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.     

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.     

Upscaled Unstructured Computational Grids for Efficient Simulation of Flow in Fractured Porous Media

Discrete fracture modeling (DFM) is currently the most promising approach for modeling of naturally fractured reservoirs and simulation of multiphase fluid flow therein. In contrast with the classical double-porosity/double permeability models, in the DFM approach all the interactions and fluid flow in and between the fractures and within the matrix are modeled in a unified manner, using the same computational grid. There is no need for computing the shape factors, which are crucial to the accuracy of the double-porosity models. We have exploited this concept in order to develop a new method for the generation of unstructured computational grids. In the new approach the geological model (GM) of the reservoir is first generated, using square or cubic grid blocks. The GM is then upscaled using a method based on the multiresolution wavelet transformations that we recently developed. The upscaled grid contains a distribution of the square or cubic blocks of various sizes. A map of the blocks’ centers is then used with an optimized Delauney triangulation method and the advancing-front technique, in order to generate the final unstructured triangulated grid suitable for use in any general reservoir simulator with any number of fluid phases. The new method also includes an algorithm for generating fractures that, contrary to the previous methods, does not require modifying their paths due to the complexities that may arise in spatial distribution of the grid blocks. It also includes an effective partitioning of the simulation domain that results in large savings in the computation times. The speed-up in the computations with the new upscaled unstructured grid is about three orders of magnitude over that for the initial GM. Simulation of waterflooding indicates that the agreement between the results obtained with the GM and the upscaled unstructured grid is excellent. The method is equally applicable to the simulations of multiphase flow in unfractured, but highly heterogeneous, reservoirs.     

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.      

A New Mathematical Model for Force Gravity Drainage in Fractured Porous Media

In force gas/oil gravity drainage process in fractured porous media, gas is flowing in both matrix and fractures leading to produce a finite gas pressure gradient. Consequently, viscous force plays an important role for displacing matrix oil toward fractures in addition to gravity force that is required to be modeled appropriately. A new analytical model for estimation of steady state oil saturation distribution with assumption of fixed gas pressure gradient throughout the matrix is presented. Moreover, based on some results of this analytical model a different numerical formulation is developed to predict the performance of oil production process. Comparison of the results obtained from this numerical model with the results of a conventional simulator demonstrates that the newly developed model can be applied with satisfactory accuracy. Numerical simulations show that the viscous displacement in fractured porous media can reduce the capillary threshold height, and thus it suggests the force gravity drainage as a favorable production mechanism when the matrix length is close to the threshold height.     

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.     

A general treatment for non-Darcy film condensation within a porous medium in the presence of gravity and forced flow

Non-Darcy film condensation over a vertical flat plate within a porous medium is considered. The Forchheimer extended Darcy model is adopted to account for the non-Darcy effects on film condensation in the presence of both gravity and externally forced flow. A general similarity transformation is proposed upon introducing a modified Peclet number based on the total velocity of condensate, resulting from both gravitational force and externally forced flow. This general treatment makes it possible to obtain all possible similarity solutions including the asymptotic results in the four different limiting regimes, namely, Darcy forced convection regime, Forchheimer forced convection regime, Darcy body force predominant regime and Forchheimer body force predominant regime. Appropriate dimensionless groups for distinguishing these asymptotic regimes are found to be the micro-scale Grashof and Reynolds numbers based on the square root of the permeability of the porous medium. Correspondingly, the non-Darcy effect on the heat transfer rate are investigated in terms of these micro-scale dimensionless numbers.     

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.     

Implications of Grain-Scale Mineralogical Heterogeneity for Radionuclide Transport in Fractured Media

The geological disposal of nuclear waste is based on the multi-barrier concept, comprising various engineered and natural barriers, to confine the radioactive waste and isolate it from the biosphere. Some of the planned repositories for high-level nuclear waste will be hosted in fractured crystalline rock formations. The potential of these formations to act as natural transport barriers is related to two coupled processes: diffusion into the rock matrix and sorption onto the mineral surfaces available in the rock matrix. Different in situ and laboratory experiments have pointed out the ubiquitous heterogeneous nature of the rock matrix: mineral surfaces and pore space are distributed in complex microstructures and their distribution is far from being homogeneous (as typically assumed by Darcy-scale coarse reactive transport models). In this work, we use a synthetically generated fracture–matrix system to assess the implications of grain-scale physical and mineralogical heterogeneity on cesium transport and retention. The resulting grain-scale reactive transport model is solved using high-performance computing technologies, and the results are compared with those derived from two alternative models, denoted as upscaled models, where mineral abundance is averaged over the matrix volume. In the grain-scale model, the penetration of cesium into the matrix is faster and the penetration front is uneven and finger-shaped. The analysis of the cesium breakthrough curves computed at two different points in the fracture shows that the upscaled models provide later first-arrival time estimates compared to the grain-scale model. The breakthrough curves computed with the three models converge at late times. These results suggest that spatially averaged upscaled parameters of sorption site distribution can be used to predict the late-time behavior of breakthrough curves but could be inadequate to simulate the early behavior.     

