Numerically quantifying energy loss caused by squirt flow

In interconnected microcracks, or in microcracks connected to spherical pores, the deformation associated with the passage of mechanical waves can induce fluid flow parallel to the crack walls, which is known as squirt flow. This phenomenon can also occur at larger scales in hydraulically interconnected mesoscopic cracks or fractures. The associated viscous friction causes the waves to experience attenuation and velocity dispersion. We present a simple hydromechanical numerical scheme, based on the interface-coupled Lamé–Navier and Navier–Stokes equations, to simulate squirt flow in the frequency domain. The linearized, quasi-static Navier–Stokes equations describe the laminar flow of a compressible viscous fluid in conduits embedded in a linear elastic solid background described by the quasi-static Lamé–Navier equations. Assuming that the heterogeneous model behaves effectively like a homogeneous viscoelastic medium at a larger spatial scale, the resulting attenuation and stiffness modulus dispersion are computed from spatial averages of the complex-valued, frequency-dependent stress and strain fields. An energy-based approach is implemented to calculate the local contributions to attenuation that, when integrated over the entire model, yield results that are identical to those based on the viscoelastic assumption. In addition to thus validating this assumption, the energy-based approach allows for analyses of the spatial dissipation patterns in squirt flow models. We perform simulations for a series of numerical models to illustrate the viability and versatility of the proposed method. For a 3D model consisting of a spherical crack embedded in a solid background, the characteristic frequency of the resulting P-wave attenuation agrees with that of a corresponding analytical solution, indicating that the dissipative viscous flow problem is appropriately handled in our numerical solution of the linearized, quasi-static Navier–Stokes equations. For 2D models containing either interconnected cracks or cracks connected to a circular pore, the results are compared with those based on Biot's poroelastic equations of consolidation, which are solved through an equivalent approach. Overall, our numerical simulations and the associated analyses demonstrate the suitability of the coupled Lamé–Navier and Navier–Stokes equations and of Biot's equations for quantifying attenuation and dispersion for a range of squirt flow scenarios. These analyses also allow for delineating numerical and physical limitations associated with each set of equations.     

Coupled simulation of transient bed evolution in alluvial channels

Abstract

In dealing with the transient sediment transport problem, the commonly used uncoupled model may not be suitable. The uncoupling technique is intended to separate the physical coupling phenomenon of water flow and sediment transport into two independent processes. Very often, as a result, severe numerical oscillation and solution instability problems appear in the simulation of transient sediment transport in alluvial channels. The coupled model, which simultaneously solves water flow continuity, momentum and sediment continuity equations, gives fewer numerical oscillation and solution instability problems. In this article, a coupled model using a matrix double-sweep method to solve the system of nonlinear algebraic equations has been developed. Several test runs designed on the basis of a schematic model have been performed. The numerical oscillation and solution instability problems have been investigated through a comparison with those obtained from an uncoupled model. Based on the proposed case studies, it can be concluded that, for transient bed evolution, the performance of the coupled model is much better than that of the uncoupled model. The numerical oscillation is reduced and the solution is more stable. This newly developed coupled model was also applied to the Cho-Shui River in Taiwan. This application study implied that the effect of the peaky flood wave propagation on the bed evolution could be simulated better by the coupled model than by the uncoupled model.     

Fluid dispersion effects on density-driven thermohaline flow and transport in porous media

This study introduces the dispersive fluid flux of total fluid mass to the density-driven flow equation to improve thermohaline modeling of salt and heat transports in porous media. The dispersive fluid flux in the flow equation is derived to account for an additional fluid flux driven by the density gradient and mechanical dispersion. The coupled flow, salt transport and heat transport governing equations are numerically solved by a fully implicit finite difference method to investigate solution changes due to the dispersive fluid flux. The numerical solutions are verified by the Henry problem and the thermal Elder problem under a moderate density effect and by the brine Elder problem under a strong density effect. It is found that increment of the maximum ratio of the dispersive fluid flux to the advective fluid flux results in increasing dispersivity for the Henry problem and the brine Elder problem. The effects of the dispersive fluid flux on salt and heat transports under high density differences and high dispersivities are more noticeable than under low density differences and low dispersivities. Values of quantitative indicators such as the Nusselt number, mass flux, salt mass stored and maximum penetration depth in the brine Elder problem show noticeable changes by the dispersive fluid flux. In the thermohaline Elder problem, the dispersive fluid flux shows a considerable effect on the shape and the number of developed fingers and makes either an upwelling or a downwelling flow in the center of the domain. In conclusion, for the general case that involves strong density-driven flow and transport modeling in porous media, the dispersive fluid flux should be considered in the flow equation.     

