Parametric Study of One-Dimensional Solute Transport in Deformable Porous Media

Formulation for the effect of dissipation of excess pore water pressure on one-dimensional advective-diffusive transport of solutes in clays is presented. The formulation is based on the effect of the rate of consolidation or swelling and excess pore pressure or suction dissipation on transient, nonlinear advective component of transport through clay. One partial differential equation is presented for advective diffusive transport that is dependent upon soil/solute properties and transient hydraulic head gradient, which is calculated from the Terzaghi consolidation equation. Finite difference method is used to solve the system of partial differential equations for consolidation and solute transport. Four hypothetical cases are evaluated to demonstrate the effect of consolidation under loading and swelling under hydraulic gradient on advective-diffusive transport and breakthrough in single and double drainage clay layer. The results show that consolidation in doubly drained clay impacts concentration profiles, but does not significantly impact breakthrough of the diffusive flux. Consolidation under single drainage conditions, significantly impacts the diffusional flux. When drainage path is the same as the diffusional flux, consolidation accelerates transport and breakthrough time can be less than 5% of the diffusional breakthrough time under no consolidation. Swelling under hydraulic gradient application can either accelerate or retard the advective diffusive flux, depending upon the ratio of the effective diffusion coefficient relative to the coefficient of consolidation. Higher the effective diffusion coefficient and lower the coefficient of consolidation result in an increase in the effect of pressure dissipation on transport.     

Improving Fluid Flow in Clarifiers Using a Highly Porous Media

The theory behind ideal sedimentation tanks assumes that the fluid moves in uniform flow. Numerous studies have shown numerous nonuniform flow patterns, which explains why the solids removal efficiency of real clarifiers does not match theory predictions. This problem gets worse when the influent flow rate exceeds what the clarifier was designed to handle. This research shows that introducing a highly porous bed of “dendrite” fibers into clarifiers designed for the pulp and paper industry removed some of the nonuniformities as shown in a residence time distribution (RTD). These clarifiers have RTDs that are similar to their waste treatment counterparts. So, it is expected that the new technology will have similar effects in waste treatment systems. The bed acts as a resistor to nonaxial flow, reducing radial and angular components of velocity. It is also shown that the greatest effect on the bulk flow patterns occurs when the bed is positioned such that all of the overflow passes through it. Increasing the bed thickness also increases the effect. Analysis of these results was performed with a new model for RTDs based on the Weibull distribution, which is mathematically similar to the equation for a mixed flow RTD.     

Modeling of Flow in Three-Dimensional, Multizone, Anisotropic Porous Media with Weakly Singular Integral Equation Method

A symmetric Galerkin boundary element method is developed for modeling steady-state Darcy’s flow in three-dimensional porous media. The proposed technique is capable of treating a nonhomogeneous medium that consists of several regions possessing different permeabilities and may contain a surface of discontinuity such as impermeable seals. The key governing equations are established based on a pair of weakly singular weak-form integral equations for the fluid pressure and the fluid flux. The crucial feature of those integral equations are that they are completely regularized such that all involved kernels are only weakly singular and that they are applicable to a medium possessing generally anisotropic permeability. A final system of governing integral equations is obtained in a symmetric form and validity of all involved integrals only requires continuity of the pressure boundary data; as a consequence, continuous interpolations can be employed everywhere in the numerical approximation. To accurately capture the jump of the fluid pressure in the local region near the boundary of the discontinuity surface, special tip elements are employed. To further enhance accuracy and computational efficiency of the method, special integration quadrature is adopted to treat both weakly singular and nearly singular integrals and an interpolation strategy is utilized to evaluate the kernels for anisotropic permeability. As demonstrated by various numerical experiments, the current method yields highly accurate results with only weak dependence on mesh refinement.     

Extended Boussinesq Equations for Water-Wave Propagation in Porous Media

This paper presents a new Boussinesq-type model equations for describing nonlinear surface wave motions in porous media. The mathematical model based on perturbation approach reported by Hsiao et al. is derived. The drag force and turbulence effect suggested by Sollitt and Cross are incorporated for observing the flow behaviors within porous media. Additionally, the approach of Chen for eliminating the depth-dependent terms in the momentum equations is also adopted. The model capability on an applicable water depth range is satisfactorily validated against the linear wave theory. The nonlinear properties of model equations are numerically confirmed by the weakly nonlinear theory of Liu and Wen. Numerical experiments of regular waves propagating in porous media over an impermeable submerged breakwater are performed and the nonlinear behaviors of wave energy transfer between different harmonics are also examined.     

