Four-Component Gas/Oil Displacements in One Dimension: Part II: Analytical Solutions for Constant Equilibrium Ratios

In this paper we derive closed form solutions for composition paths (integral curves of eigenvectors) for three- and four-component systems with constant equilibrium K-values, and we report example solutions that describe gas injection processes. Extensions to systems with an arbitrary number of components are discussed.Author for correspondence: e-mail:wangy2@bp.com     

Four-Component Gas/Water/Oil Displacements in One Dimension: Part III,Development of Miscibility

The structure of the system of conservation laws in fully compositional three-phase, four-component flow is examined for the first time. Two of the eigenvalues can be found analytically for this flow regime, regardless of the equation of state used to model the phase behavior. A cubic equation of state is used to calculate the gas/oil phase behavior and Henry’s law is used to represent the partitioning of hydrocarbons between the hydrocarbon and water phases. Sample analytical solutions are found for Riemann problems modeling the injection of carbon-dioxide and water into an oil reservoir. Finally, the structure of the solutions at the minimum miscibility pressure (MMP) for the hydrocarbon system is studied. We show that when water is injected simultaneously with gas at the MMP, multicontact miscible displacement of the oil by the injected gas develops only if the fraction of water in the injected fluid is below a critical value. For water fractions above the critical value, water flowing at a high velocity forces the composition path of the solution to remain in the three-phase region in which two hydrocarbon phases are present. We study both condensing and vaporizing gas drives to demonstrate that this result is general.     

Four-component gas/water/oil displacements in one dimension: part II,example solutions

In this article analytical solutions are constructed for a system of conservation laws modeling compositional flow of four components in three phases using the method of characteristics (MOC). Every component partitions between all phases present, and the equilibrium volume ratios of the components in each phase are fixed. Riemann problems modeling the injection of carbon dioxide and water in a depleted oil reservoir are studied, and the sensitivity of the solutions to changes in boundary conditions is analyzed. Finally the MOC solutions are compared to simulated displacements.     

Spectral Analysis of the Balance Equation of Ground Water Hydrology

The spectral analysis of the balance equation of ground water flow, associated to an asymptotic expansion of the conductivity (K) and head (h) fields, permits to show that the high wave number components of the source terms, F, and of the conductivity, K, are attenuated when h is computed as solution to the balance equation. This has important consequences on the inverse mapping (h, F) → K: in fact it is not possible to recover in a reliable way the high wave number components of K, because small errors on the corresponding components of h are amplified so that they can hidden the true signal.     

Four-Component Gas/Oil Displacements in One Dimension: Part I: Global Triangular Structure

This paper presents an analysis of the mathematical structure of three-component and four-component gas displacements. The structure of one-dimensional flows in which components partition between two phases is governed by the geometry of a set of equilibrium tie lines. We demonstrate that for systems of four components, the governing mass conservation laws for the displacement can be represented by an eigenvalue system whose coefficient matrix has a global triangular structure, which is defined in the paper, for only specific types of phase behavior. We show that four-component systems exhibit global triangular structure if and only if (1) tie lines meet at one edge of the quaternary phase diagram or (2) if tie lines lie in planes. For such systems, shock and rarefaction surfaces coincide and are planes. We prove that systems are of category (2) if equilibrium ratios (K-values) are independent of mixture composition. In particular, for such systems shock and rarefaction curves will coincide. We also show that for systems with variable K-values, the rarefaction surfaces are almost planar in a precise sense, which is described in the paper. Therefore, systems with variable K-values may be well approximated by assuming shock and rarefaction surfaces do coincide. For these special systems the construction of solutions for one-dimensional, two-phase flow with phase behavior simplifies considerably. In Part II, we describe an application of these ideas to systems in which K-values are constant.     

Effects of slip velocity and surface acoustic wave on the laminar flow stability

The stability of slip flows when a surface acoustic wave (SAW) propagating along the walls of a microchannel in the laminar flow regime is investigated. The governing equation which was derived by considering the weakly nonlinear coupling between the deformable wall and streaming slip flow is linearized and then the eigenvalue problem is solved by a numerical code together with the associated interface and slip velocity boundary conditions. The value of the critical Reynolds number was found to be near 1,441 for a Knudsen number being 0.001 (associated with a physical parameter K ₀ characterizing the SAW effect) which is much smaller than the static-wall case for conventional pressure-driven flows.     

