Effect of Material Anisotropy on the Onset of Convective Flow in Three-Dimensional Fluid-Saturated Faults

In order to investigate the effect of material anisotropy on convective instability of three-dimensional fluid-saturated faults, an exact analytical solution for the critical Rayleigh number of three-dimensional convective flow has been obtained. Using this critical Rayleigh number, effects of different permeability ratios and thermal conductivity ratios on convective instability of a vertically oriented three-dimensional fault have been examined in detail. It has been recognized that (1) if the fault material is isotropic in the horizontal direction, the horizontal to vertical permeability ratio has a significant effect on the critical Rayleigh number of the three-dimensional fault system, but the horizontal to vertical thermal conductivity ratio has little influence on the convective instability of the system, and (2) if the fault material is isotropic in the fault plane, the thermal conductivity ratio of the fault normal to plane has a considerable effect on the critical Rayleigh number of the three-dimensional fault system, but the effect of the permeability ratio of the fault normal to plane on the critical Rayleigh number of three-dimensional convective flow is negligible.     

Analysis of steady‐state heat transfer through mid‐crustal vertical cracks with upward throughflow in hydrothermal systems

We conduct a theoretical analysis of steady‐state heat transfer problems through mid‐crustal vertical cracks with upward throughflow in hydrothermal systems. In particular, we derive analytical solutions for both the far field and near field of the system. In order to investigate the contribution of the forced advection to the total temperature of the system, two concepts, namely the critical Peclet number and the critical permeability of the system, have been presented and discussed in this paper. The analytical solution for the far field of the system indicates that if the pore‐fluid pressure gradient in the crust is lithostatic, the critical permeability of the system can be used to determine whether or not the contribution of the forced advection to the total temperature of the system is negligible. Otherwise, the critical Peclet number should be used. For a crust of moderate thickness, the critical permeability is of the order of magnitude of 10^?20 m², under which heat conduction is the overwhelming mechanism to transfer heat energy, even though the pore‐fluid pressure gradient in the crust is lithostatic. Furthermore, the lower bound analytical solution for the near field of the system demonstrates that the permeable vertical cracks in the middle crust can efficiently transfer heat energy from the lower crust to the upper crust of the Earth. Copyright © 2002 John Wiley & Sons, Ltd.     

Theoretical investigation of convective instability in inclined and fluid-saturated three-dimensional fault zones

The convective instability of pore-fluid flow in inclined and fluid-saturated three-dimensional fault zones has been theoretically investigated in this paper. Due to the consideration of the inclined three-dimensional fault zone with any values of the inclined angle, it is impossible to use the conventional linear stability analysis method for deriving the critical condition (i.e., the critical Rayleigh number) which can be used to investigate the convective instability of the pore-fluid flow in an inclined three-dimensional fault zone system. To overcome this mathematical difficulty, a combination of the variable separation method and the integration elimination method has been used to derive the characteristic equation, which depends on the Rayleigh number and the inclined angle of the inclined three-dimensional fault zone. Using this characteristic equation, the critical Rayleigh number of the system can be numerically found as a function of the inclined angle of the three-dimensional fault zone. For a vertically oriented three-dimensional fault zone system, the critical Rayleigh number of the system can be explicitly derived from the characteristic equation. Comparison of the resulting critical Rayleigh number of the system with that previously derived in a vertically oriented one has demonstrated that the characteristic equation of the Rayleigh number is correct and useful for investigating the convective instability of pore-fluid flow in the inclined three-dimensional fault zone system. The related numerical results from this investigation have indicated that: (1) the convective pore-fluid flow may take place in the inclined three-dimensional fault zone; (2) if the height of the fault zone is used as the characteristic length of the system, a decrease in the inclined angle of the inclined fault zone stabilizes the three-dimensional fundamental convective flow in the inclined three-dimensional fault zone system; (3) if the thickness of the stratum is used as the characteristic length of the system, a decrease in the inclined angle of the inclined fault zone destabilizes the three-dimensional fundamental convective flow in the inclined three-dimensional fault zone system; and that (4) the shape of the inclined three-dimensional fault zone may affect the convective instability of pore-fluid flow in the system.     