Effects of chemical reactions on density-dependent fluid flow: on the numerical formulation and the development of instabilities

A three-dimensional, reactive numerical flow model is developed that couples chemical reactions with density-dependent mass transport and fluid flow. The model includes equilibrium reactions for the aqueous species, kinetic reactions between the solid and aqueous phases, and full coupling of porosity and permeability changes that result from precipitation and dissolution reactions in porous media. A one-step, global implicit approach is used to solve the coupled flow, transport and reaction equations with a fully implicit upstream-weighted control volume discretization. The Newton–Raphson method is applied to the discretized non-linear equations and a block ILU-preconditioned CGSTAB method is used to solve the resulting Jacobian matrix equations. This approach permits the solution of the complete set of governing equations for both concentration and pressure simultaneously affected by chemical and physical processes. A series of chemical transport simulations are conducted to investigate coupled processes of reactive chemical transport and density-dependent flow and their subsequent impact on the development of preferential flow paths in porous media. The coupled effects of the processes driving flow and the chemical reactions occurring during solute transport is studied using a carbonate system in fully saturated porous media. Results demonstrate that instability development is sensitive to the initial perturbation caused by density differences between the solute plume and the ambient groundwater. If the initial perturbation is large, then it acts as a “trigger” in the flow system that causes instabilities to develop in a planar reaction front. When permeability changes occur due to dissolution reactions occurring in the porous media, a reactive feedback loop is created by calcite dissolution and the mixed convective transport of the system. Although the feedback loop does not have a significant impact on plume shape, complex concentration distributions develop as a result of the instabilities generated in the flow system.     

Hybrid upwind discretization of nonlinear two-phase flow with gravity

Multiphase flow in porous media is described by coupled nonlinear mass conservation laws. For immiscible Darcy flow of multiple fluid phases, whereby capillary effects are negligible, the transport equations in the presence of viscous and buoyancy forces are highly nonlinear and hyperbolic. Numerical simulation of multiphase flow processes in heterogeneous formations requires the development of discretization and solution schemes that are able to handle the complex nonlinear dynamics, especially of the saturation evolution, in a reliable and computationally efficient manner. In reservoir simulation practice, single-point upwinding of the flux across an interface between two control volumes (cells) is performed for each fluid phase, whereby the upstream direction is based on the gradient of the phase-potential (pressure plus gravity head). This upwinding scheme, which we refer to as Phase-Potential Upwinding (PPU), is combined with implicit (backward-Euler) time discretization to obtain a Fully Implicit Method (FIM). Even though FIM suffers from numerical dispersion effects, it is widely used in practice. This is because of its unconditional stability and because it yields conservative, monotone numerical solutions. However, FIM is not unconditionally convergent. The convergence difficulties are particularly pronounced when the different immiscible fluid phases switch between co-current and counter-current states as a function of time, or (Newton) iteration. Whether the multiphase flow across an interface (between two control-volumes) is co-current, or counter-current, depends on the local balance between the viscous and buoyancy forces, and how the balance evolves in time. The sensitivity of PPU to small changes in the (local) pressure distribution exacerbates the problem. The common strategy to deal with these difficulties is to cut the timestep and try again. Here, we propose a Hybrid-Upwinding (HU) scheme for the phase fluxes, then HU is combined with implicit time discretization to yield a fully implicit method. In the HU scheme, the phase flux is divided into two parts based on the driving force. The viscous-driven and buoyancy-driven phase fluxes are upwinded differently. Specifically, the viscous flux, which is always co-current, is upwinded based on the direction of the total-velocity. The buoyancy-driven flux across an interface is always counter-current and is upwinded such that the heavier fluid goes downward and the lighter fluid goes upward. We analyze the properties of the Implicit Hybrid Upwinding (IHU) scheme. It is shown that IHU is locally conservative and produces monotone, physically-consistent numerical solutions. The IHU solutions show numerical diffusion levels that are slightly higher than those for standard FIM (i.e., implicit PPU). The primary advantage of the IHU scheme is that the numerical overall-flux of a fluid phase remains continuous and differentiable as the flow regime changes between co-current and counter-current conditions. This is in contrast to the standard phase-potential upwinding scheme, in which the overall fractional-flow (flux) function is non-differentiable across the boundary between co-current and counter-current flows.     

