Transient flow of slightly compressible fluids through double-porosity,double-permeability systems — A state-of-the-art review

The theory of transient flow of slightly compressible fluids through naturally fractured reservoirs based on the double porosity conceptualization is summarized. The main achievements in the theory of fluid flow in leaky aquifer systems which are closely related with the double-porosity, double-permeability problems are also addressed. The main emphasis of this review is the analytical treatment of these problems.     

Immiscible displacement in vertically fractured reservoirs

A dual-porosity model is defined for saturated, two-phase, compressible, immiscible flow in a vertically fractured reservoir or aquifer. This model allows detailed simulation of the matrix-fracture interaction as well as the matrix flow itself. This is accomplished by directly coupling the matrix and fracture systems along the vertical faces of the matrix blocks, incorporating gravitational effects directly, and simulating flow inside the block. Thus fluid segregation due to gravitational effects and heterogeneities can be simulated. We show that our model can be derived via homogenization techniques. The model (in incompressible form for simplicity of exposition) is then approximated by a computationally efficient finite difference scheme. Calculations are presented to show the convergence of the scheme and to indicate the behavior of the model as a function of several parameters.     

Explicit Rate–Time Solutions for Modeling Matrix–Fracture Flow of Single Phase Gas in Dual-Porosity Media

One of the widely used methods for modeling matrix–fracture fluid exchange in naturally fractured reservoirs is dual porosity approach. In this type of modeling, matrix blocks are regarded as sources/sinks in the fracture network medium. The rate of fluid transfer from matrix blocks into fracture medium may be modeled using shape factor concept (Warren and Root, SPEJ 3:245–255, 1963); or the rate–time solution is directly derived for the specific matrix geometry (de Swaan, SPEJ 16:117–122, 1976). Numerous works have been conducted to study matrix–fracture fluid exchange for slightly compressible fluids (e.g. oil). However, little attention has been taken to systems containing gas (compressible fluid). The objective of this work is to develop explicit rate–time solutions for matrix–fracture fluid transfer in systems containing single phase gas. For this purpose, the governing equation describing flow of gas from matrix block into fracture system is linearized using pseudopressure and pseudotime functions. Then, the governing equation is solved under specific boundary conditions to obtain an implicit relation between rate and time. Since rate calculations using such an implicit relation need iterations, which may be computationally inconvenient, an explicit rate–time relation is developed with the aid of material balance equation and several specific assumptions. Also, expressions are derived for average pseudopressure in matrix block. Furthermore, simplified solutions (originated from the complex general solutions) are introduced applicable in infinite and finite acting flow periods in matrix. Based on the derived solutions, expressions are developed for shape factor. An important observation is that the shape factor for gas systems is the same as that of oil bearing matrix blocks. Subsequently, a multiplier is introduced which relates rate to matrix pressure instead of matrix pseudopressure. Finally, the introduced equations are verified using a numerical simulator.     

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.     

Derivation of the Kazemi–Gilman–Elsharkawy Generalized Dual Porosity Shape Factor

Kazemi et al. (SPE Reserv Eng 7(2):219–227, 1992) suggested an empirical matrix-fracture transfer function, verified based on experimental data of Mattax and Kyte (Trans AIME 225(15):177–184, 1962), to model fluid flow in naturally fractured dual porosity petroleum reservoirs using a dual-porosity numerical simulator. Their generalized shape factor should be valid for all possible irregular matrix blocks. The factor is calculated based on the volume of the matrix block, the surface open to flow in all directions and the distances of these surfaces to the centre of the matrix block. The summation is done over all open surfaces of a matrix block. Kazemi et al. (1992) showed that for rectangles and cylinders the formula reduces to the well-known forms of the shape factor. By the time, many authors indicated the validity of the formula, but no theoretical proof was offered for that so far. This study derives the Kazemi et al. (1992) shape factor using control volume finite difference discretization on the fracture-matrix dual continuum. The matrix blocks are handled as Voronoi polyhedra. The derivation is given for both isotropic and tensorial matrix permeability. Based on this derivation the authors conclude that the Kazemi et al. (SPE Reserv Eng 7(2):219–227, 1992) formula is exact under pseudo-steady-state conditions within the dual continuum mathematical concept of natural fractured dual porosity systems.     

