3D Kriging Research Articles

Subsurface temperature maps in French sedimentary basins: new data compilation and interpolation

Abstract Assessment of the underground geothermal potential requires the knowledge of deep temperatures (1–5 km). Here, we present new temperature maps obtained from oil boreholes in the French sedimentary basins. Because of their origin, the data need to be corrected, and their local character necessitates spatial interpolation. Previous maps were obtained in the 1970s using empirical corrections and manual interpolation. In this study, we update the number of measurements by using values collected during the last thirty years, correct the temperatures for transient perturbations and carry out statistical analyses before modelling the 3D distribution of temperatures. This dataset provides 977 temperatures corrected for transient perturbations in 593 boreholes located in the French sedimentary basins. An average temperature gradient of 30.6°C/km is obtained for a representative surface temperature of 10°C. When surface temperature is not accounted for, deep measurements are best fitted with a temperature gradient of 25.7°C/km. We perform a geostatistical analysis on a residual temperature dataset (using a drift of 25.7°C/km) to constrain the 3D interpolation kriging procedure with horizontal and vertical models of variograms. The interpolated residual temperatures are added to the country-scale averaged drift in order to get a three dimensional thermal structure of the French sedimentary basins. The 3D thermal block enables us to extract isothermal surfaces and 2D sections (iso-depth maps and iso-longitude cross-sections). A number of anomalies with a limited depth and spatial extension have been identified, from shallow in the Rhine graben and Aquitanian basin, to deep in the Provence basin. Some of these anomalies (Paris basin, Alsace, south of the Provence basin) may be partly related to thick insulating sediments, while for some others (southwestern Aquitanian basin, part of the Provence basin) large-scale fluid circulation may explain superimposed cold and warm anomalies.

Development of a three-dimensional model of sedimentary texture in valley-fill deposits of Central Valley, California, USA

A three-dimensional (3D) texture model was developed to help characterize the aquifer system of Central Valley, California (USA), for a groundwater flow model. The 52,000-km2 Central Valley aquifer system consists of heterogeneous valley-fill deposits. The texture model was developed by compiling and analyzing approximately 8,500 drillers’ logs, describing lithologies up to 950 m below land surface. The lithologic descriptions on the logs were simplified into a binary classification of coarse- and fine-grained. The percentage of coarse-grained sediment, or texture, was then computed for each 15-m depth interval. The model was developed by 3D kriging of the percentage of coarse-grained deposits onto a 1.6-km spatial grid at 15-m depth intervals from land surface down to 700 m below land surface. The texture model reflects the known regional, spatial, and vertical heterogeneity in the aquifer system. The texture model correlates to sediment source areas, independently mapped geomorphic provinces, and factors affecting the development of alluvial fans, thus demonstrating the utility of using tcdrillers’ logs as a source of lithologic information. The texture model is upscaled to a layered groundwater flow model for use in defining the hydraulic properties of the aquifer system.

Hydraulic conductivity heterogeneity of a local deltaic aquifer system from the kriged 3D distribution of hydrofacies from borehole logs, Valcartier, Canada

Summary The deltaic aquifer system of the Valcartier sector in Quebec, Canada, is part of a quaternary valley fill contaminated by dissolved trichloroethene (TCE). The objective of our study is to define the aquifer system heterogeneity that should influence TCE transport. Heterogeneity is defined by the distribution of both hydrofacies and hydraulic conductivity ( K ). Hydrofacies are defined as lithologic facies with distinctive hydraulic conductivity ranges. Our approach was developed to take advantage of the abundant stratigraphic and lithologic data provided by borehole logs (7000 m logged from 430 locations). Four site-specific deltaic hydrofacies were defined on the basis of lithologic descriptions, supported by data from grain size analyses, slug tests and cone penetration tests. Each hydrofacies includes a group of geologic facies found in borehole log descriptors to which an initial mean horizontal conductivity K H is associated based on slug tests. Borehole logs were converted to hydrofacies proportions over 5-m intervals to provide 1350 data points. The spatial distribution of hydrofacies was interpolated by three successive interconnected 3D kriging steps using the new technique of “imbricated kriging”. Global dual kriging is directly carried out on the 3D grid of a numerical model. Finally, the proportions of hydrofacies were used to estimate horizontal ( K H ) and vertical ( K V ) hydraulic conductivity fields using generalized means for layered media. Final K H values assigned to the hydrofacies are calibrated by comparison with 2D trends in K H shown by slug tests. This approach also provides an estimated vertical K V field with a spatially varying proportion to K H , rather than a fixed anisotropy ratio. Imbricated kriging does not preserve the statistical variability of fine scale hydrofacies distribution representative of geological variability. However, the approach provides K H and K V estimates over a 3D numerical grid that are coherent with hydrofacies distribution, which in this case control K H and K V .

