Stability of frontal displacement of fluids in a porous medium in the presence of interphase mass transfer and phase transitions

To preserve the stability of the front relative to small perturbations when one fluid is displaced by another the pressure gradient must decrease on crossing the front in the direction of displacement. Initially, this criterion was established for the piston displacement of fluids [1, 2], and later in the case of two-phase flow of immiscible fluids in porous media for the displacement front corresponding to the saturation jump in the Buckley—Leverett problem [3, 4]. Below it is shown that the same stability criterion remains valid for flows in porous media accompanied by interphase mass transfer and phase transitions [5, 6]. Processes of these kinds are encountered in displacing oil from beds using active physicochemical or thermal methods [7] and usually reduce to pumping into the bed a slug (finite quantity) of reagent after which a displacing agent (water or gas) is forced in. The slug volume may be fairly small, especially when expensive reagents are employed, and, accordingly, in these cases the question of the stability of displacement is one of primary importance. These active processes are characterized by the formation in the displacement zone of multiwave structures which, in the large-scale approximation (i.e., with capillary, diffusion and nonequilibrium effects neglected), correspond to discontinuous distributions of the phase saturations and component concentrations [5–10]. It is shown that the stability condition for a plane front, corresponding to a certain jump, does not depend on the type of jump [11, 12] and for a constant total flow is determined, as in simpler cases, by the relation between the total phase mobilities at the jump. An increase in total flow in the direction of displacement is destabilizing, while a decrease has a stabilizing influence on the stability of the front. Other trends in the investigation of the stability of flows in porous media are reviewed in [13].Translated fron Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 98–103, March–April, 1986.     

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.     

Two-phase jet flows in porous media

An analytical model describing the development of the filtration instability of the displacement front of fluids with different viscosities in a porous medium with account for capillary forces is proposed. A set of laboratory experiments on viscous fluid displacement from a porous medium is carried out. To describe the observable flows the model deals with the characteristic profile of the mean water saturation along the flow rather than with the curves of relative phase permeabilities of the fluids. The analytical model developed well describes the results of the laboratory modeling and the data of an actual oil field operation.     

Barothermal effect in the displacement of oil from a porous medium

The formation of the temperature field due to the barothermal effect when oil is displaced from a porous medium by water is investigated in the piston displacement and two-phase flow approximations. The approach of the displacement front to the outlet from the porous medium leads to a sharp increase in temperature and the temperature anomalies are observed to depend on the saturation.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 104–109, May–June, 1992.     

Effect of the capillary forces on the moisture saturation distribution during the thawing of a frozen soil

Heterogeneous water-air mixture flows in the presence of capillary forces are investigated. It is shown that for moderate pressure and temperature gradients the distribution of the water saturation function can be determined from a nonlinear differential equation with a coefficient dependent on the porous medium parameters, the water viscosity, and the capillary pressure. The water-air mixture flow behind the ice melting front is considered.     

Pore-Scale Investigation of Two-Phase Flows in Three-Dimensional Digital Models of Natural Sandstones

The results of numerical simulation of the processes of two-phase flow through a porous medium in three-dimensional digital models of the porous space of three natural sandstone samples are given. The calculations are carried out using the lattice Boltzmann equations and the digital field gradient model over a wide range of the capillary numbers and the viscosity ratios of injected and displaced fluids. The conditions of flow through a porous medium with capillary fingering, viscous fingering and with stable displacement front are revealed.     

Instability of a liquid–vapor phase transition front in inhomogeneous wettable porous media

The stability of vertical flows through a horizontally extended two-dimensional region of a porous medium is considered in the case of presence of a phase transition front. It is shown that the plane steady-state phase transition front may have several steady-state positions in the wettable porous medium and the necessary condition of their existence is obtained. The spectral stability of the plane phase transition interface is investigated. It is found that in the presence of capillary forces exerted on the phase transition front in the wettable medium the plane front can be destabilized on the mode with both infinite and zero wavenumbers (short- and long-wave instabilities); the short-wave instability can then exist even in the case of the sole steady-state position of the front.     