Temporal Moment Analysis for Volatile Organic Compounds in Dual-Porosity Porous Media: Loss Fractions and Effective Parameters

Using transform techniques, analytical expressions for potential losses by volatilization and degradation are developed for several organic compounds in dual porosity porous media. A sensitivity analysis is conducted to study the importance of different physical/chemical processes on volatilization, degradation, and leaching losses. To obtain estimates for overall solute behavior, expressions for effective Péclet numbers and degradation rates for organic contaminants are presented using method of moments. Results indicate that large fractions of many organic compounds are likely to volatilize into the atmosphere for sandy and clayey soils under typical flow conditions. It is found that nondimensional degradation influences both advective and dispersive effects. Thus, the Péclet number from effective-parameter equation tends to be enhanced when the nondimensional degradation is rather high. The simple expressions for moments and effective parameters can be used as screening tools to assess the behavior of volatile compounds in vadoze zone of soils.     

Turbulent Flow over a Channel with Fluid-Saturated Porous Bed

The characteristics of fully developed turbulent flow in a hybrid domain channel, which consists of a clear fluid region and a porous bed, are examined numerically using a model based on the macroscopic Reynolds-averaged Navier–Stokes equations. By adopting the classical continuity interface conditions, the present model treats the hybrid domain problem with a single domain approach, and the simulated results are noted to coincide with the existing experimental data and microscopic data. The effects of porosity ? and Darcy number Da on the flow properties over and inside the porous bed are further investigated in the selected ranges of 0.6 ? ? ? 0.8, and 1.6×10?4 ? Da ? 1.6×10?2. It has been demonstrated that the presence of the porous bed causes the significant reduction of the flow velocities inside the clear fluid region relative to that of a smooth impermeable bed, and also reduces the magnitude of the integral constant B of the velocity logarithmic distributions from its traditional value 5.25. Moreover, turbulent shear stress within the upper part of the porous bed increases significantly with the porosity ? and Darcy number Da. The thickness of turbulence penetration remains proportional to the values of porosity ? and Darcy number Da.     

Turbulent Flow Over and Within a Porous Bed

The characteristics of turbulent flow in open channels with a porous bed are studied numerically and experimentally. The “microscopic” approach is followed, by which the Reynolds-averaged Navier-Stokes equations are solved numerically in conjunction with a low-Re k-ε turbulence model above and within the porous bed. The latter is represented by a bundle of cylindrical rods of certain diameter and spacing, resulting in permeability K ranging from 5.5490×10?7 to 4.1070×10?4?m2 and porosity ? from 0.4404 to 0.8286. Mean velocities and turbulent stresses are measured for ? = 0.8286 using hot-film anemometry. Emphasis is given to the effect of Darcy number Da on the flow properties over and within the porous region. Computed and experimental velocities in the free flow are shown to decrease with increasing Da due to the strong momentum exchange near the porous medium/free flow interface and the corresponding penetration of turbulence into the porous layer for highly permeable beds. Computed discharge indicates the significant reduction of the channel capacity, compared to the situation with an impermeable bed. On the contrary, laminar flow computations, along with analytical solutions and measurements, indicate opposite effects of the porous medium on the free flow.     

Viscous Flow Past a Porous Spherical Shell—Effect of Stress Jump Boundary Condition

Using the stress jump boundary condition for the tangential stresses at the porous liquid interface along with the continuity of the velocity components and normal stress, the uniform viscous flow past a porous spherical shell with external radius r1, internal radius r2 is studied. The flow inside the porous region is governed by Brinkman equation. The flow in the liquid region is governed by the Stokes equation. The flow field is computed by matching the boundary conditions at the porous-fluid interface. The effect of stress jump coefficient β on the flow field is very much felt. An increase in the drag with permeability is found for different R, different ratio of r1/r2, and also a change in magnitude of the drag for different values of stress jump coefficient β is observed. Also, the variation of torque and shear stress with permeability and the stress jump coefficient is discussed.     

