Discrete fracture model for simulating waterflooding processes under fracturing conditions

In our study, we develop a model for simulating fracturing processes in a poroelastic medium. The proposed approach combines the discrete fracture model enriched with contact plane mechanics. The model captures mechanical interactions of fractures and a deformable medium, fluid, and heat transfer in fractures and in a porous medium. Both effects of poroelasticity and thermoelasticity are accounted in our model. Mass and heat conservation equations are approximated by the finite volume method, and mechanical equilibrium equations are discretized by means of the Galerkin finite element approach. Two‐dimensional grid facets between 3‐dimensional finite elements are considered as possible fracture surfaces. Most of these facets are inactive from the beginning and are activated throughout the simulation. A fracture propagation criterion, based on Irwin's approach, is verified on each nonlinear iteration. When the criterion is satisfied, additional contact elements are added into finite element and discrete fracture model formulations respectively. The proposed approach allows modeling of existing natural and artificially created fractures within one framework. The model is tested on single‐ and multiple‐phase fluid flow examples for both isothermal and thermal conditions and verified against existing semianalytical solutions. The applicability of the approach is demonstrated on an example of practical interests where a sector model of an oil reservoir is simulated with different injection and production regimes.     

A two-dimensional poroelastic displacement discontinuity method for hydraulic fracture simulation

The coupled diffusion and deformation phenomena induced by the opening of a hydraulic fracture in a poroelastic medium are modelled using an extension of the displacement discontinuity method. This method consists in distributing fluid source and poroelastic displacement discontinuity singularities along the locus of the fracture, and over time. As an illustration, the problem of a suddenly pressurized crack in an infinite formation is examined.     

Finite element analysis of land subsidence above depleted reservoirs with pore pressure gradient and total stress formulations

The solution of the poroelastic equations for predicting land subsidence above productive gas/oil fields may be addressed by the principle of virtual works using either the effective intergranular stress, with the pore pressure gradient regarded as a distributed body force, or the total stress incorporating the pore pressure. In the finite element (FE) method both approaches prove equivalent at the global assembled level. However, at the element level apparently the equivalence does not hold, and the strength source related to the pore pressure seems to generate different local forces on the element nodes. The two formulations are briefly reviewed and discussed for triangular and tetrahedral finite elements. They are shown to yield different results at the global level as well in a three‐dimensional axisymmetric porous medium if the FE integration is performed using the average element‐wise radius. A modification to both formulations is suggested which allows to correctly solve the problem of a finite reservoir with an infinite pressure gradient, i.e. with a pore pressure discontinuity on its boundary. Copyright © 2001 John Wiley & Sons, Ltd.     

Thermal effects on fluid flow and hydraulic fracturing from wellbores and cavities in low-permeability formations

The coupled heat-fluid-stress problem of circular wellbore or spherical cavity subjected to a constant temperature change and a constant fluid flow rate is considered. Transient analytical solutions for temperature, pore pressure and stress are developed by coupling conductive heat transfer with Darcy fluid flow in a poroelastic medium. They are applicable to low permeability porous media suitable for liquid-waste disposal and also simulating reservoir for enhanced oil recovery, where conduction dominates the heat transfer process. A full range of solutions is presented showing separately the effects of temperature and fluid flow on pore pressure and stress development. It is shown that injection of warm fluid can be used to restrict fracture development around wellbores and cavities and generally to optimize a fluid injection operation. Both the limitations of the solutions and the convective flow effect are addressed. Copyright © 1999 John Wiley & Sons, Ltd.     

