The Transient Flow Analysis of Fluid in a Fractal, Double-Porosity Reservoir

The mathematical model for transient fluid flow in porous media is based in general on mass conservation principle. Because of the small compressibility of formation fluid, the quadratic term of pressure gradient is always ignored to linearize the non-linear diffusion equation. This may result in significant errors in model prediction, especially at large time scale. In order to solve this problem, it may be necessary to keep the quadratic term in the non-linear equations. In our study, the quadratic term is reserved to fully describe the transient fluid flow. Based on this rigorous treatment, the mathematical models are established to analyze the transient flow behavior in a double porosity, fractal reservoir with spherical and cylindrical matrix. In addition, Laplace transformation method is employed to solve these mathematical models and the type curves are provided to analyze the pressure transient characteristics. This study indicates that the relative errors in calculated pressure caused by ignoring the quadratic term may amount to 10?% in a fractal reservoir with double porosity, which can??t be neglected in general for fractal reservoirs with double porosity at large time scale.     

Analysis on nonlinear effect of unsteady percolation in the inhomogeneous shale gas reservoir

The nonlinear effects of unsteady multi-scale shale gas percolation,such as desorption,slippage,diffusion,pressure-dependent viscosity,and compressibility,are investigated by numerical simulation.A new general mathematical model of the problem is built,in which the Gaussian distribution is used to describe the inhomogeneous intrinsic permeability.Based on the Boltzmann transformation,an efficient semi-analytical method is proposed.The problem is then converted into a nonlinear equation in an integral form for the pressure field,and a related explicit iteration scheme is constructed by numerical discretization.The validation examples show that the proposed method has good convergence,and the simulation results also agree well with the results obtained from both numerical and actual data of two vertical fractured test wells in the literature.Desorption,slippage,and diffusion have significant influence on shale gas flows.The accuracy of the usual technique that the product of viscosity and compressibility is approximated as its value at the average formation pressure is examined.     

The symmetrical adhesive contact problem for circular elastic cylinders

The rolling contact problem involving circular cylinders is at the heart of numerous industrial processes, and critical to any elastohydrodynamic lubrication analysis is an accurate knowledge of the associated contact pressure for the static dry problem. In a recent article [1] the authors have obtained new horizontal pressure distributions, both exact and approximate for various problems involving the symmetrical contact of circular elastic cylinders. In [1] it is assumed that only the circumferential horizontal displacement is prescribed in the contact region while the vertical circumferential displacement is left arbitrary and is assumed to take on whatever value is predicted by the deformation. The advantage of this assumption is that the problem reduces to a single singular integral equation which by transformations can be simplified to an integral equation involving the standard finite Hilbert transform. Here we consider the more general displacement boundary value problem within the contact region, and to be specific we examine the problem with zero vertical circumferential displacement and prescribed horizontal circumferential displacement. The solution of this problem involves two coupled singular integral equations for the horizontal and vertical pressure distributions. Basic equations and some approximate analytical solutions are obtained for symmetrical contact of circular elastic cylinders by both parallel plates and circular cylinders which are either rigid or elastic. Numerical results for the approximate analytical solutions are given for contact by rigid parallel plates and rigid circular cylinders.     

Transient Flow of a Compressible Fluid in a Connected Layered Permeable Medium

We develop a semi-analytical model of transient fluid flow in a 2D layered permeable medium with cross-flow between adjacent layers. It is shown that the pressure satisfies a diffusion equation to leading order, even when the non-linear term and gravity are included in the mathematical model. The solution is based on an analytical expression in the transform domain for the fluid pressure in terms of interfacial flux functions; the algorithm to compute the flux functions accepts an arbitrary number of formation layers. We show some benchmark tests that validate the general model; the model is then applied to an example derived from experiments. Numerical experiments confirm the significance of the cross-flow in a particular scaling of the ratio of permeabilities and quantify the influence of the various physical parameters.     

