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Using conformal mapping techniques, we derive analytical expressions for the shape of a propagating finger in a rectangular channel in homogeneous porous media, in the absence of interfacial tension, but in the presence of gravity, acting in a direction transverse to the direction of displacement. The gravity finger propagates either along the top or the bottom boundaries of the channel, depending on the density contrast between displacing and displaced fluids. Thus, the model describes the respective cases of gravity override or gravity underrunning, which occur when a lighter fluid phase displaces a heavier one and vice-versa. The solution is expressed in terms of the finger thickness, which is a free parameter in this model. When gravity is neglected, the solution reduces to the classical solution of the Saffman–Taylor finger. Numerical illustrations are provided to examine the sensitivity of the finger geometry to the various parameters, following the scaling theory of Brener et al. (1991).

We use an approach based on invasion percolation in a gradient (IPG) to describe the displacement patterns that develop when a fluid spreads on an impermeable boundary in a porous medium under the influence of gravity (buoyancy) forces in a drainage process. The approach is intended to simulate applications, such as the spreading of a DNAPL in the saturated zone and of a NAPL in the vadose zone on top of an impermeable layer, or the classical problems of gravity underruning and gravity override in reservoir engineering. As gravity acts in a direction transverse to the main displacement direction, a novel form of IPG develops. We study numerically the resulting patterns for a combination of transverse and parallel Bond numbers and interpret the results using the concepts of gradient percolation. A physical interpretation in terms of the capillary number, the viscosity ratio and the gravity Bond number is also provided. In particular, we consider the scaling of the thickness of the spreading gravity ‘tongue’, for the cases of gravity‐dominated and viscous‐unstable displacements, and of the propagating front in the case of stabilized displacement at relatively high rates. It is found that the patterns have percolation (namely fractal‐like) characteristics, which cannot be captured by conventional continuum equations. These characteristics will affect, for example, mass transfer and must be considered in the design of remediation processes.

We use pore‐network simulations to study the dependence of the critical gas saturation in solution‐gas 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 mass‐transfer driven processes), the critical gas saturation, Sgc, follows a power‐law scaling with the final nucleation fraction (fraction of sites activated), fq. For 3‐D processes, this relation reads Sgc∼fq0.16, indicating a sensitive dependence of Sgc to fq at very small values of fq.

We combine an integral equation formulation with a hodograph transformation to solve self-similar problems describing the unconfined flow of groundwater with variable inlet conditions. A class of new semi-analytical solutions is obtained for both rectilinear and radial flow geometries. The solutions are in general agreement with those derived by Barenblatt, although there are some discrepancies for the case of radial flow. The formulation presented provides additional analytical insight, and for computational purposes is simpler than Barenblatt's. In addition, the method proposed can be successfully used for the solution of a host of other nonlinear problems that admit self-similarity.

Large-scale averaging is important for the description of displacement processes in heterogeneous porous media. For immiscible displacement, a key objective is the determination of effective (pseudo) capillary pressure and phase permeabilities. Present continuum models rely on volume averaging, mixture theories, and homogenization methods, typically under the premise of capillary control. However, such methods are intrinsically unable to provide the local saturation distribution, which is needed for the computation of effective flow properties. In this paper, paralleling pore-level approaches, a percolation method is proposed for the derivation of large-scale properties in a drainage process at low flow rates. Percolation concepts are applied to a macroscopically heterogeneous region with a random and uncorrelated permeability distribution. Appropriate modifications of ordinary and invasion percolation, and percolation with trapping are developed for the determination of the large scale averages. Analytical results are also presented for certain simple cases. We show that for drainage at conditions of local capillary control, when a local description is possible, the large-scale capillary pressure curve is a nontrivial average of the individual curves. Large scale capillary trapping is predicted and a corresponding large-scale trapped saturation is calculated. Large-scale phase premeabilities are also derived. It is found that capillary heterogeneity renders a system more strongly wet in a macroscopic sense. The results are applicable to drainage, but not to imbibition, where a different mode of penetration holds.

Effects of capillary heterogeneity at the macroscopic scale have previously been analyzed for static conditions or in the context of outflow end-effects. This paper presents a systematic study for the case of one-dimensional, steady-state flow, that complements recent work on transient displacement. We consider the saturation response to various forms of heterogeneity. Included are analytical results for certain model cases, some general results, and numerical solutions for variously correlated spatial variations. The sensitivity to process parameters, such as rate, heterogeneity length scale and correlation, is studied. Physical interpretations are offered and potential applications in the estimation of heterogeneity are discussed.

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.