Fracture-Flow-Enhanced Matrix Diffusion in Solute Transport Through Fractured Porous Media

Over the past few decades, significant progress of assessing chemical transport in fractured rocks has been made in laboratory and field investigations as well as in mathematic modeling. In most of these studies, however, matrix diffusion on fracture–matrix surfaces is considered as a process of molecular diffusion only. Mathematical modeling based on this traditional concept often had problems in explaining or predicting tracer transport in fractured rock. In this article, we propose a new conceptual model of fracture-flow-enhanced matrix diffusion, which correlates with fracture-flow velocity. The proposed model incorporates an additional matrix-diffusion process, induced by rapid fluid flow along fractures. According to the boundary-layer theory, fracture-flow-enhanced matrix diffusion may dominate mass-transfer processes at fracture–matrix interfaces, where rapid flow occurs through fractures. The new conceptual model can be easily integrated with analytical solutions, as demonstrated in this article, and numerical models, as we foresee. The new conceptual model is preliminarily validated using laboratory experimental results from a series of tracer breakthrough tests with different velocities in a simple fracture system. Validating of the new model with field experiments in complicated fracture systems and numerical modeling will be explored in future research.     

Evidence of Multi-Process Matrix Diffusion in a Single Fracture from a Field Tracer Test

Compared to values inferred from laboratory tests on matrix cores, many field tracer tests in fractured rock have shown enhanced matrix diffusion coefficient values (obtained using a single-process matrix-diffusion model with a homogeneous matrix diffusion coefficient). To investigate this phenomenon, a conceptual model of multi-process matrix diffusion in a single-fracture system was developed. In this model, three matrix diffusion processes of different diffusion rates were assumed to coexist: (1) diffusion into stagnant water and infilling materials within fractures, (2) diffusion into a degraded matrix zone, and (3) further diffusion into an intact matrix zone. The validity of the conceptual model was then demonstrated by analyzing a unique tracer test conducted using a long-time constant-concentration injection. The tracer-test analysis was conducted using a numerical model capable of tracking the multiple matrix-diffusion processes. The analysis showed that in the degraded zone, a diffusion process with an enhanced diffusion rate controlled the steep rising limb and decay-like falling limb in the observed breakthrough curve, whereas in the intact matrix zone, a process involving a lower diffusion rate affected the long-term middle platform of slowly increasing tracer concentration. The different matrix-diffusion-coefficient values revealed from the field tracer test are consistent with the variability of matrix diffusion coefficient measured for rock cores with different degrees of fracture coating at the same site. By comparing to the matrix diffusion coefficient calibrated using single-process matrix diffusion, we demonstrated that this multi-process matrix diffusion may contribute to the enhanced matrix-diffusion-coefficient values for single-fracture systems at the field scale.     

MHD Peristaltic Transport of a Jeffery Fluid in a Channel With Compliant Walls and Porous Space

Here an attempt has been made to investigate the magnetohydrodynamic (MHD) flow of a non-Newtonian fluid filling the porous space in a channel with compliant walls. Constitutive equations of a Jeffery fluid are used in the mathematical modeling. The flow is created due to sinusoidal traveling waves on the channel walls. The resulting problem is solved analytically and series solution for a stream function is derived. The effects of pertinent flow parameters are discussed through graphs.     

Fully Developed Mixed Convection Flow in a Vertical Channel Containing Porous and Fluid Layer with Isothermal or Isoflux Boundaries

An analysis of fully developed combined free and forced convective flow in a fluid saturated porous medium channel bounded by two vertical parallel plates is presented. The flow is modeled using Brinkman equation model. The viscous and Darcy dissipation terms are also included in the energy equation. Three types of thermal boundary conditions such as isothermal–isothermal, isoflux–isothermal, and isothermal–isoflux for the left–right walls of the channel are considered. Analytical solutions for the governing ordinary differential equations are obtained by perturbation series method. In addition, closed form expressions for the Nusselt number at both the left and right channel walls are derived. Results have been presented for a wide range of governing parameters such as porous parameter, ratio of Grashof number and Reynolds number, viscosity ratio, width ratio, and conductivity ratio on velocity, and temperature fields. It is found that the presence of porous matrix in one of the region reduces the velocity and temperature.     