An Improvement on Modeling of Forced Gravity Drainage in Dual Porosity Simulations Using a New Matrix-Fracture Transfer Function

In fractured oil reservoirs, the gravity drainage mechanism has great potentials to higher oil recovery in comparison with other mechanisms. Recently, the forced gravity drainage assisted by gas injection has also been considered; however, there are few comprehensive studies in the literature. Dual porosity model, the most common approach for simulation of fractured reservoirs, uses transfer function concept to represent the fluid exchange between matrix and its neighborhood fractures. This study compares the results of different available transfer functions with those of fine grid simulations when forced gravity drainage contributes to oil production from a single matrix block. These comparisons can lead to a more sophisticated formulation including the interplay of capillary, gravity and viscous forces. As a result, a new matrix-fracture transfer function can be developed and its results can be tested against the results of fine-grid simulations. Moreover, the reliability of this model for simulation of forced gravity drainage has been demonstrated by performing some sensitivity analysis.     

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.     

Two-phase flow for a horizontal well penetrating a naturally fractured reservoir with edge water injection

This paper examines the two-phase flow for a horizontal well penetrating a naturally fractured reservoir with edge water injection by means of a fixed streamline model. The mathematical model of the vertical two-dimensional flow or oil-water for a horizontal well in a medium with double-porosity is established, and whose accurate solutions are obtained by using the characteristic method. The saturation distributions in the fractured system and the matrix system as well as the formula of the time of water free production are presented. All these results provide a theoretical basis and a computing method for oil displacement by edge water from naturally fractured reservoirs.     

Effects of Fracture Boundary Conditions on Matrix-fracture Transfer Shape Factor

The matrix-fracture transfer shape factor is one of the important parameters in modeling naturally fractured reservoirs. Four decades after Warren and Root (1963, SPEJ, 245–255.) introduced the double porosity concept and suggested a relation for it, this parameter is still not completely understood. Even for a single-phase flow problem, investigators report different shape factors. This study shows that for a single-phase flow in a particular matrix block, the shape factor that Warren and Root defined is not unique and depends on the pressure in the fracture and how it changes with time. We use the Laplace domain analytical solutions of the diffusivity equation for different geometries and different boundary conditions to show that the shape factor depends on the fracture pressure change with time. In particular, by imposing a constant fracture pressure as it is typically done, one obtains the shape factor that Lim and Aziz (1995, J. Petrolean Sci. Eng. 13, 169.) calculated. However, other shape factors, similar to those reported in other studies are obtained, when other boundary conditions are chosen. Although, the time variability of the boundary conditions can be accounted for by the Duhamel’s theorem, in practice using large time-steps in numerical simulations can potentially introduce large errors in simulation results. However, numerical simulation models make use of a stepwise approximation of this theorem. It is shown in this paper that this approximation could lead to large errors in matrix-fracture transfer rate if large time-steps are chosen.     

低渗透裂缝性油气藏非稳态窜流因子研究简

深入分析了常定窜流因子和现有的非稳态因子对于低渗透基质-裂缝窜流的不适应问题,根据对23 个低渗岩心渗流曲线的拟合分析结果,引入了变渗透率系数,并在此基础上推导了非线性扩散方程,也就是低渗透基质-裂缝系统间窜流的控制微分方程;经过无量纲化处理,引入了动边界条件,分别利用积分方法和矩方法导出了非线性扩散方程在前后两个阶段的近似解析解,并在此基础上构建了非稳态窜流因子的表达形式;新的窜流因子可以有效应用于低渗透基质-裂缝系统间的窜流计算,有限元数值模拟的结果验证了新因子在计算非线性非稳态窜流方面的准确性与可靠性.     

