TWO－PHASE FLOW FOR A HORIZONTAL WELL PENETRATING A NATURALLY FRACTURED RESERVOIR WITH EDGE WATER INJECTION

ＴＷＯ－ＰＨＡＳＥＦＬＯＷＦＯＲＡＨＯＲＩＺＯＮＴＡＬＷＥＬＬＰＥＮＥＴＲＡＴＩＮＧＡＮＡＴＵＲＡＬＬＹＦＲＡＣＴＵＲＥＤＲＥＳＥＲＶＯＩＲＷＩＴＨＥＤＧＥＷＡＴＥＲＩＮＪＥＣＴＩＯＮＧｕｏＤａｌｉ（郭大立）ＬｉｕＣｉｑｕｎ（刘慈群）（Ｒｅｃｅｉｖｃ...     

TWO－PHASE FLOW FOR A HORIZONTAL WELL PENETRATING A NATURALLY FRACTURED RESERVOIR WITH EDGE WATER INJECTION

ＴＷＯ－ＰＨＡＳＥＦＬＯＷＦＯＲＡＨＯＲＩＺＯＮＴＡＬＷＥＬＬＰＥＮＥＴＲＡＴＩＮＧＡＮＡＴＵＲＡＬＬＹＦＲＡＣＴＵＲＥＤＲＥＳＥＲＶＯＩＲＷＩＴＨＥＤＧＥＷＡＴＥＲＩＮＪＥＣＴＩＯＮＧｕｏＤａｌｉ（郭大立）ＬｉｕＣｉｑｕｎ（刘慈群）（Ｒｅｃｅｉｖｅ...     

井筒气液两相流对致密气压裂水平井试井的影响

多段压裂水平井技术是目前开采致密气最常用的方法之一,在致密气压裂水平井试井测试中常常伴随着一定的产水量,井筒气液两相流会增加井筒流体的流动阻力,加大井筒流体流动对试井解释的影响.为了明确井筒气液两相流对致密气藏压裂水平井试井的影响,提高产水致密气压裂水平井的试井解释精度,建立了一种井筒气液两相流与地层渗流耦合的试井模型,采用数值方法对模型进行求解,获得了考虑井筒气液两相流的压裂水平井试井理论曲线、压力场分布及裂缝产量分布.研究结果表明:井筒气液两相流会增加试井理论曲线中压力和压力导数值,造成靠近入窗点的压力扩散要快于远离入窗点的压力扩散,引起靠近入窗点的裂缝产量要高于远离入窗点的裂缝产量.现场实例分析进一步说明,不考虑井筒两相流可能会对产水压裂水平井的试井解释结果产生很大误差,主要表现为水平井筒假设为无限大导流能力会使得拟合得到的表皮系数偏大,将测试点视为入窗点会使得拟合得到的原始地层压力偏小.所建立的考虑井筒两相流的压裂水平井试井模型为产水致密气井试井资料的正确解释提供了重要技术保障.      

An Extended-FEM Model for CO2 Leakage Through a Naturally Fractured Cap-Rock During Carbon Dioxide Sequestration

In this paper, a numerical model is developed for the assessment of carbon dioxide transport through naturally fractured cap-rocks during CO₂ sequestration in underground aquifers. The cap-rock contains two types of fracture with different length scales: micro-cracks (fissures) and macro-cracks (faults). The effect of micro-cracks is incorporated implicitly by modifying the intrinsic permeability tensor of porous matrix, while the macro-cracks are modeled explicitly using the extended finite element method (X-FEM). The fractured porous medium is decomposed into the porous matrix and fracture domain, which are occupied with two immiscible fluid phases, water and CO₂. The flow inside the matrix domain is governed by the Darcy law, while the flow within the fracture is modeled using the Poiseuille flow. The mass conservation law is fulfilled for each fluid phase at both porous matrix and fracture domain; moreover, the mass exchange between the matrix and fracture is guaranteed through the integral formulation of mass conservation law. Applying the X-FEM technique, the explicit representation of macro-cracks is modeled by enriching the standard finite element approximation space with an enrichment function. Finally, several numerical examples of CO₂ injection into a brine aquifer below a naturally fractured cap-rock are modeled in order to investigate the effects of cracks’ aperture and orientation as well as the temperature of aquifer and the depth of injection on the leakage pattern of the carbon dioxide through the cap-rock.