Atmospheric boundary layer over steep surface waves

Turbulent air-sea interactions coupled with the surface wave dynamics remain a challenging problem. The needs to include this kind of interaction into the coupled environmental, weather and climate models motivate the development of a simplified approximation of the complex and strongly nonlinear interaction processes. This study proposes a quasi-linear model of wind-wave coupling. It formulates the approach and derives the model equations. The model is verified through a set of laboratory (direct measurements of an airflow by the particle image velocimetry (PIV) technique) and numerical (a direct numerical simulation (DNS) technique) experiments. The experiments support the central model assumption that the flow velocity field averaged over an ensemble of turbulent fluctuations is smooth and does not demonstrate flow separation from the crests of the waves. The proposed quasi-linear model correctly recovers the measured characteristics of the turbulent boundary layer over the waved water surface.     

Time–space fractional governing equations of one‐dimensional unsteady open channel flow process: Numerical solution and exploration

Although fractional integration and differentiation have found many applications in various fields of science, such as physics, finance, bioengineering, continuum mechanics, and hydrology, their engineering applications, especially in the field of fluid flow processes, are rather limited. In this study, a finite difference numerical approach is proposed to solve the time–space fractional governing equations of 1‐dimensional unsteady/non‐uniform open channel flow process. By numerical simulations, results of the proposed fractional governing equations of the open channel flow process were compared with those of the standard Saint‐Venant equations. Numerical simulations showed that flow discharge and water depth can exhibit heavier tails in downstream locations as space and time fractional derivative powers decrease from 1. The fractional governing equations under consideration are generalizations of the well‐known Saint‐Venant equations, which are written in the integer differentiation framework. The new governing equations in the fractional‐order differentiation framework have the capability of modelling nonlocal flow processes both in time and in space by taking the global correlations into consideration. Furthermore, the generalized flow process may possibly shed light on understanding the theory of the anomalous transport processes and observed heavy‐tailed distributions of particle displacements in transport processes.     

The dynamic response of fluid-saturated porous materials with application to seismically induced soil liquefaction

The numerical simulation of liquefaction phenomena in fluid-saturated porous materials within a continuum-mechanical framework is the aim of this contribution. This is achieved by exploiting the Theory of Porous Media (TPM) together with thermodynamically consistent elasto-viscoplastic constitutive laws. Additionally, the Finite Element Method (FEM) besides monolithic time-stepping schemes is used for the numerical treatment of the arising coupled multi-field problem. Within an isothermal and geometrically linear framework, the focus is on fully saturated biphasic materials with incompressible and immiscible phases. Thus, one is concerned with the class of volumetrically coupled problems involving a potentially strong coupling of the solid and fluid momentum balance equations and the algebraic incompressibility constraint. Applying the suggested material model, two important liquefaction-related incidents in porous media dynamics, namely the flow liquefaction and the cyclic mobility, are addressed, and a seismic soil–structure interaction problem to reveal the aforementioned two behaviors in saturated soils is introduced.     

Nonlinear seismic response analysis of earth dams

The objective of this paper is to propose a general and efficient numerical procedure for analysing the dynamic response of geotechnical structures, which are considered as both nonlinear and two phase systems. In Section 2, the appropriate coupled dynamic field equations for the response of a two-phase soil system are briefly reviewed. The finite element spatial discretization of the field equations is described and time integration for the resulting nonlinear semi-discrete finite element equations is discussed. In Section 3, iterative techniques are examined for the solution of the global nonlinear system of finite element equations. A large amount of computational effort is expended in the iterative phase of the solution and so the iterative procedure used must be both reliable and efficient. The performance of three iterative procedure is examined: Newton Raphson, Modified Newton Raphson and Quasi-Newton methods, including BGFS and Broyden updates. Finally, in Section 4, the elasto-plastic earthquake response analysis of a two phase nonhomogeneous earth dam is presented. Extensive documentation exists¹ for the particular problem selected including recorded earthquake motions at the base and crest of the dam. The results of the numerical calculations are compared to the recorded response of the dam.     

