Stochastic analysis of immiscible displacement in porous media

The interface of two immiscible fluids flowing in porous media may behave in an unstable fashion. This instability is governed by the pore distribution, differential viscosity and interface tension between the two immiscible fluids. This study investigates the factors that control the interface instability at the wetting front. The development of the flow equation is based on the mass balance principle, with boundary conditions such as the velocity continuity and capillary pressure balance at the interface. By assuming that the two-phase fluids in porous media are saturated, a covariance function of the wetting front position is derived by stochastic theory. According to those results, the unstable interface between two immiscible fluids is governed by the fluid velocity and properties such as viscosity and density. The fluid properties that affect the interface instability are expressed as dimensionless parameters, mobility ratio, capillary number and Bond number. If the fluid flow is driven by gravitational force, whether the interface undergoes upward displacement or downward displacement, the variance of the unstable interface decreases with an increasing mobility ratio, increases with increasing capillary number, and decreases with increasing Bond number. For a circumstance in which fluid flow is horizontal, our results demonstrate that the capillary number does not influence the generation of the unstable interface.     

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.     

On unsteady motion of a rotating fluid bounded by flat porous plates

Summary In this paper, an investigation is made of the unsteady flow generated in a viscous, incompressible and homogeneous fluid bounded by (i) an infinite horizontal porous plate atz=0, or (ii) two infinite horizontal porous plates atz=0 andz=D. The fluid together with the plate(s) is in a state of solid body rotation with a constant angular velocity about the z-axis normal to the plate(s), and additionally, the plate(s) performs non-torsional elliptic harmonic oscillations in its (their) own plane(s). A uniform suction or injection is introduced in the configurations through the porous plate(s) and its influence on the unsteady flow and the associated boundary layers is examined. The unsteady flow field as well as the associated boundary layers is obtained explicitly. In contrast to the unsteady rotating flow without suction, solutions of the present problem with suction exhibit no resonant phenomena. It is shown that the suction is responsible for making the boundary layers thinner and for the elimination of the resonant phenomena. It is confirmed that the velocity field and the associated multiple boundary layers are significantly modified by suction. Physical significances of the mathematical results are discussed. Several limiting cases of interest are recovered from this analysis. The initial value problem for both the configurations is exactly solved by the Heaviside operational calculus combined with the theory of residues.     

On a monotonicity preserving Eulerian–Lagrangian localized adjoint method for advection–diffusion equations

Eulerian–Lagrangian localized adjoint methods (ELLAMs) provide a general approach to the solution of advection-dominated advection–diffusion equations allowing large time steps while maintaining good accuracy. Moreover, the methods can treat systematically any type of boundary condition and are mass conservative. However, all ELLAMs developed so far suffer from non-physical oscillations and are usually implemented on structured grids. In this paper, we propose a finite volume ELLAM which incorporates a novel correction step rendering the method monotone while maintaining conservation of mass. The method has been implemented on fully unstructured meshes in two space dimensions. Numerical results demonstrate the applicability of the method for problems with highly non-uniform flow fields arising from heterogeneous porous media.     

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 note on benchmarking of numerical models for density dependent flow in porous media

Verification of numerical models for density dependent flow in porous media (DDFPM) by the means of appropriate benchmark problems is a very important step in developing and using these models. Recently, Infinite Horizontal Box (IHB) problem was suggested as a possible benchmark problem for verification of DDFPM codes. IHB is based on Horton–Rogers–Lapwood (HRL) problem. Suitability of this problem for the benchmarking purpose has been investigated in this paper. It is shown that the wavelength of instabilities fails to be a proper criterion to be considered for this problem. However, the threshold of instability formation has been found to be appropriate for benchmarking purpose.     