Modeling fluid injection in fractures with a reservoir simulator coupled to a boundary element method

We describe an algorithm for modeling saturated fractures in a poroelastic domain in which the reservoir simulator is coupled with a boundary element method. A fixed stress splitting is used on the underlying fractured Biot system to iteratively couple fluid and solid mechanics systems. The fluid system consists of Darcy’s law in the reservoir and is computed with a multipoint flux mixed finite element method, and a Reynolds’ lubrication equation in the fracture solved with a mimetic finite difference method. The mechanics system consists of linear elasticity in the reservoir and is computed with a continuous Galerkin method, and linear elasticity in the fracture is solved with a weakly singular symmetric Galerkin boundary element method. This algorithm is able to compute both unknown fracture width and unknown fluid leakage rate. An interesting numerical example is presented with an injection well inside of a circular fracture.     

注水井泄压对井壁围岩应力场的影响

注水井泄压过程中的套管损坏现象在中国许多油田十分普遍,根据油藏流体渗流与地应力场耦合作用理论,建立了套管承载计算的力学模型,利用有限元方法,通过数值模拟的手段研究了泄压过程中井壁围岩应力场的变化规律以及油层孔隙压力的变化规律。根据计算结果提出,控制注水井泄压流量是有效防治泄压过程中套损的一种有效手段,并给出了具体的计算方法,从而,为油田现场预防和减少套管损坏提供了理论依据。     

Semianalytical modeling on 3D stress redistribution during hydraulic fracturing stimulation and its effects on natural fracture reactivation

The hydraulic fracturing technique has been widely applied in many fields, such as the enhanced geothermal systems (EGS), the improvement of injection rates for geologic sequestration of CO₂, and for the stimulations of oil and gas reservoirs. The key points for the success of hydraulic fracturing operations in unconventional resources are to accurately estimate the redistribution of pore pressure and stresses around the induced fracture and predict the reactivations of preexisting natural fractures. The pore pressure and stress regime around hydraulic fracture are affected by poroelastic and thermoelastic phenomena as well as by fracture opening compression. In this work, a comprehensive semi-analytical model is used to estimate the stress and pore pressure distribution around an injection-induced fracture from a single well in an infinite reservoir. The model allows the leak-off distribution in the formation to be three-dimensional with the pressure transient moving ellipsoidically outward into the reservoir from the fracture surface. The pore pressure and the stress changes in three dimensions at any point around the fracture caused by poroelasticity, thermoelasticity, and fracture compression are investigated. With Mohr-Coulomb failure criterion, we calculate the natural fracture reactivations in the reservoir. Then, two case studies of constant water injection into a hydraulic fracture are presented. This work is of interest in the interpretation of microseismicity in hydraulic fracturing and in the estimation of the fracture spacing for hydraulic fracturing operations. In addition, the results from this study can be very helpful for the selection of stimulated wells and further design of the refracturing operations.     

Three-dimensional fracture propagation under specified well-bore pressure

Results are presented for three-dimensional planar fracture propagation in the presence of variations in confining stress, reservoir properties and frac-fluid viscosity when the well-bore pressure is constant throughout the frac-job. Solid rock deformation during the frac-job is described by the dislocation method with singular integral equations, and the fluid flow in the fracture is represented by the finite element method. A convergence study is undertaken on the three-dimensional numerical model, and good agreement is obtained with results from laboratory tests and from the simpler circular model.     

Transient infinite elements for seepage problems in infinite media

In this paper, a time-dependent infinite element which can be used to simulate transient seepage problems in infinite media is presented. The hydraulic head distribution function of the element has been derived in detail and the property matrices of the element have been well formulated. Since both space and time variables are used in the course of constructing the hydraulic head distribution function of the element, the present infinite element can be referred to as a transient one. Using the present infinite element to model the far field of a system, the mechanism of transient seepage problems in infinite media can be rigorously simulated because the property matrices of the element are evaluated at any time of interest in the analysis. Since explicit expressions can be written for the property matrices of the infinite element, they may be evaluated quite easily and this can be carried out by writing a simple subroutine in a computer program. In order to examine the accuracy and efficiency of the present infinite element, both a one-dimensional (ID) transient seepage problem in a semi-infinite medium and a 2D transient seepage problem in a full plane have been solved using the finite and infinite element technique. It has been demonstrated that the present infinite element is very useful for the numerical simulation of transient seepage problems in infinite media.     