     

Dual Porosity and Dual Permeability Modeling of Horizontal Well in Naturally Fractured Reservoir

This article is the first investigation on the dual permeability flow issue for horizontal well-production in a naturally fractured dual-porosity reservoir. Based on the inter-porosity flow from matrix system to fracture system and treating the media directly connected with horizontal wellbore as matrix and fracture systems, we established a model of horizontal well-production and then solved the model using some modern mathematical methods, such as Laplace integral transformation, separation of variables, eigenvalue, and eigenfunction. Later in the article, we obtained the standard log–log type curves using numerical simulation and analyzed the transient flow behavior thoroughly, which showed it is dual porosity and dual permeability flow behavior. The numerical simulation results showed that there are obvious differences between dual permeability and single permeability models. The dual permeability flow behavior accelerates energy supplement during production and reduces the classical matrix-fracture (V-shaped) response. We also showed that type curves characteristics are affected by external boundary conditions, the parameter κ, ω _f and λ _mf, etc. The research results show that our model would be a good semi-analytical model supplied to users. Because the single permeability modeling ignores the direct fluid supply from matrix to wellbore, we recommend using the dual permeability modeling to make well testing and rate decline interpretation in real case studies.     

A new computation model of fractured horizontal well coupling with reservoir

Based on Green's functions and Newman's product principle, pressure drop formula was derived for considering simultaneous production of fractures and horizontal wellbore in unsteady state. A reservoir/fractured horizontal well coupling model is developed for finite conductivity condition that can be solved by the combination of quasi‐Newton method and PSO (Particle Swarm Optimization) algorithm. The solution of a practical example shows that lots of factors can affect the productivity of the fractured horizontal well. The number of the fractures has an optimizing range. Different fractures have different flow rates and the production in the central fracture is the lowest. The distribution of production rate in the wellbore has a wave‐like shape due to the influence of fractures. The closer fractures are the lower flow rate in well segments. Considering the horizontal wellbore production, the production rates in the fractures are asymmetric and the corresponding pressure drop curve is smoother. Copyright © 2010 John Wiley & Sons, Ltd.     

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...     

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...     

多尺度嵌入式离散裂缝模型模拟方法

天然裂缝性油藏和人工压裂油藏内裂缝形态多样,分布复杂,传统的离散裂缝模型将裂缝作为基岩网格的边界,采用非结构化网格进行网格划分,其划分过程复杂,计算量大。嵌入式离散裂缝模型划分网格时不需要考虑油藏内的裂缝形态,只需对基岩系统进行简单的网格剖分,可以大大降低网格划分的复杂度,从而提高计算效率。然而,在油藏级别的数值模拟和人工压裂裂缝下的产能分析中,仍然存在计算量巨大、模拟时间过长的问题。本文提出嵌入式离散裂缝模型的多尺度数值计算格式,使用多尺度模拟有限差分法研究嵌入式离散裂缝模型渗流问题。通过在粗网格上求解局部流动问题计算多尺度基函数,多尺度基函数可以捕捉裂缝与基岩间的相互关系,反映单元内的非均质性,因此该方法既有传统尺度升级法的计算效率,又可以保证计算精度,数值结果表明这是一种有效的裂缝性油藏数值模拟方法。     

One-Dimensional Matrix-Fracture Transfer in Dual Porosity Systems with Variable Block Size Distribution