Laminar flow through a channel with uniformly porous walls of different permeability

Summary The steady laminar flow of a viscous incompressible fluid through a two-dimensional channel, having fluid sucked or injected with different velocities through its uniformly porous parallel walls is considered. A solution for small suction Reynolds number has been given by the authors in a previous paper. The purpose of this paper is to present a solution valid for large Reynolds numbers for the cases of (i) suction at both walls, and (ii) suction at one wall and injection at the other. A technique of matching outer and inner expansions is used to obtain an asymptotic solution for both of these cases. Further a perturbation solution for the case of suction at one wall and injection at the other is obtained by choosing the difference between two wall velocities as the perturbation parameter. Both asymptotic and perturbation solutions are confirmed by exact numerical solutions. As expected, the resulting solutions show the presence of the usual suction boundary layers in both types of flow considered in this paper.     

Analytical Solution of Compression,Free Swelling and Electrical Loading of Saturated Charged Porous Media

Analytical solutions are derived for one-dimensional consolidation, free swelling and electrical loading of a saturated charged porous medium. The governing equations describe infinitesimal deformations of linear elastic isotropic charged porous media saturated with a mono-valent ionic solution. From the governing equations a coupled diffusion equation in state space notation is derived for the electro-chemical potentials, which is decoupled introducing a set of normal parameters, being a linear combination of the eigenvectors of the diffusivity matrix. The magnitude of the eigenvalues of the diffusivity matrix correspond to the time scales for Darcy flow, diffusion of ionic constituents and diffusion of electrical potential.     

Investigation of the Relative Motion of a Viscous Fluid and a Porous Medium Using the Brinkman Equations

Exact solutions are obtained for the following three problems in which the Brinkman filtration equations are used: laminar fluid flow between parallel plane walls, one of which is rigid while the other is a plane layer of saturated porous medium, motion of a plane porous layer between parallel layers of viscous fluid, and laminar fluid flow in a cylindrical channel bounded by an annular porous layer.     

Comparison of Three FE-FV Numerical Schemes for Single- and Two-Phase Flow Simulation of Fractured Porous Media

We benchmark a family of hybrid finite element–node-centered finite volume discretization methods (FEFV) for single- and two-phase flow/transport through porous media with discrete fracture representations. Special emphasis is placed on a new method we call DFEFVM in which the mesh is split along fracture–matrix interfaces so that discontinuities in concentration or saturation can evolve rather than being suppressed by nodal averaging of these variables. The main objective is to illustrate differences among three discretization schemes suitable for discrete fracture modeling: (a) FEFVM with volumetric finite elements for both fractures and porous rock matrix, (b) FEFVM with lower dimensional finite elements for fractures and volumetric finite elements for the matrix, and (c) DFEFVM with a mesh that is split along material discontinuities. Fracture discontinuities strongly influence single- and multi-phase fluid flow. Continuum methods, when used to model transport across such interfaces, smear out concentration/saturation. We show that the new DFEFVM addresses this problem producing significantly more accurate results. Sealed and open single fractures as well as a realistic fracture geometry are used to conduct tracer and water-flooding numerical experiments. The benchmarking results also reveal the limitations/mesh refinement requirements of FE node-centered FV hybrid methods. We show that the DFEFVM method produces more accurate results even for much coarser meshes.     

Mixed Convection in a Vertical Porous Channel

A numerical study of mixed convection in a vertical channel filled with a porous medium including the effect of inertial forces is studied by taking into account the effect of viscous and Darcy dissipations. The flow is modeled using the Brinkman–Forchheimer-extended Darcy equations. The two boundaries are considered as isothermal–isothermal, isoflux–isothermal and isothermal–isoflux for the left and right walls of the channel and kept either at equal or at different temperatures. The governing equations are solved numerically by finite difference method with Southwell–Over–Relaxation technique for extended Darcy model and analytically using perturbation series method for Darcian model. The velocity and temperature fields are obtained for various porous parameter, inertia effect, product of Brinkman number and Grashof number and the ratio of Grashof number and Reynolds number for equal and different wall temperatures. Nusselt number at the walls is also determined for three types of thermal boundary conditions. The viscous dissipation enhances the flow reversal in the case of downward flow while it counters the flow in the case of upward flow. The Darcy and inertial drag terms suppress the flow. It is found that analytical and numerical solutions agree very well for the Darcian model. An erratum to this article is available at .     

