Mathematical Geology

ABSTRACT: 3D geometric modelling is a powerful tool to better understand geology. It allows to check and validate the consistency of the separate 1D or 2D data interpretations. Building a 3D model is also a way to share and communicate a geological view. ...

Henrik Ibsen's concern in A doll's house (1879) is the exposition of what he deems social evils and shortcomings, consequently leaving his protagonist's future, after she famously slams the door on her married life, to the audience's speculation. His open ending thus exemplifies Barthes's seminal argument against unquestioned authorial authority in 'The death of the author' (1967), reasoning that what matters is the impression that a text makes on its destination, i.e., the reader. This paper looks into the take of one such reader on Ibsen's text. Lucas Hnath, who provides a sequel to Ibsen's play, daringly calling it A doll's house, part 2 (2017), interprets Nora's venturing into the outer world in terms of career and financial success. He begins his play with a knock on the same door ushering in, supposedly fifteen years later, the now established and self-assured feminist novelist Nora. As the two plays thus form a unified ontology, they showcase the Formalist postulate of the recurrent devices and the binding autonomous interrelations of literature. The themes that Ibsen starts are continued and elaborated by Hnath, who offers an extensive portrayal of various perceptions of Nora's decision to leave her family. He does so through the tactic of the discussion, originally introduced by Ibsen, which goes counter to the latter's general style of the well-made play. As New

Three methods are presented for determining paleo heat-flux from fission scar tracks in detrital apatite collected from wells. The annealing of thejission scar tracks is modelled using timetemperature integrals assuming first-order kinetics in agreement with the laboratory experiments over the last f$een years. The three methods invert the present-day fission-track well data along the burial paths of the sedimentary units. Thejirst two methods involve inverting the depth-dependent (a) areal track-number density pergm of uranium and (b) mean track length to obtain an estimate of the temporal dependence of the paleo heat-flux Since these methods tend to overlook short tracks (method (a)), and be biased towards longer tracks (method @) ) , they tend toprovidepaleo heat-flux variations which are too high and too low, respectively. The uncertainty in the determination ofpaleo heat-flux variation is also quite high, as is shown by using data @om two wells: Kambara-1 and Cumnga-1 in the NW Canning Basin, Australia. The third method inverts all of the individual distributions of track lengths at each sampled depth in a well. The sensitivity of this method is much higher than methods (a) or (b). In fact, we are able to determine not only the rate of change of heat-flux with time but, for thejirst time, variations in the direction of change of heat-flw allowing us to quantify heating and cooling events, as opposed to just heating or cooling events. In addition, we are able to determine the rate constant for track annealing and, in line with qualitative arguments advanced by Gleadow et a l (1983), we find that the rate constant appropriate to the multi-million year time-scale of subsurfce apatite track annealing is lower, by a factor of about three, than the value advocated from short-lived (on a geologic time-scale) laboratory experiments. Detailed track data from two wells, Kambara-1 and Curringa-I, yield essentially identical values for the rate constant, but very dixerent thermal history behaviours.

The accuracy of area estimations was evaluated for the ore fields within the East Pacific polymetallic nodules deposit licenced to the Interoceanmetal Joint Organization (IOM). The ore fields were contoured using various cut-off values of nodules abundance. It was found that accuracies of both the contouring and the area estimations of high-grade ore fields are low, if based solely on the results of ocean floor sampling. Comparison was made of ore fields areas estimated using two parameters: the minimum abundance of nodules and the minimum abundance of nickel equivalent. An attempt was made to apply the geostatistical methods to evaluate the accuracy of estimations of ore fields areas. The difficulties were highlighted of the usage of geostatistical methods in exploration of polymetallic nodules deposits due to irregular sampling grids.

This paper presents an overview and synthesis of an extensive research effort to characterize and quantify scale invariances in the morphology and evolution of braided rivers. Braided rivers were shown to exhibit anisotropic spatial scaling (self-affinity) in their morphology, implying a statistical scale invariance under appropriate rescaling of the axes along and perpendicular to the main direction of flow. The

