈em〉water-wet〈/em〉 mineral surfaces and non-affine tracer. A good repeatability was observed. In the triaxial test program, unaged 〈em〉water-wet〈/em〉 and aged 〈em〉mixed-wet〈/em〉 samples were hydrostatically loaded to 1.5 times yield stress so stiffness and strength could be determined. The samples were kept at the same stress level over time to monitor the volumetric creep. After a stagnant flow period of 15 days, MgCl〈sub〉2〈/sub〉 brine and seawater were flushed through the samples so the oil production and ion concentration of the effluent water could be obtained. The combined observations of the bulk volume, oil volume and estimated solid volume (from effluent analyses) enabled us to calculate pore volume and thereby oil saturation with time. The 〈em〉mixed-wet〈/em〉 samples were found to be stiffer and stronger than the 〈em〉water-wet〈/em〉 samples, and when the stress was kept at 1.5 times yield the creep curves overlapped. During the flow-through period, the changes in ion composition are insensitive to the presence of oil, and ongoing water weakening for 〈em〉mixed-wet〈/em〉 samples is the same as in the 〈em〉water-wet〈/em〉 samples. Further, we found that oil was only produced during the first 2–3 pore volumes (PVs) injected. Afterwards, no oil was produced even though the chemical reactions took place and pore volume reduced.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉A micro-continuum simulation framework is proposed to study the complex pore-scale dynamics associated with hydrocarbon recovery from shale gas. The model accounts for the presence of immiscible fluid phases and for transport mechanisms in the nanoporous structures including slip flow, adsorption, surface and Knudsen diffusion. We employ the concept of sub-grid models to simulate the transport phenomena in shale gas. Specifically, we use high-resolution FIB–SEM images that provide information on the spatial distribution of the minerals, resolved pore space, and sub-resolution porous regions. The model is used to investigate several production scenarios at the pore-scale. In one setting, the organic matter is in direct contact with a micro-crack; in the other setting, clay regions are sandwiched between the organic matter and the “open” crack. The simulations show that it is important to account for the presence of multiple immiscible fluid phases because they can play a critical role in hydrocarbon production from shale-gas formations both in terms of production rate and in terms of residual mass of hydrocarbon. Moreover, we show that, because of wettability conditions, the rate of hydrocarbon recovery, as well as the ultimate recovery, depends strongly on the spatial distribution of the kerogen and clay in the vicinity of the micro-cracks.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉This study numerically analyzed the Arrhenius activation energy effect on free convection about a permeable horizontal cylinder in porous media. The surface of the horizontal cylinder is maintained at uniform wall temperature and uniform wall concentration. Non-similar transformed governing equations are solved by Keller box method. Comparisons with previous works showed good agreement. Numerical data of the Nusselt number and the Sherwood number are presented for dimensionless reaction rate, temperature difference parameter, fitted rate constant, dimensionless activation energy, blowing/suction parameter, dimensionless coordinate, buoyancy ratio, and Lewis number. Generally, the Nusselt (Sherwood) number reduces (increases) with decreasing dimensionless activation energy or increasing dimensionless reaction rate, temperature difference parameter, and fitted rate constant.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉In this paper, we derive a pore-scale model for permeable biofilm formation in a two-dimensional pore. The pore is divided into two phases: water and biofilm. The biofilm is assumed to consist of four components: water, extracellular polymeric substance (EPS), active bacteria, and dead bacteria. The flow of water is modeled by the Stokes equation, whereas a diffusion–convection equation is involved for the transport of nutrients. At the biofilm–water interface, nutrient transport and shear forces due to the water flux are considered. In the biofilm, the Brinkman equation for the water flow, transport of nutrients due to diffusion and convection, displacement of the biofilm components due to reproduction/death of bacteria, and production of EPS are considered. A segregated finite element algorithm is used to solve the mathematical equations. Numerical simulations are performed based on experimentally determined parameters. The stress coefficient is fitted to the experimental data. To identify the critical model parameters, a sensitivity analysis is performed. The Sobol sensitivity indices of the input parameters are computed based on uniform perturbation by ± 10% of the nominal parameter values. The sensitivity analysis confirms that the variability or uncertainty in none of the parameters should be neglected.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Evaluating the anisotropy of transport parameters in rocks is important for various applications, such as reservoir engineering and rock mechanics. Owing to their anisotropic pore structures, the tortuosity, constrictivity, and pore size distribution of rocks are often anisotropic in nature, which in turn affect the permeability and diffusivity. However, it has still not been determined whether the permeability and diffusivity are anisotropic in the same manner. This study used experiments and numerical modeling to examine the effect of the pore structure on the permeability and diffusivity anisotropies of rocks. The experimental results showed a clear difference in the anisotropy ratios of the permeability (〈em〉k〈/em〉〈sup〉⊥〈/sup〉/〈em〉k〈/em〉〈sup〉‖〈/sup〉) and diffusivity (〈em〉D〈/em〉〈span〉 〈sub〉e〈/sub〉 〈sup〉⊥〈/sup〉 〈/span〉/〈em〉D〈/em〉〈span〉 〈sub〉e〈/sub〉 〈sup〉‖〈/sup〉 〈/span〉) for Berea sandstone, which is the de facto standard porous sandstone. The analysis results from micro-focus X-ray computed tomography and simulation with the lattice Boltzmann method supported the experimental difference in anisotropy ratios. In the analysis and simulation, the relation between the minimum cross-sectional porosity area and characteristic pressure gradient was estimated. The analysis results suggest that the minimum cross-sectional porosity areas that influence the permeability anisotropy are too large to physically induce anisotropic NaCl diffusion, and thus, the diffusivity of Berea sandstone is nearly isotropic.