Nonlinear two-point flux approximation for modeling full-tensor effects in subsurface flow simulations

Subsurface flow models can exhibit strong full-tensor anisotropy due to either permeability or grid nonorthogonality effects. Upscaling procedures, for example, generate full-tensor effects on the coarse scale even for cases in which the underlying fine-scale permeability is isotropic. A multipoint flux approximation (MPFA) is often needed to accurately simulate flow for such systems. In this paper, we present and apply a different approach, nonlinear two-point flux approximation (NTPFA), for modeling systems with full-tensor effects. In NTPFA, transmissibility (which provides interblock connections) is determined from reference global flux and pressure fields for a specific flow problem. These fields can be generated using either fully resolved or approximate global simulations. The use of fully resolved simulations leads to an NTPFA method that corresponds to global upscaling procedures, while the use of approximate simulations gives a method corresponding to recently developed local–global techniques. For both approaches, NTPFA algorithms applicable to both single-scale full-tensor permeability systems and two-scale systems are described. A unified framework is introduced, which enables single-scale and two-scale problems to be viewed in a consistent manner. Extensive numerical results demonstrate that the global and local–global NTPFA techniques provide accurate flow predictions over wide parameter ranges for both single-scale and two-scale systems, though the global procedure is more accurate overall. The applicability of NTPFA to the simulation of two-phase flow in upscaled models is also demonstrated.     

Analysis and comparison of mixed finite element and multipoint flux approximation methods for homogeneous media

We consider discretization on quadrilateral grids of an elliptic operator occurring, for example, in the pressure equation for porous-media flow. In a realistic setting – with non-orthogonal grid, and anisotropic, heterogeneous permeability – special discretization techniques are required. Mixed finite element (MFE) and multipoint flux approximation (MPFA) are two methods that can handle such situations. Previously, a framework for analytical comparison of MFE and MPFA in special cases has been suggested. A comparison of MFE and MPFA-O (one of two main variants of MPFA) for isotropic, homogeneous permeability on a uniformly distorted grid was also performed. In the current paper, we utilize the suggested framework in a slightly different manner to analyze and compare MFE, MPFA-O and MPFA-U (the second main variant of MPFA). We reconsider the case previously analyzed. We also consider the case of generally anisotropic, homogeneous permeability on an orthogonal grid.     

Inversion of multiple thermal indicators: Quantitative methods of determining paleoheat flux and geological parameters. I. Theoretical development for paleoheat flux

The quantitative inversion of present-day downhole thermal indicator information enables (a) assessment of the effective paleoheat flux recorded by thermal indicators, and (b) determination of geological parameters related to the dynamical burial history of sedimentary strata. The logic of the general inversion scheme underlying seven thermal indicators, viz. vitrinite reflectance, sterane, and/or hopane isomers, sterane aromatization, optical rotation,³⁹Ar/⁴⁰Ar, pollen translucency, and fission scar tracks in apatite is presented, and their capability for resolving paleoheat flux is noted. In a second paper, determination of chemical and physical parameters, as well as several other quantities of geological interest (such as the sizes and erosion times of multiple unconformities, stratigraphic ages, paleo-overpressure, igneous intrusion timing, overthrust timing and frictional heat generation, fault and slump timing, thermal effects due to radioactive layer emplacement, and salt emplacement and dissolution timing) are discussed. Such events impact on both burial and thermal histories of sedimentary layers.Resolution problems are discussed also in the second paper, as is an attractive scheme called thermal indicator tomography for sorting out degrees of resolution, precision, and uniqueness of the relevant geological parameters for each process in combination with determination of the effective paleoheat flux. Detailed statements on the mechanics of implementing the tomographic approach are provided.     

Constitutive modeling of inherent anisotropy in a strain space multiple mechanism model for granular materials

