Interaction of multiple courses of wave‐induced fluid flow in layered porous media

Different theoretical and laboratory studies on the propagation of elastic waves in layered hydrocarbon reservoir have shown characteristic velocity dispersion and attenuation of seismic waves. The wave‐induced fluid flow between mesoscopic‐scale heterogeneities (larger than the pore size but smaller than the predominant wavelengths) is the most important cause of attenuation for frequencies below 1 kHz. Most studies on mesoscopic wave‐induced fluid flow in the seismic frequency band are based on the representative elementary volume, which does not consider interaction of fluid flow due to the symmetrical structure of representative elementary volume. However, in strongly heterogeneous media with unsymmetrical structures, different courses of wave‐induced fluid flow may lead to the interaction of the fluid flux in the seismic band; this has not yet been explored. This paper analyses the interaction of different courses of wave‐induced fluid flow in layered porous media. We apply a one‐dimensional finite‐element numerical creep test based on Biot's theory of consolidation to obtain the fluid flux in the frequency domain. The characteristic frequency of the fluid flux and the strain rate tensor are introduced to characterise the interaction of different courses of fluid flux. We also compare the behaviours of characteristic frequencies and the strain rate tensor on two scales: the local scale and the global scale. It is shown that, at the local scale, the interaction between different courses of fluid flux is a dynamic process, and the weak fluid flux and corresponding characteristic frequencies contain detailed information about the interaction of the fluid flux. At the global scale, the averaged strain rate tensor can facilitate the identification of the interaction degree of the fluid flux for the porous medium with a random distribution of mesoscopic heterogeneities, and the characteristic frequency of the fluid flux is potentially related to that of the peak attenuation. The results are helpful for the prediction of the distribution of oil–gas patches based on the statistical properties of phase velocities and attenuation in layered porous media with random disorder.     

Impact of NAPL architecture on interphase mass transfer: A pore network study

Interphase mass transfer in porous media is commonly modeled using Sherwood number expressions that are developed in terms of fluid and porous medium properties averaged over some representative elementary volume (REV). In this work the influence of sub-grid scale properties on interphase mass transfer was investigated using a two-dimensional pore network model. The focus was on assessing the impact of (i) NAPL saturation, (ii) interfacial area (iii) NAPL spatial distribution at the pore scale, (iv) grain size heterogeneity, (v) REV or domain size and (vi) pore scale heterogeneity of the porous media on interphase mass transfer. Variability of both the mass transfer coefficient that explicitly accounts for the interfacial area and the mass transfer coefficient that lumps the interfacial area was examined. It was shown that pore scale NAPL distribution and its orientation relative to the flow direction have significant impact on flow bypassing and the interphase mass transfer coefficient. This results in a complex non-linear relationship between interfacial area and the REV-based interphase mass transfer rate. Hence, explicitly accounting for the interfacial area does not eliminate the uncertainty of the mass transfer coefficient. It was also shown that, even for explicitly defined flow patterns, changing the domain size over which the mass transfer process is defined influences the extent of NAPL bypassing and dilution and, consequently, the interphase mass transfer. It was also demonstrated that the spatial variability of pore scale parameters such as pore throat diameters may result in different rates of interphase mass transfer even for the same pore size distribution index.     

Particular solutions to the problem of horizontal flow of water and air through porous media near a wetting front

Particular solutions to the problem of horizontal flow of water and air through homogeneous porous media are derived and regularity properties of the solutions are presented. It is found that a singularity occurs in the solutions at the wetting fronts. Effects of air flow on water flow are discussed.     

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.     

Direct simulations of two-phase flow on micro-CT images of porous media and upscaling of pore-scale forces

Pore-scale forces have a significant effect on the macroscopic behaviour of multiphase flow through porous media. This paper studies the effect of these forces using a new volume-of-fluid based finite volume method developed for simulating two-phase flow directly on micro-CT images of porous media. An analytical analysis of the relationship between the pore-scale forces and the Darcy-scale pressure drops is presented. We use this analysis to propose unambiguous definitions of Darcy-scale viscous pressure drops as the rate of energy dissipation per unit flow rate of each phase, and then use them to obtain the relative permeability curves. We show that this definition is consistent with conventional laboratory/field measurements by comparing our predictions with experimental relative permeability. We present single and two-phase flow simulations for primary oil injection followed by water injection on a sandpack and a Berea sandstone. The two-phase flow simulations are presented at different capillary numbers which cover the transition from capillary fingering at low capillary numbers to a more viscous fingering displacement pattern at higher capillary numbers, and the effect of capillary number on the relative permeability curves is investigated. Overall, this paper presents a new finite volume-based methodology for the detailed analysis of two-phase flow directly on micro-CT images of porous media and upscaling of the results to the Darcy scale.     

