A Similarity Solution for Oil Lens Redistribution Including Capillary Forces and Oil Entrapment

Redistribution of a LNAPL lens (oil) at the phreatic surface is described using a multi-phase flow model, with emphasis on the effect of oil entrapment by water. The flow process is analyzed under the assumption that the vertical capillary and gravitational forces balance. Vertical integration leads to explicit functions which approximate the relations between the free oil volume per unit lateral area and the vertically averaged oil relative permeability on the one hand and the vertical position of the interface between zones with either two or three phases on the other hand. A linear relation between the trapped and free oil volume per unit lateral area approximates the vertically integrated nonlinear expression for the trapped oil saturation. The resulting differential equation admits a similarity solution describing the lateral spreading of free oil and the amount and location of trapped oil. Comparison with illustrative numerical computations, which are based on the nonreduced flow model in a two-dimensional planar or axisymmetric domain, shows that the analytical solution provides a good approximation of the free oil distribution at all later times.     

On Aerodynamic Forces for Viscous Compressible Flow

The paper is aimed at reviewing and adding some new results to our recent work on a force theory for viscous compressible flows around a finite body. It has been proposed to analyze aerodynamic forces directly in terms of fluid elements of nonzero vorticity and density gradient. Let ρ denote the density, u the velocity, and ω the vorticity. It is demonstrated that for largely separated flows about bluff bodies, there are two major source elements: R _e(x) =−?u ²∇ρ·∇ϕ and V _e(x) =−u×ω·∇ϕ, where ϕ is an acyclic potential, generated by the solid body moving with unit velocity in the negative direction of the force considered. In particular, under mild conditions, the (unique) choice of ϕ enforces that the elements R _e(x) and V _e(x) decay rapidly away from the body. Four kinds of finite body are considered: a circular cylinder, a sphere, a hemi sphere-cylinder, and a delta wing of elliptic section—all in transonic-to-supersonic regimes. From an extensive numerical study carried out for these bodies, it is found that these two elements contribute to 95% or more of the total drag or lift for all the cases under consideration. Moreover, R _e(x) due to density gradient becomes progressively important relative to V _e(x) due to vorticity as the Mach number increases. The present method of force analysis enables effective analysis and assessment of relative importance of aerodynamics forces, contributed from individual flow structures. The analysis could therefore be very much useful in view of the rapid growth in numerical fluid dynamics; detailed (either local or global) flow information is often available. The paper is dedicated to Sir James Lighthill in honor of his great contributions to aeronautics on the occasion of the publication of his collected works. Received 3 January 1997 and accepted 11 April 1997     

An Approximate Analytical Solution for Counter-Current Spontaneous Imbibition

An approximate analytical solution is provided for one-dimensional, counter- current, spontaneous imbibition of a wetting phase (water) into a semi-infinite porous medium. The solution is based on the assumption that a similarity solution exists for the displacement process. This assumption, in turn, rests on the assumption that the set of relative permeability and capillary pressures curves are unique functions of saturation and do not depend on the nature of the displacement. It further rests on the assumption that the saturation at the imbibition face does not vary with time. It is demonstrated that the solution is in agreement with results obtained from experiments and also numerical analyses of these experiments. The experiments utilize cylindrical samples with the radial surface and one end-face sealed, and with counter-current imbibition occurring at the open end-face. The stage of the experiment that is modeled by the present solution is the period before the imbibition front contacts the sealed end-face. An important finding of the present analysis is that the pressure upstream of the advancing invasion front is a constant. A second, improved solution is also presented; this solution is an iterative, series solution of an integral-differential equation. It converges to a stable solution in very few terms.     

An Analytical Solution for an Exponential-Type Dispersion Process

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Analytical Solution for the Effect of Viscous Dissipation on Mixed Convection in Saturated Porous Media

A new analytical solution is introduced for the effect of viscous dissipation on mixed convection flow and heat transfer about an isothermal vertical wall embedded in Darcy and non-Darcy porous media with uniform free stream velocity. The effect of viscous dissipation on mixed convection in both regimes has been analyzed for both the aiding and opposing flows using Gebhart number, Ge _x =gx/c _p. The governing parameters are Re, Ra, Pe and Ge _x . The case of Re=0 corresponds to Darcy mixed convection region and Re/Pe is identified as the mixed convection governing parameter, Ra=0 leading to pure forced convection. A good agreement was found between the numerical and analytical solutions. It was found from the Nusselt number results that viscous dissipation lowers the heat transfer rate in both Darcy and Forchheimer flow regimes for aiding as well as opposing flows.     

