A linear stability analysis for the onset of mixed convection in a saturated porous medium through an absolute instability of both two- and three-dimensional disturbances is performed. Relevant control parameters associated with the inclined temperature gradient and the vertical throughflow are the vertical and horizontal Rayleigh numbers, \(R_{\mathrm{v}}\) and \(R_{\mathrm{h}}\), and the vertical Péclet number, \(Q_{\mathrm{v}}\), respectively. This work extends previous studies on the very same problem in two fronts. For two-dimensional disturbances, the present results do not agree with the literature for a few of the parametric conditions reported. This is caused by the collision of the convectively unstable downstream propagating branch with multiple upstream propagating branches, which generates several saddle points and, hence, makes the identification of the correct pinching point more difficult. In other words, literature results are all saddle points but not always pinching points. For three-dimensional disturbances, this issue is not present and the current results agree with the literature. On the other hand, due to the inherent difficulties associated with a three-dimensional absolute instability analysis, literature results have only been able to report the group velocities at the onset of convective instability. When their real parts are zero, transition occurs directly from stable to absolutely unstable. Otherwise, transition occurs from stable to convectively unstable first and nothing can be said about the onset of absolute instability. In this work, a novel technique recently developed by the authors allowed the identification of the onset of absolute instability under all parametric conditions investigated in the literature, extending earlier results. Doing so confirmed the dichotomy already observed in these earlier studies, i.e., the onset of absolute instability for two- and three-dimensional longitudinal modes indeed differs.

Abstract 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.

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.

Abstract We address the problem of initiation of convective motion in the case of a fluid saturated porous layer, containing a salt in solution, which is heated and salted below. We amplify the very interesting recent results of Nield and Kuznetsov and examine in detail a whole range of temperature and salt boundary conditions allowing for a combination of prescribed heat flux and temperature. The behaviour of the transition from stationary to oscillatory convection is examined in detail as the boundary conditions vary from prescribed temperature and salt concentration toward those of prescribed heat flux and salt flux.

Abstract The 4th order Darcy–Bénard eigenvalue problem for the onset of thermal convection in a 3D rectangular porous box is investigated. We start from a recent 2D model Tyvand et al. (Transp Porous Med 128:633–651, 2019) for a rectangle with handpicked boundary conditions defying separation of variables so that the eigenfunctions are of non-normal mode type. In this paper, the previous 2D model (Tyvand et al. 2019) is extended to 3D by a Fourier component with wave number k in the horizontal y direction, due to insulating and impermeable sidewalls. As a result, the eigenvalue problem is 2D in the vertical xz-plane, with k as a parameter. The transition from a preferred 2D mode to 3D mode of convection onset is studied with a 2D non-normal mode eigenfunction. We study the 2D eigenfunctions for a unit width in the lateral y direction to compare the four lowest modes \(k_m = m \pi ~(m=0,1,2,3)\), to see whether the 2D mode \((m=0)\) or a 3D mode \((m\ge 1)\) is preferred. Further, a continuous spectrum is allowed for the lateral wave number k, searching for the global minimum Rayleigh number at \(k=k_c\) and the transition between 2D and 3D flow at \(k=k^*\). Finally, these wave numbers \(k_c\) and \(k^*\) are studied as functions of the aspect ratio.

Abstract The concept of the representative elementary volume (REV) is often associated with the notion of hydrodynamic dispersion and Fickian transport. However, it has been frequently observed experimentally and in numerical pore-scale simulations that transport is non-Fickian and cannot be characterized by hydrodynamic dispersion. Does this mean that the concept of the REV is invalid? We investigate this question by a comparative analysis of the advective mechanisms of Fickian and non-Fickian dispersions and their representation in large-scale transport models. Specifically, we focus on the microscopic foundations for the modeling of pore-scale fluctuations of Lagrangian velocity in terms of Brownian dynamics (hydrodynamic dispersion) and in terms of continuous-time random walks, which account for non-Fickian transport through broad distributions of advection times. We find that both approaches require the existence of an REV that, however, is defined in terms of the representativeness of Eulerian flow properties. This is in contrast to classical definitions in terms of medium properties such as porosity, for example.

Abstract 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 \(k-\varepsilon \) 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 k and \(\varepsilon \) 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.

Abstract Carbon dioxide injection into deep saline aquifers is governed by a number of physico-chemical processes including mineral dissolution and precipitation, multiphase fluid flow, and capillary trapping. These processes can be coupled; however, the impact of fluid–rock reaction on the multiphase flow properties is difficult to study and is not simply correlated with variations in porosity. We observed the impact of rock mineral dissolution on multiphase flow properties in two carbonate rocks with distinct pore structures. Observations of steady-state \(\hbox {N}_2\)–water relative permeability and residual trapping were obtained, along with mercury injection capillary pressure characteristics. These tests alternated with eight stages in which 0.5% of the mineral volume was uniformly dissolved into solution from the rock cores using an aqueous solution with a temperature-controlled acid. Variations in the multiphase flow properties did not relate simply to changes in porosity, but corresponded to the changes in the underlying pore structure. In the Ketton carbonate, dissolution resulted in an increase in the fraction of pore volume made up by the smallest pores and a decrease in the fraction made up by the largest pores. This resulted in an increase in the relative permeability to the nonwetting phase, a decrease in the relative permeability to the wetting phase, and a modest, but systematic decrease in residual trapping. In the Estaillades carbonate, dissolution resulted in an increase in the fraction of pore volume made up by pores in the central range of the initial pore size distribution, and a corresponding decrease in the fraction made up by both the smallest and largest pores. This resulted in a decrease in the relative permeability to both the wetting and nonwetting fluid phases and no discernible impact on the residual trapping. In summary, the impact of rock matrix dissolution will be strongly dependent on the impact of that dissolution on the underlying pore structure of the rock. However, if the variation in pore structure can be observed or estimated with modelling, then it should be possible to estimate the impacts on multiphase flow properties.

The diffusive behavior of nanoparticles inside porous materials is attracting a lot of interest in the context of understanding, modeling, and optimization of many technical processes. A very powerful technique for characterizing the diffusive behavior of particles in free media is dynamic light scattering (DLS). The applicability of the method in porous media is considered, however, to be rather difficult due to the presence of multiple sources of scattering. In contrast to most of the previous approaches, the DLS method was applied without ensuring matching refractive indices of solvent and porous matrix in the present study. To test the capabilities of the method, the diffusion of spherical gold nanoparticles within the interconnected, periodic nanopores of inverse opals was analyzed. Despite the complexity of this system, which involves many interfaces and different refractive indices, a clear signal related to the motion of particles inside the porous media was obtained. As expected, the diffusive process inside the porous sample slowed down compared to the particle diffusion in free media. The obtained effective diffusion coefficients were found to be wave vector-dependent. They increased linearly with increasing spatial extension of the probed particle concentration fluctuations. On average, the slowing-down factor measured in this work agrees within combined uncertainties with literature data.

Abstract Flow, transport, mechanical, and fracture properties of porous media depend on their morphology and are usually estimated by experimental and/or computational methods. The precision of the computational approaches depends on the accuracy of the model that represents the morphology. If high accuracy is required, the computations and even experiments can be quite time-consuming. At the same time, linking the morphology directly to the permeability, as well as other important flow and transport properties, has been a long-standing problem. In this paper, we develop a new network that utilizes a deep learning (DL) algorithm to link the morphology of porous media to their permeability. The network is neither a purely traditional artificial neural network (ANN), nor is it a purely DL algorithm, but, rather, it is a hybrid of both. The input data include three-dimensional images of sandstones, hundreds of their stochastic realizations generated by a reconstruction method, and synthetic unconsolidated porous media produced by a Boolean method. To develop the network, we first extract important features of the images using a DL algorithm and then feed them to an ANN to estimate the permeabilities. We demonstrate that the network is successfully trained, such that it can develop accurate correlations between the morphology of porous media and their effective permeability. The high accuracy of the network is demonstrated by its predictions for the permeability of a variety of porous media.