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉In some gas–solid reactions, a new solid substance is produced. The product acts as a shield and prevents the collision between gas and solid reactants which further causes an incomplete reaction. If the molar volume of the new product differs from the solid reactant, the inner structure of porous media is changed as well. In this paper, we discuss such gas–solid reactions in porous media using the two-dimensional lattice gas cellular automata FHP-III model. We simulate the fluid flow and chemical reaction in different porous media. We also show the effects of porosity and morphology of the solid, and reaction probability on the reaction process. Results obtained from the simulations agree closely with the theory of gas–solid reactions and diffusion theories. Hence, the proposed model is a good choice to simulate gas–solid chemical reactions in porous media at the mesoscopic level. 〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉The Horton–Rogers–Lapwood problem with strong heterogeneity and anisotropy is examined for a simple case, namely where the heterogeneity is provided by two layers, each of which is homogeneous and isotropic in a horizontal plane. We derived a new hydrodynamic boundary condition at the interface between two different porous media and then formulated and numerically solved the eigenvalue problem to determine the critical values of the wavenumber and Rayleigh number. We found that our approach works and gives the correct result for the homogeneous situation independent of the position of the interface. We also showed that for weak heterogeneity, by using modified anisotropy parameters weighted with the two layer depths, results obtained in Kvernvold and Tyvand (J Fluid Mech 90:609–624, 〈span〉1979〈/span〉) for a single-layer problem can be used to approximate the critical Rayleigh number for the double-layer problem. We found that the agreement with the two-layer solution is best if a harmonic mean is used to define the mean permeability ratio and an arithmetic mean is utilized to define the mean conductivity ratio.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Experimental evidence shows that injecting low-salinity water during the oil recovery process can lead to an increase in the amount of oil recovered. Numerous mechanisms have been proposed to explain this effect, and, in recent years, two which have gained notable support are multicomponent ionic exchange (MIE) and pH increase. Both mechanisms involve ion exchange reactions within the thin film of water separating the oil in a reservoir from the clay minerals on the surface of the reservoir rock. Since the reactions occur on the molecular scale, an upscaled model is required in order to accurately determine the dominant mechanism using centimetre-scale experiments. In this paper, we develop the first stages of this upscaling process by modelling the pore-scale motion of an oil slug through a clay pore throat. We use a law-of-mass-action approach to model the exchange reactions occurring on the oil–water and clay–water interfaces in order to derive expressions for the surface charges as functions of the salinity. By balancing the disjoining pressure in the water film with the capillary pressure across the oil–water interface, we derive an expression for the salinity-dependent film thickness. We compare the two mechanisms by modifying an existing model for the velocity of an oil slug through a pore throat. Numerical results show that the velocity increases as the salinity decreases. The percentage increase is larger for the MIE mechanism, suggesting that MIE may be the dominant causal mechanism; however, this will vary depending on the particular clay and oil being studied.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉The powerful method of eigenfunction superposition is applied to the starting flow in a sector duct filled with a porous medium. Using analytic eigenfunctions and eigenvalues of the Helmholtz equation, the solution can be expressed in a simple series. The properties of the velocity and the transient flow rate are found to depend on the sector geometry and a porous medium factor. The starting solution is then used to construct the solution to arbitrary unsteady flows.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉We present the results of the numerical investigation of the Soret-induced convection in a ternary liquid mixture consisting of dodecane, isobutylbenzene and tetralin, taken in equal portions. The mixture is placed into a square porous cavity with rigid impermeable boundaries heated from below. The lateral boundaries are adiabatic. The problem under consideration is a model of natural hydrocarbon reservoir with porous medium, and the components of mixture are representatives of the main groups of chemical compounds comprising oil. Due to the thermodiffusion effect, dodecane and isobutylbenzene as the lighter components of this mixture with positive separation ratios are accumulated in the warmer domain of the cavity, and the heavy component, tetralin, is accumulated in the colder domain, which may lead to the development of convection. The calculations are performed for the parameters of porous medium close to the real parameters of oil fields and temperature gradient that correspond to geothermal gradient. They provide data on the temporal evolution of the characteristics of the flow and component separation. We also analyze the onset and development of single-vortex and two-vortex instability modes with the growth of the Rayleigh number 〈span〉 〈span〉\(Ra_{\mathrm{{por}}}\)〈/span〉 〈/span〉. It is found that at a certain value of the supercriticality, the stationary flow regime is replaced by the oscillatory regime. At even higher values of the Rayleigh number, the chaotic oscillations take place. The transitions between single-vortex and two-vortex flows are also observed. For porosity equal to 0.1, the two oscillatory regimes at different oscillation amplitudes are excited. With the porosity growth, the region of the existence of the oscillatory regimes becomes narrower.