Experimental investigation of bedload transport processes under unsteady flow conditions

Hydraulic engineering is usually based on theoretical analysis and/or numerical modelling simulation. As the dynamic behaviour of sediment movement under unsteady flow is still unclear, and field measurement is comparatively difficult during a large flood, prior investigations through flume experiments are required. A series of flume experiments, conducted using different inflow hydrographs without sediment supply from upstream, was carried out to investigate the sediment transport process under unsteady flow conditions. A series of triangular hydrographs were performed in the experiments. The results indicate that a temporal lag was found between the flow hydrograph peak and the sediment hydrograph peak because large size sand dunes lasted for a short period in the falling limb of the flow hydrograph. The temporal lag was found to be about equal to 6–15% of the flow hydrograph duration. Owing to the temporal lag, the total bedload yield in the rising period was less than that in the falling period. Furthermore, the measured total bedload yield in the unsteady flow experiments was larger than the predicted value, which was estimated by using the results obtained from the equivalent steady flow experiment. The peak bedload transport rate for unsteady flow conditions was also larger than the predicted value. The ratios of the measured to the predicted quantities mentioned above were found to be constant values for different shapes of hydrographs. It is, therefore, expected that the analytical results of sediment transport from equivalent steady flow can be a good reference for sediment transport under unsteady flow conditions. Copyright © 2004 John Wiley & Sons, Ltd.     

Physical splitting of nonlinear effects in high-velocity stable flow through porous media

For a high-velocity stable flow through a periodic corrugated channel representing an element of porous medium, we suggest splitting the overall nonlinear macroscopic effects into two kinds of different physical origin: a pure inertia effect produced by the convective term of Navier–Stokes equations and an inertia–viscous cross effect representing a variation of the viscous dissipation due to a streamline deformation by inertia forces. We will show that the inertia–viscous cross effects may be revealed by simulating a periodic flow, whilst the pure inertia effects are produced by the microscale flow nonperiodicity. We will reveal the individual flow law for each nonlinear component and analyze the relative role of both components numerically by using the finite element method applied to the Navier–Stokes equations. Both the pure inertia and the inertia–viscous cross effects are revealed to be exponential prior to quadratic or cubic ones. The influence of the dead volume is analyzed. The inertia–viscous cross phenomena are shown to be negligible when the flow structure is clearly nonperiodic.     

Experimental analysis of pore-scale flow and transport in porous media

A novel, non-intrusive fluorescence imaging technique has been used to quantitatively measure the pore geometry, fluid velocity, and solute concentration within a saturated, three-dimensional porous medium. Discrete numerical averages of these quantities have been made over a representative volume of the medium and used to estimate macroscopic quantities that appear in conventional continuum models of flow and transport. The approach is meant to illustrate how microscopic information can be measured, averaged, and used to characterize medium-scale processes that are typically approximated constitutively. The experimental system consisted of a clear, cylindrical column packed with clear spherical beads and a refractive index-matched fluid seeded with fluorescent tracer particles and solute dye. By illuminating the fluid within the column with a scanning planar laser beam, details of flow and concentration within the pore spaces can be quantitatively observed, allowing for three-dimensional, dimensional, time dependent information to be obtained at good resolution. In time dependent information to be obtained at good resolution. In the current experiment, volumetrically averaged velocities and void-to-volume ratios are first compared with bulk measurements of fluid flux and medium porosity. Microscopic measurements of concentration are then used to construct cross-sectionally averaged profiles, mean breakthrough curves, and direct measurements of the dispersive flux, velocity variance, and concentration variance. In turn, the dispersive flux measurements are compared with mean concentration gradients to provide a basis for confirming the Fickian dispersion model and estimating dispersion coefficients for the medium. Coefficients determined in this manner are compared with others based upon traditional length-scale arguments, mean breakthrough analyses, and curve fits with numerical simulations.     

Simulation of density-driven flow in fractured porous media

We study density-driven flow in a fractured porous medium in which the fractures are represented as manifolds of reduced dimensionality. Fractures are assumed to be thin regions of space filled with a porous material whose properties differ from those of the porous medium enclosing them. The interfaces separating the fractures from the embedding medium are assumed to be ideal. We consider two approaches: (i) the fractures have the same dimension, d, as the embedding medium and are said to be d-dimensional; (ii) the fractures are considered as (d − 1)-dimensional manifolds, and the equations of density-driven flow are found by averaging the d-dimensional laws over the fracture width. We show that the second approach is a valid alternative to the first one. For this purpose, we perform numerical experiments using finite-volume discretization for both approaches. The results obtained by the two methods are in good agreement with each other.     

A peridynamic model of flow in porous media

This paper presents a nonlocal, derivative free model for transient flow in unsaturated, heterogeneous, and anisotropic soils. The formulation is based on the peridynamic model for solid mechanics. In the proposed model, flow and changes in moisture content are driven by pairwise interactions with other points across finite distances, and are expressed as functional integrals of the hydraulic potential field. Peridynamic expressions of the rate of change in moisture content, moisture flux, and flow power are derived, as are relationships between the peridynamic and the classic hydraulic conductivities; in addition, the model is validated. The absence of spacial derivatives makes the model a good candidate for flow simulations in fractured soils and lends itself to coupling with peridynamic mechanical models for simulating crack formation triggered by shrinkage and swelling, and assessing their potential impact on a wide range of processes, such as infiltration, contaminant transport, and slope stability.     