Mixed Convection Heat Transfer in the Annulus Between Two Concentric Vertical Cylinders Using Porous Layers

A numerical investigation of the steady-state, laminar, axi-symmetric, mixed convection heat transfer in the annulus between two concentric vertical cylinders using porous inserts is carried out. The inner cylinder is subjected to constant heat flux and the outer cylinder is insulated. A finite volume code is used to numerically solve the sets of governing equations. The Darcy–Brinkman–Forchheimer model along with Boussinesq approximation is used to solve the flow in the porous region. The Navier–Stokes equation is used to describe the flow in the clear flow region. The dependence of the average Nusselt number on several flow and geometric parameters is investigated. These include: convective parameter, λ, Darcy number, Da, thermal conductivity ratio, K _r, and porous-insert thickness to gap ratio (H/D). It is found that, in general, the heat transfer enhances by the presence of porous layers of high thermal conductivity ratios. It is also found that there is a critical thermal conductivity ratio on which if the values of Kr are higher than the critical value the average Nusselt number starts to decrease. Also, it found that at low thermal conductivity ratio (K _r ≈ 1) and for all values of λ the porous material acts as thermal insulation.     

Effect of T-stress on crack growth under mixed mode I–III loading

For a crack subjected to combined mode I and III loading the influence of a T-stress is analyzed, with focus on crack growth. The solid is a ductile metal modelled as elastic–plastic, and the fracture process is represented in terms of a cohesive zone model. The analyzes are carried out for conditions of small scale yielding, with the elastic solution applied as boundary conditions on the outer edge of the region analyzed. For several combinations of the stress intensity factors K_I and K_III and the T-stress crack growth resistance curves are calculated numerically in order to determine the fracture toughness. In all situations it is found that a negative T-stress adds to the fracture toughness, whereas a positive T-stress has rather little effect. For given values of K_I and T the minimum fracture toughness corresponds to K_III = 0.     

The effect of magnetic field on two-phase liquid metal flow

In this paper, the basic equations of two-phase liquid metal flow in a magnetic field are derived, and specifically, two-phase liquid metal MHD flow in a rectangular channel is studied, and the expressions of velocity distribution of liquid and gas phases and the ratioK ₀ of the pressure drop in two-phase MHD flow to that in single-phase are derived. Results of calculation show that the ratioK ₀ is smaller than unity and decreases with increasing void fraction and Hartmann number because the effective electrical conductivity in the two-phase case decreases. The Project is supported by the National Natural Science Foundation of China.     

COLLATION AND UPWINDING FOR THERMAL FLOW IN PIPELINES: THE LINEARIZED CASE

Simulating thermal effects in pipeline flow involves solving a coupled non-linear system of first-order hyperbolic equations. The advection term has two large eigenvalues of opposite signs, corresponding to the propagation of high-speed sound waves, and one eigenvalue close to or even equal to zero, representing the much slower fluid flow velocity, which transports temperature. Standard collocation methods work well for isothermal flow in pipelines, but the stagnating eigenvalue causes difficulties when thermal effects are included. In a companion paper we formulate and analyse a new numerical method for the non-linear system which arises in thermal modelling. The new method applies to general coupled systems of non-linear first-order hyperbolic partial differential equations with one degenerate eigenvalue. In the present paper we focus on a linearized constant coefficient form of the thermal flow equations. This substantially simplifies presentation of the error analysis for the numerical scheme. We also include numerical results for the method applied to the fully non-linear system. Both the error analysis and the numerical experiments show that the difficulties that come from the application of standard collocation can be overcome by using upwinded piecewise constant functions for the degenerate component of the solution.     

Spectral methods based on the least dissipative modes for wall bounded MHD flows

We present a new approach for the Spectral Direct Numerical Simulation (DNS) of Low-Rm wall-bounded magnetohydrodynamic (MHD) flows. The novelty is that instead of using bases similar to the usual Chebyshev polynomials, which are easy to implement but incur heavy computational costs to resolve the Hartmann boundary layers that arise along the walls, we use a basis made of elements that already incorporate flow structures such as anisotropic vortices and Hartmann layers. We show that such a basis can be obtained from the eigenvalue problem of the linear part of the governing equations with the problem’s boundary conditions. Since this basis is not always orthogonal, we develop a spectral method for non-orthogonal bases. We then demonstrate the efficiency of this method on the simple case of a laminar channel flow with periodic forcing. In particular, we show that this method eliminates the computational costs incurred this Hartmann layer, and this for arbitrary high magnetic fields B. We then discuss the application of our method to nonlinear, turbulent flows for which the number of modes required to resolve the flow completely decreases strongly when B increases, instead of increasing as in the case of currently employed Chebyshev-based methods.     