A three‐dimensional integral equation model for calculating poro‐ and thermoelastic stresses induced by cold water injection into a geothermal reservoir

Poro‐mechanical and thermo‐mechanical processes change the fracture aperture and thus affect the water flow pattern in the fracture during the cold water injection into enhanced geothermal systems (EGS). In addition, the stresses generated by these processes contribute to the phenomenon of reservoir seismicity. In this paper, we present a three‐dimensional (3D) partially coupled poro‐thermoelastic model to investigate the poroelastic and thermoelastic effects of cold water injection in EGS. In the model, the lubrication fluid flow and the convective heat transfer in the fracture are modeled by the finite element method, while the pore fluid diffusion and heat conductive transfer in the reservoir matrix are assumed to be 3D and modeled by the boundary integral equation method without the need to discretize the reservoir. The stresses at the fracture surface and in the reservoir matrix are obtained from the numerical model and can be used to assess the variation of in situ stress and induced seismicty with injection/extraction. Application of the model shows that rock cooling induces large tensile stresses and increases fracture conductivity, whereas the rock dilation caused by fluid leakoff decreases fracture aperture and increases compressive total stresses around the injection zone. However, increases in pore pressure reduce the effective stresses and can contribute to rock failure, fracture slip, and microseismic activity. Copyright © 2009 John Wiley & Sons, Ltd.     

基于裂缝性致密储层关键渗流参数的逆向渗吸速度计算

在致密油藏水平井体积压裂开采过程中,压裂液通过缝网与基质接触并发生逆向渗吸作用,由于接触面积很大渗吸作用不可忽视;但目前关于表征致密储层的渗吸作用,从而研究渗吸对水平井体积压裂生产过程影响的研究尚未深入.为了解决以上问题,首先利用毛管束模型,通过考虑致密储层中边界层的特征,建立了解析的渗流参数计算表达式,用以计算致密储层的渗透率、毛管力、相渗曲线这3个关键渗流参数;同时,基于以上关键渗流参数和渗吸控制方程建立了适用于致密储层的渗吸速度计算模型;然后,将渗吸项作为源汇项加入到考虑缝网的双孔单渗模型中.最后,在真实水平井体积压裂开采过程中,耦合渗吸作用.研究表明,相比于不考虑边界层特征的致密油藏,边界层的存在将大幅度减弱储层的渗吸能力,同时也说明了在致密储层中,边界层的存在是不可忽视的,如果在渗吸计算中忽视致密储层的边界层特征会严重高估渗吸对致密储层产能的影响.      

Generalized poroelastic wellbore problem

This paper presents a novel analytical solution to the transient, z‐dependent, and asymmetric problem of an infinite wellbore drilled into a fluid‐saturated porous medium. The formulations are based on Biot's linear theory of poroelasticity, in which the dependency of poroelastic field variables to spatial coordinates as well as time domain is considered in the most general form. This gives flexibility to the solution in cases that cannot be analyzed using the conventional plane strain or symmetric models. One such case is when calculating the stress variations around an inclined wellbore where the far‐field stresses are acting over a finite vertical section. The results of our solution to this case with a three‐dimensional state of far‐field stress are used to analyze the stability of inclined wellbores passing through abnormally stressed formations. The presented solution is capable of finding expressions for fundamental solutions with stress or flow boundary conditions at the wellbore. These solutions are here adopted to analyze the pressure disturbances generated by multiprobe formation tester, a standard wireline device that is designed for downhole fluid sampling as well as estimating the directional permeabilities of subsurface earth formations. A comparison with the conventional solution for the relevant pressure diffusion equation indicates that the poroelastic effect is relatively significant in relation to the transient response of the pore pressure. Further, it is shown that the finite dimensions of sink probe would, to a great extent, contribute to the formation's pore pressure variations at its immediate proximity. Copyright © 2013 John Wiley & Sons, Ltd.     

