Dual-porosity model of rate transient analysis for horizontal well in tight gas reservoirs with consideration of threshold pressure gradient

Most researches of the threshold pressure gradient in tight gas reservoirs are experimental and mainly focus on the transient pressure response, without paying much attention to the transient rate decline. This paper establishes a dual-porosity rate transient decline model for the horizontal well with consideration of the threshold pressure gradient, which represents the non-Darcy flow in a fracture system. The solution is obtained by employing the Laplace transform and the orthogonal transform. The bi-logarithmic type curves of the dimensionless production rate and derivative are plotted by the Stehfest numerical inversion method. Seven different flow regimes are identified and the effects of the influence factors such as the threshold pressure gradient, the elastic storativity ratio, and the cross flow coefficient are discussed. The presented research could interpret the production behavior more accurately and effectively for tight gas reservoirs.     

低渗透油藏中水平井两相渗流分析

大量实验表明，在低渗透多孔介质中，渗流不再符合线性达西定律，而具有启动压力梯度，本文从Buckley-Leverret不混溶两相渗流驱替理论出发，考虑启动压力梯度的影响，对水平井两相渗流进行分析，得到了椭球形边水油藏中水平井定产时的压力分布公式和定压生产时的产量公式，以及饱和度分布公式。经图示分析得出结论：启动压力梯度不仅增大油藏的生产压差，而且使油藏的无水采收率和水驱采收率减少；增大注入量可以降低启动压力梯度对两相渗流的不利影响，从而提高水驱采收率，因此开发含启动压力梯度油藏，应小井距，大流量，大压差进行注水开发；水平井的两相渗流解析解为数值模拟中水平井的处理提供很好的方法。     

Exact analytical solutions for moving boundary problems of one-dimensional flow in semi-infinite porous media with consideration of threshold pressure gradient

By defining new dimensionless variables, nonlinear mathematical models for one-dimensional flow with unknown moving boundaries in semi-infinite porous media are modified to be solved analytically. The exact analytical solutions for both constant-rate and constant-pressure inner boundary constraint problems are obtained by applying the Green's function. Two transcendental equations for moving boundary problems are obtained and solved using the Newton-Raphson iteration. The exact analytical solutions are then compared with the approximate solutions. The Pascal's approximate formula in reference is fairly accurate for the moving boundary development under the constant-rate condition. But another Pascal's approximate formula given in reference is not very robust for constant-pressure condition problems during the early production period, and could lead to false results at the maximum moving boundary distance. Our results also show that, in presence of larger TPG, more pressure drop is required to maintain a constant-rate production. Under the constant-pressure producing condition, the flow rate may decline dramatically due to a large TPG. What's more, there exists a maximum distance for a given TPG, beyond which the porous media is not disturbed.     

A NEW METHOD FOR PREDICTION OF PRODUCTIVITY OF FRACTURED HORIZONTAL WELLS BASED ON NON-STEADY FLOW

Hydraulic fracturing technologies of horizontal well are important ways to develop oil-gas field with low permeability. Productivity forecast of fractured horizontal wells is a difficult problem of hydraulic fracturing technologies. Basing on non-steady flow of fractures fluid during production, applying potential function principles, superposition principle and mathematical method for solving, coupling of seepage flow in the formation and pipe flow in the well bore, a new model on multi-fracture interference productivity forecast of fractured horizontal well is established in this article. The results indicate the coincidence rate between this model and practice is high. The pressure loss in the horizontal well bore has definite influence on the production status of fractured horizontal wells. The productions of different fractures in horizontal well bore are unequal, the productions of outer fractures are higher than middle fractures; the pressure in the well bore shows an uneven distribution, the pressure declines gradually from finger tip to heel end. Asymmetry of fractures may make productivity of fractured horizontal wells decline. The conclusions are instructive in designing fractured horizontal well for low permeability reservoir.     

致密油藏体积压裂水平井不稳定压力分析

水平井结合体积压裂改造成为目前开发致密油藏的关键技术。考虑到致密油藏多尺度渗流(基质、天然裂缝和水力裂缝)及多种流动机理(启动压力梯度和应力敏感)相互耦合的复杂渗流过程,该文建立了综合考虑致密油藏启动压力梯度和应力敏感的体积压裂水平井不稳定压力分析模型。通过Laplace变换和摄动变换等求解得到了拉氏空间中的井底压力,应用Stehfest数值反演绘制了相应的压力动态响应曲线。研究结果表明:压力响应曲线可以划分为6个流动阶段,模型的退化结果与Ozkan模型一致,验证了该文模型的正确性,并在此基础上进行了影响因素分析。该文研究结果可为致密油藏体积压裂水平井的不稳定压力分析提供一定的理论依据。     

THE PRESSURE TRANSIENT ANALYSIS OF DEFORMATION OF FRACTAL MEDIUM

The assumption of constant rock properties in pressure-transient analysis of stress-sensitive reservoirs can cause significant errors in the estimation of temporal and spatial variation of pressure. In this article, the pressure transient response of the fractal medium in stress-sensitive reservoirs was studied by using the self-similarity solution method and the regular perturbation method. The dependence of permeability on pore pressure makes the flow equation strongly nonlinear. The nonlinearities associated with the governing equation become weaker by using the logarithm transformation. The perturbation solutions for a constant pressure production and a constant rate production of a linear-source well were obtained by using the self-similarity solution method and the regular perturbation method in an infinitely large system, and inquire into the changing rule of pressure when the fractal and deformation parameters change. The plots of typical pressure curves were given in a few cases, and the results can be applied to well test analysis.     

