Pressure Transient Analysis for Multi-stage Fractured Horizontal Wells in Shale Gas Reservoirs

This article presents the PTA on the multi-stage fractured horizontal well in shale gas reservoirs incorporating desorption and diffusive flow in the matrix. Currently, most PTA models are simply based on Darcy flow both in natural fractures and matrix without considering the mechanisms of desorption and diffusion in shale matrix. Source function and Laplace transform with the numerical discrete method are employed to solve the mathematical model. The solution is presented in the Laplace domain so that the wellbore storage effect and skin factor can be easily incorporated by convolution. Type curves are plotted with Stehfest algorithm and different flow regimes are identified. The presented model could be used to interpret pressure signals more accurately for shale gas reservoirs.     

Modeling Water Leak-off Behavior in Hydraulically Fractured Gas Shale under Multi-mechanism Dominated Conditions

Fracturing-fluid leak-off in fractured gas shale is a complex process involving multiple pore/fluid transports and interactions. However, water leak-off behavior has not been modeled comprehensively by considering the multi-pores and multi-mechanisms in shale with existing simulators. In this paper, we present the development of a comprehensive multi-mechanistic, multi-porosity, and multi-permeability water/gas flow model that uses experimentally determined formation properties to simulate the fracturing-fluid leak-off of hydraulically fractured shale gas wells. The multi-mechanistic model takes into account water transport driven by hydraulic convection, capillary and osmosis, gas transport caused by hydraulic convection, and salt ion transport caused by advection and diffusion. The multi-porosity includes hydraulic fracture millipores, organic nanopores, clay nanopores, and other inorganic micropores. The multi-permeability model accounts for all the important processes in shale system, including gas adsorption on the organics’ surface, multi-mechanistic clay/other inorganic mineral mass transfer, inorganic mineral/hydraulic fracture mass transfer, and injection from a hydraulically fractured wellbore. The dynamic water saturation and pressure profiles within clay and other inorganic matrices are compared, revealing the leak-off behavior of water in rock media with different physicochemical properties. In sensitivity analyses, cases with different clay membrane efficiency, volume proportion of source rock, connate water salinity, and saturation are considered. The impacts of shale properties on water fluxes through wellbore, hydraulic fracture and matrix, and the total injection and leak-off volumes of the well during the treatment of hydraulic fracturing are investigated. Results show that physicochemical properties in both organic and inorganic matrices affect the water leak-off behavior.     

An Object-Based Shale Permeability Model: Non-Darcy Gas Flow,Sorption, and Surface Diffusion Effects

Shale samples consist of two major components: organic matter (OM) and inorganic mineral component (iOM). Each component has its distinct pore network topology and morphology, which necessitates generating a model capable of distinguishing the two media. We constructed an object-based model using the OM and iOM composition of shale samples. In the model, we integrated information such as OM population and size distribution, as well as its associated pore-size distribution. For the iOM part, we used mineralogy and pore-size information derived from X-ray diffraction, scanning electron microscopy, and nitrogen sorption measurements. Our proposed model results in millimeter-scale 2D realizations of shale samples by honoring OM and mineral types, their compositions, shapes, and size distributions. The model can capture heterogeneities smaller than 1 mm. We studied the effects of different gas flow processes and found that Knudsen diffusion and gas slippage dominate the flow, but surface diffusion has little impact on total gas flow.     

页岩气藏的双重介质-离散裂缝模型

考虑页岩气藏开发中渗流的多尺度效应,提出了一个基于裂缝-孔隙双重介质的离散裂缝模型．在该模型中,基质、天然裂缝和人工压裂裂缝采用各自控制方程独立计算,不同介质之间通过流量交换相互关联．为分析模型可靠性,分别和基于渗透率粗化及压裂裂缝导流能力无穷大的模型对比．数值算例显示,伴随着网格细分,该模型与精确渗透率粗化模型具有相同计算精度,两者收敛速度均较快,但该模型易推广到多相流动问题,而等压模型对产量将有所高估．研究了地质参数和工艺参数对气井产量的影响规律．计算结果表明天然裂缝渗透率及基质孔隙扩散系数对产气速率有着重要影响,产气速率伴随着人工压裂裂缝导流能力、长度以及数目的增加而增加,但是增加幅度会逐步趋缓．     

Flow Mechanism and Simulation Approaches for Shale Gas Reservoirs: A Review

Transport in Porous Media - The past two decades have borne remarkable progress in our understanding of flow mechanisms and numerical simulation approaches of shale gas reservoir, with much larger...     

