Modified Markov Chain Monte Carlo Method for Dynamic Data Integration Using Streamline Approach

In this paper, the Markov Chain Monte Carlo (MCMC) approach is used for sampling of the permeability field conditioned on the dynamic data. The novelty of the approach consists of using an approximation of the dynamic data based on streamline computations. The simulations using the streamline approach allows us to obtain analytical approximations in the small neighborhood of the previously computed dynamic data. Using this approximation, we employ a two-stage MCMC approach. In the first stage, the approximation of the dynamic data is used to modify the instrumental proposal distribution. The obtained chain correctly samples from the posterior distribution; the modified Markov chain converges to a steady state corresponding to the posterior distribution. Moreover, this approximation increases the acceptance rate, and reduces the computational time required for MCMC sampling. Numerical results are presented.     

Extended Probability Perturbation Method for Calibrating Stochastic Reservoir Models

Calibrating a stochastic reservoir model on large, fine-grid to hydrodynamic data requires consistent methods to modify the petrophysical properties of the model. Several methods have been developed to address this problem. Recent methods include the Gradual Deformation Method (GDM) and the Probability Perturbation Method (PPM). The GDM has been applied to pixel-based models of continuous and categorical variables, as well as object-based models. Initially, the PPM has been applied to pixel-based models of categorical variables generated by sequential simulation. In addition, the PPM relies on an analytical formula (known as the tau-model) to approximate conditional probabilities. In this paper, an extension of the PPM to any type of probability distributions (discrete, continuous, or mixed) is presented. This extension is still constrained by the approximation using the tau-model. However, when applying the method to white noises, this approximation is no longer necessary. The result is an entirely new and rigorous method for perturbing any type of stochastic models, a modified PPM employed in similar manner to the GDM.     

A Distance-based Prior Model Parameterization for Constraining Solutions of Spatial Inverse Problems

Spatial inverse problems in the Earth Sciences are often ill-posed, requiring the specification of a prior model to constrain the nature of the inverse solutions. Otherwise, inverted model realizations lack geological realism. In spatial modeling, such prior model determines the spatial variability of the inverse solution, for example as constrained by a variogram, a Boolean model, or a training image-based model. In many cases, particularly in subsurface modeling, one lacks the amount of data to fully determine the nature of the spatial variability. For example, many different training images could be proposed for a given study area. Such alternative training images or scenarios relate to the different possible geological concepts each exhibiting a distinctive geological architecture. Many inverse methods rely on priors that represent a single subjectively chosen geological concept (a single variogram within a multi-Gaussian model or a single training image). This paper proposes a novel and practical parameterization of the prior model allowing several discrete choices of geological architectures within the prior. This method does not attempt to parameterize the possibly complex architectures by a set of model parameters. Instead, a large set of prior model realizations is provided in advance, by means of Monte Carlo simulation, where the training image is randomized. The parameterization is achieved by defining a metric space which accommodates this large set of model realizations. This metric space is equipped with a “similarity distance” function or a distance function that measures the similarity of geometry between any two model realizations relevant to the problem at hand. Through examples, inverse solutions can be efficiently found in this metric space using a simple stochastic search method.     

Method of Numerical Modeling for Constitutive Relations of Clay

INTRODUCTIONThe mechanical responses of soils are more com-plicated compared with metals.By comparing thephysical and mechanical properties of the metals withsoils,Lade(1988)found that there are17differentpoints between the metals and soils,which differfrom metals inthree basic mechanical characteristics:pressure sensitivity,dilatancy,and dependence ofstress path.Wang(2004)proposed a principle of theinteraction between plastic volume and shear strains,that is,there are two relatively inde…     

Hydrogeological and Gasgeochemical Earthquake Precursors – A Review for Application

Even if earthquake precursory signals can be identified, how can they be useful? This paper investigates relationships among the attributes of 229 proposed earthquake related gasgeochemical and hydrogeological precursory signals, and applies these results to improve future earthquake prediction strategies. Sub-groups of these reported signals and relationships between sub-groups are established using parameters, including earthquake magnitude, signal duration, precursory time, and epicentral distance to the monitoring site (original studies are used wherever possible to improve data quality). A strong correlation (r=0.86) between signal duration and precursory time was identified. This suggests a relationship between the investigated precursory signals and tectonic processes related to the referenced earthquakes. Moreover, these signals are categorized into four groups, reflecting differences in monitoring station densities, measurement methods and physical processes related to signal occurrence: (a) radon exhalation from the earth’s crust, (b) exhalation of other gases (helium, argon and others), (c) temporal variation in water level or discharge of springs and (d) temporal variation in temperature and dissolved ions in the water of the monitoring sites. In addition, boundary functions are used to separate signal group subsets. Finally, it is shown how these boundary functions can be used in the context of an earthquake prediction strategy by identifying potential minimum magnitudes and maximum epicentral distances from the monitoring site.     