Mapping Three‐Dimensional Water‐Quality Data in the Chesapeake Bay Using Geostatistics1

Abstract: Data interpretation and visualization software tools with geostatistical capabilities were adapted, customized, and tested to assist the Chesapeake Bay Program in improving its water‐quality modeling protocols. Tools were required to interpolate, map, and visualize three‐dimensional (3D) water‐quality data, with the capability to determine estimation errors. Components of the software, originally developed for ground‐water modeling, were customized for application in estuaries. Additional software components were developed for retrieval, and for pre‐ and post‐ processing of data. The Chesapeake Bay Program uses the 3D mapped data for input to the Bay water‐quality model that projects the future health of the Bay and its tidal tributary system. In determining water‐quality attainment criteria, 3D kriging estimation errors are needed as a statistical measure of uncertainty. Furthermore, given the high cost of installing and operating new monitoring stations, geostatistical techniques can assist the Chesapeake Bay Program in the identification of suitable data collection locations. Following the evaluation, selection, and development of the software components phase, 3D ordinary kriging techniques with directional semi‐variograms to account for anisotropy were successfully demonstrated for mapping 3D fixed station water‐quality data, such as dissolved oxygen and salinity. Additionally, an improved delineation tool was implemented to simulate the upper and lower pycnocline boundary surfaces allowing the segregation of the interpolated 3D data into three separate zones for a better characterization of the pycnocline layer.

3D interpolation of grain size distributions in the upper 5 m of the channel bed of three lower Rhine distributaries

This article compares two methods to interpolate grain size distributions to a 3D model (grid-size 25 × 25 × 0.2 m) of the upper 5 m of the riverbed of three lower Rhine distributaries in The Netherlands. These models have been used as input for morphological predictions of the effects of maintenance and restructuring plans of the riverbed. Both methods use field data from seismic surveys and vibrocore borings to construct a 2.5 D layer model. Full grain size distributions within grid cells of 25 × 25 × 0.2 m have been obtained by interpolation within layers, using 3D Kriging, thereby analyzing spatial variability over different scales. In the first method measured grain size distributions have been interpolated. The second method uses measured grain size distributions and synthetic grain size distributions from estimated soil sample parameters (silt content, gravel content, sand median). The combination of synthetic and measured GSD strongly enhances the spatial variation in the interpolation result and prevents a misrepresentation of lithologies in the model.

Conditional 3D simulation of lithofacies with 2D seismic data

This paper presents a new methodology and the corresponding GSLIB-type program to integrate 2D average information, such as vertically-averaged lithofacies proportions, into estimation/simulation of 3D lithofacies distributions. Seismic information, available in 2D, is used in the cokriging of the 2D average lithofacies proportions. A data set related to a siltstonedolomite carbonate reservoir in W. Texas is used for case study. Three steps were taken: first, a co-located cokriging is used to incorporate 2D seismic information into the estimation of vertical lithofacies proportions; second, indicator kriging is used to derive the facies conditional probabilities at each of the 3D simulation grid node; this 3D indicator kriging uses the hard well data and the previously estimated vertical proportions; last, a p-field simulation algorithm is used to draw simulated lithofacies indicators from the previously obtained distributions. Results showed that even with limited well data, the input vertical lithofacies proportions (which carry the seismic information) is honored quasi-exactly.

Spatial interpolation of soil salinity and sodicity for a saline soil in Southern Alberta

Soil salinity and sodicity data are usually represented as contour maps based on spatially interpolated data. This study was conducted to assess two commonly used spatial interpolation methods (moving average and weighted moving average) and three forms of two-dimensional (2D) and three-dimensional (3D) kriging. Soil was sampled from two depths on a 25-m grid at 40 sites in a saline and sodic soil of Southern Alberta. Samples were analyzed for electrical conductivity (ECe) and sodium absorption ratio (SARe) of saturation paste extracts. Methods were compared by cross-validation using the mean absolute difference (MAD) between estimated and observed values. Values of MAD resulting from 3D kriging were inferior to those obtained from moving averages. Moving average methods were very sensitive to the number of neighbors used in the estimation. For kriging methods, however, precision remained stable for more than about 10–12 neighbors. Three-dimensional ordinary kriging was the most precise and robust interpolator with a MAD value of 9.74 dS m−1 for ECe. For SARe the lowest MAD value of 10.55 was obtained with 3D ordinary co-kriging. Data representation produced by 3D kriging allowed easy identification of ECe spatial structure. Key words: Spatial interpolation, kriging, moving average, soil, salinity, sodicity