Boundary integral formulation and two-dimensional fundamental solutions for dynamic behavior analysis of unsaturated soils

In this paper the coupled equations governing the dynamic behavior of unsaturated soils are derived based on the poromechanics theory within the framework of the suction-based mathematical model presented by Gatmiri (1997) [Gatmiri B. Analysis of fully coupled behavior of unsaturated porous medium under stress, suction and temperature gradient. Final report of CERMES-EDF, 1997] and Gatmiri et al. (1998) [Gatmiri B, Delage P, Cerrolaza M, UDAM: a powerful finite element software for the analysis of unsaturated porous media. Adv Eng Software 1998; 29(1): 29–43]. In this formulation, the solid skeleton displacements, water pressure and air pressure are presumed to be independent variables. The Boundary Integral formulations as well as fundamental solutions for such a dynamic u–p_w–p_a theory are presented in this paper for the first time. The boundary integral equations are derived via the use of the weighted residuals method in a way that permits an easy discretization and implementation in a Boundary Element code. Also, the associated two dimensional (2D) fundamental solutions for such deformable porous medium with linear elastic behavior are derived in Laplace transform domain using the method of Hörmander. Finally, some numerical results are presented to show the accuracy of the proposed solutions. The derived results are verified analytically by comparison with the previously introduced corresponding fundamental solutions in elastodynamic limiting case.     

A high-order time-domain transmitting boundary for cylindrical wave propagation problems in unbounded saturated poroelastic media

Based on the u–p formulation of Biot equation with an assumption of zero permeability coefficient, a high-order transmitting boundary is derived for cylindrical elastic wave propagation in infinite saturated porous media. By this transmitting boundary the total stresses on the truncated boundaries of a numerical model, such as a finite element model, are replaced by a set of spring, dashpot and mass elements, with some additionally introduced auxiliary degrees of freedom. The transmitting boundaries are incorporated into the DIANA SWANDYNE II program and an unconditionally stable implicit time integration algorithm is adopted. Despite the assumption made in the derivation of the transmitting boundary, numerical examples show that it can provide highly accurate results for cylindrical elastic wave propagation problems in infinite saturated porous medium in case the u–p formulation is applicable. Although the direct applications of the proposed transmitting boundary to general two dimensional wave problems in infinite saturated porous media are not highly accurate, acceptable accuracy can still be achieved by placing the transmitting boundary at relatively large distance from the wave source.     

Solution of a Green’s function for a saturated porous medium in a half-space subjected to a torsional force

Based on the solutions of the Green’s function for a saturated porous medium obtained by the authors,and using transformation of axisymmetric coordinates,Sommerfeld integrals and superposition of the influence field on a free surface,the authors have obtained displacement solutions of a saturated porous medium subjected to a torsional force in a half-space.The relationship curves of the displacement solutions and various parameters(permeability,frequency,etc.)under action of a unit of torque are also given in this paper.The results are consistent with previous Reissner’s solutions,where a two-phase medium decays to a single-phase medium.The solution is useful in solving relevant dynamic problems of a twophase saturated medium in engineering.     

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.     

An improved gray lattice Boltzmann model for simulating fluid flow in multi-scale porous media

A lattice Boltzmann (LB) model is proposed for simulating fluid flow in porous media by allowing the aggregates of finer-scale pores and solids to be treated as ‘equivalent media’. This model employs a partially bouncing-back scheme to mimic the resistance of each aggregate, represented as a gray node in the model, to the fluid flow. Like several other lattice Boltzmann models that take the same approach, which are collectively referred to as gray lattice Boltzmann (GLB) models in this paper, it introduces an extra model parameter, n_s, which represents a volume fraction of fluid particles to be bounced back by the solid phase rather than the volume fraction of the solid phase at each gray node. The proposed model is shown to conserve the mass even for heterogeneous media, while this model and that model of Walsh et al. (2009) [1], referred to the WBS model thereafter, are shown analytically to recover Darcy–Brinkman’s equations for homogenous and isotropic porous media where the effective viscosity and the permeability are related to n_s and the relaxation parameter of LB model. The key differences between these two models along with others are analyzed while their implications are highlighted. An attempt is made to rectify the misconception about the model parameter n_s being the volume fraction of the solid phase. Both models are then numerically verified against the analytical solutions for a set of homogenous porous models and compared each other for another two sets of heterogeneous porous models of practical importance. It is shown that the proposed model allows true no-slip boundary conditions to be incorporated with a significant effect on reducing errors that would otherwise heavily skew flow fields near solid walls. The proposed model is shown to be numerically more stable than the WBS model at solid walls and interfaces between two porous media. The causes to the instability in the latter case are examined. The link between these two GLB models and a generalized Navier–Stokes model [2] for heterogeneous but isotropic porous media are explored qualitatively. A procedure for estimating model parameter n_s is proposed.     

