Large-scale averaging analysis of multiphase flow in fractured reservoirs

The method of large-scale averaging is applied to derive and analyze a dual-porosity model of multiphase flow in naturally fractured reservoirs. The dual-porosity model contains the usual equations based on Darcy's law, and the coupling terms representing the fluid transfer between the matrix and the fractures. Both quasisteady and transient closure schemes are considered to obtain and analyze the fracture and matrix permeability tensors and the fluid transfer terms. The techniques developed here are not restricted to regular geometric fractures. Computational work aimed at showing the implications of the theory behind the derivation of the present dual-porosity model is also described. In particular, comparisons among the dual-porosity model, the single porosity model, and other dual-porosity models are presented through numerical experiments.     

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...     

Simulation of Pressure Transient Behavior for Asymmetrically Finite-Conductivity Fractured Wells in Coal Reservoirs

Based on Fick’s law in matrix and Darcy flow in cleats and hydraulic fractures, a new semi-analytical model considering the effects of boundary conditions was presented to investigate pressure transient behavior for asymmetrically fractured wells in coal reservoirs. The new model is more accurate than previous model proposed by Anbarci and Ertekin, SPE annual technical conference and exhibition, New Orleans, 27–30 Sept 1998 because new model is expressed in the form of integral expressions and is validated well through numerical simulation. (1) In this paper, the effects of parameters including fracture conductivity, coal reservoir porosity and permeability, fracture asymmetry factor, sorption time constant, fracture half-length, and coalbed methane (CBM) viscosity on bottomhole pressure behavior were discussed in detail. (2) Type curves were established to analyze both transient pressure behavior and flow characteristics in CBM reservoir. According to the characteristics of dimensionless pseudo pressure derivative curves, the process of the flow for fractured CBM wells was divided into six sub-stages. (3) This paper showed the comparison of transient steady state and pseudo steady state models. (4) The effects of parameters including transfer coefficient, wellbore storage coefficient, storage coefficient of cleat, fracture conductivity, fracture asymmetry factor, and rate coefficient on the shape of type curves were also discussed in detail, indicating that it is necessary to keep a bigger fracture conductivity and fracture symmetry for enhancing well production and reducing pressure depletion during the hydraulic fracturing design.     

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.     

Time-Dependent Shape Factors for Uniform and Non-Uniform Pressure Boundary Conditions

Matrix–fracture transfer functions are the backbone of any dual-porosity or dual-permeability formulation. The chief feature within them is the accurate definition of shape factors. To date, there is no completely accepted formulation of a matrix–fracture transfer function. Many formulations of shape factors for instantly-filled fractures with uniform pressure distribution have been presented and used; however, they differ by up to five times in magnitude. Based on a recently presented transfer function, time-dependent shape factors for water imbibing from fracture to matrix under pressure driven flow are proposed. Also new matrix–fracture transfer pressure-based shape factors for instantly-filled fractures with non-uniform pressure distribution are presented in this article. These are the boundary conditions for a case for porous media with clusters of parallel and disconnected fractures, for instance. These new pressure-based shape factors were obtained by solving the pressure diffusivity equation for a single phase using non-uniform boundary conditions. This leads to time-dependent shape factors because of the transient part of the solution for pressure. However, approximating the solution with an exponential function, one obtains constant shape factors that can be easily implemented in current dual-porosity reservoir simulators. The approximate shape factors provide good results for systems where the transient behavior of pressure is short (a case commonly encountered in fractured reservoirs).     

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.     

On Vaporizing Water Flow in Hot Sub-Vertical Rock Fractures

Water injection into unsaturated fractured rock at above-boiling temperatures gives rise to complex fluid flow and heat transfer processes. Examples include water injection into depleted vapor-dominated geothermal reservoirs, and emplacement of heat-generating nuclear wastes in unsaturated fractured rock. We conceptualize fractures as two-dimensional heterogeneous porous media, and use geostatistical techniques to generate synthetic permeability distributions in the fracture plane. Water flow in hot high-angle fractures is simulated numerically, taking into account the combined action of gravity, capillary, and pressure forces, and conductive heat transfer from the wall rocks which gives rise to strong vaporization. In heterogeneous fractures boiling plumes are found to have dendritic shapes, and to be subject to strong lateral flow effects. Fractures with spatially-averaged homogeneous permeabilities tend to give poor approximations for vaporization behavior and liquid migration patterns. Depending on water flow rates, rock temperature, and fracture permeability, liquid water can migrate considerable distances through fractured rock that is at above-boiling temperatures and be only partially vaporized.     

