Upscaled modeling of well singularity for simulating flow in heterogeneous formations

Subsurface flows are affected by geological variability over a range of length scales. The modeling of well singularity in heterogeneous formations is important for simulating flow in aquifers and petroleum reservoirs. In this paper, two approaches in calculating the upscaled well index to capture the effects of fine scale heterogeneity in near-well regions are presented and applied. We first develop a flow-based near-well upscaling procedure for geometrically flexible grids. This approach entails solving local well-driven flows and requires the treatment of geometric effects due to the nonalignment between fine and coarse scale grids. An approximate coarse scale well model based on a well singularity analysis is also proposed. This model, referred to as near-well arithmetic averaging, uses only the fine scale permeabilities at well locations to compute the coarse scale well index; it does not require solving any flow problems. These two methods are systematically tested on three-dimensional models with a variety of permeability distributions. It is shown that both approaches provide considerable improvement over a simple (arithmetic) averaging approach to compute the coarse scale well index. The flow-based approach shows close agreement to the fine scale reference model, and the near-well arithmetic averaging also offers accuracy for an appropriate range of parameters. The interaction between global flow and near-well upscaling is also investigated through the use of global fine scale solutions in near-well scale-up calculations.     

Numerical solutions for flow in porous media

A numerical approach is proposed to model the flow in porous media using homogenization theory. The proposed concept involves the analyses of micro‐true flow at pore‐level and macro‐seepage flow at macro‐level. Macro‐seepage and microscopic characteristic flow equations are first derived from the Navier–Stokes equation at low Reynolds number through a two‐scale homogenization method. This homogenization method adopts an asymptotic expansion of velocity and pressure through the micro‐structures of porous media. A slightly compressible condition is introduced to express the characteristic flow through only characteristic velocity. This characteristic flow is then numerically solved using a penalty FEM scheme. Reduced integration technique is introduced for the volumetric term to avoid mesh locking. Finally, the numerical model is examined using two sets of permeability test data on clay and one set of permeability test data on sand. The numerical predictions agree well with the experimental data if constraint water film is considered for clay and two‐dimensional cross‐connection effect is included for sand. Copyright © 2003 John Wiley & Sons, Ltd.     

Meso‐scale particle modeling of concrete deterioration caused by alkali‐aggregate reaction

A meso‐scale particle model is presented to simulate the expansion of concrete subjected to alkali‐aggregate reaction (AAR) and to analyze the AAR‐induced degradation of the mechanical properties. It is the first attempt to evaluate the deterioration mechanism due to AAR using the discrete‐element method. A three‐phase meso‐scale model for concrete composed of aggregates, mortar and the interface is established with the combination of a pre‐processing approach and the particle flow code, PFC2D. A homogeneous aggregate expansion approach is applied to model the AAR expansion. Uniaxial compression tests are conducted for the AAR‐affected concrete to examine the effects on the mechanical properties. Two specimens with different aggregate sizes are analyzed to consider the effects of aggregate size on AAR. The results show that the meso‐scale particle model is valid to predict the expansion and the internal micro‐cracking patterns caused by AAR. The two different specimens exhibit similar behavior. The Young's modulus and compressive strength are significantly reduced with the increase of AAR expansion. The shape of the stress–strain curves obtained from the compression tests clearly reflects the influence of internal micro‐cracks: an increased nonlinearity before the peak loading and a more gradual softening for more severely affected specimens. Similar macroscopic failure patterns of the specimens under compression are observed in terms of diagonal macroscopic cracks splitting the specimen into several triangular pieces, whereas localized micro‐cracks forming in slightly affected specimens are different from branching and diffusing cracks in severely affected ones, demonstrating different failure mechanisms. Copyright © 2012 John Wiley & Sons, Ltd.     

