Darcy Permeability Research Articles

Experimental method to determine anisotropic permeability of hollow fiber membrane bundles

The hydrodynamic flow resistance at low Reynolds numbers through hollow fiber membrane bundles is modeled, e.g. for computational fluid dynamics (CFD), as Darcy permeability K following the porous media approach. Experimental determination of the permeability of a particular hollow fiber bundle arrangement (fiber angle, diameter, and distance) is limited to one-directional flow measurements due to problems preparing representative elementary volume (REV) and following permeability is usually assumed to be isotropic. This ignores the fact that hollow fibers are usually arranged in an organized manner, or fiber bundle, and that resistance to flow is assumedly anisotropic. In this study, we developed a method of creating scaled-up REVs of almost any kind of geometry (by rapid prototyping) that can be measured in multi-dimensional directions within a permeameter. By means of a dimensional analysis the measured anisotropic flow resistances can be expressed in dimensionless friction factors, which can then be transformed into directional permeabilities on an arbitrary scale. The method was developed on two different fiber angle configurations (24° and 90°) in staggered and in-line fiber arrangements that are usually applied as blood oxygenator fibers. As a result, maximum anisotropy of the permeabilities were proven for flow along the fibers compared to cross-flow to the fibers with Kmax/Kmin = 4.77 for the 24° fiber layout. A comparison between the implementation of anisotropic vs. the commonly assumed isotropic permeabilities promises an improvement, especially in a more accurate estimation of stagnation zones in such modules.

On the examination of the Darcy permeability of soft fibrous porous media; new correlations

In this paper, we report a novel experimental approach to investigate the compression-dependent Darcy permeability of soft porous media. Especially, we are proposing new correlations that describe the change of the permeability of random fibrous porous media as a function of its compression. A special device was developed that consisted of a rectangular flow channel with adjustable gap thickness ranging from 3mm to 20mm. Air was forced through the thin gap filled with testing fibrous materials. By measuring the flow rate and the pressure gradient, we have successfully obtained the Darcy permeability of different fibrous porous materials at different compression ratios. Theoretical or semi-empirical models have been compared with the experimental results, indicating various degrees of disagreement. The new correlations were then proposed which fit with experimental data very well. The study presented herein provides a useful approach to evaluate the change of the permeability of fibrous porous media as a function of its compression. It will be valuable for examining fluid flow in fibrous porous media where the permeability is difficult to be measured directly. This kind of porous media widely exists in biological systems.

Method of Calculating the Fluid Permeability of Machined Skin-Covered Porous Sheets from Experimental Flow Data

Abstract Porous polymer sheets created by the “solid state microcellular foaming process” possess an impermeable solid skin on the boundary. In order to experimentally determine the internal structure's fluid permeability, the skin is removed by machining a pattern of holes or channels so that fluid flow can be established when applying a pressure difference across the faces. In this work, a fluid flow model for this particular geometry assuming a Newtonian fluid and isotropic internal porous structure is established. This yields formulae used to compute the structure's Darcy permeability from experimental flow data. Experimental verification of the model is provided for a sample of Polyetherimide (PEI) nanoporous polymer sheet whose skin was drilled in various hole patterns, thus establishing the mathematical foundation for future work in investigating this class of porous materials.

Understanding Oil and Gas Flow Mechanisms in Shale Reservoirs Using SLD–PR Transport Model

