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We consider miscible displacement between parallel plates in the absence of diffusion, with a concentration-dependent viscosity. By selecting a piecewise viscosity function, this can also be considered as 'three-fluid' flow in the same geometry. Assuming symmetry across the gap and based on the lubrication ('equilibrium') approximation, a description in terms of two quasi-linear hyperbolic equations is obtained. We find that the system is hyperbolic and can be solved analytically, when the mobility profile is monotonic, or when the mobility of the middle phase is smaller than its neighbours. When the mobility of the middle phase is larger, a change of type is displayed, an elliptic region developing in the composition space. Numerical solutions of Riemann problems of the hyperbolic system spanning the elliptic region, with small diffusion added, show good agreement with the analytical outside, but an unstable behaviour inside the elliptic region. In these problems, the elliptic region arises precisely at the displacement front. Crossing the elliptic region requires the solution of essentially an eigenvalue problem of the full higher-dimensional model, obtained here using lattice BGK simulations

The sustained propagation of combustion fronts in porous media is a necessary condition for the success of in situ combustion for oil recovery. Compared to other recovery methods, in situ combustion involves the complexity of exothermic reactions and temperature-dependent chemical kinetics. In the presence of heat losses, the possibility of ignition and extinction also exists. In this paper, we address some of these issues by studying the properties of forward combustion fronts propagating at a constant velocity in the presence of heat losses. We extend the analytical method used in smoldering combustion [7], to derive expressions for temperature and concentration profiles and the velocity of the combustion front, under both adiabatic and non-adiabatic conditions. Heat losses are assumed to be relatively weak and they are expressed using two modes: 1) a convective type, using an overall heat transfer coefficient; and, 2) a conductive type, for heat transfer by transverse conduction to infinitely large surrounding formations. In their presence we derive multiple steady-state solutions with stable low and high temperature branches, and an unstable intermediate branch. Conditions for self-sustaining front propagation are investigated as a function of injection and reservoir properties. The extinction threshold is expressed in terms of the system properties. An explicit expression is also obtained for the effective heat transfer coefficient in terms of the reservoir thickness and the front propagation speed. This coefficient is not only dependent on the thermal properties of the porous medium but also on the front dynamics.

We study two asymptotic regimes of unstable miscible displacements in porous media, in the two limits, where a permeability-modified aspect ratio, RL=L/H(kv/kh)1/2, becomes large or small, respectively. The first limit is known as transverse (or vertical) equilibrium, the second leads to the problem of non-communicating layers (the Dykstra–Parsons problem). In either case, the problem reduces to the solution of a single integro-differential equation. Although at opposite limits of the parameter RL, the two regimes coincide in the case of equal viscosities, M=1. By comparison with high-resolution simulation we investigate the validity of these two approximations. The evolution of transverse averages, particularly under viscous fingering conditions, depends on RL. We investigate the development of a model to describe viscous fingering in weakly heterogeneous porous media under transverse equilibrium conditions, and compare with the various existing empirical models (such as the Koval, Todd–Longstaff and Fayers models).

We show that for reaction-transport processes with fast kinetics (in the limit of thermodynamic equilibrium), conventional volume averaging for determining effective kinetic parameters applies only when the macroscopic variable approaches its equilibrium value. Even under such conditions, computing the effective mass transfer coefficient requires solving an eigenvalue problem, which couples the local microstructure with the global. Two examples, one involving a simple advection–dissolution problem and another a drying problem in a pore network, illustrate the theoretical predictions. Similar considerations apply for the case of finite kinetics, when the macroscale concentration approaches an equilibrium value. In that case, the effective kinetic parameter is not equal to the local, as typically assumed, but it becomes a function of the local Thiele modulus.

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 review notable advances published in the year 1999 in the area of phase change and phase growth in porous media. Phase equilibria thermodynamics, particularly in micropores, and growth kinetics, emphasizing the pore-network structure, are highlighted. Advances reported include the effects of confinement in phase transitions in micropores, and of the pore microstructure in the growth and dissolution of gas and liquid phases with applications ranging from capillary condensation to drying to gas evolution to condensation.

