The Local Analysis of Changing Force Balances in Immiscible Incompressible Two-Phase Flow

The balance of viscous, capillary and gravity forces strongly affects two-phase flow through porous media and can therefore influence the choice of appropriate methods for numerical simulation and upscaling. A strict separation of the effects of these various forces is not possible due to the nature of the nonlinear coupling between the various terms in the transport equations. However, approximate prediction of this force balance is often made by calculation of dimensionless quantities such as capillary and gravity numbers. We present an improved method for the numerical analysis of simulations which recognises the changing balance of forces – in both space and time – in a given domain. The classical two-phase transport equations for immiscible incompressible flow are expressed in two forms: (i) the convection–diffusion-gravity (CDG) formulation where convection and diffusion represent viscous and capillary effects, respectively, (ii) the oil pressure formulation where the viscous effects are attributed to the product of mobility difference and the oil pressure gradient. Each formulation provides a different perspective on the balance of forces although the two forms are equivalent. By discretising the different formulations, the effect of each force on the rate of change of water saturation can be calculated for each cell, and this can be analysed visually using a ternary force diagram. The methods have been applied to several simple models, and the results are presented here. When model parameters are varied to determine sensitivity of the estimators for the balance of forces, the CDG formulation agrees qualitatively with what is expected from physical intuition. However, the oil pressure formulation is dominated by the steady-state solution and cannot be used accurately. In addition to providing a physical method of visualising the relative magnitudes of the viscous, gravity and capillary forces, the local force balance may be used to guide our choice of upscaling method.     

Compositional gravity drainage 1. Equilibrium solutions and controlling Bond numbers for a two-phase,three-component system

The amount of wetting phase that is recovered by gravity drainage is determined by an interplay of gravitational and capillary forces. The relative importance of those forces is often expressed in terms of a Bond number. For compositional gravity drainage, where the initial and displacing fluids are not in chemical equilibrium, there is no single Bond number, as phases that form during a displacement will be associated with a different interfacial tension and density for each equilibrium tie line encountered as the compositions change during flow. We study vertical compositional displacements to determine how the Bond numbers of the initial and displacing fluids control the ultimate recovery. We find analytical solutions to the capillary/gravity equilibrium for a simplified model three-component, two-phase system. The equilibrium phase composition versus distance profiles are different than those predicted from standard viscous dominated displacements. We calculate the recovery as a function of the Bond numbers of the initial and displacing phases, and the degree of diffusion for this simple system. We discuss the important role of molecular diffusion in the ultimate recovery for condensing displacements. Finally, we find that the simple numerical average of the Bond numbers provides a reasonable estimate of an effective Bond number for calculating the retained wetting phase for many compositional gravity drainages.     

Three-Phase Flow Modelling Using Pore-Scale Capillary Pressures and Relative Permeabilities for Mixed-Wet Media at the Continuum-Scale

When regions of three-phase flow arise in an oil reservoir, each of the flow parameters, i.e. capillary pressures and relative permeabilities, are generally functions of two phase saturations and depend on the wettability state. The idea of this work is to generate consistent pore-scale based three-phase capillary pressures and relative permeabilities. These are then used as input to a 1-D continuum core- or reservoir-scale simulator. The pore-scale model comprises a bundle of cylindrical capillary tubes, which has a distribution of radii and a prescribed wettability state. Contrary to a full pore-network model, the bundle model allows us to obtain the flow functions for the saturations produced at the continuum-scale iteratively. Hence, the complex dependencies of relative permeability and capillary pressure on saturation are directly taken care of. Simulations of gas injection are performed for different initial water and oil saturations, with and without capillary pressures, to demonstrate how the wettability state, incorporated in the pore-scale based flow functions, affects the continuum-scale displacement patterns and saturation profiles. In general, wettability has a major impact on the displacements, even when capillary pressure is suppressed. Moreover, displacement paths produced at the pore-scale and at the continuum-scale models are similar, but they never completely coincide.     

