MODEL AND METHOD OF WELL TEST ANALYSIS FOR WELLS WITH VERTICAL FRACTURE

NomenclatureB———volumecoefficientofformationfluid,dimensionless;C———wellborestoragecoefficient,m3 MPa;Cft ———comprehensivecompressioncoefficientoffluidinfracture ,1 MPa;Ct ———comprehensivecompressioncoefficientofformationfluid,1 MPa ;h———heightofformation ,m ;K———formationpermeability ,μm2 ;Kf ———fracturepermeability ,μm2 ;KfWf ———fractureconductivity ,μm2 ·cm ;Lf ———halfoffracturelength,m ;N———thetotalnumberofselectedtimepoint,dimensionless;P———for…     

Numerical well test for well with finite conductivity vertical fracture in coalbed

A new well test model is developed for the hydraulic fractured well in coalbed by considering the following aspects： methane desorption phenomena, finite conductivity vertical fractures, and asymmetry of the fracture about the well. A new parameter is introduced to describe the storage of the fracture, which is named as a combined fracture storage. Another new concept called the fracture asymmetry coefficient is used to define the asymmetry of the fracture about the well. Finite element method （FEM） is used to solve the new mathematical model. The well test type curves and pressure fields are obtained and analyzed. The effects of the combined fracture storage, desorption factor, fracture conductivity, and fracture asymmetry coefficient on the well test type curves are discussed in detail. In order to verify the new model, a set of field well test data is analyzed.     

A triangular quasi-conforming finite element for transient dynamic analysis

A type of 3 node triangular element is constructed by the Quasi-conforming method, which may be used to solve the equation of a type of inverse problem of wave propagation after Laplace transformation Δu−A ² u=0. The strains in the element are approximated by an exponential function and the string-net function between neighbouring elements is approximated by one dimensional general solution of the equation. Furthermore the strain field satisfies the equation, and therefore in the derivation of the element formulation, no shape function is needed. In this sense, it is a kind of hybrid element. Compared with the ordinary linear triangular element, the new one features higher precision with coarse meshes. Some numerical tests are presented. The project is supported by the National Natural Science Foundation of China.     

页岩水力压裂中多簇裂缝扩展的全耦合模拟

水平井和水力压裂是页岩气开发中的关键技术。对水力压裂中多簇裂缝同时扩展的物理过程进行了数值模拟。采用扩展有限元法(XFEM)模拟岩石中裂缝沿着任意路径扩展,采用有限体积法(FVM)模拟裂缝中流体的流动,并且考虑井筒中流体流动以及在各簇裂缝间的流量动态分配。通过牛顿迭代对全耦合物理过程进行数值求解,重点研究了初始长度不同的两条裂缝的扩展过程,证明较大的射孔摩阻能促进两条裂缝的同时扩展。并通过算例证明了本方法的精度和有效性。     

A numerical method for solving equilibrium problems of masonry-like solids

This paper proposes a numerical method for the solution of equilibrium problems of solids which do not support tension. Some boundary-value problems are solved numerically and the solution obtained is compared to the exact one.

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Sommario In questo lavoro viene proposto un metodo numerico per la soluzione di problemi di equilibrio di solidi non resistenti a trazione. Vengono successivamente risolti numericamente alcuni problemi di equilibrio e la soluzione ottenuta è confrontata con quella esatta.

     

Numerical investigation of dual-porosity model with transient transfer function based on discrete-fracture model

Based on the characteristics of fractures in naturally fractured reservoir and a discrete-fracture model, a fracture network numerical well test model is developed.Bottom hole pressure response curves and the pressure field are obtained by solving the model equations with the finite-element method. By analyzing bottom hole pressure curves and the fluid flow in the pressure field, seven flow stages can be recognized on the curves. An upscaling method is developed to compare with the dual-porosity model(DPM). The comparisons results show that the DPM overestimates the inter-porosity coefficient λ and the storage factor ω. The analysis results show that fracture conductivity plays a leading role in the fluid flow. Matrix permeability influences the beginning time of flow from the matrix to fractures. Fractures density is another important parameter controlling the flow. The fracture linear flow is hidden under the large fracture density.The pressure propagation is slower in the direction of larger fracture density.     