Abstract The emergence of hydrocarbons within shale as a major recoverable resource has sparked interest in fluid transport through these tight mudstones. Recent studies suggest the importance to recovery of microfracture networks that connect localized zones with large organic content to the inorganic matrix. This paper presents a joint modeling and experimental study to examine the onset, formation, and evolution of microfracture networks as shale matures. Both the stress field and fractures are simulated and imaged. A novel laboratory-scale, phase-field fracture propagation model was developed to characterize the material failure mechanisms that play a significant role during the shale maturation process. The numerical model developed consists of coupled solid deformation, pore pressure, and fracture propagation mechanisms. Benchmark tests were conducted to validate model accuracy. Laboratory-grade gelatins with varying Young’s modulus were used as scaled-rock analogs in a two-dimensional Hele-Shaw cell apparatus. Yeast within the gelatin generates gas in a fashion analogous to hydrocarbon formation as shale matures. These setups allow study and visualization of host rock elastic-brittle fracture and fracture network propagation mechanisms. The experimental setup was fitted to utilize photoelasticity principles coupled with birefringence properties of gelatin to explore visually the stress field of the gelatin as the fracture network developed. Stress optics image analysis and linear elastic fracture mechanics (LEFM) principles for crack propagation were used to monitor fracture growth for each gelatin type. Observed and simulated responses suggest gas diffusion within and deformation of the gelatin matrix as predominant mechanisms for energy dissipation depending on gelatin strength. LEFM, an experimental estimation of principal stress development with fracture growth, at different stages was determined for each gelatin rheology. The interplay of gas diffusion and material deformation determines the resulting frequency and pattern of fractures. Results correlate with Young’s modulus. Experimental and computed stress fields reveal that fractures resulting from internal gas generation are similar to, but not identical to, type 1 opening mode.

Abstract New experimental and numerical techniques constitute the major recent advancements in the study of flow through porous media. However, a model that duly links the micro- and macroscales of this phenomenon is still lacking. Therefore, the present work describes a new, analytical model suitable for both Darcian and post-Darcian flow. Unlike its predecessors, most of which are based on empirical assessments or on some derivation of the Navier–Stokes equations, the presented model employed the principle of maximum entropy, along with a reduced number of premises. Nevertheless, it is compatible with classic expressions, such as Darcy’s and Forchheimer’s laws. Also, great adherence to previously published experimental results was observed. Moreover, the developed model allows for the delimitation of Darcian and post-Darcian regimes. It enabled the determination of a probabilistic distribution function of flow velocities within the pore space. Further, it bestowed richer interpretations of the physical meanings of principal flow parameters. Finally, through a new quantity called the entropy parameter, the proposed model may serve as a bridge between experimental and numerical findings both at the micro- and macroscales. Therefore, the present research yielded an analytical, entropy-based model for flow through porous media that is sufficiently general and robust to be applied in several fields of knowledge. Graphic Abstract

Abstract 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 CO2 at high pressure where CO2 becomes supercritical. Stability of foam has been studied in laboratory and field tests, and has confirmed long-term stability of foam.

Abstract The mobility of gas is greatly reduced when the injected gas is foamed. The reduction in gas mobility is attributed to the reduction in gas relative permeability and the increase in gas effective viscosity. The reduction in the gas relative permeability is a consequence of the larger amount of gas trapped when foam is present while the increase in gas effective viscosity is explicitly a function of foam texture. Therefore, understanding how foam is generated and subsequent trapped foam behavior is of paramount importance to modeling of gas mobility. In this paper, we push the envelope to enlighten our decisions of which descriptions are most physical to foam flow in porous media regarding both the flowing foam fraction and the rate of generation. We use a statistical pore network interwoven with the invasion percolation with memory algorithm to model foam flow as a drainage process and investigate the dependence of the flowing foam fraction on the pressure gradient and to shed light on foam generation mechanisms. A critical snap-off probability is required for strong foam to emerge in our network. The pressure gradient and, hence, the gas mobility reduction are very low below this critical snap-off probability. Above this snap-off probability threshold, we find that the steady-state flowing lamellae fraction scales as \((\nabla \tilde{p})^{0.19}\) in 2D lattices and as \((\nabla \tilde{p})^{0.32}\) in 3D lattices. Results obtained from our network were convolved with percolation network scaling ideas to compare the probabilities of snap-off and lamella division mechanisms in the network during the initial gas displacement at the leading edge of the gas front. At this front, during strong foam flow, lamella division is practically nonexistent in 2D lattices. In 3D lattices, lamella division occurs, but the probability of snap-off is always greater than the probability of lamella division.

Abstract Surfactant-alternating-gas (SAG) is a favored method of foam injection, in part because of excellent gas injectivity. However, liquid injectivity is usually very poor in SAG. We report a core-flood study of liquid injectivity under conditions like those near an injection well in SAG application in the field, i.e., after a prolonged period of gas injection following foam. We inject foam [gas (nitrogen) and surfactant solution] into a 17-cm-long Berea core at temperature of 90 °C with 40 bar back pressure. Pressure differences are measured and supplemented with CT scans to relate water saturation to mobilities. Liquid injectivity directly following foam is very poor. During prolonged gas injection following foam, a collapsed-foam region forms near the inlet and slowly propagates downstream, in which water saturation is reduced. This decline in liquid saturation reflects in part liquid evaporation, also pressure-driven flow and capillary effects on the core scale. In the collapsed-foam region, liquid mobility during subsequent liquid injection is much greater than downstream, and liquid sweeps the entire core cross section rather than a single finger. Mobility in the region of liquid fingering is insensitive to the quality of foam injected before gas and the duration of the period of gas injection. This implies that at the start of liquid injection in a SAG process in the field, there is a small region very near the well, crucial to injectivity, substantially different from that further out, and not described by current foam models. The results can guide the development of a model for liquid injectivity based on radial propagation of the various banks seen in the experiments.

Abstract We study the generation and flow of foam through rough-walled, fractured marble rocks that mimic natural fracture systems in carbonate reservoirs. Flow was isolated to the fracture network because of the very low rock permeability of the marble samples and foam generated in situ during co-injection of surfactant solution and gas. The foam apparent viscosities were calculated at steady pressure gradients for a range of gas fractions, and similar to foam flow in porous media, we identified two flow regimes for foam flow in fractures: a high-quality flow regime only dependent on liquid velocity and a low-quality flow regime determined by the gas and liquid velocities. Variations in local fluid saturation during co-injection were visualized and quantified using positron emission tomography combined with computed tomography.

Abstract Foam generation and transport in porous media are a proven method to improve the sweep efficiency of a flooding fluid in enhanced oil recovery process and increase the effectiveness of a treatment fluid in well intervention procedures. Foam in the porous media is often generated using surfactant alternating gas or co-injection. Although these operations result in good incremental production, the profit losses could be high due to surfactant retention and lack of water injection facilities in the target fields. One way of reducing foam generation operations expenses is by injecting the surfactant solution disperse throughout the gas phase in a process called “disperse foam.” Core-flooding experimental results have shown that disperse foam techniques reduce the surfactant retention and increase cumulative oil production. This increase means that not only the foam is being generated but also it is blocking the high mobility channels and enhancing the sweep efficiency. Additionally, the operational implementation in field operations is very simple and reduces significantly operational costs of the process. Because few laboratory core-flooding tests and field pilots have been executed using the disperse foam technique, there is a high level of uncertainty associated with the method. Besides, the models reported in the literature do not account for all the associated phenomena, including the surfactant droplets transfer between the gas and liquid phases, and the lamellae stability at low water saturation. For this reason, the development of a mechanistic disperse foam model is key to understand the phenomena associated with “disperse foam” operations. In this work, we use a previous foam mechanistic model to develop a disperse foam model that includes the physicochemical mechanisms of the foaming process a core scale. The model accounts for the foamer mass transference between the gas and liquid phases in a non-equilibrium state with a particle interception model, also accounts for the reversible and irreversible surfactant adsorption on the rock surface in dynamic conditions with a first-order kinetic model, and includes foam generation, coalescence and, transport using a population balance mechanistic model.