Numerical modelling of multiphase flow in porous media

The simultaneous flow of immiscible fluids in porous media occurs in a wide variety of applications. The equations governing these flows are inherently nonlinear, and the geometries and material properties characterizing many problems in petroleum and groundwater engineering can be quite irregular. As a result, numerical simulation often offers the only viable approach to the mathematical modelling of multiphase flows. This paper provides an overview of the types of models that are used in this field and highlights some of the numerical techniques that have appeared recently. The exposition includes discussions of multiphase, multispecies flows in which chemical transport and interphase mass transfers play important roles. The paper also examines some of the outstanding physical and mathematical problems in multiphase flow simulation. The scope of the paper is limited to isothermal flows in natural porous media; however, many of the special techniques and difficulties discussed also arise in artificial porous media and multiphase flows with thermal effects.     

Similarity solution of axisymmetric flow in porous media

Applications of the axisymmetric Boussinesq equation to groundwater hydrology and reservoir engineering have long been recognised. An archetypal example is invasion by drilling fluid into a permeable bed where there is initially no such fluid present, a circumstance of some importance in the oil industry. It is well known that the governing Boussinesq model can be reduced to a nonlinear ordinary differential equation using a similarity variable, a transformation that is valid for a certain time-dependent flux at the origin. Here, a new analytical approximation is obtained for this case. The new solution,, which has a simple form, is demonstrated to be highly accurate.     

Conservation equations for nonisothermal flow in porous media

This paper presents the mass, momentum and energy equations that can be applied to nonisothermal flow in porous media. These equations are derived by taking a suitable volume average of the microscopic equations. The resulting macroscopic equations are then appropriate for experimental comparison.     

Model coupling for multiphase flow in porous media

Numerical models for flow and transport in porous media are valid for a particular set of processes, scales, levels of simplification and abstraction, grids etc. The coupling of two or more specialised models is a method of increasing the overall range of validity while keeping the computational costs relatively low. Several coupling concepts are reviewed in this article with a focus on the authors’ work in this field. The concepts are divided into temporal and spatial coupling concepts, of which the latter is subdivided into multi-process, multi-scale, multi-dimensional, and multi-compartment coupling strategies. Examples of applications for which these concepts can be relevant include groundwater protection and remediation, carbon dioxide storage, nuclear-waste disposal, soil dry-out and evaporation processes as well as fuel cells and technical filters.     

A new scheme for numerical modelling of flow and transport processes in 3D fractured porous media

The objective of this work is to develop a new numerical approach for the three-dimensional modelling of flow and transient solute transport in fractured porous media which would provide an accurate and efficient treatment of 3D complex geometries and inhomogeneities. For this reason, and in order to eliminate as much as possible the number of degrees of freedom, the fracture network, fractures and their intersections, are solved with a coupled 2D–1D model while the porous matrix is solved independently with a 3D model. The interaction between both models is accounted for by a coupling iterative technique. In this way it is possible to improve efficiency and reduce CPU usage by avoiding 3D mesh refinements of the fractures. The approach is based on the discrete-fracture model in which the exact geometry and location of each fracture in the network must be provided as an input. The formulation is based on a multidimensional coupling of the boundary element method-multidomain (BEM-MD) scheme for the flow and boundary element dual reciprocity method-multidomain (BE-DRM-MD) scheme for the transport. Accurate results and high efficiency have been obtained and are reported in this paper.     

Random maze models of flow through porous media

Summary This paper discusses a class of stochastic models of flow through porous media in which the randomness is attached to the structure of the medium rather than to the flow path. These models are obtained by generalizing an earlier model available in the literature where a regular crystal was taken in which bonds (flow channels) were dammed in a random fashion, yielding a random maze. The hydraulic properties of general models of this type are calculated; in particular, it is shown that they exhibit the phenomenon of dispersion whereby the factor of dispersion turns out to be a linear function of the percolation velocity.     

Nonequilibrium effects in models of three-phase flow in porous media

In this paper we extend to three-phase flow the nonequilibrium formalism proposed by Barenblatt and co-workers for two-phase porous media flow. The underlying idea is to include nonequilibrium effects by introducing a pair of effective water and gas saturations, which are linked to the actual saturations by a local evolution equation. We illustrate and analyze how nonequilibrium effects lead to qualitative and quantitative differences in the solution of the three-phase flow equations.     

Effective parameters for flow in saturated heterogeneous porous media

Effective parameters for flow in saturated porous media are obtained via Taylor-Aris-Brenner moment analysis considering both periodic as well as stationary porous medium properties. It is assumed that a slug is instantaneously introduced into an unbounded, anisotropic porous medium having a compressible matrix, and that the correlation length of the local hydraulic conductivity and specific storage fluctuations is smaller than the correlation length of hydraulic head fluctuations (gradually varying flow). It is shown that the effective specific storage is equal to its volume average. The effective hydraulic conductivity is derived by a small-perturbation analysis and it is shown to consist of its volume average and of a second term which accounts for the ‘small’ local conductivity fluctuations.     