基于离散裂缝的多段压裂水平井数值试井模型及应用

水平井压裂技术已经成为开发低渗透油气藏、页岩气藏和致密气场等非常规油气藏的关键技术。基于离散裂缝模型,对裂缝进行简化,建立了二维多段压裂水平井有限导流数值试井模型,利用有限元方法求解模型,获得多段压裂水平井试井理论曲线和压力场特征。分析表明:多段压裂水平井的试井理论曲线一共分为七个阶段:井筒储存段、裂缝线性流段、裂缝-地层双线性流段、裂缝干扰段、地层线性流段、系统径向流段和边界作用段,其中裂缝-地层双线性流段和裂缝干扰是其典型特征。分析了裂缝数量、裂缝间距、裂缝不对称、裂缝不等长和裂缝部分缺失等因素对试井理论曲线的影响,结果表明:裂缝数量和裂缝间距对试井理论曲线的影响最大。较多的裂缝、较大裂缝间距、对称的裂缝和等长的裂缝有利于降低压裂水平井井底的流动阻力,提高产能。将建立的数值试井模型应用于四川盆地一口多段压裂水平井的压力恢复测试的数值试井解释,结果表明:本文建立的模型可以较好的拟合压力恢复测试数据,可以获得裂缝的导流能力和裂缝长度,为压裂效果评价和压裂设计提供指导。      

基于摄动法的不确定性有限元模型修正方法研究

开展了考虑不确定性的有限元模型修正方法的研究。基于摄动法推导了待修正参数均值和协方差矩阵的迭代格式,其中协方差的迭代格式包括是否考虑试验数据与修正参数之间相关性的两种形式。在理论研究基础上开展数值仿真研究,实现了不确定性有限元模型修正的摄动法,并研究了试验数据样本数量对修正误差的影响。仿真结果表明,该方法适用于解决系统参数与试验数据存在不确定性的模型修正问题,试验样本数量对待修正参数标准差的修正精度影响较大;忽略试验模态参数与待修正参数不确定性之间的相关性,能够避免计算二阶灵敏度矩阵,在保证修正结果准确性的前提下减少计算量。     

气藏压裂水平井非稳态流动分析

应用Green函数和Newman积原理,建立了压裂水平井与气藏耦合的非稳态流动模型,并给出求解方法。所建立的模型可应用多种约束条件,考虑了井筒压降和加速度的影响,并适用于各向异性气藏。实例计算表明：水平井在早期非稳态阶段的流量比拟稳态阶段大得多。早期非稳态流动阶段,各条裂缝之间未发生干扰,流量的大小与裂缝条数成正比例关...     

Dynamic production prediction and parameter identification for gas well with vertical fracture

ＮｏｍｅｎｃｌａｔｕｒｅＢ—ｖｏｌｕｍｅｃｏｅｆｆｉｃｉｅｎｔｏｆｆｏｒｍａｔｉｏｎｆｌｕｉｄ ,ｄｉｍｅｎｓｉｏｎｌｅｓｓ ;Ｃｆｔ—ｃｏｍｐｒｅｈｅｎｓｉｖｅｃｏｍｐｒｅｓｓｉｎｇｃｏｅｆｆｉｃｉｅｎｔｏｆｆｌｕｉｄｉｎｆｒａｃｔｕｒｅ ,(ＭＰａ) - 1;Ｃｐ—ｗｅｉｇｈｔｏｆｓｉｎｇｌｅｌａｙｅｒｐｒｏｐｐａｎｔｐｅｒｕｎｉｔｆｒａｃｔｕｒｅａｒｅａ ,ｋｇ/ｍ2 ;Ｃｔ —ｃｏｍｐｒｅｈｅｎｓｉｖｅｃｏｍｐｒｅｓｓｉｎｇｃｏｅｆｆｉｃｉｅｎｔｏｆｆｏｒｍａｔｉｏｎｆｌｕｉｄ ,(ＭＰａ) - 1;ｄＢＨＮ —Ｂｒ…     

Effective numerical methods for elasto-plastic contact problems with friction

Several effective numerical methods for solving the elasto-plastic contact problems with friction are presented. First, a direct substitution method is employed to impose the contact constraint conditions on condensed finite element equations, thus resulting in a reduction by half in the dimension of final governing equations. Second, an algorithm composed of contact condition probes and elasto-plastic iterations is utilized to solve the governing equation, which distinguishes two kinds of nonlinearities, and makes the solution unique. In addition, Positive-Negative Sequence Modification Method is used to condense the finite element equations of each substructure and an analytical integration is introduced to determine the elasto-plastic status after each time step or each iteration, hence the computational efficiency is enhanced to a great extent. Finally, several test and practical examples are presented showing the validity and versatility of these methods and algorithms. The Project Supported by National Natural Science Foundation of China.     