Abstract CO2 injection is one of the most promising techniques to enhance oil recovery. However, an unfavorable mobility ratio, reservoir heterogeneity and gravity segregation can reduce the macroscopic sweep efficiency. In situ foaming of injected CO2 is the method that has the most potential for improving sweep efficiency based on controlling CO2 mobility. This study investigates the foaming behavior of N,N,N′-trimethyl-N′-tallow-1,3-diaminopropane (DTTM) surfactant with CO2 in a transparent porous microflow model with natural rock pore structures. It focuses on the effect of the salinity induced non-Newtonian behavior of DTTM solution on foam propagation. The performance of foams stabilized by 0.5 wt% DTTM solution over the viscosity range from 0.71 (at 5 wt% NaCl) to 41 cp (at 20 wt% NaCl) was compared with conventional polymer-enhanced foams whose liquid phase contained a commonly used foaming surfactant, C15–18 Internal Olefin Sulfonate (C15–18 IOS) and a hydrolyzed polyacrylamide. Such comparisons have also provided insight into the respective impacts of liquid phase viscosification by worm-like surfactant micelles and polymer on foam texture associated with its rheological characteristics. It was found that at low aqueous phase viscosity (injection liquid viscosity of 0.71 cp) the maximum achievable viscosity of DDTM foam was around 1000 cp, which was 80 times IOS stabilized foam. The interfacial tension of DTTM was higher than that of IOS, resulting coarser foam texture and higher individual lamella resistance. An increase in DTTM solution viscosity by a factor of 33 decreased foam generation and viscosity for gas injection. This was not observed for the simultaneous injection of gas and DTTM solution. Overall, the effect of liquid phase viscosity on transient foam behavior during gas injection is similar for both DTTM and IOS regardless of the difference in the nature of viscosifying agents (WLM vs 3330 s polymer). An increase in gas injection pressure without liquid injection delayed foam propagation and reduced the magnitude of foam viscosity. The results from this study indicated that DTTM surfactant is an important alternative to commercially available polymers that have been used to enhance foam performance in porous media. This particular surfactant type also overcomes several disadvantages of polymers such as limited temperature and salinity tolerance, shear degradation, and filtering in low permeability formations.

A laboratory study of principal immiscible gas flooding schemes is reported. Very well-controlled experiments on continuous gas injection, water-alternating-gas (WAG) and alkaline–surfactant–foam (ASF) flooding were conducted. The merits of WAG and ASF compared to continuous gas injection were examined. The impact of ultra-low oil–water (o/w) interfacial tension (IFT), an essential feature of the ASF scheme along with foaming, on oil mobilisation and displacement of residual oil to waterflood was also assessed. Incremental oil recoveries and related displacement mechanisms by ASF and WAG compared to continuous gas injection were investigated by conducting CT-scanned core-flood experiments using n-hexadecane and Bentheimer sandstone cores. Ultimate oil recoveries for WAG and ASF at under-optimum salinity (o/w IFT of 10−1 mN/m) were found to be similar [60 ± 5% of the oil initially in place (OIIP)]. However, ultimate oil recovery for ASF at (near-)optimum salinity (o/w IFT of 10−2 mN/m) reached 74 ± 8% of the OIIP. Results support the idea that WAG increases oil recovery over continuous gas injection by drastically increasing the trapped gas saturation at the end of the first few WAG cycles. ASF flooding was able to enhance oil recovery over WAG by effectively lowering o/w IFT (< 10−1 mN/m) for oil mobilisation. ASF at (near-)optimum salinity increased clean oil fraction in the production stream over under-optimum salinity ASF.

Abstract Flow of nitrogen foam stabilized by alpha olefin sulfonate (C14-16 AOS) was studied in a natural sandstone porous media using X-ray Computed Tomography. Foam was generated by a simultaneous injection of gas and surfactant solution into a porous medium initially saturated with the surfactant solution. It was found that the foam undergoes a transition from a weak to a strong state at a characteristic gas saturation of Sgc = 0.75 ± 0.02. This transition coincided with a substantial reduction in foam mobility by a two-order of magnitude and also with a large reduction in overall water saturation to as low as 0.10 ± 0.02. Foam mobility transition was interpreted by the surge of yield stress as gas saturation exceeded the Sgc. We proposed a simple power-law functional relationship between yield stress and gas saturation. The proposed rheological model captured successfully the mobility transition of foams stabilized by different surfactant concentrations and for different core lengths.

Abstract Foam is to be used as a blocking agent for confining a pollutant source zone and avoid spreading in an aquifer. To this end, it is necessary to determine where injected foam flows and stays inside a porous medium. This study examines the use of electrical resistivity tomography for this purpose. Foam is injected in a large-scale 3D heterogeneous porous medium (0.84 × 0.84 × 0.84 m). During the injection, electrical resistivity tomography measurements are performed. We show that combining a large number of measurements with inversion techniques allows for the monitoring of a foam front in 3D during the injection process.

Nano-remediation is a promising in situ remediation technology. It consists in injecting reactive nanoparticles (NPs) into the subsurface for the displacement or the degradation of contaminants. However, due to the poor mobility control of the reactive nanoparticle suspension, the application of nano-remediation has some major challenges, such as high mobility of the particles, which may favor override of the contamination, and particle aggregation, which can lead to a limited distance of influence. Previous experimental studies show the potential of combining nano-remediation with foam flooding to overcome these issues. However, in order to design and optimize the process, a model which couples nanoparticle and foam transport is necessary. In this paper, a mechanistic model to describe the transport of NPs with and by a foam is presented. The model considers the delivery of nanoscale zero-valent iron (nZVI) and accounts for the processes of aggregation, attachment/detachment, and generation/destruction. Simulations show that when NPs are dispersed in the liquid phase, even in the presence of a foam, they may travel much slower than the NPs carried by the foam bubbles. This is because the nanoparticles in suspension are affected by the attachment onto the rock walls and straining at the pore throats. When the nanoparticle surface is, instead, modified in order to favor their adsorption onto the gas bubbles, NPs are carried by the foam without retardation, except for the small fraction suspended in the liquid phase. Moreover, very stable high-quality foam (\(f_\mathrm{g}\)), i.e., 80–90 vol% of gas, can be attained using properly surface-modified nZVI (i.e., a nanoparticle-stabilized foam), allowing a significant reduction of water for the operation, while increasing the efficiency of nZVI delivery, even in a low-permeability medium within the shallow subsurface.

Abstract Foam is promising for the remediation of non-aqueous phase liquids (NAPL) source zones; however, the production of foam and its behavior in porous media are poorly understood. A methodology for the selection of surfactants suitable for foam production applied to NAPL remediation was developed. Two criteria were initially used for surfactant selection: foamability as evaluated by the Ross–Miles test and interfacial tension reduction measured with the pendant drop method. Three promising surfactants were identified and used in sand column tests: Genapol LRO because it produced the highest foam height in the Ross–Miles test, Ammonyx Lo which exhibited the lowest interfacial tension with p-xylene and had the second highest foam height, and Tomadol 900 because it showed intermediate results in both tests. Viscosity was found to be proportional to foamability. Genapol LRO produced a foam so viscous that it destabilized by the end of the experiment. Ammonyx Lo produced a less viscous foam but with a stable front. Tomadol 900 produced an unstable foam with poor viscosity. Results from column tests gave indications of optimal conditions needed to produce a stable and viscous foam front.

