Seismic wave propagation in a viscoelastic porous solid saturated by viscous liquid

A general solution is deduced of the differential equations describing the propagation of elastic waves in a dissipative liquid-filled viscoelastic porous solid. The velocities of three existing waves have been expressed in convenient form using the moduli of the solid phase and by introducing the frequency-dependent equivalent mass densities. The solution is then used to examine some of the phenomena which arise when each of the three-body waves, in turn, are incident on a traction-free plane boundary. Analytic expressions for the reflection coefficients are obtained. Numerical calculations have been made, for a particular model, in case of incidentP _I wave. Effect of viscoelasticity and viscosity on the reflection coefficients has also been exhibited.     

Using Grain‐Size Distribution Methods for Estimation of Air Permeability

Knowledge of air permeability (k_a) at dry conditions is critical for the use of air flow models in porous media; however, it is usually difficult and time consuming to measure k_a at dry conditions. It is thus desirable to estimate k_a at dry conditions from other readily obtainable properties. In this study, the feasibility of using information derived from grain‐size distributions (GSDs) for estimating k_a at dry conditions was examined. Fourteen GSD‐based equations originally developed for estimating saturated hydraulic conductivity were tested using k_a measured at dry conditions in both undisturbed and disturbed river sediment samples. On average, the estimated k_a from all the equations, except for the method of Slichter, differed by less than ± 4 times from the measured k_a for both undisturbed and disturbed groups. In particular, for the two sediment groups, the results given by the methods of Terzaghi and Hazen‐modified were comparable to the measured k_a. In addition, two methods (e.g., Barr and Beyer) for the undisturbed samples and one method (e.g., Hazen‐original) for the undisturbed samples were also able to produce comparable k_a estimates. Moreover, after adjusting the values of the coefficient C in the GSD‐based equations, the estimation of k_a was significantly improved with the differences between the measured and estimated k_a less than ±4% on average (except for the method of Barr). As demonstrated by this study, GSD‐based equations may provide a promising and efficient way to estimate k_a at dry conditions.     

Parameterization of the Ångström–Prescott formula based on machine learning benefit estimation of reference crop evapotranspiration with missing solar radiation data

Accurately estimated reference evapotranspiration (ET₀) is essential to regional water management. The FAO recommends coupling the Penman–Monteith (P-M) model with the Ångström–Prescott (A-P) formula as the standard method for ET₀ estimation with missing R_s measurements. However, its application is usually restricted by the two fundamental coefficients (a and b) of the A-P formula. This paper proposes a new method for estimating ET₀ with missing R_s by combining machine learning with physical-based P-M models (PM-ET₀). The benchmark values of the A-P coefficients were first determined at the daily, monthly, and yearly scales, and further evaluated in R_s and ET₀ estimates at 80 national R_s measuring stations. Then, three empirical models and four machine-learning methods were evaluated in estimating the A-P coefficients. Machine learning methods were also used to estimate ET₀ (ML-ET₀) to compare with the PM-ET₀. Finally, the optimal estimation method was used to estimate the A-P coefficients for the 839 regular weather stations for ET₀ estimation without R_s measurement for China. The results demonstrated a descending trend for coefficient a from northwest to southeast China, with larger values in cold seasons. However, coefficient b showed the opposite distribution as the coefficient a. The FAO has recommended a larger a but a smaller b for southeast China, which produced the region's largest R_s and ET₀ estimation errors. Additionally, the A-P coefficients calibrated at the daily scale obtained the best estimation accuracy for both R_s and ET₀, and slightly outperformed the monthly and yearly coefficients without significant difference in most cases. The machine learning methods outperformed the empirical methods for estimating the A-P coefficients, especially for the sites with extreme values. Further, ML-ET₀ outperformed the PM-ET₀ with yearly A-P coefficients but underperformed those with daily and monthly ones. This study indicates an exciting potential for combining machine learning with physical models for estimating ET₀. However, we found that using the A-P coefficients with finer time scales is unnecessary to deal with the missing R_s measurements.     