Most of the developed models for fractured reservoirs assume ideal matrix block size distribution. This assumption may not be valid in reality for naturally fractured reservoirs and possibly lead to errors in prediction of production from the naturally fractured reservoirs especially during a transient period or early time production from the matrix blocks. In this study, we investigate the effect of variable block size distribution on one- dimensional flow of compressible fluids in fractured reservoirs. The effect of different matrix block size distributions on the single phase matrix-fracture transfer is studied using a recently developed semi-analytical approach. The proposed model is able to simulate fluid exchange between matrix and fracture for continuous or discrete block size distributions using probability density functions or structural information of a fractured formation. The presented semi-analytical model demonstrates a good accuracy compared to the numerical results. There have been recent attempts to consider the effect of variable block size distribution in naturally fractured reservoir modeling for slightly compressible fluids with a constant viscosity and compressibility. The main objective of this study is to consider the effect of variable block size distribution on a one-dimensional matrix-fracture transfer function for single-phase flow of a compressible fluid in fractured porous media. In the proposed semi-analytical model, the pressure variability of viscosity and isothermal compressibility is considered by solving the nonlinear partial differential equation of compressible fluid flow in the fractured media. The closed form solution provided can be applied to flow of compressible fluids with variable matrix block size distribution in naturally fractured gas reservoirs.     

Semi-Analytical Solutions for a Partially Penetrated Well with Wellbore Storage and Skin Effects in a Double-Porosity System with a Gas Cap

We have studied the effect of a constant top pressure on the pressure transient analysis of a partially penetrated well in an infinite-acting fractured reservoir with wellbore storage and skin factor effects. Semi-analytical solutions of a two-dimensional diffusivity equation have been obtained by using successive applications of the Laplace and modified finite Fourier sine transforms. Both pseudo-steady-state and transient exchanges between the matrix and the fractures have been considered. Solutions are presented that can be used to generate type curves for pressure transient analysis or can be used as a forward model in parameter estimation. The presented analysis has applications in well testing of fractured aquifers and naturally fractured oil reservoirs with a gas cap.     

考虑诱导缝多段压裂水平井非均质改造压力动态模型研究

为了准确模拟致密油藏水平井大规模压裂形成复杂裂缝网络系统和非均质储层井底压力变化,建立考虑诱导缝矩形非均质储层多段压裂水平井不稳定渗流数学模型,耦合裂缝模型与储层模型得到有限导流裂缝拉普拉斯空间井底压力解,对两种非均质储层模型分别利用数值解、边界元和已有模型验证其准确性.基于压力导数曲线特征进行流动阶段划分和参数敏感性分析,得到以下结果:和常规压裂水平井井底压力导数曲线相比较,理想模式下,考虑诱导缝影响时特有的流动阶段是综合线性流阶段、诱导缝向压裂裂缝“补充”阶段、储层线性流动阶段和拟边界控制流阶段.诱导缝条数的增加加剧了综合线性流阶段的持续时间,降低了流体渗流阻力,早期阶段压力曲线越低;当诱导缝与压裂裂缝导流能力一定时,裂缝导流能力越大,线性流持续时间越长;当所有压裂裂缝不在一个区域时,沿井筒方向两端区域低渗透率弱化了低渗区域诱导缝流体向压裂裂缝“补充”阶段,因此,沿井筒方向两端区域渗透率越低,早期阶段压力曲线越高;当所有压裂裂缝在一个区域时,渗透率变化只影响径向流阶段之后压力曲线形态,外区渗透率越低,早期径向流阶段之后压力曲线越高.通过实例验证,表明该模型和方法的实用性和准确性.     

Acid Leakoff Mechanism in Acid Fracturing of Naturally Fractured Carbonate Oil Reservoirs

In acid fracturing, excessive acid leakoff is thought to be the main reason that limits fracture propagation and live acid penetration distance. Since most carbonates are naturally fractured, we developed a new model in this paper to simulate acid leakoff into a naturally fractured carbonate oil reservoir during acid fracturing. Our model incorporates the acid-rock reaction, fracture width variation due to rock dissolution on the fractured surfaces, and fluid flow in naturally fractured carbonate oil reservoirs. Given the information of the reservoir, injected acid, and pressure in the hydraulic facture and the reservoir, the model predicts acid leakoff with time. In this study, we found that acid leakoff mechanism in naturally fractured carbonates is much different from that in reservoirs without natural fractures. Widened natural fractures by acid-rock reaction act as high-conductivity conduits allowing leakoff acid to penetrate deeper into the formation, resulting in serious leakoff. Wide natural fractures have a dominant effect on acid leakoff compared to micro-fractures and matrix.     