More and more attention has been paid to the oil and gas flow mechanisms in shale reservoirs. The solid–fluid interaction becomes significant when the pores are in the nanoscale. The interaction changes the fluid’s physical properties and leads to different flow mechanisms in shale reservoirs from those in conventional reservoirs. By using a Simplified Local Density–Peng Robinson transport model, we consider the density and viscosity profiles, which result from solid–fluid interaction. Gas rarefaction effect is negligible at high pressure, so we assume it is viscous flow. Considering the density- and viscosity-changing effects, we proposed a slit permeability model. The velocity profiles are obtained by this newly established model. This proposed model is validated by matching the density profile and velocity profile from molecular dynamic simulation. Then, the effects of pressure and pore size on gas and oil flow mechanisms are also studied in this work. The results show that both gas and oil exhibit enhanced flow rates in nanopores. Gas-phase flow in nanopores is dominated by the density-changing effect (adsorption), while the oil-phase flow is mainly controlled by the viscosity-changing effect. Both gas and oil permeability quickly decrease to the Darcy permeability when the slit aperture becomes large. The results reported in this work are representative and should significantly help us understand the mechanisms of oil and gas flow in shale reservoirs.

Network modeling of liquid flow in Yanchang shale

The pore geometry and slip conditions are two principal characteristics of liquid flow in shale noncircular nanopores. We have considered the pore shape and liquid slippage on the pore interior wall in pore-network models of the Yanchang shale in China. Multiple scanning electron microscopy (SEM) images of the Yanchang shale samples were analyzed to document and summarize various pore shapes and sizes. Fluid-flow behavior in noncircular pores significantly deviates from the flow behavior in circular pores used in many current models. In addition, liquid flow simulations in conventional reservoir systems assume a no-slip flow boundary condition, which is unacceptable in nanoscale pores associated with the Yanchang shale. We studied liquid permeability in the Yanchang shale by developing realistic pore-network models based on nitrogen sorption data and SEM image analysis. The input data to the developed pore-network models include pore-size distribution, total organic carbon, porosity, surface area, pore geometry, and liquid slip length. We found that the effective permeability based on the realistic multigeometry model is significantly higher than the predicted Darcy permeability. The apparent permeability discrepancy between Darcy’s law and the realistic slip-corrected multigeometry model is higher for shale samples with an increasing number of noncircular pores. We also found that neglecting the liquid slip and noncircularity of pores significantly underestimates apparent permeability in Yanchang shale samples.

A research study of the intra-nanopore methane flow law

In this research study, a self-developed gas seepage experimental system was used to carry out experiments on methane flow with alumina films under different rarefaction degrees conditions, in order to examine the intra-nanopore methane flow law. Then, based on the experimental results under the conditions which considered the slip boundary, a new intra-nanopore methane flow equation with a correction term was established. The change law of the ratio between the equivalent permeability and the Darcy permeability, Kp, was studied under the conditions of different rarefaction degrees. The research results showed that the alumina film was a type of ideal material for the study of methane flow law in nanoscale pores. It was found that, with the increase of the rarefaction degree, the ratio between the equivalent permeability and the Darcy permeability, Kp, gradually increased. With the increase of the rarefaction degree, multiple flow mechanisms were found to exist in the methane flow, including Darcy flow, slip flow, transition flow, and free molecular flow. The research results provided theoretical support for clearly revealing the flow mechanisms of methane and shale gases.

A biphasic approach for the study of lift generation in soft porous media

Lift generation in highly compressible porous media under rapid compression continues to be an important topic in porous media flow. Although significant progress has been made, how to model different lifting forces during the compression process remains unclear. This is mainly because the input parameters of the existing theoretical studies, including the Darcy permeability of the porous media and the viscous damping coefficient of its solid phase, were manually adjusted so as to match the experimental data. In the current paper, we report a biphasic approach to experimentally and theoretically treat this limitation. Synthetic fibrous porous materials, whose permeability were precisely measured, were subsequently exposed to sudden impacts using a porous-walled cylinder-piston apparatus. The obtained time-dependent compression of the porous media, along with the permeability data, was applied in two different theoretical models to predict the pore pressure generation, a plug flow model and a consolidation model [Q. Wu et al., J. Fluid Mech. 542, 281 (2005a)]. Comparison between the theory and the experiments on the pore pressure distribution proved the validity of the consolidation model. Furthermore, a viscoelastic model, containing a nonlinear spring in conjunction with a linear viscoelastic generalized Maxwell mechanical module, was developed to characterize the solid phase lifting force. The model matched the experimental data very well. The paper presented herein, as one of the series studies on this topic, provides an important biphasic approach to characterize different forces that contribute to the lift generation in a soft porous medium under rapid compression.

