Saturated steam relative permeabilities of unconsolidated porous media

This paper presents a method and describes an experimental device for determining the steam-water relative permeabilites of unconsolidated porous media. The experimental conditions are as close as possible to those of geothermal reservoirs. The relative permeabilities have been obtained at 180 and 150?C. Their variations versus liquid saturation are quite classical. The air-water relative permeabilities have been measured also at room temperature. The values obtained under these three conditions are almost identical. However, the air-water relative permeability differs slightly from that of steam at 180 and 150?C. We think this discrepancy is acceptable in practice, as it is easier to determine the relative permeabilities for an air-water flow at room temperature than for a steam-water flow at high temperature and pressure.     

A Model for Multiphase and Multicomponent Flow in Porous Media,Built on the Diffuse-Interface Assumption

This article presents a new theory for flow in porous media of a mixture of nonreacting chemical components. In the examples considered, these components are hydrocarbons and water. The model presented assumes that porosity is constant and uniform, and that the wetting properties of the medium are nearly neutral. The flow equations are obtained by starting with the balance equations (mass, momentum, and energy) at pore level, and averaging them over a large number of pores, using the diffuse interface assumption then the methods of irreversible thermodynamics, thus obtaining, among other things, the collective convective velocity and the component-wise diffusive velocities as functions of the component densities. When the simplification of uniform temperature is introduced, the flow equations are of the Cahn–Hilliard type (with an extra term accounting for gravitation) where the thermodynamic function is the Helmholtz free energy per unit volume of the mixture. There are no relative permeabilities. Also, the set of equations is complete in the sense that no flash calculations are necessary, phase segregation being part of the calculation. The numerical examples considered are: (i) phase segregation in a gravitational field and (ii) coning where the initial state is fully segregated.     

An experimental study of steam injection into a uniform water flow through porous media

An experimental study of steam injection into a porous media was carried out in a 2-dimensional plane porous channel. The steam was injected into a uniform downward water flow in a vertically aligned porous channel. The steam-water interface was carefully observed to understand the underlying physics. Two steam injection rate bounds were found for a given water flow rate and water subcooling. The upper bound is the steam flow rate at which the steam zone grows without limit and the lower bound is the steam flow rate at which a steam zone is just initiated. The bounds were determined experimentally for a porous channel with different permeabilities and thermal conductivities. For large particle size, chaotic oscillation of steam water interface was observed. The oscillation is believed to enhance heat and momentum transfer mechanisms. The steam zone size and shape were measured to evaluate heat transfer characteristics. The average Nusselt number is presented in terms of steam and water Reynolds numbers and the Stefan number.     

Contributions to Theoretical/Experimental Developments in Shock Waves Propagation in Porous Media

Macroscopic balance equations of mass, momentum and energy for compressible Newtonian fluids within a thermoelastic solid matrix are developed as the theoretical basis for wave motion in multiphase deformable porous media. This leads to the rigorous development of the extended Forchheimer terms accounting for the momentum exchange between the phases through the solid-fluid interfaces. An additional relation presenting the deviation (assumed of a lower order of magnitude) from the macroscopic momentum balance equation, is also presented. Nondimensional investigation of the phases' macroscopic balance equations, yield four evolution periods associated with different dominant balance equations which are obtained following an abrupt change in fluid's pressure and temperature. During the second evolution period, the inertial terms are dominant. As a result the momentum balance equations reduce to nonlinear wave equations. Various analytical solutions of these equations are described for the 1-D case. Comparison with literature and verification with shock tube experiments, serve as validation of the developed theory and the computer code.A 1-D TVD-based numerical study of shock wave propagation in saturated porous media, is presented. A parametric investigation using the developed computer code is also given.     

Finite element solution of multi-dimensional two-phase flow through porous media with arbitrary heating conditions

Mechanistic models for flow regime transitions and drag forces proposed in an earlier work are employed to predict two-phase flow characteristics in multi-dimensional porous layers. The numerical scheme calls for elimination of velocities in favor of pressure and void fraction. The momentum equations for vapor and liquid then can be reduced to a system of two partial differential equations (PDEs) which must be solved simultaneously for pressure and void fraction.

Solutions are obtained both in two-dimensional cartesian and in axi-symmetric coordinate systems. The porous layers in both cases are composed of regions with different permeabilities. The finite element method is employed by casting the PDEs in their equivalent variational forms. Two classes of boundary conditions (specified pressure and specified fluid fluxes) can be incorporated in the solution. Volumetric heating can be included as a source term. The numerical procedure is thus suitable for a wide variety of geometry and heating conditions. Numerical solutions are also compared with available experimental data.     

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.     

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.     

Study of the effect of capillary pressure on the permeability of porous media embedded with a fractal-like tree network

In this paper, the analytical expressions for permeability of (both saturated and unsaturated) porous media embedded with a fractal-like tree network are presented based on fractal theory and technique when the capillary pressure is taken into account. Both the dimensionless effective permeability and the relative permeability of the composites, which are defined as porous media embedded with a fractal-like tree network in this work, are derived and found to be a function of saturation, the capillary pressure and microstructural parameters of the networks. The relative permeabilities predicted by the present fractal model are compared with the available experimental data and a fair agreement between them is found.     