A limit form of the equations for immiscible displacement in a fractured reservoir

We study a model for simulating the flow of an immiscible displacement (waterflooding) of one incompressible fluid by another in a naturally fractured petroleum reservoir when the matrix blocks are quite small. This model is equivalent to a transformed one for immiscible flow in an unfractured reservoir with a reduced saturation and a saturation-dependent porosity. Existence and uniqueness of classical solutions are established. We present some numerical results and a comparison with a single porosity model.     

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.     

Gravitational forces in dual-porosity systems: II. Computational validation of the homogenized model

Three models are considered for single component, single phase flow in naturally fractured porous media. The microscopic model holds on the Darcy scale, and it is considered to govern the system. The macroscopic, dual-porosity model was derived in Part I of this work from the microscopic model by two-scale mathematical homogenization. In this paper, we show that the dual-porosity model predicts well the behavior of the microscopic model by comparing their computed solutions in certain reasonable test cases. Homogenization gives a complex formula for a key parameter in the dual-porosity model; herein a simple approximation to this formula is presented. The third model considered is a single-porosity model with averaged parameters. It is shown that this type of model cannot predict the behavior of the microscopic flow.This work was supported in part by the National Science Foundation and by the State of Texas Governor's Energy Office.     

Hydroelastic impact of a horizontal floating plate with forward speed

The hydroelastic responses of a horizontal plate impacting with the water at both forward and downward speeds are investigated theoretically. The longitudinal bending behavior of a horizontal elastic plate is approximated by the behavior of longitudinal strips represented as an Euler-beam model. A simplified method of hydroelastic responses of the plate is extended to the cases with forward speed and compressive force, for which the hydrodynamic pressure is found by solving a two-dimensional boundary value problem based on the linearized wave theory. In order to validate the theoretical model, a fully-coupled algorithm in LS-DYNA and the available experimental measurements are used for the predictions of the hydrodynamic pressure and deformations of the horizontal plates impacting with water at vertical velocities. The effects of the forward speed and compressive force which can occur at the bottom of ship ships, are investigated theoretically for the plates with different edge boundary conditions. The critical values of the forward speed and longitudinal compression are discussed regarding the plates with various longitudinal lengths.     

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.     

各向异性双重介质垂直裂缝井两相流体渗流

建立了各向异性双重介质中垂直裂缝井两相渗流数学模型,用有限差分法求得了其解,进行了算例分析。得到了含水饱和度沿径向、最大及最小渗透率方向的分布和水驱油前缘等饱和度面随时间的变化,分别讨论了各向异性、吸渗作用对这种介质中两相流体渗流的影响。结果表明：渗透率较在的方向,前缘推进速度较快,油井见水较早;吸渗使前缘推进速度减慢,使油井见水晚。注水开发这种油藏时,见水前注入率不能太大以充分发挥吸渗作用,存在最佳注入率。这对于水力压裂垂直裂缝井注水开发碳酸盐岩裂缝性油藏有指导意义。     

Analytical Modeling of Flow Behavior for Wormholes in Naturally Fractured–Vuggy Porous Media

Acidizing technology has been widely applied when developing naturally fractured–vuggy reservoirs. So testing and evaluating acidizing wells’ pressure behavior become necessary for further improving the wells’ performance. Analyzing transient pressure data can estimate some key reservoir parameters. Generally speaking, carbonate minerals are usually composed of dolomite and calcite which are easy to be dissolved by hydrochloric acid which is often used to react with the rock to create a high conductivity channel, namely wormhole. Pressure transient behavior in fractured–vuggy reservoirs has been studied for many years; however, the models of acidizing wells with wormholes were not reported in previous studies. This article presented an analytical model for wormholes in naturally fractured–vuggy carbonate reservoirs, and wormholes solutions were obtained through point sink integral method. The results were validated accurately by comparing with previous results and numerical simulation. Then in this paper, type curves were established to recognize the flow characteristics, and flow was divided into six flow regimes comprehensively. The calculative results showed that the characteristics of type curves were influenced by inter-porosity flow factor, wormhole number, fluids capacitance coefficient. We also showed that the pressure behavior was affected by the angles between wormholes, and the pressure depletion increased as the angle decreased, because the wormholes were closer, their interaction became stronger. At the end, a reservoir example was showed to demonstrate the methodology of new type curve analysis.