An empirical theory for gravitationally unstable flow in porous media

In this paper, we follow a similar procedure as proposed by Koval (SPE J 3(2):145–154, 1963) to analytically model CO₂ transfer between the overriding carbon dioxide layer and the brine layer below it. We show that a very thin diffusive layer on top separates the interface from a gravitationally unstable convective flow layer below it. Flow in the gravitationally unstable layer is described by the theory of Koval, a theory that is widely used and which describes miscible displacement as a pseudo two-phase flow problem. The pseudo two-phase flow problem provides the average concentration of CO₂ in the brine as a function of distance. We find that downstream of the diffusive layer, the solution of the convective part of the model, is a rarefaction solution that starts at the saturation corresponding to the highest value of the fractional-flow function. The model uses two free parameters, viz., a dilution factor and a gravity fingering index. A comparison of the Koval model with the horizontally averaged concentrations obtained from 2-D numerical simulations provides a correlation for the two parameters with the Rayleigh number. The obtained scaling relations can be used in numerical simulators to account for the density-driven natural convection, which cannot be currently captured because the grid cells are typically orders of magnitude larger than the wavelength of the initial fingers. The method can be applied both for storage of greenhouse gases in aquifers and for EOR processes using carbon dioxide or other solvents.     

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.     

Theoretical and numerical analyses of convective instability in porous media with upward throughflow

Exact analytical solutions have been obtained for a hydrothermal system consisting of a horizontal porous layer with upward throughflow. The boundary conditions considered are constant temperature, constant pressure at the top, and constant vertical temperature gradient, constant Darcy velocity at the bottom of the layer. After deriving the exact analytical solutions, we examine the stability of the solutions using linear stability theory and the Galerkin method. It has been found that the exact solutions for such a hydrothermal system become unstable when the Rayleigh number of the system is equal to or greater than the corresponding critical Rayleigh number. For small and moderate Peclet numbers (Pe ⩽ 6), an increase in upward throughflow destabilizes the convective flow in the horizontal layer. To confirm these findings, the finite element method with the progressive asymptotic approach procedure is used to compute the convective cells in such a hydrothermal system. Copyright © 1999 John Wiley & Sons, Ltd.     

Numerical diagnosis of tropical cyclogenesis based on a hypothesis of helical self-organization of moist convective atmospheric turbulence

A novel diagnosis for tropical cyclogenesis is presented by examining helical self-organization of moist convective atmospheric turbulence in a rotating, non-homogeneous atmosphere. Our original research approach employed near-cloud-resolving numerical simulations, which allows quantitative diagnosis of cyclogenesis when the primary and secondary circulations in a forming hurricane vortex become linked by deep rotating cumulonimbus cores—Vortical Hot Towers (VHTs). It is shown here how the generated linkage makes the nascent vortex an integral helical system and allows a positive energetic feedback between the circulations that, with adequate moisture fluxes from the underlying sea surface to maintain convective instability, provides a self-sustaining amplification process on the system-scale circulation. The performed investigation suggests that diagnoses using helicity may not only provide an answer to the important question of when will cyclogenesis commence, given a favorable tropical environment, but will help develop a universally accepted definition of tropical cyclogenesis that does not yet exist.     

Evidence for deep groundwater flow and convective heat transport in mountainous terrain, Delta County, Colorado, USA

The Tongue Creek watershed lies on the south flank of Grand Mesa in western Colorado, USA and is a site with 1.5 km of topographic relief, heat flow of 100 mW/m², thermal conductivity of 3.3 W m^–1 °C^–1, hydraulic conductivity of 10^-8 m/s, a water table that closely follows surface topography, and groundwater temperatures 3–15°C above mean surface temperatures. These data suggest that convective heat transport by groundwater flow has modified the thermal regime of the site. Steady state three-dimensional numerical simulations of heat flow, groundwater flow, and convective transport were used to model these thermal and hydrological data. The simulations provided estimates for the scale of hydraulic conductivity and bedrock base flow discharge within the watershed. The numerical models show that (1) complex three-dimensional flow systems develop with a range of scales from tens of meters to tens of kilometers; (2) mapped springs are frequently found at locations where contours of hydraulic head indicate strong vertical flow at the water table, and; (3) the distribution of groundwater temperatures in water wells as a function of surface elevation is predicted by the model.     