Experimental and numerical analysis of steam jet pump

Steam jet pump is the best choice for pumping radioactive and hazardous liquids because it has no moving parts and so no maintenance. However, the physics involved is highly complicated because of the mass, momentum and energy transfer between the phases involved. In this study the characteristics of SJP are studied both experimentally and numerically to pump water using saturated steam. In the experimental study the static pressure, temperature along the length of the steam jet pump and the steam and water flow rates are recorded. The three dimensional numerical study is carried out using the Eulerian two-phase flow model of Fluent 6.3 software and the direct-contact condensation model developed previously. The experimental and CFD results, of axial static pressure and temperature, match closely with each other. The mass ratio and suction lift are calculated from experimental data and it is observed that the mass ratio varies from 10 to 62 and the maximum value of suction lift is 2.12 m under the conditions of the experiment.     

Flow field and thermal characteristics in a model of a tangentially fired furnace under different conditions of burner tripping

Tangentially fired furnaces are vortex-combustion units and are widely used in steam generators of industrial plants. The present study provides a numerical investigation of the problem of turbulent reacting flows in a model furnace of a tangentially fired boiler. The importance of this problem is mainly due to its relation to large boiler furnaces used in thermal power plants. In the present work, calculation of the flow field, temperature and species concentration-contour maps in a tangentially-fired model furnace are provided. The safety of these furnaces requires that the burner be tripped (its fuel is cut off) if the flame is extinguished. Therefore, the present work provides an investigation of the influence of number of tripped burners on the characteristics of the flow and thermal fields. The details of the flow, thermal and combustion fields are obtained from the solution of the conservation equations of mass, momentum and energy and transport equations for scalar variables in addition to the equations of the turbulence model. Available experimental measurements were used for validating the calculation procedure. The results show that the vortex created due to pressure gradient at the furnace center only influenced by tripping at least two burners. However, the temperature distributions are significantly distorted by tripping any of the burners. Regions of very high temperature close to the furnace walls appear as a result of tripping the fuel in one or two of the burners. Calculated heat flux along the furnace walls are presented.     

Rapoport-leas two-phase flow model for anisotropic porous media

The Rapoport-Leas mathematical model of two-phase flow is generalized to include the case of anisotropic porous media. The formula for the capillary pressure, which specifies the relationship between the phase pressures, contains a scalar function of a vector argument. In order to determine the scalar function, the capillary pressure tensor and the tensor inverse to the tensor of characteristic linear dimensions are introduced. The capillary pressure is determined by the contraction of the second-rank tensors with a unit vector collinear to the phase pressure gradients, also assumed to be collinear. It is shown that the saturation function introduced for isotropic porous media (Leverett function) can be generalized to include anisotropic media and is now determined by a fourth-rank tensor. Generalized expressions for the Leverett and relative phase permeability functions are given for orthotropic and transversely isotropic media with account for the hysteresis of the phase permeabilities and capillary pressure.     

Evolution of the balance equations in saturated thermoelastic porous media following abrupt simultaneous changes in pressure and temperature

A mathematical model is developed for saturated flow of a Newtonian fluid in a thermoelastic, homogeneous, isotropic porous medium domain under nonisothermal conditions. The model contains mass, momentum and energy balance equations. Both the momentum and energy balance equations have been developed to include a Forchheimer term which represents the interaction at the solid-fluid interface at high Reynolds numbers. The evolution of these equations, following an abrupt change in both fluid pressure and temperature, is presented. Using a dimensional analysis, four evolution periods are distinguished. At the very first instant, pressure, effective stress, and matrix temperature are found to be disturbed with no attenuation. During this stage, the temporal rate of pressure change is linearly proportional to that of the fluid temperature. In the second time period, nonlinear waves are formed in terms of solid deformation, fluid density, and velocities of phases. The equation describing heat transfer becomes parabolic. During the third evolution stage, the inertial and the dissipative terms are of equal order of magnitude. However, during the fourth time period, the fluid's inertial terms subside, reducing the fluid's momentum balance equation to the form of Darcy's law. During this period, we note that the body and surface forces on the solid phase are balanced, while mechanical work and heat conduction of the phases are reduced.     

