Multilevel Monte Carlo methods using ensemble level mixed MsFEM for two-phase flow and transport simulations

In this paper, we propose multilevel Monte Carlo (MLMC) methods that use ensemble level mixed multiscale methods in the simulations of multiphase flow and transport. The contribution of this paper is twofold: (1) a design of ensemble level mixed multiscale finite element methods and (2) a novel use of mixed multiscale finite element methods within multilevel Monte Carlo techniques to speed up the computations. The main idea of ensemble level multiscale methods is to construct local multiscale basis functions that can be used for any member of the ensemble. In this paper, we consider two ensemble level mixed multiscale finite element methods: (1) the no-local-solve-online ensemble level method (NLSO); and (2) the local-solve-online ensemble level method (LSO). The first approach was proposed in Aarnes and Efendiev (SIAM J. Sci. Comput. 30(5):2319-2339, 2008) while the second approach is new. Both mixed multiscale methods use a number of snapshots of the permeability media in generating multiscale basis functions. As a result, in the off-line stage, we construct multiple basis functions for each coarse region where basis functions correspond to different realizations. In the no-local-solve-online ensemble level method, one uses the whole set of precomputed basis functions to approximate the solution for an arbitrary realization. In the local-solve-online ensemble level method, one uses the precomputed functions to construct a multiscale basis for a particular realization. With this basis, the solution corresponding to this particular realization is approximated in LSO mixed multiscale finite element method (MsFEM). In both approaches, the accuracy of the method is related to the number of snapshots computed based on different realizations that one uses to precompute a multiscale basis. In this paper, ensemble level multiscale methods are used in multilevel Monte Carlo methods (Giles 2008a, Oper.Res. 56(3):607-617, b). In multilevel Monte Carlo methods, more accurate (and expensive) forward simulations are run with fewer samples, while less accurate (and inexpensive) forward simulations are run with a larger number of samples. Selecting the number of expensive and inexpensive simulations based on the number of coarse degrees of freedom, one can show that MLMC methods can provide better accuracy at the same cost as Monte Carlo (MC) methods. The main objective of the paper is twofold. First, we would like to compare NLSO and LSO mixed MsFEMs. Further, we use both approaches in the context of MLMC to speedup MC calculations.     

Microsommite: crystal chemistry, phase transitions, Ising model and Monte Carlo simulations

Microsommite, ideal formula [Na₄K₂(SO₄)] [Ca₂Cl₂][Si₆Al₆O₂₄], is a rare feldspathoid that occurs in volcanic products of Vesuvius. It belongs to the cancrinite–davyne group of minerals, presenting an ABAB… stacking sequence of layers that contain six-membered rings of tetrahedra, with Si and Al cations regularly alternating in the tetrahedral sites. The structure was refined in space group P6₃ to R=0.053 by means of single-crystal X-ray diffraction data. The cell parameters are a=22.161?Å=√3a _dav, c=5.358?Å=c _dav; Z=3. The superstructure arises due to the long-range ordering of extra-framework ions within the channels of the structure. This ordering progressively decreases with rising temperature until it is completely lost and microsommite transforms into davyne. The order–disorder transformation has been monitored in several crystals by means of X-ray superstructure reflections and the critical parameters T _c?≈?750?°C and β?≈?0.12 were obtained. The kinetics of the ordering process were followed at different temperatures and the activation energy was determined to be about 125?kJ?mol^?1. The continuous order–disorder phase transition in microsommite has been discussed on the basis of a two-dimensional Ising model in a triangular lattice with nn (nearest neighbours) and nnn (next-nearest neighbours) interactions. Such a model was simulated using a Monte Carlo technique. The theoretical model well matches the experimental data; two phase transitions were indicated by the simulated runs: at low temperature only one of the three sublattices begins to disorder, whereas the second transition involves all three sublattices.     