Radial Flow in a Bounded Randomly Heterogeneous Aquifer

Flow to wells in nonuniform geologic formations is of central interest to hydrogeologists and petroleum engineers. There are, however, very few mathematical analyses of such flow. We present analytical expressions for leading statistical moments of vertically averaged hydraulic head and flux under steady-state flow to a well that pumps water from a bounded, randomly heterogeneous aquifer. Like in the widely used Thiem equation, we prescribe a constant pumping rate deterministically at the well and a constant head at a circular outer boundary of radius L. We model the natural logarithm Y = lnT of aquifer transmissivity T as a statistically homogeneous random field with a Gaussian spatial correlation function. Our solution is based on exact nonlocal moment equations for multidimensional steady state flow in bounded, randomly heterogeneous porous media. Perturbation of these nonlocal equations leads to a system of local recursive moment equations that we solve analytically to second order in the standard deviation of Y. In contrast to most stochastic analyses of flow, which require that log transmissivity be multivariate Gaussian, our solution is free of any distributional assumptions. It yields expected values of head and flux, and the variance–covariance of these quantities, as functions of distance from the well. It also yields an apparent transmissivity, T _a, defined as the negative ratio between expected flux and head gradient at any radial distance. The solution is supported by numerical Monte Carlo simulations, which demonstrate that it is applicable to strongly heterogeneous aquifers, characterized by large values of log transmissivity variance. The two-dimensional nature of our solution renders it useful for relatively thin aquifers in which vertical heterogeneity tends to be of minor concern relative to that in the horizontal plane. It also applies to thicker aquifers when information about their vertical heterogeneity is lacking, as is commonly the case when measurements of head and flow rate are done in wells that penetrate much of the aquifer thickness. Potential uses include the analysis of pumping tests and tracer test conducted in such wells, the statistical delineation of their respective capture zones, and the analysis of contaminant transport toward fully penetrating wells.     

Dynamic interaction of a spherical radiator in a fluid-filled cylindrical borehole within a poroelastic formation

Harmonic acoustic radiation from a modally oscillating spherical source positioned at the center of a fluid-filled cylindrical cavity embedded within a fluid-saturated porous elastic formation is studied in an exact manner. The formulation utilizes the Biot theory of dynamic poroelasticity along with the cylindrical to spherical wave-field transformations, and the pertinent boundary conditions to obtain a closed-form series solution. The analytical results are illustrated with a numerical example in which the spherical source, with its polar axis oriented along the main axis of a water-filled borehole and embedded within a water-saturated Ridgefield sandstone formation, is excited in vibrational modes of various orders. The magnitude of the reflected component of acoustic pressure along the axis of the borehole for a pulsating (n = 0), an oscillating (n = 1), and also a multipole (n = 0–3) spherical source as a function of the excitation frequency is calculated and discussed for representative values of the parameters characterizing the system. Special attention is paid to the effects of source excitation frequency, size, surface velocity profile, and internal impedance as well as borehole interface permeability condition on the reflected pressure magnitudes. Limiting cases are considered and fair agreements with well-known solutions are obtained.     

Exact solutions for nonlinear transient flow model including a quadratic gradient term

The models of the nonlinear radial flow for the infinite and finite reservoirs including a quadratic gradient term were presented. The exact solution was given in real space for flow equation including quadratic gradiet term for both constant-rate and constant pressure production cases in an infinite system by using generalized Weber transform.Analytical solutions for flow equation including quadratic gradient term were also obtained by using the Hankel transform for a finite circular reservoir case. Both closed and constant pressure outer boundary conditions are considered. Moreover, both constant rate and constant pressure inner boundary conditions are considered. The difference between the nonlinear pressure solution and linear pressure solution is analyzed. The difference may be reached about 8% in the long time. The effect of the quadratic gradient term in the large time well test is considered.     

用三维弹性力学方法研究圆板的稳定问题

用三维弹性力学方法研究任意边界条件圆板的轴对称稳定问题，利用Ｈ变换和Ｓｔｏｃｋｅｓ变换，导出位移函数及其偏导数的一种新型双重极数式，并由数学弹性定理论的基本方程和边界条件建立的特征方程，求得最小临界载荷的精确解，文末以简支圆板为例进行数字计算，结果表明：在弹性失稳范围内，三维弹性力学方法求得的临界载荷略低于经典理论的结果，对于薄板的弹性稳定问题，经典板理论有足够的精度。     

RIGIDITY OF A CONCAVE TEMPLATE OF RIGHT ANGLE TRAPEZOID CROSS SECTION WITH A CIRCULAR APERTURE

采用线弹性平板理论,对均匀、连续、各向同性材料制成的变横截面为直角梯形的圆孔凹模板刚度计算理论进行了研究. 首先建立力学和数学计算模型,其次,针对承受垂直冲裁力的圆形孔凹模板,建立了用挠度表示的三阶变系数常微分方程,并给出了新的边界条件,然后采用半逆解法求解,进行弯曲问题的挠度计算,进而确定刚度计算理论,最后进行实例分析.     