The drying of liquid-saturated porous media is typically approached using macroscopic continuum models involving phenomenological coefficients. Insight on these coefficients can be obtained by a more fundamental study at the pore- and pore-network levels. In this paper, we present a model based on a pore-network representation of porous media that accounts for various processes at the pore-scale. These include mass transfer by advection and diffusion in the gas phase, viscous flow in liquid and gas phases and capillary effects at the gas–liquid menisci in the pore throats. We consider isothermal drying in a rectilinear horizontal geometry, with no-flow conditions in all but one boundary, at which a purge gas is injected at a constant rate. The problem is mainly characterized by two dimensionless parameters, a diffusion-based capillary number, Ca , and a Peclet number, Pe , in addition to the various geometrical parameters of the pore network. Results on the evolution of the liquid saturation, the trapped liquid islands and the drying rate are obtained as a function of time and the dimensionless parameters. The importance of trapped liquid islands on screening mass transfer to the continuous liquid cluster is emphasized. For fixed parameter values, the drying front does not in general obey invasion percolation rules. However, as drying progresses, and depending on the relative magnitude of the capillary and Peclet numbers, a transition to a percolation-controlled problem occurs. Effects of capillarity and mass transfer on saturation profiles and drying rates are discussed. The results are then used to discuss upscaling to continuum models.

Despite significant progress made in recent years, a fundamental understanding of immiscible displacements at the macroscale is lacking. In this paper we use a version of percolation theory, based on invasion percolation in a gradient, to connect drainage processes at the pore-network scale with the displacement at the macroscale. When the mobility ratio M is sufficiently small, the displacement is stabilized and can be described by invasion percolation in a stabilizing gradient. In the opposite case, it has common features with invasion percolation in a destabilizing gradient. A diagram delineating the regimes of fully developed drainage is developed. The transition between stabilized displacement and fingering is controlled by the change of the sign of the gradient of the percolation probability, and the transition boundary is described by a scaling law involving the capillary number and the viscosity ratio. We review recent work for random networks and extend the method to correlated pore networks. As the regimes of stabilized displacement are also those for which conventional theories (such as the Buckley–Leverett equation) are expected to apply, the phase diagram helps to delineate their validity.

We model the evaporation of a stagnant liquid from an initially filled block to a flowing gas stream. The motivation for this problem arises from applications in the drying of porous media, when the pressure is low, and in the recovery of oil from fractured reservoirs by gas injection, when the pressure is high. A similarity solution is developed which accounts for diffusion in both phases. Diffusion in the liquid phase can be important in high-pressure applications, where the gas may dissolve in the liquid phase. The motion of the interface and the evaporation rates are calculated as a function of the various thermodynamic parameters for systems of interest, including n-alkanes, methane, nitrogen, or carbon dioxide. The effect of counter-diffusion is shown to slow the evaporation process, although not by an order of magnitude, in typical cases.

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.

Summary Many carbonate rocks have pore features that are large at the core plug scale. Conventional laboratory assessment of recovery behavior in these carbonates can be unreliable. Pore-network modeling offers an approach to improve our analysis of flow and displacement in such heterogeneous rocks. This paper reports on a systematic approach to link pore model inputs and laboratory data to a calibration methodology, and the subsequent use of a porenetwork model to match laboratory corefloods in a heterogeneous dolomite. It is found that spatial correlation effects, determined from core samples using thin section and computed tomography (CT) data, are important for a satisfactory match.

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.

