A finite-volume discretization for deformation of fractured media

Simulating the deformation of fractured media requires the coupling of different models for the deformation of fractures and the formation surrounding them. We consider a cell-centered finite-volume approach, termed the multi-point stress approximation (MPSA) method, which is developed in order to discretize coupled flow and mechanical deformation in the subsurface. Within the MPSA framework, we consider fractures as co-dimension one inclusions in the domain, with the fracture surfaces represented as line pairs in 2D (face pairs in 3D) that displace relative to each other. Fracture deformation is coupled to that of the surrounding domain through internal boundary conditions. This approach is natural within the finite-volume framework, where tractions are defined on surfaces of the grid. The MPSA method is capable of modeling deformation, considering open and closed fractures with complex and nonlinear relationships governing the displacements and tractions at the fracture surfaces. We validate our proposed approach using both problems, for which analytical solutions are available, and more complex benchmark problems, including comparison with a finite-element discretization.     

非均质裂隙介质中重非水相流体运移

由于裂隙介质具有强烈非均质性,使得重非水相流体(DNAPLs)在裂隙介质中的运移行为较孔隙介质更为复杂.基于指示模拟算法模拟裂隙介质的二元变量(粗糙面接触和裂隙开口),综合模拟退火算法构建裂隙渗透率随机场.采用T2VOC模拟DNAPLs在裂隙介质的运移,探讨粗糙面接触的相关长度、各向异性比、倾角以及非均质性程度对DNAPLs运移分布的影响.数值分析结果表明,粗糙面接触的相关性越差,DNAPLs污染范围越大;粗糙面接触的各向异性比和倾角增大,将导致更多DNAPLs残留;而相关长度和各向异性比增大以及倾角减小,都会引起DNAPLs锋面运移速率增大;随着渗透率非均质性增强,会导致局部粗糙面接触上蓄积的DNAPLs饱和度增大,运移速率减小.     

多孔介质中含均相/非均相反应传质过程数值模拟

结合化学反应方程式,并应用多孔多相介质溶混污染物输运过程的数值模型,对多孔多相介质中含均相/非均相化学反应传质过程进行了数值模拟。化学反应主要包含均相快速/慢速和非均相快速/慢速等5种化学反应过程,溶质输运行为的控制机制主要考虑对流、扩散及降解、吸附等。基于原有的隐式特征线Galerkin离散化的有限元方法,求解模型控制方程的边值初值问题,求解过程中把均相化学反应物质中按照反应物和生成物分开,非均相反应物质按照固相和液相分开,对均相反应物及非均相液相物质浓度耦合求解,而均相生成物和非均相固相物质独立求解。使方程组按照其不同类型进行分类,同时可减少未知数的个数。对于含有非线性内状态变量的右端项进行迭代求解。数值例题结果验证了所提出的数值方法的有效性、计算精度和稳定性。     

In search of earthquake-related hydrologic and chemical changes along Hayward Fault

Flow and chemical measurements have been made about once a month, and more frequently when required, since 1976 at two springs in Alum Rock Park in eastern San Jose, California, and since 1980 at two shallow wells in eastern Oakland in search of earthquake-related changes. All sites are on or near the Hayward Fault and are about 55 km apart. Temperature, electric conductivity, and water level or flow rate were measured in situ with portable instruments. Water samples were collected for later chemical and isotopic analyses in the laboratory. The measured flow rate at one of the springs showed a long-term decrease of about 40% since 1987, when a multi-year drought began in California. It also showed several increases that lasted a few days to a few months with amplitudes of 2.4 to 8.6 times the standard deviations above the background rate. Five of these increases were recorded shortly after nearby earthquakes of magnitude 5.0 or larger, and may have resulted from unclogging of the flow path and increase of permeability caused by strong seismic shaking. Two other flow increases were possibly induced by exceptionally heavy rainfalls. The water in both wells showed seasonal temperature and chemical variations, largely in response to rainfall. In 1980 the water also showed some clear chemical changes unrelated to rainfall that lasted a few months; these changes were followed by a magnitude 4 earthquake 37 km away. The chemical composition at one of the wells and at the springs also showed some longer-term variations that were not correlated with rainfall but possibly correlated with the five earthquakes mentioned above. These correlations suggest a common tectonic origin for the earthquakes and the anomalies. The last variation at the affected well occurred abruptly in 1989, shortly before a magnitude 5.0 earthquake 54 km away.     