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Numerical experiment involving both moisture and solute transport predictions is performed to estimate the hydrochemical characteristics of unsaturated porous soil. The moisture and solute transport in the soil are described by the flow and advection–dispersion transport equations. These transport equations are solved by the spectral element method, which is based on Legendre–Gauss–Lobatto quadrature rule and the fully implicit time scheme using the modified Picard iterative procedure constructed with the standard chord slope approximation. The estimation of hydraulic and solute transport parameters has been conducted using the Levenberg–Marquardt method. The goals of the inverse problem were to develop soil hydrochemical characteristics estimation strategies based on combined two of the following functional cost measurements: moisture content, pressure head, hydraulic conductivity, cumulative outflow, and solute concentration. The performance of the inverse algorithm was evaluated using the coefficient of determination, the root-mean-square error, and the relative error analysis which provide an optimal scheme for parameters estimation. The spectral element method was shown to provide good results with negligible error when compared to analytical values. The obtained results indicate excellent agreement of the method for estimating hydraulic and transport parameters with negligible relative error when compared estimated parameters and true values. The choice and the order of combination of objective functions affect crucially the inverse solution especially in case of large hydrochemical parameters estimates.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Wavelet transforms (WTs) constitute a class of powerful tools for spectral and local analysis of data, such as time series, geophysical data, and well logs, as well as modeling of various problems in oil reservoirs, and in particular upscaling their geological models. The application to upscaling has, however, been limited to the reservoir models that are represented by regular computational grids with blocks or cells with regular shapes, such as squares and cubes. The most natural structure of the computational grid is, however, one with irregular and unequal cells, distributed spatially with stochastic orientations. Such a grid is also the ideal computational model for, for example, fractured reservoirs as it allows inclusion of fractures with spatially distributed orientations. In this paper, we propose a generalization of the WT approach to upscaling by developing a new model of a reservoir based on irregular graphs that make it possible to use the WTs for upscaling the geological model highly efficiently. To do so, we first define a computational grid representing a reservoir as a graph and its adjacency matrix and, then, introduce graph WTs using the concept of 〈em〉lifting〈/em〉, utilized in classical signal processing and its extension to graphs. The application of the lifting-based graph WT to upscaling is then developed. The result is an algorithm that may be applied to upscaling of 〈em〉any〈/em〉 unstructured geological model represented by a computational grid in which the multiresolution graph WT is applied directly to the spatial distribution of the permeabilities (or other suitable properties). Examples in which the geological model is represented by the Voronoi tessellations are described, and simulation of waterflooding with such graph networks is carried out in order to demonstrate the accuracy and efficiency of the new method.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Reactive transport in fractured media is conceptualized as a multi-scale problem that couples a pore-scale component, which comprises Navier–Stokes flow, multi-component transport and aqueous equilibrium in the fracture, and a Darcy-scale component, which comprises multi-component diffusive transport, aqueous equilibrium and mineral reactions in the porous matrix. The model that implements this multi-scale approach builds on an existing pore-scale model and is able to capture complex fracture geometries with the embedded-boundary method. The embedded boundary acts as the interface between pore- and Darcy-scale domains. Adaptive mesh refinement is used to match resolutions at the interface while using coarser resolution away from the interface when not needed in the Darcy-scale domain. The new model is validated and then compared to results from a pore-scale model. Multi-scale model results are shown to be equivalent to pore-scale results under diffusion-controlled reactions in the pore scale and very fast dissolution in the Darcy scale. The multi-scale model provides a more accurate solution for a given resolution as it effectively sets the equilibrium concentrations as boundary conditions. The multi-scale model is capable to capture flow channelization observed in an experimental fractured core and, at the same time, limitations in the dissolution of calcite by diffusive transport through an altered porous layer. Discrepancies in effluent calcium concentrations between the multi-scale results and results from a reduced-dimension Darcy-scale model for this fractured core experiment are attributed to the solution of the flow field and the gradients that develop inside the fracture. Discrepancies in effluent magnesium concentrations exemplify the limitations of the approach because the multi-scale model requires calibration of reactive surface areas as Darcy-scale continuum models.