Conditional simulations of water–oil flow in heterogeneous porous media

This study is an extension of the stochastic analysis of transient two-phase flow in randomly heterogeneous porous media (Chen et al. in Water Resour Res 42:W03425, 2006), by incorporating direct measurements of the random soil properties. The log-transformed intrinsic permeability, soil pore size distribution parameter, and van Genuchten fitting parameter are treated as stochastic variables that are normally distributed with a separable exponential covariance model. These three random variables conditioned on given measurements are decomposed via Karhunen–Loève decomposition. Combined with the conditional eigenvalues and eigenfunctions of random variables, we conduct a series of numerical simulations using stochastic transient water–oil flow model (Chen et al. in Water Resour Res 42:W03425, 2006) based on the KLME approach to investigate how the number and location of measurement points, different random soil properties, as well as the correlation length of the random soil properties, affect the stochastic behavior of water and oil flow in heterogeneous porous media.     

Using Eigenpattern Analysis to Constrain Seasonal Signals in Southern California

— Earthquake fault systems are now thought to be an example of a complex nonlinear system (Bak, et al., 1987; Rundle and Klein, 1995). The spatial and temporal complexity of this system translates into a similar complexity in the surface expression of the underlying physics, including deformation and seismicity. Here we show that a new pattern dynamic methodology can be used to define a unique, finite set of deformation patterns for the Southern California Integrated GPS Network (SCIGN). Similar in nature to the empirical orthogonal functions historically employed in the analysis of atmospheric and oceanographic phenomena (Preisendorfer, 1988), the method derives the eigenvalues and eigenstates from the diagonalization of the correlation matrix using a Karhunen-Loeve expansion (KLE) (Fukunaga, 1970; Rundle et al., 2000; Tiampo et al., 2002). This KLE technique may be used to determine the important modes in both time and space for the southern California GPS data, modes that potentially include such time-dependent signals as plate velocities, viscoelasticity, and seasonal effects. Here we attempt to characterize several of the seasonal vertical signals on various spatial scales. These, in turn, can be used to better model geophysical signals of interest such as coseismic deformation, viscoelastic effects, and creep, as well as provide data assimilation and model verification for large-scale numerical simulations of southern California.     

等吸力三轴剪切试验中非饱和黄土强度变形及水量变化特性

通过饱和原状黄土常规三轴试验和非饱和原状黄土等吸力三轴试验研究吸力和净围压对非饱和黄土强度变形的影响,并用HUANG等、胡冉等和方祥位等提出的土水特征曲线模型分析剪切过程中排水规律。研究结果表明：等吸力下原状土样固结剪切体积变形随净围压增大而增大;等净围压下原状土样固结剪切体积变形量随吸力的增大基本呈减小趋势。p-q平面内饱和土CSL线逐渐超过低吸力下非饱和土CSL线,原因在于随着p值增大,相对于非饱和土,饱和土孔隙比越来越小,较小孔隙比对抗剪强度的贡献逐渐大于非饱和土吸力对抗剪强度的贡献。应用固结稳定的数据拟合出HUANG等和胡冉等提出的模型参数,并预测剪切过程中的排水量,发现剪切速率对排水量有影响,这两种模型适用于剪切速率慢,排水充分的情况,而方祥位等提出的模型对不同剪切速率会拟合出不同的参数。     

Approximate recovery of coseismic deformation from Taiwan strong-motion records

Since 1990, digital strong-motion accelerographs and global positioning system (GPS) instruments have been widely deployed in the Taiwan region (Shin et al. 2003; Yu et al. 2001). The 1999 Chi-Chi, Mw 7.6 earthquake and the 2003 Chengkung, Mw 6.8 earthquake were well recorded by both digital accelerographs and GPS instruments. These data offer a good opportunity to determine coseismic displacements from strong-motion records and to compare the results with those derived from GPS measurements. As noted by Boore (2001), a double integration of the acceleration data often leads to unreasonable results, and baseline corrections are therefore required in most cases before the integration. Based on the works of Iwan et al. (1985) and Boore (2001), we developed an improved method for baseline correction and validated it using an extensive set of data from shake-table tests of a known “step” displacement on 249 accelerographs. Our baseline correction method recovered about 97% of the actual displacement from the shake-table data. We then applied this baseline correction method to compute coseismic displacements from the strong-motion data of the Chi-Chi and Chengkung earthquakes. Our results agree favorably with the coseismic displacements determined by the GPS measurements at nearby sites. The ratio of seismic to geodetic displacement varies from 0.78 to 1.41, with an average of about 1.05.     