Influence of Considering the Sorption Effect in the Betti-Maxwell Reciprocal Theorem on Gas Transport Capacity in Unconventional Reservoirs

Transport in Porous Media - Pore bulk modulus in continuum apparent permeability models was generally treated as an independent parameter or deduced from the Betti-Maxwell reciprocal theorem...     

Matrix Heterogeneity Effects on Gas Transport and Adsorption in Coalbed and Shale Gas Reservoirs

In coalbeds and shales, gas transport and storage are important for accurate prediction of production rates and for the consideration of subsurface greenhouse gas sequestration. They involve coupled fluid phenomena in porous medium including viscous flow, diffusive transport, and adsorption. Standard approach to describe gas–matrix interactions is deterministic and neglects the effects of local spatial heterogeneities in porosity and material content of the matrix. In this study, adopting weak-noise and mean-field approximations and using a statistical approach in spectral domain, matrix heterogeneity effects are investigated in the presence of non-equilibrium adsorption with random partition coefficient. It is found that the local heterogeneities can generate non-trivial transport and kinetic effects which retard gas release from the matrix and influence the ultimate gas recovery adversely. Macro-transport shows 1/[1 + N _Pe /(1 + N _Pe )] dependence on the Péclet number, and persists at the diffusive ultra-low permeability limit. Macro-kinetics is directly related to Thiele modulus by the following expression: N _Th /(1 + 2N _Pe ). It leads to trapping of gas in the adsorbed phase during its release from the matrix, and to an adsorption threshold during the gas uptake by the matrix. Both effects are proportional to the initially available adsorbed gas amount and becomes more pronounced with the increasing variance of the porosity field. Consequently, a new upscaled deterministic gas mass balance is proposed for practical purposes. Numerical results are presented showing free and adsorbed gas distributions and fractional gas sorption curves for unipore coal matrix exhibiting Gaussian porosity distribution. This study is a unique approach for our further understanding of the coalbeds and gas shales, and it is important for the development of sound numerical gas production and sequestration models.     

Effects of Convection and Fracture Boundary Conditions on Heat Transfer Shape Factor in Fractured Geothermal Reservoirs

The yield stress fluids porosimetry method (YSM) was recently presented as a simple and non-toxic potential alternative to the extensively used mercury intrusion porosimetry (MIP). The success of YSM heavily relies on the choice of an appropriate yield stress fluid to be injected through the investigated porous medium. In previous works, xanthan gum aqueous solutions were used due to their ability to exhibit a pseudo-yield stress without substantial levels of unwanted thixotropy or viscoelasticity. Given that YSM is based on the existence of a yield stress, the accuracy of the obtained pore size distribution (PSD) crucially depends on the capacity of the injected fluid to emulate the shear rheology of a yield stress fluid. However, this capacity has still not been fully assessed in the case of xanthan gum solutions. Neither has the robustness of YSM with regard to errors in the determination of the shear-rheology parameters of the injected fluid been analysed. The shear viscosity of polymer solutions is known to be deeply influenced by polymer concentration. For these reasons, a first objective of this work is to evaluate the effect of polymer concentration on the accuracy of PSDs obtained by YSM when using xanthan gum solutions as injected fluids in laboratory experiments. To do so, xanthan gum solutions with different polymer concentrations were injected through analogous samples of a sintered silicate and the obtained PSDs were compared to the results of standard MIP. Moreover, the sensitivity of YSM to errors in the experimental determination of the shear-rheology parameters was also investigated through numerical experiments. The results of the present work permitted to gain further insight into the viability of YSM as an efficient alternative to MIP.     