The general solution for axial symmetrical bending of nonhomogeneous circular plates resting on an elastic foundation

In this paper,a new method,the exact analytic method,is presented on the basis of stepreduction method.By this method,the general solution for the bending of nonhomogenouscircular plates and circular plates with a circular hole at the center resting,on an elastfcfoundation is obtained under arbitrary axial symmetrical loads and boundary conditions.The uniform convergence of the solution is proved.This general solution can also be applieddirectly to the bending of circular plates without elastic foundation.Finally,it is onlynecessary to solve a set of binary linear algebraic equation.Numerical examples are givenat the end of this paper which indicate satisfactory results of stress resultants anddisplacements can be obtained by the present method.     

轴对称横观各向同性热弹性圆柱的分解

为了得到精确的应力场、位移场、温度场,将扭转圆轴的精化理论研究方法推广到轴对称横观各向同性热弹性圆柱。利用Bessel函数以及轴对称横观各向同性热弹性圆柱的通解,给出了轴对称横观各向同性热弹性圆柱的分解定理。根据柱面齐次边界条件获得了精确的精化方程,精化方程可以分解为一阶方程、超越方程、温度方程,从而将横观各向同性热弹性圆柱的轴对称问题分解为轴向拉压问题、超越问题、热-应力耦合问题。超越部分对应端部自平衡情况,可以清晰地了解到端部应力分布对内部应力场的影响,热-应力耦合部分对应无外加应力场时圆柱内部因温度变化引起的热应力。     

General steady-state solutions for transversely isotropic thermoporoelastic media in three dimensions and its application

This paper presents a set of 3D general solutions for thermoporoelastic media for the steady-state problem. By introducing two displacement functions, the equations governing the elastic, pressure and temperature fields are simplified. The operator theory and superposition principle are then employed to express all the physical quantities in terms of two functions, one of which satisfies a quasi–Laplace equation and the other satisfies a differential equation of the eighth order. The generalized Almansi's theorem is used to derive the displacements, pressure and temperature in terms of five quasi-harmonic functions for various cases of material characteristic roots. To show its practical significance, an infinite medium containing a penny-shaped crack subjected to mechanical, pressure and temperature loads on the crack surface is given as an example. A potential theory method is employed to solve the problem. One integro-differential equation and two integral equations are derived, which bear the same structures to those reported in literature. For a penny-shaped crack subjected to uniformly distributed loads, exact and complete solutions in terms of elementary functions are obtained, which can serve as a benchmark for various kinds of numerical codes and approximate solutions.     

Problems of nonlinear flow through porous media in regions with curved boundaries

The investigation of two-dimensional problems of nonlinear flow through porous media in regions with given curved boundaries involves considerable mathematical difficulties due to the fact that neither the hodograph transformation nor transition to the plane of the complex potential leads to a simplification of the problem. Attention is drawn to a transformation which in the case of circular boundaries maps the flow region onto a canonical domain (half-plane), while the transformed system, which remains, in general, nonlinear, has a much simpler structure than the initial system. For a previously introduced model of the resistance law it is shown that it is possible to construct exact solutions of the problem of an eccentric well within a circular supply contour and the problem of a circular group of wells in an annular (circular) reservoir. The dependence of the dimensionless flow rate on the eccentricity is found and the boundaries of the stagnant zones are determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 103–108, January–February, 1989.     

Solution of the laminar duct flow problem by a boundary integral equation method

Summary A boundary integral equation method is proposed for approximate numerical and exact analytical solutions to fully developed incompressible laminar flow in straight ducts of multiply or simply connected cross-section. It is based on a direct reduction of the problem to the solution of a singular integral equation for the vorticity field in the cross section of the duct. For the numerical solution of the singular integral equation, a simple discretization of it along the cross-section boundary is used. It leads to satisfactory rapid convergency and to accurate results. The concept of hydrodynamic moment of inertia is introduced in order to easily calculate the flow rate, the main velocity, and the fRe-factor. As an example, the exact analytical and, comparatively, the approximate numerical solution of the problem of a circular pipe with two circular rods are presented. In the literature, this is the first non-trivial exact analytical solution of the problem for triply connected cross section domains. The solution to the Saint-Venant torsion problem, as a special case of the laminar duct-flow problem, is herein entirely incorporated.     

污染区对部分打开井井底压力响应的影响

钻井造成的污染区对部分打开井的影响不同于裸眼井. 为了分析污染区对部分打开井井底压力响应的影响,建立了一种部分打开井的二维轴对称渗流模型,模型考虑了真实的污染区以及储层渗透率各向异性特征.利用有限元数值方法对模型进行求解,获得了部分打开井的井底压力响应及储层压力分布. 根据压力响应及压力分布特征,将部分打开井的压力响应过程划分为5 个流动阶段,其中早期局部径向流动和椭球流动是该类井最典型的特征. 对污染区的影响分析表明:传统方法中的表皮系数S 并不等于污染区引起的机械表皮系数S_d;无量纲井筒储存系数不能与机械表皮系数组合. 修正了传统方法中部分打开井的井底压力公式,验证了部分打开井的总表皮计算公式,为该类井的井底压力响应解释及产能预测提供理论指导.      

