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The dimensional character of permeability in anisotropic porous media, that is,its dimension or dimensional equation, is an information that allows setting thedimensionless groups that govern the solution of the flow equation in terms ofhydraulic potential patterns. However, employing the dimensional basis {L, M,T} (length, mass, time), the dimensionless groups containing the anisotropic per-meability do not behave as independent monomials that rule the solutions. Inthis work, the contributions appearing in the literature on the dimensional char-acter of permeability are discussed and a new approach based on discriminatedand general dimensional analysis is presented. This approach leads to the emer-gence of a new and accurate dimensionless group,kxkyl∗2yl∗2x, a ratio of permeabilitiescorrected by the squared value of an aspect factor, beingl∗xandl∗ytwo arbitrarylengths of the domain in the directions that are indicated in their subscripts. Spe-cific values of this lengths, which we name ‘hidden characteristic lengths’, arealso discussed in this article. To check the validity of this dimensionless group,numericalsimulationsoftwoillustrative2-Dseepagescenarioshavebeensolved. The authors would like to thank SéNeCa Foundation for the support given to this research and for the scholarships awarded to Martínez-Moreno E. to carry out her doctoral thesis.

A computation analysis is performed to study double-diffusive natural convection in a right-angle trapezoidal cavity packed with a porous medium. The horizontal top and bottom boundaries are insulated and impermeable. The vertical left sidewall is kept at a constant heat flux and high concentration, whereas the temperature and concentration of the mixture at the inclined sidewall are held at lower temperatures and lower concentration. The dimensionless nonlinear system is solved by employing finite difference method along Successive Under Relaxation technique. The findings are compared and validated with the existing literature for the Darcy flow driven through a single buoyancy effect (difference in density is only due to temperature variations) in a porous square enclosure. The numerical results are expressed in the form of dimensionless temperature, concentration, streamlines, isotherms and iso-concentrations, local and average Nusselt and Sherwood numbers. It is demonstrated that the Rayleigh number and buoyancy parameter enhance both surface heat and concentration rates.

We consider a micropolar fluid flow in a media perforated by periodically distributed obstacles of size e. A non-homogeneous boundary condition for microrotation is considered: the microrotation is assumed to be proportional to the rotation rate of the velocity on the boundary of the obstacles. The existence and uniqueness of solution is analyzed. Moreover, passing to the limit when e tends to zero, an analogue of the classical micropolar Darcy law in the theory of porous media is derived.

We study the homogenization of a slow viscous two-phase incompressible flow in a domain consisting of a free fluid domain, a porous medium, and the interface between them. We take into account the capillary forces on the fluid-fluid interfaces. We construct boundary layers describing the flow at the interface between the free fluid and the porous medium. We derive a macroscopic model with a viscous two-phase fluid in the free domain, a coupled Darcy law connecting two-phase velocities in the porous medium, and boundary conditions at the permeable interface between the free fluid domain and the porous medium.

Analysis of porous fins for their higher heat transfer in comparison with solid fins with identical volumes has attracted significant attention. In this paper, a two-dimensional thermal analysis of a porous fin having variable thermal conductivity coefficient is performed using finite difference method. Heat transfer through porous media is simulated using passage velocity from Darcy's model. The thermal conductivity of the solid phase is considered as a linear function of temperature. It is found that the temperature profile of the fin is completely two-dimensional even for high Rayleigh and Darcy numbers (Ra = 103∼104, Da = 0.01), because the temperature changes significantly along the transverse axis especially for lower Rayleigh and Darcy numbers. Also, the effects of important nondimensional parameters such as Rayleigh and Darcy numbers, porosity, Nusselt, thermal conductivity, and aspect ratio on the temperature profile are investigated. The results demonstrate that the temperature distribution is strongly dependent on the Rayleigh and Darcy numbers.

As is known, Darcy's model for fluid flows in isotropic homogeneous porous media gives rise to singularities in the velocity field for essentially two-dimensional flow configuration, like flows over corners. Considering this problem from the modelling viewpoint, the present study aims at removing this singularity, which cannot be regularized via conventional generalizations of the Darcy model, like Brinkman's equation, without sacrificing Darcy's law itself for unidirectional flows where its validity is well established experimentally. The key idea is that, as confirmed by a simple analogy, the permeability of a porous matrix with respect to flow is not a constant independent of the flow but a function of the flow field (its scalar invariants), decreasing as the curvature of the streamlines increases. This introduces a completely new class of models where the flow field and the permeability field are linked and, in particular problems, have to be found simultaneously. This article is protected by copyright. All rights reserved.