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉We describe a simple but effective stochastic method to model the void structure of a single fracture in a form of voxel representation. A fracture void is delineated by two bounding wall surfaces that are separated by some distance (i.e. the local aperture) at each location on the medial surface that lies within the fracture void and serves as a model reference frame. The three surface height fields are generated based on four parameters (mean, standard deviation and two spatial correlation lengths) for each field and two parameters (coefficient and synergistic length) for the spatial correlation between the fracture walls. Testing of generated models demonstrates that not only are the model fracture apertures spatially correlated and characterized as a Gaussian field, but also the two fracture walls are closely correlated, with a similar shape and/or height. With respect to fracture apertures, three quantities, i.e. the mean aperture, roughness and anisotropy, can be derived from the fracture models to describe fracture morphology. The effect of model fractures on fluid flow is investigated in order to establish the relationship between fracture permeability and the three morphological quantities, revealing a way to avoid the significant estimation error associated with the use of the cubic law. 〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Two distinct but interconnected approaches can be used to model diffusion in fluids; the first focuses on dynamics of an individual particle, while the second deals with collective (effective) motion of (infinitely many) particles. We review both modeling strategies, starting with Langevin’s approach to a mechanistic description of the Brownian motion in free fluid of a point-size inert particle and establishing its relation to Fick’s diffusion equation. Next, we discuss its generalizations which account for a finite number of finite-size particles, particle’s electric charge, and chemical interactions between diffusing particles. That is followed by introduction of models of molecular diffusion in the presence of geometric constraints (e.g., the Knudsen and Fick–Jacobs diffusion); when these constraints are imposed by the solid matrix of a porous medium, the resulting equations provide a pore-scale representation of diffusion. Next, we discuss phenomenological Darcy-scale descriptors of pore-scale diffusion and provide a few examples of other processes whose Darcy-scale models take the form of linear or nonlinear diffusion equations. Our review is concluded with a discussion of field-scale models of non-Fickian diffusion.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉We present a general novel technique to monitor saturation changes on small rock samples of only 15 mm in diameter and 20 mm in length for the purpose of assessing the kinetics of spontaneous imbibition processes. With a fully 3〈em〉D〈/em〉 imbibition configuration involving countercurrent flows through all faces of the sample, the method is based on an NMR technique in which the sole oil phase present within the sample is monitored. The experimental method is fast for two reasons that are (1) the possibility to perform accurate measurements on tiny samples and (2) the adoption of a 3〈em〉D〈/em〉 flow geometry. The kinetics of oil desaturation during spontaneous imbibition is analyzed with the help of an analytical 3〈em〉D〈/em〉 diffusion model, according to which the kinetics is proportional to the value of a “capillary” diffusion coefficient. For the purpose of demonstrating our methodology, we used this technique to compare the spontaneous imbibition of restored sandstone miniplugs from a sandstone reservoir, with and without alkali in the imbibing brine. The imbibition kinetics was quantified as capillary diffusion coefficient values. The studied case results revealed mixed impacts of alkali on the spontaneous imbibition kinetics, involving both a brine–oil interfacial tension change and a wettability alteration of the rock, the latter requiring further investigation beyond the scope of this article.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉We consider two unsteady free convection flows of a Bingham fluid when it saturates a porous medium contained within a vertical circular cylinder. The cylinder is initially at a uniform temperature, and such flows are then induced by suddenly applying either a new constant temperature or a nonzero heat flux to the exterior surface. As time progresses, heat conducts inwards and this may or may not overcome the yield threshold for flow. For the constant temperature case, flow begins immediately should the parameter, Rb, which is a nondimensional yield parameter, be sufficiently large. The ultimate fate, though, is full immobility as the cylinder eventually tends towards a new constant temperature. For the constant heat flux case, the fluid remains immobile but will begin to flow eventually should Rb be sufficiently large. The two cases have different critical values for Rb.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Stress dependency of permeability of porous rocks is described by means of a theoretical elastic cylindrical pore-shell model. This model is developed based on a bundle of elastic capillary tubes representation of the preferential flow paths formed in heterogeneous porous rocks. The radial displacement caused in tubes by the pore fluid pressure applied over the surface of the elastic cylindrical flow tubes is expressed by a Lamé-type equation. The radial displacement is incorporated into the Kozeny–Carman relationship to determine the variation of the permeability of porous rocks by variation of the pore fluid pressure. The solution of this equation yields a semi-analytical equation which provides accurate correlations of the stress dependency of the permeability data of porous rocks. The errors associated with the previous formulation of this problem by Zhu et al. (Transp Porous Med 122:235–252, 〈span〉2018〈/span〉) are explained in view of the present formulation.