An Analytical Solution for Capillary Gravity Drainage with Dominant Viscous Forces

The development of the capillary fringe during gravity drainage has a significant influence on saturation and pressure distributions in porous formations (Sarkarfarshi et al. in Int J Greenh Gas Control 23:61–71, 2014). This paper introduces an analytical solution for gravity drainage in an axisymmetric geometry with significant capillary pressure. The drainage process results from the injection of a lighter and less viscous injectant into a porous medium saturated with a heavier and more viscous pore fluid. If the viscous force dominates the capillary and the buoyancy forces, then the flow regime is approximated by differential equations and the admissible solution comprises a front shock wave and a trailing simple wave. In contrast to existing analytical solutions for capillary gravity drainage problems (e.g., Nordbotten and Dahle in 47(2) 2011; Golding et al. in J Fluid Mech 678:248–270 2011), this solution targets the saturation distribution during injection at an earlier point in time. Another contribution of this analytical solution is the incorporation of a completely drained flow regime close to the injection well. The analytical solution demonstrates the strong dependency of the saturation distribution upon relative permeability functions, gas entry capillary pressure, and residual saturation. The analytical results are compared to results from a commercial reservoir engineering software package (\(\hbox {CMG } \hbox {STARS}^{\mathrm{TM}}\)).     

A Model of Buoyancy-Driven Two-Phase Countercurrent Fluid Flow

We seek simple analytical solutions in a model of gas flow driven by a combination of buoyancy, viscous, and capillary forces. Traveling-wave solutions describe propagation of the top and bottom of the gas plume. The top of the plume has low gas saturation, but propagates much faster than the bottom. The theoretical maximum of the velocity of propagation of the top of the plume provides a simple conservative estimate of the time until plume evolution will dramatically slow down. A sequence of rarefaction and traveling-wave solutions characterizes the transition zones between the top and bottom stable regions. The analytical results are applied to studying carbon dioxide flow caused by leaks from deep geological formations used for CO₂ storage. The results are also applicable for modeling flow of natural gas leaking from seasonal gas storage, or for modeling of secondary hydrocarbon migration.     

Relative Permeabilities for Strictly Hyperbolic Models of Three-Phase Flow in Porous Media

Traditional mathematical models of multiphase flow in porous media use a straightforward extension of Darcys equation. The key element of these models is the appropriate formulation of the relative permeability functions. It is well known that for one-dimensional flow of three immiscible incompressible fluids, when capillarity is neglected, most relative permeability models used today give rise to regions in the saturation space with elliptic behavior (the so-called elliptic regions). We believe that this behavior is not physical, but rather the result of an incomplete mathematical model. In this paper we identify necessary conditions that must be satisfied by the relative permeability functions, so that the system of equations describing three-phase flow is strictly hyperbolic everywhere in the saturation triangle. These conditions seem to be in good agreement with pore-scale physics and experimental data.     

An Analysis of Three-Phase Pore Occupancies and Relative Permeabilities in Porous Media with Variable Wettability

In three-phase flow, the macroscopic constitutive relations of capillary pressure and relative permeability as functions of saturation depend in a complex manner on the underlying pore occupancies. These three-phase pore occupancies depend in turn on the interfacial tensions, the pore sizes and the degree of wettability of the pores, as characterised by the cosines of the oil–water contact angles. In this work, a quasi-probabilistic approach is developed to determine three-phase pore occupancies in media where the degree of wettability varies from pore to pore. Given a set of fluid and rock properties, a simple but novel graphical representation is given of the sizes and oil–water contact angles underlying three-phase occupancies for every allowed combination of capillary pressures. The actual phase occupancies are then computed using the contact angle probability density function. Since a completely accessible porous medium is studied, saturations, capillary pressures, and relative permeabilities are uniquely related to the pore occupancies. In empirical models of three-phase relative permeability it is of central importance whether a phase relative permeability depends only on its own saturation and how this relates to the corresponding two-phase relative permeability (if at all). The new graphical representation of pore sizes and wettabilities clearly distinguishes all three-phase pore occupancies with respect to these saturation-dependencies. Different types of saturation-dependencies may occur, which are shown to appear in ternary saturation diagrams of iso-relative permeability curves as well, thus guiding empirical approaches. However, for many saturation combinations three-phase and two-phase relative permeabilities can not be linked. In view of the latter, the present model has been used to demonstrate an approach for three-phase flow modelling on the basis of the underlying pore-scale processes, in which three-phase relative permeabilities are computed only along the actual flow paths. This process-based approach is used to predict an efficient strategy for oil recovery by simultaneous water-alternating-gas (SWAG) injection.     

