Analytical solution to one-dimensional consolidation in unsaturated soils

This paper presents an analytical solution of the one-dimensional consolidation in unsaturated soil with a finite thickness under vertical loading and confinements in the lateral directions. The boundary contains the top surface permeable to water and air and the bottom impermeable to water and air. The analytical solution is for Fredlund＇s one-dimensional consolidation equation in unsaturated soils. The transfer relationship between the state vectors at top surface and any depth is obtained by using the Laplace transform and Cayley-Hamilton mathematical methods to the governing equations of water and air, Darcy＇s law and Fick＇s law. Excess pore-air pressure, excess pore-water pressure and settlement in the Laplace-transformed domain are obtained by using the Laplace transform with the initial conditions and boundary conditions. By performing inverse Laplace transforms, the analytical solutions are obtained in the time domain. A typical example illustrates the consolidation characteristics of unsaturated soil from an- alytical results. Finally, comparisons between the analytical solutions and results of the finite difference method indicate that the analytical solution is correct.     

Theoretical analysis on hydraulic transient resulted by sudden increase of inlet pressure for laminar pipeline flow

Hydraulic transient, which is resulted from sudden increase of inlet pressure for laminar pipeline flow, is studied. The partial differential equation, initial and boundary conditions for transient pressure were constructed, and the theoretical solution was obtained by variable-separation method. The partial differential equation, initial and boundary conditions for flow rate were obtained in accordance with the constraint correlation between flow rate and pressure while the transient flow rate distribution was also solved by variable-separation method. The theoretical solution conforms to numerical solution obtained by method of characteristics (MOC) very well.     

Analytical solution of a double moving boundary problem for nonlinear flows in one-dimensional semi-infinite long porous media with low permeability简

Based on Huang＇s accurate tri-sectional nonlin- ear kinematic equation （1997）, a dimensionless simplified mathematical model for nonlinear flow in one-dimensional semi-infinite long porous media with low permeability is presented for the case of a constant flow rate on the inner boundary. This model contains double moving boundaries, including an internal moving boundary and an external mov- ing boundary, which are different from the classical Stefan problem in heat conduction： The velocity of the external moving boundary is proportional to the second derivative of the unknown pressure function with respect to the distance parameter on this boundary. Through a similarity transfor- mation, the nonlinear partial differential equation （PDE） sys- tem is transformed into a linear PDE system. Then an ana- lytical solution is obtained for the dimensionless simplified mathematical model. This solution can be used for strictly checking the validity of numerical methods in solving such nonlinear mathematical models for flows in low-permeable porous media for petroleum engineering applications. Finally, through plotted comparison curves from the exact an- alytical solution, the sensitive effects of three characteristic parameters are discussed. It is concluded that with a decrease in the dimensionless critical pressure gradient, the sensi- tive effects of the dimensionless variable on the dimension- less pressure distribution and dimensionless pressure gradi- ent distribution become more serious; with an increase in the dimensionless pseudo threshold pressure gradient, the sensi- tive effects of the dimensionless variable become more serious; the dimensionless threshold pressure gradient （TPG） has a great effect on the external moving boundary but has little effect on the internal moving boundary.     

FINITE DIFFERENCE METHOD OF TRANSINET NONLINEAR FREE SURFACE WAVE PROBLEMS

A finite difference method is developed for computing the two-dimensional trantsientpotential flow generated by an impulse on the free surface.Both the dynamic and kinematicfree surface conditions are considered in nonlinear version.the primary features of thepresent paper include the use of special coordinates transformations so that the geometry ofthe flow field is transformed into a time-invariant region,presents an iteration process.bywhich the velocity potential is computed as the solution of a Poisson equation.theapplication of fast Fourier transform(FFT)technique results in a tri-diagonal system ofequations which can be readily solved by the Thomas algorithm,the computing time issignificantly reduced.Thus an efficient technique for handling the transient potentialproblems is well justified.The feasibility of the present method has been verified by twoexamples including different initial disturbances respectively.     