Multi-scale modeling of three-phase–three-component processes in heterogeneous porous media

Flow and transport processes in porous media occur on different spatial and temporal scales and may also be locally different. Additionally, the structure of the porous medium itself generally shows a high dependence on the spatial scale.     

The acoustic signature of fluid flow in complex porous media

Effective medium approximations for the frequency-dependent and complex-valued effective stiffness tensors of cracked/porous rocks with multiple solid constituents are developed on the basis of the T-matrix approach (based on integral equation methods for quasi-static composites), the elastic–viscoelastic correspondence principle, and a unified treatment of the local and global flow mechanisms, which is consistent with the principle of fluid mass conservation. The main advantage of using the T-matrix approach, rather than the first-order approach of Eshelby or the second-order approach of Hudson, is that it produces physically plausible results even when the volume concentrations of inclusions or cavities are no longer small. The new formulae, which operates with an arbitrary homogeneous (anisotropic) reference medium and contains terms of all order in the volume concentrations of solid particles and communicating cavities, take explicitly account of inclusion shape and spatial distribution independently. We show analytically that an expansion of the T-matrix formulae to first order in the volume concentration of cavities (in agreement with the dilute estimate of Eshelby) has the correct dependence on the properties of the saturating fluid, in the sense that it is consistent with the Brown–Korringa relation, when the frequency is sufficiently low. We present numerical results for the (anisotropic) effective viscoelastic properties of a cracked permeable medium with finite storage porosity, indicating that the complete T-matrix formulae (including the higher-order terms) are generally consistent with the Brown–Korringa relation, at least if we assume the spatial distribution of cavities to be the same for all cavity pairs. We have found an efficient way to treat statistical correlations in the shapes and orientations of the communicating cavities, and also obtained a reasonable match between theoretical predictions (based on a dual porosity model for quartz–clay mixtures, involving relatively flat clay-related pores and more rounded quartz-related pores) and laboratory results for the ultrasonic velocity and attenuation spectra of a suite of typical reservoir rocks.     

Grid convergence of variable-density flow simulations in discretely-fractured porous media

Grid convergence in space and time of variable-density flow in fractured-porous rock is systematically assessed. Convergence of the flow simulation is attained using both uniform and adaptive time-stepping. This contrasts to variable-density flow in unfractured porous media where grid convergence variable-density flow problems is almost never achieved. At high discretization levels, the number of fingers in fractured-porous rock is no longer influenced by spatial-temporal grid discretization, which is not the case in unfractured porous media. However, similar to unfractured porous media, the number of fingers in fractured-porous media varies at low discretization levels. Simulated convective pattern and penetration depth of the dense plume in fractured rock depend more on spatial discretization than on temporal discretization. The appropriate spatial-temporal grid is then used to examine some aspects of mixed convection in fractured-porous rock, characterized by the mixed convection number M. The critical mixed convection number M_c = 46 represents the transition between forced and free convection in fractured porous media, which is much higher than M_c = 1 in unfractured porous media. Thus, for mixed convective flow problems, the value of M_c is not a sufficient indicator to predict the convective mode (free convection-forced convection), and the presence of vertical fractures must be included in the prediction of convective flow modes.     

Laplace-transform analytic element solution of transient flow in porous media

A Laplace-transform analytic element method (LT-AEM) is described for the solution of transient flow problems in porous media. Following Laplace transformation of the original flow problem, the analytic element method (AEM) is used to solve the resultant time-independent modified Helmholtz equation, and the solution is inverted numerically back into the time domain. The solution is entirely general, retaining the mathematical elegance and computational efficiency of the AEM while being amenable to parallel computation. It is especially well suited for problems in which a solution is required at a limited number of points in space–time, and for problems involving materials with sharply contrasting hydraulic properties. We illustrate the LT-AEM on transient flow through a uniform confined aquifer with a circular inclusion of contrasting hydraulic conductivity and specific storage. Our results compare well with published analytical solutions in the special case of radial flow.     

Adaptive finite element simulation of Stokes flow in porous media

The Stokes problem describes flow of an incompressible constant-viscosity fluid when the Reynolds number is small so that inertial and transient-time effects are negligible. The numerical solution of the Stokes problem requires special care, since classical finite element discretization schemes, such as piecewise linear interpolation for both the velocity and the pressure, fail to perform. Even when an appropriate scheme is adopted, the grid must be selected so that the error is as small as possible. Much of the challenge in solving Stokes problems is how to account for complex geometry and to capture important features such as flow separation. This paper applies adaptive mesh techniques, using a posteriori error estimates, in the finite element solution of the Stokes equations that model flow at pore scales. Different selected numerical test cases associated with various porous geometrics are presented and discussed to demonstrate the accuracy and efficiency of our methodology.