ON THE THEORY OF EVOLUTION OF RIVER NETS

Abstract

An investigation is made of the growth of river basins based on the two current models of the structure of such basins: the cyclic model and the random-graph model. Based on the assumption that the networks grow allometrically, the structure of the network that evolves has been calculated based upon the above two models. Nature seems to favor the random-graph model.     

An improved gray lattice Boltzmann model for simulating fluid flow in multi-scale porous media

A lattice Boltzmann (LB) model is proposed for simulating fluid flow in porous media by allowing the aggregates of finer-scale pores and solids to be treated as ‘equivalent media’. This model employs a partially bouncing-back scheme to mimic the resistance of each aggregate, represented as a gray node in the model, to the fluid flow. Like several other lattice Boltzmann models that take the same approach, which are collectively referred to as gray lattice Boltzmann (GLB) models in this paper, it introduces an extra model parameter, n_s, which represents a volume fraction of fluid particles to be bounced back by the solid phase rather than the volume fraction of the solid phase at each gray node. The proposed model is shown to conserve the mass even for heterogeneous media, while this model and that model of Walsh et al. (2009) [1], referred to the WBS model thereafter, are shown analytically to recover Darcy–Brinkman’s equations for homogenous and isotropic porous media where the effective viscosity and the permeability are related to n_s and the relaxation parameter of LB model. The key differences between these two models along with others are analyzed while their implications are highlighted. An attempt is made to rectify the misconception about the model parameter n_s being the volume fraction of the solid phase. Both models are then numerically verified against the analytical solutions for a set of homogenous porous models and compared each other for another two sets of heterogeneous porous models of practical importance. It is shown that the proposed model allows true no-slip boundary conditions to be incorporated with a significant effect on reducing errors that would otherwise heavily skew flow fields near solid walls. The proposed model is shown to be numerically more stable than the WBS model at solid walls and interfaces between two porous media. The causes to the instability in the latter case are examined. The link between these two GLB models and a generalized Navier–Stokes model [2] for heterogeneous but isotropic porous media are explored qualitatively. A procedure for estimating model parameter n_s is proposed.     

A dual percolation model for predicting the connectivity of fractured porous media

This paper presents a dual-percolation model coupling the percolation theory and the fracture percolation theory to study the conductivity of the fractured porous media. The Monte-Carlo method is used in the numerical simulation. First an appropriate computing scale by considering the calculation precision and elapsed time together is validated. Then, two parameters, A ₀ and D are presented in this model to determine the conductivity of the media. Generally the media can be blocked by itself in the condition of D > 2. However, the increase of pore connection and the randomness of fracture direction may release the selfblockage, increase the conductivity and make the dual porous media dissipated. A few long fractures can play a great role in the connection of media.     

Calibrating Lattice Boltzmann flow simulations and estimating uncertainty in the permeability of complex porous media

A common way to simulate fluid flow in porous media is to use Lattice Boltzmann (LB) methods. Permeability predictions from such flow simulations are controlled by parameters whose settings must be calibrated in order to produce realistic modelling results. Herein we focus on the simplest and most commonly used implementation of the LB method: the single-relaxation-time BGK model. A key parameter in the BGK model is the relaxation time τ which controls flow velocity and has a substantial influence on the permeability calculation. Currently there is no rigorous scheme to calibrate its value for models of real media. We show that the standard method of calibration, by matching the flow profile of the analytic Hagen-Poiseuille pipe-flow model, results in a BGK-LB model that is unable to accurately predict permeability even in simple realistic porous media (herein, Fontainebleau sandstone). In order to reconcile the differences between predicted permeability and experimental data, we propose a method to calibrate τ using an enhanced Transitional Markov Chain Monte Carlo method, which is suitable for parallel computer architectures. We also propose a porosity-dependent τ calibration that provides an excellent fit to experimental data and which creates an empirical model that can be used to choose τ for new samples of known porosity. Our Bayesian framework thus provides robust predictions of permeability of realistic porous media, herein demonstrated on the BGK-LB model, and should therefore replace the standard pipe-flow based methods of calibration for more complex media. The calibration methodology can also be extended to more advanced LB methods.     

ORGANISATION DE L'E. G. I. G. Avril 1957

Abstract

The “non-Darcian” behaviour of water in soils is discussed considering the coupling that can arise between the different transport phenomena. In general, it is shown that there is a reason to expect that a difference in osmotic pressure will arise when solutions flow through porous media, and that due to that difference the resulting flow be related in a non-linear fashion to the applied pressure difference. Several numerical examples are presented illustrating this behaviour.     

A review of numerical simulation of hydrothermal systems / Examen de la simulation numérique des systèmes hydrothermiques

Abstract

Many advances in simulating single and two-phase fluid flow and heat transport in porous media have recently been made in conjunction with geothermal energy research. These numerical models reproduce system thermal and pressure behaviour and can be used for heat-transport problems other than those associated with geothermal energy development, such as high-level radioactive waste disposal and heat-storage projects. Although these models are general, additional research is necessary before they can be applied to certain site-specific problems that are concerned with additional processes, such as mass transport and flow in fractured media.     