On the flow of a viscous fluid driven along a channel by suction at porous walls

This paper is a theoretical treatment of the flow of a viscous incompressible fluid driven along a channel by steady uniform suction through porous parallel rigid walls. Many authors have found such flows when they are symmetric, steady and two-dimensional, by assuming a similarity form of solution due to Berman in order to reduce the Navier-Stokes equations to a nonlinear ordinary differential equation. We generalise their work by considering asymmetric flows, unsteady flows and three-dimensional perturbations. By use of numerical calculations, matched asymptotic expansions for large values of the Reynolds number, and the theory of dynamical systems, we find many more exact solutions of the Navier-Stokes equations, examine their stability, and interpret them. In particular, we show that most previously found steady solutions are unstable to antisymmetric two-dimensional disturbances. This leads to a pitchfork bifurcation, stable asymmetric steady solutions, a Hopf bifurcation, stable time-periodic solutions, stable quasi-periodic solutions, phase locking and chaos in succession as the Reynolds number increases.     

Conjugate heat transfer inside a porous channel

Analytical and numerical analyses have been performed for fully developed forced convection in a fluid-saturated porous medium channel bounded by two parallel plates. The channel walls are assumed to be finite in thickness. Conduction heat transfer inside the channel wall is also accounted and the full problem is treated as a conjugate heat transfer problem. The flow in the porous material is described by the Darcy–Brinkman momentum equation. The outer surfaces of the solid walls are treated as isothermal. A temperature dependent volumetric heat generation is considered inside the solid wall only. Analytical expressions for velocity, temperature, and Nusselt number are obtained after simplifying and solving the governing differential equations with reasonable approximations. Subsequent results obtained by numerical calculations show an excellent agreement with the analytical results.     

The Solution by the Wave Curve Method of Three-Phase Flow in Virgin Reservoirs

There are two goals of this study. The first is to provide an introduction to the wave curve method for finding the analytic solution of a porous medium injection problem. Similar to fractional and chromatographic flow theory, the wave curve method is based on the method of characteristics, but it is applicable to an expanded range of physical processes in porous medium flow. The second goal is to solve injection problems for immiscible three-phase flow, as described by Corey’s model, in which a mixture of gas and water is injected into a porous medium containing oil and irreducible water. In particular we determine, for any choice of the phase viscosities, the proportion of the injected fluids that maximizes recovery around breakthrough time. Numerical simulations are performed to compare our solutions for Corey’s model with those of other models. For the injection problems we consider, solutions for Corey’s model are very similar to those for Stone’s model, despite the presence of an elliptic region in the latter; and they are very different from those for the Juanes-Patzek model, which preserves strict hyperbolicity. A nice feature of our analytical method is that it facilitates explaining both differences and similarities among the solutions for the three models considered.     

Analytical Study of Fluid Flow and Heat Transfer during Forced Convection in a Composite Channel Partly Filled with a Brinkman–Forchheimer Porous Medium

In this paper, the problem of fully developed forced convection in a parallel-plate channel partly filled with a homogeneous porous material is considered. The porous material is attached to the walls of the channel, while the center of the channel is occupied by clear fluid. The flow in the porous material is described by a nonlinear Brinkman–Forchheimer-extended Darcy equation. Utilizing the boundary-layer approach, analytical solutions for the flow velocity, the temperature distribution, as well as for the Nusselt number are obtained. Dependence of the Nusselt number on several parameters of the problem is extensively investigated.     

Asymptotic solutions for the asymmetric flow in a channel with porous retractable walls under a transverse magnetic field

The self-similarity solutions of the Navier-Stokes equations are constructed for an incompressible laminar flow through a uniformly porous channel with retractable walls under a transverse magnetic field. The flow is driven by the expanding or contracting walls with different permeability. The velocities of the asymmetric flow at the upper and lower walls are different in not only the magnitude but also the direction. The asymptotic solutions are well constructed with the method of boundary layer correction in two cases with large Reynolds numbers, i.e., both walls of the channel are with suction, and one of the walls is with injection while the other one is with suction. For small Reynolds number cases, the double perturbation method is used to construct the asymptotic solution. All the asymptotic results are finally verified by numerical results.     

Flow instability in a curved porous channel formed by two concentric cylindrical surfaces

Stability of laminar flow in a curved channel formed by two concentric cylindrical surfaces is investigated. The channel is occupied by a fluid saturated porous medium; the flow in the channel is driven by a constant azimuthal pressure gradient. The momentum equation takes into account two drag terms: the Darcy term that describes friction between the fluid and the porous matrix, and the Brinkman term, which allows imposing the no-slip boundary condition at the channel walls. An analytical solution for the basic flow velocity is obtained. Numerical analysis is carried out using the collocation method to investigate the onset of instability leading to the development of a secondary motion in the form of toroidal vortices. The dependence of the critical Dean number on porosity and the channel width is analyzed.     