Convective instabilities in a variable viscosity fluid cooled from above

Whether in the mantle or in magma chambers, convective flows are characterized by large variations of viscosity. We study the influence of the viscosity structure on the development of convective instabilities in a viscous fluid which is cooled from above. The upper and lower boundaries of the fluid are stress-free. A viscosity dependence with depth of the form ν₀ + ν₁ exp(?γ.z) is assumed. After the temperature of the top boundary is lowered, velocity and temperature perturbations are followed numerically until convective breakdown occurs. Viscosity contrasts of up to 10⁷ and Rayleigh numbers of up to 10⁸ are studied.For intermediate viscosity contrasts (around 10³), convective breakdown is characterized by the almost simultaneous appearance of two modes of instability. One involves the whole fluid layer, has a large horizontal wavelength (several times the layer depth) and exhibits plate-like behaviour. The other mode has a much smaller wavelength and develops below a rigid lid. The “whole layer” mode dominates for small viscosity contrasts but is suppressed by viscous dissipation at large viscosity contrasts.For the “rigid lid” mode, we emphasize that it is the form of the viscosity variation which determines the instability. For steep viscosity profiles, convective flow does not penetrate deeply in the viscous region and only weak convection develops. We propose a simple method to define the rigid lid thickness. We are thus able to compute the true depth extent and the effective driving temperature difference of convective flow. Because viscosity contrasts in the convecting region do not exceed 100, simple scaling arguments are sufficient to describe the instability. The critical wavelength is proportional to the thickness of the thermal boundary layer below the rigid lid. Convection occurs when a Rayleigh number defined locally exceeds a critical value of 160–200. Finally, we show that a local Rayleigh number can be computed at any depth in the fluid and that convection develops below depth z_r (the rigid lid thickness) such that this number is maximum.The simple similarity laws are applied to the upper mantle beneath oceans and yield estimates of 5 × 10¹⁵?5 × 10¹⁶ m² s^?1 for viscosity in the thermal boundary layer below the plate.     

Relationship between the transition frequency of local fluid flow and the peak frequency of attenuation

Local fluid flow (LFF) at the mesoscopic scale is the main dissipation mechanism of seismic waves in heterogeneous porous media within the seismic frequency band. LFF is easily influenced by the structure and boundary conditions of the porous media, which leads to different behaviors of the peak frequency of attenuation. The associated transition frequency can provide detailed information about the trend of LFF; therefore, research on the transition frequency of LFF and its relationship with the peak frequency of the corresponding attenuation (i.e., inverse of quality factor) facilitates the detailed understanding of the effect of inner structures and boundary conditions in porous media. In this study, we firstly obtain the transition frequency of fluid flux based on Biot’s theory of poroelasticity and the fast Fourier transform algorithm in a sample containing one repeating unit cell (RUC). We then analyze changes of these two frequencies in porous media with different porous properties. Finally, we extend our analysis to the influence of the undrained boundary condition on the transition frequency and peak frequency in porous media with multiple RUCs. This setup can facilitate the understanding of the effect from the undrained boundary condition. Results demonstrate that these two frequencies have the same trend at low water saturation, but amplitude variations differ between the frequencies as the amount of saturation increases. However, for cases of high water saturation, both the trend and the amplitude variation of these two frequencies fit well with each other.     

Three‐dimensional models of reservoir sediment and effects on the seismic response of arch dams

The important effects of bottom sediments on the seismic response of arch dams are studied in this paper. To do so, a three‐dimensional boundary element model is used. It includes the water reservoir as a compressible fluid, the dam and unbounded foundation rock as viscoelastic solids, and the bottom sediment as a two‐phase poroelastic domain with dynamic behaviour described by Biot's equations. Dynamic interaction among all those regions, local topography and travelling wave effects are taken into account. The results obtained show the important influence of sediment compressibility and permeability on the seismic response. The former is associated with a general change of the system response whereas the permeability has a significant influence on damping at resonance peaks. The analysis is carried out in the frequency domain considering time harmonic excitation due to P and S plane waves. The time‐domain results obtained by using the Fourier transform for a given earthquake accelerogram are also shown. The possibility of using simplified models to represent the bottom sediment effects is discussed in the paper. Two alternative models for porous sediment are tested. Simplified models are shown to be able to reproduce the effects of porous sediments except for very high permeability values. Copyright © 2004 John Wiley & Sons, Ltd.     

Hot springs mediate spatial exchange of heat and mass in the enclosed sediment domain: A stability perspective