Double Diffusion-Driven Convective Instability of Three-Dimensional Fluid-Saturated Geological Fault Zones Heated from Below

We conduct a theoretical analysis to investigate the double diffusion-driven convective instability of three-dimensional fluid-saturated geological fault zones when they are heated uniformly from below. The fault zone is assumed to be more permeable than its surrounding rocks. In particular, we have derived exact analytical solutions to the total critical Rayleigh numbers of the double diffusion-driven convective flow. Using the corresponding total critical Rayleigh numbers, the double diffusion-driven convective instability of a fluid-saturated three-dimensional geological fault zone system has been investigated. The related theoretical analysis demonstrates that: (1) The relative higher concentration of the chemical species at the top of the three-dimensional geological fault zone system can destabilize the convective flow of the system, while the relative lower concentration of the chemical species at the top of the three-dimensional geological fault zone system can stabilize the convective flow of the system. (2) The double diffusion-driven convective flow modes of the three-dimensional geological fault zone system are very close each other and therefore, the system may have the similar chance to pick up different double diffusion-driven convective flow modes, especially in the case of the fault thickness to height ratio approaching 0. (3) The significant influence of the chemical species diffusion on the convective instability of the three-dimensional geological fault zone system implies that the seawater intrusion into the surface of the Earth is a potential mechanism to trigger the convective flow in the shallow three-dimensional geological fault zone system.     

The potential for convection and implications for geothermal energy in the Perth Basin, Western Australia

Convection of groundwater in aquifers can create areas of anomalously high temperature at shallow depths which could be exploited for geothermal energy. Temperature measurements in the Perth Basin (Western Australia) reveal thermal patterns that are consistent with convection in the Yarragadee Aquifer. This observation is supported by Rayleigh number calculations, which show that convection is possible within the range of aquifer thickness, geothermal gradient, salinity gradient and permeability encountered in the Yarragadee Aquifer, assuming that the aquifer can be treated as a homogeneous anisotropic layer. Numerical simulations of convection in a simplified model of the Yarragadee Aquifer show that: (1) the spacing of convective upwellings can be predicted from aquifer thickness and permeability anisotropy; (2) convective upwellings may be circular or elongate in plan view; (3) convective upwellings create significant temperature enhancements relative to the conductive profile; (4) convective flow rates are similar to regional groundwater flow rates; and (5) convection homogenises salinity within the aquifer. Further work is required to constrain the average horizontal and vertical permeability of the Yarragadee Aquifer, to assess the validity of treating the aquifer as a homogeneous anisotropic layer, and to determine the impact of realistic aquifer geometry and advection on convection.     

Lateral force and centroid location caused by horizontal and vertical surcharge strip loads on a cross‐anisotropic backfill

This work presents analytical solutions for determining lateral force (force per unit length) and centroid location caused by horizontal and vertical surcharge surface loads acting on a cross‐anisotropic backfill. The surcharge loading types are point load, line load, uniform strip load, upward linear‐varying strip load, upward nonlinear‐varying strip load, downward linear‐varying strip load, and downward nonlinear‐varying strip load. The planes of cross‐anisotropy are assumed parallel to the backfill ground surface. The proposed solutions, derived by integrating the lateral stress solutions (Int. J. Numer. Anal. Meth. Geomech. 2005; 29 :1341–1361), do not exist in literature. Clearly, the type and degree of material anisotropy, loading distance from the retaining wall, and loading types markedly impact the proposed solutions. Two examples are utilized to illustrate the type and degree of soil anisotropy, and the loading types on the lateral force and centroid location in the isotropic/cross‐anisotropic backfills generated by the horizontal and vertical uniform, upward linear‐varying and upward nonlinear‐varying strip loads. The parametric study results demonstrate that the lateral force and centroid location accounting for soil anisotropy, loading distance from the retaining wall, dimension of the loading strip, and loading directions and types differ significantly from those estimated using existing isotropic solutions. The derived solutions can be added to other lateral pressures, such as earth pressure or water pressure, required for stability and structural analysis of a retaining wall. Additionally, they can simulate realistically actual surcharge loading problems in geotechnical engineering when backfill materials are cross‐anisotropic. Copyright © 2007 John Wiley & Sons, Ltd.     

