Flow In Homogeneous Porous Media Research Articles

Upscaled dynamic capillary pressure for two-phase flow in porous media

A closed expression for the average pressure difference (often called the macroscopic dynamic capillary pressure in the literature) is proposed for two-phase, Newtonian, incompressible, isothermal and creeping flow in homogeneous porous media. This upscaled equation complements the average equations for mass and momentum transport derived in a previous article. Consistently with this work, the expression is derived employing a simplified version of the volume-averaging method that makes use of elements of the adjoint method and Green's formula. The resulting equation for the average pressure difference is novel, as it shows that this quantity is controlled by the pressure gradient (and body forces) in each phase, as well as interfacial effects, and is applicable to situations in which the fluid–fluid interface is not necessarily at its steady position. The effective-medium quantities associated with the sources are all obtained from the solution of a single adjoint (or closure) problem to be solved on a (periodic) unit cell representative of the process. The average pressure difference predicted by the derived expression is validated through excellent comparisons with direct numerical simulations performed in a model porous structure.

Macroscopic model for unsteady generalized Newtonian fluid flow in homogeneous porous media

A macroscopic model for unsteady incompressible isothermal non-Newtonian flow in homogeneous porous media, taking into account inertial and slip effects at solid–fluid interfaces, is derived in this work. The development is carried out considering a general Newton’s law of viscosity for the fluid phase. Using the classical volume averaging method, the seepage velocity is shown to be solenoidal. The macroscopic momentum equation is derived in the Laplace domain, employing a simplified version of the volume averaging method, which calls upon Green’s formulas and adjoint problems for Green’s function pairs for the velocity and pressure. In the Laplace domain, the macroscopic momentum equation takes the form of Darcy’s law corrected by a term that accounts for the initial flow condition. Once transformed back into the time domain, this equation provides the macroscopic velocity that depends on two terms. The first one is under the form of a time convolution between the macroscopic pressure gradient and the time derivative of an apparent permeability tensor. The second one is a memory term that accounts for the effect of the initial flow conditions. These two effective quantities are determined from the solution of a single closure problem that naturally results from the derivations. The model is consistent with the unsteady model in the Newtonian case and simplifies to the steady versions of some non-Newtonian macroscopic flow models. The macroscopic model is validated with pore-scale simulations performed in 2D model porous structures, considering a Carreau fluid. The impact of inertia and non-Newtonian effects on the dynamics of the macroscopic coefficients is highlighted.

Macroscopic Model for Unsteady Generalized Newtonian Fluid Flow in Homogeneous Porous Media

Macroscopic Model for Unsteady Generalized Newtonian Fluid Flow in Homogeneous Porous Media

Equation of State’s Crossover Enhancement of Pseudopotential Lattice Boltzmann Modeling of CO2 Flow in Homogeneous Porous Media

The original Shan-Chen’s pseudopotential Lattice Boltzmann Model (LBM) has continuously evolved during the past two decades. However, despite its capability to simulate multiphase flows, the model still faces challenges when applied to multicomponent-multiphase flows in complex geometries with a moderately high-density ratio. Furthermore, classical cubic equations of state usually incorporated into the model cannot accurately predict fluid thermodynamics in the near-critical region. This paper addresses these issues by incorporating a crossover Peng–Robinson equation of state into LBM and further improving the model to consider the density and the critical temperature differences between the CO2 and water during the injection of the CO2 in a water-saturated 2D homogeneous porous medium. The numerical model is first validated by analyzing the supercritical CO2 penetration into a single narrow channel initially filled with H2O, depicting the fundamental role of the driving pressure gradient to overcome the capillary resistance in near one and higher density ratios. Significant differences are observed by extending the model to the injection of CO2 into a 2D homogeneous porous medium when using a flat versus a curved inlet velocity profile.