Summary We report on a boundary effect associated with the preferential propagation of viscous fingers along no-flow boundaries in miscible displacements in two-dimensional (2D) rectilinear geometries near conditions of transverse equilibrium (TE) (large values of the generalized aspect ratio RL). It is shown that this effect intensifies with an increase in the mobility ratio, and diminishes with increased permeability disorder, and as the system departs from conditions of TE. The effect disappears altogether if periodic boundary conditions are used. We show that this effect results from the violation of the no-slip condition at no-flow boundaries, associated with Darcy's law. It must be considered in simulations of viscous fingering in relatively homogeneous media, near conditions of TE, particularly when gravity segregation occurs.

Summary This paper presents a new approach for the estimation of the large-scale correlation function of reservoir heterogeneity from the analysis of the pressure transient response of multiple wells. The approach is based on the theory of small fluctuations of the logarithm of the permeability and models the response of the "ensemble-average pressure", obtained by averaging the pressure response of multiple well tests. A non-local difFusivity equation for the ensemble-average pressure, which incorporates directly the permeability correlation function, is obtained, from the analysis of which the permeability semi-variogram of the reservoir can be constructed in principle. We consider a particular application to 1-D geometries, where a type-curve is also derived for the case of an exponential correlation function. Numerical simulations results support the applicability of the method.

Motivated by the problem of finding the minimum threshold path (MTP) in a lattice of elements with random thresholds ${\mathrm{\ensuremath{\tau}}}_{\mathrm{i}}$, we propose a new class of invasion processes, in which the front advances by minimizing or maximizing the measure ${\mathrm{S}}_{\mathrm{n}}$=${\ensuremath{\sum}}_{\mathrm{i}}$${\mathrm{\ensuremath{\tau}}}_{\mathrm{i}}^{\mathrm{n}}$ for real n. This rule assigns long-time memory to the invasion process. If the rule minimizes ${\mathrm{S}}_{\mathrm{n}}$ (case of minimum penalty), the fronts are stable and connected to invasion percolation in a gradient [J. P. Hulin, E. Clement, C. Baudet, J. F. Gouyet, and M. Rosso, Phys. Rev. Lett. 61, 333 (1988)] but in a correlated lattice, with invasion percolation [D. Wilkinson and J. F. Willemsen, J. Phys. A 16, 3365 (1983)] recovered in the limit |n|=\ensuremath{\infty}. For small n, the MTP is shown to be related to the optimal path of the directed polymer in random media (DPRM) problem [T. Halpin-Healy and Y.-C. Zhang, Phys. Rep. 254, 215 (1995)]. In the large n limit, however, it reduces to the backbone of a mixed site-bond percolation cluster. The algorithm allows for various properties of the MTP and the DPRM to be studied. In the unstable case (case of maximum gain), the front is a self-avoiding random walk.

Asymptotic solutions are developed for miscible displacements at Stokes flow conditions between parallel plates or in a cylindrical capillary, at large values of the geometric aspect ratio. The single integro-differential equation obtained is solved numerically for different values of the Peclet number and the viscosity ratio. At large values of the latter, the solution consists of a symmetric finger propagating in the middle of the gap or the capillary. Constraints on conventional convection-dispersion-equation approach for studying miscible instabilities in planar Hele–Shaw cells are obtained. The asymptotic formalism is next used to derive—in the limit of zero diffusion— a hyperbolic equation for the cross-sectionally averaged concentration, the solution of which is obtained by analytical means. This solution is valid as long as sharp shock fronts do not form. The results are compared with recent numerical simulations of the full problem and experiments of miscible displacement in a narrow capillary.

We study miscible fluid displacement in heterogeneous porous media, in the presence of gravity, where the heterogeneity consists of multiple layers. Miscible fluids of different density and viscosity contrasts are used, such that the displacement is hydrodynamically stable. Depending on the values of the control parameters, the overall fluid displacement exhibits either an enhancement of the effect of heterogeneity or a damping. For a sufficiently large viscosity and/or density contrast, the effect of heterogeneity is overcome and the multi-layer system behaves as a single homogeneous layer. We determine experimentally an associated phase diagram and compare our data to a theoretical model valid for systems of large aspect ratio.