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Flooding of coastal areas with seawater often leads to density stratification. The stability of the density-depth profile in a porous medium initially saturated with a fluid of density 〈span〉 〈span〉\(\rho _\mathrm{f}\)〈/span〉 〈/span〉 after flooding with a salt solution of higher density 〈span〉 〈span〉\(\rho _\mathrm{s}\)〈/span〉 〈/span〉 is analyzed. The standard convection/diffusion equation subject to the so-called Boussinesq approximation is used. The depth of the porous medium is assumed to be infinite in the analytical approaches and finite in the numerical simulations. Two cases are distinguished: the laterally unbounded 〈span〉 〈span〉\({{{\mathbf {{\small {\uppercase {case~A}}}}}}}\)〈/span〉 〈/span〉 and the laterally bounded 〈span〉 〈span〉\({{{\mathbf {{\small {\uppercase {case~B}}}}}}}\)〈/span〉 〈/span〉. The ratio of the diffusivity and the density difference 〈span〉 〈span〉\((\rho _\mathrm{s} - \rho _\mathrm{f})\)〈/span〉 〈/span〉 induced gravitational shear flow is an intrinsic length scale of the problem. In the unbounded 〈span〉 〈span〉\({{{\mathbf {{\small {\uppercase {case~A}}}}}}}\)〈/span〉 〈/span〉, this geometric length scale is the only length scale and using it to write the problem in dimensionless form results in a formulation with Rayleigh number 〈span〉 〈span〉\(R = 1\)〈/span〉 〈/span〉. In the bounded 〈span〉 〈span〉\({{{\mathbf {{\small {\uppercase {case~B}}}}}}}\)〈/span〉 〈/span〉, the lateral geometry provides another length scale. Using this geometrical length scale to write the problem in dimensionless form results in a formulation with a Rayleigh number 〈em〉R〈/em〉 given by the ratio of the geometric and intrinsic length scales. For both 〈span〉 〈span〉\({{{\mathbf {{\small {\uppercase {case~A}}}}}}}\)〈/span〉 〈/span〉 and 〈span〉 〈span〉\({{{\mathbf {{\small {\uppercase {case~B}}}}}}}\)〈/span〉 〈/span〉, the well-known Boltzmann similarity solution provides the ground state. Three analytical approaches are used to study the stability of this ground state, the first two based on the linearized perturbation equation for the concentration and the third based on the full nonlinear equation. For the first linear approach, the surface spatial density gradient is used as an approximation of the entire background density profile. This results in a crude estimate of the 〈span〉 〈span〉\(L^2\)〈/span〉 〈/span〉-norm of the concentration showing that the perturbation at first grows, but eventually decays in time. For the other two approaches, the full ground-state solution is used, although for the second linear approach subject to the restriction that the ground state slowly evolves in time (the so-called frozen profile approximation). Just like the ground state, the resulting eigenvalue problems can be written in terms of the Boltzmann variable. The linearized stability approach holds only for infinitesimal small perturbations, whereas the nonlinear, variational energy approach is not subject to such a restriction. The results for all three approaches can be expressed in terms of Boltzmann 〈span〉 〈span〉\(\sqrt{t}\)〈/span〉 〈/span〉 transformed relationships between the system Rayleigh number and perturbation wave number. The results of the linear and nonlinear approaches are surprisingly close to each other. Based on the system Rayleigh number, this allows delineation of systems that are unconditionally stable, marginally stable, or transiently unstable. These analytical predictions are confirmed by direct two-dimensional numerical simulations, which also show the details of the transient instabilities as function of the wave number for 〈span〉 〈span〉\({{{\mathbf {{\small {\uppercase {case~A}}}}}}}\)〈/span〉 〈/span〉 and the wave number and Rayleigh number for 〈span〉 〈span〉\({{{\mathbf {{\small {\uppercase {case~B}}}}}}}\)〈/span〉 〈/span〉. A numerical example of the effect of a layer with low permeability is also shown. Using typical values of the physical parameters, the analytical and numerical results are interpreted in terms of dimensional length and time scales. In particular, an explicit stability criterion is given for vertical column experiments.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉The classic models describing the hygric mass transfers inside porous materials seem unsuitable in the case of bio-based materials. They are based on the assumption of instantaneous local equilibrium between relative humidity and water content (Künzel in Simultaneous heat and moisture transport in building components—one- and two-dimensional calculation using simple parameters, Fraunhofer IRB Verlag, Stuttgart, 〈span〉1995〈/span〉). These two parameters evolve according to the diffusive fluxes following the sorption isotherms. This study shows that it leads to predicting much shorter times of stabilization than those experimentally obtained. A new approach is presented here; it is free from the local instantaneous equilibrium introducing a local kinetics to describe the transformation of water from vapor state to liquid state and vice versa. The local kinetics of sorption is coupled with the well-known hysteresis phenomenon. It is adjusted from bibliographic data (Collet et al. in Energy Build 62:294–303, 〈span〉2013〈/span〉) giving mass evolution of three hemp concretes under adsorption/desorption conditions. 1D cylindrical simulations allow an excellent fitting on the experiments. Finally, a semiempirical model is proposed, allowing to determine the kinetics parameters more easily. 〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉The objective of this work is to study the applicability of various machine learning algorithms for the prediction of some rock properties which geoscientists usually define due to special laboratory analysis. We demonstrate that these special properties can be predicted only basing on routine core analysis (RCA) data. To validate the approach, core samples from the reservoir with soluble rock matrix components (salts) were tested within 100 + laboratory experiments. The challenge of the experiments was to characterize the rate of salts in cores and alteration of porosity and permeability after reservoir desalination due to drilling mud or water injection. For these three measured characteristics, we developed the relevant predictive models, which were based on the results of RCA and data on coring depth and top and bottom depths of productive horizons. To select the most accurate machine learning algorithm, a comparative analysis has been performed. It was shown that different algorithms work better in different models. However, two-hidden-layer neural network has demonstrated the best predictive ability and generalizability for all three rock characteristics jointly. The other algorithms, such as support vector machine and linear regression, also worked well on the dataset, but in particular cases. Overall, the applied approach allows predicting the alteration of porosity and permeability during desalination in porous rocks and also evaluating salt concentration without direct measurements in a laboratory. This work also shows that developed approaches could be applied for the prediction of other rock properties (residual brine and oil saturations, relative permeability, capillary pressure, and others), of which laboratory measurements are time-consuming and expensive.