Shift from transport limited to supply limited sediment concentrations with the progression of monsoon rains in the Upper Blue Nile Basin

Long‐term erosion monitoring data in the Ethiopian highlands are only available from the Soil Conservation Research Program (SCRP) watersheds including the Anjeni watershed. The 113 ha Anjeni watershed was established in 1984 and fanya juu terraces were installed in 1986. Runoff and erosion data are available from three different plot sizes and at the watershed outlet. The objective of this study was to investigate how erosion processes and sediment rating parameters vary with plot size and the progression of the rainy monsoon phase. We analyzed runoff and sediment loss data from 40 plots and the watershed outlet. The dataset included erosion data from 24 newly constructed plots (3 m length) during the rainy monsoon phase of 2012 and 2013, and 16 long‐term plots (with 12, 16, 22, and 24% slopes and 3, 15 and 30 m lengths) and the watershed outlet during the period between 1986 to 1990. Sediment concentration (C) was fitted to runoff (Q) using a power regression equation (C = aQ^b). Sediment concentration and yield increased with increasing plot length from 3 m to 15 m, but sediment yield decreased as plot length increased to 30 m.The coefficients (a and b) were affected by plot size and the progression of the rainy monsoon phase. As plot size increases, the a value increased, while the b value decreased. Greater a values were observed during the beginning of the monsoon phase, while values of b were greater towards the end of the monsoon phase. Overall findings suggest that erosion from cultivated fields is primarily controlled by transport limitations at the beginning of the monsoon phase, while towards the end of the monsoon phase, as surface covers emerge, sediment availability will be reduced, and thus sediment source would be a limitation. Copyright © 2016 John Wiley & Sons, Ltd.     

随机介质非均匀地质体尺度研究(英文)

为了定量地反映复杂非均匀介质非均匀地质体的尺度大小,本文利用统计学方法建立了能够很好地描述复杂非均匀介质特征的随机介质模型,模型参量自相关长度描述了非均匀介质横向和纵向上非均匀体的平均尺度。基于所建立的随机介质模型通过速度的横向变化和速度标准差分别探讨了自相关长度与非均匀体尺度之间的关系。对速度横向变化的研究表明:随机介质内速度具有一定均值和方差并呈随机扰动特征;随着模型自相关长度的增大,非均匀体尺度也随之增大。通过速度标准差的研究得出自相关长度与非均匀体尺度之间关系的拟合公式,利用此公式可以定量地获取非均匀体尺度的实际大小。     

Dimensionless critical shear stress evaluation from flume experiments using different gravel beds

Flume experiments were conducted using four different gravel beds (D₅₀ + 12–39 mm) and a range of marked particles (10–65 mm). The shear stresses were evaluated from friction velocities, when initial movement of marked particles occurred. Two kinds of equations were produced: first for the threshold of initial movement, and second for generalized movement. Equations of the type 0_c + a(D_i/D₅₀)^b, as proposed by Andrews (1983) are applicable even if the material is relatively well sorted. However, the values of a and b are lower (respectively 0·050 and -0·70) for initial movement. Generalized movement requires a higher shear stress (a + 0·068 and b + -0·80). D₉₀ of the bed material and y₀ (the bed roughness parameter) were also used as reference values in place of D₅₀. They produced lower values than in natural streams, mainly owing to the fact that the material used in the flume is better sorted: clusters are less well developed and the bed roughness is lower.     

Comparison of theory and experiments for dispersion in homogeneous porous media

Modeling dispersion in homogeneous porous media with the convection–dispersion equation commonly requires computing effective transport coefficients. In this work, we investigate longitudinal and transverse dispersion coefficients arising from the method of volume averaging, for a variety of periodic, homogeneous porous media over a range of particle Péclet (Pe_p) numbers. Our objective is to validate the upscaled transverse dispersion coefficients and concentration profiles by comparison to experimental data reported in the literature, and to compare the upscaling approach to the more common approach of inverse modeling, which relies on fitting the dispersion coefficients to measured data. This work is unique in that the exact microscale geometry is available; thus, no simplifying assumptions regarding the geometry are required to predict the effective dispersion coefficients directly from theory. Transport of both an inert tracer and non-chemotactic bacteria is investigated for an experimental system that was designed to promote transverse dispersion. We highlight the occurrence of transverse dispersion coefficients that (1) depart from power-law behavior at relatively low Pe_p values and (2) are greater than their longitudinal counterparts for a specific range of Pe_p values. The upscaling theory provides values for the transverse dispersion coefficient that are within the 98% confidence interval of the values obtained from inverse modeling. The mean absolute error between experimental and upscaled concentration profiles was very similar to that between the experiments and inverse modeling. In all cases the mean absolute error did not exceed 12%. Overall, this work suggests that volume averaging can potentially be used as an alternative to inverse modeling for dispersion in homogeneous porous media.     

