Analysis of Multiphase Non-Darcy Flow in Porous Media

Recent laboratory studies and analyses (Lai et al. Presented at the 2009 Rocky Mountain Petroleum Technology Conference, 14–16 April, Denver, CO, 2009) have shown that the Barree and Conway model is able to describe the entire range of relationships between flow rate and potential gradient from low- to high-flow rates through porous media. A Buckley and Leverett type analytical solution is derived for non-Darcy displacement of immiscible fluids in porous media, in which non-Darcy flow is described using the Barree and Conway model. The comparison between Forchheimer and Barree and Conway non-Darcy models is discussed. We also present a general mathematical and numerical model for incorporating the Barree and Conway model in a general reservoir simulator to simulate multiphase non-Darcy flow in porous media. As an application example, we use the analytical solution to verify the numerical solution for and to obtain some insight into one-dimensional non-Darcy displacement of two immiscible fluids with the Barree and Conway model. The results show how non-Darcy displacement is controlled not only by relative permeability, but also by non-Darcy coefficients, characteristic length, and injection rates. Overall, this study provides an analysis approach for modeling multiphase non-Darcy flow in reservoirs according to the Barree and Conway model.     

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.     

Miscible displacement from fractured porous media

A model of miscible displacement of incompressible fluids from a fractured porous medium is proposed. The model describes the process of displacement of oil by solvents, the cycling process of displacement of aliphatic hydrocarbon gas by dry gas at low repressions on the formation, and other processes of single-phase multicomponent displacement from fractured porous media. Problems relating to the pumping of a neutral admixture and admixture slugs through a fractured porous reservoir are solved.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 100–110, November–December, 1989.The authors are grateful to K. S. Basniev, A. K. Kurbanov, V. I. Maron, and M. I. Shvidler for useful discussions.     

Inertial and Second Grade Effects on Macroscopic Equations for the Flow of a Fluid in a Porous Medium

Homogenization techniques are used to upscale from pore to laboratory or field scale viscous and second grade nonNewtonian flow in a porous medium. Nonlinear forms of Darcy's law are obtained and analysed under a series of symmetry properties. The general case of displacement of one of these fluids by another with different properties is considered and a linear stability analysis is performed.     

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.     

Dynamics of non-Newtonian fluid interfaces in a porous medium: Compressible fluids

This paper deals with the illustration of rheological effects of non-Newtonian displacing and displaced fluids on the dynamics of a moving interface in a porous medium. Both fluids are considered to be compressible. Specifically, the rheological effects are shown and discussed on the interface position and on its velocity in terms of rheological parameters of power-law fluids. The approximate analytical solutions are obtained for the boundary and initial conditions of practical interest, from which an optimal policy of injection can be found to control the dynamics of a moving interface in oil displacement mechanism.     

Emulsion flow through a porous medium with allowance for interphase mass transfer between the components

A new model of the flow of two miscible, mutually-insoluble fluids in a porous medium with the formation of an emulsion and adsorption of the fluid components on the skeleton is proposed. The model takes into account the effect of interphase mass transfer on the emulsion dynamics and the active porosity. A continuous general solution of the one-dimensional model and the problem of breakdown of a discontinuity is constructed. The flow regimes generated in displacement problems which depend on the shape of the adsorption isotherms and the densities of the fluid components are considered. The time dependence of the production rate is constructed for frontal displacement regimes and for displacement regimes with the formation of a zone of mixing (Riemann wave) of the initial reservoir and injected fluids. These functions coincide, at least qualitatively, with the experimental data [1] indicating an initial increase in production rate even against a background of falling reservoir pressure, transition through a maximum, and subsequent decline. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 77–88, January–February, 1997. The work was carried out with financial support from the Russian Foundation for Fundamental Research (project No. 96-01-00991).     