Fully Nonlinear Boussinesq-Type Equations for Waves and Currents over Porous Beds

The paper introduces a complete set of Boussinesq-type equations suitable for water waves and wave-induced nearshore circulation over an inhomogeneous, permeable bottom. The derivation starts with the conventional expansion of the fluid particle velocity as a polynomial of the vertical coordinate z followed by the depth integration of the vertical components of the Euler equations for the fluid layer and the volume-averaged equations for the porous layer to obtain the pressure field. Inserting the kinematics and pressure field into the Euler and volume-averaged equations on the horizontal plane results in a set of Boussinesq-type momentum equations with vertical vorticity and z-dependent terms. A new approach to eliminating the z dependency in the Boussinesq-type equations is introduced. It allows for the existence and advection of the vertical vorticity in the flow field with the accuracy consistent with the level of approximation in the Boussinesq-type equations for the pure wave motion. Examination of the scaling of the resistance force reveals the significance of the vertical velocity to the pressure field in the porous layer and leads to the retention of higher-order terms associated with the resistance force. The equations are truncated at O(μ4), where μ = measure of frequency dispersion. An analysis of the vortical property of the resultant equations indicates that the energy dissipation in the porous layer can serve as a source of vertical vorticity up to the leading order. In comparison with the existing Boussinesq-type equations for both permeable and impermeable bottoms, the complete set of equations improve the accuracy of potential vorticity as well as the damping rate. The new equations retain the conservation of potential vorticity up to O(μ2). Such a property is desirable for modeling wave-induced nearshore circulation but is absent in existing Boussinesq-type equations.     

Intake Shape Factors for Entry Points in Porous Media with Transversely Isotropic Hydraulic Conductivity

The intake shape factors for spherical and disc-shaped entry points located within a hydraulically transversely isotropic porous medium of infinite extent are presented in exact closed form. Implications of these results on a methodology for the in situ evaluation of the principal hydraulic conductivities are discussed.     

Laminar Water Wave and Current Passing Over Porous Bed

The problem of the dynamic interaction of water waves, current, and a hard poroelastic bed is dealt with in this study. Finite-depth homogeneous water with harmonic linear water waves passing over a semi-infinite poroelastic bed is investigated. In order to reveal the importance of viscous effect for different bed forms, viscosity of water is considered herein. In a boundary layer correction approach, the governing equations of the poroelastic material are decoupled without losing physical generality. The contribution of pressure effect and shear effect to the hard poroelastic bed, which is a valuable indication to the mechanism of ripple formation, is clarified in the present study. This approach will be helpful in saving time and storage capacity when it is applied to numerical computation.     

Turbulent Boundary Layer Flows Above a Porous Surface Subject to Flow Injection

A new analytical expression for velocity profile in a fully developed turbulent boundary layer above a porous surface subject to flow injection is derived by solving the coupled Reynolds equations and turbulent kinetic energy equation. The advection of turbulent kinetic energy is considered during the derivation, whereas the earlier studies have neglected it. The new solution reduces to the universal logarithmic law in the case of no flow injection. For the small injection, the solution can be expanded into a series form in terms of the normalized injection velocity. The leading order terms are found to be equivalent to those in the earlier works in which the advection of turbulent kinetic energy has been neglected in the derivation. The new solution can provide more accurate prediction of bed shear stress for a wide range of flow injection rate, fluid type (e.g., from air to water), and surface roughness. On the other hand, the earlier theories may significantly underestimate bed shear stress under high injection rates.     

Capillary Fringe and Unsaturated Flow in a Porous Reservoir Bank

Steady, 2D Darcian seepage from a zero-depth reservoir into a homogeneous porous bank is studied analytically. Using the Green-Ampt assumption for hydraulic conductivity as a function of pressure and the Vedernikov model for the tension-saturated zone, a free boundary problem with the capillary fringe spread along the bank surface is solved. Accurate direct calculations are made for the extent of the capillary rise along a vertical and horizontal bank. Unsaturated flow from a vertical phreatic surface into a bank with either an impermeable or isobaric vertical soil slope is analyzed in terms of the Philip model. Explicit expressions for the Darcian velocity components, stream function, and/or Kirchhoff potential are presented. The flow topology and characteristics are shown to depend strongly on the boundary condition at the soil surface.     

Real Porous Media: Local Geometry and Transports

Our recent progress in the analysis and reconstruction of real porous media are surveyed and some emphasis is put on the two-field and on the grain reconstruction methods. Then, after a short summary of our previous analyses of transports, two recent developments are presented, namely, the determination of the resistivity index and of the dispersion tensor in multiphase flow.     

Breakthrough Curves and Simulation of Virus Transport through Fractured Porous Media

In this paper, an advective dispersive virus transport equation, including first-order adsorption and an inactivation constant, is used for simulating the movement of viruses in fractured porous media. The implicit finite-difference numerical technique is used to solve the governing equations for viruses in the fractured porous media. In this work, the focus is (1)?to investigate the transport processes of the movement of viruses in both fractured rock and porous rock without fracture and (2)?to simulate the experimental data of biocolloids through a fractured aquifer model. It is seen that movement of the contaminant is faster in the fractured rock than in the porous rock formation. Higher values of diffusion coefficient, matrix porosity, mass transfer constant, and inactivation rate reduce both temporal and spatial virus concentrations in the fracture. Also, experimental data of biocolloids in the fractured aquifer model with constant and time-dependent inactivation rates were simulated successfully.     