Study of forward–backward solute dispersion profiles in a semi-infinite groundwater system

ABSTRACT

Forward–backward solute dispersion with an intermediate point source in one-dimensional semi-infinite homogeneous porous media is studied in this paper. Solute transport under sorption conditions, first-order decay and zero-order production terms are included. The first type of boundary condition is taken as a constant point source at an intermediate point from where forward and backward solute dispersion is examined. The Laplace transform method is adopted to solve the governing equation analytically. All the analytical results are obtained in graphical form to investigate the forward–backward solute transport in porous media for various hydrological input data. The graphical nature of the analytical solution is compared with numerical data taken from existing literature and similar results are obtained. Also, numerical solution of the governing equation is obtained by the Crank-Nicolson finite difference scheme and validated with the analytical solution, which demonstrates good agreement between them. Accuracy of the solution is also observed by using RMSE.     

Multicomponent competitive monovalent cation exchange in hierarchical porous media with multimodal reactive mineral facies

This paper presents a stochastic model for multicomponent competitive monovalent cation exchange in hierarchical porous media. Reactive transport in porous media is highly sensitive to heterogeneities in physical and chemical properties, such as hydraulic conductivity (K), and cation exchange capacity (CEC). We use a conceptual model for multimodal reactive mineral facies and develop a Eulerian-based stochastic theory to analyze the transport of multiple cations in heterogeneous media with a hierarchical organization of reactive minerals. Numerical examples investigate the retardation factors and dispersivities in a chemical system made of three monovalent cations (Na⁺, K⁺, and Cs⁺). The results demonstrate how heterogeneity influences the transport of competitive monovalent cations, and highlight the importance of correlations between K and CEC. Further sensitivity analyses are presented investigating how the dispersion and retardation of each cation are affected by the means, variances, and integral scales of K and CEC. The volume fraction of organic matter is shown to be another important parameter. The Eulerian stochastic framework presented in this work clarifies the importance of each system parameters on the migration of cation plumes in formations with hierarchical organization of facies types. Our stochastic approach could be used as an alternative to numerical simulations for 3D reactive transport in hierarchical porous media, which become prohibitively expensive for the multicomponent applications considered in this work.     

Statistical characteristics of flow as indicators of channeling in heterogeneous porous and fractured media

We introduce two new channeling indicators D_ic and D_cc based on the Lagrangian distribution of flow rates. On the basis of the participation ratio, these indicators characterize the extremes of both the flow-tube width distribution and the flow rate variation along flow lines. The participation ratio is an indicator biased toward the larger values of a distribution and is equal to the normalized ratio of the square of the first-order moment to the second-order moment. Compared with other existing indicators, they advantageously provide additional information on the flow channel geometry, are consistently applicable to both porous and fractured media, and are generally less variable for media generated using the same parameters than other indicators. Based on their computation for a broad range of porous and fracture permeability fields, we show that they consistently characterize two different geometric properties of channels. D_ic gives a characteristic scale of low-flow zones in porous media and a characteristic distance between effectively flowing structures in fractured cases. D_cc gives a characteristic scale of the extension of high-flow zones in porous media and a characteristic channel length in fractured media. D_ic is mostly determined by channel density and permeability variability. D_cc is, however, more affected by the nature of the correlation structure like the presence of permeability channels or fractures in porous media and the length distribution in fracture networks.     

A high-order time-domain transmitting boundary for cylindrical wave propagation problems in unbounded saturated poroelastic media

Based on the u–p formulation of Biot equation with an assumption of zero permeability coefficient, a high-order transmitting boundary is derived for cylindrical elastic wave propagation in infinite saturated porous media. By this transmitting boundary the total stresses on the truncated boundaries of a numerical model, such as a finite element model, are replaced by a set of spring, dashpot and mass elements, with some additionally introduced auxiliary degrees of freedom. The transmitting boundaries are incorporated into the DIANA SWANDYNE II program and an unconditionally stable implicit time integration algorithm is adopted. Despite the assumption made in the derivation of the transmitting boundary, numerical examples show that it can provide highly accurate results for cylindrical elastic wave propagation problems in infinite saturated porous medium in case the u–p formulation is applicable. Although the direct applications of the proposed transmitting boundary to general two dimensional wave problems in infinite saturated porous media are not highly accurate, acceptable accuracy can still be achieved by placing the transmitting boundary at relatively large distance from the wave source.     

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.     

Visualization experiments of biodegradation in porous media and calculation of the biodegradation rate

Biodegradation in porous media is studied with carefully controlled and well-characterized experiments in model porous media constructed of etched glass. Porous media of this type allow visual observation of the phenomena that take place at pore scale. An aqueous solution of five organic pollutants (toluene, phenol, o-cresol, naphthalene and 1,2,3-trimethylbenzene) was used as a model NAPL (representing creosote). The bacteria used were Pseudomonas fluorescens, which are indigenous (even predominant) in many contaminated soils. The maximum aqueous concentrations of the specific organic substances, below which biodegradation becomes possible, were determined as a function of temperature from toxicity experiments. Visualization experiments were made under various flow velocities and organic loadings to study the morphology and thickness of the biofilm as a function of the pore size and the distance from the entrance, and the efficiency of biodegradation. The efficiency of biodegradation decreased as the aqueous concentration of NAPL at the inlet increased and/or as the flow velocity increased. The thickness of biofilm decreased as the distance from the inlet increased and/or the pore diameter decreased. A quasi-steady-state theoretical model of biodegradation was used to calculate the values of the mesoscopic biochemical rates and to predict the profile of NAPL concentration in the porous medium and the thickness of biofilm in pores. The agreement between experimental data and model predictions is quite satisfactory.     