Prediction of Darcy Permeability and Forchheimer’s Coefficient of Porous Structures Relevant to Regenerator of a Stirling Cryocooler Using a Correlation Based Method

A regenerative heat exchanger is the most vital component in the design of a Stirling cryocooler. Computational Fluid Dynamics (CFD) is the best technique for the design and prediction of the performance of a regenerator. The reliability of the simulation results depend on the accuracy of the Darcy permeability [Formula: see text] and Forchheimer’s inertial coefficient [Formula: see text] used for modeling the momentum transfer in porous media. Usually these coefficients are calculated from pressure drop data obtained from experiment. Because of the requirement of sophisticated equipments for the measurement and analysis of data, experimental study becomes expensive. This paper proposes a friction factor correlation-based method for the prediction of directional permeability and Forchheimer’s inertial coefficient of wire mesh structures relevant to Stirling cryocooler. The friction factor for the flow of helium through #325, #400 and #635 SS wire matrices with porosities of 0.6969, 0.6969 and 0.6312 are calculated using standard correlations and compared with the friction factor given by Clearman et al. based on steady flow experimental study. The friction factor obtained from Blass and Tong/London correlations are in agreement with that of Clearman et al. The viscous and inertial resistances are calculated from the friction factor obtained from Blass and Tong/London correlations. Using these values, the regenerator was modeled as a porous media in Fluent. From the steady flow simulation, pressure drop at different mass flow rates (0.08–1.44[Formula: see text]g/s) is obtained. The maximum deviation of predicted pressure drop from the reported experimental data is 15.14%. The Darcy permeability [Formula: see text] and Forchheimer’s inertial coefficient [Formula: see text] obtained from correlation-based method was used for modeling the oscillatory flow of helium through a #400 regenerator. The pressure amplitude and phase at regenerator exit were obtained at different frequencies. The average deviation of predicted pressure amplitude from the experimental data is 15.83%. The model could predict the phase angle also accurately. Therefore, the proposed method can be used to calculate the hydrodynamic parameters of woven wire screen matrices applied to Stirling cryocoolers.

Low- and high-order accurate boundary conditions: From Stokes to Darcy porous flow modeled with standard and improved Brinkman lattice Boltzmann schemes

The present contribution focuses on the accuracy of reflection-type boundary conditions in the Stokes–Brinkman–Darcy modeling of porous flows solved with the lattice Boltzmann method (LBM), which we operate with the two-relaxation-time (TRT) collision and the Brinkman-force based scheme (BF), called BF-TRT scheme. In parallel, we compare it with the Stokes–Brinkman–Darcy linear finite element method (FEM) where the Dirichlet boundary conditions are enforced on grid vertices. In bulk, both BF-TRT and FEM share the same defect: in their discretization a correction to the modeled Brinkman equation appears, given by the discrete Laplacian of the velocity-proportional resistance force. This correction modifies the effective Brinkman viscosity, playing a crucial role in the triggering of spurious oscillations in the bulk solution. While the exact form of this defect is available in lattice-aligned, straight or diagonal, flows; in arbitrary flow/lattice orientations its approximation is constructed. At boundaries, we verify that such a Brinkman viscosity correction has an even more harmful impact. Already at the first order, it shifts the location of the no-slip wall condition supported by traditional LBM boundary schemes, such as the bounce-back rule.For that reason, this work develops a new class of boundary schemes to prescribe the Dirichlet velocity condition at an arbitrary wall/boundary-node distance and that supports a higher order accuracy in the accommodation of the TRT-Brinkman solutions. For their modeling, we consider the standard BF scheme and its improved version, called IBF; this latter is generalized in this work to suppress or to reduce the viscosity correction in arbitrarily oriented flows. Our framework extends the one- and two-point families of linear and parabolic link-wise boundary schemes, respectively called B-LI and B-MLI, which avoid the interference of the Brinkman viscosity correction in their closure relations.The performance of LBM and FEM is thoroughly evaluated in three benchmark tests, which are run throughout three distinctive permeability regimes. The first configuration is a horizontal porous channel, studied with a symbolic approach, where we construct the exact solutions of FEM and BF/IBF with different boundary schemes. The second problem refers to an inclined porous channel flow, which brings in as new challenge the formation of spurious boundary layers in LBM; that is, numerical artefacts that arise due to a deficient accommodation of the bulk solution by the low-accurate boundary scheme. The third problem considers a porous flow past a periodic square array of solid cylinders, which intensifies the previous two tests with the simulation of a more complex flow pattern. The ensemble of numerical tests provides guidelines on the effect of grid resolution and the TRT free collision parameter over the accuracy and the quality of the velocity field, spanning from Stokes to Darcy permeability regimes. It is shown that, with the use of the high-order accurate boundary schemes, the simple, uniform-mesh-based TRT-LBM formulation can even surpass the accuracy of FEM employing hardworking body-fitted meshes.