水闸闸室结构三维样条边界元分析

用三维样边界元法分析水闸闸室结构。底板，闸墩和载水墙等为其子结构，交通桥，工作桥和胸墙等处理为内部支撑。地基和边载可以是任意的，只要能给定地表位移面力关系。在各种工况下，不论是设置平板门还是弧形门，是平底板是反拱底板，即使在稀疏剖分下也能给出高精度的位移场，应力场和地基反力场。     

边界元法与光弹性实验相结合的应力分析法

本文介绍了边界无法与光弹性实验相结合进行应力分析的方法。叙述了用边界无法求解主应力的基本原理。结合光弹性实验等差线条纹图,可分离出主应力。     

变系数渗流场的正交各向异性边界元分析

本文给出了计算复杂渗流场的简化方法.首先由加权余量法得到变系数渗流问题的积分方程.再将区域及边界剖分,对各单元使用形心处的渗透系数而化为正交各向异性问题.最后用正交各向异性问题的基本解,得到对应于边界单元法的线性代数方程组.算例表明方法可靠.     

虚边界元法的应用及其求解方法

由弹性力学问题的虚边界元方法出发，给出若干算例，对基在接触，塑性，蠕变等非线性问题中的应用，做了进一步探讨，提出了相应的求解方案。     

随机边界元法在薄板可靠性分析中的应用

本文利用Taylor级数展开，将结构的材料随机性化为一个等效随机外载荷来处理，从而利用相应同一结构确定性问题的基本解，分别建立关于响应的均值和偏差的边界积分方程；结合一阶二次矩法，应用随机边界元法分析了材料弹性常数的随机性和薄板厚度不均匀性对薄板结构可靠性的影响，对计算结果分析看出；（1）对于同一种单元划分和同一相关长度，可靠性指数保持了对具体相关模式的少敏感性；（2）对于同一种单元划分和同一相关模式，可靠性指数随相关长度的增大而减小。     

三维弹性力学边界元多域缩聚法分析

采用超参非连续元离散三维弹性力学问题边界积分方程，借助三角极坐标变换方法处理奇异积分。将超参非连续元用于多域边界元分析，解决了自由度约束问题。提出了二次缩聚的概念，提高了多域缩聚边界元法的求解效率。通过数值算例表明了本文方法的可行性和有效性。     

域内具有支承及集中质量的薄板自由振动的边界元分析

本文基于薄板小挠度弯曲问题的基本解,建立了任意边界条件、域内具有支承及附有征意个集中质量的薄板自山振动的边界积分方程,文中计算了若干算例,其精度是实际工程中所允许的。     

流固偶合运动的边界元法

本文给出了流固偶合运动(包括物体散射辐射及偶合运动)的边界元法理论和应用.对于散射问题,求出了物体引起的散射势及入射波作用于物体的载荷.对于辐射问题,求出了辐射势及物体在流体中运动的附加质量和附加阻尼.偶合问题包括求其中包含的散射势和辐射势以及作用于物体之上的散射力、物体的附加质量、附加阻尼、物体在入射波作用下的运动.在偶合运动问题中,本文采取了边界积分方程与物体在流体中的运动方程联立求解的方法,并将其运用到边界元法的数值过程中.所编制的程序有较高的精度.最后给出了数值计算结果与理论解的比较.     

两斜交圆柱壳应力分析的边界元法

本文研究了两斜交圆柱壳的边界元法。提出了用板的基本解叠加级数形式的修正项构成圆柱壳的基本解,提高了计算精度,缩短了计算时间。对区域积分进行特殊处理,从而避免了内部网格的划分,大大减少了数据准备工作量和占机内存。并编制了FORTRAN计算程序,进行了数值计算。     

非线性Reissner板杂交元及边界元解

首先改进文[1]的互补变分原理。再建议一种较为普遍性的方法,导出精确的边界积分方程。最后给出变分有限元及边界元解,算例证实有限元格式及迭代方式有效。