Analysis of Cryptosporidium parvum oocyst transport in porous media

Cryptosporidium parvum is a protozoan parasite, transmitted through aqueous environments in the form of an oocyst. In this study, a transport model into which sorption, filtration and inactivation mechanisms are incorporated is applied to simulate laboratory column data, and the suitability of a kinetic model to describe the C. parvum oocyst transport and removal in porous media is compared with an equilibrium model. The kinetic model is applied to simulate previous column experimental data and successfully simulates the concentration peak; the late time tailing effect appeared in the breakthrough curves, indicating that the kinetic model is more suitable than the equilibrium one at simulating the fate and transport of the oocysts in porous media. Simulation illustrates that sorption causes retardation along with a tailing in the breakthrough curve. Additionally, filtration acts as a major mechanism of removing the oocysts from the aqueous phase, whereas the role of inactivation in reducing the viable oocyst concentration is minimal. Copyright © 2004 John Wiley & Sons, Ltd.     

Field scale impacts of spatially correlated relative permeability in heterogeneous multiphase systems

Two-dimensional numerical simulations of two-phase (DNAPL-water) flow in spatially correlated random fields demonstrate the influence of nonwetting phase (NWP) relative permeability–saturation (k_r,N–S_W) relationships correlated to porous media intrinsic permeability (k). Both the volume of porous media invaded by the NWP and the length of time during which the NWP is migrating are under predicted if k_r,N–k correlation is not accounted for in the model formulation. Not accounting for the k_r,N–k correlation resulted in under predicting the volume of porous media invaded by up to approximately 10%, which is likely not significant for many practical applications. However, not accounting for the k_r,N–k correlation resulted in under predicting field scale migration times by up to a factor of 4, which is likely significant in that the migration times are on the order of years to several decades for the DNAPL (1,2-DCE) considered in this study. The under prediction of migration times was greater for lower permeability aquifers.     

Finite volume model for reactive transport in fractured porous media with distance- and time-dependent dispersion

Abstract

There are very few studies of fractured porous media that use distance- and time-dependent dispersion models, and, to the best of our knowledge, none which compare these with constant dispersion models. Therefore, in this study, the behaviour of temporal and spatial concentration profiles with distance- and time-dependent dispersion models is investigated. A hybrid finite volume method is used to solve the governing equations for these dispersion models. The developed numerical model is used to study the effects of matrix diffusion coefficient, groundwater velocity and matrix and fracture retardation factor on concentration profiles in the application of constant, distance-dependent and time-dependent dispersion models. In addition, an attempt is made to evaluate the applicability of these dispersion models by using the models to simulate experimental data. It was found that a better fit to the observed data is obtained in the case of distance- and time-dependent dispersion models as compared to the constant dispersion model. Thus, these numerical experiments indicate that distance- and time-dependent dispersion models have better simulation potential than the constant dispersion model.     

On results of the surface wave analyses in poroelastic media by means of the Simple Mixture Model and the Biot model

This note is devoted to an overview of the results on the number, velocity and attenuation of surface waves in two-component saturated poroelastic media. The author's numerical results which have been obtained using the Simple Mixture Model are compared to some results of other authors who carried out either an analysis by means of the classical Biot model for poroelastic media or experiments. It is shown that the Simple Mixture Model is a simplification of the Biot model. Due to its simpler form the Simple Mixture Model is more suitable for a complex analysis which depends on several parameters. Thus, the Simple Mixture Model is the only model for which, simultaneously, either the dependence on the frequency, ω, and the dependence on the bulk permeability, π (boundary porous medium/vacuum) or the dependence on the frequency, ω, and the dependence on the surface permeability, α (boundary porous medium/fluid) has been considered.     

Seismic reflection and transmission coefficients at an air-water interface of saturated porous soil

Based on the modified Biot＇s theory of two-phase porous media, a study was presented on seismic reflection and transmission coefficients at an air-water interface of saturated porous soil media. The major differences between air-saturated soils and water-saturated soils were theoretically discussed, and the theoretical formulas of reflection and transmission coefficients at an air-water interface were derived. The characteristics of propagation and attenuation of elastic waves in air-saturated soils were given and the relations among the frequency, the angle of incidence and the reflection, transmission coefficients were analyzed by using numerical methods. Numerical results show that the propagation characteristic of the wave in air-saturated soils is great different from that in water-saturated soils. The frequency and the angle of incidence can have great influences on the reflection and transmission coefficients at interface. Some new cognition about the wave propagation is obtained and the study suggests that we may carefully pay attention to the influence of air on the dynamic analysis of seismic wave.