Two-phase brine mixtures in the geothermal context and the polymer flood model

Two-phase mixtures of hot brine and steam are important in geothermal reservoirs under exploitation. In a simple model, the flows are described by a parabolic equation for the pressure with a derivative coupling to a pair of wave equations for saturation and salt concentration. We show that the wave speed matrix for the hyperbolic part of the coupled system is formally identical to the corresponding matrix in the polymer flood model for oil recovery. For the class ofstrongly diffusive hot brine models, the identification is more than formal, so that the wave phenomena predicted for the polymer flood model will also be observed in geothermal reservoirs.Roman Symbols A,B coefficient matrices (5) - c(x,t) salt concentration (primary dependent variable) - C(p, s, c, q _t) wave speed matrix (6) - f source term (5) - g acceleration due to gravity (constant) - h _b(p, c) brine specific enthalpy - h _v(p) vapour specific enthalpy - j conservation flux (1) - k absolute permeability (constant) - k _b(s), k_v(s) relative permeabilities of the brine and vapour phases - K conductivity - p(x,t) pressure (primary dependent variable) - q volume flux (Darcy velocity) (3) - s(x,t) brine saturation (primary dependent variable) - t time (primary independent variable) - T=T _sat(p) saturation temperature - u _b(p, c) brine specific internal energy - u _m T rock matrix specific internal energy - u _v(p) vapour specific internal energy - U(x, t) shock velocity - x space (primary independent variable) Greek Symbols porosity (constant) - _b(p, c) brine dynamic viscosity - _v(p) vapour dynamic viscosity - (p, s, c) conservation density (1) - _b(p, c) brine density - _v(p) vapour density Suffixes b brine - m rock matrix - t total - v vapour - S salt - M mass - E energy     

Heat transfer by conduction,natural convection and radiation across a rectangular cellular structure

This paper describes results on the effects of wall conduction and radiation heat exchange among surfaces on laminar natural convection heat transfer in a two-dimensional rectangular cavity modelling a cellular structure. Parametric heat transfer calculations have been performed, and numerical results are presented in graphical and tabular form. Local and average Nusselt numbers along the cavity walls are reported for a range of parameters of physical interest. The findings suggest that the local or the average Nusselt number is one of many parameters that control conjugate heat transfer problems. The results indicate that natural convection heat transfer in the cavity is reduced by heat conduction in the walls and radiation exchange among surfaces. The results obtaibed for the total heat transfer rate through the system using the two-dimensional model are compared with those based on a one-dimensional model.     

Analysis of one-dimensional horizontal two-phase flow in geothermal reservoirs

In the absence of capillarity the single-component two-phase porous medium equations have the structure of a nonlinear parabolic pressure (equivalently, temperature) diffusion equation, with derivative coupling to a nonlinear hyperbolic saturation wave equation. The mixed parabolic-hyperbolic system is capable of substaining saturation shock waves. The Rankine-Hugoniot equations show that the volume flux is continuous across such a shock. In this paper we focus on the horizontal one-dimensional flow of water and steam through a block of porous material within a geothermal reservoir. Starting from a state of steady flow we study the reaction of the system to simple changes in boundary conditions. Exact results are obtainable only numerically, but in some cases analytic approximations can be derived. When pressure diffusion occurs much faster than saturation convection, the numerical results can be described satisfactorily in terms of either saturation expansion fans, or isolated saturation shocks. At early times, pressure and saturation profiles are functionally related. At intermediate times, boundary effects become apparent. At late times, saturation convection dominates and eventually a steady-state is established. When both pressure diffusion and saturation convection occur on the same timescale, initial simple shock profiles evolve into multiple shocks, for which no theory is currently available. Finally, a parameter-free system of equations is obtained which satisfactorily represents a particular case of the exact equations.     

Enhanced conduction in steady,vertical flows of water,steam and carbon dioxide in a porous medium

Transport in Porous Media - The steady, vertical transport of water, energy and carbon dioxide in a two-phase porous medium is analysed when capillarity can be ignored. Such flows are possible only...     

Dissipation matrix and artificial heat conduction for Godunov‐type schemes of compressible fluid flows

This paper aims to reassess the Riemann solver for compressible fluid flows in Lagrangian frame from the viewpoint of modified equation approach and provides a theoretical insight into dissipation mechanism. It is observed that numerical dissipation vanishes uniformly for the Godunov‐type schemes in the sense that associated dissipation matrix has zero determinant if an exact or approximate Riemann solver is used to construct numerical fluxes in the Lagrangian frame. This fact connects to some numerical defects such as the wall‐heating phenomenon and start‐up errors. To cure these numerical defects, a traditional numerical viscosity is added, as well as the artificial heat conduction is introduced via a simple passage of the Lax–Friedrichs type discretization of internal energy. Copyright © 2016 John Wiley & Sons, Ltd.     