The principal thesis of Dr. Rohlf's paper is that, in order to extract information from a shape, that shape must be measured completely. Our perspective is that "total" shapes of irregular two-dimensional objects (i.e. not regular polygons nor simple conics) cannot be discussed fruitfully. Such shapes can be thought to consist of an indefinite number of attributes, not all of which are necessarily of interest. To us, instead, the major task in shape analysis is to determine the most information-rich components of shape and use variation in the value of such components from object to object in order to better understand underlying process(es) responsible for shape variability. Therefore, to us the only way to evaluate the efficacy of one or another shape measurement procedure is to test it in the context of a well-designed experiment that has nontrivial objectives. Rohlf, for instance, is concerned that an analysis consisting of a set of Fourier amplitudes is misleading if phase information is not included simultaneously. If this were so, much research based on x-ray diffraction and radio astronomy would be of little worth. We, on the other hand, have published papers wherein the response of the amplitude of a single harmonic (shown objectively to be the most information-rich) is related to detrital provenance or, in the case of a foraminiferan, a function of latitude. Lack of completeness seems to not have been a problem in many studies in morphometry wherein variation of simple length-width ratios has produced worthwhile results. Finally, no characterization of a two-dimensional shape of a fossil or sand grain can ever be "complete" inasmuch as these are derived by projection from threedimensional objects. How does Rohlf and company know that the "essence" of the shape is captured in a paltry two dimensions? To be sure, incompleteness can result in ambiguity or missed opportunities, but we certainly are not powerless to avoid such perils.

The principal thesis of Dr. Rohlf's paper is that, in order to extract information from a shape, that shape must be measured completely. Our perspective is that "total" shapes of irregular two-dimensional objects (i.e. not regular polygons nor simple conics) cannot be discussed fruitfully. Such shapes can be thought to consist of an indefinite number of attributes, not all of which are necessarily of interest. To us, instead, the major task in shape analysis is to determine the most information-rich components of shape and use variation in the value of such components from object to object in order to better understand underlying process(es) responsible for shape variability. Therefore, to us the only way to evaluate the efficacy of one or another shape measurement procedure is to test it in the context of a well-designed experiment that has nontrivial objectives. Rohlf, for instance, is concerned that an analysis consisting of a set of Fourier amplitudes is misleading if phase information is not included simultaneously. If this were so, much research based on x-ray diffraction and radio astronomy would be of little worth. We, on the other hand, have published papers wherein the response of the amplitude of a single harmonic (shown objectively to be the most information-rich) is related to detrital provenance or, in the case of a foraminiferan, a function of latitude. Lack of completeness seems to not have been a problem in many studies in morphometry wherein variation of simple length-width ratios has produced worthwhile results. Finally, no characterization of a two-dimensional shape of a fossil or sand grain can ever be "complete" inasmuch as these are derived by projection from threedimensional objects. How does Rohlf and company know that the "essence" of the shape is captured in a paltry two dimensions? To be sure, incompleteness can result in ambiguity or missed opportunities, but we certainly are not powerless to avoid such perils.

Analysis o f empirical data considered to be mixtures of a finite number of end members has been a topic o f increasing interest recently. The algorithms EXTENDED CABFAC and QMODEL by Klovan and Miesch (1976J represent a satisfactory solution to this problem if pure end members are captured within the data set or if the composition of "true" end members are known a priori. Where neither condition is satisfied, the composition of "external" end members can, under certain conditions, be deduced from the structure of the data. Described herein is an algorithm termed EXTENDED QMODEL which defines feasible end members which are "closest" to the data envelope.

Analysis o f empirical data considered to be mixtures of a finite number of end members has been a topic o f increasing interest recently. The algorithms EXTENDED CABFAC and QMODEL by Klovan and Miesch (1976J represent a satisfactory solution to this problem if pure end members are captured within the data set or if the composition of "true" end members are known a priori. Where neither condition is satisfied, the composition of "external" end members can, under certain conditions, be deduced from the structure of the data. Described herein is an algorithm termed EXTENDED QMODEL which defines feasible end members which are "closest" to the data envelope.

The ability to test for similarities and differences among families of shapes by closed-form Fourier expansion is greatly enhanced by the concept of homology. Underlying this concept is the assumption that each term of a Fourier series, when compared to the same term in another series, represents the "same thing". A method that ensures homology is one which minimizes the "centering error," as reflected in the first harmonic term of the Fourier expansiono The problem is to chose a set of edge points derived from a much larger, but variable, number of edge points such that a valid homologous Fourier series can be calculated. Methods are reviewed and criteria given to define a "proper,' solution. An algorithm is presented which takes advantage of the fact that minimization of the "error term" can be accomplished by minimizing the distance between the origin of the polar coordinate system in the calculation of the Fourier series and the shape centroid. The use of this algorithm has produced higher quality solutions for quartz grain provenance studies.

This paper proposes an extension of a single coupled Markov chain model to characterize heterogeneity of geological formations, and to make conditioning on any number of well data possible. The methodology is based on the concept of conditioning a Markov chain on the future states. Because the conditioning is performed in an explicit way, the methodology is efficient in terms of computer time and storage. Applications to synthetic and field data show good results.

The program as published (Howarth, 1971) could give rise to slight scaring errors in the production of large maps owing to the use of an integer value in the calculation of the x-and y-axis scaling factors in the subroutine PSCALE. A truncation error would arise in subsequent calculation of the true-scaleplot-scale conversions via the subroutines RCONVR and PCONVR. The following modifications to the program are suggested to remove this source of error.