Physics‐Informed Deep Neural Networks for Learning Parameters and Constitutive Relationships in Subsurface Flow Problems

AbstractWe present a physics‐informed deep neural network (DNN) method for estimating hydraulic conductivity in saturated and unsaturated flows governed by Darcy's law. For saturated flow, we approximate hydraulic conductivity and head with two DNNs and use Darcy's law in addition to measurements of hydraulic conductivity and head to train these DNNs. For unsaturated flow, we approximate unsaturated conductivity function and capillary pressure with DNNs and train these DNNs using measurements of capillary pressure and the Richards equation. Because it is difficult to measure unsaturated conductivity in the field, we assume that no measurements of unsaturated conductivity are available. The proposed approach enforces the partial differential equation (PDE) (Darcy or Richards equation) constraints by minimizing the PDE residual at select points in the simulation domain. We demonstrate that physics constraints increase the accuracy of DNN approximations of sparsely observed functions and allow for training DNNs when no direct measurements of the functions of interest are available. For the saturated conductivity estimation problem, we show that the physics‐informed DNN method is more accurate than the state‐of‐the‐art maximum a posteriori probability method. For the unsaturated flow in homogeneous porous media, we find that the proposed method can accurately estimate the pressure‐conductivity relationship based on the capillary pressure measurements only, even in the presence of measurement noise.

Flow-Area Relations in Immiscible Two-Phase Flow in Porous Media

We present a theoretical framework for immiscible incompressible two-phase flow in homogeneous porous media that connects the distribution of local fluid velocities to the average seepage velocities. By dividing the pore area along a cross-section transversal to the average flow direction up into differential areas associated with the local flow velocities, we construct a distribution function that allows us not only to re-establish existing relationships between the seepage velocities of the immiscible fluids but also to find new relations between their higher moments. We support and demonstrate the formalism through numerical simulations using a dynamic pore-network model for immiscible two-phase flow with two- and three-dimensional pore networks. Our numerical results are in agreement with the theoretical considerations.

Thermophoretic effects on instabilities of nanoflows in porous media

Instabilities of nanoparticle-laden non-isothermal flows in homogeneous porous media are investigated. The study is conducted for two representative systems; a hot fluid displacing a cold one (HDC) and a cold fluid displacing a hot one (CDH). The effects of Brownian diffusion and of thermophoresis, representing the average motion of the nanoparticles as a result of temperature gradients, are analysed. In the HDC case, the synergetic Brownian and thermophoretic effects induce a migration of nanoparticles towards the cold fluid and tend systematically to enhance the instability. In particular, because of these combined effects, an initially stable displacement can actually be destabilized. In the CDH case however, Brownian diffusion still acts towards the transport of nanoparticles downstream into the hot fluid while thermophoresis tends to resist such migration. These counteracting effects lead to the generation of local accumulations of nanoparticles at the front and engender the development of local stable regions in the flow. These stable regions hinder the growth of the instabilities, especially those of backward-developing fingers. It is concluded that, in this case, thermophoresis acts against Brownian diffusion and results in less unstable displacements compared to flows where thermophoresis is absent. This effect, exclusively associated with thermophoresis, will not be observed in nanoparticle-free non-isothermal displacements. Finally, it is found that the main effects of Brownian diffusion and thermophoresis arise mainly from their contributions to nanoparticle transport while their effects on the energy balance are negligible and can be disregarded.