Lubricated squeezing flow of thin slabs of wheat flour dough: comparison of results at constant plate speed and constant extension rates

Lubricated squeezing flow experiments on wheat flour dough have, until now, mostly been performed in constant plate speed mode (CPS), i.e. at a permanently increasing extension rate. We have compared the results obtained under the CPS and constant extension rate (CER) modes using one of the very few commercial rheometers that allow operation in the CER mode. In both cases, and at any constant biaxial strain, a power law could be fitted to the stress versus extension rate data, the “consistency index” (K) increasing continuously with the strain and the “flow behaviour index” (n) being constant only up to a low strain value (≈0.25) and then decreasing. When compared to the CER mode, the CPS mode produced higher K and n values. For wheat flour doughs, an increase in K with extension may be associated to a strain-hardening phenomenon but the roles of viscoelasticity and lubricant thinning are discussed.     

Effect of pressure-dependent slip on flow curve multiplicity

Various microstructural pictures for slip at polymer/solid interfaces lead to relations which have a region where multiple values of slip velocity are predicted for the same shear stress. This leads to the expectation of multivalued flow curves, which has been verified in specific cases by numerous researchers. We study the effect of pressure dependence on flow curve multiplicity using a simple multivalued slip relation to model the phenomena of hysteresis and spurt flow in polymer extrusion. A continuation technique is used to trace out the boundaries of the region of flow curve multiplicity as pressure drop and die length to diameter (L/D) ratio are changed. Results for Newtonian, shear thinning and viscoelastic constitutive equations show that, despite the multivalued nature of the slip model, multiplicity (and thus hysteresis) is absent at high L/D.  For the sake of completeness, we also carry out time-dependent simulations at constant piston speed taking fluid compressibility into account. These simulations show that oscillations in the pressure drop and exit volumetric flow rate result only if the system is operated in the multiplicity region of the steady state flow curve, in agreement with the results of similar simulations by researchers using various multivalued slip models without pressure dependence. The results demonstrate that a multivalued slip model does not guarantee multiplicity in the flow curve for the constant pressure drop operation, nor oscillations for constant piston speed operation. Received: 18 August 1997 Accepted: 30 March 1998     

Optimal anisotropic three-phase conducting composites: Plane problem

The paper establishes tight lower bound for effective conductivity tensor K₁ of two-dimensional three-phase conducting anisotropic composites and defines optimal microstructures. It is assumed that three materials are mixed with fixed volume fractions and that the conductivity of one of the materials is infinite. The bound expands the Hashin–Shtrikman and translation bounds to multiphase structures, it is derived using a combination of translation method and additional inequalities on the fields in the materials; similar technique was used by Nesi, 1995, Cherkaev, 2009 for isotropic multiphase composites. This paper expands the bounds to the anisotropic composites with effective conductivity tensor K₁. The lower bound of conductivity (G-closure) is a piece-wise analytic function of eigenvalues of K₁, that depends only on conductivities of components and their volume fractions. Also, we find optimal microstructures that realize the bounds, developing the technique suggested earlier by Albin et al., 2007a, Cherkaev, 2009. The optimal microstructures are laminates of some rank for all regions. The found structures match the bounds in all but one region of parameters; we discuss the reason for the gap and numerically estimate it.     

Contribution of point defects to the temperature anomaly of the yield stress in single crystals of the Ni<Subscript>3</Subscript>Ge alloy

The temperature dependence of the yield stress τ_* Ni ₃ Ge single crystals is studied. The temperature dependence τ_*(T) in the high-temperature region (above 420 K) is found to be conditioned by thermally activated accumulation of the density of non-screw components of superdislocations. Interaction of point defects with edge dislocations and its effect on the temperature anomaly of the yield stress in Ni ₃ Ge single crystals are analyzed. The calculated results are found to agree with experimental data. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 154–161, July–August, 2007.     