Numerical analysis of thermal fracturing in subsurface cold water injection by finite element methods

Integration of poromechanics and fracture mechanics plays an important role in understanding a series of thermal fracturing phenomena in subsurface porous media such as cold water flooding for enhanced oil recovery, produced‐water reinjection for waste disposal, cold water injection for geothermal energy extraction, and CO₂ injection for geosequestration. Thermal fracturing modeling is important to prevent the potential risks when fractures propagate into undesired zones, and it involves the coupling of heat transfer, mass transport, and stress change as well as the fracture propagation. Analytical method, finite element method, and finite difference method as well as boundary element method have been used to perform the thermal fracturing modeling considering different degrees and combinations of coupling. In this paper, extended finite element method is employed for the thermal fracturing modeling in a fully coupled fashion with remeshing avoided, and the stabilized finite element method is employed to account for the convection‐dominated heat transfer in the fracturing process with numerical oscillation circumvented. With the thermal fracturing model, a hypothetical numerical experiment on cold water injection into a deep warm aquifer is conducted. Results show that parameters such as injection rate, injection temperature, aquifer stiffness, and permeability can affect the fracture development in different ways and extended finite element method and stabilized finite element method provide effective tools for thermal fracturing simulation. Copyright © 2012 John Wiley & Sons, Ltd.     

基于ABAQUS平台的水力裂缝扩展有限元模拟研究

随着扩展有限元理论的深入研究,利用扩展有限元方法模拟水力压裂具有了一定的可操作性。相比于常规有限元方法,XFEM方法具有计算结果精度高和计算量小的优点。但是,如何模拟射孔孔眼、如何模拟流体与岩石相互作用以及分析水力裂缝的扩展规律仍然是难题。以研究水力压裂裂缝扩展规律为目的,建立了岩石多孔介质应力平衡方程、流体渗流连续性方程和边界条件。通过有限元离散化方法对耦合方程矩阵进行处理。通过富集函数定义初始裂缝（射孔孔眼）,选择最大主应力及损伤变量D分别作为裂缝起裂和扩展判定准则,利用水平集方法模拟水力裂缝扩展过程。数值模拟结果显示：增加射孔方位角、压裂液排量和减小水平地应力差,起裂压力上升;黏度对起裂压力无明显影响。增加射孔方位角、压裂液排量、黏度和减小水平地应力差值有助于裂缝宽度的增加。增加水平地应力差值、压裂液排量和减小射孔方位角以及压裂液黏度有助于裂缝长度增加,反之亦然。基于ABAQUS的水力裂缝扩展有限元法可对不同井型和诸多储层物性参数及压裂施工参数进行分析,且裂缝形态逼真,裂缝面凹凸程度清晰,结果准确。此研究可作为一种简便有效研究水力压裂裂缝扩展规律的方法为油田水力压裂设计与施工提供参考与依据。     

稠油油藏热-流体-力学耦合模型研究及应用

在稠油油藏开发中,由于原油黏度高一般采用注蒸汽开采。基于连续介质力学理论,考虑了热对流的影响,建立了蒸汽注入条件下对地层压力、变形和有效应力定量评价的改进的热-流体-力学耦合数学模型;应用全隐式顺序Galerkin有限元数值解方案对注蒸汽井热-流体-力学耦合过程进行了数值模拟。计算结果表明,由于地层流体高温膨胀引起的高压力,会导致位移、变形,甚至地层结构破裂;温度对地层破裂形态的影响显著。研究表明,这个改进模型比传统模型更加准确、实用,可以用于蒸汽注入速率优化和油藏、地质、热-流动力反应、地应力水平等综合作用效应研究。     

哈得逊东河砂岩油藏外围水平井合理配产影响因素分析

哈得逊东河砂岩油藏外围水平井目前舍水上升较快,产油量出现明显递减.研究以有效抑制外围底水锥进和产量递减、提高原油采收率为目的,通过对比该油藏水平井生产情况与油层中夹层的发育程度及位置、水平井井轨迹和油水粘度比的关系,深入分析了影响该油藏外围水平井合理配产的因素,并利用了油藏数值模拟方法加以论证.研究认为,夹层发育较多、垂向渗透能力相对较低且夹层位于油水界面之上时,水平井配产可较高;水平井位于油层顶部,避水高度较高时,水平井配产可较高;油水粘度比较低时,水平井配产可较高.     