Mathematical modeling of intraperitoneal drug delivery: simulation of drug distribution in a single tumor nodule

The intraperitoneal (IP) administration of chemotherapy is an alternative treatment for peritoneal carcinomatosis, allowing for higher intratumor concentrations of the cytotoxic agent compared to intravenous administration. Nevertheless, drug penetration depths are still limited to a few millimeters. It is thus necessary to better understand the limiting factors behind this poor penetration in order to improve IP chemotherapy delivery. By developing a three-dimensional computational fluid dynamics (CFD) model for drug penetration in a tumor nodule, we investigated the impact of a number of key parameters on the drug transport and penetration depth during IP chemotherapy. Overall, smaller tumors showed better penetration than larger ones, which could be attributed to the lower IFP in smaller tumors. Furthermore, the model demonstrated large improvements in penetration depth by subjecting the tumor nodules to vascular normalization therapy, and illustrated the importance of the drug that is used for therapy. Explicitly modeling the necrotic core had a limited effect on the simulated penetration. Similarly, the penetration depth remained virtually constant when the Darcy permeability of the tissue changed. Our findings illustrate that the developed parametrical CFD model is a powerful tool providing more insight in the drug transport and penetration during IP chemotherapy.

Fluid flow through replicated microcellular materials in the Darcy-Forchheimer regime

We extend here a “bottleneck” flow model derived earlier for incompressible fluids flowing under creeping flow conditions [Despois, J. and Mortensen, A: Acta Materialia 53 (2005) 1381] to flow regimes where inertial losses are no longer negligible, causing the governing flow law to deviate from Darcy's law and become the Darcy-Forchheimer law. The proposed law is compared with measurements of the Darcian permeability KD and of the Forchheimer coefficient C in forced-flow of air through microcellular aluminium made by the replication process. The geometrical features of the cellular medium are varied in terms of volume fraction of porosity (in the range of 0.66–0.86) and the average cell diameter from (108–425 μm). As found previously in measurements with water, the Darcy permeability of the foams for airflow is also reasonably well captured by the model. In the Forchheimer-regime the model gives good quantitative agreement with data if one assumes that the amount of air kinetic energy that is dissipated when passing across each bottleneck linking one pore to its neighbour along the fluid flow path corresponds to the difference, in a stream of constant cross-sectional area, between a uniform fluid velocity profile and the non-uniform profile that is created by the no-slip condition along the window boundary.