In many natural environments, such as in underwater hot springs and hydrothermal vents, thermal gradients are accompanied with changes in the concentration of chemical compounds transported to the seawater, causing the so-called double-diffusive, mixed convection. To study the physical scenarios in such systems, a vertical channel filled with a porous medium saturated with saline water is considered. The motion in the sediment-filled channel is induced by two buoyancy forces and an external pressure gradient, similar to the situation in a vent with an upward flow direction. The fluid flow has been modeled by an extended Darcy model, and the flow instability mechanisms have been studied numerically. The linear stability analysis is performed considering a wide range of Darcy number (Da = 10⁻⁵ -10⁻⁸). The instability boundary curve showed three distinct dynamic regimes: (i) Rayleigh-Taylor (R-T), (ii) log-log non-linear variation, and (iii) log-log linear variation. The domain of different regimes were sensitive to external pressure gradient as well as permeability. Similar to cross-diffusive natural convection in pure viscous fluids, a linear relationship between logarithmic absolute values of critical thermal Rayleigh number (∣Ra_T∣) and solute Rayleigh number (Ra_C) is found in the third regime. Based on the permeability, for any solute Rayleigh number (Ra_C), there existed a minimum value of Reynolds number (Re), below which R-T type of instability appeared. Above this minimum value, the instability was due to two buoyancy forces, known as buoyant instability. Simulations of secondary flow via energy analysis demonstrated the development of complex dynamics at the critical state in all three regimes characterized by transition of multi to uni-cellular structures and vice verse.     

Effect of fracture pressure depletion regimes on the dual-porosity shape factor for flow of compressible fluids in fractured porous media

A precise value of the matrix-fracture transfer shape factor is essential for modeling fluid flow in fractured porous media by a dual-porosity approach. The slightly compressible fluid shape factor has been widely investigated in the literature. In a recent study, we have developed a transfer function for flow of a compressible fluid using a constant fracture pressure boundary condition [Ranjbar E, Hassanzadeh H, Matrix-fracture transfer shape factor for modeling flow of a compressible fluid in dual-porosity media. Adv Water Res 2011;34(5):627-39. doi:10.1016/j.advwatres.2011.02.012]. However, for a compressible fluid, the consequence of a pressure depletion boundary condition on the shape factor has not been investigated in the previous studies. The main purpose of this paper is, therefore, to investigate the effect of the fracture pressure depletion regime on the shape factor for single-phase flow of a compressible fluid. In the current study, a model for evaluation of the shape factor is derived using solutions of a nonlinear diffusivity equation subject to different pressure depletion regimes. A combination of the heat integral method, the method of moments and Duhamel’s theorem is used to solve this nonlinear equation. The developed solution is validated by fine-grid numerical simulations. The presented model can recover the shape factor of slightly compressible fluids reported in the literature. This study demonstrates that in the case of a single-phase flow of compressible fluid, the shape factor is a function of the imposed boundary condition in the fracture and its variability with time. It is shown that such dependence can be described by an exponentially declining fracture pressure with different decline exponents. These findings improve our understanding of fluid flow in fractured porous media.     

Intermittent convection

Abstract

A theoretical analysis of pseudo two-dimensional, finite-amplitude, thermal convection is made for an infinite Prandtl number fluid which is subjected to a constant heat flux out of the top boundary and insulated at the bottom. For large Rayleigh numbers the convective flow becomes intermittent and the system is characterized by the following cyclic process: the formation of a thermal boundary layer by diffusion, the instability of this layer when it becomes sufficiently thick, the destruction of the layer by the convective flow, the dying down of the convection, and the reforming of the thermal boundary layer by diffusion. The periodicity and the horizontal wave number of the intermittent convective flow are found to be independent of the depth of the fluid layer but depend on the rate of cooling and the properties of the fluid.     

Reflection and transmission of inhomogeneous waves in a composite porous solid saturated by two immiscible fluids

This paper is concerned with reflection and transmission of a plane, elastic, and inhomogeneous wave striking obliquely at some discontinuity inside a porous medium composed of two distinct solids and saturated by two immiscible fluids. It is found that four P‐ and two SV‐waves are reflected, whereas four P‐ and two SV‐waves are transmitted at the interface. All reflected and transmitted waves are inhomogeneous in nature and specified with different directions of propagation and attenuation vectors. An expression for the Umov–Poynting energy flux vector is derived for the system. Continuity of energy flux along normal to the interface gives 12 required boundary conditions. Expressions of amplitude and energy ratios of various reflected and transmitted waves are derived. Variations in amplitude and energy coefficients of reflected and transmitted waves with angle of incidence are numerically studied for a porous matrix composed of shaley sandstone and clay, saturated with water and oil. The effects of change in oil saturation and volume fraction of clay are also observed on amplitude ratios. Numerical simulation reveals that the change in sign in the difference of capillary pressure across the interface causes jump in the values of amplitude ratios of all waves.     