Two approaches to modeling unstable flow and mixing of variable density fluids in porous media

Density variation of aqueous phase fluids flowing in a porous medium, resulting from spatial and temporal variation of solute concentration, often gives rise to unstable flow, and therefore has a significant effect on solute transport. Studies on simulating unstable flow and mixing of variable density fluids in seemingly homogeneous porous media are rare. In this study, a three-dimensional (3-D) and a one-dimensional (1-D) model were developed to simulate unstable flow and mixing in a vertical, nominally 1-D system. In the 3-D model, the fluid flow and solute transport equations were solved numerically with a very fine spatial discretization. The 1-D numerical model was derived from a theoretical model to simulate the flow and mixing of fluids with variable density and viscosity at the field scale. To evaluate the models, simulated results were compared with experimental data from displacement experiments in a vertical sand column. The results show that the 1-D model provides fairly good prediction of breakthrough curves and that the 3-D model is able to qualitatively simulate breakthrough curves for highly unstable flow and mixing.Contribution from the Alabama Agric. Exp. Sta. as AAES Journal No. 3-955037.     

Dynamics of non-Newtonian fluid interfaces in a porous medium: Incompressible fluids

This paper concerns the applications of frontal advance theory to the dynamics of a moving flat interface in a porous medium, when both displacing and displaced fluids are of power law behaviour. The rheological effects of non-Newtonian behaviour of these fluids on the interface position and its velocity are numerically illustrated and discussed with regard to the practical implications in oil displacement mechanisms. The results obtained should be useful in finding an optimal policy of injection in order to control the dynamics of the moving interface in field projects of enhanced oil recovery floods.     

On viscoelastic effects in non-Newtonian steady flows through porous media

This paper presents a mathematical model for describing approximately the viscoelastic effects in non-Newtonian steady flows through a porous medium. The rheological behaviour of power law fluids is considered in the Maxwell model of elastic behaviour of the fluids. The equations governing the steady flow through porous media are derived and an analytical solution of these equations in the case of a simple flow system is obtained. The conditions for which the viscoelastic effects may become observable from the pressure distribution measurements are shown and expressed in terms of some dimensionless groups. These have been found to be relevant in the evaluation of viscoelastic effects in the steady flow through porous media.     

Theoretical and Experimental Investigations of Miscible Displacement in Fractured Porous Media

In this paper a mathematical model for miscible displacement in fractured porous media is developed. The model takes into account mechanisms of mass transfer between fracture and matrix. The model is normalized by using the dimensionless parameters, which characterize the process, and the analytical solutions of the resulting system of equations are provided by utilizing the method of characteristics. For comparison the results of model with experimental results, laboratory displacement tests have been performed in fractured systems under miscible displacement. The porous media used were cylindrical Asmari cores from Iranian reservoirs containing an artificially vertical fracture. Normal heptane and kerosene were two miscible fluid used. There is very good agreement between experiments and model prediction.     

Stability of vertical miscible displacements with developing density and viscosity gradients

Viscous fingering and gravity tonguing are the consequences of an unstable miscible displacement. Chang and Slattery (1986) performed a linear stability analysis for a miscible displacement considering only the effect of viscosity. Here the effect of gravity is included as well for either a step change or a graduated change in concentration at the injection face during a downward, vertical displacement. If both the mobility ratio and the density ratio are favorable (the viscosity of the displacing fluid is greater than the viscosity of the displaced fluid and, for a downward vertical displacement, the density of the displacing fluid is less than the density of the displaced fluid), the displacement will be stable. If either the mobility ratio or the density ratio is unfavorable, instabilities can form at the injection boundary as the result of infinitesimal perturbations. But if the concentration is changed sufficiently slowly with time at the entrance to the system, the displacement can be stabilized, even if both the mobility ratio and the density ratio are unfavorable. A displacement is more likely to be stable as the aspect ratio (ratio of thickness to width, which is assumed to be less than one) is increased. Commonly the laboratory tests supporting a field trial use nearly the same fluids, porous media, and displacement rates as the field trial they are intended to support. For the laboratory test, the aspect ratio may be the order of one; for the field trial, it may be two orders of magnitude smaller. This means that a laboratory test could indicate that a displacement was stable, while an unstable displacement may be observed in the field.     