An attempt has been made to study the entropy generation analysis of couple stress fluid flow in an annulus between two concentric rotating vertical cylinders. There is a porous lining attached to the inside of an outer cylinder. The flow is under the influence of a radial magnetic field. The flow in the annular gap is caused by rotation of the cylinders. The Stokes couple stress flow model is employed. The flow in the porous sleeve is governed by Darcy's law. The velocity, temperature, entropy generation number, Bejan number, wall shear stress and heat transfer rate at the inner and outer cylinders are obtained numerically by employing a finite difference scheme with vanishing of couple stresses on the boundary. The effect of relevant parameters on the flow and entropy generation rate are discussed and depicted through graphs.

According to the objectives of the research group 1498, this paper deals with degradation effects in concrete structures that are caused by cyclic flexural loading. The goal is to determine their influence on the fluid transport processes within the material on the basis of experimental results and numerical simulations. The overall question was, to which extent the ingress of externally supplied alkalis and subsequently an alkali-silica reaction are affected by such modifications in the microstructure. Degradation in the concrete microstructure is characterized by ultrasonic wave measurements as well as by microscopic crack analysis. Furthermore, experiments on the penetration behavior of water into the investigated materials were performed. The penetration behavior into predamaged concrete microstructures was examined by the classical Karsten tube experiment, nuclear magnetic resonance method, and time domain reflectometry techniques. In order to create an appropriate model of the material's degradation on the water transport, the Darcy law was applied to describe the flow in partially saturated concrete. Material degradation is taken into account by an effective permeability that is dependent on the state of degradation. This effective permeability is obtained by the micromechanical homogenisation of the flow in an Representative Elementary Volume (REV) with distributed ellipsoidal microcracks embedded in a porous medium. The data gained in the microscopic crack analysis is used as input for the micromechanical model. Finite element simulations for unsaturated flow using the micromechanical model were compared with the experimental results showing good qualitative and quantitative agreement.

We present a new a priori stability and convergence analysis for the local discontinuous Galerkin method applied to the instationary Darcy problem formulated on a d-dimensional polytope with nonhomogeneous Neumann and Dirichlet boundary conditions. In addition to including a spatially and temporally varying permeability tensor into all estimates, the utilized analysis technique produces a convergence order result for the primary and the flux variables. The only stabilization in the proposed scheme is represented by a penalty term in the primary unknown, and our analysis provides some insights into the role played by this particular choice of stabilization. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1374–1394, 2017

We consider a non-stationary incompressible non-Newtonian flow in a thin porous medium of thickness e which is perforated by periodically distributed solid cylinders of size ae. The viscosity is supposed to obey the power law with flow index 32

We consider a non-stationary Stokes system in a thin porous medium Ωϵ of thickness ϵ which is perforated by periodically solid cylinders of size aϵ. We are interested here to give the limit behavior when ϵ goes to zero. To do so, we apply an adaptation of the unfolding method. Time-dependent Darcy's laws are rigorously derived from this model depending on the comparison between aϵ and ϵ. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we study a mathematical model of nonlinear thermoelastic wave propagation in fluid-saturated porous media, considering memory effect in the heat propagation. In particular, we derive the governing equations in one dimension by using the Gurtin–Pipkin theory of heat flux history model and specializing the relaxation function in such a way to obtain a fractional Erdelyi–Kober integral. In this way, we obtain a nonlinear model in the framework of time-fractional thermoelasticity, and we find an explicit analytical solution by means of the invariant subspace method. A second memory effect that can play a significant role in this class of models is parametrized by a generalized time-fractional Darcy law. We study the equations obtained also in this case and find an explicit traveling wave type solution. Copyright © 2016 John Wiley & Sons, Ltd.