Permanent Fronts in Two-Phase Flows in a Porous Medium

A family of exact solutions for a model of a one-dimensional horizontal flow of two immiscible, incompressible fluids in a porous medium, including the effects of capillary pressure, is obtained analytically by solving the governing singular parabolic nonlinear diffusion equation. Each solution has the form of a permanent front propagating with a constant velocity. It is shown that, for every propagation velocity, there exists a set of permanent fronts all of which are moving with this velocity in an inflowing wetting–outflowing non-wetting flow configuration. Global bifurcations of this set, with the front velocity as a bifurcation parameter, are investigated analytically and numerically in detail in the case when the permeabilities and the capillary pressure are linear functions of the wetting phase saturation. Main results for the nonlinear Brooks–Corey model are also presented. In both models three global bifurcations occur. By using a geometric dynamical system approach, the nonlinear stability of the permanent fronts is established analytically. Based on the permanent front solutions, an interpretation of the dynamics of an arbitrary front of finite extent in the model is given as follows. The instantaneous upstream (downstream) velocity of an arbitrary non-quasistationary front is equal to the velocity of a permanent front whose shape coincides up to two leading orders with the instantaneous shape of the non-quasistationary front at the upstream (respectively, downstream) location. The upstream and downstream locations of the front undergo instantaneous translations governed by modified nonsingular hyperbolic equations. The portion of the front in between these locations undergoes a diffusive redistribution governed by a nonsingular nonlinear parabolic diffusion equation. We have proposed a numerical approach based on a parabolic–hyperbolic domain decomposition for computing non-quasistationary fronts.     

Percolation model of two-phase flow through porous media

A physical model of the process of two-phase flow of immiscible fluids through a porous medium is developed and used to make an analytical calculation of the dependence of the relative phase permeabilities on the saturation of the medium by one of the phases. The theory is compared qualitatively with experiment for a model capillary radius frequency function and quantitatively with numerical calculations made on a computer. In both cases good agreement is obtained. The pressure dependences of the phase permeabilities are analyzed. The question of residual saturation with the wetting fluid after completion of the displacement process is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 88–95, January–February, 1987.     

Influence of the structure of a porous medium on the residual gas saturation in the case of capillary displacement by a liquid

A model of a porous medium consisting of randomly branching conical pores is used to investigate the quasistatic displacement of gas by a wetting liquid without application of an external pressure. Allowance is made for the circumstance that in the capillary process all the pores have at least one-sided permeability for the liquid phase. An expression is obtained that relates the residual gas saturation to the parameters which characterize the structure of the pores and the wetting properties of the system. Two new characteristics of the pore space are introduced — the branching parameter and the opening angle of the pores — and the influence of these parameters on the residual saturation is investigated. It is shown that for individual classes of natural media the residual gas saturation depends only on the porosity and the contact angle of wetting.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 128–133, September–October, 1981.     

A Parametric Model for Predicting Relative Permeability-Saturation-Capillary Pressure Relationships of Oil–Water Systems in Porous Media with Mixed Wettability

A parametric two-phase, oil–water relative permeability/capillary pressure model for petroleum engineering and environmental applications is developed for porous media in which the smaller pores are strongly water-wet and the larger pores tend to be intermediate- or oil-wet. A saturation index, which can vary from 0 to 1, is used to distinguish those pores that are strongly water-wet from those that have intermediate- or oil-wet characteristics. The capillary pressure submodel is capable of describing main-drainage and hysteretic saturation-path saturations for positive and negative oil–water capillary pressures. At high oil–water capillary pressures, an asymptote is approached as the water saturation approaches the residual water saturation. At low oil–water capillary pressures (i.e. negative), another asymptote is approached as the oil saturation approaches the residual oil saturation. Hysteresis in capillary pressure relations, including water entrapment, is modeled. Relative permeabilities are predicted using parameters that describe main-drainage capillary pressure relations and accounting for how water and oil are distributed throughout the pore spaces of a porous medium with mixed wettability. The capillary pressure submodel is tested against published experimental data, and an example of how to use the relative permeability/capillary pressure model for a hypothetical saturation-path scenario involving several imbibition and drainage paths is given. Features of the model are also explained. Results suggest that the proposed model is capable of predicting relative permeability/capillary pressure characteristics of porous media mixed wettability.     

Flow in a fractured medium with fractal fracture geometry

A model of a fractured porous medium in which the fracture system forms a fractal with Hausdorff-Bezikovich dimension d is proposed. The fractal is immersed in a saturated porous medium with the dimension D (D d, D=2, 3). The rock skeleton is assumed to be nondeformable. The system of flow equations is written out for cylindrically (D=2) and spherically (D=3) symmetrical flows. When D=d the model reduces to the well-known Barenblatt-Zheltov model. Certain particular solutions, which make it possible to determine the phenomenological parameters of the model experimentally, are obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 66–70, September–October, 1990.     

Boiling in Porous Media: Model and Simulations

We present a modelization of the heat and mass transfers within a porous medium, which takes into account phase transitions. Classical equations are derived for the mass conservation equation, whereas the equation of energy relies on an entropy balance adapted to the case of a rigid porous medium. The approximation of the solution is obtained using a finite volume scheme coupled with the management of phase transitions. This model is shown to apply in the case of an experiment of heat generation in a porous medium. The vapor phase appearance is well reproduced by the simulations, and the size of the two-phase region is correctly predicted. A result of this study is the evidence of the discrepancy between the air – water capillary and relative permeability curves and water – water vapor ones.     