蒙特卡罗模拟在区域地震滑坡灾害评价中应用

2004年10月23日,日本新泻中越地区发生史上最强地震之一,震级达到Mw6.6。这次地震诱发上千个规模大小不一、形式各样的滑坡,造成一定的灾害损失和人员伤亡。因此,为了有效地避免和减轻这方面的灾害,有必要有效地、科学地预测和评价地震诱发滑坡的空间发生规律。为此以一定的地区作为研究对象,提出了蒙特卡罗随机模拟和Newmark滑动体位移分析法在区域地震斜坡失稳概率分析中的应用。该法充分考虑到岩土参数和相关地震参数空间变异性,结合地理信息系统空间分析的优势,以网格模型为基础,分析该区斜坡滑动体位移分布规律,并以2cm为失稳临界值探讨网格斜坡破坏的概率。实例证明,所提出的模型有效地预测了滑坡发生的空间分布规律。     

Probabilistic modeling of seismically triggered landslides using Monte Carlo simulations

The 2004 Chuetsu earthquakes of Niigata (Japan) triggered numerous landslides, and the most widespread types of landslides were highly disrupted, relatively shallow slides and soil (debris) flows. This paper presented a method to evaluate slope instability using Newmark displacement on a pixel-by-pixel basis in a given area. The proposed method was able to integrate Newmark displacement modeling and Monte Carlo simulations within geographical information systems. In the modeling, an empirical attenuation relationship was utilized to calculate Arias intensity over this study area, and the variability of geotechnical parameters was taken into account to calculate coseismic landslide displacement. Before deriving the displacement from related inputs, the Monte Carlo simulations ran 1,500 times and generated 1,500 displacement values for each grid cell, and then means and standard deviations of displacement were calculated and probabilistic distributions can be obtained. Finally, given 10 cm as a threshold value of displacement, estimated probabilities of displacement exceeding 10 cm were shown as a map of seismic landslide hazards. The resulting hazard map was classified into four categories from very low to high level.     

Determining optimum number of geotechnical testing samples using Monte Carlo simulations

Knowing how many samples to test to adequately characterize soil and rock units is always challenging. A large number of tests decrease the uncertainty and increase the confidence in the resulting values of design parameters. Unfortunately, this large value also adds to project costs. This paper presents a method to determine the number of samples as a function of the coefficient of variation. If, as in the case of a reliability-based design, the resistance factors are a function of the coefficient of variation of the measurements, then lowering the coefficient of variation (COV) can result in lowering of the resistance factor resulting in a less conservative design. In this study, laboratory samples were isolated by specific unified soil classification system soil type and general site location. A distribution was fitted for each of the geotechnical parameters measured. For each scenario, groups of 2, 3, 4, 5, 10, 15, 20, 30, 50, and 100 random samples were generated by using Monte Carlo simulations from the fitted distributions. For each group, the variability was calculated in terms of the COV. In all cases, the COV decreased as the sample size increased. However, the rate of decrease for the COV was the greatest at a low number of samples; it becomes increasingly smaller at a higher number of samples.     

蒙特卡罗法在势流计算中的应用研究

针对水动力学实际问题多存在复杂几何边界的状况,提出了用不规则游动网格求解偏微分方程的蒙特卡罗法,建立了相应的随机游动模型。选择具有复杂自由面的堰闸流动问题作为算例,验证了新方法的正确性。与有限元法相比,蒙特卡罗法解势流等线性问题时更灵活,可以根据需要,单独计算流动区域内任意一点的未知物理量,且所用计算容量较少。     

A perturbation stochastic finite-element method for groundwater flow models based on an undetermined-coefficients approach

A new perturbation technique and finite-element method, which incorporates an undetermined-coefficients approach, is proposed to conduct stochastic simulation for problems of groundwater flow. Formulas of the mean value and variance of groundwater head are derived to exclude the process of computing the first and second-order derivatives of groundwater head with respect to stochastic variables (coefficient of transmissibility, storage coefficient, etc.) in the calculation process of common finite element methods. For ordinary stochastic groundwater problems, the statistical properties of groundwater head can be easily obtained by using the proposed method, which is especially efficient for the stochastic problems that have fewer input stochastic variables. A stochastic numerical simulation of a two-dimensional confined-groundwater flow problem has been conducted to validate effectiveness and determine the limitations of the proposed method. The results illustrate that the accuracy and efficiency of the method proposed are satisfactory, and that the computing time is shorter when there is a small number of input stochastic variables.     

