Finite element analysis of steady-state natural convection problems in fluid-saturated porous media heated from below

In this paper, a progressive asymptotic approach procedure is presented for solving the steady-state Horton–Rogers–Lapwood problem in a fluid-saturated porous medium. The Horton–Rogers–Lapwood problem possesses a bifurcation and, therefore, makes the direct use of conventional finite element methods difficult. Even if the Rayleigh number is high enough to drive the occurrence of natural convection in a fluid-saturated porous medium, the conventional methods will often produce a trivial non-convective solution. This difficulty can be overcome using the progressive asymptotic approach procedure associated with the finite element method. The method considers a series of modified Horton–Rogers–Lapwood problems in which gravity is assumed to tilt a small angle away from vertical. The main idea behind the progressive asymptotic approach procedure is that through solving a sequence of such modified problems with decreasing tilt, an accurate non-zero velocity solution to the Horton–Rogers–Lapwood problem can be obtained. This solution provides a very good initial prediction for the solution to the original Horton–Rogers–Lapwood problem so that the non-zero velocity solution can be successfully obtained when the tilted angle is set to zero. Comparison of numerical solutions with analytical ones to a benchmark problem of any rectangular geometry has demonstrated the usefulness of the present progressive asymptotic approach procedure. Finally, the procedure has been used to investigate the effect of basin shapes on natural convection of pore-fluid in a porous medium. © 1997 by John Wiley & Sons, Ltd.     

基于粒径分维值检验的松散介质渗透系数空间估值——以河北平原为例

为了克服松散含水介质非均质性各向异性对介质渗透系数的求取所带来的困难,以水动力条件为线索,利用互相关法寻找同一时代地层及同一时代地表水的主、次流方向.在主流方向上,建立了石家庄-束鹿-衡水段介质颗粒粒径随距离递变的数学模型,并以此为基础推求无钻孔地段的粒径.同时,运用分形理论,计算了中、上更新统地层主流方向平均粒径空间分布的分维值,并将其作为检验粒径估值正确与否的判据.通过不同岩性的渗透系数-深度模型,实现了用平均粒径预测渗透系数的全过程.     

横观各向同性含液饱和多孔介质黏弹性瞬态反应

在考虑横观各向同性含液饱和多孔介质固体骨架和流体可压缩性以及固体骨架的黏弹性特征下,基于横观各向同性含液饱和多孔介质u-w形式的三维动力控制方程,以固相位移u、液相相对位移w为基本未知量,综合运用Laplace变换、双重Fourier变换等方法,在直角坐标系下通过引入中间变量,将六元2阶动力控制方程组化为两组各含4个未知变量的常微分方程组,给出了直角坐标系下横观各向同性含液饱和多孔介质三维黏弹性动力反应的积分形式一般解;作为理论推导的验证,通过引入初始条件和边界条件,对横观各向同性含液饱和多孔介质半空间黏弹性瞬态反应问题进行了求解。解答的退化验证表明,所推导的理论解是正确的。     

有效压力对低渗透多孔介质孔隙度、渗透率的影响

以低渗透多孔介质为研究对象,通过实验得出孔隙度、渗透率随有效压力变化曲线。研究表明,流体在低渗透多孔介质中渗流时,流固耦合效应十分显著。这是因为低渗透多孔介质的孔隙很小,而孔隙度的微小变化,都会对渗透率产生大的影响,因此低渗透介质随有效应力的变化十分明显。由于流固耦合效应的存在,在低渗透油气藏的开发中,不宜采用降压开采,进行油藏数值模拟时,也应进行油藏流固耦合数值模拟。      

Dynamics of unsaturated poroelastic solids at finite strain

We derive the governing equations for the dynamic response of unsaturated poroelastic solids at finite strain. We obtain simplified governing equations from the complete coupled formulation by neglecting the material time derivative of the relative velocities and the advection terms of the pore fluids relative to the solid skeleton, leading to a so‐called u^s ? p^w ? p^a formulation. We impose the weak forms of the momentum and mass balance equations at the current configuration and implement the framework numerically using a mixed finite element formulation. We verify the proposed method through comparison with analytical solutions and experiments of quasi‐static processes. We use a neo‐Hookean hyperelastic constitutive model for the solid matrix and demonstrate, through numerical examples, the impact of large deformation on the dynamic response of unsaturated poroelastic solids under a variety of loading conditions. Copyright © 2011 John Wiley & Sons, Ltd.     

Stochastic analysis of bounded unsaturated flow in heterogeneous aquifers: Spectral/perturbation approach

This paper describes a stochastic analysis of steady state flow in a bounded, partially saturated heterogeneous porous medium subject to distributed infiltration. The presence of boundary conditions leads to non-uniformity in the mean unsaturated flow, which in turn causes non-stationarity in the statistics of velocity fields. Motivated by this, our aim is to investigate the impact of boundary conditions on the behavior of field-scale unsaturated flow. Within the framework of spectral theory based on Fourier–Stieltjes representations for the perturbed quantities, the general expressions for the pressure head variance, variance of log unsaturated hydraulic conductivity and variance of the specific discharge are presented in the wave number domain. Closed-form expressions are developed for the simplified case of statistical isotropy of the log hydraulic conductivity field with a constant soil pore-size distribution parameter. These expressions allow us to investigate the impact of the boundary conditions, namely the vertical infiltration from the soil surface and a prescribed pressure head at a certain depth below the soil surface. It is found that the boundary conditions are critical in predicting uncertainty in bounded unsaturated flow. Our analytical expression for the pressure head variance in a one-dimensional, heterogeneous flow domain, developed using a nonstationary spectral representation approach [Li S-G, McLaughlin D. A nonstationary spectral method for solving stochastic groundwater problems: unconditional analysis. Water Resour Res 1991;27(7):1589–605; Li S-G, McLaughlin D. Using the nonstationary spectral method to analyze flow through heterogeneous trending media. Water Resour Res 1995; 31(3):541–51], is precisely equivalent to the published result of Lu et al. [Lu Z, Zhang D. Analytical solutions to steady state unsaturated flow in layered, randomly heterogeneous soils via Kirchhoff transformation. Adv Water Resour 2004;27:775–84].     