A Model to Estimate Heat Efficiency in Steam-Assisted Gravity Drainage by Condensate and Initial Water Flow in Oil Sands

Steam-assisted gravity drainage (SAGD) is one of the most popular approaches for oil sands recovery. In this study, an accurate and simple heat efficiency calculation model is presented by modeling transient heat transfer involving the flow of both hot condensate and mobile initial water in oil sands. A 2-D numerical simulation procedure is proposed to calculate temperature distribution ahead of the steam chamber edge in SAGD. A new heat efficiency model is proposed to estimate heat efficiency in SAGD by using the temperature distribution obtained from the 2-D numerical simulation. Results indicate that convection dominates the heat transfer ahead of the steam chamber edge which is induced by condensate and initial water flow in oil sands. On the basis of these studies, abundant analysis shows that the heat efficiency of SAGD can be improved by using moderate steam injection pressures for the reservoir scenarios from 1 to 10 darcy permeability.

Apparent Permeability and Gas Flow Behavior in Carboniferous Shale from the Qaidam Basin, China: An Experimental Study

Shale can act as an unconventional gas reservoir with low permeability and complex seepage characteristics. Study of the apparent permeability and percolation behavior of shale gas is important in understanding the permeability of shale reservoirs, to evaluate formation damage, to develop gas reservoirs, and to design wells. This study simulated methane percolation at 298.15 K under inlet pressures ranging from 0.2 to 4 MPa and a constant outlet pressure of 0.1 MPa to investigate shale gas percolation behavior and apparent permeability. Five representative shale cores from the Carboniferous Hurleg and Huitoutala formations in the eastern Qaidam Basin, China, were analyzed. Each experiment measured the volume flow rate of methane and the inlet pressure. Pseudopressure approach was used to analyze high-velocity flow in shale samples, and apparent permeability at different pressures was calculated using the traditional method. A nonlinear apparent permeability model that considers diffusion and slippage is established from theory and experimental data fitting, and the shale gas flow characteristics affected by slippage and diffusion are analyzed. The results indicate that the pseudopressure formulation that considers the effect of gas properties on high-velocity flow produces a more accurate linear representation of the experimental data. The apparent gas permeability of shale consists of contributions from Darcy permeability, slippage, and diffusion. The apparent permeability and gas flow behavior in the studied shales strongly depended on pressure. The diffusion contribution increased greatly as pressure decreased from 2 to 0.2 MPa, and the smaller the shale permeability, the greater the relative contribution of diffusion flow. At pressures greater than 2 MPa, slip flow contributes $$\sim $$ 20% of the total flux, Darcy flow contributes up to 70%, and diffusion makes only a minor contribution. This study provides useful information for future studies of the mechanism of shale gas percolation and the exploration and development of Qaidam Basin shale gas specifically.

Effect of Fluid Bypassing on the Experimentally Obtained Darcy and Non-Darcy Permeability Parameters of Ceramic Foam Filters

Ceramic foam filters (CFFs) are used to remove solid particles and inclusions from molten metal. In general, molten metal which is poured on the top of a CFF needs to reach a certain height to build the required pressure (metal head) to prime the filter. To estimate the required metal head, it is necessary to obtain permeability coefficients using permeametry experiments. It has been mentioned in the literature that to avoid fluid bypassing, during permeametry, samples need to be sealed. However, the effect of fluid bypassing on the experimentally obtained pressure gradients seems not to be explored. Therefore, in this research, the focus was on studying the effect of fluid bypassing on the experimentally obtained pressure gradients as well as the empirically obtained Darcy and non-Darcy permeability coefficients. Specifically, the aim of the research was to investigate the effect of fluid bypassing on the liquid permeability of 30, 50, and 80 pores per inch (PPI) commercial alumina CFFs. In addition, the experimental data were compared to the numerically modeled findings. Both studies showed that no sealing results in extremely poor estimates of the pressure gradients and Darcy and non-Darcy permeability coefficients for all studied filters. The average deviations between the pressure gradients of the sealed and unsealed 30, 50, and 80 PPI samples were calculated to be 57.2, 56.8, and 61.3 pct. The deviations between the Darcy coefficients of the sealed and unsealed 30, 50, and 80 PPI samples found to be 9, 20, and 31 pct. The deviations between the non-Darcy coefficients of the sealed and unsealed 30, 50, and 80 PPI samples were calculated to be 59, 58, and 63 pct.