An experimental study of global instabilities due to the tidal (elliptical) distortion of a rotating elastic cylinder

Abstract

Broad band secondary instability of elliptical vortex motion has been proposed as a principal source of shear-flow turbulence. Here experiments on such instability in an elliptical flow with no shear boundary layer are described. This is made possible by the mechanical distortion in the laboratory frame of a rotating fluid-filled elastic cylinder. One percent ellipticity of a 10 cm diameter cylinder rotating once each second can give rise to an exponentially-growing mode stationary in the laboratory frame. In first order this mode is a sub-harmonic parametric Faraday instability. The finite-amplitude equations represent angular momentum transfer on an inertial time scale due to Reynolds stresses. The growth of this mode is not limited by boundary friction but by detuning and centrifugal stabilization. On average, a generalized Richardson number achieves a marginal value through much of the evolved flow. However, the characteristic flow is intermittent with the cycle: rapid growth, stabilizing momentum transfer from the mean flow, interior re-spin up, and then again. Data is presented in which, at large Reynolds numbers, seven percent ellipticity causes a fifty percent reduction in the kinetic energy of the rotating fluid. In the geophysical setting, this tidal instability in the earth's interior could be inhibited by sub-adiabatic temperature gradients. A near adiabatic region greater than 10 km in height would permit the growth of tidally destabilized modes and the release of energy to three-dimensional disturbances. Such disturbances might play a central role in the geodynamo and add significantly to overall tidal dissipation.     

含两项不混合/单项黏性流体孔隙介质分界面上反透射系数特征研究

弹性孔隙介质分界面上的反透射系数特征,在岩性划分、流体识别、储层边界判识等方面有重要的应用.本文研究上层为含两项不混合黏性流体孔隙介质、下层为含单项黏性流体孔隙介质分界面上的反透射理论.首先根据两种孔隙介质分界面上的能量守恒得到边界条件,再将波函数、位移、应力与应变关系代入边界条件,推导出完全连通孔隙情况下,第一类纵波入射到孔隙介质分界面上的反透射系数方程.通过建立砂岩孔隙介质模型,分别分析不同孔隙流体类型、不同含油饱和度及不同入射角情况下,各类波的反透射系数特征.研究表明,第二、三类纵波反透射系数数值比第一类纵波小多个数量级,且两者对入射角的变化不敏感,但对孔隙流体性质、含油饱和度的变化较敏感,而横波反透射系数特征恰好与此相反;第一类纵波反透射系数特征比较复杂,入射角、孔隙流体的性质及含油饱和度的变化都对其产生影响.不同孔隙流体弹性物性的差异、孔隙介质中含油饱和度的变化及不同入射角引起垂向和切向应力分量的变化都会影响各类波的反透射系数特征,分析这些特征可以为研究储层含油气性提供理论基础.     

基于横向各向同性BISQ方程的弹性波传播数值模拟

Biot流动和喷射流动是含流体多孔隙介质中流体流动的两种重要力学机制. 近年来，利用同时处理这两种力学机制的BISQ(Biot-Squirt)模型，弹性波衰减和频散的问题已被广泛研究；然而基于BISQ方程的波场数值模拟尚未见到公开的报道．本文从BISQ方程出发，利用交错网格方法对横向各向同性孔隙介质中不同频率和相界情况，以及双层介质中的弹性波传播进行数值模拟，研究了在同时考虑两种流动机制作用情况下地震波和声波的传播特性及传播过程中出现的各种波动现象．      

Steady hydromagnetic viscous flow in an annulus with porous walls

Summary An exact solution of electrically conducting viscous incompressible flow in an annulus with porous walls under an external radial magnetic field is obtained when the motion is due to both longitudinal motion of the inner boundary and a constant axial pressure gradient, and the fluid injection rate at one wall is equal to the fluid withdrawal rate at the other. The fluid may be injected at the outer wall and sucked at the inner or vice versa. The solution for the hydromagnetic flow between two flat plates has also been obtained as a limiting case of the annulus problem.     

Numerical simulation of elastic wave propagation based on the transversely isotropic BISQ equation

Introduction More real models are being developed by the modern seismology. As we all know, the earth is not a simple elastic body. Oil and gas reservoir, ground surface, seashore zone, sea bottom layer, etc, are porous solid media with fluids. It has been confirmed that there are two main fluid flow mechanisms in these media (Dvorkin, Nur, 1993), i.e., the Biot flow mechanism (Biot, 1956, 1962) based on the macroscopic property and the Squirt-flow mechanism (Mavko, Nur, 1979) based on the …