Velocity variations in carbonate rocks due to dual porosity and wave-induced fluid flow

Stiffness variations in carbonates may be described as resulting from different concentrations of flat compliant pores or cracks, which can have a significant effect on the effective stiffness and acoustic properties (e.g., velocities and attenuations) of dry as well as saturated carbonates, although they carry extremely little porosity. As shown in this paper, the effects of dual porosity and wave-induced fluid flow or pore pressure communication may also play a significant role. On the basis of a previously published T-matrix approach to model the effective viscoelastic properties of cracked porous media, we illustrate the (frequency-dependent) effects of wave-induced fluid flow (mainly squirt flow) or pore pressure communication for a model structure consisting of a mixture of fluid-saturated porous grains and fluid-saturated cavities (vugs, etc.) that are embedded in a solid matrix associated with carbonates. We assume that the pores within the porous grains are decoupled from the pores in the solid matrix (and possibly saturated with different fluids) but that each pore system at the micro and/or mesoscale may or may not be connected. For each of four different connectivity models, we present numerical results for four different cases of microstructure (that emphasize the importance of cracks and flat compliant pores). Our numerical results indicate that the velocity and attenuation spectra of carbonates vary significantly, even when the crack density and all other volume concentrations are constant.     

Density estimation of two-phase flow with multiscale and randomly perturbed data

In this paper, we describe an efficient approach for quantifying uncertainty in two-phase flow applications due to perturbations of the permeability in a multiscale heterogeneous porous medium. The method is based on the application of the multiscale finite element method within the framework of Monte Carlo simulation and an efficient preprocessing construction of the multiscale basis functions. The quantities of interest for our applications are the Darcy velocity and breakthrough time and we quantify their uncertainty by constructing the respective cumulative distribution functions. For the Darcy velocity we use the multiscale finite element method, but due to lack of conservation, we apply the multiscale finite volume element method as an alternative for use with the two-phase flow problem. We provide a number of numerical examples to illustrate the performance of the method.     

Directions of preferential flow in a hillslope soil, 1. Quasi‐steady flow

Preferred infiltration is mainly perceived as vertically down whereas subsurface storm flow is thought to occur parallel to slopes. The transition from vertical to lateral flow in a layered hillslope soil is the focus of the contribution. Transient flow is assumed to move as a wetting front. Three time‐domain reflectometry (TDR) wave‐guides, each 0·15 m long, were mounted in the shape of a truncated tetrahedron with its peak pointing down. Each wave‐guide focuses the front velocity along its axis. The three front‐velocity vectors are decomposed into their x, y and z components, which are then assembled to the resultant velocity vector. The volume density flux of preferred flow is the product of the front velocity and the mobile water content. The latter is the amplitude of transient soil moisture measured with each wave‐guide. The resultant vector of the volume flux density is computed similarly to the velocity vector. The experimental approach allows for the rapid assessment of transient flows without relying on the variation of water potentials. The experiments indicate that the directions of the resultant vectors of velocity and volume flux density can be estimated if the moisture variations of the three TDR wave‐guides are strongly correlated during the passing of the wetting front. Copyright © 2004 John Wiley & Sons, Ltd.     

Nonequilibrium statistical mechanical derivation of a nonlocal Darcy's Law for unsaturated/saturated flow

As illustrated variously by wetting and drying scanning curves, flow in unsaturated porous media is inherently nonlocal. This nonlocality is also manifest in hysteresis in the classical Darcy conductivity. It is the authors' belief that most current theories of unsaturated/saturated flow are often inadequate, as they do not account for spatial nonlocality and memory. Here we provide a fundamental theory in which nonlocality of the flow constitutive theory is a natural consequence of force balances. The results are derived from general principles in statistical physics and under appropriate limiting conditions, the classical Darcy's Law is recovered for saturated flow. A notable departure in this theory from other nonlocal flow theories is that a classical Darcy type equation on a small scale need not exist.     