Theoretical and Experimental Investigations of Miscible Displacement in Fractured Porous Media

In this paper a mathematical model for miscible displacement in fractured porous media is developed. The model takes into account mechanisms of mass transfer between fracture and matrix. The model is normalized by using the dimensionless parameters, which characterize the process, and the analytical solutions of the resulting system of equations are provided by utilizing the method of characteristics. For comparison the results of model with experimental results, laboratory displacement tests have been performed in fractured systems under miscible displacement. The porous media used were cylindrical Asmari cores from Iranian reservoirs containing an artificially vertical fracture. Normal heptane and kerosene were two miscible fluid used. There is very good agreement between experiments and model prediction.     

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...     

悬浮体力学——流体力学与胶体科学交叉的新兴学科

本文介绍了交叉新兴学科悬浮体力学的概貌.该学科发生发展在低Re数流体力学与胶体科学的交叉边缘处.它具有很广泛的应用价值,也有许多吸引人的重要理论课题.近年来在这领域中的研究工作规模十分巨大.对这学科有兴趣的人们而言,这是十分令人鼓舞的时期.在给出了本学科的基本特征和主要研究内容之后,对于相互作用着的低Re数、低Stk数、高Kn数悬浮粒子的运动学,对于在均匀悬浮体中的相对布朗扩散,对于非均匀单分散与多分散悬浮体中的绝对布朗扩散,对于稀释单分散和多分散悬浮体中的重力沉降,对于悬浮粒子的布朗碰并、重力碰并和纯变形场碰并,以及对于高Pcclet数与低Pcclet数下的悬浮体的有效粘性等问题,本文都作了简要的论述.     

Modeling of Heat and Mass Transfer in a Fractured Porous Medium

While fractured formations are possibly the most important contributors to the production of oil worldwide, modeling fractured formations with rigorous treatments has eluded reservoir engineers in the past. To date, one of the most commonly used fractured reservoir models remains the one that was suggested by Warren and Root nearly four decades ago. In this paper, a new model for fractures embedded in a porous medium is proposed. The model considers the Navier-Stokes equation in the fracture (channel flow) while using the Brinkman equation for the porous medium. Unlike the previous approach, the proposed model does not require the assumption of orthogonality of the fractures (sugar cube assumption) nor does it impose incorrect boundary conditions for the interface between the fracture and the porous medium. Also, the transfer coefficient between the fracture and matrix interface does not need to be specified, unlike the cases for which Darcy's law is used. In order to demonstrate the usefulness of the approach, a two-dimensional model of a fractured formation is developed and numerical simulation runs conducted.

The proposed model is derived through a series of finite element modeling runs for various cases using the Navier-Stokes equation in the channel while maintaining the Brinkman equation in the porous medium. Various cases studied include different fracture orientations, fracture frequencies, and thermal and solutal constraints. The usefulness of the proposed model in modeling complex formations is discussed. Finally, a series of numerical runs also provided validity of the proposed model for the cases in which thermal and solutal effects are important. Such a study of double diffusive phenomena, coupled with forced convection, in the context of fractured formations has not been reported before.     

An experimental study of the effective thermal conductivity of a sheared suspension of rigid spheres

An experimental study was conducted to measure the effective thermal conductivity of a sheared suspension of rigid spherical particles. The objective was to verify the theoretical prediction of Leal (1973) for a dilute suspension undergoing shear at low particle Peclet number, and to extend the range of the experiments to conditions beyond the scope of the theory. Surprisingly, reasonable agreement with the theoretical prediction was observed even for suspensions of moderate concentrations (volume fraction ≤ 0.25) and higher Peclet numbers [Pe 0(1)]. The trend of the data, however, verifies the obvious fact that the theory does not completely describe the transport behavior at higher concentrations and Peclet numbers. The range of quantitative applicability of Leal's result is apparently only for Pe < 0.01 and < 0.01, but the changes in the effective thermal conductivity in this domain were too small to be measured in our apparatus.     

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.     