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.     

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．     

Pressure transient response of stochastically heterogeneous fractured reservoirs

A stochastic model for flow through inhomogeneous fractured reservoirs of double porosity, based on Barenblattet al.'s continuum approach, is presented. The fractured formation is conceptualized as an interconnected fracture network surrounding porous blocks, and amenable to the continuum approach. The block permeability is negligible in comparison to that of the fractures, and therefore the reservoir permeability is represented by the permeability of the fracture network. The fractured reservoir inhomogeneity is attributed to the fracture network, while the blocks are considered homogeneous. The mathematical model is represented by a coupled system of partial differential random equations, and a general solution for the average and for the correlation moments of the fracture pressure are obtained by the Neumann expansion (or Adomian decomposition). The solution for pressure is represented by an infinite series and an approximate solution for radial flow, is obtained by retaining the first two terms of the series. The purpose of this investigation is to get an insight on the pressure behavior in inhomogeneous fractured reservoirs and not to obtain type curves for determination of reservoir properties, which owing to the nonuniqueness of the solution, is impossible. For the present analysis we assumed an ideal reservoir with cylindrical symmetric inhomogeneity around the well. Fractured rock reservoirs being practically inhomogeneous, it is of interest to compare the pressure behavior of such reservoirs, with Warren and Roots's solution for (ideal) homogeneous reservoirs, used as a routine for determining the fractured reservoir characteristic parameters and, using the results of well tests. The comparison of the results show that inhomogeneous and homogeneous reservoirs exhibit a similar pressure behavior. While the behavior is identical, the same drawdown or a build-up pressure curve may be fitted by different characteristic dimensionless parameters and, when attributed to an inhomogeneous or a homogeneous reservoir. It is concluded that the ambiguity in determining the fractured reservoir and, makes questionable the usefulness of determination of these parameters. Computations were also carried out to determine the correlation between the fracture pressure at the well and the fracture pressure at different points.     

裂缝性低渗透油藏流-固耦合理论与数值模拟

根据裂缝性低渗油藏的储层特征,建立适合裂缝性砂岩油藏渗流的等效连续介质模型。将渗流力学与弹塑性力学相结合,建立裂缝性低渗透油藏的流-固耦合渗流数学模型,并给出其数值解.通过数值模拟对一实际井网开发过程中孔隙度、渗透率的变化以及开发指标进行计算,并和刚性模型以及双重介质模型的计算结果进行了分析比较.     

A New Non-Darcy Flow Model for Low-Velocity Multiphase Flow in Tight Reservoirs

The pore and pore-throat sizes of shale and tight rock formations are on the order of tens of nanometers. The fluid flow in such small pores is significantly affected by walls of pores and pore-throats. This boundary layer effect on fluid flow in tight rocks has been investigated through laboratory work on capillary tubes. It is observed that low permeability is associated with large boundary layer effect on fluid flow. The experimental results from a single capillary tube are extended to a bundle of tubes and finally to porous media of tight formations. A physics-based, non-Darcy low-velocity flow equation is derived to account for the boundary layer effect of tight reservoirs by adding a non-Darcy coefficient term. This non-Darcy equation describes the fluid flow more accurately for tight oil reservoir with low production rate and low pressure gradient. Both analytical and numerical solutions are obtained for the new non-Darcy flow model. First, a Buckley–Leverett-type analytical solution is derived with this non-Darcy flow equation. Then, a numerical model has been developed for implementing this non-Darcy flow model for accurate simulation of multidimensional porous and fractured tight oil reservoirs. Finally, the numerical studies on an actual field example in China demonstrate the non-negligible effect of boundary layer on fluid flow in tight formations.     

Effect of a periodic action on average flow in an inhomogeneous liquid-saturated porous medium

The problem of the average flow of a viscous incompressible fluid saturating a stationary porous incompressible matrix under a periodic action is considered. It is shown that a spatial inhomogeneity of the medium porosity leads to an average fluid flow, quadratically dependent on the action amplitude, in the direction of increase in porosity. In particular, this means that wave action on an oil reservoir could lead to fluid flow on the interfaces from low-porosity,weakly permeable collector regions into high-porosity regions, for example, to flow from blocks to fractures in fractured porous reservoirs, which makes it possible to enhance oil production. It is shown that in the presence of a constant pressure gradient the flow component generated by a periodic action can provide a substantial proportion of the total flow, especially on the boundaries with low-porosity strata or blocks. With increase in amplitude this may significantly exceed the component associated with the constant pressure gradient.     