Rates of development of tafoni in the Moenkopi and Kaibab formations in Meteor Crater and on the Colorado Plateau,northeastern Arizona

Tafoni are pits formed by non‐uniform weathering in otherwise uniform rock. Two equations have been proposed for the rate of development of tafoni, both based on 2000‐year‐old outcrops from the coast of Japan. We have taken tafoni measurements from the Meteor Crater, Arizona, and vicinity that extend the equations back at least 50 000 years. As reported in earlier studies, we found pit depth to be the best tafone parameter to measure. The size of the pit decreases significantly with increasing inclination of the rock surface; however, the size of the pit can vary greatly for other reasons. In some cases the measurements are statistically significantly different between two stations taken from contiguous areas of similar inclination and aspect in an apparently homogeneous bed. It is clear, however, that over tens of thousands of years tafoni enlarge significantly. Our data are generally log‐normal and all are markedly heteroscedastic. The 1991 equation proposed by Matsukura and Matsuoka does not fit our data. The 1996 equation proposed by Sunamura provides a better fit. We propose a sigmoidal equation D = b1 + e^(b2+(b3/t)) where D is the depth, t is the age, and b1, b2 and b3 vary with lithology. This new equation fits our data far better than the earlier published equations. Copyright © 2002 John Wiley & Sons, Ltd.     

Implications of next generation attenuation ground motion prediction equations for site coefficients used in earthquake resistant design

Proposals are developed to update Tables 11.4‐1 and 11.4‐2 of Minimum Design Loads for Buildings and Other Structures published as American Society of Civil Engineers Structural Engineering Institute standard 7‐10 (ASCE/SEI 7–10). The updates are mean next generation attenuation (NGA) site coefficients inferred directly from the four NGA ground motion prediction equations used to derive the maximum considered earthquake response maps adopted in ASCE/SEI 7–10. Proposals include the recommendation to use straight‐line interpolation to infer site coefficients at intermediate values of (average shear velocity to 30‐m depth). The NGA coefficients are shown to agree well with adopted site coefficients at low levels of input motion (0.1 g) and those observed from the Loma Prieta earthquake. For higher levels of input motion, the majority of the adopted values are within the 95% epistemic‐uncertainty limits implied by the NGA estimates with the exceptions being the mid‐period site coefficient, F_v, for site class D and the short‐period coefficient, F_a, for site class C, both of which are slightly less than the corresponding 95% limit. The NGA data base shows that the median value of 913 m/s for site class B is more typical than 760 m/s as a value to characterize firm to hard rock sites as the uniform ground condition for future maximum considered earthquake response ground motion estimates. Future updates of NGA ground motion prediction equations can be incorporated easily into future adjustments of adopted site coefficients using procedures presented herein. Published 2014. This article is a U.S. Government work and is in the public domain in the USA. Earthquake Engineering & Structural Dynamics published by John Wiley & Sons Ltd.     

Regional daily baseflow prediction

The purposes of this study are to identify the bias of applying the analysis of a log–log plot of baseflow and to derive an equation to describe successive regional mean baseflow. The function ?dQ/dt = a Q^b has been used to describe baseflow in many studies that obtain the values of a and b from the log–log plot. According to analysis in this study, the value of 1 can be assigned to b in two boundary conditions, but the parameter a is proved to be related to the depth of water table and starting time of recession and thus different values of a may be found for different recession events. This paper points out that no single regression line can be obtained by plotting all baseflow data on a log–log diagram. Instead, there should be parallel lines, and each for a recession event. It implies that no single set of parameters a and b can be applied to predict baseflow. Thus, a new equation describing the relationship between three successive mean baseflows was derived in this study. The bias in the analysis of the log–log plot and the ability of the derived equation to predict baseflow were verified for five watersheds in Taiwan. Results indicate that the formula of mean baseflow prediction can provide reasonable estimates of flows with a leading time of 6 days. Furthermore, stream flows of the Tonkawa creek watershed in USA were used to verify that using average flows can result in better predictions than using instantaneous flows. Copyright © 2004 John Wiley & Sons, Ltd.     