The short movie Nora is a remarkable example of the gender roles in the film industry. This paper attempts to analyse gender-induced elements resided in the short-movie by analysing the relationship between Nora and her husband, Torvald from a deconstructivist approach. This analysis tries to figure out the roles endowed to women and men in the society by looking at characters' features and dynamics. Additionally, this analysis will give a chance to media & literacy students to work on a critical movie from a different perspective. Nora A Doll's House is a three-play act written by Henrik Ibsen and it covers a number of topics such as women roles in a patriarchal society and monetary issues about the modern world.

The uncertainty of classification in discriminant analysis may result from the original characteristics of the phenomena studied, the approach of inferring population parameters, and the credibility of the parameters which are estimated by geologist or statistician. A credibility function and a significance function are proposed. Both can be used to appraise the uncertainty of classification. The former is involved with

This research presents an application of specific types of a Markov chain model (called the coupled Markov chain model) to characterize subsurface heterogenity. This model incorprates soft geological data in additional to the available hard data to charactrize subsurface heterogenity conditioned on a number of boreholes. Data from resarch site in the Netherlands are used to evaluate the performance of the coupled Markov chain model (CMC). The model is used to study the effect of conditioning on number of boreholes on reducing the uncertainty of the subsurface heterogenity at the site.

Phenolic chemicals form multi-substance systems. They can pollute aquifers by the leacheate from tar residues and their concentrations which vary in space and time and require quantitative models of composition, spatial distribution and temporal alteration of the contaminants for monitoring and remediation. Mathematical models for groundwater flow, contaminant transport or mass balances and numerous unspecific multivariate-statistical methods cannot be applied if there are only few observation points within comparatively small polluted industrial sites. (5) subdivision of the studied area regarding its multivariate properties; and (6) interpretative temporal analysis. This approach is demonstrated with groundwater contamination at a brownfield site.

Phenolic chemicals form multi-substance systems. They can pollute aquifers by the leacheate from tar residues and their concentrations which vary in space and time and require quantitative models of composition, spatial distribution and temporal alteration of the contaminants for monitoring and remediation. Mathematical models for groundwater flow, contaminant transport or mass balances and numerous unspecific multivariate-statistical methods cannot be applied if there are only few observation points within comparatively small polluted industrial sites. However, multi-dimensional heuristc models of pattern recognition can be used to study attribute patterns, object pattern, the spatial distribution of object classes, and the temporal development of objects. A proved sequence of investigative steps includes: (1) determination of contaminant association patterns by cluster analysis; (2) visualization of spectral analysis of polluting constituents; (3) determination of classes of pollution from cluster dendrograms; (4) descriptive class characteristics; (5) subdivision of the studied area regarding its multivariate properties; and (6) interpretative temporal analysis. This approach is demonstrated with groundwater contamination at a brownfield site.

Quantifying risk in the absence of hard data presents a significant challenge. Onshore mining of the diamondiferous linear beach deposit along the south western coast of Namibia has been ongoing for more than 80 years. A historical delineated campaign from the 1930s to 1960s used coast perpendicular trenches spaced 500 m apart, comprising a total of 26 000 individual samples, to identify 6 onshore raised beaches. These linear beaches extend offshore and are successfully mined in water depths deeper than 30 m. There is, however, a roughly 4 km wide submerged coast parallel strip adjacent to the mostly mined out onshore beaches for which no real hard data is available at present. The submerged beaches within the 4 km coast parallel strip hold great potential for being highly diamondiferous. To date hard data is not yet available to quantify or validate this potential. The question is how to obtain sufficient hard data within the techno economic constraints to enable a resource with an acceptable level of confidence to be developed. The work presented in this thesis illustrates how virtual orebodies (VOBs) are created based on geological proxies in order to have a basis to assess and rank different sampling and drilling strategies. Paper I demonstrates the challenge of obtaining a realistic variogram that can be used in variogrambased geostatistical simulations. Simulated annealing is used to unfold the coastline and improve the detectable variography for a number of the beaches. Paper II shows how expert opinion interpretation is used to supplement sparse data that is utilised to create an indicator simulation to study the presence and absence of diamondiferous gravel. When only the sparse data is used the resultant simulation is unsuitable as a VOB upon which drilling strategies can be assessed. Paper III outlines how expert opinion hand sketches are used to create a VOB. The composite probability map based on geological proxies is adjusted using a grade profile based on adjacent onshore data before it is seeded with stones and used as a VOB for strategy testing. Paper IV illustrates how the Nachman model based on a Negative Binomial Distribution (NBD) is used to predict a minimum background grade by considering only the zero proportions (Z p ) of the grade data. v In the realm of creating spatial simulations that can serve as VOBs it is very difficult to attempt to quantify uncertainty when no hard data is available. In the absence of hard data, geological proxies and expert opinion are the only inputs that can be used to create VOBs. Subsequently these VOBs are used as a base to be analysed in order to evaluate and rank different sampling and drilling strategies based on techno economic constraints. VOBs must be updated and reviewed as hard data becomes available after which sampling strategies should be reassessed. During early stage exploration projects the Z p of sample results can be used to predict a minimum background grade and rank different targets for further sampling and valuation. The research highlights the possibility that multi point statistics (MPS) can be used. Higher order MPS should be further investigated as an additional method for creating VOBs upon which sampling strategies can be assessed. A grateful word of thanks to the following people: My supervisor Chris Prins for your appreciation of the west coast including physically withstanding gale force winds and teaching me how to incubate problems My co-supervisor Richard Minnitt for your kind guidance and wisdom John Ward and Brian Bluck for opening my eyes to the west coast geology narrative The Namdeb Mineral Resource Department for sharing the realization that as we try to define this resource a big part of ourselves is being defined by it Mitford, Riaan and Inge for making the timeline of this work possible Malcolm, David, Andy, Ina, Kean, Scott, Christien and Kumar for your encouragement and shifting my boundaries The Futree community garden and Gertrude Kalipi: thank you for showing me that if you try enough different things partial success is imminent Emila and Ben for joining the journey through potty training and learning how to read and write Jurgen for being so different to me and loving me so much vii slice of wind in the most vibrant moment of a raging sandstorm the cutting wind scorching all surrounds turns its head and looks at a framed instant of calmness within itself 1.