Nous avons etudie, theoriquement et a l'aide de simulations numeriques, les correlations spatiales des profils de saturation obtenus au cours de l'invasion d'un milieu poreux par une phase non mouillante (drainage). La saturation (concentration volumique) de la phase envahissante est moyennee dans la direction perpendiculaire a l'ecoulement ; cette saturation est ensuite etudiee dans des conditions ou les effets capillaires sont dominants. Ce processus est bien decrit par la Percolation d'Invasion. Pour un milieu sans correlation, on montre que la structure des correlations approche un mouvement Brownien fractionnaire (fractional Brownian motion (fBm)) d' exposant de Hurst H = (D-1)/2, ou D est la dimension fractale de l' amas de percolation. Suffisamment loin du seuil de percolation, la structure des correlations du profil de saturation est decrite par un bruit blanc. Des mesures de bruits au cours d'experiences de deplacement a l'aide d'une technique acoustique ont ete effectuees. A 2-D, la theorie est confirmee par des simulations. A 3-D, la difference entre theorie d'une part et simulations et experience d'autre part est attribuee a des effets de taille finie. Les resultats sont ensuite generalises au cas de la percolation correlee.

We study the stability of the downdip displacement in a porous medium of a dense and viscous fluid by a lighter and less viscous fluid, and vice versa. Conventional predictions based on a Long-Wave (LW) theory lead to identical instability thresholds for the two flows. A Short-Wave (SW) analysis suggests that instability sets in earlier than the LW predictions and that the two thresholds are different in the two flows. Using an acoustic technique in 3D displacements, we have determined these thresholds and the instability diagram. Our data support the SW approach.

We study the effect of fluid contact area on viscous coupling in the parallel flow of immiscible fluids in a porous media geometry. We consider flow on opposite sides of a planar interface, consisting of alternating solid and open (slit) segments. We use the analytical solution of Tio and Sadhal [15] to derive explicit expressions for viscous coupling in terms of the fractional area of contact between the fluids and the viscosity ratio,M. ForM=1, the coefficient matrix obtained is symmetric showing that Onsager's relations are satisfied. In this case, the resulting viscous coupling is typically very small, in agreement with recent experimental results. Lattice gas simulations forM=1 using theBGK model support the theoretical results and show that viscous coupling further diminishes as the wall thickness increases. Assuming the same configuration, analytical results are next derived forM≠1. The results confirm an existing reciprocity relation between the off-diagonal terms. Viscous coupling remains small.

Motivated by a problem in foam generation and propagation in porous media, we develop necessary and sufficient conditions for a more general quasilinear hyperbolic problem, involving reactions of finite rate, to admit as solution a steady state following the initial equilibrium state. It is found that for such a solution to exist (and to be physically well-posed), it is necessary that the generalized flux vector F and the generalized concentration vector C be colinear, namely that F = A(C)C, where A(C) is a scalar function. If an additional condition for the overlap of characteristics is valid, the two conditions are sufficient and necessary.

Mass-transfer driven growth of a single gas cluster in a porous medium under the application of a supersaturation in the far field is examined. We discuss the growth pattern and its growth rate. Contrary to compact (spherical) growth in the bulk, growth patterns in porous media are disordered and vary from percolation to diffusion-limited aggregation (DLA) as the cluster size increases. At conditions of low supersaturation, scaling laws for the boundaries that delineate these patterns and of the corresponding growth rates are derived. In three dimensions (3D), it is found that the cluster grows as ${\mathit{R}}_{\mathit{g}}$\ensuremath{\sim}${\mathit{t}}_{\mathit{f}}^{1/(\mathit{D}}$-1), where ${\mathit{D}}_{\mathit{f}}$ is the pattern fractal dimension (\ensuremath{\approxeq}2.50 for percolation or DLA). A similar result involving logarithmic corrections is found for 2D. These results generalize the classical scaling ${\mathit{R}}_{\mathit{g}}$\ensuremath{\sim}${\mathit{t}}^{1/2}$ to fractal clusters.