Finite element analysis for the unsteady nearshore circulation due to wave-current interaction (I)—numerical model

In this paper, a numerical model for predicting the unsteady nearshore circulation due to wave-current interaction was proposed. In addition to the traditional continuity, momentum and energy equations, the dispersion and refraction relations were included in the governing equations. Moreover, the effects of lateral shears, wind, radiation and bottom stresses were analysed in the governing equations. Therefore, we expect that this model may more completely and exactly reflect the law of wave-current interaction. In part (II) we will adopt the selective lumping two-step explicit finite element method to solve the model, and some examples will be presented.     

Flow field modelling near a well with a conductive fracture

In the technology of oil recovery the oil production rate can be increased by generation of a vertical sand-filld conductive fracture on the wall of the well. Oil diffuses through the conductive fracture to the well. In this paper the seepage flow and isothermal deformation fields in both the formation and fracture and the oil production rate at the well are studied by modelling the formation as an infinite poroelastic medium saturated with a one-phase compressible fluid. The fracture is treated as a one-dimensional poroelastic medium. Darcy flows are considered in both the formation and fracture. The plane strain condition is imposed. Our solution is obtained numerically by a finite element method based on a variational principle. The accuracy of the analysis is studied by comparison of the numerical solutions of some problems with their analytical solutions. Since we are dealing with the transient flow problem of an infinite region, an extrapolation technique is employed to find the finite element solution. The production rate of a well with the conductive fracture is compared with that of a well without the conductive fracture.     

连续曲线箱梁模型试验研究

建立一两跨连续曲线箱梁有机玻璃试验模型。二跨连续箱梁模型的跨径为45cm 45cm,分别进行了在集中荷载、均布荷载作用下的模型试验研究。垂直集中荷载采用杠杆法加载,均布荷载采用特制条形砝码搁置加载。采用YJ-25静态电阻应变仪、平衡箱,并用半桥测量,温度自补偿方法测定应变值。各种测试值均取分级荷载下读数的平均值,测量得到了沿横截面的应力、应变分布规律。用有限段法和有限元法对该模型进行了应力计算,与试验结果比较吻合较好,实验验证了理论的正确性。     

A combined finite volumes - finite elements method for a low-Mach model

In this paper, we develop a finite volumes - finite elements method based on a time splitting to simulate some low-Mach flows. The mass conservation equation is solved by a vertex-based finite volume scheme using a τ-limiter. The momentum equation associated with the compressibility constraint is solved by a finite element projection scheme. The originality of the approach is twofold. First, the state equation linking the temperature, the density, and the thermodynamic pressure is imposed implicitly. Second, the proposed combined scheme preserves the constant states, in the same way as a similar one previously developed for the variable density Navier-Stokes system. Some numerical tests are performed to exhibit the efficiency of the scheme. On the one hand, academic tests illustrate the ability of the scheme in term of convergence rates in time and space. On the other hand, our results are compared to some of the literature by simulating a transient injection flow as well as a natural convection flow in a cavity.     

Numerical analysis of complex flow of fibre suspensions in a viscoelastic fluid

Research efforts made so far to simulate fibre suspension flows are limited to fibre suspensions in Newtonian fluids. Though short fibre composites are mostly made of polymers, lack of suitable constitutive equations for fibre suspensions in viscoelastic fluids frustrates attempts to simulate flows of these suspensions. A preliminary work done by the author led to a constitutive equation for semiconcentrated fibre suspensions in the Oldroyd-B fluid. This paper describes the mathematical formulation of the flow problems for such a suspension and the numerical procedure to solve them. Some numerical results of flow past a sphere in a tube are also presented. This paper is supported by the University of Melbourne, the National Natural Science Foundation of China and Zhejiang Province.     

A three-dimensional transient two-phase flow analysis with a density pertubation finite volume method

In this paper, we present the analysis of a full three dimensional transient two-phase flow with strong disequilibria. This analysis uses a finite volume scheme with a recently developed density perturbation method. The application test case is a three dimensional extension of a one-dimensional experiment on blowdown pipe. The results are stable with mesh refinements and show realistic three-dimensional effects.     

一种新的金属压缩试验方法的有限元分析

对一种新的金属压缩试验方法进行了有限元数值分析，论证了这种方法的可行性和有性，并且对不同尺寸的试件进行了有限元计算．     

A comparison of spatial grids for numerical modelling of flows in near-coastal seas

Three-dimensional algorithms for the numerical computation of flows caused by tides or meteorological forcing are developed for four of Arakawa's spatial grid types using a spectral method in the vertical dimension. Three of the grids, in which the velocity components are computed at the same grid points, offer potential advantages over the commonly used C-grid. The computed results from the four grids are compared for three test problems based on the linearized hydrodynamical equations. It is concluded that the B-grid provides a viable alternative to the C-grid, with significant advantages when a spectral method is used.