低渗透多孔介质渗流动边界模型的解析与数值解

考虑启动压力梯度的低渗透多孔介质非达西渗流模型属于强非线性动边界问题, 分别利用相似变量变换方法和基于空间坐标变换的有限差分方法, 对内边界变压力情况下、考虑启动压力梯度的一维低渗透多孔介质非达西渗流动边界模型进行了精确解析与数值求解研究. 研究结果表明:该动边界模型存在唯一的精确解析解, 且所求得的精确解析解可严格验证数值解的正确性;且当启动压力梯度值趋于零时, 非达西渗流动边界模型的精确解析解将退化为达西渗流情况下的精确解析解. 由求解结果作出的非零无因次启动压力梯度下的地层压力分布曲线表现出紧支性特点, 其与达西渗流模型的有显著不同. 因此, 研究低渗透多孔介质中非稳态渗流问题时, 应该考虑动边界的影响. 研究内容完善了低渗透多孔介质的非达西渗流力学理论, 为低渗透油气藏开发的试井解释与油藏数值模拟技术提供了理论基础.     

煤层气在非饱和水流阶段的非定常渗流摄动解

煤层甲烷由煤层的割理裂隙系统流入生产井一般经历：单相水流、非饱和流和气、水两相饱和流三个阶段，在非饱和流阶段，储层压力降至临界解吸压力之后，储存在煤基质中的吸附气体少量被解吸出来形成互不连续的气泡并阻止水的流动，含气量尚未达到饱和程度。同时煤层甲烷运移包含渗流场、变形场和应力场的动态耦合过程。本文考虑渗流过程中水-气两相不溶混流体与固体耦合作用，建立了非饱和水流阶段非定常渗流问题的流固耦合数学模型，对该强非线性一维数学模型采用摄动法和积分变换法进行解析求解，并讨论了其压力动态特性，分析了压力随饱和度S及时间t变化的规律和气相及耦合作用的影响，这些研究对煤层气、石油和天然气的开采等地下工程领域具有一定的指导意义。     

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。     

低渗透煤层气有限导流垂直裂缝井压力动态分析

作为典型的致密多孔介质,煤岩储层已被证实存在启动压力梯度.根据煤层气垂直裂缝井的双线性流动机制,综合考虑启动压力梯度和井筒储存效应的影响,建立了一个新的低渗透煤层气有限导流垂直裂缝井双线性流动数学模型,采用Laplace变换和Stehfest数值反演方法对数学模型进行了求解,并分析了无因次启动压力梯度等参数对无因次井底压力及其导数曲线的影响规律.分析结果表明:典型的低渗透煤层气垂直裂缝井双线性流动曲线可划分为早期续流段、双线性流段、煤层线性流段、过渡流段和煤层边界线性流段5个特征阶段,其中由于启动压力梯度存在的影响,无因次井底压力及其导数曲线自煤层线性流段开始出现明显上翘,且启动压力梯度值越大,曲线上翘趋势越明显;此外,煤层边界线性流段呈现为单位斜率的直线,而非1/2斜率的线性流段直线.这些结果表现出启动压力梯度对低渗透煤层气垂直裂缝井双线性流动的影响,可用于指导现场煤层气井试井分析.     

动边界双重介质油藏低速非达西渗流试井模型

裂缝性油藏中基质岩块的渗透率一般很低,大量岩心测试实验证实在基质岩块内的液体渗流和在一定含水饱和度下的气体渗流将偏离达西渗流,往往出现低速非达西渗流,表现出启动压力梯度以及流体流动边界不断向外扩展等特殊现象。本文充分考虑启动压力梯度与动边界的影响,建立了微可压缩双重介质油藏低速非达西渗流的试井数学模型,对时间和空间变量...     

TRANSIENT WELL TEST MODEL FOR THE WELL WITH GAS AND WATER DISTRIBUTED IN COALBED

煤层气是一种高效清洁的非常规天然气资源,其开采过程是一个排水降压采气的过程. 由于煤层气主要是以吸附态的形式存在于煤层中,当煤层压力降低到临界解吸压力以下时煤层气从煤层中解吸出来并与水一起采出,因此煤层中流体是气水两相分布的. 本文根据煤层气藏排采过程中的解吸特征,通过考虑气水两相分布的渗透率关系,提出了一种与解吸区域大小相关的煤层气井不稳定试井模型. 该模型较好地描述了煤层气排采过程中煤层内气水的流动状态,采用分区模式对气水两相进行描述. 通过有限体积方法求解了所建立的试井模型,计算得到了煤层气井气水两相分布不稳定试井理论曲线,分析了煤层气解吸系数、解吸复合半径、气水饱和度分布等对试井理论曲线的影响.     

海陆过渡相页岩气藏不稳定渗流数学模型

海陆过渡相页岩常与煤层和砂岩呈互层状产出, 储层连续性较差、横向变化快、非均质性强, 水力压裂技术是其获得经济产量的关键手段. 然而, 目前缺乏有效的海陆过渡相页岩气藏不稳定渗流数学模型, 对其渗流特征分析及储层参数评价不利. 针对这一问题, 首先建立海陆过渡相页岩气藏压裂直井渗流数学模型, 其次采用径向复合模型来反映强非均质性, 采用Langmuir等温吸附方程来描述气体的解吸和吸附, 分别采用双重孔隙模型和边界元模型模拟天然裂缝和水力裂缝, 建立并求解径向非均质的页岩气藏压裂直井不稳定渗流数学模型, 分析海陆过渡相页岩气藏不稳定渗流特征, 并进行数值模拟验证和模型分析应用. 分析结果表明, 海陆过渡相页岩气藏不稳定渗流特征包括流动早期阶段、双线性流、线性流、内区径向流、页岩气解吸、内外过渡段、外区径向流及边界控制阶段. 将本模型应用在海陆过渡相页岩气试井过程中, 实际资料拟合效果较好, 其研究成果可为同类页岩气藏的压裂评价提供一些理论支撑, 具有较好应用前景.      