Temperature-driven groundwater convection in cold climates

The aim was to study density-driven groundwater flow and analyse groundwater mixing because of seasonal changes in groundwater temperature. Here, density-driven convection in groundwater was studied by numerical simulations in a subarctic climate, i.e. where the water temperature was <4 °C. The effects of soil permeability and groundwater temperature (i.e. viscosity and density) were determined. The influence of impermeable obstacles in otherwise homogeneous ground was also studied. An initial disturbance in the form of a horizontal groundwater flow was necessary to start the convection. Transient solutions describe the development of convective cells in the groundwater and it took 22 days before fully developed convection patterns were formed. The thermal convection reached a maximum depth of 1.0 m in soil of low permeability (2.71 · 10^?9 m²). At groundwater temperature close to its density maximum (4 °C), the physical size (in m) of the convection cells was reduced. Small stones or frost lenses in the ground slightly affect the convective flow, while larger obstacles change the size and shape of the convection cells. Performed simulations show that “seasonal groundwater turnover” occurs. This knowledge may be useful in the prevention of nutrient leakage to underlying groundwater from soils, especially in agricultural areas where no natural vertical groundwater flow is evident. An application in northern Sweden is discussed.     

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     

Fluid-dynamics driving saline water in the North East German Basin

In several areas of the North German Basin, saline water comes close to, or even reaches the surface. Available data from wells indicate that brine stratification is under unstable conditions in the deeper underground. In order to analyse the possible transport mechanisms, 3D thermohaline simulations have been carried out for two different scenarios. The 3D regional model (230×330 km) indicates that salty water is driven to the surface by hydrostatical forces from the surrounding highlands. In addition, a smaller scale model (10×10 km) has been constructed with a grid resolution accounting for possible convective flow. The results indicate that convective flow may play a dominant role in areas with minor topography. In summary, the complex pattern of near surface occurrences of saline water probably results from the interaction of hydrostatic and thermal forces.     

On the time scales of nonlinear instability in miscible displacement porous media flow

In this paper, we analyze the time scales associated with instable fingering induced by density contrasts in miscible displacement porous media flow. We perform numerical simulations of a two-dimensional domain with boundaries that are closed to flow and identify the three regimes of the dynamics, namely the development of a stable diffusive boundary layer, the onset and growth of instabilities, and the fully nonlinear dynamics. Special focus is given to the onset of the fully nonlinear regime. The results are generic in the sense that there are no parameters in the non-dimensional model problem. Large ensembles are studied and an error estimate is given based on the combined effect of numerical errors and sampling errors. The nonlinear time scales show a dependence on the size of initial perturbations. We estimate this size for three formations used for CO₂ storage and find that the onset of enhanced convective mixing is considerably delayed compared with the linear onset time.     

长江与嘉陵江交汇区水流结构的数值模拟

针对天然河流交汇区域复杂的地形条件及水流运动特性,采用水气两相流三维数值模型,对长江与嘉陵江交汇区水流运动进行深入研究,分析了交汇区域分离区、剪切层、流速场及螺旋度的变化特性。研究结果表明:长江与嘉陵江交汇的水流分离区形状受地形影响明显,随着水位的增加向右岸移动,剪切层整体呈一个扭曲的曲面;干支流原有的环流在交汇之后重新汇合,左岸未出现明显的环流,右岸逆时针的环流有减弱趋势,交汇区域纵向流速呈现高速与低速带分布特征。此外,长江和嘉陵江在交汇口下游螺旋度呈现左侧为负、右侧为正的对称分布,水流结构表现出逐渐形成双螺旋流的趋势,其中左侧的螺旋流逆时针运动,右侧的螺旋流顺时针运动。     