Estimation of Pressure-Dependent Diffusive Permeability of Coal Using Methane Diffusion Coefficient: Laboratory Measurements and Modeling

Gas diffusion in coal is critical for the prediction of coalbed methane (CBM) production, especially for the late-stage CBM reservoir when both gas pressure and permeability are relative low. Using only Darcy permeability to evaluate the quality of gas transport might not be effective. Diffusive flow can be the dominant flow at low reservoir pressures. In this work, the methane diffusion coefficient was measured for pulverized San Juan sub-bituminous and Pittsburgh bituminous coal samples using the classic unipore model and particle method. The diffusion coefficient results showed a negative correlation with pressure, as reported previously. The significance of diffusion flow is strongly related to the rate of the methane ad-/desorption process, more severely in the low pressure range [<2 MPa (280 psi)]. The measured diffusion coefficient can be converted to the equivalent permeability. This equivalent permeability, termed diffusive permeability, can be considered as the contribution of diffusion flow, in...

Poly(N‐isopropylacrylamide) gel‐based macroporous monolith for continuous‐flow recovery of palladium(II) ions

ABSTRACTMacroporous monoliths, composed of thermoresponsive, tertiary‐aminated, and crosslinking monomers, were prepared for continuous‐flow separation of palladium(II) ions. N‐Isopropylacrylamide was required to form the porous structure in the monoliths, indicating that the mechanism of porous structure formation involved polymerization‐induced phase separation of the poly(N‐isopropylacrylamide) gel. Tertiary‐aminated monoliths showed adsorption selectivity for palladium(II) ions in hydrochloric media, compared with copper(II) ions. The maximum capacities of the monoliths with tertiary amine contents of 10, 20, 30, and 70 mol % for palladium(II) ions were 0.6, 1.1, 1.3, and 2.3 mmol/g, respectively. Darcy's permeabilities of water through the macroporous monolith were 10−14 to 10−13 m2, and those were comparable to that through a commercially available membrane filter with a pore size of several micrometers. In the continuous‐flow process, the macroporous monolith with tertiary amine selectively adsorbed palladium(II) ions in the coexistence of copper(II) ions with 10 times higher concentration than the palladium(II) ions. The palladium(II) ions were eluted from the macroporous monolith, and the concentration of palladium(II) ions in the eluate was up to 45 times of that in the feed solution. The average enrichment factor and total recovery percentage of palladium(II) ions were 8.7 times and 95%, respectively. © 2016 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2017, 134, 44385.

Darcy Permeability of Hollow Fiber Membrane Bundles Made from Membrana Polymethylpentene Fibers Used in Respiratory Assist Devices.

Hollow fiber membranes (HFMs) are used in blood oxygenators for cardiopulmonary bypass or in next generation artificial lungs. Flow analyses of these devices is typically done using computational fluid dynamics (CFD) modeling HFM bundles as porous media, using a Darcy permeability coefficient estimated from the Blake-Kozeny (BK) equation to account for viscous drag from fibers. We recently published how well this approach can predict Darcy permeability for fiber bundles made from polypropylene HFMs, showing the prediction can be significantly improved using an experimentally derived correlation between the BK constant (A) and bundle porosity (ε). In this study, we assessed how well our correlation for A worked for predicting the Darcy permeability of fiber bundles made from Membrana polymethylpentene (PMP) HFMs, which are increasingly being used clinically. Swatches in the porosity range of 0.4 to 0.8 were assessed in which sheets of fiber were stacked in parallel, perpendicular, and angled configurations. Our previously published correlation predicted Darcy within ±8%. A new correlation based on current and past measured permeability was determined: A = 497ε - 103; using this correlation measured Darcy permeability was within ±6%. This correlation varied from 8% to -3.5% of our prior correlation over the tested porosity range.