On the behavior of mud floc size distribution: model calibration and model behavior

In this paper, we study a population balance equation (PBE) where flocs are distributed into classes according to their mass. Each class i contains i primary particles with mass m _p and size L _p. All differently sized flocs can aggregate, binary breakup into two equally sized flocs is used, and the floc??s fractal dimension is d ₀?=?2, independently of their size. The collision efficiency is kept constant, and the collision frequency derived by Saffman and Turner (J Fluid Mech 1:16?C30, 1956) is used. For the breakup rate, the formulation by Winterwerp (J Hydraul Eng Res 36(3):309?C326, 1998), which accounts for the porosity of flocs, is used. We show that the mean floc size computed with the PBE varies with the shear rate as the Kolmogorov microscale, as observed both in laboratory and in situ. Moreover, the equilibrium mean floc size varies linearly with a global parameter P which is proportional to the ratio between the rates of aggregation and breakup. The ratio between the parameters of aggregation and breakup can therefore be estimated analytically from the observed equilibrium floc size. The parameter for aggregation can be calibrated from the temporal evolution of the mean floc size. We calibrate the PBE model using mixing jar flocculation experiments, see Mietta et al. (J Colloid Interface Sci 336(1):134?C141, 2009a, Ocean Dyn 59:751?C763, 2009b) for details. We show that this model can reproduce the experimental data fairly accurately. The collision efficiency ?? and the ratio between parameters for aggregation and breakup ?? and E are shown to decrease linearly with increasing absolute value of the ??-potential, both for mud and kaolinite suspensions. Suspensions at high pH and different dissolved salt type and concentration have been used. We show that the temporal evolution of the floc size distribution computed with this PBE is very similar to that computed with the PBE developed by Verney et al. (Cont Shelf Res, 2010) where classes are distributed following a geometrical series and mass conservation is statistically ensured. The same terms for aggregation and breakup are used in the two PBEs. Moreover, we argue, using both PBEs, that bimodal distributions become monomodal in a closed system with homogeneous sediment, even when a variable shear rate is applied.     

饱和多孔微极介质的波动方程及其势函数方程

土是由一定尺寸大小颗粒所构成的多孔介质,具有明显的颗粒特性,当土颗粒间的孔隙被流体（如水或油）充满时则成为饱和土.利用微极理论和Biot波动理论的研究成果,把饱和土中多孔固体骨架部分近似地视为微极介质,孔隙中的流体部分视为质点介质,获得饱和多孔微极介质的弹性波动方程.借鉴Greetsma理论,建立了饱和多孔微极介质弹性本构方程力学参数与相应单相介质弹性参数的相互关系,使饱和多孔微极介质弹性波动方程中的物理参数具有明确的物理意义,易于在试验中确定.运用场论理论把饱和多孔微极介质的波动方程简化为势函数方程,建立了饱和多孔微极介质中五种弹性波的弥散方程,数值分析了五种简谐体波在无限饱和多孔微极介质中的传播特性. 结果表明,P1波、P2波和剪切S1波的波速弥散曲线与经典饱和多孔介质基本相同,当频率小于临界频率ω0时旋转纵波θ波和横波S2波不存在,当频率大于临界频率ω0时,θ波和S2波的传播速度随频率增加而减小.     

Modeling non-equilibrium mass transport in biologically reactive porous media

We develop a one-equation non-equilibrium model to describe the Darcy-scale transport of a solute undergoing biodegradation in porous media. Most of the mathematical models that describe the macroscale transport in such systems have been developed intuitively on the basis of simple conceptual schemes. There are two problems with such a heuristic analysis. First, it is unclear how much information these models are able to capture; that is, it is not clear what the model's domain of validity is. Second, there is no obvious connection between the macroscale effective parameters and the microscopic processes and parameters. As an alternative, a number of upscaling techniques have been developed to derive the appropriate macroscale equations that are used to describe mass transport and reactions in multiphase media. These approaches have been adapted to the problem of biodegradation in porous media with biofilms, but most of the work has focused on systems that are restricted to small concentration gradients at the microscale. This assumption, referred to as the local mass equilibrium approximation, generally has constraints that are overly restrictive. In this article, we devise a model that does not require the assumption of local mass equilibrium to be valid. In this approach, one instead requires only that, at sufficiently long times, anomalous behaviors of the third and higher spatial moments can be neglected; this, in turn, implies that the macroscopic model is well represented by a convection–dispersion–reaction type equation. This strategy is very much in the spirit of the developments for Taylor dispersion presented by Aris (1956). On the basis of our numerical results, we carefully describe the domain of validity of the model and show that the time-asymptotic constraint may be adhered to even for systems that are not at local mass equilibrium.     