A double-porosity model for a fractured aquifer with non-Darcian flow in fractures

Abstract

Non-Darcian flow in a finite fractured confined aquifer is studied. A stream bounds the aquifer at one side and an impervious stratum at the other. The aquifer consists of fractures capable of transmitting water rapidly, and porous blocks which mainly store water. Unsteady flow in the aquifer due to a sudden rise in the stream level is analysed by the double-porosity conceptual model. Governing equations for the flow in fractures and blocks are developed using the continuity equation. The fluid velocity in fractures is often too high for the linear Darcian flow so that the governing equation for fracture flow is modified by Forcheimer's equation, which incorporates a nonlinear term. Governing equations are coupled by an interaction term that controls the quasi-steady-state fracture—block interflow. Governing equations are solved numerically by the Crank-Nicolson implicit scheme. The numerical results are compared to the analytical results for the same problem which assumes Darcian flow in both fractures and blocks. Numerical and analytical solutions give the same results when the Reynolds number is less than 0.1. The effect of nonlinearity on the flow appears when the Reynolds number is greater than 0.1. The higher the rate of flow from the stream to the aquifer, the higher the degree of nonlinearity. The effect of aquifer parameters on the flow is also investigated. The proposed model and its numerical solution provide a useful application of nonlinear flow models to fractured aquifers. It is possible to extend the model to different types of aquifer, as well as boundary conditions at the stream side. Time-dependent flow rates in the analysis of recession hydrographs could also be evaluated by this model.     

A coupled fluid layer–rigid disk–poroelastic half-space vibration problem

This paper proposes a coupled fluid layer–foundation–poroelastic half-space vibration model to study how still water affects foundations operating underwater. As an example, we consider the problem of the vertical vibration of a rigid disk on a poroelastic half-space covered by a fluid layer having a finite depth. The solution of the disk vibration problem is obtained using the boundary conditions at the free surface of the fluid layer and the boundary conditions at the fluid layer–poroelastic medium interface. The solution is expressed in terms of dual integral equations that are converted into Fredholm integral equations of the second kind and solved numerically. Selected numerical results for the vertical dynamic impedance coefficient are examined based on different water depths, poroelastic materials, disk permeabilities and frequencies of excitation. Based on the numerical results, it is proposed that the hydrodynamic pressure caused by the foundation vibration is the intrinsic reason that the existence of a fluid layer has such a great effect on the dynamic characteristics of the foundation. In many cases, the hydrodynamic pressure caused by the foundation vibration cannot be ignored when designing dynamic underwater foundations. These results are helpful in understanding the dynamic response of foundations under still water without water waves, such as foundations in pools, lakes and reservoirs.     

A model based on Hirano-Exner equations for two-dimensional transient flows over heterogeneous erodible beds

In order to study the morphological evolution of river beds composed of heterogeneous material, the interaction among the different grain sizes must be taken into account. In this paper, these equations are combined with the two-dimensional shallow water equations to describe the flow field. The resulting system of equations can be solved in two ways: (i) in a coupled way, solving flow and sediment equations simultaneously at a given time-step or (ii) in an uncoupled manner by first solving the flow field and using the magnitudes obtained at each time-step to update the channel morphology (bed and surface composition). The coupled strategy is preferable when dealing with strong and quick interactions between the flow field, the bed evolution and the different particle sizes present on the bed surface. A number of numerical difficulties arise from solving the fully coupled system of equations. These problems are reduced by means of a weakly-coupled strategy to numerically estimate the wave celerities containing the information of the bed and the grain sizes present on the bed. Hence, a two-dimensional numerical scheme able to simulate in a self-stable way the unsteady morphological evolution of channels formed by cohesionless grain size mixtures is presented. The coupling technique is simplified without decreasing the number of waves involved in the numerical scheme but by simplifying their definitions. The numerical results are satisfactorily tested with synthetic cases and against experimental data.     

A two-grid method for coupled free flow with porous media flow

This paper presents a two-grid method for solving systems of partial differential equations modelling incompressible free flow coupled with porous media flow. This work considers both the coupled Stokes and Darcy as well as the coupled Navier-Stokes and Darcy problems. The numerical schemes proposed are based on combinations of the continuous finite element method and the discontinuous Galerkin method. Numerical errors and convergence rates for solutions obtained from the two-grid method are presented. CPU times for the two-grid algorithm are shown to be significantly less than those obtained by solving the fully coupled problem.     

On the Coupling Between Channel Level and Surface Ground-Water Flows

This paper is devoted to a mathematical analysis of some general models of mass transport and other coupled physical processes developed in simultaneous flows of surface, soil and ground waters. Such models are widely used for forecasting (numerical simulation) of a hydrological cycle for concrete territories. The mathematical models that proved a more realistic approach are obtained by combining several mathematical models for local processes. The water-exchange models take into account the following factors: Water flows in confined and unconfined aquifers, vertical moisture migration allowing earth surface evaporation, open-channel flow simulated by one-dimensional hydraulic equations, transport of contamination, etc. These models may have different levels of sophistication. We illustrate the type of mathematical singularities which may appear by considering a simple model on the coupling of a surface flow of surface and ground waters with the flow of a line channel or river.     