Layered clouds in the Indian monsoon region

Contrary to the prevalent belief that tropical region is characterized by convective clouds rather than by layer clouds, we have suggested that deep convective clouds occur on meso-scale, but layer clouds occur on larger synoptic-scale with a relatively small region of deep convective clouds. Sustenance of deep convective clouds is inhibited by the presence of inertio-gravity waves, which have alternating layers of upward and downward motion in the vertical. We have also shown that inertio-gravity waves generate regions of relatively strong horizontal velocity, vertically separated by layers of relatively weak horizontal velocity. Layers of strong horizontal velocity are created by inertio-gravity wave system through convergence of vertical flux of horizontal momentum. We have also suggested that horizontal convergence/divergence of moisture flux is generated by inertio-gravity waves, giving rise to vertically alternating layers of high/low humidity, and visible or sub-visible clouds. Layers of high humidity become layers of strong radar reflectivity at frequency of 53 MHz at which MST Radar at Gadanki, near Tirupati, India, operates. These observations, more than 2,50,000 in number, for vertical grid points, spread over all the months of the year, have helped us, among other observations, to arrive at these conclusions. Further, the analysis suggests that the main source of strong MST radar reflectivity is not mechanical turbulence as is commonly believed.     

The helicity of the velocity field for cellular convection in a rotating layer

The helicity of a cellular convective flow in a horizontal layer of a compressible fluid (gas) heated from below and rotating about the vertical axis is studied using finite-difference numerical simulations. The medium is assumed to be polytropically stratified. A thermal perturbation that produces a system of Bénard-type hexagonal convection cells is introduced at the initial time. Next, the cells are deformed by the action of the Coriolis force; however, at some stage of the evolution, the flow is nearly steady (at later times, the cells break down). For given Rayleigh and Prandtl numbers, the velocity-field helicity for this stage averaged over the layer increases with decreasing polytrope index (i.e., with increasing the curvature of the static entropy profile) and has a maximum at a certain rotational velocity of the layer. Numerical simulations of such quasi-ordered convective flows should reduce the uncertainties in estimates of the helicity, a quantity important for the operation of MHD dynamos.     

南海西南次海盆扩张期热液循环与洋壳增生的数值模拟

洋壳厚度受多方面因素的影响，前人大多关注地幔温度、地幔源成分等岩石圈深部因素，很少关注岩石圈浅层的热液循环对洋壳厚度的影响。利用基于有限元的数值模拟手段，对扩张期不同背景(洋中脊、拆离断层)、不同扩张速率的热液循环与洋壳增生的关系进行研究。结果表明：洋壳增生达到稳定前，热液循环导致理论洋壳厚度发生阶段性减薄，减薄量随时间改变，并且推迟了上地幔中熔融体出现的时间；当洋壳增生达到稳定后，热液循环下产生的理论洋壳厚度反而比无热液循环的更厚。结合洋壳增生过程中对流热通量的变化分析，在洋壳增生前期的上地幔温度低，驱动热液循环的热源小，产生的对流热通量相对较小且不稳定，热液循环缓慢冷却上地幔顶部的温度，进而推迟上地幔初始熔融的时间，减弱上地幔的熔融，并造成一定时间阶段内的生成理论洋壳比正常理论洋壳厚度更薄；当洋壳增生达到稳定后，对流热通量达到最大并稳定，热液循环持续快速的冷却上地幔顶部温度，导致上地幔深部的热向上地幔顶部补给，反而增大了上地幔顶部的温度和熔融量，进而增大了理论洋壳厚度。随着扩张速率的增大，理论洋壳厚度增大，对流热通量增大，热液循环导致的洋壳阶段性减薄的最大减薄量也增大，阶段性减薄的时间缩短。结合南海西南次海盆的洋壳结构特征分析：两条横跨南海西南次海盆的地震剖面显示，海盆内存在异常薄的洋壳区域，并且两条地震剖面的最薄洋壳厚度相差0. 85 km,推测海盆内异常薄洋壳和不同扩张时期的最薄洋壳厚度差异受到扩张期热液循环阶段性减薄洋壳作用的影响。     

