A Steady-State Upscaling Approach for Immiscible Two-Phase Flow

The paper presents a model for computing rate-dependent effective capillary pressure and relative permeabilities for two-phase flow, in 2 and 3 space-dimensions. The model is based on solving the equations for immiscible two-phase flow at steady-state, accounting for viscous and capillary forces, at a given external pressure drop. The computational performance of the steady-state model and its accuracy is evaluated through comparison with a commercial simulator ECLIPSE. The properties of the rate-dependent effective relative permeabilities are studied by way of computations using the developed steady-state model. Examples presented show the dependence of the effective relative permeabilities and capillary pressures, which incorporate the effects of fine scale wettability heterogeneity, on the external pressure drop, and thereby on the dimensionless macro-scale capillary number. The effective relative permeabilities converge towards the viscous limit functions as the capillary number tends to infinity. Special cases, when the effective relative permeabilities are rate-invariant, are also studied. The applicability of the steady-state upscaling algorithm in dynamic displacement situations is validated by comparing fine-gridded simulations in heterogeneous reservoirs against their homogenized counterparts. It is concluded that the steady-state upscaling method is able to accurately predict the dynamic behavior of a heterogeneous reservoir, including small scale heterogeneities in both the absolute permeability and the wettability.     

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.     

New criteria for the validity of steady-state upscaling

We present new dimensionless criteria to determine the validity of steady-state upscaling techniques in the limit that capillary (capillary limit, CL) or viscous (viscous limit, VL) forces dominate flow in a simple, layered geological system. We begin by identifying a suit of dimensionless groups which characterize the balance of capillary and viscous forces, then use numerical experiments to determine empirically the threshold values of these dimensionless groups for which each upscaling method is valid. Our criteria capture the effects of capillary trapping and are valid regardless of fluid mobility, wettability, or end-point saturation. They can be used to determine the reservoir conditions for which each upscaling method is valid. Previous studies have used a single dimensionless number to characterize the balance of forces, so have failed to properly identify the range of validity. We apply our new criteria to explain cases when the upscaling methods have been observed to do unexpectedly well or poorly. We also demonstrate that the CL method can be valid for a wider range of reservoir conditions than previously thought, particularly in mixed- and oil-wet systems where capillary trapping is minimal.     

Dynamics of the Water–Oil Front for Two-Phase, Immiscible Flow in Heterogeneous Porous Media. 1 – Stratified 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 on the dynamics of the front. In particular, when the mobility ratio is favorable, the viscous coupling 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 first paper, we focus on flows in stratified reservoirs, with stratification parallel to the mean flow. Our results allow to predict the evolution of large Fourier mode of the front, and the emergence of a stationary front, for favorable mobility ratios. Numerical experiments confirm our predictions. Our approach is applied to downscaling. Extension of our theory to isotropic media is presented in the companion paper.     

A New Mathematical Model for Force Gravity Drainage in Fractured Porous Media

In force gas/oil gravity drainage process in fractured porous media, gas is flowing in both matrix and fractures leading to produce a finite gas pressure gradient. Consequently, viscous force plays an important role for displacing matrix oil toward fractures in addition to gravity force that is required to be modeled appropriately. A new analytical model for estimation of steady state oil saturation distribution with assumption of fixed gas pressure gradient throughout the matrix is presented. Moreover, based on some results of this analytical model a different numerical formulation is developed to predict the performance of oil production process. Comparison of the results obtained from this numerical model with the results of a conventional simulator demonstrates that the newly developed model can be applied with satisfactory accuracy. Numerical simulations show that the viscous displacement in fractured porous media can reduce the capillary threshold height, and thus it suggests the force gravity drainage as a favorable production mechanism when the matrix length is close to the threshold height.     

Surface tension and buoyancy-driven flow in a non-isothermal liquid bridge

The Navier–Stokes–Boussinesq equations governing the transport of momentum, mass and heat in a non-isothermal liquid bridge with a temperature-dependent surface tension are solved using a vorticity-stream-function formulation together with a non-orthogonal co-ordinate transformation. The equations are discretized using a pseudo-unsteady semi-implicit finite difference scheme and are solved by the ADI method. A Picard-type iteration is adopted which consists of inner and outer iterative processes. The outer iteration is used to update the shape of the free surface. Two schemes have been used for the outer iteration; both use the force balance normal to the free surface as the distinguished boundary condition. The first scheme involves successive approximation by the direct solution of the distinguished boundary condition. The second scheme uses the artificial force imbalance between the fluid pressure, viscous and capillary forces at the free surface which arises when the boundary condition for force balance normal to the surface is not satisfied. This artificial imbalance is then used to change the surface shape until the distinguished boundary condition is satisfied. These schemes have been used to examine a variety of model liquid bridge situations including purely thermocapillary-driven flow situations and mixed thermocapillary- and bouyancy-driven flow.     