Transient response of a circular cavity in a poroelastic medium

This paper considers the transient response of a pressurized long cylindrical cavity in an infinite poroelastic medium. To obtain transient solutions, Biot's equations for poroelastodynamics are specialized for this problem. A set of exact general solutions for radial displacement, stresses, pore pressure and discharge are derived in the Laplace transform space by using analytical techniques. Solutions are presented for three different types of prescribed transient radial pressures acting on the surface of a permeable as well as an impermeable cavity surface. Time domain solutions are obtained by inverting Laplace domain solutions using a reliable numerical scheme. A detailed parametric study is presented to illustrate the influence of poroelastic material parameters and hydraulic boundary conditions on the response of the medium. Comparisons are also presented with the corresponding ideal elastic solutions to portray the poroelastic effects. It is noted that the maximum radial displacement and hoop stress at the cavity surface are substantially higher than the classical static solutions and differ considerably from the transient elastic solutions. Time histories and radial variations of displacement, hoop stress, pore pressure and fluid discharge corresponding to a cavity in two representative poroelastic materials are also presented.     

A numerical procedure for simulation of hydraulically-driven fracture propagation in poroelastic media

A procedure for numerical approximation to two-dimensional, hydraulically-driven fracture propagation in a poroelastic material is described. The method uses a partitioned solution procedurè to solve a finite element approximation to problems described by the theory of poroelasticity, in conjunction with a finite difference approximation for modelling fluid flow along the fracture. An equilibrium fracture model based on a generalized, Dugdale–Barenblatt concept is used to determine the fracture dimensions. An important feature is that the fracture length is a natural product of the solution algorithm. Two example problems verify the accuracy of the numerical procedure and a third example illustrates a fully-coupled simulation of fracture propagation. Photographs taken from a high-performance engineering workstation provide insight into the nature of the coupling among the physical phenomena.     

Non-linear analysis of geomechanical problems using coupled finite and infinite elements

In modeling of many geomechanics problems such as underground openings, soil-foundation structure interaction problems, and in wave propagation problems through semi-infinite soil medium the soil is represented as a region of either infinite or semi-infinite extent. Numerical modeling of such problems using conventional finite elements involves a truncation of the far field in which the infinite boundary is terminated at a finite distance. In these problems, appropriate boundary conditions are introduced to approximate the solution of the infinite or semi-infinite boundaries as closely as possible. However, the task of positioning the finite boundary in conventional finite element discretization and the definition of the boundary and its conditions is very delicate and depends on the modeller's skill and intuition. Moreover, such a choice is influenced by the size of the domain to be discretized. Consequently, the dimensions of the global matrices and the time required for solution of the problem will increase considerably and also selection of the arbitrary location of truncated boundary may lead to erroneous result. In order to over come these problems, mapped infinite elements have been developed by earlier researchers (Simoni and Schrefier, 1987). In the present work the applicability of infinite element technique is examined for different geomechanics problems. A computer program INFEMEP is developed based on the conventional finite element and mapped infinite element technique. It is then validated using selected problems such as strip footing and circular footing. CPU time taken to obtain solutions using finite element approach and infinite element approach was estimated and presented to show the capability of coupled modeling in improving the computational efficiency. Mesh configurations of different sizes were used to explore the enhancement of both computational economy and solution accuracy achieved by incorporation of infinite elements to solve elastic and elasto-plastic problems in semi-infinite/finite domain as applied to geotechnical engineering. © Rapid Science Ltd. 1998