Direct Statistical Simulation of a Supersonic Flow of a Binary Mixture of Rarefied Gases around a Transversely Positioned Cylinder

A supersonic flow of a binary mixture of gases in a wide range of rarefaction (from a flow with a Knudsen number K _n = 0.1 to a free-molecular flow) around a cylinder is studied by means of direct statistical Monte Carlo simulations (DSMC method). The influence of a small fraction of heavy particles in a light gas flow on the region of significant nonequilibrium near the cylinder and on the heat flux is considered.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 5, pp. 53–59, September–October, 2005.     

A Note on the Apparent Conductivity of Stratified Porous Media in Unsaturated Steady Flow Above a Water Table

We aim at deriving the apparent unsaturated conductivity (AUC) K ^(ap), defined as the ratio between the mean flux and the mean head gradient in a stratified vadose zone above the water table. This is achieved for steady flow generated by a constant infiltrating flux applied at the soil surface. By adopting the first-order approximation in the two parameters of the conductivity curve, and under a few additional simplifying assumptions, we were able to analytically compute K ^(ap). It is shown that this latter varies between K ^(ap) ≡ K _H (the harmonic mean) at the water table, and K ^(ap) ≡ K _ef (the effective conductivity in gravitational flow) far above the water table. Profiles of the AUC are illustrated, and the impact of parameters values is discussed.     

Predictive Pore-Scale Modeling of Single and Multiphase Flow

We show how to predict flow properties for a variety of rocks using pore-scale modeling with geologically realistic networks. The pore space is represented by a topologically disordered lattice of pores connected by throats that have angular cross-sections. We successfully predict single-phase non-Newtonian rheology, and two and three-phase relative permeability for water-wet media. The pore size distribution of the network can be tuned to match capillary pressure data when a network representation of the system of interest is unavailable. The aim of this work is not simply to match experiments, but to use easily acquired data to estimate difficult to measure properties and to predict trends in data for different rock types or displacement sequences.     

Deformations and momentum-energy complexes

Summary Let be a simply connected region of an Einstein-Riemann space that is filled by the trajectories of a time-oriented normal congruence K. The most general form of the momentum-energy tensor is determined under the assumption that we know the deformation of the metric tensor under transport along K. The time-like eigenvector and associated eigenvalue of this momentum-energy tensor are examined.This paper is part of the RAND-Sponsored-Research studies in Epitactic Cosmography.     

Flow development of xanthan solutions in capillary rheometers

Flow experiments through capillaries with 0.2% xanthan in aqueous solution and 0.1 N NaCl brine were carried out to study the influence of the molecular conformation on the flow development at relatively low shear rates, from 20s^–1 to 400s^–1. Capillaries with a wide range of length-to-diameter ratios, L/D = 4.5 to 1015 were used.The apparent viscosity as a function of L/D at a constant shear rate shows a continuous decrement of the viscosity as L/D increases, until an asymptotic value is reached. The decrement in the apparent viscosity is partially explained in terms of slip. It was found that slip is a function of L/D as well as shear stress, i.e., slip develops during flow, thus inducing spatial anisotropy in the fluid until a stable state is reached. However, the substantial difference in apparent viscosity between short capillaries and capillaries longer than 300 D may be attributed to dominant elongational flow due to the contraction in the small capillaries and slip in long capillaries.The flow in a sufficiently long capillary can be divided in four regions rather than three, as is usually assumed. In the first region, which corresponds to the entry, elongational and shear flow coexist and elongational flow dominant. In the second region, end effects and slip development are coupled. In the third region the flow is fully developed and end effects are negligible. However, the fluid shows physical characteristics different from those of the fluid at rest, as a consequence of prior slip development. The fourth zone is the exit region in which the velocity rearranges due to the change of boundary conditions. The length of each region depends on the conformation of the macromolecules and shear rate. In addition, it was found that the stiffness of xanthan increases with the increase of the ionic strength.Finally, a performance of Bagley's analysis in the whole range of L/D studied showed that the use of the Bagley correction is not a reliable way to correct for end effects when the flow is not fully developed and/or in the presence of slip.Dedicated to Arthur S. Lodge at the occasion of his 70th birthday and his retirement from the University of Wisconsin.