This text focuses essentially on capillary equilibria and capillary flows (visco-capillary hydrodynamics) in the presence of solids and in porous media. Several types of immiscible two-phase equilibria and flow systems are examined at various scales, in the presence of solid frames (cylindrical tubes and planar joints), and inside the pore space of porous soils, rocks, and other materials. In the case of porous media, this leads to studying capillary effects in unsaturated water flow, and immiscible two-phase flow, in homogeneous, heterogeneous, random and/or stratified porous media. Several problems of porous media hydrodynamics influenced by capillarity are treated in the last chapters of Vol. 1, and then, will be treated at more length, along with case studies, in the subsequent Vol. 2.

Summary In this paper, we analyze discontinuous finite volume methods for the stationary Stokes–Darcy problem that models coupled fluid flow and porous media flow. The discontinuous finite volume methods are combinations of finite volume method and discontinuous Galerkin method with three interior penalty types (incomplete symmetric, nonsymmetric, and symmetric), briefly, using discontinuous functions as trial functions in the finite volume method. Optimal error estimates in broken H1 norm are obtained for the three discontinuous finite volume methods. Optimal error estimates in the standard L2 norm are derived for the symmetric interior penalty discontinuous finite volume method. Numerical experiments are presented to confirm the theoretical results with non-matching meshes across the common interface of Stokes region and Darcy region. Copyright © 2015 John Wiley & Sons, Ltd.

© 2015 Society of Chemical Industry and John Wiley & Sons, Ltd. Most large-scale flow and transport simulations for geologic carbon sequestration (GCS) applications are carried out using simulators that solve flow equations arising from Darcy's law. Recently, the computational advantages of invasion-percolation (IP) modeling approaches have been presented. We show that both the Darcy's-law- and the gravity-capillary balance solved by IP approaches can be derived from the same multiphase continuum momentum equation. More specifically, Darcy's law arises from assuming creeping flow with no viscous momentum transfer to stationary solid grains, while it is assumed in the IP approach that gravity and capillarity are the dominant driving forces in a quasi-static two-phase (or more) system. There is a long history of use of Darcy's law for large-scale GCS simulation. However, simulations based on Darcy's law commonly include significant numerical dispersion as users employ large grid blocks to keep run times practical. In contrast, the computational simplicity of IP approaches allows large-scale models to honor fine-scale hydrostratigraphic details of the storage formation which makes these IP models suitable for analyzing the impact of small-scale heterogeneities on flow. However, the lack of time-dependence in the IP models is a significant disadvantage, while the ability of Darcy's law to simulate a range of flows from single-phase- and pressure-gradient-driven flows to buoyant multiphase gravity-capillary flow is a significant advantage. We believe on balance that Darcy's law simulations should be the preferred approach to large-scale GCS simulations.

Non-Darcy flow is an important characteristic of gas well and its accurate description will increase accuracy of productivity evaluation and prediction. In this article, we study the influence of non-Darcy flow boundary on production prediction. Based on flow theory in porous media, we discuss the models of non-Darcy flow boundary with different Reynolds numbers, and propose the criteria judging non-Darcy flow: non-Darcy critical permeability (kc), non-Darcy critical flowing pressure (pc), non-Darcy critical rate (qgc). Then, non-Darcy skin-depended factors are derived with different Reynolds number, which reflect the influence of true skin on non-Darcy flow. Finally, we study the influence of these factors on productivity prediction and well testing. The results show that non-Darcy flow boundary may be nearby wellbore radius in low permeability reservoirs, whose influence could be considered as a negative skin which will decrease non-Darcy flow. And, compared with the open flow capacity of convenient binomial deliverability equation and Darcy equation, the open flow capacity of the equation that considers the influence of non-Darcy flow boundary is just between that of the two equations, which will verify the influence of the non-Darcy flow boundary. What's more, the non-Darcy boundary influences the open flow capacity significantly. And, non-Darcy skin-depended factor will become larger with the increase of skin. Finally, when the designed rates of well testing should be larger than the critical rates, the points of test will be on a perfectly straight line or else the points will not. The research of this article will increase the accuracy of productivity evaluation and prediction in low permeability reservoirs. © 2015 American Institute of Chemical Engineers Environ Prog, 34: 1822–1831, 2015