A linear dynamic model for a saturated porous medium

A linear isothermal dynamic model for a porous medium saturated by a Newtonian fluid is developed in the paper. In contrast to the mixture theory, the assumption of phase separation is avoided by introducing a single constitutive energy function for the porous medium. An important advantage of the proposed model is it can account for the couplings between the solid skeleton and the pore fluid. The mass and momentum balance equations are obtained according to the generalized mixture theory. Constitutive relations for the stress, the pore pressure are derived from the total free energy accounting for inter-phase interaction. In order to describe the momentum interaction between the fluid and the solid, a frequency independent Biot-type drag force model is introduced. A temporal variable porosity model with relaxation accounting for additional attenuation is introduced for the first time. The details of parameter estimation are discussed in the paper. It is demonstrated that all the material parameters in our model can be estimated from directly measurable phenomenological parameters. In terms of the equations of motion in the frequency domain, the wave velocities and the attenuations for the two P waves and one S wave are calculated. The influences of the porosity relaxation coefficient on the velocities and attenuation coefficients of the three waves of the porous medium are discussed in a numerical example.     

Numerical simulation of flow boiling for organic fluid with high saturation temperature in vertical porous coated tube

A semi-analytical model is developed for the prediction of flow boiling heat transfer inside vertical porous coated tubes. The model assumes that the forced convection and nucleate boiling coexist together in the annular flow regime. Conservations of mass, momentum, and energy are used to solve for the liquid film thickness and temperature. The heat flux due to nucleate boiling consists of those inside and outside micro-tunnels. To close the equations, a detailed analysis of various forces acting on the bubble is presented to predict its mean departure diameter. The active nucleation site density of porous layer is determined from the pool boiling correlation by introducing suppression factor. The flow boiling heat transfer coefficients of organic fluid (cumene) with high saturation temperature in a vertical flame-spraying porous coated tube are studied numerically. It is shown that the present model can predict most of the experimental values within ±20%. The numerical results also indicate that the nucleate boiling contribution to the overall heat transfer coefficient decreases from 50% to 15% with vapor quality increasing from 0.1 to 0.5.     

Mixing and Mass Transfer in Multicontact Miscible Displacements

In this work, we investigate the accuracy of some physical models that are frequently used to describe and interpret dispersive mixing and mass transfer in compositional reservoir simulation. We have designed a quaternary analog fluid system (alcohol?Cwater?Chydrocarbon) that mimics the phase behavior of CO₂-hydrocarbon mixtures at high pressure and temperature. A porous medium was designed using PolyTetraFlouroEthylene (PTFE) materials to ensure that the analog oil acts as the wetting phase, and the properties of the porous medium were characterized in terms of porosity, permeability and dispersivity. Relative permeability and interfacial tension (IFT) measurements were also performed to delineate interactions between the fluid system and the porous medium. The effluent concentrations from two-component first-contact miscible (FCM) displacement experiments exhibit a tailing behavior that is attributed to imperfect sweep of the porous medium: A feature that is not captured by normal dispersion models. To represent this behavior in displacement calculations, we use dual-porosity (DP) models including mass transfer between flowing and stagnant porosities. Two 4-component two-phase displacement experiments were performed at near-miscible and multicontact miscible (MCM) conditions and the effluent concentrations were interpreted by numerical calculations. We demonstrate that the accuracy of our displacement calculations relative to the experimental observations is sensitive to the selected models for dispersive mixing, mass transfer between flowing and stagnant porosities, and IFT scaling of relative permeability functions. We also demonstrate that numerical calculations substantially agree with the experimental observations for some physical models with limited need for model parameter adjustment. The combined experimental and modeling effort presented in this work identifies and explores the impact of a set of physical mechanisms (dispersion and mass transfer) that must be upscaled adequately for field-scale displacement calculations in DP systems.     

AN ALGEBRAIC MULTIGRID METHOD FOR COUPLED THERMO-HYDRO-MECHANICAL PROBLEMS

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Wave propagation in fractured porous media

A theory of wave propagation in fractured porous media is presented based on the double-porosity concept. The macroscopic constitutive relations and mass and momentum balance equations are obtained by volume averaging the microscale balance and constitutive equations and assuming small deformations. In microscale, the grains are assumed to be linearly elastic and the fluids are Newtonian. Momentum transfer terms are expressed in terms of intrinsic and relative permeabilities assuming the validity of Darcy's law in fractured porous media. The macroscopic constitutive relations of elastic porous media saturated by one or two fluids and saturated fractured porous media can be obtained from the constitutive relations developed in the paper. In the simplest case, the final set of governing equations reduce to Biot's equations containing the same parameters as of Biot and Willis.Now at Izmir Institute of Technology, Anafartalar Cad. 904, Basmane 35230, Izmir, Turkey.     

Effects of Temperature-Dependent Viscosity on Forced Convection Inside a Porous Medium

Considering the exponential viscosity–temperature relation, effect of temperature-dependent viscosity on forced convection of a liquid through a porous medium, bounded by isoflux parallel plates, is investigated numerically based on the general model of momentum transfer. Local effects of viscosity variation on the distribution of velocity and temperature are analyzed. Moreover, global aspects of the problem are investigated where corrections are proposed for total pressure drop and the fully developed Nusselt number, in the form of out/in viscosity ratio. Results are obtained over a wide range of permeabilities from clear (of solid material) fluid to very low permeability, where for constant properties one expects a nearly slug flow.