Monica—A computer algorithm for solving boundary value problems using the double-hardening constitutive law Monot: II. Algorithm validation

The computer algorithm MONICA¹, which incorporates the double-hardening constitutive law Monot², has been tested on a large number of numerical examples. This paper summarizes the validation process, which has included consideration of both single element stress paths and multi-element boundary value problems. These results show MONICA to be effective for analysing practical problems using a soil model of the double-hardening type. They also demonstrate the ability of Monot to model a wide range of material responses. Problems involving pore pressure changes are identified as an area in which this form of analysis is of particular use^3,4.     

Implementation of hierarchical single surface δ0 and δ1 models in finite element procedure

Hierarchical single surface (HISS) δ₀ and δ₁ models are briefly discussed. An algorithm for the implementation of δ₀ and δ₁ models in a finite element code and the associated subroutines, written in FORTRAN 77, are presented. The algorithm is general and can be adopted for other computer codes with minor changes. Detailed steps and subroutines are included for easy implementation of the models in specific codes. Two example problems are also presented to enable the user to verify the implementation.     

Application of the equations of radioactive growth and decay to geochronological models and explicit solution of the equations by Laplace transformation

A recently developed method of pore-fluid age determination assumes secular equilibrium in the ²³⁸U decay chain. The efficacy of this approximation is investigated using computer evaluations of the equations that give the time evolution of the ²³⁸U decay chain, i.e. the solution of the equations of radioactive growth and decay. This analysis is performed considering two alternative geochemical scenarios to that of secular equilibrium — only ²³⁸U present initially and ²³⁸U and ²³⁴U present initially. In addition, the effects of the ²³⁵U decay chain are also determined in a similar fashion. These particular examples were chosen to show that more sophisticated geochronological models for many dating applications can be developed using such computer calculations. To facilitate such analyses, a solution of the equations of radioactive growth and decay for an arbitrary initial condition is derived using the Laplace transformation method and matrix algebra. Other solutions — both general and special — that are found in some well-known textbooks are reviewed.     

Large-scale stochastic linear inversion using hierarchical matrices

Stochastic inverse modeling deals with the estimation of functions from sparse data, which is a problem with a nonunique solution, with the objective to evaluate best estimates, measures of uncertainty, and sets of solutions that are consistent with the data. As finer resolutions become desirable, the computational requirements increase dramatically when using conventional solvers. A method is developed in this paper to solve large-scale stochastic linear inverse problems, based on the hierarchical matrix (or ? ² matrix) approach. The proposed approach can also exploit the sparsity of the underlying measurement operator, which relates observations to unknowns. Conventional direct algorithms for solving large-scale linear inverse problems, using stochastic linear inversion techniques, typically scale as ??(n ² m+nm ²), where n is the number of measurements and m is the number of unknowns. We typically have n ? m. In contrast, the algorithm presented here scales as ??(n ² m), i.e., it scales linearly with the larger problem dimension m. The algorithm also allows quantification of uncertainty in the solution at a computational cost that also grows only linearly in the number of unknowns. The speedup gained is significant since the number of unknowns m is often large. The effectiveness of the algorithm is demonstrated by solving a realistic crosswell tomography problem by formulating it as a stochastic linear inverse problem. In the case of the crosswell tomography problem, the sparsity of the measurement operator allows us to further reduce the cost of our proposed algorithm from ??(n ² m) to $\mathcal {O}(n^{2} \sqrt {m} + nm)$ . The computational speedup gained by using the new algorithm makes it easier, among other things, to optimize the location of sources and receivers, by minimizing the mean square error of the estimation. Without this fast algorithm, this optimization would be computationally impractical using conventional methods.     

A virtual element method for the two-phase flow of immiscible fluids in porous media

A primal C⁰-conforming virtual element discretization for the approximation of the bidimensional two-phase flow of immiscible fluids in porous media using general polygonal meshes is discussed. This work investigates the potentialities of the Virtual Element Method (VEM) in solving this specific problem of immiscible fluids in porous media involving a time-dependent coupled system of non-linear partial differential equations. The performance of the fully discrete scheme is thoroughly analysed testing it on general meshes considering both a regular problem and more realistic benchmark problems that are of interest for physical and engineering applications.