堆石料级配缩尺方法对其室内试验结果的影响

对同一条现场级配曲线通过缩尺方法缩制成不同的试验模拟级配,进行了密度、力学和渗流特性的对比试验。试验结果显示：全采用等量替代法缩尺后由于小于5 mm含量保持不变,粗、细颗粒充填关系不理想,对应于密度和力学特性最差,渗透系数最大,随着混合法中相似级配法的使用,小于5 mm含量逐渐增加,粗、细颗粒充填关系得到明显改善,缩尺后的密度和力学特性逐渐增加,却带来渗透系数的逐渐减小。目前国内相关规程、规范对级配缩尺方法并没有做具体、明确的规定,有必要通过大量室内和现场对比试验,总结出室内科学的缩尺方法并建立反映缩尺效应的经验公式。     

Computational homogenization of uncoupled consolidation in micro‐heterogeneous porous media

Variationally consistent homogenization is exploited for the analysis of transient uncoupled consolidation in micro‐heterogeneous porous solids, whereby the classical approach of first‐order homogenization for stationary problems is extended to transient problems. Homogenization is then carried out in the spatial domain on representative volume elements (RVE), which are introduced in quadrature points in standard fashion. Along with the classical averages, a higher‐order conservation quantity is obtained. An iterative FE²‐algorithm is devised for the case of nonlinear permeability and storage coefficients, and it is applied to pore pressure changes in asphalt‐concrete (particle composite). Various parametric studies are carried out, in particular, with respect to the influence of the ‘substructure length scale’ that is represented by the size of the RVE's. Copyright © 2009 John Wiley & Sons, Ltd.     

Implementation of a Locally Conservative Numerical Subgrid Upscaling Scheme for Two-Phase Darcy Flow

We present a locally mass conservative scheme for the approximation of two-phase flow in a porous medium that allows us to obtain detailed fine scale solutions on relatively coarse meshes. The permeability is assumed to be resolvable on a fine numerical grid, but limits on computational power require that computations be performed on a coarse grid. We define a two-scale mixed finite element space and resulting method, and describe in detail the solution algorithm. It involves a coarse scale operator coupled to a subgrid scale operator localized in space to each coarse grid element. An influence function (numerical Greens function) technique allows us to solve these subgrid scale problems independently of the coarse grid approximation. The coarse grid problem is modified to take into account the subgrid scale solution and solved as a large linear system of equations posed over a coarse grid. Finally, the coarse scale solution is corrected on the subgrid scale, providing a fine grid representation of the solution. Numerical examples are presented, which show that near-well behavior and even extremely heterogeneous permeability barriers and streaks are upscaled well by the technique.     

Numerical Calculation of Equivalent Cell Permeability Tensors for General Quadrilateral Control Volumes

A new method for upscaling fine scale permeability fields to general quadrilateral-shaped coarse cells is presented. The procedure, referred to as the conforming scale up method, applies a triangle-based finite element technique, capable of accurately resolving both the coarse cell geometry and the subgrid heterogeneity, to the solution of the local fine scale problem. An appropriate averaging of this solution provides the equivalent permeability tensor for the coarse scale quadrilateral cell. The general level of accuracy of the technique is demonstrated through application to a number of flow problems. The real strength of the conforming scale up method is demonstrated when the method is applied in conjunction with a flow-based gridding technique. In this case, the approach is shown to provide results that are significantly more accurate than those obtained using standard techniques.     

One‐dimensional analytical solution for mesoscopic flow induced damping in a double‐porosity dual‐permeability material