Symmetry properties of macroscopic transport coefficients in porous media

We report on symmetry properties of tensorial effective transport coefficients characteristic of many transport phenomena in porous systems at the macroscopic scale. The effective coefficients in the macroscopic models (derived by upscaling (volume averaging) the governing equations at the underlying scale) are obtained from the solution of closure problems that allow passing the information from the lower to the upper scale. The symmetry properties of the macroscopic coefficients are identified from a formal analysis of the closure problems and this is illustrated for several different physical mechanisms, namely, one-phase flow in homogeneous porous media involving inertial effects, slip flow in the creeping regime, momentum transport in a fracture relying on the Reynolds model including slip effects, single-phase flow in heterogeneous porous media embedding a porous matrix and a clear fluid region, two-phase momentum transport in homogeneous porous media, as well as dispersive heat and mass transport. The results from the analysis of these study cases are summarized as follows. For inertial single-phase flow, the apparent permeability tensor is irreducibly decomposed into its symmetric (viscous) and skew-symmetric (inertial) parts; for creeping slip-flow, the apparent permeability tensor is not symmetric; for one-phase slightly compressible gas flow in the slip regime within a fracture, the effective transmissivity tensor is symmetric, a result that remains valid in the absence of slip; for creeping one-phase flow in heterogeneous media, the permeability tensor is symmetric; for two-phase flow, we found the dominant permeability tensors to be symmetric, whereas the coupling tensors do not exhibit any special symmetry property; finally for dispersive heat transfer, the thermal conductivity tensors include a symmetric and a skew-symmetric part, the latter being a consequence of convective transport only. A similar result is achieved for mass dispersion. Beyond the physical mechanisms under consideration in the present work, the reported technique can be viewed as a general methodology applicable to any type of upscaled model obtained by volume averaging.

Inertial effects in cyclic time‐dependent displacement flows in homogeneous porous media

Miscible displacements in homogeneous porous media are investigated through numerical simulations for cyclic time‐dependent velocities considering inertial effects. It is found that the velocity's models, cycle period, and amplitude have significant impacts on the flow, however the effects greatly depend on whether inertia is considered or not. Generally, cyclic extraction‐injection (E/I) processes are always less unstable than constant injection and their injection‐extraction (I/E) counterpart, regardless of the strength of inertia. Instabilities of E/I processes are reduced as the velocity period or amplitude increases, and these attenuating effects increase with the Reynolds number. However, in the case of I/E processes, the effects of the period and the amplitude are strongly dependent on inertia. For non‐inertial cases, larger periods or amplitudes lead to more unstable flows, while for their inertial counterparts, the effects are found to be non‐monotonic. In particular, as the period increases, instabilities first decrease and then increase. A period‐stabilizing range is identified in which the displacements of the time‐dependent injection velocities are actually less unstable than those of the constant injection flow. When the period is further increased to the period‐destabilizing range, the flows tend to become more unstable. Moreover, the velocity amplitude can attenuate or enhance the instabilities depending on whether the period is within the period‐stabilizing or period‐destabilizing range. Furthermore, stronger inertial effects are found to greatly reduce the instabilities and to expand the period‐stabilizing range.

Structural optimization of porous media for fast and controlled capillary flows.

A general quantitative model of capillary flow in homogeneous porous media with varying cross-sectional sizes is presented. We optimize the porous structure for the minimization of the penetration time under global constraints. Programmable capillary flows with constant volumetric flow rate and linear evolution of flow distance to time are also obtained. The controlled innovative flow behaviors are derived based on a dynamic competition between capillary force and viscous resistance. A comparison of dynamic transport on the basis of the present design with Washburn's equation is presented. The regulation and maximization of flow velocity in porous materials is significant for a variety of applications including biomedical diagnostics, oil recovery, microfluidic transport, and water management of fabrics.

Numerical Simulations of Gravity Driven Reversible Reactive Flows in Homogeneous Porous Media

The effect of reversibility on the instability of a miscible vertical reactive flow displacement is examined. A model, where densities and/or viscosities mismatches between the reactants and the chemical product trigger instability, is adopted. The problem is governed by the continuity equation, Darcy’s law, and the convection-diffusion-reaction equations. The problem is formulated and solved numerically using a combination of the highly accurate spectral methods based on Hartley’s transform and the finite-difference technique. Nonlinear simulations were carried out for a variety of parameters to analyse the effects of the reversibility of the chemical reaction on the development of the flow under different scenarios of the frontal instability. In general, faster attenuation in the development and growth of the instability is reported as the reversibility of the chemical reaction increases. However, it was observed that reversibility is capable of triggering instability for particular choices of the densities and viscosities mismatches. In addition, the effect of the reversibility in enhancing the instability was illustrated by presenting the total relative contact area between the reactants and the product.