Typical models for the representation of naturally fractured systems generally rely on the double-porosity Warren-Root model or on random arrays of fractures. However, field observations have demonstrated the existence of multiple length scales in a variety of naturally fractured media. Present models fail to capture this important property of self-similarity. We first use concepts from the theory of fragmentation and from fractal geometry to construct numerically a network of fractures that exhibits self-similar behavior over a range of scales. The method is a combination of fragmentation concepts and the iterated function system approach and allows for great flexibility in the development of patterns. Next, numerical simulation of unsteady single-phase flow in such networks is described. It is found that the pressure transient response of finite fractals behaves according to the analytical predictions of Chang and Yortsos (1990) provided that there exists a power law in the mass-radius relationship around the test well location. Finite size effects can become significant and interfere with the identification of the fractal structure. The paper concludes by providing examples from actual well tests in fractured systems which are analyzed using fractal pressure transient theory.

We study theoretically and by simulation the spatial correlation of the fluctuations of the transversely averaged fraction of the invading phase in [ital invasion] [ital percolation]. For an uncorrelated lattice, the profile has the correlation structure of the trace of [ital fractional] [ital Brownian] [ital motion] (FBM) with [ital Hurst] exponent [ital H]=[ital D][minus]2 for 3D percolation and [ital H]=[ital D][minus]1.5 for 2D percolation, where [ital D] is the fractal dimension of the percolation cluster. Extension to correlated percolation shows similar results, with the FBM signal displaying [ital persistent] correlations ([ital H][gt]0.5) in 3D and [ital antipersistent] correlations ([ital H][lt]0.5) in 2D percolation. Experiments of nonwetting fluid invasion in porous media confirm the theory in 3D.

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.

A model describing the generation, transport, and extraction of gas in a landfill has been developed. In the model the landfill gas is assumed to be an equimolar mixture of CH 4 and CO 2 . The waste is thought to comprise three classes of materials differing in their degree of biodegradability but all following first-order biodegradation kinetics. The model describes changes in the total pressure and not the partial pressures of the individual components. It predicts the pressure field inside a landfill, where gas extraction occurs from a number of wells placed at arbitrary points. The wells are assumed to be one-dimensional line sinks with uniform gas extraction rates. When the walls of the landfill are impermeable to flow, the model equations can be solved analytically. The full problem describing a landfill with permeable walls is treated numerically. The effect of the various parameters, such as gas-extraction rates, generation rates, and permeabilities are discussed.

We consider drainage processes in model geometries that represent a matrix block-fracture system. Flow visualization in etched glass micromodels was carried out for various pairs of fluids, injection rates (capillary numbers), and viscosity ratio values. It was found that for a flow rate below a critical value (capillary number threshold) displacement occurs only in the fracture, with matrix invasion occurring after the critical value is exceeded. The experiments were modeled with the use of a pore network simulator based on meniscus displacement. Numerical and experimental results were compared and found to be in good agreement. A theory for the critical capillary number and the invasion process was subsequently developed. The process shares many of the features of invasion in homogeneous systems, with the notable exceptions that the role of capillary pressure is played here by the capillary number, and that the process is dynamic rather than quasi-static. Effective relative permeabilities for the matrix-fracture system were also calculated. Contrary to homogeneous systems, these curves depend on the mobility ratio. The results find application to fractured systems or to systems containing high permeability streaks.

We investigate effects of capillary heterogeneity induced by variations in permeability in the direction of displacement in heterogeneous porous media under drainage conditions. The investigation is three-pronged and uses macroscopic simulation, based on the standard continuum equations, experiments with the use of an acoustic technique and pore network numerical models. It is found that heterogeneity affects significantly the saturation profiles, the effect being stronger at lower rates. A good agreement is found between the continuum model predictions and the experimental results based on which it can be concluded that capillary heterogeneity effects in the direction of displacement act much like a body force (e.g. gravity). A qualitative agreement is also found between the continuum approach and the pore network numerical models, which is expected to improve when finite size effects in the pore network simulations diminish. The results are interpreted with the use of invasion percolation concepts.