An investigation of an Emden-Fowler equation from thin film flow

A third-order ordinary differential equation (ODE) for thin film flow with both Neumann and Dirichlet boundary conditions is transformed into a second-order nonlinear ODE with Dirichlet boundary conditions.Numerical solutions of the nonlinear second-order ODE are investigated using finite difference schemes.A finite difference formulation to an Emden-Fowler representation of the second-order nonlinear ODE is shown to converge faster than a finite difference formulation of the standard form of the second-order nonlinear ODE.Both finite difference schemes satisfy the von Neumann stability criteria.When mapping the numerical solution of the second-order ODE back to the variables of the original third-order ODE we recover the position of the contact line.A nonlinear relationship between the position of the contact line and physical parameters is obtained.     

A hybrid vertex-centered finite volume/element method for viscous incompressible flows on non-staggered unstructured meshes

This paper proposes a hybrid vertex-centered finite volume/finite element method for solution of the two dimensional (2D) incompressible Navier-Stokes equations on unstructured grids.An incremental pressure fractional step method is adopted to handle the velocity-pressure coupling.The velocity and the pressure are collocated at the node of the vertex-centered control volume which is formed by joining the centroid of cells sharing the common vertex.For the temporal integration of the momentum equations,an implicit second-order scheme is utilized to enhance the computational stability and eliminate the time step limit due to the diffusion term.The momentum equations are discretized by the vertex-centered finite volume method (FVM) and the pressure Poisson equation is solved by the Galerkin finite element method (FEM).The momentum interpolation is used to damp out the spurious pressure wiggles.The test case with analytical solutions demonstrates second-order accuracy of the current hybrid scheme in time and space for both velocity and pressure.The classic test cases,the lid-driven cavity flow,the skew cavity flow and the backward-facing step flow,show that numerical results are in good agreement with the published benchmark solutions.     

THE METHOD OF CONTINUOUS DISTRIBUTION OF SINGULARITIES TO TREAT THE STOKES FLOW OF THE ARBITRARY OBLATE AXISYMMETRICAL BODY

This paper deals with the Stokes flow of the arbitrary oblate axisymmetrical body bymeans of constant density and quadratic distribution function approximation for themethod of continuous distribution of singularities.The Sampson spherical infinite series arechosen as fundamental singularities.The convergence.accuracy and range of application ofboth two approximations are examined by the unbounded Stokes flow past the oblatespheroid.It is demonstrated that the drag factor and pressure distribution both conformwith the exact solution very well.Besides.the properties,accuracy and the range ofapplication are getting better with the improving of the approximation of the distributionfunction.As an example of the arbitrary oblate axisymmetrical bodies,the Stokes flow ofthe oblate Cassini oval are calculated by these two methods and the results are convergentand consistent.Finally,with the quadratic distribution approximation the red blood cell.which has physiologic meaning.is considered and for the first time     

High-order boundary conditions for the problems of Laplace equation in infinite region and their application

The high-order boundary conditions for the problems of Laplace equation in infiniteregion have been developed.The improvement in accuracy for numerical solution isachieved by imposing the high-order boundary conditions on the exterior boundary of areduced finite region in which the numerical method is used.So both the computing effortsand the required storage in computer are reduced.The numerical examples show that thelst-order boundary condition approaches to the exact boundary condition and it is clearlysuperior to the traditional boundary condition and the2nd-order boundary condition.     

Semi-analytical solution to one-dimensional consolidation in unsaturated soils

In this paper, a series of semi-analytical solutions to one-dimensional consolidation in unsaturated soils are obtained. The air governing equation by Fredlund for unsaturated soils consolidation is simplified. By applying the Laplace transform and the Cayley-Hamilton theorem to the simplified governing equations of water and air, Darcy＇s law, and Fick＇s law, the transfer function between the state vectors at top and at any depth is then constructed. Finally, by the boundary conditions, the excess pore-water pressure, the excess pore-air pressure, and the soil settlement are obtained under several kinds of boundary conditions with the large-area uniform instantaneous loading. By the Crump method, the inverse Laplace transform is performed, and the semi-analytical solutions to the excess pore-water pressure, the excess pore-air pressure, and the soils settlement are obtained in the time domain. In the case of one surface which is permeable to air and water, comparisons between the semi-analytical solutions and the analytical solutions indicate that the semi-analytical solutions are correct. In the case of one surface which is permeable to air but impermeable to water, comparisons between the semi-analytical solutions and the results of the finite difference method are made, indicating that the semi-analytical solution is also correct.     