Surface wave propagation in a transversely isotropic elastic layer overlying a liquid saturated porous solid half-space and lying under the uniform layer of liquid

Dispersion of Rayleigh-type surface wave is studied in a homogeneous transversely isotropic elastic layer overlying a nondissipative liquid-saturated porous solid half-space and lying under a uniform layer of homogeneous liquid. The frequency equation in the form of ninth-order determinant is obtained.Special cases have been deduced by reducing the depth of the layers to zero and by changing the transverse isotropic layer to an isotropic layer. Dispersion curves for the phase velocity have been plotted for a particular model.     

On the Use of Crushed Recycled Glass Instead of Silica Sand in Dual‐Media Filters

Crushed recycled glass was evaluated as an alternative to silica sand in dual‐media filters. Pilot scale inline filtration experiments were carried out using raw waters from three different water sources with turbidities between 6.0 and 14.0 NTU. Two physically identical filter columns were operated in parallel in the experiments. One filter consisted of 62.5 cm silica sand and 41.5 cm anthracite coal, whereas the other filter contained 62.5 cm crushed recycled glass plus 41.5 cm anthracite coal. The total bed depth was 104 cm for both filters. The properties of the media were as follows: Glass effective size = 0.77 mm, uniformity coefficient = 1.41. Sand effective size = 0.79 mm, uniformity coefficient = 1.33. Coal effective size = 1.45 mm, uniformity coefficient = 1.39. Experiments were repeated five times as follows: (i) Without the use of a coagulant, (ii–iii) with 5 and 10 mg/L of alum, and (iv–v) with 5 and 10 mg/L of ferric chloride. The filtration rate used was 11.5 m/h. Turbidity, particle counts, and head losses were measured and compared as functions of time. The following were observed: (i) Effluent turbidities and particle counts of the two filters were very close, i.e., essentially the same effluent quality was obtained when crushed glass was used instead of silica sand. (ii) In the majority of the tests, the filter with crushed glass generated both a smaller clean‐bed head loss and smaller clogging head losses than those of the filter containing sand. It is concluded that crushed glass may be a good alternative to silica sand in dual‐media filtration.     

Novel insights on intensity and typology of direct human-nature interactions in protected areas through passive crowdsourcing

Recent advances in geotagging, sharing and automatically analyzing online content from Social Networking Sites (SNS) offer unprecedented opportunities for the analysis of human-nature interactions. Previous studies in this field, however, offer limited insights regarding the benefits of automated content analysis especially at large scales, biases arising from the selection of SNS sources, and the predictive power of visitation models based on SNS data. We explore quantitative and qualitative aspects related to intensity, interests and sentiments associated with on-site experiences in 568 protected areas in Israel and the Palestinian Authority. We analyze counts and content of >100,000 photographs and tweets from four different SNSs, calibrate visitation models and predict visitation in unmonitored sites, cluster sites based on the typology of human-nature interactions reflected in online photographs, and characterize the polarity of sentiments associated with experiences in individual sites and clusters thereof. We find benefits in combining data from multiple sources and controlling for biases related to sites’ photogenicity and type of human-nature interactions. Our results suggest that current best estimates of visitation in unmonitored sites underestimate by 39% the actual number of visits. We discuss how the techniques and findings in this study are applicable in the broader context of the management and conservation of sites of environmental or cultural interest.     

Lattice-Boltzmann simulations of the capillary pressure–saturation–interfacial area relationship for porous media

Hysteresis in the relationship between capillary pressure (P_c), wetting phase saturation (S_w) and nonwetting–wetting interfacial area per volume (a_nw) is investigated using multiphase lattice-Boltzmann simulations of drainage and imbibition in a glass bead porous system. In order to validate the simulations, the P_c–S_w and a_nw–S_w main hysteresis loops were compared to experimental data reported by Culligan et al. [Culligan KA, Wildenschild D, Christensen BS, Gray WG, Rivers ML, Tompson AB. Interfacial area measurements for unsaturated flow through porous media. Water Resour Res 2004;40:W12413]. In general, the comparison shows that the simulations are reliable and capture the important physical processes in the experimental system. P_c–S_w curves, a_nw–S_w curves and phase distributions (within the pores) show good agreement during drainage, but less satisfactory agreement during imbibition. Drainage and imbibition scanning curves were simulated in order to construct P_c–S_w–a_nw surfaces. The root mean squared error (RMSE) and mean absolute error (MAE) between drainage and imbibition surfaces was 0.10 mm⁻¹ and 0.03 mm⁻¹, respectively. This small difference indicates that hysteresis is virtually nonexistent in the P_c–S_w–a_nw relationship for the multiphase system studied here. Additionally, a surface was fit to the main loop (excluding scanning curves) of the drainage and imbibition P_c–S_w–a_nw data and compared to the surface fit to all of the data. The differences between these two surfaces were small (RMSE = 0.05 mm⁻¹ and MAE = 0.01 mm⁻¹) indicating that the P_c–S_w–a_nw surface is adequately represented without the need for the scanning curve data, which greatly reduces the amount of data required to construct the non-hysteretic P_c–S_w–a_nw surface for this data.