块石路基层中对流过程与温度特性关系试验研究

利用块石层调控路基地温是主动冷却路基的一种重要措施,块石层内部对流换热过程是冻土工程应用研究的重要内容之一,块石层中由低压或温差引起的空气流动的实际测定一直是其中亟待解决的问题。通过高精度微风速探测仪器,首次获得由温差引起的块石层空气流动特征。研究发现：块石层中对流过程和温度变化过程相互影响,在加速区和稳定区实现快速降温之后呈现缓慢降温过程;块石层内温差大于空气对流的启动温差时,自然对流速度和温差具有线性相关性;随着降温过程在相同温差条件下具有不同的对流特性;块石层上表面温度波动对内部温度波动的影响随深度而减弱,温度延迟性与深度呈线性相关性,块石层不同层位温度变化的非对称性由不同的传热方式和传热效率导致;在热传导期块石层存在热传导影响范围,为保证降温效果块石层厚度需大于热传导范围。研究成果为理论分析和数值模拟提供直接的有力支持以及对该种措施降温机制的进一步认识和改进都具有重要意义。     

Influence of fluid flow on the regional thermal field: results from 3D numerical modelling for the area of Brandenburg (North German Basin)

We analyse the effect of fluid flow on the recent thermal field for the Brandenburg region (North German Basin) which is strongly affected by salt structures. The basin fill is modified by a thick layer of mobilized salt (Zechstein, Upper Permian) that decouples the overburden from deeper parts of the lithosphere and is responsible for thermal anomalies since salt has a distinctly higher thermal conductivity than the surrounding sediments and is impermeable to fluid flow. Numerical simulations of coupled fluid flow and heat transfer are carried out to investigate the influence of fluid flow on the shallow temperature field above the Zechstein salt, based on the finite element method. A comparison of results from conductive and coupled modelling reveals that the temperature field down to the low-permeable Triassic Muschelkalk is influenced by fluids, where the shallow low-permeable Tertiary Rupelian-clay is absent. Overall cooling is induced by forced convective forces, the depth range of which is controlled by the communication pathways between the different aquifers. Moreover, buoyancy-induced effects are found in response to temperature-dependent differences in the fluid density where forced convective forces are weak. The range of influence is controlled by the thickness and the permeability of the permeable strata above the Triassic Muschelkalk. With increasing depth, thermal conduction mainly controls the short-wavelength pattern of the temperature distribution, whereas the long-wavelength pattern results from interaction between the highly conductive crust and low-conductive sediments. Our results provide generic implications for basins affected by salt tectonics.     

Three-dimensional benchmark for variable-density flow and transport simulation: matching semi-analytic stability modes for steady unstable convection in an inclined porous box

This benchmark for three-dimensional (3D) numerical simulators of variable-density groundwater flow and solute or energy transport consists of matching simulation results with the semi-analytical solution for the transition from one steady-state convective mode to another in a porous box. Previous experimental and analytical studies of natural convective flow in an inclined porous layer have shown that there are a variety of convective modes possible depending on system parameters, geometry and inclination. In particular, there is a well-defined transition from the helicoidal mode consisting of downslope longitudinal rolls superimposed upon an upslope unicellular roll to a mode consisting of purely an upslope unicellular roll. Three-dimensional benchmarks for variable-density simulators are currently (2009) lacking and comparison of simulation results with this transition locus provides an unambiguous means to test the ability of such simulators to represent steady-state unstable 3D variable-density physics.     

块石路基对流特性实验研究

利用块石层调控冻土路基地温是冻土工程主动冷却路基的一种重要措施,并在青藏铁路重大工程建设中得到广泛应用.通过块石层中空气流动特性的深入研究,对其结构优化和效能提升具有重要意义,而不同温差条件下块石层中空气的流动特性,以及微风速的实际测定一直是其中亟待解决的问题.通过高精度微风速探测,首次获得封闭条件下块石层的空气流动特征.结果表明:在边界温度波动条件下,块石层内的降温过程与其内部自然对流过程密切相关,在对流作用下快速完成降温过程;升温过程则在相对较长时段内主要由内部的热传导换热完成,由此导致温度变化曲线非对称这一特殊现象的出现,并随深度增加而愈加明显.块石层上下端面的温差是对流发生及强度的关键控制因素,当温差达到一定量值时流速随温差的增加而加大,同时空气温度的变化也会对空气流动过程产生影响;块石层内整个空气流动过程是一个传递过程,也是热量传递的过程.该实验研究结果将对该种工程措施机理的进一步认识和改进、相关模拟计算参数的选取,以及在冻土区高等级公路等重大工程建设中的进一步有效应用均具有重要意义.     