Heterogeneities, such as fractures and cracks, are ubiquitous in porous rocks. Mesoscopic heterogeneities, that is, heterogeneities on length scales much larger than typical pore size but much smaller than the wavelength, are increasingly believed to be responsible for significant wave energy loss in the seismic frequency band. When a compressional wave stresses a material containing mesoscopic heterogeneities, the more compliant parts of the material (e.g., fractures and cracks) respond with a greater fluid pressure than the stiffer portions (e.g., matrix pores). The induced fluid flow, resulting from the pressure gradients developed on such scale, is called mesoscopic flow. In the present study, the double‐porosity dual‐permeability model is adopted to incorporate mesoscopic heterogeneities into rock models to account for the attenuation of wave energy. Based on the model, the damping effect due to mesoscopic flow in a one‐dimensional porous structure is investigated. Analytical solutions for several boundary‐value problems are obtained in the frequency domain. The dynamic responses of infinite and finite porous layer are examined. Numerical calculations show that the damping effect of mesoscopic flow is significant on the pore pressure response and the resulting effective stress. For the displacement, the effect is seen only at the very low frequency range or near the resonance frequencies. Copyright © 2017 John Wiley & Sons, Ltd.     

An upscaling method for one‐phase flow in heterogeneous reservoirs. A weighted output least squares (WOLS) approach

In this paper we study the problem of determining the effective permeability on a coarse scale level of problems with strongly varying and discontinuous coefficients defined on a fine scale. The upscaled permeability is defined as the solution of an optimization problem, where the difference between the fine scale and the coarse scale velocity field is minimized. We show that it is not necessary to solve the fine scale pressure equation in order to minimize the associated cost‐functional. Furthermore, we derive a simple technique for computing the derivatives of the cost‐functional needed in the fix‐point iteration used to compute the optimal permeability on the coarse mesh. Finally, the method is illustrated by several analytical examples and numerical experiments. This revised version was published online in July 2006 with corrections to the Cover Date.     

Large‐scale poroelastic fractured reservoirs modeling using the fast multipole displacement discontinuity method

An effective approach to modeling the geomechanical behavior of the network and its permeability variation is to use a poroelastic displacement discontinuity method (DDM). However, the approach becomes rather computationally intensive for an extensive system of cracks, particularly when considering coupled diffusion/deformation processes. This is because of additional unknowns and the need for time‐marching schemes for the numerical integration. The Fast Multipole Method (FMM) is a technique that can accelerate the solution of large fracture problems with linear complexity with the number of unknowns both in memory and CPU time. Previous works combining DDM and FMM for large‐scale problems have accounted only for elastic rocks, neglecting the fluid leak‐off from the fractures into the matrix and its influence on pore pressure and stress field. In this work we develop an efficient geomechanical model for large‐scale natural fracture networks in poroelastic reservoirs with fracture flow in response to injection and production operations. Accuracy and computational performance of the proposed method with those of conventional poroelastic DDM are compared through several case studies involving up to several tens of thousands of boundary elements. The results show the effectiveness of the FMM approach to successfully evaluate field‐scale problems for the design of exploitation strategies in unconventional geothermal and petroleum reservoirs. An example considering faults reveals the impact of reservoir compartmentalization because of sealing faults for both geomechanical and flow variables under elastic and poroelastic rocks. Copyright © 2015 John Wiley & Sons, Ltd.     

Pore scale study of permeability and tortuosity for flow through particulate media using Lattice Boltzmann method

In this paper, Lattice Boltzmann method (LBM) has been used to study the effects of permeability and tortuosity on flow through saturated particulate media and identify the relationships between permeability and tortuosity with other parameters such as particles diameter, grain specific surface, and porosity. LBM is a simple kinematic model that can incorporate the essential physics of microscopic and mesoscopic processes involved in flow through granular soils. The obtained results indicate that the 2D LB model, due to its inherent theoretical advantages, is capable of demonstrating that the porosity and specific surface are the most influential parameters in determining the intrinsic permeability of granular media. The obtained results show that particles' size diameter has a two‐fold effect on the coefficient of permeability: one is through specific surface and the other is by tortuosity factor. Numerical study also reveals that tortuosity of granular soils decreases almost linearly with increasing the porosity. Copyright © 2010 John Wiley & Sons, Ltd.     