Decoupling Conditions for Elasto-plastic Consolidation Question Based on Numerical Modeling Method

INTRODUCTION Theconsolidationofsaturatedsoilisanimpor tantsubjectingeomechanics.Biot(1941)proposed thethree dimensionalconsolidationtheory,whichis basedontheprincipleofeffectivestress,continuity conditionsandequilibriumequations.Theremarka blesuccessofthistheoryistheanalysisofthetime dependenteffectandcouplingeffectofsoilandpore water.Thus,thetheoryisaclassictheoryinsatu ratedsoilstatistics.Itwasalmostunfeasibletoperformanalysesof thistheorywithoutthedevelopmentofhigh speed computersandnu…     

Latent Variable Modeling for Integrating Output from Multiple Climate Models

Numerical models of atmosphere–ocean circulation are widely used to understand past climate and to project future climate change. Although the same laws of physics, chemistry, and fluid dynamics govern any general circulation model, each model’s formulations and parameterizations are different, yielding different projections. Notwithstanding, models within an ensemble will have varying degrees of similarity for different outputs of interest. Multi-model ensembles have been used to increase forecast skill by using simple or weighted averages where weights have been obtained by considering factors such as estimated model bias and consensus with other models (Giorgi and Mearns, J. Clim. 15:1141–1158, 2002, Geophys. Res. Lett. 30:1629–1632, 2003; Tebaldi et al., Geophys. Res. Lett. 31:L24213, 2004, J. Clim. 18:1524–1540, 2005). This paper considers an alternative view of multi-model ensembles. For use with the North American Regional Climate Change Assessment Program (NARCCAP), multivariate statistical models are employed to characterize modes of similarity within the members of an ensemble. Specifically, we propose a spatially-correlated latent variable model which facilitates the exploration of when, where, and how regional climate models are similar, and what factors best predict observed locations of model convergence.     

Fitting Data Under Omnidirectional Noise: A Probabilistic Method for Inferring Petrophysical and Hydrologic Relations

While geophysicists recognize that all measurements of media properties are subject to noise, methods for fitting petrophysical relations commonly employ regression-based formulas, which assume no error in one of the properties being related (the “independent" variable). To derive a more rigorous method for fitting such relations, we take a probabilistic viewpoint on the problem of fitting petrophysical relations to sample correlation data. Unlike prior approaches, we take into account the fact that noise is present in both measured properties. Under basic assumptions, we derive a new objective function for such problems which is proportional to the data likelihood and which can be used for both model parameter optimization and model selection. We present several numerical experiments outlining the utility of our method, and compare results of our method against results of other commonly used methods, such as kriging, regression, and total distance minimization. The results of our applications using the maximum likelihood technique appear visually accurate, and we also provide quantitative comparisons of performance that suggest the method produces more desirable results.     

Numerical Method for Modeling the Constitutive Relationship of Sand under Different Stress Paths

Overonehundredconstitutivemodelsforsand havebeenproposed,howeveronlyafewmodelsare widelyappliedinengineering.Therearetwochief reasonsforthis.(1)Sandiscomposedofgeological materialsandsituatedinaneutralenvironmentthat hassufferedvariousactions.Therefore,itisdifficult todescribethecomplexbehaviorofsoilsbyusinga simplemathematicalmodelwithonlyafewparame ters.(2)Therealsoexistsabasicparadoxintradi tionalmodelingmethods.Oneaspectofthemodelsis neededtodescribetheconstitutiverelationofsoilac curatel…     

BGHMS: A Geohistory Modeling System for Cratonic Basins

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ODSIM: An Object-Distance Simulation Method for Conditioning Complex Natural Structures

Stochastic simulation of categorical objects is traditionally achieved either with object-based or pixel-based methods. Whereas object-based modeling provides realistic results but raises data conditioning problems, pixel-based modeling provides exact data conditioning but may lose some features of the simulated objects such as connectivity. We suggest a hybrid dual-scale approach to combine both shape realism and strict data conditioning. The procedure combines the distance transform to a skeleton object representing coarse-scale structures, plus a classical pixel-based random field and threshold representing fine-scale features. This object-distance simulation method (ODSIM) uses a perturbed distance to objects and is particularly appropriate for modeling structures related to faults or fractures such as karsts, late dolomitized rocks, and mineralized veins. We demonstrate this method to simulate dolomite geometry and discuss strategies to apply this method more generally to simulate binary shapes.     