Extraction of physically realistic pore network properties from three-dimensional synchrotron X-ray microtomography images of unconsolidated porous media systems

This paper presents application of a series of algorithms used to extract pore network structure from high-resolution three-dimensional synchrotron microtomography images of unconsolidated porous media systems. These algorithms are based on the three-dimensional skeletonization that simplifies the pore space to networks in the form of nodes connected to paths. Dilation algorithms were developed to generate inscribed spheres on the nodes and paths of the medial axis to represent pore-bodies and pore-throats of the network, respectively. The end result is a physically representative pore network structure, i.e. three-dimensional spatial distribution (i.e. x-, y-, and z-coordinates) of pore-bodies and pore-throats, pore-body size distribution, pore-throat size distribution, and the connectivity. Systems analyzed in this study include different glass bead systems and natural marine sand. The media ranged in size from 0.123 to 1.0 mm, while the image volumes ranged between 7.7 and 108.9 mm³. In addition to extracting the pore network structure, the porosity, specific surface area, and representative elementary volume analysis on the porosity were calculated. Spatial correlation between pore-body sizes in the network was investigated using semivariograms and integral scale concepts. The impact of resolution on the calculated property was also investigated.

In this work, we show that microtomography is an effective tool to non-destructively extract the structure of many systems. The quality of the datasets depends on photon energy, photon flux, size of the sample, type of the sample, and size of the sample ‘features’. Results show that the developed method of extracting pore network structure is applicable to ideal and natural porous media systems. The impact of resolution on the quantification of the network structure properties varies in its significance based on feature size of the system and the properties being calculated. Therefore, a thorough resolution sensitivity analysis should be carried out to determine the degree of error associated with a system imaged at a given resolution.     

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.     

Micro-computed tomography pore-scale study of flow in porous media: Effect of voxel resolution

A fundamental understanding of flow in porous media at the pore-scale is necessary to be able to upscale average displacement processes from core to reservoir scale. The study of fluid flow in porous media at the pore-scale consists of two key procedures: Imaging - reconstruction of three-dimensional (3D) pore space images; and modelling such as with single and two-phase flow simulations with Lattice-Boltzmann (LB) or Pore-Network (PN) Modelling. Here we analyse pore-scale results to predict petrophysical properties such as porosity, single-phase permeability and multi-phase properties at different length scales. The fundamental issue is to understand the image resolution dependency of transport properties, in order to up-scale the flow physics from pore to core scale. In this work, we use a high resolution micro-computed tomography (micro-CT) scanner to image and reconstruct three dimensional pore-scale images of five sandstones (Bentheimer, Berea, Clashach, Doddington and Stainton) and five complex carbonates (Ketton, Estaillades, Middle Eastern sample 3, Middle Eastern sample 5 and Indiana Limestone 1) at four different voxel resolutions (4.4 µm, 6.2 µm, 8.3 µm and 10.2 µm), scanning the same physical field of view. Implementing three phase segmentation (macro-pore phase, intermediate phase and grain phase) on pore-scale images helps to understand the importance of connected macro-porosity in the fluid flow for the samples studied. We then compute the petrophysical properties for all the samples using PN and LB simulations in order to study the influence of voxel resolution on petrophysical properties. We then introduce a numerical coarsening scheme which is used to coarsen a high voxel resolution image (4.4 µm) to lower resolutions (6.2 µm, 8.3 µm and 10.2 µm) and study the impact of coarsening data on macroscopic and multi-phase properties. Numerical coarsening of high resolution data is found to be superior to using a lower resolution scan because it avoids the problem of partial volume effects and reduces the scaling effect by preserving the pore-space properties influencing the transport properties. This is evidently compared in this study by predicting several pore network properties such as number of pores and throats, average pore and throat radius and coordination number for both scan based analysis and numerical coarsened data.     

