The Motion of Long Bubbles in a Network of Tubes

The motion and interaction of discrete bubbles in porous materials is studied numerically using a network model. The goal is to extend analytical results for the motion of bubbles through a single straight tube to a more realistic geometry for porous materials, modeled here as a planar network of straight tubes of different radii. The problem is characterized by two dimensionless parameters, the capillary number (Ca) and the volume fraction of bubbles (); results are characterized by determining the effective permeability of the network and the mean residence time of bubbles in the material. The simulations indicate that at low volume fraction most of the bubbles follow a limited number of high-flow pathways through the network. In this case the predictions of our simulations can be approximated by a simple analytical model. Bubbles interact with each other because their presence changes the local resistance to flow in individual tubes. As increases, interactions between individual bubbles become important resulting in a wider range of residence times in the porous material.     

Real-Space Renormalization Estimates for Two-Phase Flow in Porous Media

We present a spatial renormalization group algorithm to handle immiscibletwo-phase flow in heterogeneous porous media. We call this algorithmFRACTAM-R, where FRACTAM is an acronym for Fast Renormalization Algorithmfor Correlated Transport in Anisotropic Media, and the R stands for relativepermeability. Originally, FRACTAM was an approximate iterative process thatreplaces the L × L lattice of grid blocks, representing the reservoir,by a (L/2) × (L/2) one. In fact, FRACTAM replaces the original L× L lattice by a hierarchical (fractal) lattice, in such a way thatfinding the solution of the two-phase flow equations becomes trivial. Thistriviality translates in practice into computer efficiency. For N=L ×L grid blocks we find that the computer time necessary to calculatefractional flow F(t) and pressure P(t) as a function of time scales as N^1.7 for FRACTAM-R. This should be contrasted with thecomputational time of a conventional grid simulator N^2.3. The solution we find in this way is an accurateapproximation to the direct solution of the original problem.     

Upscaling of Channel Systems in Two Dimensions Using Flow-Based Grids

A methodology for the gridding and upscaling of geological systems characterized by channeling is presented. The overall approach entails the use of a flow-based gridding procedure for the generation of variably refined grids capable of resolving the channel geometry, a specialized full-tensor upscaling method to capture the effects of permeability connectivity, and the use of a flux-continuous finite volume method applicable to full tensor permeability fields and non-orthogonal grids. The gridding and upscaling procedures are described in detail and then applied to several two-dimensional systems. Significant improvement in the accuracy of the coarse scale models, relative to that obtained using uniform Cartesian coarse scale models, is achieved in all cases. It is shown that, for some systems, improvement results from the use of the flow-based grid, while in other cases the improvement is mainly due to the new upscaling method.     

A tensor estimator for the homogenization of absolute permeability

An estimator for an effective permeability tensor based on one-phase incompressible flow is presented. Effective large-scale permeability tensors are well approximated by rough approximations to the fine-scale pressure. The estimator works for all kinds of heterogeneous reservoirs and is fairly independent of boundary conditions.     

General Anisotropic Effective Medium Theory for the Effective Permeability of Heterogeneous Reservoirs

One of the techniques to calculate the effective property of a heterogeneous medium is the effective medium theory. The present paper presents a general mathematical formulation for the effective medium approximation using a self-consistent choice of the effective permeability, to apply it to the case of a general anisotropic 2D medium and to the case of a 3D isotropic medium with randomly oriented ellipsoidal inclusions. The 2D results are compared with analytical results and with a homogenization technique with good result. The 3D correlations are used to derive percolation thresholds in two-phase systems with a large permeability contrast, which are compared to numerical results from the literature, also with good results.     

On the use of conventional cocurrent and countercurrent effective permeabilities to estimate the four generalized permeability coefficients which arise in coupled,two-phase flow

In the case of coupled, two-phase flow of fluids in porous media, the governing equations show that there are four independent generalized permeability coefficients which have to be measured separately. In order to specify these four coefficients at a specific saturation, it is necessary to conduct two types of flow experiments. The two types of flow experiments used in this study are cocurrent and countercurrent, steady-state permeability experiments. It is shown that, by taking this approach, it is possible to define the four generalized permeability coefficients in terms of the conventional cocurrent and countercurrent effective permeabilities for each phase. It is demonstrated that a given generalized phase permeability falls about midway between the conventional, cocurrent effective permeability for that phase, and that for the countercurrent flow of the same phase. Moreover, it is suggested that the conventional effective permeability for a given phase can be interpreted as arising out of the effects of two types of viscous drag: that due to the flow of a given phase over the solid surfaces in the porous medium and that due to momentum transfer across the phase 1-phase 2 interfaces in the porous medium. The magnitude of the viscous coupling is significant, contributing at least 15% to the total conventional cocurrent effective permeability for both phases. Finally, it is shown that the nontraditional generalized permeabilities which arise out of viscous coupling effects cannot equal one another, even when the viscosity ratio is unity and the surface tension is zero.     

采动诱发岩体移动破坏过程数值模拟研究

本文基于岩石介质的宏观非线性主要是由非均质性和各向异性造成的, 应用新的数值计算软件RFPA(2D), 对采动引起岩体失稳破坏的全过程进了数值模拟研究。     

Transient Flow of a Compressible Fluid in a Connected Layered Permeable Medium

We develop a semi-analytical model of transient fluid flow in a 2D layered permeable medium with cross-flow between adjacent layers. It is shown that the pressure satisfies a diffusion equation to leading order, even when the non-linear term and gravity are included in the mathematical model. The solution is based on an analytical expression in the transform domain for the fluid pressure in terms of interfacial flux functions; the algorithm to compute the flux functions accepts an arbitrary number of formation layers. We show some benchmark tests that validate the general model; the model is then applied to an example derived from experiments. Numerical experiments confirm the significance of the cross-flow in a particular scaling of the ratio of permeabilities and quantify the influence of the various physical parameters.     

On the Convection in a Porous Medium with Inclined Temperature Gradient and Vertical Throughflow. Part II. Absolute and Convective Instabilities,and Spatially Amplifying Waves

This article presents a new methodology to estimate the effective permeability of random fractured media of any anisotropy containing both microfractures and a large number of long fractures crosscutting the representative volume element. The fractures are replaced by fictitious permeable materials for which the tangential permeability is deduced from a Poiseuille flow. A self-consistent scheme is proposed to derive the macroscopic permeability. On the one hand, the contribution of long fractures to the effective permeability writes by simple superposition of the fracture tangential permeabilities. On the other hand, the contribution of microfractures needs to resort to auxiliary problems requiring the computation of second-order Hill (or Eshelby) tensors related to ellipsoids embedded in an anisotropic matrix, for which a complete procedure is detailed. The effect of the microfracture normal permeability is put in evidence in the upscaling scheme and analyzed. In particular, it is shown that it must be chosen large enough to allow the connections between families. Examples are finally developed and compared to numerical simulations in the 2D case.