对水文时间序列混沌特征参数估计问题的讨论

水文过程到底是不是低维混沌过程一直是个有争议的问题。相关文献在混沌特征参数估计中存在不少问题,包括:时延量估计的主观性很强,不同研究者的估计值差别很大;在关联维估计中,很多研究者有意无意地忽略了一个基本原则,即只有在关联维估计图上存在明确的标度区的情况下才能准确判断存在有限关联维;很多研究者在计算水文时间序列的关联维时仍采用原始的G-P计算公式,而没有采用Theiler提的修正公式,从而可能误将相点在时序上的相关性当做一种状态空间几何特征,造成关联维估计错误;国内相关研究中还普遍存在序列长度偏短的问题,对这些问题进行了讨论并给出了相应的结论。     

Effect of chemical additives on the dynamics of grinding media in wet ball mill grinding

Recent papers on the role of grinding aids in wet ball mills indicate that certain polymeric aids favorably act by altering pulp rheology, leading to improved production rates at high pulp densities. Analysis of recent data indicates that there can be an improvement in terms of the production of fine particles per unit time, but on the other hand, the production per unit of expended energy remains the same. Our experiments indicate that there is a critical pulp viscosity above which the grinding media tend to centrifuge because the balls cannot be detached from the mill wall during the time of rotation. Under these conditions the mill power draft decreases and a smaller amount of fines is generated. The polymeric additives are able to reduce the pulp viscosity to a value less than the critical value so that the balls no longer centrifuge and the mill draws full power. From an analysis of the force field acting on a ball inside the mill, this critical viscosity can be estimated and the results are in agreement with experiment.     

Boundary element analysis of contaminant transport in fractured porous media

A two-dimensional boundary integral method to analyse the flow of contaminant in fractured media having a two- or three-dimensional orthogonal fracture network is presented. The method assumes that the fractures provide the paths of least resistance for transport of contaminants while the matrix, because of its low permeability, acts as ‘storage blocks’ into which the contaminant diffuses. Laplace transform is used to eliminate the time variable in the governing equation in order to facilitate the formulation of a boundary integral equation in the Laplace transform space. Conventional boundary element techniques are applied to solve for the contaminant concentrations at specified locations in the spatial domain. The concentration in the time domain is then obtained by using an efficient inversion technique developed by Talbot. The method is able to analyse the behaviour of waste repositories which have diminishing concentration due to the mass transport of the contaminant into the surrounding fractured media.     

Finite element simulation of fluid flow in fractured rock media

Fluid dynamics models are used by the petroleum industry to model single- and/or multi-phase flow within fractured rock formations, in order to facilitate extraction of fluids such as oil and natural gas, and in other areas of engineering to study groundwater flow, as well as to estimate contaminant seepage and transport. In this paper, the numerical modelling software Comsol is used to simulate air and water flow through a specimen of granite with a single vertical fracture subjected to triaxial loading conditions. The intent of the model is to simulate triaxial test findings on a rock specimen with a natural fracture. Fluid flow is simulated at various confining and inlet pressures using the cubic law. Model results were in good agreement with laboratory findings. Pressure distribution along the fracture and across the specimen are as expected with a near linear pressure distribution along the length of the fracture. A drawdown effect on pressure distribution across the specimen in the vicinity of the fracture is also observed. Pressure gradient was largely uniform; however, some localised zones of high gradient along the fracture are observed.     

A model for the constant-head pumping test conducted in vertically fractured media

An analytical model for the constant-head pumping test is developed for a partially penetrating well that has a finite thickness skin, and intersects a single vertical fracture. In the model, the fracture is fully confined and flow occurs only in the fracture. The model is developed using Laplace transform and finite Fourier transform methods. The model is to be used for analysing well test data from vertically fractured media and for verifying numerical models. Dimensionless curves are used to study the effects of a finite thickness skin and a partially penetrating wellbore. In the presence of a finite thickness skin, or a partially penetrating wellbore, a typical flow response for the constant-head pumping test has three distinct periods of flow corresponding to small-, intermediate- and large-time. Small- and large-time approximations are presented for the model. For tests where the wellbore is fully penetrating, or the partial penetration ratio is known, these approximations can be used to analyse field data.     