Displacement of a Newtonian fluid by a non-Newtonian fluid in a porous medium

This paper presents an analytical Buckley-Leverett-type solution for one-dimensibnal immiscible displacement of a Newtonian fluid by a non-Newtonian fluid in porous media. The non-Newtonian fluid viscosity is assumed to be a function of the flow potential gradient and the non-Newtonian phase saturation. To apply this method to field problems a practical procedure has been developed which is based on the analytical solution and is similar to the graphic technique of Welge. Our solution can be regarded as an extension of the Buckley-Leverett method to Non-Newtonian fluids. The analytical result reveals how the saturation profile and the displacement efficiency are controlled not only by the relative permeabilities, as in the Buckley-Leverett solution, but also by the inherent complexities of the non-Newtonian fluid. Two examples of the application of the solution are given. One application is the verification of a numerical model, which has been developed for simulation of flow of immiscible non-Newtonian and Newtonian fluids in porous media. Excellent agreement between the numerical and analytical results has been obtained using a power-law non-Newtonian fluid. Another application is to examine the effects of non-Newtonian behavior on immiscible displacement of a Newtonian fluid by a power-law non-Newtonian fluid.     

A mixture theory analysis for the surface-wave propagation in an unsaturated porous medium

The mixture theory is employed to the analysis of surface-wave propagation in a porous medium saturated by two compressible and viscous fluids (liquid and gas). A linear isothermal dynamic model is implemented which takes into account the interaction between the pore fluids and the solid phase of the porous material through viscous dissipation. In such unsaturated cases, the dispersion equations of Rayleigh and Love waves are derived respectively. Two situations for the Love waves are discussed in detail: (a) an elastic layer lying over an unsaturated porous half-space and (b) an unsaturated porous layer lying over an elastic half-space. The wave analysis indicates that, to the three compressional waves discovered in the unsaturated porous medium, there also correspond three Rayleigh wave modes (R1, R2, and R3 waves) propagating along its free surface. The numerical results demonstrate a significant dependence of wave velocities and attenuation coefficients of the Rayleigh and Love waves on the saturation degree, excitation frequency and intrinsic permeability. The cut-off frequency of the high order mode of Love waves is also found to be dependent on the saturation degree.     

Large-eddy simulation of fluid mixing in tee with sintered porous medium

Mixing processes of hot and cold fluids in a tee with and without sintered copper spheres are simulated by FLUENT using the large-eddy simulation (LES) turbulent flow model and the sub-grid scale (SGS) Smagorinsky-Lilly (SL) model with buoyancy. Comparisons of numerical results of the two cases with and without sintered copper spheres show that the porous medium significantly reduces velocity and temperature fluctuations because the porous medium can effectively restrict the fluid flow and enhance heat transfer. The porous medium obviously increases the pressure drop in the main duct. The porous medium reduces the power spectrum density (PSD) of temperature fluctuations in the frequency range from 1 Hz to 10 Hz.     

Visualization of the effects of buoyancy on liquid-liquid displacements in vertically-aligned porous medium cells

Conclusions The results of this work suggest that the use of two-dimensional porous medium cells aligned in the vertical plane affords a promising experimental technique for studying the very significant effects that buoyancy forces are capable of exerting on the stability of liquid/liquid displacement processes in porous media. The cell described here is relatively easy to use and permits a wide range of cell orientations, flow modes, injection points, and recovery points to be investigated. Detailed quantitative studies involving a wide range of fluids, flowrates, and flow modes, are currently under way and will be reported in due course.     

An overview on nonlinear porous flow in low permeability porous media

This paper gives an overview on nonlinear porous flow in low permeability porous media, reveals the microscopic mechanisms of flows, and clarifies properties of porous flow fluids. It shows that, deviating from Darcy's linear law, the porous flow characteristics obey a nonlinear law in a low-permeability porous medium, and the viscosity of the porous flow fluid and the permeability values of water and oil are not constants. Based on these characters, a new porous flow model, which can better describe low permeability reservoir, is established. This model can describe various patterns of porous flow, as Darcy's linear law does. All the parameters involved in the model, having definite physical meanings, can be obtained directly from the experiments.     