Smoothed particle hydrodynamics (SPH) is a mesh-free numerical method originally developed in the 1970s for astrophysical analysis. In the recent decades, it has been applied to a wide range of research fields, including fluid and solid mechanics, hydrodynamics and coastal and geotechnical engineering. This paper presents an application of SPH to the problem of interaction between the soil skeleton and the pore water in saturated soils. The formulation of the equations of conservation in SPH has been modified so that a drag force due to seepage is introduced and calculated using Darcy's law. Two test cases are considered for the analysis: a basic drainage of a vertical sand column due to gravity and seepage through a vertical dam (also known as the Muskat problem). Both simulations show very good agreement with the analytical solutions. The results prove the potential of SPH. Numerische Modellierung gesattigter Boden mit der Smoothed Particle Hydrodynamics – Teil 1: Grundwasserstromungsanalyse. Die Smoothed Particle Hydrodynamics (SPH) ist eine netzfreie numerische Methode, die ursprunglich in den 1970er-Jahren fur astrophysikalische Berechnungen entwickelt wurde. In den vergangenen Jahrzehnten hat sie zahlreiche Anwendungsgebiete gefunden, u. a. Fluid- und Strukturmechanik, Hydrodynamik, Kusteningenieurwesen und auch Geotechnik. Dieser Artikel zeigt eine Anwendung der SPH zur Untersuchung der Interaktion zwischen Korngerust und Porenwasser wassergesattigter Boden. Zu diesem Zweck wurde die Formulierung der Erhaltungsgleichungen um eine Stromungskraft basierend auf dem Gesetz von Darcy erweitert. Zur Validierung werden zwei Sickerstromungsprobleme betrachtet: die Entwasserung einer Bodensaule unter Gravitation und die Stromung durch einen vertikalen Damm, auch bekannt als das Muskat-Problem. Beide Simulationen haben eine sehr gute Ubereinstimmung mit den analytischen Losungen gezeigt. Die Ergebnisse bestatigen das Potential der SPH.

In this paper, a modified Reynolds equation in polar coordinates using the Phan-Thien and Tanner rheological model is combined with a modified Darcy law describing a poroelastic confining surface. The Phan-Thien and Tanner model is commonly used to describe the viscoelastic behaviour of synovial fluid, while the poroelastic model can portray articular cartilage. Together, they may provide a simulation basis for lubrication of the tribology of the knee contact under squeezing. The results obtained for the axisymmetric case are analysed and compared for certain non-Newtonian shear phenomena on the basis of works from the literature. Copyright © 2015 John Wiley & Sons, Ltd.

We present a numerical study using a discrete element method (DEM) coupled with a finite element method (FEM) at the boundary to simulate the fluid flow, geomechanical deformation, and dynamic fracturing together to enhance the sustainability analysis for geological sequestration of CO2. The fluid flow, geomechanical deformation, and fracturing due to the injection of fluid are all modeled by the bonded DEM (bonded-DEM), where fluid flow is modeled by solving the Darcy flow directly on the Lagrangian particles. Because of the high computational expense, the bonded-DEM is only used in the domain where fracturing is highly possible, namely the area near to the injection well and around the pre-existing fault. For the area far away from the high risky domain, the deformation and pressure solutions are obtained by a standard finite element method (FEM). The stress, deformation, and pressure obtained from FEM are fed back into the bonded-DEM simulations as boundary conditions that were applied to the DEM boundary particles. The proposed model has the potential to be used to evaluate the safety and sustainability of a sequestration site. By predicting the critical time when the fault is reactivated and the time when CO2 breaks through the caprock through the reactivated fault. The model also shows that the ground surface displacement can be used as an effective monitoring indicator for fracturing, fault reactivation, and CO2 breakthrough in aquifer and caprock, implying a very useful monitoring method for the safety of any sequestration site.

This paper presents the formulation of finite element methods for the numerical modeling of a poro-elastic two-phase (aggregates/mixture phase) solid. The displacement and pressure fields are decomposed, following the EAS method, into a regular part and an enhanced part. This leads to discontinuous strain and pressure gradient fields allowing to capture the jump in mechanical and hydrical properties passing through the interface between the aggregates and the mixture phase. All these enhanced fields are treated in the context of the Embedded Finite Element Method through a local enhancement of the finite element interpolations as these jumps appear. The local character of these interpolations lead after a static condensation of the enhanced fields to a problem exhibiting the same structure as common poro-elastic finite element models, but incorporating now the mechanical and hydrical properties of a two-phase solid.