泥石流堆积物中细颗粒含量与渗透系数关系试验研究

泥石流拦砂坝通常建造在沟床堆积物上。泥石流堆积物的渗透性是影响坝底扬压力的关键因素。目前关于土体渗透性的研究主要集中于无黏性粗粒土,对泥石流堆积物这种宽级配土体渗透性的研究比较缺乏。选择北川县泥石流沟床堆积物作为试验材料,通过渗透试验,确定了影响宽级配土渗透性的细颗粒上限粒径;在此基础上,通过试验研究了细颗粒含量与渗透系数的关系。结果表明,泥石流堆积物中粗颗粒仅起骨架作用,细颗粒才是决定其渗透性能的关键;显著影响宽级配土渗透性的细颗粒上限粒径为0.1 mm;细颗粒（<0.1 mm）含量与渗透系数呈负指数关系,并且当细颗粒含量超过20%以后,泥石流堆积土的渗透性趋于稳定。     

Computations of Absolute Permeability on Micro-CT Images

We apply an accurate numerical scheme to solve for Stokes flow directly on binarized three-dimensional rock images, such as those obtained by micro-CT imaging. The method imposes no-flow conditions exactly at the solid boundaries and employs an algebraic multigrid method to solve for the resultant set of linear equations. We compute the permeability of a range of consolidated and unconsolidated porous rocks; the results are comparable with those obtained using the lattice Boltzmann method and agree with experimental measurements on larger core samples. We show that the Kozeny–Carman equation can over-estimate permeability by a factor of 10 or more, particularly for the more heterogeneous systems studied. We study the existence and size of the representative elementary volume (REV) at lamina scale. We demonstrate that the REV for permeability is larger than for static properties—porosity and specific surface area—since it needs to account for the tortuosity and connectedness of the flow paths. For the carbonate samples, the REV appeared to be larger than the image size. We also study the anisotropy of permeability at the pore scale. We show that the permeability of sandpacks varies by less than 10 % in different directions. For sandstones, permeability changes by 25 % on average. However, the anisotropy of permeability in carbonates can be up to 50 %, indicating the existence of connected pores in one direction which are not connected in another.     

Influence of damage on pore size distribution and permeability of rocks

The model proposed in this article relates permeability to porosity measurements that can easily be performed in the laboratory. The pore size distribution (PSD) curve is updated with strains and damage. The updated volumetric fractions of natural pores and cracks are introduced in the expression of permeability. Contrary to classical permeability models based on PSD integrations, the model proposed in this article accounts for possible changes in the porosity modes: one mode for undamaged samples and two modes for cracked samples. The proposed approach also accounts for varying states of damage, as opposed to classical fracture network models, in which the cracks pattern is fixed. The only material parameters that are required to describe the microstructure are the lower and upper bounds of the pores size for both natural pores and cracks. All the other PSD parameters involved in the model are related to macroscopic parameters that can easily be determined in the laboratory, such as the initial void ratio. The framework proposed in this article can be used in any damage constitutive model to determine the permeability of a brittle porous medium. Drained triaxial compression tests have been simulated. Before cracks initiation, permeability decreases while the larger natural pores are getting squeezed. After the occurrence of damage, permeability grows due to the increase of cracks density. The model performs well to represent the influence of the confining pressure on damage evolution and permeability variations. Copyright © 2012 John Wiley & Sons, Ltd.     

Multi-scale modelling of the permeability evolution of fine sands during cement suspension grouting with filtration

Fluid flow during permeation grouting of fine sands with a microcement-based grout is studied by assuming that the heterogeneous medium composed of the initial granular skeleton, filtered cement and the interstitial fluid phase can be replaced by a continuous equivalent medium at the macroscopic level. Consequently, the method of Homogenization of Periodic Structures (HPS) is used to identify the effective permeability tensor evolution under the effect of cement filtration. The expression of the macroscopic permeability tensor derived through the HPS procedure is shown to depend on the permeating fluid viscosity and the geometrical arrangement of the sand grains and cement deposit within the microstructure. Numerical computations are made using various two-dimensional and three-dimensional microstructures, and the model results are confronted with grouting experiments performed on small scale columns in the laboratory.     