二维各向异性岩土介质中渗流分析的等参有限元方法

为了建筑环境的保护,软土地区基坑围护中的坑内降水与坑外水位保持常是一对矛盾的课题.而坑内外的水位差,极易形成止水结构两侧的渗流运动.对饱和各向异性、水平延伸、厚度大致相等、且存在一、二类边界时的承压含水层中的承压水二维非稳定流渗流问题,本文推导出了二维、各向异性渗流等参有限元方程,用以进行基坑围护止水结构附近渗流场分析,并给出了算例.     

A regional-scale groundwater flow model for the Leon-Chinandega aquifer,Nicaragua

Groundwater flow in the Leon-Chinandega aquifer was simulated using transient and steady-state numerical models. This unconfined aquifer is located in an agricultural plain in northwest Nicaragua. Previous studies were restricted to determining groundwater availability for irrigation, overlooking the impacts of groundwater development. A sub-basin was selected to study the groundwater flow system and the effects of groundwater development using a numerical groundwater flow model (Visual MODFLOW). Hydrological parameters obtained from pumping tests were related to each hydrostratigraphic unit to assign the distribution of parameter values within each model layer. River discharge measurements were crucial for constraining recharge estimates and reducing the non-uniqueness of the model calibration. Steady-state models have limited usefulness because of the major variation of recharge and agricultural pumping during the wet and dry seasons. Model results indicate that pumping induces a decrease in base flow, depleting river discharge. This becomes critical during dry periods, when irrigation is highest. Transient modeling indicates that the response time of the aquifer is about one hydrologic year, which allows the development of management strategies within short time horizons. Considering further development of irrigated agriculture in the area, the numerical model can be a powerful tool for water resources management.     

A local-scale groundwater flow model for groundwater resources management in Dakhla Oasis,SW Egypt

Dakhla Oasis is located in the Western Desert of Egypt. Groundwater exploited from the Nubian Sandstone aquifer is the only available water resource in this area. This resource has been heavily exploited since 1960, which has led to a substantial decline in the potentiometric surface of the aquifer. A regional numerical groundwater flow model, calibrated under unsteady-state conditions, has been developed and used to investigate the hydrodynamic impacts of different groundwater management options on the potentiometry of the aquifer. To account for local details and to allow a precise analysis of pumping and the resulting drawdown in Dakhla Oasis, a local-scale model was developed by refining the grid cells in the calibrated regional model. The local-scale model gave a detailed picture about the expected drawdown due to the different groundwater management options in the next 100 years. The simulated results indicated that the planned increase in groundwater extraction will have a major impact on groundwater flow patterns in the whole area located southwest of Dakhla Oasis.

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Resumen El oasis Dakhla está situado en el gran desierto occidental de Egipto. El agua subterránea que se extrae del acuífero de roca arenosa nubia es la única fuente de agua disponible en esta área. Este recurso ha sido explotado en medida considerable desde 1960. Este hecho he causado una disminución sustancial en la superficie potenciométrica del acuífero. Se ha desarollado un modelo de flujo de agua subterránea numérico regional calibrado bajo condiciones de estado variable. Este modelo se ha utilizado para investigar el impacto hidrodinámico que diferentes opciones de manejo de aguas subterráneas tienen sobre la potenciometría del acuífero. Se desarolló un modelo de escala local mediante el refinamiento de las células de malla en el modelo regional calibrado con el objecto de involucrar los detalles locales y de permitir un análisis preciso del bombeo y disminución del nivel de aqua resultante en el oasis Dakhla. El modelo de escala local proporcionó un imagen detallada de la disminución de nivel esperada según las diferentes opciones de manejo de agua subterranea en los próximos 100 años. Los resultados simulados indican que el incremento de extración planificado en la agua subterránea tendrá un gran impacto en el patrón de flujo de agua subterranea en toda el area ubicada al suroeste del oasis de Dakhla.