低渗透裂缝性油气藏非稳态窜流因子研究

深入分析了常定窜流因子和现有的非稳态因子对于低渗透基质-裂缝窜流的不适应问题,根据对23 个低渗岩心渗流曲线的拟合分析结果,引入了变渗透率系数,并在此基础上推导了非线性扩散方程,也就是低渗透基质-裂缝系统间窜流的控制微分方程;经过无量纲化处理,引入了动边界条件,分别利用积分方法和矩方法导出了非线性扩散方程在前后两个阶段的近似解析解,并在此基础上构建了非稳态窜流因子的表达形式;新的窜流因子可以有效应用于低渗透基质-裂缝系统间的窜流计算,有限元数值模拟的结果验证了新因子在计算非线性非稳态窜流方面的准确性与可靠性.      

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...     

Development of an Analytical Time-Dependent Matrix/Fracture Shape Factor for Countercurrent Imbibition in Simulation of Fractured Reservoirs

In dual porosity modeling of naturally fractured reservoirs, fluids exchange between the high porous matrix blocks and high permeable fracture systems is governed by transfer function. Therefore, transfer function, and specially shape factor as the main part of it, control fluids flow behavior, which certainly have significant effects on development and management plan of naturally fractured reservoirs. Also several formulations have been proposed for shape factor by a number of researchers, nearly all of them derived for expansion mechanism. But, shape factor is a phase sensitive parameter that can greatly affect results of simulation. Moreover, several shortcomings are inherent in the derived expressions of shape factor for imbibition process. The main aim of this work is to develop a new time-dependent matrix–fracture shape factor specific to countercurrent imbibition. In this study, fluid saturation distribution within a matrix block is analytically derived by solving capillary–diffusion equation under different imposed boundary conditions for the process where countercurrent imbibition is the dominant oil drive mechanism. The validity of the solutions is checked against literature experimental data (Bourbiaux and Kalaydjian, SPERE 5, 361–368, SPE 18283, 1990) and also by performing single porosity fine grid simulations. Then, the concept of analogy between the transport phenomena is employed to propose a new expression for matrix–fracture transfer function that is used to derive transient shape factor. It is illustrated in this article that time variation of imbibtion rate and shape factor can be used to diagnose different states of imbibition process. Although, the displacement process and employed approaches are completely different in this and other studies (Chang, Technical report, 1993; Kazemi and Gilman (eds.) Flow and contaminant transport in fractured rock. Academic Press, San Dieg, 1993; Zimmerman et al., Water Resour Res, 29, 2127–2137, 1993; Lim and Aziz, J Pet Sci Eng 13, 169–178, 1995), but we arrived at the consistent values of shape factor under limiting condition of pseudo steady state flow. This means that after establishment of pseudo steady state, shape factor is only controlled by matrix geometry regardless of the displacement process, i.e., expansion or imbibition mechanism, However, shape factor is completely phase sensitive and process dependent during unsteady and late-transient states. Finally, boundary condition dependency of shape factor is investigated.     

Diffusion Mass Transfer in Miscible Oil Recovery: Visual Experiments and Simulation

Co-and counter-current type transfers due to diffusion and -free- convection caused by the buoyant forces between fracture and matrix were studied experimentally using 2-D glass-bead models. Mineral oil and kerosene were used as the displaced phase. The model saturated with oil was exposed to solvent phase (pentane) under static conditions (no flow in fracture) to mimic matrix-fracture interaction during gas or liquid solvent injection in naturally fractured reservoirs. Displacement fronts and patterns were analyzed and quantified using fractal techniques to obtain correlations between the fractal properties and displacement type. Displacements resulted in a mixture of bulk diffusion and -free- convection mainly depending on the interaction type (co- or counter-current), oil type, and displacement direction (horizontal and vertical). Conditions yielding different types of displacement patterns were identified. Finally, a stochastic model that was inspired from invasion percolation and diffusion limited aggregation algorithms was developed for the horizontal displacement cases. The experimental observations were matched to the displacement patterns obtained through the stochastic modeling.     