Cyclic hysteretic flow in porous medium column: model, experiment, and simulations

A periodic vertical movement of the groundwater table results in a subsequent cyclic response of the water content and pressure profiles in the vadose zone. The sequence of periodic wetting and drying processes can be affected by hysteresis effects in this zone. A one-dimensional saturated/unsaturated flow model based on Richards’ equation and the Mualem (Soil Sci. 137 (1984) 283) hysteresis model is formulated which can take into account multi-cycle hysteresis effects in the relation between capillary pressure and water content. The numerical integration of the unsaturated flow equation is based on a Galerkin-type finite element method. The flow domain is discretised by finite elements with linear shape functions. Simulations start with static water content and pressure profiles, which correspond to either a boundary drying or wetting retention curve. To facilitate the numerical solution of the hysteretic case an implicit non-iterative procedure was chosen for the solution of the nonlinear differential equation. Laboratory experiments were performed with a vertical sand column by imposing a high frequency periodic pressure head at the lower end of the column. The total water volume in the column, and the periodic water content profile averaged over time were measured. The boundary drying and wetting curves of the relation between water content and capillary pressure were determined by independent experiments. The simulations of the experimental conditions show a clear effect of the hysteresis phenomenon on the water content profile. The simulations with hysteresis agree well with the measurements. Computed dimensionless water content profiles are presented for different oscillation frequencies with and without consideration of hysteresis.     

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 numerical method for two-phase flow in fractured porous media with non-matching grids

We propose a novel computational method for the efficient simulation of two-phase flow in fractured porous media. Instead of refining the grid to capture the flow along the faults or fractures, we represent the latter as immersed interfaces, using a reduced model for the flow and suitable coupling conditions. We allow for non matching grids between the porous matrix and the fractures to increase the flexibility of the method in realistic cases. We employ the extended finite element method for the Darcy problem and a finite volume method that is able to handle cut cells and matrix-fracture interactions for the saturation equation. Moreover, we address through numerical experiments the problem of the choice of a suitable numerical flux in the case of a discontinuous flux function at the interface between the fracture and the porous matrix. A wrong approximate solution of the Riemann problem can yield unphysical solutions even in simple cases.     

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.     

Efficient fully-coupled solution techniques for two-phase flow in porous media: Parallel multigrid solution and large scale computations

This paper is concerned with the fast resolution of nonlinear and linear algebraic equations arising from a fully implicit finite volume discretization of two-phase flow in porous media. We employ a Newton-multigrid algorithm on unstructured meshes in two and three space dimensions. The discretized operator is used for the coarse grid systems in the multigrid method. Problems with discontinuous coefficients are avoided by using a newly truncated restriction operator and an outer Krylov-space method. We show an optimal order of convergence for a wide range of two-phase flow problems including heterogeneous media and vanishing capillary pressure in an experimental way. Furthermore, we present a data parallel implementation of the algorithm with speedup results.     

Numerical modeling of two-phase flow in heterogeneous permeable media with different capillarity pressures

Contrast in capillary pressure of heterogeneous permeable media can have a significant effect on the flow path in two-phase immiscible flow. Very little work has appeared on the subject of capillary heterogeneity despite the fact that in certain cases it may be as important as permeability heterogeneity. The discontinuity in saturation as a result of capillary continuity, and in some cases capillary discontinuity may arise from contrast in capillary pressure functions in heterogeneous permeable media leading to complications in numerical modeling. There are also other challenges for accurate numerical modeling due to distorted unstructured grids because of the grid orientation and numerical dispersion effects. Limited attempts have been made in the literature to assess the accuracy of fluid flow modeling in heterogeneous permeable media with capillarity heterogeneity. The basic mixed finite element (MFE) framework is a superior method for accurate flux calculation in heterogeneous media in comparison to the conventional finite difference and finite volume approaches. However, a deficiency in the MFE from the direct use of fractional flow formulation has been recognized lately in application to flow in permeable media with capillary heterogeneity. In this work, we propose a new consistent formulation in 3D in which the total velocity is expressed in terms of the wetting-phase potential gradient and the capillary potential gradient. In our formulation, the coefficient of the wetting potential gradient is in terms of the total mobility which is smoother than the wetting mobility. We combine the MFE and discontinuous Galerkin (DG) methods to solve the pressure equation and the saturation equation, respectively. Our numerical model is verified with 1D analytical solutions in homogeneous and heterogeneous media. We also present 2D examples to demonstrate the significance of capillary heterogeneity in flow, and a 3D example to demonstrate the negligible effect of distorted meshes on the numerical solution in our proposed algorithm.     

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.