低渗油层压裂水平井两相流研究

依据压裂水平井不同流动区域的流动规律, 将压裂水平井的渗流分为裂缝中的高速非达西流 动区、裂缝控制影响的椭圆渗流区、远离裂缝的基质非达西渗流区, 考虑启动压力梯度的影 响, 对压裂水平井两相渗流进行了分析, 得到了低渗透油层压裂水平井的产量公式. 研究结 果表明, 裂缝的导流能力越大, 压裂水平井的产量越高. 但随着开采时间的增加, 其产量递减幅度越 大; 压裂裂缝长度越小, 压裂水平井的初始产量越高. 但随着生产时间的推移, 压裂裂缝的 长度越大, 产量的递减幅度越小; 中间裂缝长, 两翼裂缝短的情况下, 压裂水平井的产量最 高.     

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.     

Flow of Coal-Bed Methane to a Gallery

Coal-seam methane reservoirs have a number of unique feature compared to conventional porous or fractured gas reservoirs. We propose a simplified mathematical model of methane movement in a coal seam taking into account the following features: a relatively regular cleat system, adsorptive methane storage, an extremely slow mechanism of methane release from the coal matrix into cleats and a significant change of permeability due to desorption.Parameters of the model have been combined into a few dimensionless complexes which are estimated to an order of magnitude. The simplicity of the model allows us to fully investigate the influence of each parameter on the production characteristics of the coal seam. We show that the reference time of methane release from the coal matrix into cleats – the parameter which is most poorly investigated – may have a critical influence on the overall methane production.     

Two problems of convective diffusion to the surfaces of poorly streamlined bodies

Problems of diffusion to particles of nonspherical shape at large Peclet numbers have been analyzed in many papers (see [1–7], for example). The solution of the problem of mass exchange of an ellipsoidal bubble at low Reynolds numbers was obtained in [1] while the solution at high Reynolds numbers was obtained in [2, 3]. In [4] an expression is given for the diffusional flux to the surface of a solid ellipsoidal particle over which a uniform Stokes stream flows. Generalization to the case of particles of arbitrary shape was done in [5, 6], while generalization to any number of critical lines on the surface of the body was done in [7, 8]. The two-dimensional problem of steady convective diffusion to the surface of a body of arbitrary shape is analyzed in the approximation of a diffusional boundary layer (ADBL). The simple analytical expressions obtained are more suitable for practical calculations than those in [5-8], since they allow one to determine at once, in the same coordinate system in which the field of flow over the particle was analyzed, the value of the diffusional flux to its surface (from the corresponding hydrodynamic characteristics). The plane problem of the diffusion to an elliptical cylinder in a uniform Stokes stream is solved. The problems of the diffusion to a plate of finite dimensions (in the plane case) and a disk (in the axisymmetric case) whose planes are normal to the direction of the incident stream are considered. It is shown that, in contrast to the results known earlier (see [4, 6-15], for example), where the total diffusional flux was proportional to the cube root of the Peclet number, here it is proportional to the one-fourth power.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 104–109, November–December, 1978.The authors thank Yu. P. Gupalo, Yu. S. Ryazantsev, and Yu. A. Sergeev for a useful discussion.     

Dynamics of dual-particles settling under gravity

As a first step towards understanding particle–particle interaction in fluid flows, the motion of two spherical particles settling in close proximity under gravity in Newtonian fluids was investigated experimentally for particle Reynolds numbers ranging from 0.01 to 2000. It was observed that particles repel each other for Re>0.1 and that the separation distance of settling particles is Reynolds number dependent. At lower Reynolds numbers, i.e. for Re<0.1, particles settling under gravity do not separate.The orientation preference of two spherical particles was found to be Reynolds number dependent. At higher Reynolds numbers, the line connecting the centres of the two particles is always horizontal, regardless of the way the two particles are launched. At lower Reynolds numbers, however, the particle centreline tends to tilt to an arbitrary angle, even of the two particles are launched in the horizontal plane. Because of the tilt, a side migration of the two particles was found to exist. A linear theory was developed to estimate the side migration velocity. It was found that the maximum side migration velocity is approximately 6% of the vertical settling velocity, in good agreement with the experimental results.Counter-rotating spinning of the two particles was observed and measured in the range of Re=0–10. Using the linear model, it is possible to estimate the influence of the tilt angle on the rate of rotation at low Reynolds numbers. Dual particles settle faster than a single particle at small Reynolds numbers but not at higher Reynolds numbers, because of particle separation. The variation of particle settling velocity with Reynolds number is presented. An equation which can be used to estimate the influence of tilt angle on particle settling velocity at low Reynolds number is also derived.