The modelling of spatial uncertainty in attributes of geological phenomena is frequently based on the stochastic simulation of Gaussian random fields. This paper presents a generalization of the sequential Gaussian simulation method founded upon the group decomposition of the posterior probability density function of a stationary and ergodic Gaussian random field into posterior probability densities of a set of groups of nodes of size ν. The method, which is termed "generalized sequential Gaussian simulation on group size ν," relies computationally on sharing the neighborhood of adjacent nodes and simulates groups of ν nodes at a time, instead of the traditional node-by-node simulation. The new method is computationally efficient and suitable for simulation on large grids of nodes. Results suggest that, in terms of computing cost (time), an optimal size ν of a group is about 80% of the optimal neighborhood used for sequential Gaussian simulation and that computation can be up to 50 times faster than the regular sequential Gaussian method, with little loss in accuracy. The effectiveness of the method is assessed by using a general measure of accuracy, "screen-effect approximation loss" (SEA loss), defined herein as the mean-square difference between the simulated value posterior to the information in the neighborhood and the simulated value posterior to all information, and shown to be determined by the corresponding posterior variances. The results presented show that both the exponential and the spherical models perform well and can meet the less-than 5% relative SEA loss requirement for any grid setup using a relatively small neighborhood. The Gaussian covariance model was found to have a relatively high relative SEA loss in most cases.

Sequential simulation of a continuous variable usually requires its transformation into a binary or a Gaussian variable, giving rise to the classical algorithms of sequential indicator simulation or sequential Gaussian simulation. Journel (1994) showed that the sequential simulation of a continuous variable, without any prior transformation, succeeded in reproducing the covariance model, provided that the simulated values are drawn from

We discuss a petroleum discovery model that greatly simplifies the approach initiated by Barouch and Kaufman (1976) in which exploration is viewed as a sampling without replacement process, and the probability of discovery of a pool is proportional to its size. Calculations that formerly required lengthy Monte Carlo simulations have been reduced to compact formulas.

We show that the Gibbs Sampler in the Gaussian case is closely linked to linear fixed point iterations. In fact stochastic linear iterations converge toward a stationary distribution under the same conditions as the classical linear fixed point one. Furthermore the covariance matrices are shown to satisify a related fixed point iteration, and consequently the Gibbs Sampler in the gaussian case corresponds to the classical Gauss-Seidel iterations on the inverse of the covariance matrix, and the stochastic over-relaxed Gauss-Seidel has the same limiting distribution as the Gibbs Sampler. Then an efficient method to simulate a gaussian vector is proposed. Finally numerical investigations are performed to understand the effect of the different strategies such as the initial ordering, the blocking and the updating order for iterations. The results show that in a geostatistical context the rate of convergence can be improved significantly compared to the standard case.