Analysis of characteristics of pressure curve in gas-condensate wells

This paper utilizes a flow equation with a sink item that describes the characteristics of pressure-time chart when the pressure is higher than the maximum condensate pressure.We have established a sink item to show the influence of accumulation of condensate liquid according to Duhamet Principle of Superposition,and introduced two coefficients for it:condensing strength R_D and condensing relaxation timeλ_D.This paper gives the principle and the quantitative expression of the well pressure influenced by condensate function in the flow equation.An analytical solution for an infinite system is obtained(constant rate).These results can be used to analyse the unsteady flow test of constant production.     

考虑二次梯度项及动边界的双重介质低渗透油藏流动分析

在传统试井模型的非线性偏微分方程中根据弱可压缩流体的假设,忽略了二次梯度项,对于低渗透油藏这种方法是有疑问的.低渗透问题一个显著的特点就是流体的流动边界随着时间不断向外扩展.为了更好地研究双重介质低渗透油藏中流体的流动问题,考虑了二次梯度项及活动边界的影响,同时考虑了低渗透油藏的非达西渗流特征,建立了双重介质低渗透油藏流动模型.采用Douglas-Jones预估-校正差分方法获得了无限大地层定产量生产时模型的数值解,分别讨论了不同参数变化时压力的变化规律及活动边界随时间的传播规律,还分析了考虑和忽略二次梯度项影响时模型数值解之间的差异随时间的变化规律,做出了典型压力曲线图版,这些结果可用于实际试井分析.     

Effect of quadratic pressure gradient term on a one-dimensional moving boundary problem based on modified Darcy’s law

A relatively high formation pressure gradient can exist in seepage flow in low-permeable porous media with a threshold pressure gradient, and a significant error may then be caused in the model computation by neglecting the quadratic pressure gradient term in the governing equations. Based on these concerns, in consideration of the quadratic pressure gradient term, a basic moving boundary model is constructed for a one-dimensional seepage flow problem with a threshold pressure gradient. Owing to a strong nonlinearity and the existing moving boundary in the mathematical model, a corresponding numerical solution method is presented. First, a spatial coordinate transformation method is adopted in order to transform the system of partial differential equations with moving boundary conditions into a closed system with fixed boundary conditions; then the solution can be stably numerically obtained by a fully implicit finite-difference method. The validity of the numerical method is verified by a published exact analytical solution. Furthermore, to compare with Darcy’s flow problem, the exact analytical solution for the case of Darcy’s flow considering the quadratic pressure gradient term is also derived by an inverse Laplace transform. A comparison of these model solutions leads to the conclusion that such moving boundary problems must incorporate the quadratic pressure gradient term in their governing equations; the sensitive effects of the quadratic pressure gradient term tend to diminish, with the dimensionless threshold pressure gradient increasing for the one-dimensional problem.     

基于渗透率连续变化的低渗透多孔介质非线性渗流模型

基于低渗透多孔介质渗透率的渐变理论,确定了能精确描述低渗透多孔介质渗流特征的非线性运动方程,并通过实验数据拟合．验证了非线性运动方程的有效性。非线性渗流速度关于压力梯度具有连续-阶导数,方便于工程计算;由此建立了低渗透多孔介质的单相非线性径向渗流数学模型,并巧妙采用高效的Douglas-Jones预估一校正有限差分方法求得了其数值解。数值结果分析表明：非线性渗流模型为介于拟线性渗流模型和达西渗流模型之间的一种中间模型或理想模型,非线性渗流模型和拟线性渗流模型均存在动边界;拟线性渗流高估了启动压力梯度的影响,使得动边界的移动速度比实际情况慢得多;非线性越强,地层压力下降的范围越小,地层压力梯度越陡峭,影响地层压力的敏感性减弱,而影响地层压力梯度的敏感性增强。     