SUMMARY Considering the effect of non-Darcy flow, the perturbation theory and normal mode method are introduced to analyze the linear stability of one-dimensional non-Darcy flow of gases in broken rocks. A stability criterion for linear systems is obtained theoretically when the steady states of pressure and velocity fields are perturbed, and the effects of the physical parameters on the linear governing system are studied theoretically and numerically. It is pointed out that the deviation coefficient from Darcy's law plays an important role in the governing system; the increasing absolute value of deviation coefficient from Darcy's law stabilizes the system, and the numerical results are shown graphically. Copyright © 2015 John Wiley & Sons, Ltd.

The coupled problem for a generalized Newtonian Stokes flow in one domain and a generalized Newtonian Darcy flow in a porous medium is studied in this work. Both flows are treated as a first-order system in a stress-velocity formulation for the Stokes problem and a volumetric flux-hydraulic potential formulation for the Darcy problem. The coupling along an interface is done using the well-known Beavers–Joseph–Saffman interface condition. A least squares finite element method is used for the numerical approximation of the solution. It is shown that under some assumptions on the viscosity the error is bounded from above and below by the least squares functional. An adaptive refinement strategy is examined in several numerical examples where boundary singularities are present. Due to the nonlinearity of the problem a Gauss–Newton method is used to iteratively solve the problem. It is shown that the linear variational problems arising in the Gauss–Newton method are well posed. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1150–1173, 2015

In this article, we consider the coupled Navier–Stokes and Darcy problem with the Beavers–Joseph interface condition. With suitable restrictions of physical parameters α and ν, we prove the existence and local uniqueness of a weak solution. Then we propose a coupled finite element scheme and a decoupled and linearized scheme based on two-grid finite element. Under suitable further restrictions, their optimal error estimates are obtained. Finally numerical experiments indicate the validity of the theoretical results as well as the efficiency and effectiveness of the decoupled and linearized two-grid algorithm. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1009–1030, 2015

Summary In this work, we investigate the main pumping parameters that influence a fluid-driven fracture in cohesive poroelastic and poroelastoplastic weak formations. These parameters include the fluid viscosity and the injection rate. The first parameter dominates in the mapping of the propagation regimes from toughness to viscosity, whereas the second parameter controls the storage to leak-off dominated regime through diffusion. The fracture is driven in weak permeable porous formation by injecting an incompressible viscous fluid at the fracture inlet assuming that the fracture propagates under plane strain conditions. Fluid flow in the fracture is modeled by lubrication theory. Pore fluid movement in the porous formation is based on the Darcy law. The coupling follows the Biot theory, whereas the irreversible rock deformation is modeled with the Mohr–Coulomb yield criterion with associative flow rule. Fracture propagation criterion is based on the cohesive zone approach. Leak-off is also considered. The investigation is performed numerically with the FEM to obtain the fracture opening, length, and propagation pressure versus time. We demonstrate that pumping parameters influence the fracture geometry and fluid pressures in weak formations through the viscous fluid flow and the diffusion process that create back stresses and large plastic zones as the fracture propagates. It is also shown that the product of the propagation velocity and fluid viscosity, µv that appears in the scaling controls the magnitude of the plastic zones and influences the net pressure and fracture geometry. These findings may explain partially the discrepancies in net pressures between field measurements and conventional model predictions for the case of weak porous formation. Copyright © 2014 John Wiley & Sons, Ltd.

Summary This paper presents a numerical scheme for fluid-particle coupled discrete element method (DEM), which is based on poro-elasticity. The motion of the particles is resolved by means of DEM. While within the proposition of Darcian regime, the fluid is assumed as a continuum phase on a Eulerian mesh, and the continuity equation on the fluid mesh for a compressible fluid is solved using the FEM. Analytical solutions of traditional soil mechanics examples, such as the isotropic compression and one-dimensional upward seepage flow, were used to validate the proposed algorithm quantitatively. The numerical results showed very good agreement with the analytical solutions, which show the correctness of this algorithm. Sensitivity studies on the effect of some influential factors of the coupling scheme such as pore fluid bulk modulus, volumetric strain calculation, and fluid mesh size were performed to display the accuracy, efficiency, and robustness of the numerical algorithm. It is revealed that the pore fluid bulk modulus is a critical parameter that can affect the accuracy of the results. Because of the iterative coupling scheme of these algorithms, high value of fluid bulk modulus can result in instability and consequently reduction in the maximum possible time-step. Furthermore, the increase of the fluid mesh size reduces the accuracy of the calculated pore pressure. This study enhances our current understanding of the capacity of fluid-particle coupled DEM to simulate the mechanical behavior of saturated granular materials. Copyright © 2014 John Wiley & Sons, Ltd.