Analytical expressions for three different modes in harmonic motion of sliding structures

The problem of response of sliding structures subjected to harmonic support motions is considered. The periodic motions, consisting of the three different modes, stick-stick, stick-slip and slip-slip, are a significant part of the responses to harmonic excitations. Assuming the periodicity of the motion, the condition of the initiation of the slip-slip motion is derived. Then analytical expressions for the occurrence of the periodic motions are obtained without integrating the equations of motion. The accuracy of the analytical expressions is confirmed by comparison with numerical results herein obtained. It is observed that the numerical results are sensitive to the starting and ending times of slip motions.     

Hydrodynamic modelling for groundwater flow through permeable reactive barriers

Hydrodynamic modelling for analysis of groundwater flow through permeable reactive barriers (PRBs) is addressed in this paper. Permeable reactive barriers constitute an emerging technology for in situ remediation of groundwater contamination and have many advantages over the traditional ex situ treatment methods. The transport domains during groundwater flow through PRBs often may involve free‐flow or non‐porous sections. To model the fluid mobility efficiently in such situations, the free and porous flow zones (PRBs) must be studied in conjunction with each other. The present paper is devoted to the analysis of groundwater flow through combined free flow domains and PRBs. The free‐flow regime is modelled using the Navier–Stokes equations whereas the permeable barriers are simulated by either the Darcy or the Brinkman equation. In order to couple the governing equations of motions, well‐posed mathematical formulations of matching boundary conditions are prescribed at the interface between the free‐groundwater‐flow zones and the permeable barriers. Combination of the Navier–Stokes equations with the Brinkman equation is more straightforward owing to their analogous forms. However, the Navier–Stokes and Darcy equations are incompatible mathematically and cannot be linked directly. The problem is resolved in this paper by invoking validated hydrodynamical expressions for describing the flow behaviour at the interfaces between free‐flow and porous zones. Three schemes for the analyses of fluid flow in combined domains are applied to the case of groundwater flow through permeable reactive barriers and different model results are compared. Copyright © 2002 John Wiley & Sons, Ltd.     

Analysis of a numerical solution to the one-dimensional stochastic convection equation

Under certain conditions the concentration and flux of a substance moving in a stochastic flow field are described by the stochastic convection equation. A numerical method for solving the one-dimensional problem is studied here. The differential operator is replaced by a discrete linear operator based on finite differences. The resulting system of stochastic equations is then replaced by a system of equations whose solution is the mean concentration or mean flux. This final system is analysed and conditions for a stable numerical solution are obtained. Finally, numerical examples are given and are compared to an approximate analytical solution to the stochastic convection equation.     

A numerical method for simulating non-Newtonian fluid flow and displacement in porous media

Flow and displacement of non-Newtonian fluids in porous media occurs in many subsurface systems, related to underground natural resource recovery and storage projects, as well as environmental remediation schemes. A thorough understanding of non-Newtonian fluid flow through porous media is of fundamental importance in these engineering applications. Considerable progress has been made in our understanding of single-phase porous flow behavior of non-Newtonian fluids through many quantitative and experimental studies over the past few decades. However, very little research can be found in the literature regarding multi-phase non-Newtonian fluid flow or numerical modeling approaches for such analyses.For non-Newtonian fluid flow through porous media, the governing equations become nonlinear, even under single-phase flow conditions, because effective viscosity for the non-Newtonian fluid is a highly nonlinear function of the shear rate, or the pore velocity. The solution for such problems can in general only be obtained by numerical methods.We have developed a three-dimensional, fully implicit, integral finite difference simulator for single- and multi-phase flow of non-Newtonian fluids in porous/fractured media. The methodology, architecture and numerical scheme of the model are based on a general multi-phase, multi-component fluid and heat flow simulator — TOUGH2. Several rheological models for power-law and Bingham non-Newtonian fluids have been incorporated into the model. In addition, the model predictions on single- and multi-phase flow of the power-law and Bingham fluids have been verified against the analytical solutions available for these problems, and in all the cases the numerical simulations are in good agreement with the analytical solutions. In this presentation, we will discuss the numerical scheme used in the treatment of non-Newtonian properties, and several benchmark problems for model verification.In an effort to demonstrate the three-dimensional modeling capability of the model, a three-dimensional, two-phase flow example is also presented to examine the model results using laboratory and simulation results existing for the three-dimensional problem with Newtonian fluid flow.