     

Efficient multigrid solution of boundary element models of cracks and faults

We consider the multigrid solution of a discretized singular integral equation whose solution represents the displacement discontinuity distribution across a pressurized crack or the slip on a fault subjected to a prescribed shear stress. The multigrid technique reduces the operation count for a crack model having N degrees of freedom from O(N³) operations for standard stationary iterative methods to O(N²) operations. In the numerical simulations performed the multigrid approach proves to be extremely efficient even for small values of N. We use Fourier analysis to determine the spectral properties of the coarse grid correction process and the effect of a number of different interpolation operators on the multigrid algorithm. We show that the multigrid technique can be combined with the process of lumping remote influences to yield an algorithm that involves O(N) operations. The performance of multigrid iteration in a nonlinear environment is explored by considering seams filled with nonlinear material. For this purpose a segmented multigrid algorithm is developed that allows for different seam constitutive relations to be used along different line segments. The O(N²) operation count characteristic of linear multigrid iteration is shown to persist in this nonlinear environment and the segmented multigrid approach is shown to provide significant computational savings for values of N well within the range for typical problems that occur in practice.     

Sulfoarsenide complexes of gold in hydrothermal solutions in the formation of gold ore deposits (thermodynamic modeling)

The composition and conditions of the formation of gold sulfoarsenide complexes were studied by means of the solution of inverse problems of physicochemical modeling on the basis of the SELEKTOR software, with the computer analysis of the experimental data on Au dissolution in the presence of orpiment at 200–300°C and 1 kbar pressure. It was shown that sulfoarsenide complexes of gold formed in sulfurous-arsenious metalliferous hydrothermal solutions quantitatively prevailed in acidic and near-neutral solutions in the presence of As. The stability of the H₂AuAsS ₃ ⁰ sulfoarsenide complex and of its AuAsS ₂ ⁰ deprotonated analogue depends on the concentration of arsenic in the system, just as the ratio of sulfoarsenide and hydrosulfide complexes of gold. The productive metalliferous generations of sulfides in pyrite-arsenopyrite parageneses are deposited involving gold sulfoarsenides.     

An algorithm andFortran-iv program for large-scaleQ-mode factor analysis and calculation of factor scores

An algorithm andFortran-iv computer program,cabfac, forQ-mode factor analysis is described. The program will accept up to 1500 items and 50 variables on a moderate-size computer.     

A study on one-dimensional consolidation of layered systems

A general analytical solution, which is more explicit than the one given by Schiffman and Stein,¹ for the problem of one-dimensional consolidation of layered soils is presented. A relevant computer program is developed and the computed results on some examples are included. From these results, an in-depth study on the one-dimensional consolidation behaviour of layered systems is then made. It is demonstrated through these examples that both the solution technique and the computer program developed are very efficient. It is found that the effects of coefficients of permeability and volume compressibility of soil on the consolidation of layered systems are different and cannot be embodied into the coefficient of consolidation of soil. The stiffness of soil layer also plays an important role on the rate of consolidation.     

History matching on the Imperial College fault model using parallel tempering

The history-matching inverse problem from petroleum engineering is analysed using the Imperial College fault model. This fault model produces a challenging inverse problem and is designed to show some of the problems which can occur whilst performing history-matching calculations on complicated geologies. It is shown that there can be multiple distinct geologies which match the history data. Furthermore, it is shown that the maximum-a-posteriori estimate does not correspond to the true geology in some cases. Both of these statements are corroborated via numerical examples where the parameter spaces are ?, ?³, ?⁷ and ?¹³. In addition, it is shown that the number of matches which agree with the data increases with dimension for these examples. It is also shown that the different matches can result in different reservoir management decision which, if incorrectly taken, would incur substantial financial penalties. All of these analyses are performed in a systematic manner, where it is shown that the standard algorithms can give a misleading answer. The history-matching problem is written as a minimisation problem, and it is shown that knowledge of all of the local minima is required. This presents significant computational issues as the resulting objective function is highly nonlinear, expensive to evaluate and multimodal. Previously used algorithms have been proved to be inadequate. Parallel tempering is a method which, if run for long enough, can find all the local minima. However, as the objective is expensive, a number of algorithm modifications had to be used to ensure convergence within a reasonable time. This new information is outlined in the paper. The algorithm as implemented produced results and new insights into this problem which were not suspected before. The results produced by this algorithm for the multimodal history-matching problem are superior to all other results of which we are aware. However, a considered amount of computation time was used within this paper, so this result does not infer that the algorithm cannot be improved upon. This algorithm not only produces good results but can be applied to all other history-matching problems. We have shown that this method provides a robust route of finding multiple local optima/solutions to the inverse problem, which is of considerable benefit to the petroleum industry. Furthermore, it is an entirely parallel algorithm which is becoming computationally feasible for other history-matching problems.     