A model of thermo-diffusive mass transport in geothermal systems using a stability theory formalism

The coexistence of chemical potential gradients of solute species in a hydrothermal solution, and a temperature gradient imposed by a heat source may lead to complex coupled transport of matter and heat. If the geological material is permeable to solutes, thermal diffusion (Soret effect) may constitute an important transport mechanism that could redistribute ionic species in the fluid phase. An attempt is made to explain, by stability theory, the non-steady-state behaviour of the system and the temperature oscillations within a thin convective layer adjacent to the walls of the heat source.     

Lateral stress caused by horizontal and vertical surcharge strip loads on a cross‐anisotropic backfill

This study derives analytical solutions for estimating the lateral stress caused by horizontal and vertical surcharge strip loads resting on a cross‐anisotropic backfill. The following loading types are employed in this work: point load, line load, uniform strip load, upward linear‐varying strip load, upward nonlinear‐varying strip load, downward linear‐varying strip load and downward nonlinear‐varying strip load. The cross‐anisotropic planes are assumed to be parallel to the horizontal surface of the backfill. The solutions proposed herein have never been mentioned in previous literature, but can be derived by integrating the point load solution in a Cartesian co‐ordinate system for a cross‐anisotropic medium. The calculations by the presented solutions are quick and accurate since they are concise and systematized. Additionally, the proposed calculations demonstrate that the type and degree of material anisotropy and the horizontal/vertical loading types decisively influence the lateral stress. This investigation presents examples of the proposed horizontal and vertical strip loads acting on the surface of the isotropic and cross‐anisotropic backfills to elucidate their effects on the stress. The analytical results reveal that the stress distributions accounting for soil anisotropy and loading types are quite different from those computed from the available isotropic solutions. Restated, the derived solutions, as well as realistically simulating the actual surcharge loading circumstances, provide a good reference for the design of retaining structures for the backfill materials are cross‐anisotropic. Copyright © 2005 John Wiley & Sons, Ltd.     

An Approach to a Visual Language for Convective Fluid Flows: Visualization of Geophysical Simulations

It was known that deep within numbers and binary data from simulations of geophysical convective flows resided various patterns. Two models of convective fluid flows were being considered. One was a model of two-dimensional (768 × 256) air convection with finite Prandtl number of one and Rayleigh number of 10⁸−10¹⁰, and another was a model of three-dimensional (up to 120 × 120 × 90) mantle convection with infinite Prandtl number and Rayleigh number of 10⁶−10⁸. Clearly, phenomena existed which superceded each individual dimensionless computer model to provide a piece of information regarding actual fluid flows. The problem was how to find, prove, and communicate these patterns and phenomena for convection simulations with gigabytes of data. In a search for such an analytical and communicative tool, the alternative of visualization was considered. The need for visualization was recognized and discussed. Then, utilizing both two- and three-dimensional models of high Rayleigh number convection, basic techniques of style and content were developed. Applications of the visualization techniques were designed utilizing IBM’s Data Explorer in order to create communicative images and movies, and after the applications, the problems of data storage and transfer became apparent. Throughout the process though, it became clear how important the language of vision actually could be in the geophysics community. In a field in which words such as plumes and internal waves have in ways replaced mathematics as the basic language for science, there is a need for another resource, another language-the visualization of convective fluid flows.     

Exact solutions for one‐dimensional consolidation of single‐layer unsaturated soil

Based on the Fredlund consolidation theory of unsaturated soil, exact solutions of the governing equations for one‐dimensional consolidation of single‐layer unsaturated soil are presented, in which the water permeability and air transmission are assumed to be constants. The general solution of two coupled homogeneous governing equations is first obtained. This general solution is expressed in terms of two functions psi₁ and ψ₂, where ψ₁ and ψ₂, respectively, satisfy two second‐order partial differential equations, which are in the same form. Using the method of separation of variables, the two partial differential equations are solved and exact solutions for three typical homogeneous boundary conditions are obtained. To obtain exact solutions of nonhomogeneous governing equations with three typical nonhomogeneous boundary conditions, the nonhomogeneous boundary conditions are first transformed into homogeneous boundary conditions. Then according to the method of undetermined coefficients and exact solutions of homogenous governing equations, the series form exact solutions are put forward. The validity of the proposed exact solutions is verified against other analytical solutions in the literature. Copyright © 2011 John Wiley & Sons, Ltd.     