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉We study the impact, on the drying rate, of the presence of suspended elements, such as calcium sulfate ions, in the interstitial fluid of a porous medium. In order to single out this process in the complexity of a porous medium, we study it through drying in a simple capillary exhibiting characteristics such that it reproduces some critical aspects of drying of porous media. Another specificity of our work is that we focus on the evaporation of ionic solution at their solubility limit. We first show that in such a capillary, the drying process varies depending on the wettability characteristics. Typically, the drying rate is much smaller with hydrophobic surfaces because of the air–liquid interface which tends to withdraw inside the medium, while for hydrophilic surfaces there remains a continuous liquid film up to the entrance. Then, it appears that an ionic solution dries slower than a pure liquid, because the crystals formed along the capillary walls tend to induce a dewetting of the capillary entrance, pushing inward the first liquid–air interface from which most of the evaporation occurs. An experiment with a model colloidal suspension further illustrates this mechanism: The accumulation of solid particles along the wall forms a deposit which pushes inward the first liquid–air interface from which evaporation takes place. Finally, we look at the impact, on the drying characteristics, of the presence of different additives in the ionic solution.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Soil water evaporation plays a critical role in mass and energy exchanges across the land–atmosphere interface. Although much is known about this process, there is no agreement on the best modeling approaches to determine soil water evaporation due to the complexity of the numerical modeling scenarios and lack of experimental data available to validate such models. Existing studies show numerical and experimental discrepancies in the evaporation behavior and soil water distribution in soils at various scales, driving us to revisit the key process representation in subsurface soil. Therefore, the goal of this work is to test different mathematical formulations used to estimate evaporation from bare soils to critically evaluate the model formulations, assumptions and surface boundary conditions. This comparison required the development of three numerical models at the REV scale that vary in their complexity in characterizing water flow and evaporation, using the same modeling platform. The performance of the models was evaluated by comparing with experimental data generated from a soil tank/boundary layer wind tunnel experimental apparatus equipped with a sensor network to continuously monitor water–temperature–humidity variables. A series of experiments were performed in which the soil tank was packed with different soil types. Results demonstrate that the approaches vary in their ability to capture different stages of evaporation and no one approach can be deemed most appropriate for every scenario. When a proper top boundary condition and space discretization are defined, the Richards equation-based models (Richards model and Richards vapor model) can generally capture the evaporation behaviors across the entire range of soil saturations, comparing well with the experimental data. The simulation results of the non-equilibrium two-component two-phase model which considers vapor transport as an independent process generally agree well with the observations in terms of evaporation behavior and soil water dynamics. Certain differences in simulation results can be observed between equilibrium and non-equilibrium approaches. Comparisons of the models and the boundary layer formulations highlight the need to revisit key assumptions that influence evaporation behavior, highlighting the need to further understand water and vapor transport processes in soil to improve model accuracy.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Evaporation of a saline solution from a porous medium often leads to the precipitation of salt at the surface of the porous medium. It is commonly observed that the crystallized salt does not form everywhere at the porous medium surface but at some specific locations. This is interpreted at the signature of spatial variations in the salt concentration at the surface of the porous medium prior to the onset of crystallization. We explore numerically the link between the ion concentration spatial variations at the surface and porous medium heterogeneities considering strongly anisotropic short-range correlated permeability Gaussian fields corresponding to a vertical layering perpendicular to the top evaporative surface for the case of the evaporation–wicking situation. It is shown that the ion concentration extrema at the surfaces correspond to stagnation points with minima corresponding to divergent stagnation points and maxima to convergent stagnation points. Counter-intuitively, the ion concentration maxima are shown to correspond to permeability minima. However, the ion concentration absolute maximum does not necessarily always correspond to the permeability absolute minimum. More generally, the study emphasizes