Steady-state methods are commonly used for the determination of the relative permeabilities of single component vapor-liquid systems, such as steam-water. Even though such experiments are performed under adiabatic conditions, the interpretation of results has proved to be difficult. Reasons involve phase change, heat transfer, capillarity and rate considerations. This paper presents a systematic study of the saturation and temperature distributions in steady-state, vapor-liquid flows and examines their sensitivity to parameters such as injection rate, permeability and core length. Two sets are analyzed, one pertaining to the simultaneous injection of a two-phase mixture (Sanchez and Schechter, 1987) and another involving phase change (liquid to vapor) within the core (Miller, 1951). End effects and, in general, effects associated with capillary heterogeneity are also analyzed. Solution trajectories in the temperature-saturation plane are constructed. They allow a unified approach to the two problems (two-phase or single-phase injection), which are shown to lie on different parts of the same trajectory. Minimum conditions for the development of in-situ evaporation are developed. Then, the process sensitivity to parameter values is examined for steady-state two-phase injection and relative permeability estimation. It is found that a plateau region develops, within which the inlet saturation is approximately constant, although corresponding saturation profiles are not necessarily flat. Numerical estimates show that the estimation error involved increases with decreasing enthalpy and decreasing permeability.

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.

Capillary effects can be important in immiscible flow in heterogeneous media, particularly at low capillary numbers. Experiments are performed to study steady-state saturation profiles during drainage in 3D porous media with sharp changes in permeability. It is found that the saturation response is closely linked to the permeability variation. The results are in good agreement with numerical simulations based on a macroscopic description.

For sufficiently large viscosity ratios and injection rates, miscible displacements in a vertical Hele–Shaw cell at high rates become unstable, leading to three-dimensional (3D) fingering patterns. Below the instability threshold, the base state is 2D in the form of a “tongue” of constant thickness. We apply the long wave Saffman–Taylor stability analysis to find an expression for the threshold of instability as a function of the viscosity ratio and the injection rate. The results are in agreement with the experimental data.

We consider the parallel flow of two immiscible fluids in a Hele-Shaw cell. The evolution of disturbances on the fluid interfaces is studied both theoretically and experimentally in the large-capillary-number limit. It is shown that such interfaces support wave motion, the amplitude of which for long waves is governed by a set of KdV and Airy equations. The waves are dispersive provided that the fluids have unequal viscosities and that the space occupied by the inner fluid does not pertain to the Saffman-Taylor conditions (symmetric interfaces with half-width spacing). Experiments conducted in a long and narrow Hele-Shaw cell appear to validate the theory in both the symmetric and the non-symmetric cases.

Summary The Buckley-Leverett (BL) equation has been routinely used for the description of immiscible displacement in laboratory Dperations and for parameter estimation and prediction. Recent advances, however, have led to a serious challenge of the validity of many of the traditional theories. For the case of BL displacement, most studies have examined heterogeneity effets as they relate to. viscous fingering. In this paper, we investigate the effects of capillary heterogeneity induced by variations in permeability in the direction of displacement. We study a variety of heterogeneity profiles, both uncorrelated and correlated spatially. We find that capillary heterogeneity significantly affects the saturation distributions, which closely follow the heterogeneity variation. The saturation response is stronger at lower rates, at more unfavorable mobility, under drainage conditions, and for smaller spatial correlations. The results are in agreement with available experimental data for primary drainage and secondary imbibition. They may be useful in the interpretation of saturation profiles and the identification of heterogeneity.