The potential for free and mixed convection in sedimentary basins

 Free thermal convection and mixed convection are considered as potential mechanisms for mass and heat transport in sedimentary basins. Mixed convection occurs when horizontal flows (forced convection) are superimposed on thermally driven flows. In cross section, mixed convection is characterized by convection cells that migrate laterally in the direction of forced convective flow. Two-dimensional finite-element simulations of variable-density groundwater flow and heat transport in a horizontal porous layer were performed to determine critical mean Rayleigh numbers for the onset of free convection, using both isothermal and semi-conductive boundaries. Additional simulations imposed a varying lateral fluid flux on the free-convection pattern. Results from these experiments indicate that forced convection becomes dominant, completely eliminating buoyancy-driven circulation, when the total forced-convection fluid flux exceeds the total flux possible due to free convection. Calculations of the thermal rock alteration index (RAI=q·∇T) delineate the patterns of potential diagenesis produced by fluid movement through temperature gradients. Free convection produces a distinct pattern of alternating positive and negative RAIs, whereas mixed convection produces a simpler layering of positive and negative values and in general less diagenetic alteration. Received, January 1999/Revised, June 1999/Accepted, July 1999     

Thermal effects on fluid flow and hydraulic fracturing from wellbores and cavities in low-permeability formations

The coupled heat-fluid-stress problem of circular wellbore or spherical cavity subjected to a constant temperature change and a constant fluid flow rate is considered. Transient analytical solutions for temperature, pore pressure and stress are developed by coupling conductive heat transfer with Darcy fluid flow in a poroelastic medium. They are applicable to low permeability porous media suitable for liquid-waste disposal and also simulating reservoir for enhanced oil recovery, where conduction dominates the heat transfer process. A full range of solutions is presented showing separately the effects of temperature and fluid flow on pore pressure and stress development. It is shown that injection of warm fluid can be used to restrict fracture development around wellbores and cavities and generally to optimize a fluid injection operation. Both the limitations of the solutions and the convective flow effect are addressed. Copyright © 1999 John Wiley & Sons, Ltd.     

岩石导热热阻对裂隙对流换热的影响机制

增强型地热系统(EGS)中高温岩石与流体之间的对流换热特征一直以来是干热岩(HDR)研究的重要基础内容。岩石导热热阻对裂隙对流换热特征具有重要影响。为研究其具体影响,综合运用理论解析与数值模拟2种研究方法,通过对解析解讨论以及建立数值模型,研究两平行光滑平板之间的换热规律。结果表明：流体速度、传热边界层充分发展时,局部努塞尔特准数Nu_x为定值,与其他因素无关;局部对流换热系数h_x仅与流体热导率k和裂隙开度e有关,与其他因素无关。上下平板壁面热流恒定时,Nu_x为8.235;温度恒定时,Nu_x为7.54。然后建立多组导热热阻不同的岩石裂隙对流换热数值模型,发现岩石导热热阻增大,温度场进口段延长,对流换热系数h增大。岩石长度显著影响进口段占比,进而影响h的大小。h随着长度增大而减小;当岩石长度足够长时,进口段占比足够小,此时除k与e之外的参数对h基本没影响。并且发现实验室常用岩石长度为100 mm,而典型EGS工程中裂隙长度是米级的,建议室内实验重视岩石长度对裂隙对流换热特征的影响。     

Assessment of the Design of an Experimental Cover with Capillary Barrier Effect Using 4 Years of Field Data