A Probabilistic Approach for Identifying Independent Remote Compressions in an Intraplate Region: The Iberian Chain (Spain)

The discrimination between distinct remote compressions and multiple local stress deviations within a single compressive stress field has been carried out in the central-eastern Iberian Chain, by using structural criteria, computing palaeostress directions, identifying and ‘filtering’ stress deviations, and analysing time relationships. A probabilistic analysis based on a systematic comparison of real and expected frequencies of coexistence of two compressions is applied by means of the χ ² Test. This allows us to identify those tectonic compressions that behave as independent events from the probabilistic point of view. The results suggest that among five initially defined compression directions only three can be considered as representative of distinct (although partially superposed) externally applied intraplate stress fields: Iberian s.l. (NE-SW), Betic s.l. (NW-SE), and Pyrenean (N-S to NNE-SSW).     

Anisotropic Mean Traveltime Curves: A Method to Estimate Anisotropic Parameters from 2D Transmission Tomographic Data

We present the mathematical deduction and properties of the mean traveltime curves for homogeneous elliptical anisotropic media. These curves generalize their isotropic counterparts which have been introduced in the past as a simple data quality analysis technique at the pre-inversion stage for 2D transmission experiments, allowing the inference of prior velocity models to gain stability at the tomographic inversion. Also, the anisotropy parameters (maximum velocity, anisotropic direction and ratio) are shown to affect the shape of these curves. The degree of asymmetry of the anisotropic mean traveltime curves (displacement of the mean time and standard deviation minima from the middle of the gathering line) is related to the direction of anisotropy which can then be visually estimated. Least squares’ fitting of the anisotropic theoretical models to their experimental counterparts is an effective method to estimate at the pre-inversion stage a macroscopic elliptical anisotropic velocity model, valid at the scale of the experiment, and able to match the experimental mean traveltime distribution. Sensitivity analysis has shown that the mean curve is less prone to errors than the standard deviation curve. Parameter identification from the standard deviation curve becomes unstable for noise levels higher than 5%; data errors produce smearing of the value of the estimated anisotropy ratio and wrong directions of anisotropy biased towards zero degrees. Also, identification from the mean traveltime curve becomes stable when the maximum velocity is well constrained. Finally, this methodology is illustrated with the application to the Grimsel data set. Performing MTC analysis is always recommended since it does not need high numerical requirements, and as shown in the sensibility analysis section, errors in data can be misinterpreted as geological anisotropies. J.L. Fernández Martínez is a visiting professor at UC Berkeley, Department of Civil and Environmental Eng., CA 94720-1710.     

A Liquidus Thermobarometer for Modeling of the Magnetite–Melt Equilibrium

A liquidus thermobarometer for the magnetite–silicate melt equilibrium was obtained by processing of a sample of 93 experimental equilibria of magnetite with basic melts using the methods of multidimensional statistics. The equations reproduce experimental data in a wide range of basic compositions, temperatures, and pressures with low error. Verification of the thermobarometers shows that the maximum error in liquidus temperature reproduction does not exceed ±7°C. The level of cumulative magnetite in the vertical sections of the Tsypringa, Kivakka, and Burakovskii intrusions is predicted with errors from ±10 to ±50 m.     

A Hamilton–Jacobi Framework for Modeling Folds in Structural Geology

A novel mathematical framework for modeling folds in structural geology is presented. All the main fold classes from the classical literature: parallel folds, similar folds, and other fold types with convergent and divergent dip isogons are modeled in 3D space by linear and non-linear first-order partial differential equations. The equations are derived from a static Hamilton–Jacobi equation in the context of isotropic and anisotropic front propagation. The proposed Hamilton–Jacobi framework represents folded geological volumes in an Eulerian context as a time of arrival field relative to a reference layer. Metric properties such as distances, gradients (dip and strike), curvature, and their spatial variations can then be easily derived and represented as three-dimensional continua covering the whole geological volume. The model also serves as a basis for distributing properties in folded geological volumes.     