Vapor-liquid counterflow in porous media arises in processes such as heat pipes, oil recovery and geothermal systems. Previous studies analysed these phenomena in separate contexts. This paper presents a unified description from which previous models result as limiting cases. The analysis includes capillarity, heat conduction, and Kelvin effects. The importance of each term to various processes is examined. Significantly, it is found that the critical heat flux is not constant but increases with decreasing permeability. A threshold permeability is identified below which steady states may not exist. Analogous conclusions are reached regarding liquid-dominated geothermal systems. 24 refs., 15 figs.

The onset of viscous instability and the subsequent fingering have been mostly studied under conditions of a sharp mobility contrast. In stable miscible flows in porous media, however, mobility gradients of variable extent can develop due to hydrodynamic dispersion. Graded mobility and density profiles will affect instabilities, either by mitigating (in the case of a monotonic variation) or by enhancing (nonmonotonic variation) the rates of growth. The mitigating effect was exploited by Claridge and Gorrel and Homsy for the optimal design of graded mobility banks. Hickernell and Yortsos showed that in the linear stability limit, the growth rate is controlled by the maximum of the logarithmic derivative of the mobility profile. We present an experimental study of miscible displacements in porous media to study effects of mobility gradients in viscous instability. We take advantage of molecular diffusion in miscible displacements to create a mobility gradient zone and subsequently initiate the instability b...

Capillary effects can be important in immiscible flows in heterogeneous media, particularly at low capillary numbers (Ca). We present experiments and simulations of slow drainage in 3-D porous media, either homogeneous and in the presence of buoyancy or heterogeneous and in its absence. An acoustic technique allows for an accurate study of the 3-D fronts and the cross-over region. Our results suggest that both cases can be described by invasion percolation in a gradient. Both front tails scale with the corresponding Bond numbers as σft≈B−47 in agreement with the theory. An analogous scaling for viscous effects is also given. The noise of these fronts are found correlated in the form of a fractional Brownian motion (fBm) of a Hurst exponent H≈.5. At higher Ca, experiments performed in 3-D porous media with sharp changes in permeability, exhibit a saturation profile response closely linked to the permeability variations. This viscous response to heterogeneity provides an opportunity to investigate and determine correlated (even at all scales, i.e. fBm), permeability fields.

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.

Summary We present a formulation for a fractal fracture network embedded into a Euclidean matrix. Single-phase flow in the fractal object is described by an appropriate modification of the diffusivity equation. The system's pressure-transient response is then analyzed in the absence of matrix participation and when both the fracture network and the matrix participate. The results obtained extend previous pressure-transient and well-testing methods to reservoirs of arbitrary (fractal) dimensions and provide a unified description for both single- and dual-porosity systems. Results may be used to identify and model naturally fractured reservoirs with multiple scales and fractal properties.

Drying in porous structures is simulated with a 2-D pore network model that accounts for various processes at the pore-scale (mass transfer by advection and diffusion in the gas phase, viscous flow in liquid and gas phases and capillary effects at the gas-liquid interface). We further study the effect of capillarity-driven viscous flow through macroscopic liquid films. It is shown that film flow is a major transport mechanism in drying of porous media, its effect being dominant when capillarity controls the process, which is the case in typical applications.

In this paper we present numerical results obtained with a pore-network model for the drying of porous media that accounts for various processes at the pore scale. These include mass transfer by advection and diffusion in the gas phase, viscous flow in the liquid and gas phases and capillary effects at the liquid-gas interface. We extend our work by studying the effect of capillarity-induced flow in macroscopic liquid films that form at the pore walls as the liquid-gas interface recedes. A mathematical model that accounts for the effect of films on the drying rates and phase distribution patterns is presented. It is shown that film flow is a major transport mechanism in the drying of porous materials, its effect being dominant when capillarity controls the process, which is the case in typical applications.