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Résumé Loasis de Dakhla est située dans la partie occidentale du désert dÉgypte. La seule source deau disponible dans la région provient de lexploitation de laquifère gréseux de Nubian. Cette ressource a largement été exploitée depuis 1960, résultant en la diminution significative de la surface piézométrique de laquifère. Un modèle numérique découlement régional de leau souterraine, calibré en régime transitoire, a été développé et utilisé afin détudier les impacts hydrodynamiques de différents scénarios de gestion de laquifère sur la surface piézométrique. Afin dinclure les particularités locales et de réaliser une analyse précise du pompage et du rabattement qui y est associé dans lOasis de Dakhla, un modèle local découlement a été développé en raffinant le maillage dans le modèle régional préalablement calibré. Le modèle local découlement fournit une image détaillée du rabattement prévu pour les cent prochaines années selon les différents scénarios de gestion de laquifère. Les résultats obtenus par simulation numérique indiquent que laugmentation prévue dans lexploitation de leau souterraine aura un impact majeur sur les patrons découlement de leau souterraine dans toute la région située au sud-ouest de loasis de Dakhla.

     

Application of the two-stage Markov chain Monte Carlo method for characterization of fractured reservoirs using a surrogate flow model

In this paper, we develop a procedure for subsurface characterization of a fractured porous medium. The characterization involves sampling from a representation of a fracture’s permeability that has been suitably adjusted to the dynamic tracer cut measurement data. We propose to use a type of dual-porosity, dual-permeability model for tracer flow. This model is built into the Markov chain Monte Carlo (MCMC) method in which the permeability is sampled. The Bayesian statistical framework is used to set the acceptance criteria of these samples and is enforced through sampling from the posterior distribution of the permeability fields conditioned to dynamic tracer cut data. In order to get a sample from the distribution, we must solve a series of problems which requires a fine-scale solution of the dual model. As direct MCMC is a costly method with the possibility of a low acceptance rate, we introduce a two-stage MCMC alternative which requires a suitable coarse-scale solution method of the dual model. With this filtering process, we are able to decrease our computational time as well as increase the proposal acceptance rate. A number of numerical examples are presented to illustrate the performance of the method.     

A review of Markov Chain Monte Carlo and information theory tools for inverse problems in subsurface flow

Parameter identification is one of the key elements in the construction of models in geosciences. However, inherent difficulties such as the instability of ill-posed problems or the presence of multiple local optima may impede the execution of this task. Regularization methods and Bayesian formulations, such as the maximum a posteriori estimation approach, have been used to overcome those complications. Nevertheless, in some instances, a more in-depth analysis of the inverse problem is advisable before obtaining estimates of the optimal parameters. The Markov Chain Monte Carlo (MCMC) methods used in Bayesian inference have been applied in the last 10 years in several fields of geosciences such as hydrology, geophysics or reservoir engineering. In the present paper, a compilation of basic tools for inference and a case study illustrating the practical application of them are given. Firstly, an introduction to the Bayesian approach to the inverse problem is provided together with the most common sampling algorithms with MCMC chains. Secondly, a series of estimators for quantities of interest, such as the marginal densities or the normalization constant of the posterior distribution of the parameters, are reviewed. Those reduce the computational cost significantly, using only the time needed to obtain a sample of the posterior probability density function. The use of the information theory principles for the experimental design and for the ill-posedness diagnosis is also introduced. Finally, a case study based on a highly instrumented well test found in the literature is presented. The results obtained are compared with the ones computed by the maximum likelihood estimation approach.     