A NOVEL 3-D MODEL FOR THE WATER CRESTING IN HORIZONTAL WELLS

In the presence of bottom water, a drop in the reservoir pressure due to fluid production causes the aquifer water to expand and to flow into the reservoir. Therefore, hydrocarbon production from a well is limited by the critical flow rate. The main purpose of this study is to investigate the breakthrough time and the critical rate by using a novel 3-D horizontal well model. Based on the hypothesis that the horizontal well is located in any position of a circular reservoir with no-flow boundary on the top of the reservoir and constant pressure boundary at the bottom, the horizontal well has been regarded as an infinite conductivity line sink and then a 3-D steady-state flow model of the horizontal well is set up. A point sink pressure solution can be obtained with the Fourier transform. The result of the pressure distribution of the uniform flux horizontal well can be presented by means of the principle of superposition. According to the stable water cresting theory, this study confirms the stable height of water cresting and the critical rate. Meanwhile, it can re-confirm the breakthrough time at a specific rate. The output of a comparison between this 3-D model and the reservoir numerical simulator （Eclipse） shows the method presented here can be applied to investigate the behavior of a water cresting and to predict the breakthrough time at the bottom water driver reservoir.     

水位降落条件下非稳定渗流试验研究

水位降落时产生的非稳定渗流场对坝坡的渗流和稳定影响较大。通过砂槽模型室内试验模拟了粉砂、细砂、黏土等多种材质的边坡在水位降落过程中的非稳定渗流物理过程,分析了水位降落过程中上游边坡的渗流场特点,比较了不同边坡材料、渗透特性、坡比和水位降落速度的泄水过程中边坡渗流场自由面形式和等势线变化特点,重点研究了给水度μ、渗透系数k、库水位降落的速度v对自由面最高点、上游边坡中的孔隙水压力、坡降的影响,定量分析了水位降落时k/μv判断骤降、缓降、快降的标准,探讨了非稳定渗流浸润线的计算公式。结果表明：渗透系数小、给水度大的土坡在水位降落过程中产生的上游边坡孔隙水压力值和渗透坡降较大,非稳定流场自由面最高点位置也更高;水位降落时的上游边坡渗流方向指向坡面;浸润线最高点高度与坡前水深的比值h0/H与降落时间t/T间存在线性关系,k/μv作为判断骤缓降的指标与其线性系数呈指数关系。     

低渗透油藏渗流的差分法数值模拟

针对低渗透问题，给出了无量纲化的渗流控制方程、初始条件以及边界条件，并使用差分方法进行数值模拟。计算结果使用Gringarten-Bourdet图版进行井底压力分析，给出了一种计算动边界位置的方法，并详细讨论了动边界变化情况。     

致密油藏体积压裂水平井变窜流系数模型研究

水平井结合水力压裂技术已经成为高效开发致密油藏的有效手段。该文针对水平井经过压裂后在主裂缝周围形成的微裂缝密度随着到主裂缝的距离递减变化的特点,基于双重介质模型建立了窜流系数随着到主裂缝距离变化的体积压裂水平井线性流动模型,通过引入Airy函数及Laplace变换求得了模型的解析解,运用Stehfest数值反演技术得到了在实空间中的解,最后对定产量生产时的井底压力动态及定井底流压生产时的产能动态进行了分析。研究结果表明:井底压力及产能动态曲线可分为4个流动阶段,基质形状假设为球形及柱状对结果影响不大;窜流系数最大值越大,总系统线性流就发生得越早,双线性流阶段无因次产能就越大;窜流系数最小值越大,双线性流发生的时间越早,而且持续的时间越长,总系统线性流阶段无因次产能越大。研究结果可为致密油藏压力及产能动态的合理分析提供科学依据。     

A streamline approach for identification of the flowing and stagnant zones for five-spot well patterns in low permeability reservoirs

The mechanism of the fluid flow in low permeability reservoirs is different from that in middle-high permeability rese-rvoirs because of the existence of the Threshold Pressure Gradient (TPG). When the...     