Percolation model of two-phase equilibrium flow in a porous medium with microheterogeneous wettability

The percolation model of two-phase flow described in [1, 2] is used as a basis for examining the problem of the behavior of the characteristics of two-phase equilibrium flow in a porous medium when the capillaries have a radius distribution and differ with respect to the wettability properties of their surfaces. Analytic expressions describing the dependence of the relative phase permeability coefficients on the saturation of the medium by the displacing phase and the microinhomogeneous wettability parameters are obtained. A qualitative comparison shows the theoretical results to be consistent with the data of a direct numerical computer calculation of a grid model [3]. The effect of the microinhomogeneity parameters and the form of the capillary radius distribution function on the phase permeabilities is analyzed within the framework of the approach developed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 86–93, September–October, 1989.     

Investigation of hydrodynamic instability of CO<Subscript>2</Subscript> injection into an aquifer

The two-dimensional problem of supercritical carbon dioxide injection into an aquifer is solved. Shocks and rarefaction waves propagating in a sequence from an injection well into the formation are described within the framework of a complete nonisothermal model of flows in a porous medium. In the approximation of isothermal immiscible water and carbon dioxide flow the hydrodynamic stability of the leading displacement front is investigated for various reservoir pressures and temperatures. The parameters of unstable fronts are determined using a sufficient instability condition formulated in analytic form. The approximate analytic results are supported by the direct numerical simulation of CO₂ injection using the complete model in which thermal effects and phase transitions are taken into account.     

Nonstationary model of immiscible fluid flow through porous media

Unsteady two-phase flow through a microinhomogeneous porous medium is considered. A forest growth model — a percolation model that enables nonequilibrium effects to be taken into account — is proposed for describing the dynamics of the process. In the context of the plane problem expressions are obtained for determining the saturation and the characteristic dimensions of the stagnation zones of trapped phase behind the displacement front.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 73–80, November–December, 1993.     

Micromechanics of residual saturation formation during immiscible displacement in a porous medium

The process of displacement of a nonwetting fluid has been studied experimentally on a transparent model of a porous medium for various percolation velocities in the stable front regime, when the viscosity of the displacing fluid is greater than that of the fluid displaced. The flow structures in the final displacement regime, when the nonwetting phase is distributed in the pore space in the form of individual drops or ganglia, have been visually investigated. Imbibition is numerically modeled on a two-dimensional network model with allowance for various microdisplacement mechanisms. The effect of the initial displacing phase saturation on the magnitude and structure of the residual displaced fluid saturation is demonstrated. The fractal dimensionality of the displacement boundary is measured.Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 116–121, May–June, 1994.     

Effect of the percolation heat transfer component on the temperature field and thawing dynamics of soils

The effect of the convective heat transfer component on the temperature field and thawing front dynamics of soils is investigated for high fluid percolation velocities in the thawed zone. The steady state interchange and approximate self-similarity methods are used to obtain upper and lower bounds of the solution of the Stefan problem with a convective heat transfer component in a porous medium. From the results of the calculations conclusions are drawn concerning the accuracy and limits of applicability of the solutions obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 166–171, March–April, 1987.     

Dynamics of the Water–Oil Front for Two-Phase, Immiscible Flow in Heterogeneous Porous Media. 2 – Isotropic Media

We study the evolution of the water–oil front for two-phase, immiscible flow in heterogeneous porous media. Our analysis takes into account the viscous coupling between the pressure field and the saturation map. Although most of previously published stochastic homogenization approaches for upscaling two-phase flow in heterogeneous porous media neglect this viscous coupling, we show that it plays a crucial role in the dynamics of the front. In particular, when the mobility ratio is favorable, it induces a transverse flux that stabilizes the water–oil front, which follows a stationary behavior, at least in a statistical sense. Calculations are based on a double perturbation expansion of equations at first order: the local velocity fluctuation is defined as the sum of a viscous term related to perturbations of the saturation map, on one hand, plus the perturbation induced by the heterogeneity of the permeability field with a base-state saturation map, on the other hand. In this companion paper, we focus on flows in isotropic media. Our results predict the dynamics of the water–oil front for favorable mobility ratios. We show that the statistics of the front reach a stationary limit, as a function of the geostatistics of the permeability field and of the mobility ratio evaluated across the front. Results of numerical experiments and Monte-Carlo analysis confirm our predictions.     

Two-dimensional flow in porous strata with conductivity varying discontinuously along second-order curves

Solutions of boundary-value problems of two-dimensional flow in a porous medium are obtained on the basis of the theory of axisymmetric generalized analytic functions [1,2] and conversion formulas [3] for a broad class of strata whose conductivity changes abruptly along second-order curves. The singular points of these functions model arbitrary two-dimensional flows. In space the solutions describe the axisymmetric flow in porous media whose homogeneity interfaces are second-order surfaces of revolution. The solutions obtained are applied to new problems associated with environmental protection and the nonpolluting operation of water intakes under complex geological conditions.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.1, pp. 120–128, January–February, 1993.