Analysis of transient groundwater flow through a stochastic modelling approach

This study demonstrates the application of a stochastic procedure to enhance confidence in the predictions of numerical groundwater flow models. The objective was to assess the hydrogeological conditions and groundwater potential of a semi-confined aquifer system in Legon, Accra. The study applied the parameter randomization procedure in the Groundwater Modelling System (GMS) to evaluate model uncertainties arising from uncertainties in the data of model input parameters. A total of 2500 equally likely solutions of the numerical model were simulated through the Latin hypercube stochastic option. Regarding the size of the terrain and the number of parameters used for calibrating the model, the 2500 model realizations were adjudged to be largely sufficient to represent the possible solutions that could arise from the various permutations of the parameter data. High variances would suggest significant differences in the realizations, indicating high levels of non-uniqueness in the solutions. The 2500 solutions were compared using standard deviations computed from the differences in the estimated horizontal hydraulic conductivity, specific yield and specific storage estimates. The transient model calibrated for this domain suggests very close similarities in the model-computed aquifer parameters for all the 2500 solutions. This suggests that any of the solutions could be used to represent the conditions prevailing in the domain. Risk analyses were then performed on abstractions through all the 2500 solutions. The study shows that the domain is sustained largely by lateral subsurface inflows from adjoining areas amounting to 399,278 m³/day, with direct vertical infiltration and percolation of precipitation accounting for an average of 2450 m³/day. The analysis suggests that at this rate of input, the domain can sustain groundwater abstraction to meet all water needs in the area. In addition, the domain has demonstrated resilience against annual increment in abstraction rates by 5 % over a 20-year period at the current recharge conditions. However, increasing abstraction rates by up to 200 % of the current abstraction rates over the 20-year period (2015–2035) but with a gradual decrease in groundwater recharge by up to 10 % over the period will result in drastic lowering of the hydraulic head especially in the south-central portions of the domain. This situation might induce reversal of the current flow and potentially lead to contamination and/or aquifer dewatering in the domain.     

Analyzing reproduction of correlations in Monte Carlo simulations: application to mine project valuation

Monte Carlo (MC) simulations are extensively used to assess risk in mining ventures; however, the correlation between the inputs used to build the models is often overlooked. We observed how value-at-risk (VaR) of a mining venture was affected by running MC simulations, using two different input correlation methods: Spearman's rank correlation and copulas using Kendall's tau. The goal was to compare different correlation approaches on risk analysis associated with uncertain parameters of mining ventures and uncover which one would yield the most accurate result. Three case studies were carried out to compare correlation structures. Modelling the input variable correlations was better achieved using copulas since they were able to capture a wider range of correlations that did not make any linearity assumptions. In the case study based on MC simulations, the impact of the input correlation choice on the VaR was rather severe with an approximate 9% difference between the results obtained with Spearman's correlations and the Normal copula correlations.     

基于蒙特卡罗随机模拟的矿岩崩落块度预测研究

基于实测节理面空间几何参数,采用蒙特卡罗方法对某大型镍矿III矿区节理空间进行了模拟。采用自主研发的三维矿岩块度预测软件MAKEBLOCK对其矿岩块度分布进行了预测和分析。结果表明,该矿区大多数矿岩块体体积小于 0.2 m3,绝大多数块体形状为盘状和块状,块体等效尺寸大多在0.2 m到1.5 m之间。预测结果可以作为自然崩落法工程设计与实施的参考依据。     

Monte Carlo simulations of the dissolution of borosilicate and aluminoborosilicate glasses in dilute aqueous solutions

The aim of this study was to use Monte Carlo simulations to provide atomic-level insights into the dissolution behavior of borosilicate and aluminoborosilicate glasses in dilute aqueous solutions. In the first part of this work, the effects of different structural features, such as the presence of non-bridging oxygens (NBO) or the formation of boroxol rings, on glass dissolution were evaluated separately and led to the following conclusions. (1) The dependence of the dissolution rate on the amount of NBO was found to be linear at all Si/B ratios and the accelerating effect of NBO was shown to increase with increasing Si/B ratio. (2) The formation of boroxol rings and of clusters of boroxol rings resulted in an increase of the dissolution rate at all Si/B ratios and, again, the extent of the rate increase was strongly dependent on the Si/B ratio. (3) For aluminosilicate glasses, the implementation of the aluminum avoidance rule was found to increase the rate of dissolution relative to that obtained for a random distribution.In the second part of this work, the dissolution of the NeB glasses studied by Pierce et al. (2010) was modeled in dilute aqueous solutions. Pierce et al. concluded from their study that either the rupture of the Al-O bonds or that of the Si-O bonds was the rate-limiting step controlling the dissolution of the NeB glasses. The simulations refined this conclusion and showed that, at low B/Al ratios, the rupture of both Al-O-Si and Si-O-Si linkages contributed to the dissolution rate whereas, at high B/Al ratios, the dissolution rate was independent of the rupture of Al-O-Si linkages and was controlled by S¹ sites (silicon sites at the glass-water interface with one connection to nearest-neighbor sites) and dissolution via detachment of clusters.     