On unsteady unidirectional flows of a second grade fluid

Some properties of unsteady unidirectional flows of a fluid of second grade are considered for flows produced by the sudden application of a constant pressure gradient or by the impulsive motion of one or two boundaries. Exact analytical solutions for these flows are obtained and the results are compared with those of a Newtonian fluid. It is found that the stress at the initial time on the stationary boundary for flows generated by the impulsive motion of a boundary is infinite for a Newtonian fluid and is finite for a second grade fluid. Furthermore, it is shown that initially the stress on the stationary boundary, for flows started from rest by sudden application of a constant pressure gradient is zero for a Newtonian fluid and is not zero for a fluid of second grade. The required time to attain the asymptotic value of a second grade fluid is longer than that for a Newtonian fluid. It should be mentioned that the expressions for the flow properties, such as velocity, obtained by the Laplace transform method are exactly the same as the ones obtained for the Couette and Poiseuille flows and those which are constructed by the Fourier method. The solution of the governing equation for flows such as the flow over a plane wall and the Couette flow is in a series form which is slowly convergent for small values of time. To overcome the difficulty in the calculation of the value of the velocity for small values of time, a practical method is given. The other property of unsteady flows of a second grade fluid is that the no-slip boundary condition is sufficient for unsteady flows, but it is not sufficient for steady flows so that an additional condition is needed. In order to discuss the properties of unsteady unidirectional flows of a second grade fluid, some illustrative examples are given.     

Unsteady unidirectional micropolar fluid flow

This paper considers the unsteady unidirectional flow of a micropolar fluid, produced by the sudden application of an arbitrary time dependent pressure gradient, between two parallel plates. The no-slip and the no-spin boundary conditions are used. Exact solutions for the velocity and microrotation distributions are obtained based on the use of the complex inversion formula of Laplace transform. The solution of the problem is also considered if the upper boundary of the flow is a free surface. The particular cases of a constant and a harmonically oscillating pressure gradient are then examined and some numerical results are illustrated graphically.     

Nonlinear Pressure Diffusion in Flow of Compressible Liquids Through Porous Media

In the flow of liquids through porous media, nonlinear effects arise from the dependence of the fluid density, porosity, and permeability on pore pressure, which are commonly approximated by simple exponential functions. The resulting flow equation contains a squared gradient term and an exponential dependence of the hydraulic diffusivity on pressure. In the limiting case where the porosity and permeability moduli are comparable, the diffusivity is constant, and the squared gradient term can be removed by introducing a new variable y, depending exponentially on pressure. The published transformations that have been used for this purpose are shown to be special cases of the Cole–Hopf transformation, differing in the choice of integration constants. Application of Laplace transformation to the linear diffusion equation satisfied by y is considered, with particular reference to the effects of the transformation on the boundary conditions. The minimum fluid compressibilities at which nonlinear effects become significant are determined for steady flow between parallel planes and cylinders at constant pressure. Calculations show that the liquid densities obtained from the simple compressibility equation of state agree to within 1% with those obtained from the highly accurate Wagner-Pru?  equation of state at pressures to 20 MPa and temperatures approaching 600 K, suggesting possible applications to some geothermal systems.     

考虑二次梯度项影响的三孔双渗模型渗流特征

传统的试井模型与物质平衡方程都是不一致的.在非线性偏微分方程中,根据弱可压缩液体的假设,忽略二次梯度项,在试井较长时间内将产生误差.由于二次梯度项的存在,描述多孔介质微可压缩液体流动的压力分布的偏微分方程是非线性的.本文对三重介质渗流特征系统,保留了非线性偏微分方程中的所有项,没有忽略二次梯度项,建立了由基岩系统、裂缝系统及溶洞系统组成的考虑井筒储集和表皮效应对压力影响的三孔双渗模型.采用一阶向前差分,二阶中心差分的方法获得了此类三重介质模型的差分方程,用解非线性方程组的Newton迭代法求得了方程组的数值解.分别讨论了三重介质的参数变化时的无因次压力的变化规律,做出了典型无因次压力曲线图.     

The Flow Problem of Fluids Flow in a Fractal Reservoir with Double Porosity

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Analysis of hybrid algorithms for the Navier–Stokes equations with respect to hydrodynamic stability theory

Hybrid three‐dimensional algorithms for the numerical integration of the incompressible Navier–Stokes equations are analyzed with respect to hydrodynamic stability in both linear and nonlinear fields. The computational schemes are mixed—spectral and finite differences—and are applied to the case of the channel flow driven by constant pressure gradient; time marching is handled with the fractional step method. Different formulations—fully explicit convective term, partially and fully implicit viscous term combined with uniform, stretched, staggered and non‐staggered meshes, x‐velocity splitted and non‐splitted in average and perturbation component – are analyzed by monitoring the evolution in time of both small and finite amplitude perturbations of the mean flow. The results in the linear field are compared with correspondent solutions of the Orr–Sommerfeld equation; in the nonlinear field, the comparison is made with results obtained by other authors. Copyright © 2002 John Wiley & Sons, Ltd.