变形三重介质低渗透油藏垂直裂缝井的非线性流动分析

不仅考虑低渗透油藏具有启动压力梯度的渗流特征,还考虑应力敏感地层中介质的变形;发展了Cinco-Ley H.提出的有限导流垂直裂缝井双线性流理论,将流体在垂直裂缝与地层中形成的流动划分为两个区域—垂直裂缝中的线性流区域和变形三重介质低渗透油藏中的非线性流区域;由此建立了变形三重介质低渗透油藏有限导流垂直裂缝井的非线性流...     

页岩气藏压裂水平井试井分析

页岩气藏资源丰富,开发潜力巨大,已成为目前研究的热点.与常规气藏相比,页岩气藏运移机制复杂,流动模式呈非线性,有必要考虑页岩气的吸附解吸,天然微裂缝的应力敏感性,人工裂缝内的非达西流等非线性因素对压裂水平井压力响应的影响. 基于双重介质和离散裂缝混合模型,分别采用Langmuir等温吸附方程描述吸附解吸,渗透率指数模型描述应力敏感,Forchheimer方程描述非达西效应,建立页岩气藏压裂水平井数值试井模型. 运用伽辽金有限元法对模型进行求解.根据试井特征曲线,划分流动阶段,着重分析非线性因素对压力响应的影响.结果表明:页岩气藏压裂水平井存在压裂裂缝线性流、压裂裂缝径向流、地层线性流、系统径向流及封闭边界影响5 种流动阶段.吸附解吸的影响发生窜流之后,Langmuir吸附体积增大,拟压力导数曲线凹槽更加明显,系统径向流出现时间与压力波传播到边界时间均延迟;天然裂缝系统的应力敏感性主要影响试井曲线的晚期段,拟压力和拟压力导数曲线均表现为上翘,应力敏感效应越强,上翘幅度越大;高速非达西效应对早期段影响较大,非达西效应越强,拟压力降幅度越大,试井曲线上翘.与解析解的对比以及矿场实例验证了模型的正确性与适用性.     

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.     

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.     

Stable Propagation of Saturation Overshoots for Two-Phase Flow in Porous Media

Propagation of saturation overshoots for two-phase flow of immiscible and incompressible fluids in porous media is analyzed using different computational methods. In particular, it is investigated under which conditions a given saturation overshoot remains stable while moving through a porous medium. Two standard formulations are employed in this investigation, a fractional flow formulation and a pressure–saturation formulation. Neumann boundary conditions for pressure are shown to emulate flux boundary conditions in homogeneous media. Gravity driven flows with Dirichlet boundary conditions for pressure that model infiltration into heterogeneous media with position-dependent permeability are found to exhibit pronounced saturation overshoots very similar to those seen in experiment.     

多孔介质应力对孔隙压力影响的研究

通过大量的现场测试发现 ,含瓦斯煤岩是一个复杂的系统 ,煤体应力的变化是引起其他一系列变化的主导因素。随煤体应力的变化 ,孔隙压力呈指数形式变化 ;介质的渗透性受孔隙压力梯度的控制。     

Comprehensive Seepage Simulation of Fluid Flow in Multi-scaled Shale Gas Reservoirs

Shale gas seepage behaviour is a multi-field/-scale problem and makes transient pressure analysis a very challenging task. Non-Darcy flow in nanopores is prominent due to the broken of continuity hypothesis. Slippage effect and Knudsen diffusion are two important seepage mechanisms in nanopores, while recent studies show surface diffusion is another important transporting mechanism on surface of nanopores. Porous kerogen system contains large amounts of dissolved gas, which should not be overlooked. In this study, a comprehensive mathematical model was established by pseudo-quadruple porosity medium conception, coupling the effects of slippage flow, Knudsen diffusion, surface diffusion, ad-/desorption and gas transferring from kerogen to nanopore system, while fluid flow in fractures/macropores is described by Darcy’s law. Transient pressure behaviours of a multiple fractured horizontal well in box-shaped shale gas reservoir were studied, with nine possible flow regimes divided and parameters sensitivity analysed. Adsorbed constant and dissolved constant were defined to reflect the amount of adsorbed gas and dissolved gas, respectively. Research shows that adsorbed gas and dissolved gas are two important gas storage forms, neither of which should be neglected. The study can not only help us understand fluid flow mechanisms in nanopores from microscopic perspective, but enable us to analyse production performance and determine key operational parameters from macroscopic perspective.     

Degassing of water-solved gases in thin pipings during fast pressure drops

 A one-dimensional model is presented, which describes the transient two-phase flow in thin pipes during fast pressure drops and degassing by use of Eulerian and Lagrangian systems. The reduction in dimension is obtained by introduction of a geometry model for bubbly and slug flow regimes. The complete model includes the transient two-phase flow, bubble formation and bubble growth. The flow model predicts rising velocities of bubbles and plugs in arbitrary inclined highly accurate pipes. The mass transfer (diffusion) of the dissolved phase is calculated by the bubble growth model. The quality of the model was examined by simulation of experimental series, whereby water was depressurised from the saturation pressure of the dissolved gas mixture (air), by variation of saturation pressure, pressure gradient and pipe geometry. The results of numerical simulation fit the experimental data well. Received on 17 January 2000