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.     

Stochastic Solution for Uncertainty Propagation in Nonlinear Shallow-Water Equations

This paper presents a stochastic approach to describe input uncertainties and their propagation through the nonlinear shallow-water equations. The formulation builds on a finite-volume model with a Godunov-type scheme for its shock capturing capabilities. Orthogonal polynomials from the Askey scheme provide expansion of the variables in terms of a finite number of modes from which the mean and higher-order moments of the distribution can be derived. The orthogonal property of the polynomials allows the use of a Galerkin projection to derive separate equations for the individual modes. Implementation of the polynomial chaos expansion and its nonintrusive counterpart determines the modal contributions from the resulting system of equations. Examples of long-wave transformation over a submerged hump illustrate the stochastic approach with uncertainties represented by Gaussian distribution. Additional results demonstrate the applicability of the approach with other distributions as well. The stochastic solution agrees well with the results from the Monte Carlo method, but at a small fraction of its computing cost.     

Numerical Solution of Boussinesq Equations to Simulate Dam-Break Flows

To investigate the effect of nonhydrostatic pressure distribution, dam-break flows are simulated by numerically solving the one-dimensional Boussinesq equations by using a fourth-order explicit finite-difference scheme. The computed water surface profiles for different depth ratios have undulations near the bore front for depth ratios greater than 0.4. The results obtained by using the Saint Venant equations and the Boussinesq equations are compared to determine the contribution of individual Boussinesq terms in the simulation of dam-break flow. It is found that, for typical engineering applications, the Saint Venant equations give sufficiently accurate results for the maximum flow depth and the time to reach this value at a location downstream of the dam.     

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.     

Experimental Evidence for Slow Compressional Waves

One of the most conspicuous aspects of the Biot–Frenkel theory is the existence of the slow compressional wave. An overview is given of the various experimental techniques that were/are used to study its appearance and properties.     

Effect of Convergence on Nonlinear Flow in Porous Media

The behavior of flow through porous media has been the subject of study for a long time. The relationship relating friction factor and Reynolds number using the square root of intrinsic permeability as the characteristic length is examined for flow in porous media with converging boundaries. An experimental investigation of the effect of convergence of streamlines on the linear and nonlinear parameters for different radial flow lines in a converging permeameter for different ratios of radii of the test section is also studied. In the present case, crushed rocks of sizes 11.64 and 4.73 mm were used as media and water as fluid, to develop curves relating friction factor and Reynolds number for different radial flow lines with different ratios of radii of the test section of the permeameter.     

Damping Effect on Waves Propagating Past a Submerged Horizontal Plate and a Vertical Porous Wall

A set of analytical solutions for waves propagating past a combined submerged horizontal plate and vertical porous wall breakwater system is presented. The wave damping effect caused by the horizontal plate induced flow constriction is considered in the analysis. The velocity potentials in each fluid domain are derived based on the linear wave theory and the unknown coefficients are determined from the matching conditions using three sets of orthogonal eigenfunctions. Reflection and transmission coefficients are presented to evaluate the performance of the breakwater system. The analytical solutions in terms of the reflection and transmission coefficients as well as the hydrodynamic force on the vertical porous wall are found in good agreement with published laboratory measurements. In comparison with the solutions without taking into account the wave damping effect, the present analytical solutions significantly improve the accuracy of the wave predictions, especially for the reflected waves.     

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.     

Oblique Impact of Water Waves on Thin Porous Walls

There is a paradoxical phenomenon in earlier studies when the incoming water wave is parallel to a porous breakwater, the water wave permeates completely without regard to the largeness of the the porosity of the porous breakwater. For solving the problem of the water waves obliquely impacting upon the thin porous wall, a new boundary condition on the thin porous wall—which can remedy the above mentioned paradoxical phenomenon—is proposed based on the concept that the incident angle remains unchanged when the water wave permeates into the wall. According to this new boundary condition, an analytic solution of an oblique water wave impacting on a thin porous wall of any permeability is obtained. It is found that the above paradoxical phenomenon, as the water wave is parallel to a thin porous wall, disappears. And, as the incident angle approaches 90°, the reflection coefficient and the transmission coefficient reasonably converge to 1 and 0, respectively, while on the contrary, those in the earlier investigations converge to 0 and 1.     

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.     

Propagation of Love Waves in an Inhomogeneous Fluid Saturated Porous Layered Half-Space with Properties Varying Exponentially

Based on Biot’s theory for transversely isotropic fluid saturated porous media, the complex dispersion equation for Love waves in a transversely isotropic fluid-saturated porous layered half-space is derived with the consideration of the inhomogeneity of the layer. The equation is solved by an iterative method. Detailed numerical calculation is presented for an inhomogeneous fluid-saturated porous layer overlying a purely elastic half-space. The dispersion and attenuation of Love waves are discussed. In addition, the upper and lower bounds of Love wave speed are also explored.     