An Analytical Solution for Contaminant Transport in nonuniform Flow

We obtain an analytical solution for two-dimensional steady state mass transport in a trapezoidal embankment in a spatially varying velocity field through its replacement with a hydrologically equivalent rectangular embankment. Application of the Dupuit approximation and conform transformation allow for computation of the concentration field in the resulting rectangle in the complex potential plane. The latter allows deriving expression for the mass flow rate of contaminants, which is analogous to the Dupuit--Forchheimer discharge formula for volumetric water flow rate. Numerical simulation of advection- dispersion in the actual domain compares favorably with these analytical results, and provides limits of the ratio between transverse and longitudinal dispersivities within which the Dupuit approximation is applicable to mass transport problems.     

Moisture Absorption in Capillary Active Materials: Analytical Solution for a Multiple Step Diffusivity Function

This paper deals with the double-constraint methodology for calibration of steady-state groundwater flow models. The methodology is based on updating the hydraulic conductivity of the model domain by comparing the results of two forward groundwater flow models: a model in which known fluxes are specified as boundary conditions and a model in which known heads are specified as boundary conditions. A new zone-integrated double-constraint approach is presented by partitioning the model domain in zones with presumed constant hydraulic conductivity (soft data), and the double-constraint methodology is reformulated accordingly. The feasibility of the method is illustrated by a practical case study involving a numerical steady-state groundwater flow model with about 3 million grid blocks, subdivided into four zones corresponding to the major hydrogeological formations. The results of the zone-integrated double-constraint method for estimating the horizontal and vertical hydraulic conductivities of the zones compare favourably with a classical model calibration based on minimisation of the differences between calculated and measured heads, while the double-constraint method proves to be more robust and computationally less cumbersome.     

An Analytical Solution Method for Composite Layer Diffusion Problems with an Application in Metallurgy

An analytical solution method for composite layer diffusion in cylindrical geometry is studied, relying on local analytical solutions for the single material layers combined with a numerical solution scheme for the material boundary states. The one-layer submodel was formulated for one-dimensional geometry and constant material properties. An efficient algorithm for the arising local Sturm-Liouville eigenvalue problems is developed on the basis of the calculus of variations and the JWKB-technique known from quantum mechanics. Two example heat conduction calculations of the temperature profile over a composite solid were carried out. The results of the examples were in satisfactory agreement with those of previously published calculations using alternative methods. This, together with the computational benefits of obtaining the eigenvalue spectrum for the composite medium one layer at a time, demonstrates the feasibility of the adopted technique for situations requiring an analytical solution. The method was applied to thermal state computation for the lining of a metallurgical ladle, the results showing good consistency with thermocouple measurements.     

Analytical Solutions to Multiphase First-Contact Miscible Models with Viscous Fingering

In this paper, we analyze an empirical model of viscous fingering for three-component, two-phase, first-contact miscible flows. We present the complete range of analytical solutions to secondary and tertiary water-alternating-gas (WAG) floods. An important ingredient in the construction of analytical solutions is the presence of detached (nonlocal) branches of the Hugoniot locus, that is, curves in composition space that satisfy the Rankine–Hugoniot conditions but do not contain the reference state. We illustrate how, in water–solvent floods into a medium with mobile water and residual oil (immobile to water), the solvent front and the water Buckley–Leverett front may interact, resulting in a leading water/solvent shock that is stable to viscous fingering. The analytical solutions explain why in these miscible tertiary floods, oil and solvent often break through simultaneously. We discuss the implications of the new solutions in the design of miscible tertiary floods, such as the estimation of the optimum WAG ratio.     

An Analytical Solution for 1-D Non-Darcy Flow Through Slanting Coarse Deposits

In this paper, an analytical solution to solve 1-D partial differential equation is presented for fully developed turbulent flow through highly permeable sloping deposited porous medium. The present solution will be applicable for a wide range of slopes varying from zero to relatively steep slopes. To confirm the solution, the analytical results have been validated using two sets of experimental data including rounded and crushed material. To see the compatibility of solution, a Darcy-based form of the solution is derived and compared with proposed solution and experimental data. The results showed a satisfactory agreement with experimental records from water surface profiles through rock cavities for both rounded and crushed rock materials. Finally, it may be concluded that the proposed solution could be used to analyze water surface profiles and normal depth in such slanting permeable porous media. This solution provides a reliable realization of the flow profiles in porous materials which are widely used in open-channel flow concepts.     