Asymptotic investigation of two-phase couette flow

We consider plane and cylindrical Couette flow for a two-phase medium. The motion of the medium is described by the equations obtained in [1]. Collisions between the particles are disregarded, and their motion, in addition to the inertial forces, is determined by the pressure gradient of the carrying phase and the forces of viscous interaction between the carrying phase and the particles. We obtain simple asymptotic solutions of the indicated problems for small and large values of the dimensionless determining parameters. In a number of cases the solution has the nature of a boundary layer on solid walls.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 67–73, July–August, 1978.     

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.     

A Non-isothermal Pore Network Drying Model with Gravity Effect

The concept of immiscible displacement as an invasion percolation (IP) process driven by heat and mass transfer is used in a pore network model for convective drying of capillary porous media. The coupling between heat and mass transfer occurs at the liquid–gas interface through temperature-dependent equilibrium vapor pressure and surface tension as well as the phase change enthalpy (in evaporation and condensation). The interfacial effects due to capillary forces and gravity are combined in an invasion potential; viscous forces are neglected. Simulation results show stabilized invasion patterns and finite drying front width by the influence of gravity.     

Elliptic Regions and Stable Solutions for Three-Phase flow in Porous Media

In the limit of zero capillary pressure, solutions to the equations governing three-phase flow, obtained using common empirical relative permeability models, exhibit complex wavespeeds for certain saturation values (elliptic regions) that result in unstable and non-unique solutions. We analyze a simple but physically realizable pore-scale model: a bundle of cylindrical capillary tubes, to investigate whether the presence of these elliptic regions is an artifact of using unphysical relative permeabilities. Without gravity, the model does not yield elliptic regions unless the most non-wetting phase is the most viscous and the most wetting phase is the least viscous. With gravity, the model yields elliptic regions for any combination of viscosities, and these regions occupy a significant fraction of the saturation space. We then present converged, stable numerical solutions for one-dimensional flow, which include capillary pressure. These demonstrate that, even when capillary forces are small relative to viscous forces, they have a significant effect on solutions which cross or enter the elliptic region. We conclude that elliptic regions can occur for a physically realizable model of a porous medium, and that capillary pressure should be included explicitly in three-phase numerical simulators to obtain stable, physically meaningful solutions which reproduce the correct sequence of saturation changes.     

A variationally consistent finite element approach to the two‐fluid internal contact‐line problem

A new method for simulating incompressible viscous fluid flow involving moving internal contact lines is presented. The steady state interface shape is determined by a variationally consistent formulation of the surface tension contribution to the equations of motion adapted to the case of internal contact lines through the application of a global force balance compatibility condition that consistently removes the pressure indeterminacy. The Crouzeix–Raviart element is chosen to capture the pressure discontinuity at the two‐fluid interface. The resulting discrete equations are solved by an iterative procedure which displays fast convergence characteristics for small capillary numbers. Numerical results for the case of the steady movement of a fluid meniscus in a two‐dimensional channel are presented. Copyright © 2000 John Wiley & Sons, Ltd.     

AN UPPER BOUND ON RANDOM BUFFETING FORCES CAUSED BY TWO-PHASE FLOWS ACROSS TUBES

This paper investigates the random buffeting excitation forces that apply to tubes in two-phase cross-flows. The spectral magnitude of these forces is studied with the help of a database that includes most of the available experimental data on direct or indirect force measurements. Different fluid mixtures, namely steam–water, air–water and various types of Freon, as well as different thermohydraulic or geometrical situations are included in the database. Using a formalism similar in principle to that used successfully in the modelling of buffeting in single-phase flows, the scaling of the data is undertaken. While dynamic pressure, viscosity or surface tension are found not to be relevant parameters, gravity forces allow us to define appropriate dimensionless spectra for all cases. The meaning of these parameters as well as the effects of flow regimes or fluid mixtures are discussed. Finally, an upper bound on the magnitude of these forces, which is of practical applicability, is proposed.     

Multi-Stage Upscaling: Selection of Suitable Methods

Reservoirs are often composed of an assortment of rock types giving rise to permeability heterogeneities at a variety of length-scales. To predict fluid flow at the full-field scale, it is necessary to be aware of these different types of heterogeneity, to recognise which are likely to have important effects on fluid flow, and to capture them by upscaling. In fact, we may require a series of stages of upscaling to go from small-scales (mm or cm) to a full-field model. When there are two (or more) phases present, we also need to know how these heterogeneities interact with fluid forces (capillary, viscous and gravity). We discuss how these effects may be taken into account by upscaling. This study focusses on the effects of steady-state upscaling for viscous-dominated floods and tests carried out on a range of 2D models are described. Upscaling errors are shown to be reduced slightly by the increase in numerical dispersion at the coarse scale. We select a combination of three different upscaling methods, and apply this approach to a model of a North Sea oil reservoir in a deep marine environment. Six different genetic units (rock types) were identified, including channel sandstone and inter-bedded sandstone and mudstone. These units were modelled using different approaches, depending on the nature of the heterogeneities. Our results show that the importance of small-scale heterogeneity depends on the large-scale distribution of the rock types. Upscaling may not be worthwhile in sparsely distributed genetic units. However, it is important in the dominant rock type, especially if there is good connectivity through the unit between the injector wells (or aquifer) and the producer wells.This revised version was published online in May 2005. In the previous version one of the authors name was missing.     