This paper is devoted to a geostatistical attempt at modeling migration errors when localizing a reflector in the ground. Starting with a probabilistic velocity model and choosing the simple geometrical optics background for the wave propagation in such media, we give the expression of the errors. This may be quantified provided the covariance of the velocity field is known. Variance

Fractal modelling has been applied extensively as a means of characterizing the spatial distribution of geological phenomena that display self-similarity at differing scales of measurement. A fractal distribution exists where the number of objects exhibiting values larger than a specified magnitude displays a power-law dependence on that magnitude, and where this relationship is scale-invariant. This paper shows that a number of distributions, including power-function, Pareto, log-normal and Zipf, display fractal properties under certain conditions and that this may be used as the mathematical basis for developing fractal models for data exhibiting such distributions. Population limits, derived from fractal modelling using a summation method, are compared with those derived from more conventional probability plot modelling of stream sediment geochemical data from northeastern New South Wales. Despite some degree of subjectivity in determining the number of populations to use in the models, both the fractal and probability plot modelling have assisted in isolating anomalous observations in the geochemical data related to the occurrence of mineralisation or lithological differences between sub-catchments. Thresholds for the main background populations determined by the fractal model are similar to those established using probability plot modelling, however the summation method displays less capacity to separate out anomalous populations, especially where such populations display extensive overlap. This suggests, in the geochemical data example provided, that subtle differences in the population parameters may not significantly alter the fractal dimension.

Data of a microfossil group, the planktonic foraminifera, have been tested to determine the conformity of various real data distributions to univariate and multivariate normality and the effects that standard transformations have upon the distributions. Studies of two bivariate samples, one trivariate sample, and two quadrivariate samples of size data indicate that distributions frequently deviate greatly from multivariate normality. Univariate distributions are generally positively skewed and show a tendency for leptokurtosis. A logarithmic transformation improved both univariate and multivariate distributions but the number of distributions conformable to normality increased only slightly-from zero to one in the multivariate case and from one to four in the univariate case (totally 15 distributions). Aresine (p/lO0) ~ transformations of percentage data in two samples including 16 and 23 species, respectively, decreased highly significant deviations from multivariate normality but distributions remained greatly non-normal. Although markedly positively skewed and leptokurtic univariate distributions were improved in most instances, the number of normal distributions (two) did not change. It follows that neither of the transformations caused significant increases in the number of normal distributions but if it is assumed that the consequences of non-normality are less severe as the deviation from normality decreases, the transformations are justified.

In isotropic semivariograms, ordinary least squares can estimate nugget effect and sill by partitioning its range. By conducting simulation, a semivariogram model with previously given parameters will be estimated through bootstrap method. Least square-bootstrap (LS-Bootstrap) will be applied to estimate the parameters of the model after resampling the errors of the model. The selection of the resulting semivariogram model from bootstrap method will be affected by the number of distance lags, the precision level of the range partitions, the number of bootstrap iterations, and the given reference model. The exponential and Gaussian models are sufficiently good in the estimation for the models with the same references. Meanwhile, the estimation yielded from spherical model is quite far from the reference 5124 K. N. Sari et al. exponential and Gaussian models, with the mean square error value reaching 713. The estimation with bootstrap method which is the same as the reference model will be faster to converge with the maximum iteration of 50. Besides, bootstrap method enables to obtain the point estimates and interval estimates of the nugget effect, sill, and range parameters.

This paper presents a computer algorithm that is capable of cogenerating pairs of three‐dimensional, cross‐correlated random fields. The algorithm produces random fields of real variables by the inverse Fourier transform of a randomized, discrete three‐dimensional spectral representations of the variables. The randomization is done in the spectral domain in a way that preserves the direct power and cross‐spectral density structure. Two types of cross spectra were examined. One type specifies a linear relationship between the two fields, which produces the same correlation scales for both variables but different variances. The second cross spectrum is obtained from a specified transfer function and the two power spectra, and it produces fields with different correlation scales. For both models the degree of correlation is specified by the coherency. A delay vector can also be specified to produce an out‐of‐phase correlation between the two fields. The algorithm is very efficient comp...

Summary Over 75,000 permeability measurements were collected from a meter-scale block of Massillon sandstone, characterized by conspicuous crossbedding that forms two distinct nested scales of heterogeneity. With the aid of a gas minipermeameter, spatially exhaustive fields of permeability data were acquired at each of five different sample supports (i.e., sample volumes) from each block face. These data provide a unique opportunity to physically investigate the relationship between the multiscale cross-stratified attributes of the sandstone and the corresponding statistical characteristics of the permeability. These data also provide quantitative physical information concerning the permeability upscaling of a complex heterogeneous medium. Here, a portion of the data taken from a single block face cut normal to stratification is analyzed. The results indicate a strong relationship between the calculated summary statistics and the cross-stratified structural features visibly evident ...