The setting up of the equations for two-phase flows by the kinetic theory method

The fundamental equations for two-phase flows are deduced from the Boltzmann's equation. The collision terms are treated with a method similar to what is used in the classical kinetic theory for handling the transport properties of dense gases. It is shown that collision pressure and collision thermal flux exist in gas-particle flows in addition to the general partial pressure and partial thermal flux. Their physical natures are quite different from those of the general partial pressure and partial thermal flux. The applicability of the binary collision assumption and the molecular chaos assumption to gas-particle flows is also discussed. Finally, the equations for two-phase flows obtained by the method of the kinetic theory are compared with those obtained by average continuum models and by the model of particle clouds. The results from the kinetic theory show clearly the physical significance of various parameters and clarify some confusing concepts. Institute of Mechanics, Academia Sinica     

Two-phase pressure drop in multiple thick- and thin-orifice plates

Two-phase flow pressure changes through singularities such as sudden expansion and sudden contraction of thick- and thin-orifice plates were modeled. The modeling was based on the reversible and irreversible losses through contractions and expansions. The volume-averaged momentum equation and the reversible mechanical energy equation were used to evaluate the irreversibilities. Local slip ratios, which were necessary for the prediction of pressure drop through these singularities, were correlated from a large number of experimental data. To check the validity of the analytical predictions, an experimental test section was designed, and experiments were performed to produce benchmark pressure-drop data for thin- and thick-plate orifices. The working fluid used for these experiments was R-113. The predictive methods developed agree well with the experiments by the authors and with a wide range of two-phase flow test results obtained by others for steam-water systems. A parametric study shows the relative importance of geometry and of the flow variables such as quality, liquid-to-vapor density, and viscosity ratios on the pressure drop multipliers conventionally used in two-phase flows.     

On higher order multigrid methods with application to a geothermal reservoir model

This paper considers the multigrid iterative method applied to the solution of finite difference approximations to a linear second-order self-adjoint elliptic equation. It represents an extension of work by Dinar and Brandt. We compare two methods to obtain fourth-order convergence. The first is local error extrapolation developed by Brandt, the second is iterative improvement developed by Lindberg. This work considers non-separable problems, but only on a rectangular domain with Dirichlet boundary conditions. We consider test cases with non-smooth (i.e. discontinuous second derivatives) as well as smooth solutions. We also apply the multigrid method to an elliptic equation with non-separable coefficients which occurs in a geothermal model. In this case an analysis of the error fails to show any advantage in a fourth-order difference scheme over a second-order scheme. However, we do demonstrate that the multigrid iteration performs well on this problem. Also, this example shows that the multigrid iteration can be combined with iterative improvement to create an efficient fourth-order method for a non-separable elliptic equation which is coupled with a marching equation. Other work has found an advantage in this fourth-order scheme for a similar geothermal model.     

Two-phase flows in gun barrel: Theoretical and experimental studies

In the paper theoretical and numerical model of two-phase flow of solid granular propellant and its products of combustion in the gun barrel during interior ballistic cycle is given. Two cases are considered: base ignition of propellant charge and ignition by igniter. The theoretical model includes the balance equations of mass, momentum and energy for both phases, as well as necessary constitutive laws. The igniter efflux in the propellant chamber is obtained by incorporation in the model the two-phase flow model of igniter function. The convergent, unconditionally stable, numerical procedure is formed to solve the system of equations of the theoretical model. An original procedure of numerical grid adaptation to the flow field increase, caused by the projectile motion down the gun bore, is developed. The TWOPIB code for the computation of whole interior ballistic cycle of ammunition is developed. Four kinds of experimental investigations were carried out:igniter function in open air, flamespreading through propellant charge in the fibreglass tube during base ignition or during ignition by igniter, and firing of 100 mm APFSDS projectile. Verification of the theoretical–numerical approach by the comparison with experimental data is carried out. The great number of computational results is presented for the parameters that can not be measured, but which are necessary for more complete understanding of examined processes. The presented theoretical–numerical access enables, not only the complete optimisation of propellant charges, but more successful solutions of many interior ballistic problems.     

Numerical modeling of the single-phase Stefan problem in a layer with transparent and semitransparent boundaries

Numerical modeling of the single-phase Stefan problem in a semitransparent layer with transparent, nonabsorbing, and partially radiation-absorbing boundaries is performed. It is shown that at low temperatures of the medium, convection is a determining factor on the boundary of the irradiated sample, and at high temperatures, radiation is predominant. The absence of absorption on the boundaries of the layer leads to acceleration of the heating of the plate and considerable deceleration of melting processes. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 84–91, May–June, 2006.     

Numerical simulation of 1-D unsteady two-phase flows with shocks

In the present paper, random-choice method (RCM) and second-order GRP difference method, which are high resolution methods used for pure gas flows with shocks, are extended and employed to study the problem of one-dimensional unsteady two-phase flows. The two-phase shock wave and the flow field behind it in a dusty gas shock tube are calculated and the time-dependent change of the fiow parameters for the gas antiparticle phase are obtained. The numerical results indicate that both the two methods can give the relaxation structure of the two-phase shocks with a sharp discontinuous front and that the GRP method has the advantages of less time-consuming and higher accuracy over the RCM method.     