Seismotectonic Model and CN Earthquake Prediction in Italy

—The choice of the regions is essential in the application of the algorithm CN, therefore a seismotectonic criterion for their definition is tested. In order to take into account the geodynamic complexity characterising the Italian peninsula, we established to strictly follow the seismotectonic zones, including in each region only zones with similar seismogenic behaviour and the transitional zones connected to them. Three regions have been successfully defined in this way, corresponding approximately to the North, Centre and South of Italy. The reduction of the space-time uncertainty and the increase of the stability of prediction results obtained with this regionalisation, with respect to the previous applications of CN in Italy (Keilis-Borok et al., 1990; Costa et al., 1995, 1996), can be interpreted as a validation of the seismotectonic model.     

Transmitting boundary for water-saturated transversely isotropic strata based on the u–U formulation

A new transmitting boundary in a cylindrical coordinate system has been developed for modeling the elastic waves radiating out to an infinite boundary in water-saturated transversely isotropic soil strata over a rigid bedrock. The saturated soil strata are assumed to consist of a porous material and modeled as a transversely isotropic two-phase medium, based on the u−U formulation. The newly developed transmitting boundary is combined with the finite-elements model of the near-field region, using the same u−U formulation, and applied to the study of the dynamics of a rigid circular foundation in porous isotropic or transversely isotropic layered strata, either fully or partly saturated with water. The verification and application examples give valuable insights into new and interesting aspects of the dynamic behavior of rigid circular foundations in fully or partly saturated two-phase ground in terms of permeability, transverse anisotropy, and ground-water table level.     

A layer of an unstable seismotectonic strain as an analog of a waveguide at depths of 12–20 km in the Earth’s crust of the Tajik depression

The existence of a layer of unstable seismotectonic strain, which spatially coincides with the waveguide previously detected here at depths of 12–20 km, is established in the central part of the Tajik Depression [Lukk et al., 1970; Nersesov and Chepkunas, 1970; 1971]. This crustal layer is assumed to be weakened and saturated by fluid. The latter is supposedly achieved due to the supply of the liquid component from the bottom crust or upper mantle in the cracks and pores of the waveguide material. A near vertical pillar-like seismogenic body revealed in the work [Shevchenko et al., 2011], which penetrates the waveguide in the depth interval 0–40 km, is considered as one of the possible channels of such inflow. The detected loosened layer (the waveguide) is characterized by a signficant reduction in seismic activity. However, no such reduction is observed within the pillar-like seismogenic body at these depths. Moreover, compared to the nearest ambient environment, the upper 10–15 km of this body feature considerably increased seismic activity that manifested itself in a series of 13 strong earthquakes (with M ≥ 4.7) that occurred within the past approximately 100 years.     

Propagation of seismic waves through liquefied soils

To predict the earthquake response of saturated porous media it is essential to correctly simulate the generation, redistribution, and dissipation of excess pore water pressure during and after earthquake shaking. To this end, a reliable numerical tool requires a dynamic, fully coupled formulation for solid–fluid interaction and a versatile constitutive model. Presented in this paper is a 3D finite element framework that has been developed and utilized for this purpose. The framework employs fully coupled dynamic field equations with a u–p–U formulation for simulation of pore fluid and solid skeleton interaction and a SANISAND constitutive model for response of solid skeleton. After a detailed verification and validation of the formulation and implementation of the developed numerical tool, it is employed in the seismic response of saturated porous media. The study includes examination of the mechanism of propagation of the earthquake-induced shear waves and liquefaction phenomenon in uniform and layered profiles of saturated sand deposits.     