To carry out an efficient and effective exploitation of a slate mine, it is necessary to have detailed information about the production potential of the site. To assist us in estimating the quality of slate from a small set of drilling data within an unexploited portion of the mine, the following estimation techniques were applied: kriging, regularization networks (RN), multilayer perceptron (MLP) networks, and radial basis function (RBF) networks. Our numerical results for the test holes show that the best results were obtained using an RN (kriging) which takes into account the known anisotropy. Differing deposit configurations were obtained, depending on the method applied. Variations in the form of pockets were obtained when using a radial pattern with RBF, RN, and kriging models while a stratified pattern was obtained with the MLP model. Pockets are more suitable for a slate mine, which indicates that the selection of a technique should take account of the specific configuration of the deposit according to mineral type.

Modeling hyper-dimensional spatial variability is a complex task from both practical and theoretical standpoints. In this paper we develop a method for modeling hyper-dimensional covariance (variogram) structures using the product-sum covariance model initially developed to model spatio-temporal variability. We show that the product-sum model can be used recursively up to an arbitrarily large number of dimensions while preserving relative modeling simplicity and yielding valid covariance models. The method can be used to model variability in anisotropic conditions with multiple axes of anisotropy or when temporal evolution is involved, and thus is applicable to “full anisotropic 3D+time” situations often encountered in environmental sciences. It requires fewer assumptions than the traditional product-sum modeling approach. The new method also presents an alternative to classical approaches to modeling zonal anisotropy and requires fewer parameters to be estimated from data. We prese...

Measurement of thousands of quartz particle shapes from a variety of modern and ancient settings has led to the following conclusions concerning the origin and nature of shape frequency distributions: 1) Quartz shape frequency distributions are polydimensional and polymodal; six or more independent variables are necessary to account for more than 90 percent of shape variation, with each variable generating a polymodal frequency distribution; 2) Each source rock yields quartz particles which are represented by shape frequency distributions with 3 modes or more; mixed sediment samples are so complexly polymodal that they can be considered to be unique "fingerprints"; and 3) This provenance-produced polymodality tends to be reduced with increasing abrasion in the variables which describe the finer details of the particle shape ((higher harmonics). However, even in the most mature sands of the Saint Peter Sandstone, shape variables which describe large scale shape components (lower harmonics) persist. Thus, in the same sample, both provenance and process history (degree of abrasion) can be evaluated. A relationship between size and shape exists and must be considered in any shape analysis. The relationship is not continuous but is manifest by discrete shifts in shape frequency distributions at around 125 microns. We feel that this shift is related to provenance associated with the different rates of transport of grain sizes above and below 125 microns.

Tile confining pressure exerted by tilt, prograding Tertiary elastic wedge has caused extensive salt dqfi~rnlatitm in rile Gulf qf Mexict~. Tilt, high rates of hneral salt nlati~nl fi'om tilt" Louisiana sllelf break In the Sigshee Escarpment are t'~t7~eeted to have sig#~ficant iaq~act tm the dynamic evohltion of the surrounding sedinlentar 3" fiJrntatitnlS. Tilt' el,¢ilatitm of fluid pressure in supra-and ~uhsalt sedinlents has heen modeled in a series of pseudo-I~x'lls ahmg a pro[Jh, intersecting numerous salt sheets and strl,'tures Hff[#tore Lxmisiana. Based upon the estinlated rates of salt nloW'nlent, the tinting can hi" determined .fin insertitmldepletioa of sah in the sedinwuts as the stilt has moved through, and by, sedintents on its way has#lward An t~ecution qf a series of l-D fluid(flow ~'ompacliatl nladets has enabled estimatialt of the spatial vuriatio#t in overpressure huiht-up u'ith time; iadit'ttlillg thai ovet'pt'essltre ttll('ertainty is aholll IO 20% o#oltt tl recall o]'ahout 4(I atmospheres, with uncertainties ol1 salt thieknesx and salt speed roughly conlparahh' in importance in contrihuting to the uncertainty. Tiw quantitative behaviors suggest that the sT"wed t!] htteral stilt insertion and the thickness of the salt are the tmlin Jactors t'attsing anomalatts overpressare huihlup. ,4 qu~nltification ¢~]" the dynaalic hehavior.~ ix sign([icaat when moth'ling the tithing ¢~'potentia[ trapping ¢~ hydrocarbons beneath salt sheets and ¢!f the suhsalt overpressure to he ewectt'd in suhsah drilling.