Support vector regression to predict the performance of stabilized aggregate bases subject to wet–dry cycles

Durability is a notion that is integrated with the performance of stabilized pavement materials. Also, because it can be quantified and measured, it carries significant influence on the design of pavements. This study focuses on using support vector machine, a machine learning algorithm, in assessing the performance of stabilized aggregate bases subject to wet–dry cycles. Support Vector Regression (SVR) is a statistical learning algorithm that is applied to regression problems and is gaining popularity in pavement and geotechnical engineering. In our study, SVR was shown to be superior to the least‐squares (LS) method. Results of this study show that SVR significantly reduces the mean‐squared error (MSE) and improves the coefficient of determination (R²) compared to the widely used LS method. Copyright © 2011 John Wiley & Sons, Ltd.     

Algorithm of Calculation of Lyapounov Coefficients for Analysis of Chemical Autooscillations, as Applied to Calcite Crystallization Model

The numerical algorithm of calculation of Lyapounov coefficients (L _k) of any order is developed. The apparatus of analytical calculations is not used in this algorithm. The proposed algorithm is of use for usual computer languages and allows us to find the numerical value of L _k for any k and to make complete qualitative analyses of dynamic models on the plane.     

An interactive global optimization algorithm for geological problems

A number of problems in geology can be formulated so that they consist of optimizing a real-valued function (termed the objective function) on some interval or over some region. Many methods are available for solution if the function is unimodal within the domain of interest. Direct methods, involving only function evaluations, are particularly useful in geological problems where the objective function may be strongly nonlinear and constructed from sampled data. In practical problems, the objective function often is not unimodal. Standard optimization routines are not capable of distinguishing between local extrema or of locating the global extremum, which is the point of interest in most cases. The usual approach—trying several different starting points in the hope that the best local extremum found is the global extremum—is inefficient and unreliable. An ancillary algorithm has been developed which avoids these problems and which couples with a variety of local optimization routines. The algorithm first constructs a grid of objective function values over some feasible region. The region dimensions and grid spacings are based on specific problem considerations. First differences are then calculated for successive points along each grid line and monitored in sign only, which rapidly locates extrema. User interaction determines how many of these extrema will undergo further investigation, which is carried out by passing locations to a local optimization subroutine. The algorithm has proved successful on a number of problems. A geological example—determination of benthic mixing parameters in deep-sea sediments via minimization of stratigraphic offset between ¹⁸ O signals from two different species of planktonic foraminifera—is given. FORTRAN code is provided for the global optimization routine, a golden section search subroutine for one-dimensional objective functions, and a simplex subroutine for multidimensional problems.     

Embankment reliability versus factor of safety: Before and after slide repair

Probabilistic slope stability analyses incorporate more information regarding the input variables than do traditional deterministic analyses. Consequently, more information is contained in the results. A probabilistic method for slope stability analysis for very general problems was presented by Wolff and Harr.¹ The method allows analysis of a broad range of slope stability problems. In this Application Brief, the method is reviewed and used to illustrate the significant change in reliability of a dam slope from the time of an actual slide to the time the slide was repaired by the construction of a rock toe berm.     

大比例尺成矿预测中高精度立方格预测模型快速构建算法

立方格模型是一种常用的大比例尺三维可视化预测模型。在计算机三维图形学中,立方格模型又被称为块段模型,该模型比较适合表示非均匀体数据的空间分布,在矿山大比例尺深边部成矿预测、三维反演、矿山储量估算等领域得到广泛应用。目前,构建高精度立方格预测模型的一般方法是将已有地质体、地表、构造等三维地学对象进行栅格化。针对栅格化过程中遇到的2个关键问题,即体形的空间形态复杂和需要计算的长方体数量大（≥10⁶）,应用三维网格分割与种子填充（flood-fill）思想实现快速构建三维复杂体的立方格模型算法,其时间复杂度为O(n)。该算法的主要优点是花费相对小的空间代价高效地解决大量三角面与长方体的相交和空间查询计算问题。同时,应用该算法得到的立方格预测模型的长方体被划分为边界、内部和外部,便于开展预测要素综合模拟、矿山储量计算以及三维空间分析等其他后续地质应用。特别是将种子填充思想应用于高精度立方格模型构模,具有一定的创新性。本算法不仅适用于矢量体的快速栅格化,也适用于空间非自相交曲面（例如数字高程模型）的快速栅格化。     

Fast multipole displacement discontinuity method (FM‐DDM) for geomechanics reservoir simulations

The displacement discontinuity method (DDM) is frequently used in geothermal and petroleum applications for modeling the behavior of fractures in linear‐elastic rocks. The DDM requires O(N²) memory and O(N³) floating point operations (where N is the number of unknowns) to construct the coefficient matrix and solve the linear system of equations by direct methods. Therefore, the conventional implementation of the DDM is not computationally efficient for very large systems of cracks, often limiting its application to small‐scale problems. This work presents an approach for solving large‐scale fracture problems using the fast multipole method (FMM). The approach uses both the DDM and a kernel‐independent version of the FMM along with a preconditioned generalized minimal residual algorithm to accelerate the solution of linear systems of equations using desktop computers. Using the fundamental solutions for constant displacement discontinuity in a two‐dimensional elastic medium, several numerical examples involving fracture networks representing fractured reservoirs are treated. Numerical results show good agreement with analytical solutions and demonstrate the efficiency of the FMM implementation of the DDM for large‐scale simulations. Copyright © 2015 John Wiley & Sons, Ltd.     