The recovery of volatile oils from the matrix of fractured porous media can be significantly aided through gas injection. Within certain assumptions, this process operates much like the isothermal drying of porous media. For example, the recovery from the matrix block is controlled by mechanisms involving capillary, viscous and buoyancy forces in the liquid and the liquid-gas interfaces, and mass diffusion and convection in the gas phase. The receding of the liquid phase is followed by the formation of macroscopic liquid films along the pore walls. Films provide hydraulic conductivity between macroscopically isolated liquid clusters and help transfer liquid over significant distances in the porous matrix. In this work, we study the effect of such films. We propose a mathematical model for the capillarity-driven flow through wetting films in the context of evaporation and drying. A pore-network simulator that accounts for flow through liquid films, viscous flow in the bulk liquid phase of the clusters and mass transfer by diffusion in the gas phase is used. We study the effect of liquid films on phase distribution patterns and rates of recovery. We find that under strong capillarity conditions, the films span across the whole block, enhancing significantly liquid flow from distant clusters and improving recovery rates. An upscaling of the pore-network findings to macroscopic quantities is then presented. We comment on the extension of the method to binary mixtures and gravity effects. Our results show that flow through liquid films is a major transport mechanism in such processes.

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.

Change-of-type behavior from hyperbolic to elliptic is common to quasilinear hyperbolic systems. This issue is addressed here for the particular case of miscible flow of three fluids between two parallel plates. Change of type occurs at the leading edge of the displacement front and reflects the failing of the equilibrium assumption, necessary for the quasilinear hyperbolic formalism, at the front. To cross the elliptic region requires the solution of the full, higher-dimensionality problem, obtained here using lattice gas simulations. For the specific example, it is found that the system self-selects a front structure independent of injection conditions.

A number of problems in porous media involve the simultaneous flow of two immiscible fluids. As a result of the capillarity of the porous medium, however, the flow occurs in an intermittent way, in which phases may get trapped, immobilized, or conversely mobilized due to the flow of the other phase. This process of ganglia motion has been described in detailed past works by various authors using elaborate pore-network simulations. In this paper, we apply a different approach, in which all the hydrodynamic information of the flowing fluid and the effect of the various ganglia enters at the various ganglia interfaces using a boundary integral method. Methods of this type have been extensively used in potential theory, but have not been applied in the context of flow and mobilization of immiscible fluids in porous media. Their advantage is that the hydrodynamic problem is mapped at the ganglia interfaces and allows a fast and convenient way to describe the forces on the disconnected phase, the possible mobilization, stranding, coalescence, crowding effects, etc. We present the method development by deriving the integral equation to be solved on the fluid interfaces. We then use a numerical method for its solution and examine a variety of problems including: the condition for the mobilization of a stranded ganglion as a function of the capillary number and the ganglion shape; its subsequent motion in the random capillary environment and the rate of stranding or coalescence; effects of gravity; and the interaction between neighboring ganglia. The latter does not always enter as a crowding effect, as intuitively expected. The method finds applications to problems where the flowing phase is disconnected, as for example in foamy oil flow or the countercurrent flow of steam-water in the presence of gravity.

The sustained propagation of combustion fronts in porous media is a necessary condition for the success of an in situ combustion project for oil recovery. Compared to other recovery methods, in situ combustion involves the added complexity of exothermic reactions and temperature-dependent chemical kinetics. In the presence of heat losses, the possibility of ignition and extinction (quenching) exists. In this report, we address the properties of combustion fronts propagating at a constant velocity in the presence of heat losses.

We develop an effective continuum model to describe the heterogeneous nucleation and subsequent growth of a gas phase from a supersaturated, slightly compressible binary liquid in a porous medium, driven by solute diffusion. The evolution of the gas results from the withdrawal of the liquid at a constant rate. The model addresses two stages before the onset of bulk gas flow, nucleation and gas phase growth. We assume negligible gradients due to gravity or viscous forces, thus the critical gas saturation, which signals the onset of bulk gas flow, is only a function of the nucleation fraction. We show that the important quantities characterizing the process, such as the fraction of pores that host activated sites, the deviation from thermodynamic equilibrium, the maximum supersaturation in the system and the critical gas saturation depend crucially on the nucleation characteristics of the medium.