Numerical simulation of geological phenomena are nowadays widely used to understand them. A good example are the numerous codes developed to model mass motion in different processes. Such models are approximations to the real phenomena because the physical conditions are simplified or because the values of the parameters intervening in the modeling are known only approximately. For this reason, an obligatory step in the application of these models is the assessment of their sensitivity to changes in the parameters. Sensitivity analysis is also required to evaluate the influence of the different input variables in the outcome of the model. In general this analysis is carried out by repeated computation of the model to different sets of input variables and settings of the model. A more fruitful procedure to carry out such analysis can be achieved by automatic differentiation. In this paper we illustrate such approach through its application to the model of an eruptive plinian column. The results of the analysis allow to assess, in a systematic and efficient manner, the relevance of the different boundary values and parameters of the model. The analysis reveals the effects on velocity, temperature, gas fraction, radius and ultimately column height, by changes in their boundary value (i.e. at the vent). The effect of changes in the value of the gas constants, density of pyroclasts and specific heat of the erupted material is also evidenced. Alternatively, the results can be regarded as a mean to estimate the values of the variables for different set of boundary values without recurring to new runs of the model. Automatic differentiation is shown to be a convenient scheme to obtain derivatives of model variables. This type of analysis forms in addition the basis for further work in inversion schemes.

A three-dimensional (3D) mass transport numerical model is presented. The code is based on a particle tracking technique: the random-walk method, which is based on the analogy between the advection–dispersion equation and the Fokker–Planck equation. The velocity field is calculated by the mixed hybrid finite element formulation of the flow equation. A new efficient method is developed to handle the

Understanding multi-phase fluid flow and transport processes within aquifers, candidate reservoirs for CO 2 sequestration, and petroleum reservoirs requires understanding a diverse set of geologic properties of the aquifer or reservoir, over a wide range of spatial and temporal scales. We focus on multiphase flow dynamics with wetting (e.g., water) and non-wetting (e.g., gas or oil) fluids, with one invading another. This problem is of general interest in a number of fields and is illustrated here by considering the sweep efficiency of oil during a waterflood. Using a relatively fine-resolution grid throughout a relatively large domain in these simulations and probing the results with advanced scientific visualization tools (Reservoir Visualization Analysis [RVA]/ ParaView software) promote a better understanding of how smaller-scale features affect the aggregate behavior at larger scales. We studied the effects on oil-sweep efficiency of the proportion, hierarchical organization, and connectivity of high-permeability open-framework conglomerate (OFC) cross-sets within the multi-scale stratal architecture found in fluvial deposits. We analyzed oil production rate, water breakthrough time, and spatial and temporal distribution of residual oil saturation. As expected, the effective permeability of the reservoir exhibits large-scale anisotropy created by the organization of OFC cross-sets within unit bars, and the organization of unit bars within compound-bars. As a result, oil-sweep efficiency critically depends on the direction of the pressure gradient. However, contrary to expectations, the total amount of trapped oil due to the effect of capillary trapping does not depend on the magnitude of the pressure gradient within the examined range. Hence the pressure difference between production and injection wells does not affect sweep efficiency; although the spatial distribution of oil remaining in the reservoir depends on this value. Whether or not clusters of connected OFC span the domain affects only the absolute rate of oil production-not sweep efficiency.

Geochemical samples from part of Lake Geneva were analyzed for 29 oxides and trace elements. The variables and samples were subjected to R-and Q-mode analyses. The following techniques were applied in sequence: data transformation (normalization and standardization), data reduction (principal component and factor analysis), and automatic classification (dendrograph). The data were treated using various combinations of these techniques, and the resulting classifications evaluated by means of several criteria. The best classification of the samples is given by a cluster analysis performed on four principal components computed from standardized variables. The discriminatory power of the variables also was measured and determined to depend on their degree of intercorrelation. As a final result, the 29 original variables were reduced to four components and the sediment samples classified into four facies, leading to easily interpretable geochemical maps.

Sverdrup Basin hosts a structural petroleum play in Mesozoic clastic reservoirs. Twenty-one discovered fields (eight crude oil and 25 natural gas pools) have 294.1 Â 10 6 m 3 crude oil, and 500.3 Â 10 9 m 3 natural gas, original in-place contingent resources. We discuss and compare discovery process and volumetric assessment methods that, respectively, predict a 673.1 Â 10 6 m 3 or 698.7 Â 10 6 m 3 median crude oil resource and a 1187.4 Â 10 9 m 3 or 1202.8 Â 10 9 m 3 median natural gas resource. Both methods predict that the largest crude oil and third-largest natural gas pools are undiscovered, a result inferred to be consistent with available data and the exploration history. Volumetric assessments can precede any discoveries and they use common geoscience data inputs; however, they can be affected by data interdependencies and biases from exploratory sampling and subjective parameter estimates, particularly those affecting the number of accumulations. Discovery process methods solve for the accumulation numbers and size distribution simultaneously, accounting for sampling bias and free of data interdependencies, but only once sufficient discoveries exist. The Sverdrup Basin dataset and exploration history permit us to cross-validate volumetric and discovery process assessments and validate their predictions, for example undiscovered pool sizes, against a regional geoscience dataset. The discovery process results agree well with geoscience constraints, but the initial volumetric assessment must be restricted to predict undiscovered pool sizes consistent with the geoscience dataset. Our analysis illustrates advantages and potential pitfalls for volumetric and discovery process assessments and shows that cross-validation between methods and against available data constrains resource potentials and improves confidence in result.