Upscaling immiscible two-phase dispersed flow in homogeneous porous media: A mechanical equilibrium approach

In this work, we model immiscible two-phase dispersed flow in homogeneous porous media by upscaling the governing mass and momentum transport equations at the pore scale using the method of volume averaging. The model consists of a closed set of macroscopic equations for mass and momentum transport applicable for the dispersed and continuous phases. Furthermore, under the local mechanical equilibrium assumption, only one macroscopic equation arises for momentum transport, which resembles an extension of Darcy׳s law; whereas for mass transport, the equilibrium model reduces to the continuity equation. These macroscopic models are written in terms of effective medium coefficients that are computed from solving the associated closure problems in representative regions of the pore scale. After performing a parametric analysis, we observe that the magnitude of the longitudinal component of the permeability-like coefficient increases with the saturation and viscosity of the dispersed phase. We validated the model by comparing the predictions of the permeability coefficient with experimental data available in the literature. The results exhibit a relative error percent that ranges from 1% to 15%.

Development and application of a multi-scale k–ε model for turbulent porous medium flows

A multi-scale k–ε eddy viscosity model for turbulence in porous media is developed. When the double averaging is applied to the momentum equation, the dispersive covariance, the macro-scale and micro-scale Reynolds stresses appear and need modelling to close the equation. The conventional eddy viscosity modelling is applied to model the second moments for engineering applications. A k–ε two-equation eddy viscosity model is employed for obtaining the volume averaged Reynolds stress which consists of the macro-scale and the micro-scale Reynolds stresses. The micro-scale Reynolds stress is also independently modelled and obtained by solving another set of k and ε equations, whilst an algebraic model is developed for the dispersive covariance. The presently proposed multi-scale four equations eddy viscosity model is evaluated in developed turbulent flows in homogeneous porous media, porous wall channel flows and porous rib-mounted channel flows with satisfactory accuracy.

A High Order Method for Three Phase Flow in Homogeneous Porous Media

The modeling of three-phase uid ow has many important applications in reservoir simulation. In this paper, we introduce a high order method for sequentially solving the phase pressuresaturation formulation of three-phase ow. The sequential approach, while less stable than a fully coupled approach, is more computationally ecient. We present a discontinuous Galerkin method in space, suitable for the sequential approach due to its high-order accuracy. We consider coarser meshes with high degree polynomials to minimize the dimension of the system while maximizing accuracy. Numerical results are given for homogeneous media.

Experimental investigation on water flow in cubic arrays of spheres

One-dimensional uniform flow in homogeneous porous media was experimentally investigated. Head drop experiments were conducted in four test tubes with cubic arrays of spheres in diameter 3mm, 5mm, 8mm and 10mm. The experimental results indicate that Darcy’s law should be an approximate expression by neglecting the inertial term for flow at low velocity. Nonlinearity is attributed to inertial term in porous medium before the turbulent flow emerges. Forchheimer equation with constant coefficients can well predict the flow in porous medium. The relationship between the diameter of the particles and the coefficients a and b in the equations were verified. Different Ergun type equations were used to predict the head drop and compared to the experimental data. It shows that the Irmay equation could well predict the fluid flow in cubic arrays of spheres, while the prediction of head drop by Ergun equation was much higher than observed data. It indicates that the coefficients α and β in the Ergun type equations have certain relations with porosity or the pore structure and would vary for different medium. The discontinuity observed was interpreted by transition from steady flow to weakly turbulence and compared with previous studies.