Abstract Models for simulating foam-based displacements fall into two categories: population-balance (PB) models that derive explicitly foam texture or bubble size from pore-level mechanisms related to lamellas generation and coalescence, and steady-state semi-empirical (SE) models that account implicitly for foam texture effects through a gas mobility reduction factor. This mobility reduction factor has to be calibrated from a large number of experiments on a case-by-case basis in order to match the physical effect of parameters impacting foam flow behaviour such as fluids saturation and velocity. This paper proposes a methodology to set up steady-state SE models of foam flow on the basis of an equivalence between SE model and PB model under steady-state flow conditions. The underlying approach consists in linking foam mobility and foam lamellas density (or texture) data inferred from foam corefloods performed with different foam qualities and velocities on a series of sandstones of different permeabilities. Its advantages lie in a deterministic non-iterative transcription of flow measurements into texture data and in a separation of texture effects and shear-thinning (velocity) effects. Then, scaling of foam flow parameters with porous medium permeability is established from the analysis of calibrated foam model parameters on cores of different permeability, with the help of theoretical representations of foam flow in a confined medium. Although they remain to be further confirmed from other well-documented experimental data sets, the significance of those scaling laws is great for the assessment of foam-based enhanced oil recovery (EOR) processes because foam EOR addresses heterogeneous reservoirs. Simulations of foam displacement in a reservoir cross section demonstrate the necessity to scale foam SE models with respect to facies heterogeneity for reliable evaluation.

Abstract This study investigates how to determine the optimal supercritical CO2 foam injection strategies, in terms of total injection rate (or injection pressure, equivalently) and injection foam quality, to place injected foams deep and far into the reservoir. Two different mechanisms that limit field foam propagation, such as “conversion from strong foam to weak foam” and “gravity segregation,” are examined separately, and the results are combined together. The first is performed by using a mechanistic foam model based on bubble population balance, while the second is conducted by an analytical model (called Stone and Jenkins model) and reservoir simulations with a commercial software (CMG-STARS). Note that the gas-phase mobility, required as a key input parameter for gravity segregation simulations, is calibrated by the mechanistic model, which is a significant advance in this study. The results from both mechanisms show in general that foam propagation distance increases with increasing injection pressure or rate (which is often limited by the formation fracturing pressure) and increases with decreasing foam quality down to a certain threshold foam quality below which the distance is not sensitive to foam quality any longer. It is found that the mobilization pressure gradient (i.e., the pressure gradient above which foam films are mobilized to create a population of bubbles) plays a key role to determine the distance. Therefore, the injection of supercritical CO2 foams with lower mobilization pressure gradient should be more favored in field applications. As a step prior to real-world reservoir applications, this study deals with a relatively ideal reservoir (i.e., large homogeneous cylindrical reservoir) focusing on the steady state after foam treatment in the absence of oil.

The optimization of foam injection in porous media for enhanced oil recovery or soil remediation requires a large screening of surfactant formulations. Tests of foam stability in vials often used quick criteria to accelerate selection and ensure performance in porous media. Using a selection of surfactant formulations of different chemistry and foam behaviors, the correlation between foam in vials and in porous media is investigated. Along with foam stability, foamability which quantifies the ability to create foam is shown to play a role in the maximum apparent viscosity. This is a first evidence that foamability is a key parameter for the maximum apparent viscosity reached in a steady state of apparent viscosity. To account for the relative contribution of foamability and foam stability, a parameter is inspired from the widely accepted model of population balance. These results support a workflow based on large foam screening in a first step and sandpack experiments in a second step, prior to more representative but longer coreflood tests. Finally, these experimental data emphasize the relevance of population balance simulations as a description based on experimental measurement. Second, the flow visualization in the sandpack allows the extraction of a local velocity of the liquid in the flowing foam. This parameter gives an experimental evidence that the transition between the high-quality and low-quality regime corresponds to a change in the efficiency of foam lamellae network to transport gas concomitantly to liquid. The local liquid velocity also represents an indirect and easy measurement of flow structure, and it is shown to change from one formulation to another. This observation highlights the complex relation between local microstructure and physical chemistry of surfactants.

Abstract Many oil reservoirs are at high temperatures and contain brines of high salinity and hardness. The focus of this work is to develop robust foams stabilized by a mixture of nanoparticles and surfactants for such reservoirs. Two types of silica nanoparticles (Si-NP1, Si-NP2) with different grafted low molecular weight ligands/polymers were used. First, aqueous stability analysis of these nanoparticle dispersions were conducted at high-temperature (80 °C) and high-salinity conditions (API Brine; 8 wt% NaCl and 2 wt% CaCl2). The screened nanoparticles were used in combination with an anionic surfactant. Second, bulk foam and emulsion stability tests were performed to investigate their performance in stabilizing the air–water and oil–water interface, respectively. Third, foam flow experiments in the absence of oil were performed to characterize the foam rheology. Finally, oil displacement experiments were conducted in an in-house, custom-built 2D sand pack with flow visualization. The sand pack had two layers of different mesh size silica sand which yielded a permeability contrast of 6:1. Brine floods followed by foam floods (80% quality) were conducted, and foam flow dynamics were monitored. The grafting of low molecular weight polymers/ligands on silica nanoparticle surfaces resulted in steric stabilization under high-temperature and high-salinity conditions. Foam flow experiments revealed a synergy between Si-NP2 and surfactant in stabilizing foam in the absence of crude oil. In the oil displacement experiments in the layered sand packs, the waterflood recoveries were low (~ 33% original oil in place) due to channeling in the top high-permeability zone, leaving the bottom low-permeability zone completely unswept. Foam flooding with just the surfactant leads to a drastic improvement in sweep efficiency. It resulted in an incremental oil recovery as high as 43.3% OOIP. Different cross-flow behaviors were observed during foam flooding. Significant cross-flow of oil from low-permeability zone to high-permeability zone was observed for the case of surfactant. Conversely, the Si-NP2-surfactant blend resulted in no cross-flow from the low-permeability region with complete blocking of the high-permeability region due to the formation of in situ emulsion. Such selective plugging of high-perm zones using nanoparticles with tailored surface coating and concentration has significant potential in recovering oil from heterogeneous reservoirs.

A high-resolution (~ 1 μm) three-dimensional (3D) X-ray micro-computed tomography (μ-CT) was used to nondestructively detect pore characteristics in silicon particle preforms with different starch contents (10%, 20% and 30%) and particle sizes (20, 50 and 90 μm). A pressure infiltration equipment was made to infiltrate the preforms. The preforms were infiltrated at constant temperature (800 °C) and pressure (400 kPa) with different pressure-applied times (3, 5, 8, 11 and 15 s). Since the discrepancy between the infiltration heights measured in the experiment and calculated with the general infiltration equation was found, a modified infiltration equation was proposed considering the pore characteristic parameters from 3D μ-CT. The results of the modified infiltration equation showed good agreement with the experiment.

Abstract Under certain flow conditions, fluid flow through porous media starts to deviate from the linear relationship between flow rate and hydraulic gradient. At such flow conditions, Darcy’s law for laminar flow can no longer be assumed and nonlinear relationships are required to predict flow in the Forchheimer regime. To date, most of the nonlinear flow behavior data is obtained from flow experiments on packed beds of uniformly graded granular materials (Cu = d60/d10 < 2) with various average grain sizes, ranging from sands to cobbles. However, natural deposits of sand and gravel in the subsurface could have a wide variety of grain size distributions. Therefore, in the present study we investigated the impact of variable grain size distributions on the extent of nonlinear flow behavior through 18 different packed beds of natural sand and gravel deposits, as well as composite filter sand and gravel mixtures within the investigated range of uniformity (2.0 < Cu < 17.35) and porosity values (0.23 < n < 0.36). Increased flow resistance is observed for the sand and gravel with high Cu values and low porosity values. The present study shows that for granular material with wider grain size distributions (Cu > 2), the d10 instead of the average grain size (d50) as characteristic pore length should be used. Ergun constants A and B with values of 63.1 and 1.72, respectively, resulted in a reasonable prediction of the Forchheimer coefficients for the investigated granular materials.