If observed oil and gas field size distributions are obtained by random samplings, the fitted distributions should approximate that of the parent population of oil and gas fields. However, empirical evidence strongly suggests that larger fields tend to be discovered earlier in the discovery process than they would be by random sampling. Economic factors also can limit the number of small fields that are developed and reported. This paper examines observed size distributions in state and federal waters of offshore Texas. Results of the analysis demonstrate how the shape of the observable size distributions change with significant hydrocarbon price changes. Comparison of state and federal observed size distributions in the offshore area shows how production cost differences also affect the shape of the observed size distribution. Methods for modifying the discovery rate estimation procedures when economic factors significantly affect the discovery sequence are presented. A primary conclusion of the analysis is that, because hydrocarbon price changes can significantly affect the observed discovery size distribution, one should not be confident about inferring the form and specific parameters of the parent field size distribution from the observed distributions.

A model of the discovery process can be used to predict the size distribution of future petroleum discoveries in partially explored basins. The parameters of the model are estimated directly from the historical drilling record, rather than being determined by assump tions or analogies. The model is based on the concept of the area of influence of a drill hole, which states that the area of a basin exhausted by a drill hole varies with the size and shape of targets in the basin and with the density of previously drilled wells. It also uses the concept of discovery efficiency, which measures the rate of dis covery within several classes of deposit size. The model was tested using 25 years of historical exploration data (1949-74) from the Denver basin. Prom the trend in the discovery rate (the number of discoveries per unit area exhausted), the discovery efficiencies in each class of deposit size were estimated. Using pre-1956 discovery and drilling data, the model accurately predicted the size distribu tion of discoveries for the 1956-74 period. AREA OF INFLUENCE OF A DRILL HOLE The area exhausted by a drill hole varies with the size of the targets that can occur in the basin. A basin may be fully explored with respect to large targets but Al A2 PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING 01 2345 TARGET SIZE, IN SQUARE MILES 'A region is connected if and only if any two points in the region can be joined by a curve line that does not go outside of the region (region is in one piece).

To derive meaningful results from a geostatistical study, it is extremely important to establish the relationship between geology and variograms. However, it is not always easy to establish such a relationship, mainly because of the inadequacy of quantitative or qualitative information. Large amounts of reliable well-log porosity data and the detailed geological information from a complex carbonate reservoir located in

We derive the bias and sampling variation of the harmonic average. These expressions, based on the h)gnormal distribution, are validated using Monte Carlo and jackknife analysis of field data. The average has a positive bias. The sampling variation results suggest that, for moderate to large sample sizes, the harmonic average is no more variable than the arithmetic average if the medium is appropriately sampled.

The spatio-temporal variogram is an important factor in spatio-temporal prediction through kriging, especially in fields such as environmental sustainability or climate change, where spatio-temporal data analysis is based on this concept. However, the traditional spatiotemporal variogram estimator, which is commonly employed for these purposes, is extremely sensitive to outliers. We approach this problem in two ways in the paper. First, new robust spatio-temporal variogram estimators are introduced, which are defined as M-estimators of an original data transformation. Second, we compare the classical estimate against a robust one, identifying spatio-temporal outliers in this way. To accomplish this, we use a multivariate scalecontaminated normal model to produce reliable approximations for the sample distribution of these new estimators. In addition, we define and study a new class of M-estimators in this paper, including real-world applications, in order to determine whether there are any significant differences in the spatio-temporal variogram between two temporal lags and, if so, whether we can reduce the number of lags considered in the spatio-temporal analysis.

In this paper, we obtain a saddlepoint approximation for the small sample distribution of several variogram estimators such as the classical Matheron's estimator, some M-estimators like the robust Huber's variogram estimator, and also the α-trimmed variogram estimator. The tail probability approximation obtained is very accurate even for small sample sizes. In the approximations we consider that the observations follow a distribution close to the normal, specifically, a scale contaminated normal model. To obtain the approximations we transform the original observations into a new ones, which can be considered independent if a linearized variogram can be accepted as model for them. To check this, a goodness of fit test for a variogram model is defined in the last part of the paper.

A common problem in geostatistics is variogram estimation, in order to choose an acceptable model for kriging. Nevertheless, there is no standard method, first, to test if a particular model can be accepted as valid and, second, to choose among several competing variogram models. The problem is even more complex if, in addition, there are outliers in the data. In this paper we propose to use the distribution of some classical and robust variogram estimators to test, first, the validity of a particular model, accepting it if the p-value of the test, with this particular model as null hypothesis, is large enough and, second, to compare several competing models, choosing the model with the largest p-value among several acceptable models.