Near-Critical Gas–Liquid Flow in Porous Media: Monovariant Model, Analytical Solutions and Convective Mass Exchange Effects

We examine a class of hydrocarbon reservoirs whose thermodynamic state remains close to the critical point during the all period of reservoir exploitation. Such a situation is typical for the so-called gas–condensate systems, in which the liquid phase is formed from gas when pressure decreases. Due to proximity to critical point, the mixture contains many components which are neutral with respect to the phase state. This determines a low thermodynamic degree of freedom of the system. As the results, the mathematical flow model allows a significant reduction in the number of conservation equations, whatever the number of chemical components. In the vicinity of a well, the system may be reduced to one transport equation for saturation. This nonlinear model yields exact analytical solutions when the flow is self-similar. In more general case of flow, we develop partially linearized solutions which are shown to be sufficiently exact. The spectrum of examined cases covers the flow in a medium with a sharp heterogeneity and a sharp variation in the flow rate. A significant relative gas flow past liquid gives rise to a convective mass exchange phenomenon which appears highly different from that observed in static. In the case of a medium discontinuity, the convective mass exchange gives rise to a phenomenon of condensate saturation billow formation. A sharp variation in the flow rate leads to a hysteretic behavior of the saturation field.     

Pressure distribution in the vicinity of a circular battery of wells operating in the elastic-water-drive filtration regime

We pose the problem of determining the unsteady pressure during combined operation of a circular battery of wells (sinks) in a concentric, uniform, horizontal, circular reservoir under elasticregime conditions, which admits solution by integral-transform methods. An exact solution of the posed problem is presented. A simplified approximate solution of the problem is recommended on the bases of analysis of the exact solution and an example calculation.     

Exact solution of the multi-cracked Euler–Bernoulli column

The use of distributions (generalized functions) is a powerful tool to treat singularities in structural mechanics and, besides providing a mathematical modelling, their capability of leading to closed form exact solutions is shown in this paper. In particular, the problem of stability of the uniform Euler–Bernoulli column in presence of multiple concentrated cracks, subjected to an axial compression load, under general boundary conditions is tackled. Concentrated cracks are modelled by means of Dirac’s delta distributions. An integration procedure of the fourth order differential governing equation, which is not allowed by the classical distribution theory, is proposed. The exact buckling mode solution of the column, as functions of four integration constants, and the corresponding exact buckling load equation for any number, position and intensity of the cracks are presented. As an example a parametric study of the multi-cracked simply supported and clamped–clamped Euler–Bernoulli columns is presented.     

A general solution for Stokes flow and its application to the problem of a rigid plate translating in a fluid

A general solution for 3D Stokes flow is given which is different from, and more compact than the existing ones and more compact than them in that it involves only two scalar harmonic functions. The general solution deduced is combined with the potential theory method to study the Stokes flow induced by a rigid plate of arbitrary shape translating along the direction normal to it in an unbounded fluid.The boundary integral equation governing this problem is derived. When the plate is elliptic, exact analytical results are obtained not only for the drag force but also for the velocity distributions. These results include and complete the ones available for a circular plate. Numerical examples are provided to illustrate the main results for circular and elliptic plates. In particular, the elliptic eccentricity of a plate is shown to exhibit significant influences.     

Pressure Buildup During Supercritical Carbon Dioxide Injection From a Partially Penetrating Borehole into Gas Reservoirs

Injecting CO₂ into a subsurface formation causes a buildup of pressure in the vicinity of the injection well. While a large injection rate can reduce the cost associated with injection, an indefinitely large injection rate can result in excessive formation damage. To obtain an optimal injection rate without exceeding the safe pressure limits, one will like to have some knowledge of the transient pressure buildup characteristics resulting from a particular injection rate. While elaborate numerical simulations can provide reliable pressure buildup predictions, they require extensive knowledge about the formation, which is normally not available at the start of an injection process. To alleviate this problem, using some simplifying assumptions, we have developed a solution to predict the transient buildup of pressure resulting from injection of supercritical carbon dioxide from a partially penetrating well into a gas reservoir. The solution in space and time is first obtained in the Fourier–Laplace transform space, and then inverted back into real space (in cylindrical coordinates) and time. We use the solution to study pressure transient characteristics for different formation permeabilities and anisotropy ratios. Results obtained using the solution compared well with those from numerical simulations.