Gas migration in coal is strongly controlled by surface diffusion of adsorbed gas within the coal matrix. Surface diffusion coefficients are obtained by inverse modelling of transient gas desorption data from powdered coals. The diffusion coefficient is frequently considered to be dependent on time and initial pressure. In this article, it is shown that the pressure dependence can be eliminated by performing a joint inversion of both the diffusion coefficient and adsorption isotherm. A study of the log–log slope of desorbed gas production rate against time reveals that diffusion within the individual coal particles is a multi-rate process. The application of a power-law probability density function of diffusion rates enables the determination of a single gas diffusion coefficient that is constant in both time and initial pressure.

Abstract We present a theoretical and numerical study on the (in)stability of the interface between two immiscible liquids, i.e., viscous fingering, in angled Hele-Shaw cells across a range of capillary numbers (Ca). We consider two types of angled Hele-Shaw cells: diverging cells with a positive depth gradient and converging cells with a negative depth gradient, and compare those against parallel cells without a depth gradient. A modified linear stability analysis is employed to derive an expression for the growth rate of perturbations on the interface and for the critical capillary number (\(Ca_c\)) for such tapered Hele-Shaw cells with small gap gradients. Based on this new expression for \(Ca_c\), a three-regime theory is formulated to describe the interface (in)stability: (i) in Regime I, the growth rate is always negative, thus the interface is stable; (ii) in Regime II, the growth rate remains zero (parallel cells), changes from negative to positive (converging cells), or from positive to negative (diverging cells), thus the interface (in)stability possibly changes type at some location in the cell; (iii) in Regime III, the growth rate is always positive, thus the interface is unstable. We conduct three-dimensional direct numerical simulations of the full Navier–Stokes equations, using a phase field method to enforce surface tension at the interface, to verify the theory and explore the effect of depth gradient on the interface (in)stability. We demonstrate that the depth gradient has only a slight influence in Regime I, and its effect is most pronounced in Regime III. Finally, we provide a critical discussion of the stability diagram derived from theoretical considerations versus the one obtained from direct numerical simulations.

Abstract X-ray microcomputed tomography (X-ray μ-CT) is a rapidly advancing technology that has been successfully employed to study flow phenomena in porous media. It offers an alternative approach to core scale experiments for the estimation of traditional petrophysical properties such as porosity and single-phase flow permeability. It can also be used to investigate properties that control multiphase flow such as rock wettability or mineral topology. In most applications, analyses are performed on segmented images obtained employing a specific processing pipeline on the greyscale images. The workflow leading to a segmented image is not straightforward or unique and, for most of the properties of interest, a ground truth is not available. For this reason, it is crucial to understand how image processing choices control properties estimation. In this work, we assess the sensitivity of porosity, permeability, specific surface area, in situ contact angle measurements, fluid–fluid interfacial curvature measurements and mineral composition to processing choices. We compare the results obtained upon the employment of two processing pipelines: non-local means filtering followed by watershed segmentation; segmentation by a manually trained random forest classifier. Single-phase flow permeability, in situ contact angle measurements and mineral-to-pore total surface area are the most sensitive properties, as a result of the sensitivity to processing of the phase boundary identification task. Porosity, interfacial fluid–fluid curvature and specific mineral descriptors are robust to processing. The sensitivity of the property estimates increases with the complexity of its definition and its relationship to boundary shape.

Anisotropy is a very typical observation in the intrinsic bedding structure of coal. To study the influence of anisotropy of coal structure and stress state on the evolution of permeability, a newly developed multifunctional true triaxial geophysical apparatus was used to carry out mechanical and seepage experiments on bedded coal. The permeability and deformation of three orthogonal directions in cubic coal samples were collected under true triaxial stress. It has detected the significant permeability anisotropy, and the anisotropy is firmly determined by the bedding direction and stress state of coal. Based on the true triaxial mechanical and seepage test results, the coal with bedding was simplified to be represented by a cubic model, and the dynamic anisotropic (D-A) permeability model was derived by considering the influence of bedding and stress state. The rationality of the permeability model was verified by the experimental data. Comparing the permeability model with Wang and Zang (W–Z) model, Cui and Bustin (C–B) model and Shi and Durucan (S–D) model, it is found that the theoretical calculated values of the D-A permeability model are in better agreement with the experimental measured values, reflecting the superiority of the D-A permeability model. Based on incorporating the model of D-A permeability under the concept of multiphysics field coupling, the numerical simulation experiments of coal seam gas extraction with different initial permeability anisotropic ratios were carried out by using COMSOL multiphysics simulator. The influence of initial permeability anisotropy ratio on gas pressure distribution in coal seam during gas extraction was explored, which provides theoretical guidance for the optimization of borehole layout for gas extraction in coal mine.

Abstract A dual-continuum model can offer a practical approach to understanding first-order behaviours of poromechanically coupled multiscale systems. To close the governing equations, constitutive equations with models to calculate effective constitutive coefficients are required. Several coefficient models have been proposed within the literature. However, a holistic overview of the different modelling concepts is still missing. To address this we first compare and contrast the dominant models existing within the literature. In terms of the constitutive relations themselves, early relations were indirectly postulated that implicitly neglected the effect of the mechanical interaction arising between continuum pressures. Further, recent users of complete constitutive systems that include inter-continuum pressure coupling have explicitly neglected these couplings as a means of providing direct relations between composite and constituent properties, and to simplify coefficient models. Within the framework of micromechanics, we show heuristically that these explicit decouplings are in fact coincident with bounds on the effective parameters themselves. Depending on the formulation, these bounds correspond to end-member states of isostress or isostrain. We show the impacts of using constitutive coefficient models, decoupling assumptions and parameter bounds on poromechanical behaviours using analytical solutions for a 2D model problem. Based on the findings herein, we offer recommendations for how and when to use different coefficient modelling concepts.

We present a new statistical variance approach for characterizing heterogeneities related to pore spaces in reservoir rocks. Laboratory-based computer microtomography data for reservoir sandstone samples were acquired and processed using advanced image segmentation techniques. The samples were processed using a method based on the digital rock physics concept using the high-performance Navier–Stokes flow solver in the GeoDict commercial software package. The digitized structures were subjected to computational fluid dynamic simulations. The effects of structural matrix modifications caused by the precipitation of minerals on the porosity–permeability relationship and the characterization of the representative elementary volume were assessed. The variances of the digital flow fields were compared at the pore scale (6 µm). The algorithm for analysing variance was benchmarked using a synthetic dataset that provided artificial repetitive structural patterns at both low and high resolutions. This gave an estimate of the sensitivity of the proposed algorithm to minor inhomogeneities. Representative elementary volume variance analysis was performed by comparing the correlation coefficients for various pore–grain composition patterns with the variances of simulated mean flow velocities. Probability density functions indicate that the flow velocities and pore space geometries differed greatly for different samples. The normalized probability density functions of the mean flows shifted to higher velocities as the resolution decreased. We found that a representative elementary volume analysis was more reliably achieved by analysing the mean flow velocity variance than by analysing the pore microstructure alone.

Collection of landfill gas by horizontal perforated wells is studied. The problem combines flow through porous media in the landfill and unobstructed pipe flow in the well. Respective analytical solutions to flow equations are used in an iterative numerical procedure to solve the coupled system. Realistic landfill input parameters confirm the feasibility of estimates obtained with the model. The study identifies flow control parameters and furnishes tools to evaluate surface flux and radius of influence for this type of well.