Microscale characterisation of stochastically reconstructed carbon fiber-based Gas Diffusion Layers; effects of anisotropy and resin content

A novel process-based methodology is proposed for the stochastic reconstruction and accurate characterisation of Carbon fiber-based matrices, which are commonly used as Gas Diffusion Layers in Proton Exchange Membrane Fuel Cells. The modeling approach is efficiently complementing standard methods used for the description of the anisotropic deposition of carbon fibers, with a rigorous model simulating the spatial distribution of the graphitized resin that is typically used to enhance the structural properties and thermal/electrical conductivities of the composite Gas Diffusion Layer materials. The model uses as input typical pore and continuum scale properties (average porosity, fiber diameter, resin content and anisotropy) of such composites, which are obtained from X-ray computed microtomography measurements on commercially available carbon papers. This information is then used for the digital reconstruction of realistic composite fibrous matrices. By solving the corresponding conservation equations at the microscale in the obtained digital domains, their effective transport properties, such as Darcy permeabilities, effective diffusivities, thermal/electrical conductivities and void tortuosity, are determined focusing primarily on the effects of medium anisotropy and resin content. The calculated properties are matching very well with those of Toray carbon papers for reasonable values of the model parameters that control the anisotropy of the fibrous skeleton and the materials resin content.

A reduced basis finite element heterogeneous multiscale method for Stokes flow in porous media

A reduced basis Darcy–Stokes finite element heterogeneous multiscale method (RB-DS-FE-HMM) is proposed for the Stokes problem in porous media. The multiscale method is based on the Darcy–Stokes finite element heterogeneous multiscale method (DS-FE-HMM) introduced in Abdulle and Budáč (2015) that couples a Darcy equation solved on a macroscopic mesh, with missing permeability data extracted from the solutions of Stokes micro problems at each macroscopic quadrature point. To overcome the increasingly growing cost of repeatedly solving the Stokes micro problems as the macroscopic mesh is refined, we parametrize the microscopic solid geometry and approximate the infinite-dimensional manifold of parameter dependent solutions of Stokes problems by a low-dimensional space. This low-dimensional (reduced basis) space is obtained in an offline stage by a greedy algorithm and used in an online stage to compute the effective Darcy permeability at a cost independent of the microscopic mesh. The discretization of the parametrized Stokes problems relies on a Petrov–Galerkin formulation that allows for a stable and fast online evaluation of the required permeabilities. A priori and a posteriori estimates of the RB-DS-FE-HMM are derived and a residual-based adaptive algorithm is proposed. Two- and three-dimensional numerical experiments confirm the accuracy of the RB-DS-FE-HMM and illustrate the speedup compared to the DS-FE-HMM.

An analytical Ergun-type equation for porous foams

The empirical coefficients of the Ergun equation for granular packed beds have often been adjusted in the literature towards quantitative agreement with experimental pressure drop data of porous foams. Adjusting the coefficients of the Ergun equation is not good practice since it resembles a fudge factor approach for which the coefficients have to be adjusted for every new application. In this study an analytical Ergun-type equation is proposed for porous foams. The interstitial geometric configuration and accompanying flow conditions are remodelled to yield an equation for the pressure gradient that have different functional dependencies on porosity, than in the case of granular media. Comparison of the pressure gradient predicted by the proposed model with an Ergun type equation available in the literature enables quantification of the empirical coefficients of the latter equation in terms of porosity. It is thereby illustrated that the constant empirical coefficients relate to the pore-scale geometry and therefore have physical meaning. The proposed model results from adaptations made to existing pore-scale models resembling porous foams. The model predictions are compared to experimental data and empirical models from the literature for the Darcy permeability and non-Darcy coefficient. The satisfactory correspondence provides confidence in the analytical modelling procedure. The proposed model follows a similar trend as an empirical model proposed in the literature for which the coefficients of the Ergun equation have been adjusted for porous foams. Improvement in the predictive capability of the model is illustrated through comparison of the pressure gradient predicted by the proposed model with that of the existing models.