采用挤压边墙技术的高面板坝裂缝成因分析

为加快施工进度,国内很多高面板堆石坝都采用了挤压边墙技术,用上游坡面的挤压边墙代替传统的垫层。挤压边墙的施工特点决定了其与上覆面板之间的结构关系将与传统的面板-垫层有很大差别。虽然目前面板坝的施工技术和施工水平较以前有较大的提高,但少数工程仍然出现了比较严重的面板裂缝,而采用挤压边墙技术的面板坝施工期出现的裂缝分布规律与采用面板-垫层施工方法有明显的不同。面板裂缝的产生与面板施工期的应力变形状态密切相关,目前面板堆石坝计算中采用的整体模型方法对面板的概化较多,得到的面板应力精度较差。根据挤压边墙实际的施工特点,提出采用子模型法分析面板的应力变形,对某面板坝工程裂缝成因的分析结果表明,温降和上游坡面局部高程不平整引起的面板与挤压边墙之间的局部约束过大是导致面板裂缝的主要原因。     

温度、三轴应力条件下砂岩渗透率阶段特征分析

利用20 MN伺服控制高温高压岩体三轴试验机进行了长石细砂岩的渗透率试验,研究了恒定三轴压力和常温至600℃条件下砂岩渗透率变化特征。试验表明,随着温度增加,长石细砂岩渗透率变化规律表现为5阶段特征：①低温段,随温度升高渗透率略有下降;②阈值温度前段,达到阈值温度后,渗透率急剧增大,产生一个突跃,增加了65倍;③阈值温度后段,随着温度的继续升高,渗透率出现下降;④稳定段,渗透率达到谷值后不再下降,并稳定在一个较高水平,是室温渗透率的8倍;⑤高温段,渗透率重新增大。通过对长石细砂岩的细观结构变化及声发射事件（AE events）计数率分析后认为：渗透率的阶段性变化规律主要与内部矿物颗粒之间及其内部因局部热应力集中而诱发产生的微裂缝的开闭有关;高温条件下的长石细砂岩渗透率二次剧烈变化则主要与砂岩中部分矿物晶体颗粒成分在高温环境下的熔融、重结晶等现象有关。     

基于细观力学的混凝土劈拉破坏率相关特性研究

提出了一种高效的混凝土细观力学预处理方法,其中包括基于背景网格的骨料投放与细观三组分的识别;建立了颗粒离散元的混凝土细观力学模型。利用该模型对不同应变率下混凝土的劈拉试验进行了数值仿真,仿真结果显示出的混凝土动力强度提高的规律以及不同应变率下混凝土的破坏模式与试验结果吻合较好。研究表明,随着应变率的提高,混凝土破坏时产生更多弥散状分布的裂纹,同时有些裂纹穿过强度较高的骨料,导致混凝土在高应变率破坏过程中消耗更多的能量,是混凝土的动力强度提高的重要原因。此外,在高应变率下,力链产生更多的分叉,空间分布的不均匀性和随机性增加,因此,在高应变率下混凝土动力强度表现出更大的离散性。     

Permeability of stratified sands

Summary The permeability of graded sands was studied in the laboratory to determine the influence, if any, of stratification order on permeability (i.e. the position in which finer or coarser layers occur in the flow path). The evidence suggests that the order of stratification does not influence the order of magnitude of permeability of a sand. This order of magnitude is determined by the fine fraction of the sand. However, the order of stratification does have an influence on the finer details (i.e. variation within the same order of magnitude) of permeability. In this regard, an unsorted sand (all fractions mixed together) has the lowest permeability while a sorted one that has the fine fraction at the end of the sequence has the highest vertical permeability. It is also concluded that empirical formulae that use only the effective size, D ₁₀, to estimate permeability yield results that are comparable with those that incorporate sorting in their formulae. However, experimental results show that these formulae do not yield reliable results when used for more denser sands.