Consideration of fabric anisotropy is crucial to gaining an improved understanding of the behavior of granular materials. This paper presents a constitutive model to describe the sand behavior associated with fabric anisotropy within a framework of a strain space multiple mechanism model. In the proposed model, a second-order fabric tensor is extended by incorporating a new function that represents the effect of inherent (or initial fabric) anisotropy, along with three additional parameters: two of them, a₁ and a₂ , control the degree of anisotropy, and the second mode of inherent anisotropy can be expressed by introducing the parameter a₂ as well as the first mode by the parameter a₁ . The third parameter, θ₀ , expresses the principal direction of inherent anisotropy (eg, the normal vector direction of bedding planes relative to horizontal axis). The formulation of the dilative component of dilatancy (ie, positive dilatancy) is also extended to consider the effect of inherent anisotropy based on the interlocking mechanism. Experimental data on the complex anisotropic responses of Fraser River sand and Toyoura sand under monotonic loading is used to validate this model. The proposed model is shown to successfully capture anisotropic responses, which become contractive or dilative depending on different principal-stress directions, with a single set of anisotropy parameters; thus, the model is considered to possess the capability to simulate the anisotropic behaviors of granular materials. In addition to different loadings on the same fabric, the effects of different fabric anisotropies upon the sand behavior under the same loadings are also investigated.     

Evolution of induced fabric in a strain space multiple mechanism model for granular materials

The strain space multiple mechanism model idealizes the behavior of granular materials based on a multitude of virtual simple shear mechanisms oriented in arbitrary directions. Within this modeling framework, the virtual simple shear stress is defined as a quantity that depends on the contact distribution function as well as the normal and tangential components of inter‐particle contact forces, which evolve independently during the loading process. In other terms, the virtual simple shear stress is an intermediate quantity in the upscaling process from the microscopic level (characterized by the contact distribution and inter‐particle contact forces). The stress space fabric (i.e. the orientation distribution of the virtual simple shear stress) produces macroscopic stress through the tensorial average. Thus, the stress space fabric characterizes the fundamental and higher modes of anisotropy induced in granular materials. Comparing an induced fabric associated with the biaxial shear of plane granular assemblies obtained via a simulation using Discrete Element Method to the strain space multiple mechanism model suggests that the strain space multiple mechanism model has the capability to capture the essential features in the evolution of an induced fabric in granular materials. Copyright © 2012 John Wiley & Sons, Ltd.     

Inversion of multiple thermal indicators: Quantitative methods of determining paleoheat flux and geological parameters. II. Theoretical development for chemical, physical, and geological parameters

Using the logic for quantitative inversion of present-day downhole thermal indicators, the inversion procedure can be used to determine unknown, or poorly known, chemical and physical parameters as well as other geological quantities of interest which impact on burial history and thermal history of an evolving sedimentary basin. Some such quantities are: amount of erosion and timing of unconformities, paleo-overpressuring, stratigraphic age, timing of igneous intrusion and insertion temperature, overthrust timing and frictional heating, fault and slump timing, effects due to emplacement of a radiocative layer, and salt emplacement and dissolution timing. Combining a priori unknown values of these chemical/physical and geological parameters with unknown (a priori) paleoheat flux variations, a theoretical scheme, called thermal indicator tomography, is developed for the systematic determination of all parameters at the same time and on the same footing. Case histories will be discussed in subsequent papers in this series.     

渗透系数作为岩体卸荷分带量化指标的研究

渗透系数在一定程度上可以反映岩体内部裂隙发育程度、节理开度及卸荷程度,因此常作为岩体卸荷带划分的定量指标之一。但由于卸荷岩体的非线性特征,以及卸荷带研究方法和工程背景条件等方面的差异,各家划分卸荷带时所采用的渗透系数界限标准各有不同。本文从工程实例出发,通过对定性划分的各卸荷分带内对应钻孔段渗透系数的统计分析,以其统计平均值作为岩体卸荷分带的界线标准。实际应用表明,渗透系数作为卸荷带划分依据具有一定的优势,按渗透系数划分的卸荷带深度与其他方法综合确定的卸荷带深度基本接近。     

双重孔隙-裂隙岩体中洞室变形及强度各向异性的三维有限元分析

将确定双重孔隙-裂隙介质凝聚力及内摩擦角的方法与描述介质强度各向异性的微结构-无迹张量方法相结合,引入到三维有限元程序中。以简单的算例,通过解析解与数值解的比较,验证了所开发的有限元程序的可靠性。针对一个假定的位于被3组正交裂隙所切割的围岩中的矩形洞室,使用Mohr-Coulomb准则进行弹塑性数值模拟,就不同的工况分析了围岩中的位移、应力及塑性区的状态。计算结果显示：裂隙组的不同展布及组合使得岩体的变形及强度性质有着不同的各向异性,从而相应地导致围岩中的位移、应力及塑性区的分布与量值产生明显的差异。