The consequences of local thermal non-equilibrium (LTNE) on both stationary and oscillatory weak nonlinear stability of gravity-driven porous convection in an incompressible fluid-saturated rotating porous layer are investigated. A stability map is drawn in the Darcy–Taylor and scaled Vadasz number plane to demarcate the regions of stationary and oscillatory convection, and thereby, co-dimension-2 points are determined. It is found that the effect of increasing interphase heat transfer coefficient is to enhance the region of stationary convection and decrease the region of oscillatory convection. The complex Ginzburg–Landau equations are derived using the multi-scale method, and pitchfork and Hopf bifurcations occur at stationary and oscillatory critical Darcy–Rayleigh numbers, respectively. The linear and nonlinear oscillatory neutral curves are illustrated, and at the quartic point, the transition from supercritical to subcritical bifurcations is identified for the governing parameters. The impact of LTNE model is to enhance the region of forward bifurcation and post-transient amplitude compared to LTE case. Heat transfer is obtained in terms of Nusselt number for both stationary and oscillatory convection. The region of enhancement in heat flux for oscillatory convection in the smaller scaled Vadasz number domain with increasing Darcy–Taylor number increases with increasing interphase heat transfer coefficient and the porosity-modified conductivity ratio.

Abstract Luikov’s equations of heat and mass transfer with pressure gradient have significant applications especially in the case of intensive drying in porous media. The established analytical solution of Luikov’s equations including pressure gradient effect in a complete form is presented in this work. The existence of complex roots in the analytical solution describing intensive drying with pressure gradient was overlooked in literature. A Matlab function capable of searching and sorting out the complex eigenvalues is also showcased. Three test cases are analyzed and compared with numerical solutions: one theoretical case to emphasize the importance of complex roots in analytical solution while two cases of drying process in ceramic and barley kernel to ensure practical applicability. Excellent matching between analytical and numerical results is noticed when complex eigenvalues are included.

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.  https://doi.org/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.

Determining the time of breakthrough of injected water is important when assessing waterflood in an oil reservoir. Breakthrough time distribution for a passive tracer (for example water) in percolation porous media (near the percolation threshold) gives insights into the dynamic behavior of flow in geometrically complex systems. However, the application of such distribution to realistic two-phase displacements can be done based on scaling of all parameters. Here, we propose two new approaches for scaling of breakthrough time (characteristic times) in two-dimensional flow through percolation porous media. The first is based on the flow geometry, and the second uses the flow parameters of a representative homogenous model. We have tested the effectiveness of these two approaches using a large number of dynamic simulations. The results show significant improved distribution curves for the breakthrough (transit) time between an injector and a producer located in a heterogeneous porous medium in comparison with the previous scaling methods.

With X-ray computed tomography still being flawed as a result of limitations in terms of spatial resolution and cost, toxic mercury intrusion porosimetry (MIP) is nowadays the prevailing technique to determine PSDs of most porous media. Recently, yield stress fluids porosimetry method (YSM) has been identified as a promising clean alternative to MIP. This technique is based on the particular percolation patterns followed by yield stress fluids, which only flow through certain pores when injected at a given pressure gradient. In previous works, YSM was used to characterize natural and synthetic porous media, and the results were compared with MIP showing reasonable agreement. However, considerable uncertainty still remains regarding the characterized pore dimension with each method arising from the highly complex geometry of the interstices in real porous media. Therefore, a critical stage for the validation of YSM consists in achieving successful characterization of model porous media with well-known pore morphology and topology. With this objective in mind, a set of four packs of glass beads each with a given monodisperse bead size were characterized in the present work using different porosimetry methods: experimental YSM, numerically simulation of MIP and pore-network extraction from a 3D image. The results provided by these techniques were compared, allowing the identification of the pore dimensions being characterized in each case. The results of this research elucidate the causes of the discrepancies between the considered porosimetry methods and demonstrate the usefulness of the PSD provided by YSM when predicting flow in porous media.

Understanding the long-term evolution of coal permeability under the influence of gas adsorption-induced multiple processes is crucial for the efficient sequestration of CO2, coalbed methane extraction and enhanced coal bed methane recovery. In previous studies, coal permeability is normally measured as a function of gas pressure under the conditions of constant effective stresses, uniaxial strains and constant confining pressures. In all these experiments, an equilibrium state between coal matrix and fracture is normally assumed. This assumption has essentially excluded the effect of matrix–fracture interactions on the evolution of coal permeability. In this study, we hypothesize that the current equilibrium assumption is responsible for the discrepancy between theoretical expectations and experimental measurements. Under this hypothesis, the evolution of coal permeability is determined by the effective stress gap between coal matrix and fracture. This hypothesis is tested through an experiment of CO2 injection into a coal core under the constant effective stress. In this experiment, the effective stress in the fracture system is unchanged while the effective stress in the matrix evolves as a function of time. In the experiment, the coal permeability was measured continuously throughout the whole period of the experiment (~ 80 days). The experimental results show that the core expands rapidly at the beginning due to the gas injection-induced poroelastic effect. After the injection, the core length remains almost unchanged. But, the measured permeability declines from 60 to 0.48 μD for the first month. It rebounds slowly for the subsequent 2 months. These results indicate that the effective stress gap has a significant impact on the evolution of coal permeability. The switch of permeability from the initial reduction (the first 30 days) to rebound (the subsequent 50 days) suggests a transition of matrix deformation from nearby the fracture wall to further away area. These findings demonstrate that the evolution of coal permeability is primarily controlled by the spatial transformation of effective stresses between matrix and fracture.

Transport processes such as the dispersion and mixing of solutes are governed by the interplay of advection and diffusion, where advection acts to organise fluid streamlines and diffusion acts to randomise solute molecules. Thus, the structure and organisation of streamlines, termed the Lagrangian kinematics of the flow, is central to the understanding and modelling of these transport processes. A key question is whether the streamlines in three-dimensional (3D) Darcy flows can wander freely through the fluid domain, or whether all streamlines of the flow are organised into a series of smooth, non-intersecting two-dimensional (2D) surfaces. The existence of such a foliation of surfaces constrains the Lagrangian kinematics in a manner similar to that of 2D flows, which in turn constrains the allowable transport processes. In a series of pioneering studies, Sposito (Water Resour. Res., 30(8):2395–2401, 1994; Adv. Water Resour., 24(7):793–801, 2001) argues that steady Darcy flow in locally isotropic media gives rise to Lamb surfaces, 2D material surfaces which are spanned by both the streamlines and vortex lines (field lines of the vorticity vector) of the flow. Hence, the existence of these surfaces renders the kinematics of such 3D steady Darcy flow as two dimensions. This topological constraint strongly affects transverse mixing and dispersion because 2D steady flow fields limit the rate of deformation of fluid elements and can only admit zero hydrodynamic transverse dispersion. In this study, however, we show that Lamb surfaces are not ubiquitous to all steady Darcy flows in locally isotropic media. We derive the conditions for when Lamb surfaces exist in such Darcy flows, and discuss the implications of these findings for the transport, mixing, and dispersion of solutes.

Abstract A reliable and practical method of hygric property characterisation is a major determinant in properly analysing hygric performance of built constructions. The current empirical approach, however, yields data that are still incomplete, not fully representative, and not entirely reliable. In this study, hygric properties are therefore determined directly from pore structure information by applying stationary unsaturated pore-scale physics to model moisture storage and transport. The pore space is captured by visualisation techniques and further extracted into a discrete network of variably shaped pore bodies interconnected by pore throats. In this network, corner and surface adsorption and capillary condensation are simulated to define the moisture storage. Air entrapment is furthermore considered to determine the capillary moisture content. The resulting spatial moisture distribution in the pore network allows moisture to flow, in wet elements in the liquid phase and in dry elements in the vapour phase. The adsorbed corner and surface films moreover enable additional liquid flow. These various simultaneous flows aggregately define the transport property. A comparison to measured data validates the presented hygric property model.