Wave Attenuation over a Rigid Porous Medium on a Sandy Seabed

In this study, an analytic solution of wave interaction with a rigid porous medium above a poro-elastic sandy bottom is derived to investigate the attenuation of the surface wave and the wave-induced soil response. In the model, both inertial and damping effects of the flow are considered in the rigid porous region using the potential theory, while the consolidation theory is adopted in the sand region. A new complex dispersion relation involving parameters of the rigid porous and the poro-elastic medium is obtained. The analytic solutions are verified by some special cases, such as wave interaction with a porous structure over an impermeable bottom or wave interaction with a poro-elastic medium only. Numerical results indicate that the wave attenuation is highly dependent upon the thickness of the rigid porous layer, the soil stiffness, and their respective coefficients of permeability. Increasing the thickness of the rigid porous layer will shorten the wavelength of the surface wave regardless of the sand coarseness. The pore pressure in fine-sand is larger than in coarse sand, with both decaying with wave progression. It is also found that increasing the thickness of the rigid porous medium will effectively reduce the pore pressure in the sand. For the applications, an extended hyperbolic mild-slope equation is finally obtained, based on the basic analytic solutions. Examples of the wave height transformation over submerged permeable breakwaters on a slope sandy seabed are given. The simulated results show that the wave decay of the coarse sand seabed is larger than those of fine-sand and impermeable seabeds when waves pass after the submerged porous breakwater. The wave damping versus the friction factor for various height of the submerged breakwater is discussed.     

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.     

Pressure Waves in Porous Medium Saturated with Liquid Containing Gas Bubbles

Theoretical analyses on nonlinear pressure waves evolution in porous medium saturated with a liquid containing gas bubbles is carried out. The evolution equations for fast and slow longitudinal modes are derived for slightly nonlinear, disperse, and dissipation processes. The pressure wave distribution in gas bubble liquid-saturated porous media was investigated experimentally. It was revealed that both modes might have oscillating structure induced by bubble oscillation in the wave. It is shown that the wave damping is determined by a combined impact of heat losses due to gas cooling in the bubbles and dissipation due to longitudinal displacement of liquid and porous skeleton, both influenced by the wave. Experimental data on the velocity and structure of fast and slow modes are compared with results of theoretical modeling.     

Parameter Estimation in Finite Element Simulations of Rayleigh Waves

Spectral analysis of surface waves measurements are used to develop subsurface soil profiles or as a tool to delineate abandoned crown pillar structures. Finite element modeling of Rayleigh waves has practical application in simulating SASW measurements. Developing a reliable and accurate finite element model to simulate Rayleigh waves requires the proper mesh dimensions and attenuation parameters. This research proposes a new simplified methodology for quantifying mesh dispersion effects. In addition, methods of verifying damping ratios for numerical simulations are presented. The evaluation of an array of nodal displacements in the frequency–wave-number domain effectively illustrates mesh dispersion effects and the presence of parasitic modes of vibration. Calculations of damping ratio show that mass and stiffness damping parameters are valid within a specified frequency bandwidth. The new techniques are tested for a half-space model; however, they can be used for the analysis of layered media. In addition, equations are given for the calculation of linear Rayleigh damping. These equations satisfy the conditions of an average damping with minimum variance within the frequency bandwidth of interest.     

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.     

Propagation of Long Water Waves through Branching Channels

This paper investigates the fundamental behavior of long water waves propagating through branching channels of uniform depth and width. Numerical simulations based on the Boussinesq long wave model were carried out to investigate the effects of channel width b, effective wavelength λe, wave amplitude α, and angle θ between channel branches on wave transmission and reflection. Our results showed that the transmission and reflection of long waves through branching channels are dominantly governed by a single dimensionless parameter b/λe, whereas other parameters such as the angle between channel branches are less important. Detailed quantitative results for predicting long wave transmission and reflection in different branching channels based on the similarity parameter b/λe were determined. From these results, we have discovered some very interesting and distinct long wave behaviors in narrow and wide branching channels. To verify the numerical results, experiments were conducted in a right-angled branching channel, and the experimental results showed good agreement with the numerical predictions.     

Wave Propagation in a Pipe Pile for Low-Strain Integrity Testing

This paper presents an analytical solution methodology for a tubular structure subjected to a transient point loading in low-strain integrity testing. The three-dimensional effects on the pile head and the applicability of plane-section assumption are the main problems in low-strain integrity testing on a large-diameter tubular structure, such as a pipe pile. The propagation of stress waves in a tubular structure cannot be expressed by one-dimensional wave theory on the basis of plane-section assumption. This paper establishes the computational model of a large-diameter tubular structure with a variable wave impedance section, where the soil resistance is simulated by the Winkler model, and the exciting force is simulated with semisinusoidal impulse. The defects are classified into the change in the wall thickness and Young’s modulus. Combining the boundary and initial conditions, a frequency-domain analytical solution of a three-dimensional wave equation is deduced from the Fourier transform method and the separation of variables methods. On the basis of the frequency-domain analytic solution, the time-domain response is obtained from the inverse Fourier transform method. The three-dimensional finite-element models are used to verify the validity of analytical solutions for both an intact and a defective pipe pile. The analytical solutions obtained from frequency domain are compared with the finite-element method (FEM) results on both pipe piles in this paper, including the velocity time history, peak value, incident time arrival, and reflected wave crests. A case study is shown and the characteristics of velocity response time history on the top of an intact and a defective pile are investigated. The comparisons show that the analytical solution derived in this paper is reliable for application in the integrity testing on a tubular structure.     

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.