Summary It is well known that for a sufficiently high seepage velocity, the governing flow law of porous media is nonlinear (J. Computers & Fluids 2010; 39: 2069–2077). However, this fact has not been considered in the studies of soil-pore fluid interaction and in conventional soil mechanics. In the present paper, a fully explicit dynamic finite element method is developed for nonlinear Darcy law. The governing equations are expressed for saturated porous media based on the extension of the Biot (J. Appl. Phys. 1941; 12: 155–164) formulation. The elastoplastic behavior of soil under earthquake loading is simulated using a generalized plasticity theory that is composed of a yield surface along with non-associated flow rule. Numerical simulations of porous media subjected to horizontal and vertical components of ground motion excitations with different permeability coefficients are carried out; while computed maximum pore water pressure is specially taken into consideration to make the difference between Darcy and non-Darcy flow regimes tangible. Finally, the effect of non-Darcy flow on the evaluated liquefaction potential of sand in comparison to conventional Darcy law is examined. Copyright © 2014 John Wiley & Sons, Ltd.

In this article, we present three nonsymmetric mixed hybrid methods and compare with some recently developed reduction methods which are suitable for the single-phase Darcy flow problem with full anisotropic and highly heterogeneous permeability on general quadrilateral grids. The methods reviewed are multipoint flux approximation (MPFA), multipoint flux mixed finite element method, mixed-finite element with broken method, MPFA-type mimetic finite difference method, and symmetric mixed-hybrid finite element method. The numerical experiments of these methods on different distorted meshes are compared, as well as their differences in performance of fluxes are discussed. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1484–1506, 2014

Summary This paper presents a new heterogeneous multiscale modeling method for porous media flows. Physics at the global level is governed by one set of PDEs, while features in the solution that are beyond the resolution capacity of the global model are accounted for by the next refined set of governing equations. In this method, the global or coarse model is given by the Darcy equation, while the local or refined model is given by the Darcy–Stokes equation. Concurrent domain decomposition where global and local models are applied to adjacent subdomains, as well as overlapping domain decomposition where global and local models coexist on overlapping domains, is considered. An interface operator is developed for the case where global and local models commute along the common interface. For the overlapping decomposition, a residual-based coupling technique is developed that consistently facilitates bottom-up embedding of scale effects from the local Darcy–Stokes model into the global Darcy model. Numerical results are presented for nonoverlapping and overlapping domain decompositions for various benchmark problems. Computed results show that the hierarchically coupled models accurately account for the heterogeneity of the medium and efficiently incorporate local features into the global response. Copyright © 2014 John Wiley & Sons, Ltd.

Filtrodynamic behavior of trans-filter time-dependent pressure signals DP(t) is determined for membrane and frit filters using latex spheres of varying diameter D. Membrane data are best interpreted via a time-dependent, accreting filtration bed, based on Darcy’s law. A single parameter, permeability k, describes each membrane/particle pair. For small particles, k increases with increasing D, then becomes D-independentforlargeones.Predictablebehavior for polydisperse mixtures of small spheres is obtained. The mechanism and behavior of filtration for non-membrane metallic frits is dramatically different, and better described by a previous ‘‘characteristic loading’’ model. Use of frit and membrane filters in series allowed monitoring each filter’s separate response to particle accumulation.