Prediction of an environmental issue of mine blasting: an imperialistic competitive algorithm-based fuzzy system

Ground vibration resulting from blasting is one of the most important environmental problems at open-cast mines. Therefore, accurately approximating the blast-induced ground vibration is very significant. By reviewing the previous investigations, many attempts have been done to create the empirical models for estimating ground vibration. Nevertheless, the performance of the empirical models is not good enough. In this research work, a new hybrid model of fuzzy system (FS) designed by imperialistic competitive algorithm (ICA) is proposed for approximating ground vibration resulting from blasting at Miduk copper mine, Iran. For comparison aims, various empirical models were also utilized. Results from different predictor models were compared by using coefficient of multiple determination (R ²), variance account for and root-mean-square error between measured and predicted values of the PPVs. Results prove that the FS–ICA model outperforms the other empirical models in terms of the prediction accuracy. In other words, the FS–ICA model with R ² of 0.942 can forecast PPV better than the USBM with R ² of 0.634, Ambraseys–Hendron with R ² of 0.638, Langefors–Kihlstrom with R ² of 0.637 and Indian Standard with R ² of 0.519.     

Propagation of shock waves at the surface of heterogeneous soil grounds

The paper deals with the propagation of shock waves at the surface of soils. Heterogeneity and damping are introduced into analytical half-space solutions. The suggested model explains two phenomena, often observed with shock propagation in actual soils, that differ from the behaviour of the homogeneous half-space: the pronounced decay of the disturbances with distance and the elongation of the disturbance into a train of waves. The effects of heterogeneity and damping are discussed quantitatively. The response of footings on heterogeneous soils has been investigated by several authors. Awojobi⁴ considered the Gibson soil in which the shear modulus increases linearly with depth. Luco⁵ and Gazetas and Roesset⁶ investigated a multi-layered soil, the shear modulus being constant within each layer. Gazetas⁷, using a technique suggested by Gupta⁸ extended this method to layers with linearly varying shear modulus. Little work is available on the propagation of waves in heterogeneous bodies. Some results concerning the modes and the mode shapes in heterogeneous soils were reported by Ewing, Jardetzky and Press⁹ and Bath.¹⁰ The modes have to be superposed in an appropriate way to obtain the displacement field at the surface. This has been approximately achieved by the finite element formulations of Lysmer, ¹¹, Lysmer and Waas¹² and Waas.¹³ Auersch¹⁴ applied this method to a homogeneous layer on a rigid base. He found some dispersion of the Rayleigh wave within a narrow frequency range. Finite elements combined with discrete Laplace transforms, however, consume much computer time. Rao and Goda¹⁵ and Rao¹⁶ calculated surface vibrations of a half-space with exponentially varying shear modulus and density. Their method is similar to Lamb'S¹ procedure for the homogeneous half-space. Only one mode–the Rayleigh wave–occurs in their heterogeneous half-space. The examples show the considerable effect of heterogeneity on wave propagation. In the present paper, more general variations of the shear modulus are considered.     

Bulk organic δ13C and C/N ratios as palaeosalinity indicators within a Scottish isolation basin

Microfossils in isolation basin sediments are frequently used to reconstruct sea‐level change, but preservation problems and non‐analogue situations can limit their usefulness. Here we investigate the potential of stable carbon isotopes (δ¹³C) and C/N ratios from bulk organic matter, as an alternative proxy of salinity within isolation basin sediments from a basin in northwest Scotland. Within the Holocene sediment δ¹³C and C/N are determined largely by the mean weighted values of the predominant source of the organic material. Analysis of modern materials and comparison with the diatom record shows that the marine parts of the sequence are dominated by high δ¹³C and variable C/N. In the fresh water sequences the organic material is a mixture of both freshwater aquatic and terrestrial plant input that have relatively low δ¹³C and high C/N. The application of δ¹³C and C/N ratios in the studied basin in general follow the environmental change recorded by the diatoms and shows the potential of bulk organic matter in the investigation of salinity change in isolation basins. Copyright © 2005 John Wiley & Sons, Ltd.