the key role played by the impact of heterogeneities on the velocity field induced in the medium by the evaporation process. It is also shown that the number of ion mass fraction maxima at the porous medium surface is generally much lower than the naive prediction based on the number of correlation lengths spanning the medium. 〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉The evaporation of salt (NaCl) solutions from porous media is studied in the presence of surfactants, because surfactants are often used as cleaning agents for salt-contaminated stones. We show that, contrary to what is commonly assumed, the presence of the surfactant and the changed wetting properties do not affect the drying kinetics: The impact of the surfactants is rather that of a crystallization modifier for the salt. Upon adding a cationic or nonionic surfactant to salt solution, the drying rate is unchanged initially, but can slow down dramatically at later times due to the formation of a salt crust at the surface. When this happens, the total drying time increases compared to pure NaCl solutions without surfactants, at least for very porous stones for which the pores become completely blocked. Surprisingly, for a low-porosity stone the small pores at the surface remain open. The longer drying time for the large porosity stone increases the risk of, e.g., frost or fungal damage to the stones. Consequently, the use of surfactants in conservation treatments should be done with caution.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Evaporation of colloidal suspension in two-dimensional (2D) porous media leads to the formation of self-assembled clogging structures (SCS). The self-assembly pattern is studied with a hybrid two-phase lattice Boltzmann method incorporating non-isothermal phase change, particle transport and deposition models. During drying, particles accumulate along the liquid–vapor interface while colloidal suspension is evaporating. Upon reaching a certain local concentration threshold, the particles deposit and form a solid structure. The patterns formed by these structures are analyzed in different 2D porous media. In small porous systems of 4 pillars, the self-assembly of C-shaped and X-shaped structures is observed, which compares well with experimental bridge configurations. SCS in porous media of three different initial particle concentrations and of three different porosities are studied in larger porous systems. Simulated self-assembled clogging configurations show good qualitative matches with experimental configuration results. Particle concentration and porosity are both seen to affect the dynamic drying processes as well as the final self-assembled clogging configuration. The liquid configuration and the clogging structure affect each other mutually during drying. With initial concentration increasing from 〈em〉C〈/em〉〈sub〉0〈/sub〉 = 0.00 to 〈em〉C〈/em〉〈sub〉0〈/sub〉 = 0.16 at a given porosity 〈span〉 〈span〉\( \phi_{0} = 0.68 \)〈/span〉 〈/span〉, the average evaporation rate and porosity decrease by 21.9% and 1.9%, respectively, due to blockage of pores. With increasing initial porosity from 〈span〉 〈span〉\( \phi_{0} = 0.53 \)〈/span〉 〈/span〉 to 〈span〉 〈span〉\( \phi_{0} = 0.81 \)〈/span〉 〈/span〉 at a given concentration of 〈em〉C〈/em〉〈sub〉0〈/sub〉 = 0.16, the average evaporation rate increases by a factor of 2.9 due to larger liquid–vapor interfacial area. Also with the given concentration of 〈em〉C〈/em〉〈sub〉0〈/sub〉 = 0.16, the decrease in the porosity (0.94%, 1.9% and 2.7%) is higher for higher initial porosity (〈span〉 〈span〉\( \phi_{0} = 0.53, \, 0.68,{\text{ and }}0.81 \)〈/span〉 〈/span〉), since more particles (proportional to 〈span〉 〈span〉\( \phi_{0} \cdot C_{0} \)〈/span〉 〈/span〉) are initially present. This work opens the door for numerically assisted design of colloid-deposition-based clogging patterns.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉By its very nature, research into multi-physical processes occurring in porous and fractured media requires a collaborative approach. An interdisciplinary approach has led to the adoption of collaborative software development paradigms in this field relying on software for scientific computing as research infrastructures. The development of open-source software has become a cornerstone of computational approaches in academia and has even spawned successful business models in the commercial world. This article is geared toward readers who want to learn more about potential benefits of open-source software in porous media research and who want to familiarize themselves with typical workflows required to become an active contributor to or user of open-source solutions for porous media simulation. The article puts general principles, motivations and concepts into the specific context of experiences and lessons learned from the authors developing the open-source software projects OpenGeoSys and DuMu〈span〉 〈span〉\(^{\text {x}}\)〈/span〉 〈/span〉.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉The contribution of Robin boundaries on the onset of convection in a horizontal saturated porous layer covered by a free surface on the top is investigated here. The saturated solid matrix is assumed in a regime where the temperature profile of the solid phase differs from a fluid one. Two energy equations are adopted as a consequence of the local thermal non-equilibrium model (LTNE), and four Biot numbers are arising out of the third kind of boundaries imposed on both surfaces. The dimensionless parameters 〈em〉H〈/em〉 and 〈span〉 〈span〉\(\gamma \)〈/span〉 〈/span〉 which rule the transition from local thermal equilibrium (LTE) to non-equilibrium one or vice versa are taken into account. The cases of equal and different Biot numbers have been considered beside the asymptotic limits of LTE and LTNE one. A linear stability analysis of the basic motionless state has been performed. The perturbation terms of the main steady flows are evaluated in the form of plane waves. The eigenvalue problem is solved either analytically or numerically depending on the temperature gradient of the fluid phase. The analytical solution is handled through a dispersion relation, while the numerical one is computed by the Runge–Kutta solver combined with the shooting method. The variation in Darcy–Rayleigh number and wave number is obtained with respect to Biot numbers for all resulting cases.