Poromechanics Response of Inclined Wellbore Geometry in Fractured Porous Media

This paper presents the poromechanics/poroelastic analytical solution for stress and pore pressure fields induced by the action of drilling and/or the pressurization of an inclined/horizontal wellbore in fractured fluid-saturated porous media, or naturally fractured fluid-saturated rock formations. The model which is developed within the framework of the coupled processes in the dual-porosity/dual-permeability approach accounts for coupled isothermal fluid flow and rock/fractures deformation. The solution to the inclined/horizontal wellbore problem is derived for a wellbore drilled in an infinite naturally fractured poroelastic medium, subjected to three-dimensional in situ state of stress and pore pressure. The dual-porosity analytical solution is first reduced to the limiting single-porosity case and verified against an existing single-porosity solution. A comparison between single-porosity and dual-porosity poroelastic results is conducted and displayed in this work. Finally, wellbore stability analyses have been carried out to demonstrate possible applications of the solution.     

Experimental Investigations of Colloidal Silica Grouting in Porous Media

This paper presents the results of an experimental investigation performed to understand the processes influencing the injection of colloidal silica grout into porous media. Based on the combined analysis of grout injection pressures and the visually observed grout distribution patterns, three major processes, gelation, shear, and viscous fingering, have been identified to occur during grout injection. The results demonstrate the dynamic interplay between grout viscosity variations and the resulting flow instabilities.     

Energy Equation for Volatile Liquid Transport in Porous Media

Two energy balance equations widely used to describe simultaneous transfer of heat and mass in porous media are inconsistent with control volume energy conservation. Potential energy, enthalpy, and internal energy terms are involved in the discrepancies. Energy within a volume is properly counted as the sum of internal, potential, and kinetic energy. However, one equation uses enthalpy where internal energy should have been used. In the other, potential energy and shifts in internal energy associated with heat of wetting are not included. Energy conservation for a control volume dictates summing convective fluxes of internal, potential, and kinetic energy at the control volume surface along with conducted heat and work crossing the boundary. The pressure–volume (pv) work at the volume surface may be combined with internal energy convection so that flow of enthalpy is used in the flux term. Examples of energy change versus work input in adiabatic processes illustrate the error introduced when enthalpy rather than internal energy is used to compute control volume energy content. For porous media flows kinetic energy can be dropped. A consistent equation based on the control volume approach is presented. It includes effects due to internal energy, potential energy, heat of wetting, conducted heat, non-pv work, enthalpy, and mass flow. Substantial temperature changes due to heat of wetting have been found experimentally in a separate work. A comparison is needed of the experiments and a numerical simulation based on the new equation.     

Local Balance Unsteady Friction Model

The paper proposes the evaluation of unsteady friction by a one-dimensional local balance model. The model is applied for the case of water hammer in a single pipeline for both the downstream end and upstream end valve, and for both rapid valve closure and opening. The model is based on local balance of the friction force. Comparisons with experimental results show that the model correctly predicts the extreme values of pressure head oscillation, as well as its shape for both rapid valve closure and opening, and then overcomes the limits of previous unsteady friction models based on instantaneous acceleration. As the comparisons with experimental results can be made easily only for pressure oscillations and can be affected by dissipation mechanisms other than friction, the performance of the model is examined also by comparison with the results of a two-dimensional low-Reynolds number k–ε model.     

Analytical Solutions for Contaminant Diffusion in Double-Layered Porous Media

Analytical solutions for conservative solute diffusion in one-dimensional double-layered porous media are presented in this paper. These solutions are applicable to various combinations of fixed solute concentration and zero-flux boundary conditions (BC) applied at each end of a finite one-dimensional domain and can consider arbitrary initial solute concentration distributions throughout the media. Several analytical solutions based on several initial and BCs are presented based on typical contaminant transport problems found in geoenvironmental engineering including (1) leachate diffusion in a compacted clay liner (CCL) and an underlying stratum; (2) contaminant removal from soil layers; and (3) contaminant diffusion in a capping layer and underlying contaminated sediments. The analytical solutions are verified against numerical solutions from a finite-element method based model. Problems related to leachate transport in a CCL and an underlying stratum of a landfill and contaminant transport through a capping layer over contaminated sediments are then investigated, and the suitable definition of the average degree of diffusion is considered.