One important step in the design of inclined covers with capillary barrier effect (CCBE) is the determination of the water diversion length (DL). Numerical simulations can predict the DL more precisely than steady-state analytical solutions. Nevertheless, as simplified methods have always been part of engineering design, the application of analytical solutions with conservative boundary conditions, may allow engineers to make reasonable predictions, particularly during the pre-feasibility stage of a project. In this study, a CCBE was designed, constructed and instrumented at the Saint-Tite-des-Caps landfill, Quebec, Canada. This CCBE included a seepage control layer superimposing a sand-gravel capillary barrier. The seepage control layer was made up of deinking by-products (DBP), an industrial by-product that was previously disposed of as waste. The capillary barrier was designed using an adaptation of the Ross analytical solution and the scenario considered was that of steady-state flow during constant seepage flow applied uniformly at the top of the sand-gravel capillary barrier. Although these conditions appear simplistic, they were deemed reasonable because placement of the seepage control layer on the top of the capillary barrier led to very low suctions at the interface, thereby allowing uniform downward seepage rates, limited by the saturated hydraulic conductivity of the DBP. In this paper, a discussion about the behaviour of the cover system based on 4 years of field data from several instruments is presented. The challenge of using DBP, more precisely the settlement of the DBP layer and its impact on k _sat , is also assessed. The DL was reassessed considering the new k _sat . A discussion on the validity of employing analytical solutions to determine DL is also presented. This paper illustrates how certain variables affect the design of inclined CCBEs that include a highly compressible material as seepage control layer.     

热液源研究的重要进展和“三源”交代热液成矿学说

热液来源在很大程度上决定了热液矿床的成因。过去由于缺乏研究热液来源的有效手段使矿床成因长期争论不休。近３０年来，热液水和成矿物质来源都有较好的测定方法，如热液的δ￣（１８）Ｏ、δＤ的测定，根据围岩的δ￣（１８）Ｏ变化、热液矿床的地理位置与热液水的δ￣（１８）Ｏ和δＤ的关系、流体包裹体的成分、矿床分布与围岩含水性的关系等确定热液水的类型和来源，根据围岩中成矿金属含量的变化而确定矿质来源等。这样在热液来源方面取得了一系列重要成果，在此基础上提出了热驱动对流循环成矿模式，进而发展为“三源”交代热液成矿学说。并根据新学说总结出新的成矿规律，建立了新的在实践中有明显效果的预测方法。但这方面的研究还刚刚开始，许多重要的成矿问题尚待研究。     

Semi-analytical solutions of a contaminant transport equation with nonlinear sorption in 1D

A new method to determine semi-analytical solutions of one-dimensional contaminant transport problem with nonlinear sorption is described. This method is based on operator splitting approach where the convective transport is solved exactly and the diffusive transport by finite volume method. The exact solutions for all sorption isotherms of Freundlich and Langmuir type are presented for the case of piecewise constant initial profile and zero diffusion. Very precise numerical results for transport with small diffusion can be obtained even for larger time steps (e.g., when the Courant-Friedrichs-Lewy (CFL) condition failed).     

Precise temperature monitoring in boreholes: evidence for oscillatory convection? Part II: theory and interpretation

In the previous part of this work (Cermak, Safanda and Bodri, this volume p.MMM) we have described experimental data and quantified the heterogeneity features of the microtemperature time series. The spectral analysis and the local growth of the second moment technique revealed scaling structure of all observed time series generally similar and suggested the presence of two temperature forming processes. The longer-scale part can be attributed to the heat conduction in compositional and structural heterogeneous solid rocks, further affected by various local conditions. Short-scale temperature oscillations are produced by the intra-hole fluid convection due to inherent instability of water column filling the hole. Here we present how the observational evidence is supported by the results of the computer simulations. The exact modes of intra-hole convection may be different, ranging from quasi-periodic (“quiescent”) state to close of turbulence. As demonstrated by numerical modeling and referred on laboratory experiments, at higher Rayleigh numbers the periodic character of oscillation characteristic for “quiescent” regime is superseded by stochastic features. This so called “oscillatory” convection occurs due to instability within the horizontal boundary layers between the individual convectional cells. In spite of the fact that the basic convective cell motion is maintained and convection is characterized by slow motion, the oscillatory intra-hole flow and corresponding temperature patterns exhibit typical features of turbulence. The idea of boundary layer instability as a source of stochastic temperature fluctuations could explain many distinct features of borehole temperatures that previously cannot be interpreted.