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉In this study, we present the results of measurements of pressure drops during the flow of emulsions stabilized by carboxymethylcellulose sodium salt (NaCMC), xanthan gum (XG) and poly(ethylene oxide) (PEO) through a packed bed of glass spheres. The concentration of dispersed phase ranged from 10 to 50 vol% and consisted of flocculated droplets with diameters much smaller than the pore size. Highly flocculated emulsions with the addition of NaCMC were yield-stress fluids whose flow curve can be described by the Herschel–Bulkley equation. An empirical model was formulated for Herschel–Bulkley fluids which allows predicting pressure losses during their flow through a packed bed. In this model, the friction factor was made dependent on the Reynolds number proposed by Kembłowski and Michniewicz (Rheol Acta 18:730–739, 〈span〉1979〈/span〉. 〈span〉https://doi.org/10.1007/BF01533348〈/span〉) and generalized for yield-stress fluids. Also, a correlation was proposed which enables the prediction of values of the modified dimensionless plug size based on calculated values of the modified Herschel–Bulkley number. The viscosity curves obtained for the emulsions with added XG were described with the Carreau model. In the case of emulsions, the shift factor values necessary to calculate the shear rates depend on the concentration of the dispersed phase and the diameter of droplets. If the value of the shift factor is known, the friction factor can be determined from the Ergun equation. During the flow of the emulsion with added PEO through the packed bed, just as during the flow of the aqueous solution of this polymer, an apparent thickening region is noted. The relative increase in the apparent viscosity of the emulsion with added PEO is lower than the apparent viscosity of the aqueous PEO solution. This shows that elastic instability is suppressed by an increase in emulsion viscosity induced by the flocculation of droplets. 〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉We consider convection in a horizontal porous layer of uniform thickness which is heated from below and which is composed of two anisotropic sublayers with principal axes lying in the three coordinate directions. The aim is to determine criteria for the onset of convection by finding the critical Rayleigh number, wavenumber and roll orientation relative to the coordinate axes. The full set of nondimensional parameters has at least six members even when the sublayers are considered to be thermally isotropic, and therefore, we select some special cases in order to illuminate the type of qualitative behaviour which may be expected. One such case is where the anisotropic sublayers are identical except that one sublayer is rotated by an angle of 〈span〉 〈span〉\(90^\circ \)〈/span〉 〈/span〉 to the other. In this situation, the most unstable roll is found to lie at an angle of 〈span〉 〈span〉\(\pm \,45^\circ \)〈/span〉 〈/span〉 to the principal axes. It is also found that fluid particles exhibit a mean longitudinal flow as they circulate about the vortex axis. This drift along the vortex is balanced by an equal and opposite drift in the two neighbouring vortices. Convection in each sublayer is shown to be two dimensional even though the full flow field is three dimensional.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉A channel flow coupled with a transversal stream through porous structures is investigated numerically in this study. Velocity profiles are obtained on the pore scale and averaged to the macroscale in order to evaluate the validity of the Beavers–Joseph interface condition. For this purpose, different ratios between the velocity at the inlet of the channel and the velocity at the base of the porous structure are considered. The effects of Reynolds number, velocity ratio, and geometrical arrangement of the porous structure on the Beavers–Joseph constant are then examined. A critical velocity ratio is found where, for lower ratios, the interface momentum transfer is mainly affected by the channel flow and, for higher ratios, the flow in the porous structure governs. The uniformity of the Beavers–Joseph constant at the interface is found to decrease with increasing velocity ratios. The present results have then been compared with simulations from a coupled Navier–Stokes/Darcy–Forchheimer model.〈/p〉

Description: We demonstrate how to use numerical simulation models directly on micro-CT images to understand the impact of several enhanced oil recovery (EOR) methods on microscopic displacement efficiency. To describe the physics with high-fidelity, we calibrate the model to match a water-flooding experiment conducted on the same rock sample (Akai et al. in Transp Porous Media 127(2):393–414, 2019. 10.1007/s11242-018-1198-8). First we show comparisons of water-flooding processes between the experiment and simulation, focusing on the characteristics of remaining oil after water-flooding in a mixed-wet state. In both the experiment and simulation, oil is mainly present as thin oil layers confined to pore walls. Then, taking this calibrated simulation model as a base case, we examine the application of three EOR processes: low salinity water-flooding, surfactant flooding and polymer flooding. In low salinity water-flooding, the increase in oil recovery was caused by displacement of oil from the centers of pores without leaving oil layers behind. Surfactant flooding gave the best improvement in the recovery factor of 16% by reducing the amount of oil trapped by capillary forces. Polymer flooding indicated improvement in microscopic sweep efficiency at a higher capillary number, while it did not show an improvement at a low capillary number. Overall, this work quantifies the impact of different EOR processes on local displacement efficiency and establishes a workflow based on combining experiment and modeling to design optimal recovery processes.