Modeling of subsidence and stress-dependent hydraulic conductivity for intact and fractured porous media

Summary This study investigates the changes in deformation and stress dependent hydraulic conductivities that occur as a result of underground mining in intact and fractured porous media. The intact porous medium is assumed to be comprised of regularly packed spherical grains of uniform size. The variation in grain size or pore space due to the effect of changing intergranular stresses results in a change in rock hydraulic conductivity. A model is developed to describe the sensitivity of hydraulic conductivity to effective stresses through Hertzian contact of spherical grains. The fractured porous medium is approximated as an equivalent fracture network in which a single fracture is idealized as a planar opening having a constant equivalent thickness or aperture. Changes in fracture aperture as a result of changes in elastic deformation control the variation of hydraulic conductivity. A model is presented to illustrate the coupling between strain and hydraulic conductivity. Subsidence induced deformations that result from mining induced changes in hydraulic conductivity in both intact and fractured media. These changes are examined and compared with results from a mining case study.     

A model for moisture and heat flow in fractured unsaturated porous media

The present study investigates propagation of a cohesive crack in non‐isothermal unsaturated porous medium under mode I conditions. Basic points of skeleton deformation, moisture, and heat transfer for unsaturated porous medium are presented. Boundary conditions on the crack surface that consist of mechanical interaction of the crack and the porous medium, water, and heat flows through the crack are taken into consideration. For spatial discretization, the extended finite element method is used. This method uses enriched shape functions in addition to ordinary shape functions for approximation of displacement, pressure, and temperature fields. The Heaviside step function and the distance function are exploited as enrichment functions for representing the crack surfaces displacement and the discontinuous vertical gradients of the pressure and temperature fields along the crack, respectively. For temporal discretization, backward finite difference scheme is applied. Problems solved from the literature show the validity of the model as well as the dependency of structural response on the material properties and loading. Copyright © 2016 John Wiley & Sons, Ltd.     

Appearance of the nonlinearity from the nonlocality in diffusion through multiscale fractured porous media

We shall consider diffusion or single-phase flow in a multiscale porous medium which represents an infinite set of self-similar double-porosity media. At each scale, the medium consists of a highly permeable network of connected channels and low-permeable blocks. The characteristic scale of heterogeneity is ε at the highest level of hierarchy, wherein ε is a small parameter. The ratio between the channel and block permeability at each scale is ε ². The process analyzed is described using a diffusion equation with an oscillating multiscale diffusion parameter. The macroscale behavior is of interest. The transition to the macroscale is performed by means of the two-scale homogenization procedure. One step of averaging at each level of hierarchy leads to the appearance of the memory terms in the averaged equation. The successive averaging steps lead to progressive memory accumulation, so at each step of averaging, the macroscale model changes its type, and even the result of the second step is unknown a priori. The objective was to determine the macroscopic limit model for the infinite number of scales. By the method of induction, we obtained the macroscale model for an arbitrary number of scales and its limit for the infinite hierarchy. The limit model represents the system of two equations with memory terms. The kernel of the memory operator is the solution of a nonlinear integro-differential equation. Its solution is obtained through Laplace transform.     

Effect of treated municipal wastewater on bean growth, soil chemical properties, and chemical fractions of zinc and copper

The current study was carried out in order to investigate the short-term effect of different dilutions of wastewater on soil chemical properties, chemical fractions of zinc (Zn) and copper (Cu), and to assess the chemical buildup of heavy metal on two bean species. The experiment was performed as factorial based on completely randomized design with different dilutions of wastewater on two bean cultivars in two soil textures. The treatments consisted of irrigation with treated wastewater over all growing season, irrigation with wastewater and freshwater in equal proportions, and irrigation with well water only as control. The result showed that soil parameters are significantly affected by application of wastewater irrigation. Irrigation with wastewater increased the concentrations of organic matter, electrical conductivity, N, K, P, Fe, Cu, Zn, Mn, and Ni in soils compared to the control treatment. However, their values were all below international standards. Application of wastewater decreased soil pH and calcium carbonate equivalent. Plant tissue analysis showed increases in N, P, K, Zn, Cu, Zn, and Mn concentrations in grain and frond of beans in wastewater treatment as compared to the control. The concentrations of all elements in plants were lower than the toxic threshold. Chemical fractionations of Zn and Cu indicated that chemical forms of these metals were affected by irrigation with wastewater. Irrigating with wastewater resulted in the movement of Zn from the labile fractions towards the nonlabile fractions. In turn, mobility factor of Cu increased with application of wastewater. Maximum fresh and dry yields of beans were obtained from wastewater treatment compared to the control treatment. This study indicated that wastewater irrigation improves soil properties, plant growth, and yield without any contamination in soil and toxicity in plants.     