Trend analyses with river sediment rating curves

Sediment rating curves, which are fitted relationships between river discharge (Q) and suspended‐sediment concentration (C), are commonly used to assess patterns and trends in river water quality. In many of these studies, it is assumed that rating curves have a power‐law form (i.e. C = aQ^b, where a and b are fitted parameters). Two fundamental questions about the utility of these techniques are assessed in this paper: (i) how well to the parameters, a and b, characterize trends in the data, and (ii) are trends in rating curves diagnostic of changes to river water or sediment discharge? As noted in previous research, the offset parameter, a, is not an independent variable for most rivers but rather strongly dependent on b and Q. Here, it is shown that a is a poor metric for trends in the vertical offset of a rating curve, and a new parameter, â, as determined by the discharge‐normalized power function [C = â (Q/Q_GM)^b], where Q_GM is the geometric mean of the Q‐values sampled, provides a better characterization of trends. However, these techniques must be applied carefully, because curvature in the relationship between log(Q) and log(C), which exists for many rivers, can produce false trends in â and b. Also, it is shown that trends in â and b are not uniquely diagnostic of river water or sediment supply conditions. For example, an increase in â can be caused by an increase in sediment supply, a decrease in water supply or a combination of these conditions. Large changes in water and sediment supplies can occur without any change in the parameters, â and b. Thus, trend analyses using sediment rating curves must include additional assessments of the time‐dependent rates and trends of river water, sediment concentrations and sediment discharge. Published 2014. This article is a U.S. Government work and is in the public domain in the USA. Hydrological Processes published by John Wiley & Sons Ltd.     

Space-time correlations ofb values with stress release

A total of 1503 events for a 2-month period associated with am _N 2.6 rockburst is investigated for possible space-time correlations between low magnitude (–1.1 to –0.4)b values and several estimates of stress (static stress drop, apparent stress, and dynamic stress). Spatial variations of decreasingb values were found to be well correlated with increasing stress release estimates for time intervals prior to the rockburst and following the aftershock sequence. The strongest correlation tob value was with the dynamic stress drop, having correlation coefficients of 0.87 and 0.79 for the two intervals, respectively. The rockburst was found to actually occur at the intersection of the spatial coordinates corresponding to the largest gradient inb value. Based on these correlations, we conclude that the low magnitude seismicity is an indicator of the stress state within the rock mass, and can be used to study and forecast stress patterns in the vicinity of an impending major event. Time variations, however, did not show the same clear correlations and these are discussed in terms of departure from steady state conditions. Regardless, our results favour the use ofb values in a spatial, context rather than in a time analysis approach, and we consider thatb values provide valuable information regarding the changing stress conditions within the seismogenic volume.     

含气介质临界点、流体和骨架弹性参数的数值计算方法

目前临界孔隙度及孔隙介质的研究主要为测试分析的实验方法,如何运用数值计算方法求取孔隙流体介质临界点和组分弹性参数一直是备受关注的课题.本文提出了求取孔隙介质临界点、孔隙内流体和骨架弹性参数的数值计算公式及方法,并结合含气样品测试数据实现了这种数值计算.文章首先给出了3个包含孔隙度的线性方程,通过有机组合每个线性方程中的两个系数得到求取有关弹性参数的数值计算公式.然后,详细阐述了计算的方法和步骤以及需要注意的问题,把室内含气砂岩样品实测的综合介质密度、纵波速度和横波速度作为数值计算的输入数据,求得了临界点和流体及骨架等相关弹性参数的具体数值.通过比较分析本文数学方法求得的计算数据与实验方法测得的实测数据,表明了该数值计算公式的正确性和数值实现方法的有效性.     

A study on the Wilkins and Forchheimer equations used in coarse granular media flow

Complexity of the pore geometry and the random nature of flow velocity make it difficult to predict and represent post laminar flow through porous media. Present study experimentally investigates the applicability of Forchheimer and Wilkins equations for post laminar flow where Darcy’s law is invalid due to predominant inertial effect. It is observed that both porosity and media size have significant influence over the coefficients of the Forchheimer coefficients. To incorporate the effect of porosity and media size, behaviour of Forchheimer coefficients are investigated with hydraulic radius as characteristic length. An inversely proportional variation trend is found for all the present and earlier reported data. A new empirical relation between Forchheimer coefficients and hydraulic radius is obtained which can be universally applicable for all media size and porosity. Coefficients of the Wilkins equation are found to be non-deviating for different hydraulic radius in the present study and in the reported literature validating its applicability in predicting the non laminar flow through porous media. Further the Wilkins equation is modified after incorporating the correction factors for better applicability on the field.     