Stereological Method for Reducing Probability of Earthquake-Induced Damage in a Nuclear Waste Repository

In Sweden, spent nuclear fuel is planned to be placed in copper/iron canisters and deposited at a depth of approximately 500 m in granitic rock. Earthquakes may induce secondary shear movements in fractures intersecting canister deposition holes, thereby threatening the integrity of the canisters. The extent of a secondary movement is related to earthquake distance and magnitude and to the size of the intersecting fracture. A probability of a canister being intersected by a critically large fracture can be calculated for given fracture size and orientation distributions, assuming that no measures are taken to identify and avoid such fractures. This paper analyses a stereological method of reducing this probability through observations of fractures fully intersecting the drift tunnels overlying the deposition holes. Deposition positions located in the planar extension of such full intersections are rejected. Both exact, numerical solutions and approximate solutions to this stereological problem are derived and the correctness of the solutions is verified by simulations. Also, the cost in terms of unutilised deposition positions is calculated. The probability of critical canister/fracture intersections is a few percent for typical fracture populations determined from field observations at a candidate site for a spent nuclear fuel repository in Sweden. By applying the suggested method, it is demonstrated that this probability can be reduced by a factor of about 35 in a typical case. The expense in terms of unutilised tunnel length is around 10 percent, which is seen as reasonable.     

A New Method for Clay Mineral Analysis and Its Application in Geology

X-ray diffraction (XRD) peaks in a low-angle diffraction section of clay minerals, especially those of authigenic origin, have broadening and tailing features in shape. Using the five basic parameters, peak position, peak height, width, shape coefficient and asymmetry, to describe an XRD peak is more accurate, comprehensive and integrated than using only 3 of them, position, height and width. Following the concept of the five basic parameters of an XRD peak, the program Decoform proposed in this study provides more information in mineralogical analyses by fitting actual XRD profiles. In combination with the HW-IR plot, Decoform can be systematically and accurately used in the comprehensive analyses of crystallinity, domain size, lattice strain and quantitative phase. It is also of value for the geological investigations of diagenesis, metamorphism, basin maturity, structural stress field and so on.     

Basic Lithofacies-Succession Model for the Wumishan Cyclothems: Their Markov Chain Analysis and Regularly Vertical Stacking Patterns in the Third-Order Sequences

The Mesoproterozoic Wumishan Formation in the Jixian section of Tianjin is a succession of 3300-m-thick carbonate strata formed in a period of about 100 Ma (1310±20 Ma-1207±10 Ma). In this succession of strata, the carbonate metre-scale cyclic sequences belonging to peritidal type with an approximately symmetrical lithofacies-succession are best developed. The wide development of 1:4 stacking patterns shows that these metre-scale cyclic sequences are genetically related to the short-eccentricity cycles, which are called the Wumishan cyclothems that could truly represent sedimentary cycles. Generally, massive and thick-bedded calcareous dolomites and dolomitic limestones of stromatolite biostromes and thrombolite bioherms constitute the central part of the Wumishan cyclothems. The lower and upper parts consist of tidal flat dolostones, sandy-muddy dolostone and the top part is composed of lagoonal facies dolomitic shales with a paleosol cap. Therefore, an approximately symmetrical lithofacies-successio     

Application of Numerical Modeling for Groundwater Flow System Analysis in the Akaki Catchment,Central Ethiopia

A three dimensional steady-state finite difference groundwater flow model is used to quantify the groundwater fluxes and analyze the subsurface hydrodynamics in the Akaki catchment by giving particular emphasis to the well field that supplies water to the city of Addis Ababa. The area is characterized by Tertiary volcanics covered with thick residual and alluvial soils. The model is calibrated using head observations from 131 wells. The simulation is made in a two layer unconfined aquifer with spatially variable recharge and hydraulic conductivities under well-defined boundary conditions. The calibrated model is used to forecast groundwater flow pattern, the interaction of groundwater and surface water, and the effect of pumping on the well field under different scenarios. The result indicates that the groundwater flows regionally to the south converging to the major well field. Reservoirs and rivers play an important role in recharging the aquifer. Simulations made under different pumping rate indicate that an increase in pumping rate results in substantial regional groundwater level decline, which will lead to the drying of springs and shallow hand dug wells. Also, it has implications of reversal of flow from contaminated rivers into productive aquifers close to main river courses. The scenario analysis shows that the groundwater potential is not enough to sustain the ever-growing water demand of the city of Addis Ababa. The sensitivity and scenario analysis provided important information on the data gaps and the specific sites to be selected for monitoring, and may be of great help for transient model development. This study has laid the foundation for developing detailed predictive groundwater model, which can be readily used for groundwater management practices.