Embedded discontinuity approach for coupled hydromechanical analysis of fractured porous media

In this paper, a new continuum approach for the coupled hydromechanical analysis of fractured porous media is proposed. The methodology for describing the hydraulic characteristics invokes an enriched form of Darcy's law formulated in the presence of an embedded discontinuity. The constitutive relations governing the hydromechanical response are derived by averaging the fluid pressure gradient and the discontinuous displacement fields over a selected referential volume of the material, subject to some physical constraints. The framework incorporates an internal length scale which is explicitly embedded in the definition of gradient operators. The respective field equations are derived following the general form of balance equations in interacting continua. The conventional finite element method is then employed for the spatial discretization, and the generalized Newmark scheme is used for the temporal discretization. The proposed methodology is verified by some numerical examples dealing with a steady-state flow through fractured media as well as a time-dependent consolidation in the presence of a discontinuity.     

Long-term modeling of alteration-transport coupling: Application to a fractured Roman glass

To improve confidence in glass alteration models, as used in nuclear and natural applications, their long-term predictive capacity has to be validated. For this purpose, we develop a new model that couples geochemical reactions with transport and use a fractured archaeological glass block that has been altered for 1800 years under well-constrained conditions in order to test the capacity of the model.The chemical model considers three steps in the alteration process: (1) formation of a hydrated glass by interdiffusion, whose kinetics are controlled by a pH and temperature dependent diffusion coefficient; (2) the dissolution of the hydrated glass, whose kinetics are based on an affinity law; (3) the precipitation of secondary phases if thermodynamic saturation is reached. All kinetic parameters were determined from experiments. The model was initially tested on alteration experiments in different solutions (pure water, Tris, seawater). It was then coupled with diffusive transport in solution to simulate alteration in cracks within the glass. Results of the simulations run over 1800 years are in good agreement with archaeological glass block observations concerning the nature of alteration products (hydrated glass, smectites, and carbonates) and crack alteration thicknesses. External cracks in direct contact with renewed seawater were altered at the forward dissolution rate and are filled with smectites (400−500 μm). Internal cracks are less altered (by 1 or 2 orders of magnitude) because of the strong coupling between alteration chemistry and transport. The initial crack aperture, the distance to the surface, and sealing by secondary phases account for these low alteration thicknesses. The agreement between simulations and observations thus validates the predictive capacity of this coupled geochemical model and increases more generally the robustness and confidence in glass alteration models to predict long-term behavior of nuclear waste in geological disposal or natural glass in the environment.     

Retention properties of flow paths in fractured rock

There is no straightforward way to extrapolate solute retention properties from typical site characterisation scales to typical scales in the performance assessment of the geological disposal of nuclear wastes. Solutes diffuse much deeper into the rock matrix under performance assessment flow conditions than under site characterisation flow conditions. The modelling approach applied in this study, associated with the Äspö Task Force, enables evaluation of the contribution of the individual immobile layers to the overall retention. This makes it possible to determine the influence of the immobile zone heterogeneity on solute retention under different flow conditions. It appears that there is a significant difference between the dominating immobile retention zones on site characterisation and performance assessment scales. Fractured rock is characterised by heterogeneity and in particular a large spread of hydraulic properties. This favours formation of the preferential flow paths by leading to a few dominating transport paths. Large hydraulic features are, on average, better hydraulic conductors than smaller ones. This causes spatial scale effects for the solute retention properties. In particular, the hydraulic properties at the early parts of flow paths are more favourable to retention than those at the later parts of the flow paths.     

Semi-empirical equations for the systematic decrease in permeability with depth in porous and fractured media

Permeability loss with depth is a general trend in geological media and plays an essential role in subsurface fluid flow and solute transport. In the near surface zone where groundwater movement is active, the decrease in permeability with depth is dominated by the mechanical compaction of deformable media caused by the increase in lithostatic stress with depth. Instead of using empirical equations from statistical analysis, by considering the well-defined relationships among permeability, porosity, fracture aperture and effective stress under lithostatic conditions, new semi-empirical equations for the systematic depth-dependent permeability are derived, as well as the equations for the depth-dependent porosity in a porous medium and the depth-dependent fracture aperture in a fractured medium. The existing empirical equations can be included in the new equations as special cases under some simplification. These new semi-empirical equations perform better than previous equations to interpret the depth-dependent permeability of the Pierre Shale (with a maximum depth of approximately 4,500 m) and the granite at Stripa, Sweden (with a maximum depth of about 2,500 m).