Foam Drainage in Porous Media

In this paper we present a simple analysis of liquid drainage in foams confined in porous media. First we derive the equation for the evolution of the liquid saturation using general mass and momentum conservation arguments and phenomenological relations between the transport parameters and liquid saturation. We find an unusual foam drainage equation in which the determinant terms express the competition between the external force field, represented here by the gravity field, and capillary pressure gradient. We present analytical solutions of the drainage equation in three cases: (a) gravity forces are dominant over capillary forces, (b) capillary forces are dominant over gravity forces, and (c) capillary and gravity forces are comparable in order of magnitude.     

Relative permeabilities from two- and three-dimensional pore-scale network modelling

We present a computer study of two-phase flow in a porous medium. The porous medium is represented by an isotropic network of up to 80 000 randomly placed nodes connected by thin tubes. We then simulate two-fluid displacements in this network and are able to demonstrate the effects of viscous and capillary forces. We use the local average flow rates and pressures to calculate effective saturation dependent relative pemeabilities, fractional flows and capillary pressures. Using a radial Buckley-Leverett theory, the mean saturation profile can be inferred from the solution of the fractional flow equation, which is consistent with the computed saturation. We show that the relative permeability may be a function of both viscosity ratio and capillary number.     

Capillary Heterogeneity Trapping and Crossflow in Layered Porous Media

We examine the effect of capillary and viscous forces on the displacement of one fluid by a second, immiscible fluid across and along parallel layers of contrasting porosity, and relative permeability, as well as previously explored contrasts in absolute permeability and capillary pressure. We consider displacements with wetting, intermediate-wetting and non-wetting injected phases. Flow is characterized using six independent dimensionless numbers and a dimensionless storage efficiency, which is numerically equivalent to the recovery efficiency. Results are directly applicable to geologic carbon storage and hydrocarbon production. We predict how the capillary–viscous force balance influences storage efficiency as a function of a small number of key dimensionless parameters, and provide a framework to support mechanistic interpretations of complex field or experimental data, and numerical model predictions, through the use of simple dimensionless models. When flow is directed across layers, we find that capillary heterogeneity traps the non-wetting phase, regardless of whether it is the injected or displaced phase. However, minimal trapping occurs when the injected phase is intermediate-wetting or when high-permeability layers contain a smaller moveable volume of fluid than low-permeability layers. A dimensionless capillary-to-viscous number defined using the layer thickness rather than the more commonly used system length is most relevant to predict capillary heterogeneity trapping. When flow is directed along layers, we show that, regardless of wettability, increasing capillary crossflow reduces the distance between the leading edges of the injected phase in each layer and increases storage efficiency. This may be counter-intuitive when the injected phase is non-wetting. Crossflow has a significant impact on storage efficiency only when high-permeability layers contain a smaller moveable volume of fluid than low-permeability layers. In that case, capillary heterogeneity traps the wetting phase, regardless of whether it is the injected or displaced phase.     