含边裂纹矩形截面偏心柱最大挠度的一个解析解

本文采用Rayleigh-Ritz能量变分法，计算分析了两端铰支含边裂纹矩形截面偏心柱的弹性挠度。首先选取三角函数级数作为柱挠度的试函数；然后分别计算弹性体系弯曲变形能和裂纹引起的变形能增量以及外边势能，进而得到体系的总势能；最终根据势能驻值条件确定挠度系数，从而得到一个在裂纹截面满足变形协调条件的挠度方程级数解。在假设裂纹位于柱中间截面的基础上，进一步分析推出了最大挠度的解析公式。文中还就本文解与相应的Okamura解进行了对比分析，指出了Okamura解存在的缺点和适用范围。     

Analytical Solution for Sliding Rounded-Edge Contact

Reliable analysis of the local stress fields in the vicinity of sliding frictional contacts of engineering components is a pre-requisite for the reliable assessment of structural integrity. In this paper we present the use of Muskhelishvili potentials to derive an analytical solution for a semi-infinite punch with a rounded edge pressed against a half-pane substrate, and a general numerical solution for a kind of Cauchy integral involved in the analytical contact solution. Using these solutions, the effect of the friction coefficient on the normal traction distribution is investigated. Numerical results show that the peak normal traction value is altered in comparison with the frictionless case solution, but this variation is mild (less than 5%), provided the friction coefficient does not exceed about 0.6. Finally, the adaptation of the analytical result to the solution of practical contact problems is addressed.     

含中心裂纹压电体的二维精确解

解析地研究了含中心裂纹的压电体，它在无穷远处承受机电载荷，并在裂面上满足由Parton和Kudryavtsev以及Hao和Shen提出的绝对电边界条件。在平面应变假设下，给出其二维精确解，并提供了机械应变能释放率和裂尖能量释放率等数值结果。考虑工业应用范围之内常用的远场载荷时，由绝对电边界条件得出的能量释放率表现出明显的非线性特征及载荷相关性，而不是完全与电场无关，这一结论与Xu和Rajapakse在较小载荷下得到的规律不同。     

An Analytical Solution Evaluating Steady-State Plumes of Sequentially Reactive Contaminants

A steady state version of an analytical solution of the reactive flow and transport problem is derived. This solution allows a better understanding of the steady-state plume characteristics, and allows the prediction of the effectiveness of natural attenuation of sequentially reactive contaminants. The sequential reactions are assumed to be first order. Using spatial moments as compact information of plume characteristics, the total mass, plume centroids, and extents of each contaminant plume are derived as functions of dispersivity, velocity, and reaction rates. The steady-state plume centroids and extents are independent of stoichiometric coefficients. The solution is demonstrated for a four-species sequentially reactive transport in a one-dimensional column. The extension to three dimensions is easily made and does not change the basic functions.     

有限长滑动轴承非线性油膜力的近似解法

本文中提出了一种求解有限长径向滑动轴承非线性油膜力的近似解析方法.在滑动轴承-转子系统非线性动力行为分析中,油膜力计算模型通常采用"π"油膜假设,但是,实际工况中油膜的存在区域并非是"π"区域,运行时油膜中出现气穴,破裂成条纹状(即具有Reynolds边界条件).本文中的近似解析方法采用Reynolds边界条件,基于变分原理,运用分离变量法求解油膜的压力分布,其中油膜压力的周向分离函数通过无限长轴承的油膜压力分布获得,油膜的破裂终止位置角通过连续条件确定,轴向分离函数运用变分原理并结合周向函数求得.计算结果表明:本文中提出的方法和有限元方法的结果吻合得很好.在此基础上,分析了一些轴承参数对油膜压力分布的影响.     

应变软化土体中柱形扩孔解析

给出子考虑剪胀作用的线性软化土体中柱形孔扩张的应力、位移场解析解;根据体积平衡方程导出极限护孔压力和最大塑性区半径的解析表达式,较好地弥补了经典理论的不足;计算讨论了土体软化和剪胀特性对柱形扩孔的影响。     

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.     

An Analytical Solution for Unsteady Flow in a Phreatic Aquifer in the Case of Continuous Rise

In this study, we examine the classical problem of unsteady flow in a phreatic aquifer, induced by continuous rise of the water flux and head on its boundary. A closed-form analytical solution for the governing Boussinesq equation is derived for a semi-infinite aquifer.     

Analytical Solution Incorporating History-Dependent Processes for Quick Assessment of Capillary Trapping During CO<Subscript>2</Subscript> Geological Storage

Carbon storage in saline formations is considered as a promising option to ensure the necessary decrease of CO₂ anthropogenic emissions. Its industrial development in those formations is above all conditioned by its safety demonstration. Assessing the evolution of trapped and mobile CO₂ across time is essential in the perspective of reducing leakage risks. In this work, we focus on residual trapping phenomenon occurring during the wetting of the injected CO₂ plume. History dependent effects are of first importance when dealing with capillary trapping. We then apply the classical fractional flow theory (Buckley–Leverett type model) and include trapping and hysteresis models; we derive an analytical solution for the temporal evolution of saturation profile and of CO₂ trapped quantity when injecting water after the gas injection (“artificial imbibition”). The comparison to numerical simulations for different configurations shows satisfactory match and justifies, in the case of industrial CO₂ storage, the assumptions of incompressible flow with no consideration of capillary pressure. The obtained analytical solution allows the quick assessment of both the quantity and the location of mobile gas left during imbibition.