Hybrid nanomaterial flow and heat transport in a stretchable convergent/divergent channel:a Darcy-Forchheimer model

The flow behavior in non-parallel walls is an important factor of any physical model including cavity flow and canals, which is applicable for diverging/converging channel. The present communication explains that the flow of the hybrid nanomaterial subjected to the convergent/divergent channel has non-parallel walls. It is assumed that the hybrid nanomaterial movement is in the porous region. A Darcy-Forchheimer medium of porosity is considered to interpret the porosity features. A useful similarity function is adopted to get the strong ordinary coupled equations. Numerical solutions are achieved through the Runge-Kutta-Fehlberg(RKF) fourth-fifth order method, and they are validated with the existing results. Physical nature of the involving constraints is reported with the help of plots. It is explored that the velocity of divergent channel decreases, and convergent channel enhances for the higher solid volume faction. Further, the presence of inertia coefficient and porosity parameter amplifies the velocity at the wall.     

Unsteady flow and heat transfer of porous media sandwiched between viscous fluids

The problem of unsteady oscillatory flow and heat transfer of porous medin sandwiched between viscous fluids has been considered through a horizontal channel with isothermal wall temperatures. The flow in the porous medium is modeled using the Brinkman equation. The governing partial differential equations are transformed to ordinary differential equations by collecting the non-periodic and periodic terms. Closed-form solutions for each region are found after applying the boundary and interface conditions. The influence of physical parameters, such as the porous parameter, the frequency parameter, the periodic frequency parameter, the viscosity ratios, the conductivity ratios, and the Prandtl number, on the velocity and temperature fields is computed numerically and presented graphically. In addition, the numerical values of the Nusselt number at the top and bottom walls are derived and tabulated.     

Numerical Analysis of Coupled Stokes/Darcy Flows in Industrial Filtrations

Free flow channel confined by porous walls is a feature of many of the natural and industrial settings. Viscous flows adjacent to saturated porous medium occur in cross-flow and dead-end filtrations employed primarily in pharmaceutical and chemical industries for solid–liquid or gas–solid separations. Various mathematical models have been put forward to describe the conjugate flow dynamics based on theoretical grounds and experimental evidence. Despite this fact, there still exists a wide scope for extensive research in numerical solutions of these coupled models when applied to problems with industrial relevance. The present work aims towards the numerical analysis of coupled free/porous flow dynamics in the context of industrial filtration systems. The free flow dynamics has been expressed by the Stokes equations for the creeping, laminar flow regime whereas the flow behaviour in very low permeability porous media has been represented by the conventional Darcy equation. The combined free/porous fluid dynamical behaviour has been simulated using a mixed finite element formulation based on the standard Galerkin technique. A nodal replacement technique has been developed for the direct linking of Stokes and Darcy flow regimes which alleviates specification of any additional constraint at the free/porous interface. The simulated flow and pressure fields have been found for flow domains with different geometries which represent prototypes of actual industrial filtration equipment. Results have been obtained for varying values of permeability of the porous medium for generalised Newtonian fluids obeying the power law model. A series of numerical experiments has been performed in order to validate the coupled flow model. The developed model has been examined for its flexibility in dealing with complex geometrical domains and found to be generic in delivering convergent, stable and theoretically consistent results. The validity and accuracy of the simulated results has been affirmed by comparing with available experimental data.     

Open-channel flows and waterfalls

Steady two-dimensional free-surface flows of an inviscid incompressible fluid are studied here, using the complex potential theory. The first flow is a uniform free stream flow in a channel of finite depth. The second is similar, but terminated abruptly as a downstream flow exiting into a falling jet, with and without the effect of gravity. These problems have already been solved for polygonal walls. This paper presents an iterative process for computing flows over arbitrarily shaped channels, with and without the presence of a waterfall at the exit. This process is based on the solution of a mixed boundary problem in the unit disk. The method emphasizes the correspondence between the walls in the physical plane and in the unit disk. The numerical data agree with previously published results and extend them to arbitrary curved walls. This method yields solutions only for supercritical flows.     

Convective magnetohydrodynamic flow in a vertical channel

An analysis is presented for the combined forced and free convective magnetohydrodynamic flow in a vertical, finite rectangular channel that is subjected simultaneously to a pressure gradient and a temperature gradient. Exact solutions are found for electrically nonconducting channel walls and perfectly conducting walls. In particular, the case of heating from below is examined and discussed.