Application of a Numerical Inverse Laplace Integration Method to Surface Loading on a Viscoelastic Compressible Earth Model

Normal mode approaches for calculating viscoelastic responses of self-gravitating and compressible spherical earth models have an intrinsic problem of determining the roots of the secular equation and the associated residues in the Laplace domain. To bypass this problem, a method based on numerical inverse Laplace integration was developed by Tanaka et al. (2006, 2007) for computations of viscoelastic deformation caused by an internal dislocation. The advantage of this approach is that the root-finding problem is avoided without imposing additional constraints on the governing equations and earth models. In this study, we apply the same algorithm to computations of viscoelastic responses to a surface load and show that the results obtained by this approach agree well with those obtained by a time-domain approach that does not need determinations of the normal modes in the Laplace domain. Using the elastic earth model PREM and a convex viscosity profile, we calculate viscoelastic load Love numbers (h, l, k) for compressible and incompressible models. Comparisons between the results show that effects due to compressibility are consistent with results obtained by previous studies and that the rate differences between the two models total 10–40%. This will serve as an independent method to confirm results obtained by time-domain approaches and will usefully increase the reliability when modeling postglacial rebound.     

Focal Mechanism Solutions of the 2008 Wenchuan earthquake sequence from P-wave polarities and SH/P amplitude ratios: new results and implications

The 2008 Wenchuan earthquake, a major intraplate earthquake with M _w 7.9, occurred on the slowly deforming Longmenshan fault. To better understand the causes of this devastating earthquake, we need knowledge of the regional stress field and the underlying geodynamic processes. Here, we determine focal mechanism solutions (FMSs) of the 2008 Wenchuan earthquake sequence (WES) using both P-wave first-motion polarity data and SH/P amplitude ratio (AR) data. As P-wave polarities are more reliable information, they are given priority over SH/P AR, the latter of which are used only when the former has loose constraint on the FMSs. We collect data from three categories: (1) permanent stations deployed by the China Earthquake Administration (CEA); (2) the Western Sichuan Passive Seismic Array (WSPSA) deployed by Institute of Geology, CEA; (3) global stations from Incorporated Research Institutions for Seismology. Finally, 129 events with magnitude over M _s 4.0 in the 2008 WES are identified to have well-constrained FMSs. Among them, 83 are well constrained by P-wave polarities only as shown by Cai et al. (Earthq Sci 24(1):115–125, 2011), and the rest of which are newly constrained by incorporating SH/P AR. Based on the spatial distribution and FMSs of the WES, we draw following conclusions: (1) the principle compressional directions of most FMSs of the WES are subhorizontal, generally in agreement with the conclusion given by Cai et al. (2011) but with a few modifications that the compressional directions are WNW–ESE around Wenchuan and ENE–WSW around Qingchuan, respectively. The subhorizontal compressional direction along the Longmenshan fault from SW to NE seems to have a left-lateral rotation, which agrees well with regional stress field inverted by former researchers (e.g., Xu et al., Acta Seismol Sin 30(5), 1987; Acta Geophys Sin 32(6), 1989; Cui et al., Seismol Geol 27(2):234–242, 2005); (2) the FMSs of the events not only reflected the regional stress state of the Longmenshan region, but also were obviously controlled by the faults to some extent, which was pointed out by Cai et al. (2011) and Yi et al. (Chin J Geophys 55(4):1213–1227, 2012); (3) while the 2008 Wenchuan earthquake and some of its strong aftershocks released most of the elastic energy accumulated on the Longmenshan fault, some other aftershocks seem to occur just for releasing the elastic energy promptly created by the 2008 Wenchuan earthquake and some of its strong aftershocks. (4) Our results further suggest that the Longmenshan fault from Wenchuan to Beichuan was nearly fully destroyed by the 2008 Wenchuan earthquake and accordingly propose that there is less probability for great earthquakes in the middle part of the Longmenshan fault in the near future, although there might be a barrier to the southwest of Wenchuan and it is needed to pay some attention on it in the near future.     

On the possible correlation between the Gutenberg-Richter parameters of the frequency-magnitude relationship

The study of the Gutenberg-Richter (GR) parameters a and b has been very important to describe and characterize the seismicity over the different seismic provinces around the world. As far as we know, the possible correlation between the GR parameters a and b has not received enough attention. Bayrak et al. reported the a and b values for 27 active seismic regions around the boundaries of the main tectonic plates of the world. From these data, we found that there exists a positive correlation between the a and b parameters (R =?0.85, R² =?0.72). On the other hand, we made around 150 computer runs of a spring-block model proposed by Olami et al. (Phys Rev Lett 68(8):1244–1247, 1992). This model roughly emulates the interaction between two fault planes and it reaches a self-organized critical state. With these simulations, we also found that the a and b parameters are positively correlated. Motivated by these results, we propose an analytical demonstration that indeed a and b are positively correlated. In addition, we discuss on other possible applications of the spring-block model to actual seismicity and to frictional experiments made with sandpapers.