Computing the Longtime Behaviour of NMR Propagators in Porous Media Using a Pore Network Random Walk Model

We present a pore network model combined with a random walk algorithm allowing the simulation of molecular displacement distributions in porous media as measured by NMR. A particular feature of this technique is the ability to probe the time evolution of these distributions. The objective is to predict the displacement behaviour for time intervals larger than the experimental observation time and explore the asymptotic dispersion regime at long times. Starting from 3D micro-CT images, we computed the variance of displacement distributions of water molecules in a Fontainebleau sand and found very good agreement of the time evolution of the variance with experimental data, without fitting parameter. The model confirms a weak superdispersion in the asymptotic regime. In addition, we conclude that, since pore network models do not take into account small scale features of the porous medium (e.g., surface roughness and grain shape), the origin of the observed superdispersion is mainly due to the topology and geometry of the porous medium.     

Linear dynamic model for porous media saturated by two immiscible fluids

A linear isothermal dynamic model for a porous medium saturated by two immiscible fluids is developed in the paper. In contrast to the mixture theory, phase separation is avoided by introducing one energy for the porous medium. It is an important advantage of the model based on one energy approach that it can account for the couplings between the phases. The volume fraction of each phase is characterized by the porosity of the porous medium and the saturation of the wetting phase. The mass and momentum balance equations are constructed according to the generalized mixture theory. Constitutive relations for the stress, pore pressure are derived from the free energy function. A capillary pressure relaxation model characterizing one attenuation mechanism of the two-fluid saturated porous medium is introduced under the constraint of the entropy inequality. In order to describe the momentum interaction between the fluids and the solid, a frequency independent drag force model is introduced. The details of parameter estimation are discussed in the paper. It is demonstrated that all the material parameters in our model can be calculated by the phenomenological parameters, which are measurable. The equations of motion in the frequency domain are obtained in terms of the Fourier transformation. In terms of the equations of motion in the frequency domain, the wave velocities and the attenuations for three P waves and one S wave are calculated. The influences of the capillary pressure relaxation coefficient and the saturation of the wetting phase on the velocities and attenuation coefficients for the four wave modes are discussed in the numerical examples.     

Extension of the Darcy�CForchheimer Law for Shear-Thinning Fluids and Validation via Pore-Scale Flow Simulations

Flow of non-Newtonian fluids through porous media at high Reynolds numbers is often encountered in chemical, pharmaceutical and food, as well as petroleum and groundwater engineering, and in many other industrial applications. Under the majority of operating conditions typically explored, the dependence of pressure drops on flow rate is non-linear and the development of models capable of describing accurately this dependence, in conjunction with non-trivial rheological behaviors, is of paramount importance. In this work, pore-scale single-phase flow simulations conducted on synthetic two-dimensional porous media are performed via computational fluid dynamics for both Newtonian and non-Newtonian fluids and the results are used for the extension and validation of the Darcy?CForchheimer law, herein proposed for shear-thinning fluid models of Cross, Ellis and Carreau. The inertial parameter ?? is demonstrated to be independent of the viscous properties of the fluids. The results of flow simulations show the superposition of two contributions to pressure drops: one, strictly related to the non-Newtonian properties of the fluid, dominates at low Reynolds numbers, while a quadratic one, arising at higher Reynolds numbers, is dependent on the porous medium properties. The use of pore-scale flow simulations on limited portions of the porous medium is here proposed for the determination of the macroscale-averaged parameters (permeability K, inertial coefficient ?? and shift factor ??), which are required for the estimation of pressure drops via the extended Darcy?CForchheimer law. The method can be applied for those fluids which would lead to critical conditions (high pressures for low permeability media and/or high flow rates) in laboratory tests.