Tortuosity is an important physical characteristic of porous materials; for example, it is a critical parameter determining the effective diffusion coefficient dictating mixing between miscible fluids in porous rock structures as is relevant to enhanced gas recovery processes. Accurate measurement of tortuosity remains challenging, resulting in various definitions dictated largely by the measurement protocol applied. Here, we focus primarily on ‘diffusive’ tortuosity (τd), which is defined as the ratio of the bulk fluid diffusion coefficient to the restricted diffusion coefficient applicable to the porous media under study. Specifically, we consider carbonate rock cores ranging in permeability from 2 to 5300 mD and adapt pulsed field gradient (PFG) NMR methodology such that accurate measurements of tortuosity are obtained over a sufficiently representative length scale of the porous media. To this end, we deploy supercritical methane as a probe molecule exploiting both its high mobility and proton density. Tortuosity measurements are shown to be independent of both pressure and diffusion observation time, conclusively proving that our measurements are in the asymptotic regime in which all of the pore space is adequately sampled by the diffusing methane molecules. The resultant ‘diffusive’ tortuosity measurements (which ranged from 3.1 to 5.6) are then compared against independent electrical conductivity measurements of tortuosity using a two-electrode impedance technique applied to the carbonate samples saturated with brine solution. Agreement between the ‘diffusive tortuosity,’ as measured by PFG NMR, and ‘electrical’ tortuosity was remarkably good (within 10%), given the very different measurements techniques used, for most of the carbonate rock samples considered.

Abstract In this paper, an approximate integral equation solution for a horizontal, unsteady flow of two viscous incompressible fluids is derived. Wettability variation of the porous medium is also considered in the present study. The non-wetting phase recovery fraction is obtained in terms of dimensionless time for 1D model from the derived solution. The derived approximate recovery fraction has been compared with the existing experimental data and empirical correlations for spontaneous imbibition from water-wet medium. It is observed that the normalized water saturation profiles show sharp gradients indicating some instability of the water front. Hence, a linear stability analysis is developed to study viscous instability of the co-current spontaneous imbibition. Viscous instability phenomenon for a two-phase oil–water system is also discussed. The corresponding instability number related to co-current imbibition recovery is estimated. The present study clearly spells out the effect of the instability number on the recovery process and estimates a critical instability number for co-current spontaneous imbibition.

Abstract This paper investigates a peculiar case of thermal convection in a vertical porous prism with impermeable and partially conducting walls. We facilitate the analysis in the numerical finite-element environment alongside with analytical considerations, in special cases where direct solutions are feasible. The present eigenvalue problem results in a non-normal-mode behaviour in the horizontal cross-sectional plane. Further, it is identified that the stagnation points for the horizontal flow are displaced from the extremal points of the temperature perturbation, for both symmetric and antisymmetric eigenfunctions. In addition, the corresponding normal-mode counterparts are provided from an analogy solution. We show that the critical Rayleigh number decreases with increasing Robin parameter values for all of the investigated aspect ratios. Finally, the influence of the aspect ratio on the critical Rayleigh number for the fully conducting wall case is identified. An asymptotic benchmark case of the Robin condition is validated from well-known analytical solutions which confirm the effectiveness of the predictions made in this paper. In fact, this is the first contribution that reports a three-dimensional geometry with a two-dimensional non-normal mode.

The measurement of fluid pressure inside pores is a major challenge in experimental studies of two-phase flow in porous media. In this paper, we describe the manufacturing procedure of a micro-model with integrated fibre optic pressure sensors. They have a circular measurement window with a diameter of 260 μ m , which enables the measurement of pressure at the pore scale. As a porous medium, we used a PDMS micro-model with known physical and surface properties. A given pore geometry was produced following a procedure we had developed earlier. We explain the technology behind fibre optic pressure sensors and the procedure for integrating these sensors into a micro-model and demonstrate their utility for the measurement of pore pressure under transient two-phase flow conditions. Finally, we present and analyse results of single and two-phase flow experiments performed in the micro-model and discuss the link between small-scale fast pressure changes with pore-scale events.

Macroscale three-dimensional modeling of fluid flow in a thin porous layer under unsaturated conditions is a challenging task. One major issue is that such layers do not satisfy the representative elementary volume length-scale requirement. Recently, a new approach, called reduced continua model (RCM), has been developed to describe multiphase fluid flow in a stack of thin porous layers. In that approach, flow equations are formulated in terms of thickness-averaged variables and properties. In this work, we have performed a set of experiments, where a wet 260 - μ m -thin porous layer was placed on top of a dry layer of the same material. We measured the change of average saturation with time using a single-sided low-field nuclear magnetic resonance device known as NMR-MOUSE. We have employed both RCM and the traditional Richards equation-based models to simulate our experimental results. We found that the traditional unsaturated flow model cannot simulate experimental results satisfactorily. Very close agreement was obtained by including the dynamic capillary term as postulated by Hassanizadeh and Gray in the traditional equations. The reduced continua model was found to be in good agreement with the experimental result without adding dynamic capillarity term. Moreover, the computational effort needed for RCM simulations was one order of magnitude less than that of traditional models.

Hysteresis in the saturation versus capillary pressure curves of neutrally wettable fibrous media was simulated with a random pore network model using a Voronoi diagram approach. The network was calibrated to fit experimental air-water capillary pressure data collected for carbon fibre paper commonly used as a gas diffusion layer in fuel cells. These materials exhibit unusually strong capillary hysteresis, to the extent that water injection and withdrawal occur at positive and negative capillary pressures, respectively. Without the need to invoke contact angle hysteresis, this capillary behaviour is re-produced when using a pore-scale model based on the curvature of a meniscus passing through the centre of a toroid. The classic Washburn relation was shown to produce erroneous results, and its use is not recommended when modelling fibrous media. The important effect of saturation distribution on the effective diffusivity of the medium was also investigated for both water injection and withdrawal cases. The findings have bearing on the understanding of both capillarity in fibrous media and fuel cell design.

During infiltration of water in soil, menisci form at the interface of water, grains, and air in the pores, inducing suction due to surface tension. Due to the random distribution of interconnected pores of different sizes, characteristic of porous media, differences in suction and friction inside pores give a diffusing infiltration front. The process of infiltration is often simulated by solving Richards' equation in which the water flux is calculated with Darcy's law. Underlying Darcy's law is the assumption that the gradients in flow potential and the flow resistance due to viscous forces are independent from each other. This paper shows that these parameters are dependent and negatively correlated. A new method for calculating flows in unsaturated porous media has been developed to evaluate the impact of the covariance on infiltration predictions. The results show that the impact is significant and leads to a reduction in infiltration rate and mean friction experienced during infiltration. The method thereby provides a physical explanation for the subdiffusion observed during water infiltration in soil and is consequently expected to provide more insights into the processes of infiltration.

Coated paper is an example of a multi-layer porous medium, involving a coating layer along the two surfaces of the paper and a fibrous layer in the interior of the paper. The interface between these two media needs to be characterized in order to develop relevant modeling tools. After careful cutting of the paper, a cross section was imaged using focused ion beam scanning electron microscopy. The resulting image was analyzed to characterize the coating layer and its transition to the fibrous layer. Such image analysis showed that the coating layer thickness is highly variable, with a significant fraction of it being thinner than a minimum thickness required to keep ink from invading into the fibrous layer. The overall structure of the coating and fibrous layers observed in this analysis provide insights into how the system should be modeled, with the resulting conclusion pointing to a specific kind of multi-scale modeling approach.