内、外边界定压条件下地层压力的解

本文在内、外边界定压的条件下得到了平面单向流和平面径向流两种渗流形式在拉氏空间的解析解，利用Stehfest数值反演方法给出了地层各点的压力分布曲线，由这些曲线可以清楚地看到流体的流动由不稳态流向稳态流过渡以及最后达到稳态流的过程。本文给出的压力解可用于准确预测油井(或水井)由开井达到稳定生产所需要的最小时间，同时也是对油藏渗流力学基础理论的一个有意义的补充。     

不同库水消落方式下动水压力型滑坡变形与稳定性响应研究

水库滑坡灾害是大型水库运行调度工作中备受关注的问题。为了科学指导汛期水库运行调度中的滑坡灾害风险防控,基于库水动力作用效应及类型分析,通过大型离心模型试验和数值模拟研究了库水下降诱发滑坡变形破坏机理,重点分析了库水持续性下降、不同间歇期下降、相同间歇期多阶段下降三种消落方式下的滑坡稳定性响应规律。研究表明:在库水下降末期,随着坡面静水压力的快速消落,滑体内地下水位严重滞后于库水位变动,易产生高的动水压力、导致滑坡变形破坏。在库水位持续下降过程中,库水下降速率越大,地下水头差和水力梯度越大,则稳定性会持续下降。不同间歇期下降相比持续性下降,地下水力梯度比降低,可缓解库水位持续快速下降导致的地下水滞后及动水压力影响,稳定性下降速率会减缓。而相同间歇期多阶段下降则可进一步减小影响,稳定性降幅明显减小。     

WELL TESTING ANALYSIS FOR HORIZONTAL WELL WITH CONSIDERATION OF THRESHOLD PRESSURE GRADIENT IN TIGHT GAS RESERVOIRS

A fundamental solution for homogeneous reservoir in infinite space is derived by using the point source function with the consideration of the threshold pressure gradient. The fundamental solution of the continuous point source function is then derived based on the Green function. Various boundary conditions of the reservoirs are considered for this case and the corresponding solutions are obtained through the mirror image reflection and the principle of superimposition. The line source solution is obtained by integration. Subsequently, the horizontal-well bottom hole pressure response function for a non-linear gas flow in the homogeneous gas reservoir is obtained, and the response curve of the dimensionless bottom hole pressure and the derivative for a horizontal well in the homogeneous gas reservoir are obtained. In the end, the sensitivities of the relevant parameters are analyzed. The well test model presented in this paper can be used as the basis of the horizontal well test analysis for tight gas reservoirs.     

任意形状复合油藏压力动态的边界元分析

油气藏形状对井底压力动态响应有着显著的影响，常规的解析方法只能计算规则形状油气藏的井底压力响应。该文建立了任意形状复合油藏不稳定渗流的数学模型，并将区域分割法与边界元方法相结合，研究了任意形状复合油藏不稳定压力响应的计算方法，应用杜哈美原理考虑了表皮效应与井储系数对井底压力动态响应的影响。绘制并分析了复合油藏的典型曲线，通过分析指出油藏的边界性质会影响典型曲线后期形态，而内区形状的也可能掩盖复合油藏典型曲线的特点。     

用边界元法计算任意形状混合边界的油藏压力

建立了考虑源（汇）影响、任意形状、混合边界条件、含有不渗透区油藏稳定渗流的数学模型，采用边界元方法对其进行了求解，获得了油藏内的压力分布，绘制了相应的压力剖面图。通过对比压力剖面图和平均压力发现，由于油藏的封闭边界部分不能向生产井提供能量，致使地层能量被消耗较多，油藏平均压力降低；由于油藏内不渗透区的存在，能量在绕过其向生产井传播时要有部分能量损失，所以油藏的平均压力会降低。     

变形介质油藏渗流规律和压力特征分析

本文基于渗透率随压力呈分段的指数变化规律，建立并求解了平面径向稳定渗流方程和非线性不稳定渗流方程。研究了不可压缩液体的稳定渗流规律和弱可压缩液体的不稳定渗流规律。推导出无限大油藏非线性不稳定渗流基本微分方程的解析线源解。建立了各种边界条件下考虑井筒储存效应和表皮效应的非线性不稳定渗流数学模型，并用数值法对其求解。绘制了相应的压力及其导数典型曲数，分析了渗透率变异系数对这些典型曲线形态的影响。     

THE SIMILAR STRUCTURE OF SOLUTIONS IN FRACTAL MUITILAYER RESERVOIR INCLUDING A QUADRATIC GRADIENT TERM

When the classical nonlinear partial differential equations are used to model the fractal reservoir, based on the assumption of low compressibility fluids, the effects of the quadratic gradient term are ignored, which would be questionable for mixed gas reservoirs and low permeability reservoirs. To consider the influence of the wellbore storage, the nonlinear mathematical flow model of the fractal multilayer reservoir is built in this paper, with three kinds of outer boundaries (infinite boundaries, constant pressure boundaries and closed boundaries). Using the Laplace transform method, the solutions for the dimensionless reservoir pressure and the bottom hole pressure in the Laplace space are obtained. An analysis shows that the solutions involve similar structures even for three different kinds of outer boundaries, and can be unified as a continuous fraction. The unified expression would make it more convenient to analyze the formation parameters, which greatly facilitates the development of the well test analysis software.     

分形油藏渗流问题的精确解及动态特征

本文将油井有效半径引入分形油藏渗流问题的内边界条件之中，从而建立了分形油藏渗流问题的新模型，并在考虑到井筒讲集和表皮效应的情况下求得了外边界无限大，有界封闭和有界定压三种典型情况下分形油藏压力分布的精确解析表达式。