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.     

Deep 3D thermal modelling for the city of Berlin (Germany)

This study predicts the subsurface temperature distribution of Germany’s capital Berlin. For this purpose, a data-based lithosphere-scale 3D structural model is developed incorporating 21 individual geological units. This model shows a horizontal grid resolution of (500 × 500) m and provides the geometric base for two different approaches of 3D thermal simulations: (1) calculations of the steady-state purely conductive thermal field and (2) simulations of coupled fluid flow and heat transport. The results point out fundamentally different structural and thermal configurations for potential geothermal target units. The top of the Triassic Middle Buntsandstein strongly varies in depth (159–2,470 m below sea level) and predicted temperatures (15–95 °C), mostly because of the complex geometry of the underlying Permian Zechstein salt. The top of the sub-salt Sedimentary Rotliegend is rather flat (2,890–3,785 m below sea level) and reveals temperatures of 85–139 °C. The predicted 70 °C-isotherm is located at depths of about 1,500–2,200 m, cutting the Middle Buntsandstein over large parts of Berlin. The 110 °C-isotherm at 2,900–3,700 m depth widely crosscuts the Sedimentary Rotliegend. Groundwater flow results in subsurface cooling the extent of which is strongly controlled by the geometry and the distribution of the Tertiary Rupelian Clay. The cooling effect is strongest where this clay-rich aquitard is thinnest or missing, thus facilitating deep-reaching forced convective flow. The differences between the purely conductive and coupled models highlight the need for investigations of the complex interrelation of flow- and thermal fields to properly predict temperatures in sedimentary systems.     

Meteorological analysis of the tornado in Ciudad Acuña,Coahuila State,Mexico, on May 25, 2015

Extreme meteorological conditions favor the development of severe storms and tornadoes that may have largely impacts on the population despite its relatively short life. Tornadic severe storms have been documented around the World. In Mexico (MEX), the study of the occurrence of tornadoes and severe storms is relatively new. In this research, we have selected an event of severe tornadic storm in Ciudad Acuña, Mexico. The storm was driven by a frontal system moving southward from USA converging with a warmer moist air flux from the Gulf of Mexico. The tornado strikes on the Northeast of Mexico, in Coahuila State, on May 25, 2015. Imagery of infrared channel from GOES 13 satellite and the presence of a hook echo in radar data of May 25, 2015, indicate a supercell structure. The maximum values of radial velocity were about ?20 and 15 m s^?1. In this study, the WRF model was used in order to simulate the mesoscale meteorological conditions of the tornado. Model simulations capture atmospheric features observed in Doppler radar. The simulated storm-relative helicity values were between 400 and 500 m² s^?2. The simulated convective available potential energy values were of 3000 J kg^?1. These values were higher than values for convective storms, located over the region of Ciudad Acuña in Mexico and Del Rio in USA. The supercell was a result of high humidity and temperature gradients, conditioned by frontal activity and moisture flux intensifications from the Gulf of Mexico.

     

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.     

Numerical modeling of the flow and transport of radionuclides in heterogeneous porous media

This paper is concerned with numerical methods for the modeling of flow and transport of contaminant in porous media. The numerical methods feature the mixed finite element method over triangles as a solver to the Darcy flow equation and a conservative finite volume scheme for the concentration equation. The convective term is approximated with a Godunov scheme over the dual finite volume mesh, whereas the diffusion–dispersion term is discretized by piecewise linear conforming triangular finite elements. It is shown that the scheme satisfies a discrete maximum principle. Numerical examples demonstrate the effectiveness of the methodology for a coupled system that includes an elliptic equation and a diffusion–convection–reaction equation arising when modeling flow and transport in heterogeneous porous media. The proposed scheme is robust, conservative, efficient, and stable, as confirmed by numerical simulations.