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.     

Fracturing of viscoelastic geomaterials and application to sedimentary layered rocks

We study analytically the behavior of a viscoelastic brittle solid loaded in tension, in which fractures may grow or not depending on the amount of dissipation allowed by the viscous behavior. We highlight a threshold in extension rate, below which the solid will not be fractured. Applied to sedimentary rocks, this model shows how viscous effects can prevent fracture growth in geological formations.     

A Computational Approach to Determine Average Reservoir Pressure in a Coalbed Methane (CBM) Well Flowing Under Dominant Matrix Shrinkage Effect

Desorption of gas from coal matrix alters the pore volume of fracture network. Consequently, cleat porosity and permeability of reservoir changes as pressure depletes. The method of standard pressure analysis calculations produces incorrect results in the case of coalbed methane reservoirs producing under dominant matrix shrinkage effect. The change in cleat porosity and permeability due to shrinkage of coal matrix following gas desorption with pressure depletion invalidates the underlying assumptions made in the derivation of diffusivity equation. Consequently, equations of pseudo-steady state commonly used in conventional reservoirs no longer remain valid as the porosity and permeability values change with pressure depletion. In this paper, effort has been made to describe pseudo-steady-state flow in coalbed methane reservoirs in the form of a new equation that accounts for pressure dependency of cleat porosity and permeability due to shrinkage of coal matrix. The concept of Al-Hussainy et al. (1966) has been extended to define a new pseudo-pressure function which assimilates within itself the pressure dependence of porosity and permeability Palmer and Mansoori (1998). Equation has been used to relate the cleat porosity with pressure. The equation-based computational method suggested in this paper finds its usefulness in estimating average reservoir pressure for any known flowing bottom hole pressure and thus reducing the frequency of future pressure buildup tests. The new equation is also useful in predicting reservoir pressure under the situation when coal matrix shrinks below desorption pressure. The equation used in the computational method has been validated with the help of numerical simulator CMG-GEM.     

Lattice Boltzmann Simulation of Fluid Flow in Synthetic Fractures

Fractures play an important role in reservoir engineering as they dominate the fluid flow in the reservoir. All evidence suggests that rarely can one model flow and transport in a fractured rock consistently by treating it as a uniform or mildly nonuniform isotropic continuum. Instead, one must generally account for the highly erratic heterogeneity, directional dependence, dual or multicomponent nature and multiscale behavior of fractured rocks. As experimental methods are expensive and time consuming most of the time numerical methods are used to study flow and transport in a fractured rock. In this work, we present results of the numerical computations for single phase flow simulations through two-dimensional synthetically created fracture apertures. These synthetic rock fractures are created using different fractal dimensions, anisotropy factors, and mismatch lengths. Lattice Boltzmann method (LBM), which is a new computational approach suitable to simulate fluid flow especially in complex geometries, was then used to determine the permeability for different fractures. Regions of high velocity and low velocity flow were identified. The resulting permeability values were less than the ones obtained with the cubic law estimates. It has been found that as the mean aperture–fractal dimension ratio increased permeability increased. Moreover as the anisotropy factor increased permeability decreased. Neural network simulations were used to generalize the results.     

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.     

A numerical study of EGS heat extraction process based on a thermal non-equilibrium model for heat transfer in subsurface porous heat reservoir

With a previously developed numerical model, we perform a detailed study of the heat extraction process in enhanced or engineered geothermal system (EGS). This model takes the EGS subsurface heat reservoir as an equivalent porous medium while it considers local thermal non-equilibrium between the rock matrix and the fluid flowing in the fractured rock mass. The application of local thermal non-equilibrium model highlights the temperature-difference heat exchange process occurring in EGS reservoirs, enabling a better understanding of the involved heat extraction process. The simulation results unravel the mechanism of preferential flow or short-circuit flow forming in homogeneously fractured reservoirs of different permeability values. EGS performance, e.g. production temperature and lifetime, is found to be tightly related to the flow pattern in the reservoir. Thermal compensation from rocks surrounding the reservoir contributes little heat to the heat transmission fluid if the operation time of an EGS is shorter than 15 years. We find as well the local thermal equilibrium model generally overestimates EGS performance and for an EGS with better heat exchange conditions in the heat reservoir, the heat extraction process acts more like the local thermal equilibrium process.