On an attempt to simplify the Quartapelle–Napolitano approach to computation of hydrodynamic forces in open flows

In this paper we are interested in the Quartapelle–Napolitano approach to calculation of forces in viscous incompressible flows in exterior domains. We study the possibility of deriving a simpler formulation of this approach which might lead to a more convenient expression for the hydrodynamic force, but conclude that such a simplification is, within the family of approaches considered, impossible. This shows that the original Quartapelle–Napolitano formula is in fact “optimal” within this class of approaches.     

On the length of annular jets interacting with an ambient medium

Annular jets of immiscible fluids are the subject of intense study. Particularly topical in applications are jets in the shape of a right circular cylinder. The space within annular jets may be reduced or increased by the influence of transverse forces and also by hydrodynamic instability of the jet flow. Twisting of the jet tends to make it close up. In the present paper, a study is made of ways of obtaining annular jets with nearly cylindrical shape and the greatest cavity length possible, allowance being made for gravity, capillary pressure surface forces, a pressure difference between the two sides of the phase interface, and the interaction with the ambient medium. A study is made of the influence of the velocity of the fluid and the medium in the initial section on the shape of the joint steady axisymmetric flow of immiscible viscous phases, including the shape of the middle surface of the annular jet.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 3–11, November–December, 1983.     

Lie symmetry solutions for two-phase mass flows

We apply Lie symmetry method to a set of non-linear partial differential equations, which describes a two-phase rapid gravity mass flow as a mixture of solid particles and viscous fluid down a slope (Pudasaini, J. Geophys. Res. 117 (2012) F03010, 28 pp [1]). In order to systematically explore the mathematical structure and underlying physics of the two-phase mixture flow, we generate several similarity forms in general form and construct self-similar solutions. Our analysis generalizes the results, obtained by applying the Lie symmetry method to relatively simple single-phase pressure-driven gravity mass flows, to the two-phase mass flows that include several dominant driving forces and strong phase-interactions. Analytical and numerical solutions are presented for the symmetry-reduced homogeneous and non-homogeneous systems of equations. Analytical and numerical results show that the new models presented here can adequately describe the dynamics of two-phase debris flows, and produce observable phenomena that are consistent with the physics of the flow. The solutions are strongly dependent on the choice of the symmetry-reduced model, as characterized by the group parameters, and the physical parameters of the flows. These solutions reveal strong non-linear and distinct dynamic evolutions, and phase-interactions between the solid and fluid phases, namely the phase-heights and phase-velocities.     

Relative Permeability Analysis of Tube Bundle Models

The analytical solution for calculating two-phase immiscible flow through a bundle of parallel capillary tubes of uniform diametral probability distribution is developed and employed to calculate the relative permeabilities of both phases. Also, expressions for calculating two-phase flow through bundles of serial tubes (tubes in which the diameter varies along the direction of flow) are obtained and utilized to study relative permeability characteristics using a lognormal tube diameter distribution. The effect of viscosity ratio on conventional relative permeability was investigated and it was found to have a significant effect for both the parallel and serial tube models. General agreement was observed between trends of relative permeability ratios found in this work and those from experimental results of Singhal et al. (1976) using porous media consisting of mixtures of Teflon powder and glass beads. It was concluded that neglecting the difference between the average pressure of the non-wetting phase and the average pressure of the wetting phase (the macro-scale capillary pressure) – a necessary assumption underlying the popular analysis methods of Johnson et al. (1959) and Jones and Roszelle (1978) – was responsible for the disparity in the relative permeability curves for various viscosity ratios. The methods therefore do not account for non-local viscous effects when applied to tube bundle models. It was contended that average pressure differences between two immiscible phases can arise from either capillary interfaces (micro-scale capillary pressures) or due to disparate pressure gradients that are maintained for a flow of two fluids of viscosity ratio that is different from unity.     

Flow and instability of capillary jets interacting with the surrounding medium

The flow of an axially symmetric capillary jet of a viscous incompressible liquid in the space occupied by another liquid is investigated. The problem of stationary flow in the jet and in the surrounding medium under the action of viscosity, capillary forces, and gravity was obtained numerically. The instability problem of this flow to small perturbations in the form of running waves is stated and solved numerically. The values of the dimensionless Reynolds, Weber, and Froude numbers are explained, as well as the effect of the initial velocity profile in the jet, its instability, and subsequent jet decay into drops.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 50–59, November–December, 1978.     

Contribution of capillary forces to the volume of residual oil pillars

Equations for calculating the limiting equilibrium shapes of the residual oil pillars in homogeneous and stratified inhomogeneous reservoirs of varying wettability are written within the framework of a two-phase displacement model with allowance for capillary pressure. Comparative calculations for an individual well in a circular reservoir show that, in general, the effect of capillary pressure on the shape and volume of the pillars is not negligibly small and may reach 10% of the pore volume.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 114–123, January–February, 1991.