SUMMARYPorous media have properties with heterogeneities on several length scales. It is possible to build digital models of such properties. However these can be so detailed that a computing machine of the same power as that used to build the property model is not able to solve the uid ow equations using standard discretisation methods-storage is needed for workspace, and the discrete equations have to be solved in a reasonable time. This paper reviews averaging techniques, devised to simulate large scale features of solutions without necessarily solving all the ÿne scale equations.

The bacterial core genome is of intense interest and the volume of whole genome sequence data in the public domain available to investigate it has increased dramatically. The aim of our study was to develop a model to estimate the bacterial core genome from next-generation whole genome sequencing data and use this model to identify novel genes associated with important biological functions. Five bacterial datasets were analysed, comprising 2096 genomes in total. We developed a Bayesian decision model to estimate the number of core genes, calculated pairwise evolutionary distances (p-distances) based on nucleotide sequence diversity, and plotted the median p-distance for each core gene relative to its genome location. We designed visually-informative genome diagrams to depict areas of interest in genomes. Case studies demonstrated how the model could identify areas for further study, e.g. 25% of the core genes with higher sequence diversity in the Campylobacter jejuni and Neisseria meningitidis genomes encoded hypothetical proteins. The core gene with the highest p-distance value in C. jejuni was annotated in the reference genome as a putative hydrolase, but further work revealed that it shared sequence homology with beta-lactamase/metallo-beta-lactamases (enzymes that provide resistance to a range of broad-spectrum antibiotics) and thioredoxin reductase genes (which reduce oxidative stress and are essential for DNA replication) in other C. jejuni genomes. Our Bayesian model of estimating the core genome is principled, easy to use and can be applied to large genome datasets. This study also highlighted the lack of knowledge currently available for many core genes in bacterial genomes of significant global public health importance.

This paper proposes a hierarchical nonlinear approximation scheme for scalar-valued multivariate functions, where the main objective is to obtain an accurate approximation with using only very few function evaluations. To this end, our iterative method combines at any refinement step the selection of suitable evaluation points with kriging, a standard method for statistical data analysis. Particular improvements over previous non-hierarchical methods are mainly concerning the construction of new evaluation points at run time. In this construction process, referred to as experimental design, a flexible two-stage method is employed, where adaptive domain refinement is combined with sequential experimental design. The hierarchical method is applied to statistical data analysis, where the data is generated by a very complex and computationally expensive computer model, called a simulator. In this application, a fast and accurate statistical approximation, called an emulator, is required as a cheap surrogate of the expensive simulator. The construction of the emulator relies on computer experiments using a very small set of carefully selected input configurations for the simulator runs. The hierarchical method proposed in this paper is, for various analysed models from reservoir forecasting, more efficient than existing standard methods. This is supported by numerical results, which show that our hierarchical method is, at comparable computational costs, up to ten times more accurate than traditional non-hierarchical methods, as utilized in commercial software relying on the response surface methodology (RSM).

The vulnerability of water supplies to shortage depends on the complex interplay between streamflow variability and the management and demands of the water system. Assessments of water supply vulnerability to potential changes in streamflow require methods capable of generating a wide range of possible streamflow sequences. This paper presents a method to generate synthetic monthly streamflow sequences that reproduce the statistics of the historical record and that can express climate-induced changes in user-specified streamflow characteristics. The streamflow sequences are numerically simulated through random sampling from a parametric or a nonparametric distribution fitted to the historical data while shuffling the values in the time series until a sequence matching a set of desired temporal properties is generated. The desired properties are specified in an objective function which is optimized using simulated annealing. The properties in the objective function can be manipulated to generate streamflow sequences that exhibit climate-induced changes in streamflow characteristics such as interannual variability or persistence. The method is applied to monthly streamflow data from the Thames River at Kingston (UK) to generate sequences that reproduce historical streamflow statistics at the monthly and annual time scales and to generate perturbed synthetic sequences expressing changes in short-term persistence and interannual variability.

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Summary.The main mathematical techniques used in building geological models for input to fluid flow simulation are reviewed. The subject matter concerns the entire geological and reservoir simulation modelling workflow relating to the subsurface. To provide a realistic illustration of a complete fluid flow model, a short outline of two-phase incompressible flow through porous media is given. The mathematics of model building is discussed in a context of seismic acquisition, processing and interpretation, well logging and geology. Grid generation, geometric modelling and spatial statistics are covered in considerable detail. A few new results in the area of geostatistics are proved. In particular the equivalence of radial basis functions, general forms of kriging and minimum curvature methods is shown. A Bayesian formulation of uncertainty assessment is outlined. The theory of inverse problems is discussed in a general way, from both deterministic and statistical points of view. There is a brief discussion of upscaling. A case for multiscale geological modelling is made and the outstanding research problems to be solved in building multiscale models from many types of data are discussed.

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