Dynamics of fluid flow and heat transfer in homogeneous porous media

AbstractNon‐isothermal miscible viscous flow in homogeneous porous media is studied through nonlinear simulation of the process and analysis of breakthrough efficiency. The main focus is on the effect of heat exchange between the displacing fluid and the solid matrix on the resulting instability. Higher diffusion rate of heat in the fluids reduces the effect of lagging between the fronts due to heat exchange. Hence, sweep efficiency of the injected solvent is less sensitive to thermal lag coefficient as the diffusion ratio increases. Variation of heat sweep efficiency with thermal lag coefficient and thermal log mobility ratio has been analysed as well. © 2012 Canadian Society for Chemical Engineering

Analytical Solutions to Gas Flow Problems in Low Permeability Porous Media

In most of conventional porous media the flow of gas is basically controlled by the permeability and the contribution of gas flow due to gas diffusion is ignored. The diffusion effect may have significant impact on gas flow behavior, especially in low permeability porous media. In this study, a dual mechanism based on Darcy flow as well as diffusion is presented for the gas flow in homogeneous porous media. Then, a novel form of pseudo pressure function was defined. This study presents a set of novel analytical solutions developed for analyzing steady-state and transient gas flow through porous media including effective diffusion. The analytical solutions are obtained using the real gas pseudo pressure function that incorporates the effective diffusion. Furthermore, the conventional assumption was used for linearizing the gas flow equation. As application examples, the new analytical solutions have been used to design new laboratory and field testing method to determine the porous media parameters. The proposed laboratory analysis method is also used to analyze data from steady-state flow tests of three core plugs. Then, permeability (k) and effective diffusion coefficient (D e) was determined; however, the new method allows one to analyze data from both transient and steady-state tests in various flow geometries.

An extended stability criterion for density-driven flows in homogeneous porous media

Stability of density-driven flows is a challenging problem with current applications in major areas like energy exploration, water pollution, nuclear and oil industries. The mathematical model for such flows is a system of coupled non linear partial differential equations. To study the physical stability of the system, we consider steady-state flow and perturb the solution of the full system of equations (without Boussinesq approximation) and investigate how it evolves in time: if the solution does not grow indefinitely, the system is called stable. The perturbations are treated as being the result of sub-scale interactions between the velocity field and the solute mass. Making use of a two-scale expansion of the solution, we derived extended stability criteria that include the effects of density, viscosity and flow velocity in flow configurations aligned parallel as well as orthogonal to gravity forces. Numerical simulations with the numerical simulator d 3 f are presented to test the theoretical stability criteria.

Computation of turbulent flows over porous/fluid interfaces

This paper reports turbulence modelling for flows over porous walls whose permeability makes the flows more turbulent than those over impermeable rough walls. In order to avoid treating the complicated surface geometry of the porous wall, the strategy of the analytical wall-function (AWF) for rough wall turbulence by Suga et al (2006 Int. J. Heat. Fluid Flow 27 852–66) is employed. For the flows inside the walls, an approximation using the Brinkman equations for flows in homogeneous porous media is applied and coupled with the AWF equations. The presently proposed scheme proves its good performance for predicting turbulent channel and boundary layer flows over permeable walls.

On finite element methods for heterogeneous elliptic problems

Dealing with variational formulations of second order elliptic problems with discontinuous coefficients, we recall a single field minimization problem of an extended functional presented by Bevilacqua et al. [Bevilacqua, L., Feijóo, R.A., Rojas, L.F., 1974. A variational principle for the Laplace operator with application in the torsion of composite rods. International Journal of Solids Structuring 10, 1091–1102], which we associate with the basic idea supporting discontinuous Galerkin finite element methods. We review residual based stabilized mixed methods applied to Darcy flow in homogeneous porous media and extend them to heterogeneous media with an interface of discontinuity. For smooth interfaces, the proposed formulations preserve the continuity of the flux and exactly imposes the constraint between the tangent components of Darcy velocity on the interface. Convergence studies for a heterogeneous and anisotropic porous medium confirm the same rates of convergence predicted for homogeneous problem with smooth solutions.