We present an experimental study of dissolution-driven convection in a three-dimensional porous medium formed from a dense random packing of glass beads. Measurements are conducted using the model fluid system MEG/water in the regime of Rayleigh numbers, R a = 2000 - 5000 . X-ray computed tomography is applied to image the spatial and temporal evolution of the solute plume non-invasively. The tomograms are used to compute macroscopic quantities including the rate of dissolution and horizontally averaged concentration profiles, and enable the visualisation of the flow patterns that arise upon mixing at a spatial resolution of about ( 2 × 2 × 2 ) mm 3 . The latter highlights that under this Ra regime convection becomes truly three-dimensional with the emergence of characteristic patterns that closely resemble the dynamical flow structures produced by high-resolution numerical simulations reported in the literature. We observe that the mixing process evolves systematically through three stages, starting from pure diffusion, followed by convection-dominated and shutdown. A modified diffusion equation is applied to model the convective process with an onset time of convection that compares favourably with the literature data and an effective diffusion coefficient that is almost two orders of magnitude larger than the molecular diffusivity of the solute. The comparison of the experimental observations of convective mixing against their numerical counterparts of the purely diffusive scenario enables the estimation of a non-dimensional convective mass flux in terms of the Sherwood number, S h = 0.025 R a . We observe that the latter scales linearly with Ra, in agreement with both experimental and numerical studies on thermal convection over the same Ra regime.

The prediction of water table height in unconfined layered porous media is a difficult modelling problem that typically requires numerical simulation. This paper proposes an analytical model to approximate the exact solution based on a steady-state Dupuit-Forchheimer analysis. The key contribution in relation to a similar model in the literature relies in the ability of the proposed model to consider more than two layers with different thicknesses and slopes, so that the existing model becomes a special case of the proposed model herein. In addition, a model assessment methodology based on the Bayesian inverse problem is proposed to efficiently identify the values of the physical parameters for which the proposed model is accurate when compared against a reference model given by MODFLOW-NWT, the open-source finite-difference code by the U.S. Geological Survey. Based on numerical results for a representative case study, the ratio of vertical recharge rate to hydraulic conductivity emerges as a key parameter in terms of model accuracy so that, when appropriately bounded, both the proposed model and MODFLOW-NWT provide almost identical results.

Groundwater flow models are usually subject to uncertainty as a consequence of the random representation of the conductivity field. In this paper, we use a Gaussian process model based on the simultaneous dimension reduction in the conductivity input and flow field output spaces in order quantify the uncertainty in a model describing the flow of an incompressible liquid in a random heterogeneous porous medium. We show how to significantly reduce the dimensionality of the high-dimensional input and output spaces while retaining the qualitative features of the original model, and secondly how to build a surrogate model for solving the reduced-order stochastic model. A Monte Carlo uncertainty analysis on the full-order model is used for validation of the surrogate model.

In this paper, we develop a surrogate modelling approach for capturing the output field (e.g. the pressure head) from groundwater flow models involving a stochastic input field (e.g. the hydraulic conductivity). We use a Karhunen-Loève expansion for a log-normally distributed input field and apply manifold learning (local tangent space alignment) to perform Gaussian process Bayesian inference using Hamiltonian Monte Carlo in an abstract feature space, yielding outputs for arbitrary unseen inputs. We also develop a framework for forward uncertainty quantification in such problems, including analytical approximations of the mean of the marginalized distribution (with respect to the inputs). To sample from the distribution, we present Monte Carlo approach. Two examples are presented to demonstrate the accuracy of our approach: a Darcy flow model with contaminant transport in 2-d and a Richards equation model in 3-d.

Liquid penetration into thin porous media such as paper is often simulated using continuum-scale single-phase Darcy's law. The underlying assumption was that a sharp invasion front percolates through the layer. To explore this ambiguous assumption and to understand the controlling pore-scale mechanisms, we have developed a dynamic pore-network model to simulate imbibition of a wetting phase from a droplet into a paper coating layer. The realistic pore structures are obtained using the FIB-SEM imaging of the coating material with a minimum resolution of 3.5 nm. Pore network was extracted from FIB-SEM images using Avizo software. Data of extracted pore network are used for statistically generating pore network. Droplet sizes are chosen in the range of those applicable in inkjet printing. Our simulations show no sharp invasion front exists and there is the presence of residual non-wetting phase. In addition, penetration of different sizes of droplets of different material properties into the pore network with different pore body and pore throat sizes are performed. We have found an approximately linear decrease in droplet volume with time. This contradicts the expected t -behavior in vertical imbibition that is obtained using macroscopic single-phase Darcy's law. With increase in flow rate, transition of imbibition invasion front from percolation-like pattern to a more sharper one with less trapping of non-wetting phase is also reported. Our simulations suggest that the single-phase Darcy's law does not adequately describe liquid penetration into materials such as paper coating layer. Instead Richards equation would be a better choice.

We present an experimental and numerical study of immiscible two-phase flow of Newtonian fluids in three-dimensional (3D) porous media to find the relationship between the volumetric flow rate (Q) and the total pressure difference ([Formula: see text]) in the steady state. We show that in the regime where capillary forces compete with the viscous forces, the distribution of capillary barriers at the interfaces effectively creates a yield threshold ([Formula: see text]), making the fluids reminiscent of a Bingham viscoplastic fluid in the porous medium. In this regime, Q depends quadratically on an excess pressure drop ([Formula: see text]). While increasing the flow rate, there is a transition, beyond which the overall flow is Newtonian and the relationship is linear. In our experiments, we build a model porous medium using a column of glass beads transporting two fluids, deionized water and air. For the numerical study, reconstructed 3D pore networks from real core samples are considered and the transport of wetting and non-wetting fluids through the network is modeled by tracking the fluid interfaces with time. We find agreement between our numerical and experimental results. Our results match with the mean-field results reported earlier.

In this study, a grain-scale modelling technique has been developed to generate the capillary pressure-saturation curves for swelling granular materials. This model employs only basic granular properties such as particles size distribution, porosity, and the amount of absorbed water for swelling materials. Using this model, both drainage and imbibition curves are directly obtained by pore-scale simulations of fluid invasion. This allows us to produce capillary pressure-saturation curves for a large number of different packings of granular materials with varying porosity and/or amount of absorbed water. The algorithm is based on combining the Discrete Element Method for generating different particle packings with a pore-unit assembly approach. The pore space is extracted using a regular triangulation, with the centres of four neighbouring particles forming a tetrahedron. The pore space within each tetrahedron is referred to as a pore unit. Thus, the pore space of a particle packing is represented by an assembly of pore units for which we construct drainage and imbibition capillary pressure-saturation curves. A case study on Hostun sand is conducted to test the model against experimental data from literature and to investigate the required minimum number of particles to have a Representative Elementary Volume. Then, the capillary pressure-saturation curves are constructed for Absorbent Gelling Material particles, for different combinations of porosity values and amounts of absorbed water. Each combination yields a different configuration of pore units, and thus distinctly different capillary pressure-saturation curves. All these curves are shown to collapse into one curve for drainage and one curve for imbibition when we normalize capillary pressure and saturation values. We have developed a formula for the Van Genuchten parameter [Formula: see text] (which is related to the inverse of the entry pressure) as a function of porosity and the amount of absorbed water.

There are many situations in computational fluid dynamics which require the definition of source terms in the Navier-Stokes equations. These source terms not only allow to model the physics of interest but also have a strong impact on the reliability, stability, and convergence of the numerics involved. Therefore, sophisticated numerical approaches exist for the description of such source terms. In this paper, we focus on the source terms present in the Navier-Stokes or Euler equations due to porous media-in particular the Darcy-Forchheimer equation. We introduce a method for the numerical treatment of the source term which is independent of the spatial discretization and based on linearization. In this description, the source term is treated in a fully implicit way whereas the other flow variables can be computed in an implicit or explicit manner. This leads to a more robust description in comparison with a fully explicit approach. The method is well suited to be combined with coarse-grid-CFD on Cartesian grids, which makes it especially favorable for accelerated solution of coupled 1D-3D problems. To demonstrate the applicability and robustness of the proposed method, a proof-of-concept example in 1D, as well as more complex examples in 2D and 3D, is presented.