Linear Stability of Miscible Displacement Processes in Porous Media in the Absence of Dispersion

The linear stability of miscible displacement processes in porous media is examined in the absence of diffusion and dispersion. Bounds for the rate of growth of the disturbance are derived. The asymptotic behavior of the rate of growth as a function of the wavenumber of the disturbance and the mobility profile characteristics is obtained for both small and large wavenumbers. A closed-form solution is also presented for a particular mobility profile. It is shown that such displacement processes are linearly unstable in the case when the mobility profile contains any segments of decreasing mobility, and marginally stable in the opposite case. The effect of gravity on linear stability, in the case of compressible flows, is also briefly discussed.     

Pore-network study of the characteristic periods in the drying of porous materials

We study the periods that develop in the drying of capillary porous media, particularly the constant rate (CRP) and the falling rate (FRP) periods. Drying is simulated with a 3-D pore-network model that accounts for the effect of capillarity and buoyancy at the liquid-gas interface and for diffusion through the porous material and through a boundary layer over the external surface of the material. We focus on the stabilizing or destabilizing effects of gravity on the shape of the drying curve and the relative extent of the various drying periods. The extents of CRP and FRP are directly associated with various transition points of the percolation theory, such as the breakthrough point and the main liquid cluster disconnection point. Our study demonstrates that when an external diffusive layer is present, the constant rate period is longer.     

Numerical study of the linear stability of immiscible displacement in porous media

In this paper the linear stability of immiscible displacement in porous media is examined by numerical methods. The method of matched initial value problems is used to solve the eigenvalue problem for displacement processes pertaining to initially mobile phases. Both non capillary and capillary displacement in rectilinear flow geometries is studied. The results obtained are in agreement with recent asymptotic studies. A sensitivity analysis with respect to process parameters is carried out. Similarities and differences with the stability of Hele-Shaw flows are delineated.This is a revised version of paper SPE 13163, presented at the 59th Annual Technical Conference of the Society of Petroleum Engineers, Houston, Texas, 16–19 Sept. 1984.     

Darcian Dynamics: A New Approach to the Mobilization of a Trapped Phase in Porous Media

We develop a new approach, which we term Darcian Dynamics, to simulate two-phase (liquid-gas) flow in porous media, when the gas phase is disconnected in the form of ganglia. The method is based on the assumption of homogeneous fluid flow for the liquid, although it does allow for heterogeneous capillary thresholds due to the pore microstructure. Using techniques from potential theory, the hydrodynamic interaction between liquid and gas is expressed through an integral representation over the ganglia interfaces. We use a numerical method to solve the resulting integral equation, and explore conditions for the onset of ganglia mobilization as well as for subsequent events, such as break-up, coalescence and stranding. The interaction between the ganglia and the flowing phase is influenced by the capillary and gravity (Bond) numbers, and by geometric factors, such as size, orientation, and ganglia density. The latter effect depends on the hydrodynamic interaction in addition to the intuitively expected crowding effect.     

A Numerical Study of the Critical Gas Saturation in a Porous Medium

We use porenetwork simulations to study the dependence of the critical gas saturation in solutiongas drive processes on the geometric parameters of the porous medium. We show that for a variety of growth regimes (including global and local percolation, instantaneous and sequential nucleation, and masstransfer driven processes), the critical gas saturation, Sgc, follows a powerlaw scaling with the final nucleation fraction (fraction of sites activated), fq. For 3D processes, this relation reads Sgcfq0.16, indicating a sensitive dependence of Sgc to fq at very small values of fq.     

一般n阶特征量泛函的Euler-Lagrange方程及与定量因果原理、相对性原理和广义牛顿三定律的统一

本文获得了有各种相互作用的一般n阶特征量泛函,其耦合系数反映了不同特征量泛函之间的耦合强度.依据定量因果原理,导出了一般n阶特征量泛函的变分原理,获得了一般n阶特征量泛函的Euler-Lagrange方程,它的不同系数可拟合不同的物理现实,如从线性到任意n阶非线性物理系统,使复杂难解的任意n阶非线性物理系统变得具体可解.并获得了该对称变换下不变的m个的守恒量,以及它们之间的关系和统一描述.依据定量因果原理导出了相对性原理,证明了绝对加速参考系、牵连参考系和相对参考系的力都有来自加速度和质量变化的贡献.利用定量因果原理自然导出了广义牛顿第一定律和广义牛顿第二定律,而且还导出了一个新定律,即广义牛顿第三定律,亦即平移不变性系统合力为零定理.进而将研究结论应用于对银河系的修正引力势、分子势、夸克禁闭势等,且其结果与物理实验一致.     

A theoretical analysis of vertical flow equilibrium

The assumption of Vertical Equilibrium (VE) and of parallel flow conditions, in general, is often applied to the modeling of flow and displacement in natural porous media. However, the methodology for the development of the various models is rather intuitive, and no rigorous method is currently available. In this paper, we develop an asymptotic theory using as parameter the variable . It is rigorously shown that the VE model is obtained as the leading order term of an asymptotic expansion with respect to 1/R _L ² . Although this was numerically suspected, it is the first time that it is theoretically proved. Using this formulation, a series of special cases are subsequently obtained depending on the relative magnitude of gravity and capillary forces. In the absence of strong gravity effects, they generalize previous works by Zapata and Lake (1981), Yokoyama and Lake (1981) and Lake and Hirasaki (1981), on immiscible and miscible displacements. In the limit of gravity-segregated flow, we prove conditions for the fluids to be segregated and derive the Dupuit and Dietz (1953) approximations. Finally, we also discuss effects of capillarity and transverse dispersion.     

Mixing and reaction fronts in laminar flows

Autocatalytic reaction fronts between unreacted and reacted mixtures in the absence of fluid flow propagate as solitary waves. In the presence of imposed flow, the interplay between diffusion and advection enhances the mixing, leading to Taylor hydrodynamic dispersion. We present asymptotic theories in the two limits of small and large Thiele modulus (slow and fast reaction kinetics, respectively) that incorporate flow, diffusion, and reaction. For the first case, we show that the problem can be handled to leading order by the introduction of the Taylor dispersion replacing the molecular diffusion coefficient by its Taylor counterpart. In the second case, the leading-order behavior satisfies the eikonal equation. Numerical simulations using a lattice gas model show good agreement with the theory. The Taylor model is relevant to microfluidics applications, whereas the eikonal model applies at larger length scales.