On basic equations of turbulent swirling gas-solid flows and their application in cyclones

The basic equations of turbulent gas-solid flows are derived by using the pseudo-fluid model of particle phase with a refined two-phase turbulence model. These equations are then applied to swirling gas-particle flows for analyzing the collection efficiency in cyclone separators.     

Two exact solutions of the DPL non-Fourier heat conduction equation with special conditions

This paper presents two exact explicit solutions for the three dimensional dual-phase lag （DLP） heat conduction equation, during the derivation of which the method of trial and error and the authors＇ previous experiences are utilized. To the authors＇ knowledge, most solutions of 2D or 3D DPL models available in the literature are obtained by numerical methods, and there are few exact solutions up to now. The exact solutions in this paper can be used as benchmarks to validate numerical solutions and to develop numerical schemes, grid generation methods and so forth. In addition, they are of theoretical significance since they correspond to physically possible situations. The main goal of this paper is to obtain some possible exact explicit solutions of the dual-phase lag heat conduction equation as the benchmark solutions for computational heat transfer, rather than specific solutions for some given initial and boundary conditions. Therefore, the initial and boundary conditions are indeterminate before derivation and can be deduced from the solutions afterwards. Actually, all solutions given in this paper can be easily proven by substituting them into the governing equation.     

Finite element analysis of multicomponent two-phase flows with interphase mass and momentum transport

The application of the finite element method to multiphase flow problems with interphase mass and heat transfer is described. A general forinulation is used that determines the position of the interfacial boundary and allows for multiple solvents, differential volatilities and concentration- and temperature-dependent thermophysical properties. Species phase change and the dramatic volume change that acompanies interphase mass transfer make implementation of the theory challening, since these events lead to discontinuous velocities and concentrations at phase boundaries. These discontinuities are especially large in processes involving rapid evaporation or condensation. As examples we examine the effects of rapid drying on film and fibre formation of sol--gel materials, which are often laden with volatile species.     

Adjoint approach of the spatial sensitivity to disturbances of internal flows with free surface

A local adjoint technique is developed in order to determine the most sensitive location to perturbations of steady states near bifurcation points in the case of confined flows with free‐surface boundary. Transitions to stationary or periodic flows are studied. The method is validated by comparison of its results with those given by a time approach. It is then applied to the stability study and the feedback control of thermocapillary flows in liquid bridge. Copyright © 2005 John Wiley & Sons, Ltd.     

Experimental results for nucleating steam flow in a two-dimensional supersonic duct and comparison with theory

Results are presented of experiments conducted in a two-dimensional duct carrying a supersonic flow of condensing steam. The measurements comprised static pressure readings along the profiled surfaces of the duct and ‘fog’ droplet sizing using a light attenuation technique. Three sets of results for dry supercooled and nucleating steam flows are presented, are are compared with the predictions of a two-dimensional numerical calculation method.     

modeling of multicomponent multiphase mixture flows on the basis of the density-functional method

Examples of numerical calculations of isothermal flows of two-phase two-component mixtures based on the density-functional method are presented. Using this method, the following problems are calculated in the two-dimensional formulation: drop impact on a liquid layer, drop rupture in a Couette flowfield, wetting-angle formation for a drop on a solid surface, development of Rayleigh-Taylor and Kelvin-Helmholtz instability on a gas-liquid interface.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, 2004, pp. 101–114.Original Russian Text Copyright © 2004 by Demyanov and Dinariev.     

Time-dependent liquid metal flows with free convection and a deformable free surface

The finite element method is employed to investigate time-dependent liquid metal flows with free convection, free surfaces and Marangoni effects. The liquid circulates in a two-dimensional shallow trough with differentially heated vertical walls. The spatial formulation incorporates mixed Lagrangian approximations to the velocity, pressure, temperature and free surface position. The time integration is performed with the backward Euler and trapezoid rule methods with step size control. The Galerkin method is used to reduce the problem to a set of non-linear equations which are solved with the Newton–Raphson method. Calculations are performed for conditions relevant to the electron beam vaporization of refractory metals. The Prandtl number is 0·015 and Grashof number are in the transition range between laminar and turbulent flow. The results reveal the effects of flow intensity, surface tension gradients, mesh refinement and time integration strategy.     

气液两相流动与固壁相互作用耦合求解的研究

气液两相流动与固壁相互作用的研究是液滴撞击壁面运动研究的重要基础.以结合了VOF和Level Set两种方法优点的用于气液相界面追踪的复合Level Set-VOF方法和利用唯象分析方法建立的能够反映接触角滞后性及壁面性质对润湿过程影响的壁面润湿模型为基础,提出了气液两相流动与固壁相互作用耦合求解流程,给出了气液两相流动与固壁相互作用耦合求解过程中接触线速度的计算方法及边界条件的确定方法.通过与已有实验结果的对比,对提出的气液两相流动与固壁相互作用耦合求解方法的有效性进行了验证.