A groundwater flow model for overexploited basaltic aquifer and Bazada formation in India

Recent changes in land use practices, such as increase in orange orchards in central India, has put undue pressure on the groundwater resources. Excess withdrawal from the aquifers has resulted in groundwater table decline. The stage of groundwater development in some watersheds has reached 155.85 %, converting these into overexploited watersheds. In the present research paper, a groundwater flow model has been developed to evaluate the groundwater system in a basaltic terrain with Bazada formation. A conceptual model has been developed and calibrated for steady and transient states and the sensitivity analysis was carried out. Future predictions, for current scenario where present practices are continued and for scenario with 20 % reduction in groundwater draft have been made, to select the best strategy for mitigating the problem. The modeling results show that the decline in groundwater level in basaltic and Bazada unconfined aquifers will result into drying up (water level more than 15 m bgl) of 243 km² area by 2020. To restore the groundwater level, it is simulated that the groundwater draft rate must be reduced by 20 % for next 10 years. It may be achieved by adopting groundwater management strategies, particularly for irrigation sector.     

地下水流模型识别的分步解法

地下水流模型识别问题是综合的水文地质分析与计算过程,但常常被片面理解为反求水文地质参数问题,这种片面的认识大大妨碍了对水文地质条件的认识及对水文地质过程的理解,并由此导致一些不应有的错误结论。简要论述了地下水流模型识别的概念及问题类型,利用模型分解的方法指出地下水流模型的识别可分2步完成:一是利用附加水头场(或附加降深场)求参数;二是利用初始水头场确定边界条件及源汇项。除参数外,源汇项、初始条件、边界条件也是地下水流模型识别的重要内容。     

基于蒙特卡罗随机有限元法的三维随机渗流场研究

通过建立改进Latin超立方抽样和对偶抽样相结合的复合抽样法,以提高Monte Carlo方法的计算效率,并将其引入Monte Carlo随机有限元（MSFEM）。基于三维有限元模型,采用MCSFEM对山坪土石坝进行随机渗流场分析,研究渗透系数和水头边界条件的随机特性对渗流场的干扰,进行变异系数和抽样次数的敏感性分析。最后,对渗流场的求解量进行概型分析。研究表明：总水头势、流速及渗透体积力的变异性随着渗透系数随机性的增强而变大;复合抽样法既能有效加快Monte Carlo的收敛速度,又能降低样本间的统计相关性,说明了该方法的实用性与有效性;当渗透系数服从正态分布时,渗流场中所取结点的水头和坡降也服从正态分布。     

WEKU – a GIS-Supported stochastic model of groundwater residence times in upper aquifers for the supraregional groundwater management

 The supraregional GIS-supported stochastical model, WEKU, for the determination of groundwater residence times in the upper aquifers of large groundwater provinces is presented. Using a two-dimensional analytical model of groundwater flow, groundwater residence times are determined within two extreme cases. In the first case, maximal groundwater residence times are calculated, representing the part of groundwater, that is drained by the main surface water of a groundwater catchment area. In the second case, minimal groundwater residence times for drainage into the nearest surface water are determined. Using explicit distribution functions of the input parameters, mean values as well as potential ranges of variations of the groundwater residence times are derived. The WEKU model has been used for the determination of groundwater residence times throughout Germany. The model results – mean values and deviations of the groundwater velocity and the maximal and minimal groundwater residence times in the upper aquifers – are presented by general maps and discussed in detail. It is shown that the groundwater residence times in the upper aquifer vary regionally, differentiated between less than 1 year and more than 2000 years. Using this information, the time scales can be specified, until measures to remediate polluted groundwater resources may lead to a substantial groundwater quality improvement in the different groundwater provinces of Germany. With respect to its supraregional scale of application, the WEKU model may serve as a useful tool for the supraregional groundwater management on a state, federal or international level. Received: 15 August 1995 · Accepted: 15 October 1995