Transient Pressure Behavior of Reservoirs with Discrete Conductive Faults and Fractures

Fractures and faults are common features of many well-known reservoirs. They create traps, serve as conduits to oil and gas migration, and can behave as barriers or baffles to fluid flow. Naturally fractured reservoirs consist of fractures in igneous, metamorphic, sedimentary rocks (matrix), and formations. In most sedimentary formations both fractures and matrix contribute to flow and storage, but in igneous and metamorphic rocks only fractures contribute to flow and storage, and the matrix has almost zero permeability and porosity. In this study, we present a mesh-free semianalytical solution for pressure transient behavior in a 2D infinite reservoir containing a network of discrete and/or connected finite- and infinite-conductivity fractures. The proposed solution methodology is based on an analytical-element method and thus can be easily extended to incorporate other reservoir features such as sealing or leaky faults, domains with altered petrophysical properties (for example, fluid permeability or reservoir porosity), and complicated reservoir boundaries. It is shown that the pressure behavior of discretely fractured reservoirs is considerably different from the well-known Warren and Root dual-porosity reservoir model behavior. The pressure behavior of discretely fractured reservoirs shows many different flow regimes depending on fracture distribution, its intensity and conductivity. In some cases, they also exhibit a dual-porosity reservoir model behavior.     

裂隙岩体中非饱和渗流与运移的概念模型及数值模拟

探讨了裂隙岩体中非饱和地下水渗流与溶质运移的几种概念模型的构造及数值模拟问题 ,如裂隙网络模型、连续体模型、等效连续体模型、双孔隙度 (单渗透率 )模型、双渗透率模型、多组份连续体模型等。在裂隙岩体中 ,非饱和地下水的渗流可能只局限于岩体中的岩石组份、或裂隙网络 ,也可能在裂隙和岩石中同时发生 ;对前一种情形只需考虑单一连续体中的流动 ,而后一种情况则需要包括地下水在岩石和裂隙之间的交换。岩体中的裂隙网络往往是溶质运移的主要通道 ;但当溶质在裂隙与岩石之间的渗透和扩散是重要的运移机制时 ,就需要考虑岩石与裂隙界面处的溶质交换。为了模拟岩石与裂隙之间地下水和溶质的交换 ,就需要了解岩石与裂隙之间相互作用的模式和范围 ,使得这类问题的概念模型较单一连续体模型多了一层不确定性、其数值模拟也变得更为困难。因为在实际问题中不易、甚至根本不能判别非饱和渗流的实际形态 ,具体采用哪种模型主要取决于分析的目的和对现场数据的掌握程度。不论哪种模型都会受到模型及参数不确定性的影响 ,因此必须考虑与其他辅助模型的比较.     

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.     

Modeling the effects of capillary pressure with the presence of full tensor permeability and discrete fracture models using the mimetic finite difference method

Capillary dominated flow or imbibition—whether spontaneous or forced—is an important physical phenomena in understanding the behavior of naturally fractured water-driven reservoirs (NFR’s). When the water flows through the fractures, it imbibes into the matrix and pushes the oil out of the pores due to the difference in the capillary pressure. In this paper, we focus on modeling and quantifying the oil recovered from NFR’s through the imbibition processes using a novel fully implicit mimetic finite difference (MFD) approach coupled with discrete fracture/discrete matrix (DFDM) technique. The investigation is carried out in the light of different wetting states of the porous media (i.e., varying capillary pressure curves) and a full tensor representation of the permeability. The produced results proved the MFD to be robust in preserving the physics of the problem, and accurately mapping the flow path in the investigated domains. The wetting state of the rock affects greatly the oil recovery factors along with the orientation of the fractures and the principal direction of the permeability tensor. We can conclude that our novel MFD method can handle the fluid flow problems in discrete-fractured reservoirs. Future works will be focused on the extension of MFD method to more complex multi-physics simulations.

     

THE TRANSIENT ELLIPTIC FLOW OF POWER-LAW FLUID IN FRACTAL POROUS MEDIA

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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.