Grain Size Analysis and Permeametry for Estimating Hydraulic Conductivity in Engineered Porous Media

Grain size analysis and permeametry are common methods for estimating the hydraulic conductivity (K) of porous media. It is well known that these methods have limited accuracy when they are used to characterize natural sediments. However, hydrogeological research has increasingly introduced technologies dependent on engineered porous media that may be less problematic because complex geologic structures are eliminated in the lab and field-scale packings. The recently introduced Horizontal Reactive Media Treatment Wells (HRX® Wells), for in situ, passive remediation of groundwater is one such example. The HRX Well passively collects groundwater and directs it through a horizontal pipe packed with an engineered porous medium. In this project, grain size analysis was conducted for sand and sand-iron mixtures to estimate K using the 16 algorithms provided in the HydrogeoSieveXL2.3.2 software. The results were compared to K determined by permeametry and a field-scale column, 30 cm long and 25 cm in diameter, representing an HRX Well. The best comparability of K estimates from grain size analysis and permeametry were obtained using the USBR, Slichter, and Shepherd K estimation methods. These also showed good agreement between lab-scale and field-scale K estimations, with reproducibility within the range ±20%. This study shows that laboratory K estimations can be representative across various relevant scales, including the field-scale, for engineered porous media. This finding extends to filter packs, and other engineered porous media design methods by emphasizing and demonstrating one case of accuracy in lab-scale permeability estimation for field-scale implementations.     

APPROXIMATIONS TO SHEAR-WAVE VELOCITY AND MOVEOUT EQUATIONS IN ANISOTROPIC MEDIA1

Backus and Crampin derived analytical equations for estimating approximate phase-velocity variations in symmetry planes in weakly anisotropic media, where the coefficients of the equations are linear combinations of the elastic constants. We examine the application of similar equations to group-velocity variations in off-symmetry planes, where the coefficients of the equations are derived numerically. We estimate the accuracy of these equations over a range of anisotropic materials with transverse isotropy with both vertical and horizontal symmetry axes, and with combinations of transverse isotropy yielding orthorhombic symmetry. These modified equations are good approximations for up to 17% shear-wave anisotropy for propagations in symmetry planes for all waves in all symmetry systems examined, but are valid only for lower shear-wave anisotropy (up to 11%) in off-symmetry planes. We also obtain analytical moveout equations for the reflection of qP-, qSH-, and qSV- waves from a single interface for off-symmetry planes in anisotropic symmetry. The moveout equation consists of two terms: a hyperbolic moveout and a residual moveout, where the residual moveout is proportional to the degree of anisotropy and the spread length of the acquisition geometry. Numerical moveout curves are computed for a range of anisotropic materials to verify the analytical moveout equations.     

Paraxial approximation for poroelastic media

Amongst different existing techniques developed for absorbing boundaries in transient dynamic analysis, the paraxial approximation represents an elegant framework. This approach is extended to the case of saturated porous media, which is of vital interest in many earthquake engineering problems. With respect to the frequency content of the dynamic process, several formulations exist for modelling the two-phase behaviou of soil or rock materials. The existence of a shear wave and two dilatational waves, which are standard features of porous media, is verified for the adopted formulation. The paraxial approximation is presented in the case of the u – p Biot two-phase dynamic formulation, suitable for seismic analyses and implemented by using a finite element approach. The efficiency of these boundaries is further verified through same numerical examples.     

横向各向同性多孔介质中的地震波传播

基于各向异性多孔介质中的广义Ｂｉｏｔ理论，导出了横向各向同性多孔介质中波传播的特征方程．指出在多孔介质中有４种类型的频散和耗散波传播：准纵波ＱＰ１（快纵波）、准纵波ＱＰ２（慢纵波）、准横波ＱＳＶ和横渡ＳＨ．文中给出了４种波速度的解析表达式．数值计算频率曲线和衰减曲线与Ｓｃｈｍｉｔｔ（１９８９）用均值处理得到的结果类似．还给出了波传播过程中３种类型准体波之间的耦合系数（或称转换系数）．