Experimental investigation of two-phase laminar forced convection in a single porous tube heat exchanger is presented. The effect of Darcy, Reynolds, and Prandtl numbers on the performance of this heat exchanger during the condensation process of carbon dioxide at different test conditions were investigated. Gravel sand with different porosities is used as a porous medium. The flow in the porous medium is modeled using the Brinkman–Forchheimer-extended Darcy model. Parametric studies are also conducted to evaluate the effects of porosity and Reynolds and Prandtl numbers on the heat transfer coefficient and the friction factor. A dimensionless performance parameter is developed in order to be used in evaluating the porous tube heat exchanger based on both the heat transfer enhancement and the associated pressure drop. The study covers a wide range of inlet pressure (Pin), mass flow rate (), porosity of gravel sand (e), and Darcy number (Da) which ranged: 34.5 ≤ Pin ≤ 43 bars, 8 * 10− 5 ≤ ≤ 16 * 10− 5 kg/s, 34.9% ≤ e ≤ 44.5%, 1.6 * 10− 6 ≤ Da ≤ 5 * 10− 6, respectively. The study predicted the combined effect of the Reynolds number, Darcy number, porosity, and Prandtl number on the heat transfer and pressure drop of carbon dioxide during the condensation process in a porous medium. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21117

Perforated vertical wells are being used in the petroleum industry around the world. Perforations are the only channel between the wellbore and the reservoir. Accurate prediction of inflow performance for perforated vertical wells is of great significance to reservoir engineering analysis and perforation optimization design. The performance of a perforated vertical well depends on the fluid that enters the well and the fluid flow within the vertical section. The total pressure drop in vertical wells is dominated by gravitational effects, rather than frictional effects as in horizontal wells. Better understanding of the total pressure drop effects in vertical wellbores is helpful for petroleum engineers to design successful horizontal wells and optimize well performance. A considerable amount of analytical and numerical work has been performed on the flow performance of both perforated vertical wells and open-hole horizontal wells. Dikken presented the first semianalytical model that couples the reservoir to the wellbore with turbulent flow in the wellbore. Ouyang proposed a reservoir/wellbore-coupling model by using an analytical reservoir model and a semianalytical wellbore flow model. Ansah et al. developed a finite-element well-inflow model with an asymmetric spiral distribution of cone-shaped perforations around a wellbore to predict the flow performance of perforated vertical-well completions. Pressure-drop calculations have since been discussed in a variety of ways. Islam et al. presented a model allowing for the effect of inflow through perforation within the wellbore, but this model cannot analyze the impact of the perforation distribution on well performance. Landman et al. provided a model that allows for finding the optimal perforation distribution to increase the productivity for a horizontal wells. Hagoort studied an analytical model for single-phase Darcy flow to a single perforation in a semi-infinite porous medium. A considerable amount of analytical work has illustrated that the completion parameters are the main factors that influence the inflow profiles in vertical wells, but there has not been any investigation on the application of these to perforated vertical wells, including wellbore pressure drop. In this study, a mathematical model that couples the pressure and flow rate for perforated vertical well is presented. A basic optimization model is defined and applied to both infinite-conductivity and finite-conductivity wells. The numerical algorithm is recommended to solve the optimization models and the basic data of the “X” well are used for the calculations to help evaluate the models.

We present a method to efficiently simulate coronary perfusion in subject-specific models of the heart within clinically relevant time frames. Perfusion is modelled as a Darcy porous-media flow, where the permeability tensor is derived from homogenization of an explicit anatomical representation of the vasculature. To account for the disparity in length scales present in the vascular network, in this study, this approach is further refined through the implementation of a multi-compartment medium where each compartment encapsulates the spatial scales in a certain range by using an effective permeability tensor. Neighbouring compartments then communicate through distributed sources and sinks, acting as volume fluxes. Although elegant from a modelling perspective, the full multi-compartment Darcy system is computationally expensive to solve. We therefore enhance computational efficiency of this model by reducing the N-compartment system of Darcy equations to N pressure equations, and N subsequent projection problems to recover the Darcy velocity. The resulting 'reduced' Darcy formulation leads to a dramatic reduction in algebraic-system size and is therefore computationally cheaper to solve than the full multi-compartment Darcy system. A comparison of the reduced and the full formulation in terms of solution time and memory usage clearly highlights the superior performance of the reduced formulation. Moreover, the implementation of flux and, specifically, impermeable boundary conditions on arbitrarily curved boundaries such as epicardium and endocardium is straightforward in contrast to the full Darcy formulation. Finally, to demonstrate the applicability of our methodology to a personalized model and its solvability in clinically relevant time frames, we simulate perfusion in a subject-specific model of the left ventricle.