Modeling Non-Darcian Single- and Two-Phase Flow in Transparent Replicas of Rough-Walled Rock Fractures

While it is generally assumed that in the viscous flow regime, the two-phase flow relative permeabilities in fractured and porous media depend uniquely on the phase saturations, several studies have shown that for non-Darcian flows (i.e., where the inertial forces are not negligible compared with the viscous forces), the relative permeabilities not only depend on phase saturations but also on the flow regime. Experimental results on inertial single- and two-phase flows in two transparent replicas of real rough fractures are presented and modeled combining a generalization of the single-phase flow Darcy’s law with the apparent permeability concept. The experimental setup was designed to measure injected fluid flow rates, pressure drop within the fracture, and fluid saturation by image processing. For both fractures, single-phase flow experiments were modeled by means of the full cubic inertial law which allowed the determination of the intrinsic hydrodynamic parameters. Using these parameters, the apparent permeability of each fracture was calculated as a function of the Reynolds number, leading to an elegant means to compare the two fractures in terms of hydraulic behavior versus flow regime. Also, a method for determining the experimental transition flow rate between the weak inertia and the strong inertia flow regimes is proposed. Two-phase flow experiments consisted in measuring the pressure drop and the fluid saturation within the fractures, for various constant values of the liquid flow rate and for increasing values of the gas flow rate. Regardless of the explored flow regime, two-phase flow relative permeabilities were calculated as the ratio of the single phase flow pressure drop per unit length divided by the two-phase flow pressure drop per unit length, and were plotted versus the measured fluid saturation. Results confirm the dependence of the relative permeabilities on the flow regime. Also the proposed generalization of Darcy’s law shows that the relative permeabilities versus fluid saturation follow physical meaningful trends for different liquid and gas flow rates. The presented model fits correctly the liquid and gas experimental relative permeabilities as well as the fluid saturation.     

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.     

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.     

Simulation of three-phase displacement experiments

Three-phase displacement experiments for a water-benzyl alcohol-decane system are simulated. Literature experimental three-phase relative permeabilities for the system are used to describe the relative permeabilities in the three-phase region for different three-phase relative permeability models. Saturation trajectories and elliptical regions are mapped in the three-phase region. Simulations are performed to model displacement experiments including breakthrough and the formation of multiple shocks. The model can be used to predict the results for other displacements. In an experiment where significant gravity segregation is present, the displacement is more accurately modeled by assuming a uniform initial condition than by using the actual vertical saturation and assuming no cross flow. It is shown how different residual saturation values can be measured in the laboratory depending on the initial saturation conditions in the core. The experimental residual saturations can be significantly different than the ‘theoretical’ or model values.     

Water removal from porous media by gas injection: experiments and simulation

The flow of a saturated gas through a porous medium, partially occupied by a liquid phase, causes evaporation due to gas expansion. This process, referred to as flow-through drying, is important in a wide variety of natural and industrial applications, such as natural gas production, convective drying of paper, catalysts, fuel cells and membranes. X-ray imaging experiments were performed to study the flow-through drying of water-saturated porous media during gas injection. The results show that the liquid saturation profile and the rate of drying are dependent on the viscous pressure drop, the state of saturation of the gas and the capillary characteristics of the porous medium. During the injection of a completely saturated gas, drying occurs only due to gas expansion. Capillary-driven flow from regions of high saturation to regions of low saturation lead to more uniform saturation profiles. During the injection of a dry gas, a drying front develops at the inlet and propagates through the porous medium. The experimental results are compared with numerical results from a continuum model. A good agreement is found for the case of sandstone. The comparison is less satisfactory for the experiments with limestone.     

Numerical simulation of the sedimentation of an elliptic body in an incompressible viscous fluid

In this note we discuss the application of a methodology combining distributed Lagrange multiplier based fictitious domain techniques, finite element approximations and operator splitting, to the numerical simulation of the motion of an elliptic body falling in a Newtonian incompressible viscous fluid. The motion of the body is driven by the hydrodynamical forces and gravity. As qualitatively expected, the elliptic body rotates so that its broad side tends to be perpendicular to the flow direction.     

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.     

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.