The prediction of physicochemical rock properties in subsurface models regularly suffers from uncertainty observed at the submeter scale. Although at this scale – which is commonly termed the lithofacies scale – the physicochemical variability plays a critical role for various types of subsurface utilization, its dependence on syndepositional and postdepositional processes is still subject to investigation.

The impact of syndepositional and postdepositional geological processes, including depositional dynamics, diagenetic compaction and chemical mass transfer, onto the spatial distribution of physicochemical properties in siliciclastic media at the lithofacies scale is investigated in this study. We propose a new workflow using two cubic rock samples where eight representative geochemical, thermophysical, elastic and hydraulic properties are measured on the cubes' faces and on samples taken from the inside. The scalar fields of the properties are then constructed by means of spatial interpolation. The rock cubes represent the structurally most homogeneous and most heterogeneous lithofacies types observed in a Permian lacustrine-deltaic formation that deposited in an intermontane basin. The spatiotemporal controlling factors are identified by exploratory data analysis and geostatistical modeling in combination with thin section and environmental scanning electron microscopy analyses.

Sedimentary structures are well preserved in the spatial patterns of the negatively correlated permeability and mass fraction of

The multifarious patterns observed in this study emphasize the importance of high-resolution sampling in order to properly model the variability present in a lithofacies-scale system. Following this, the physicochemical variability observed at the lithofacies scale might nearly cover the global variability in a formation. Hence, if the local variability is not considered in full-field projects – where the sampling density is usually low – statistical correlations and, thus, conclusions about causal relationships among physicochemical properties might be feigned inadvertently.

The utilization of the subsurface in disciplines such as petroleum reservoir engineering, geothermal heat extraction, mining, carbon capture and storage or nuclear waste disposal requires highly accurate spatial predictions of relevant physical or geochemical properties in order to assess the economic feasibility of a target region

In sedimentary bodies, the spatial distribution of the properties is mainly controlled by depositional and diagenetic processes

With proper knowledge of the statistical and causal relationships among physicochemical rock properties at different scales, prognostic property models can be significantly enhanced by the integration of small-scale uncertainty into upscaling or conditional simulation algorithms

In order to quantify the spatial variability and the multidimensional relationships among physicochemical properties at the 3D lithofacies scale, the quasi-continuous scalar fields of two rock cubes are modeled by means of spatial interpolation, which is constrained by laboratory measurements. The rock cubes have volumes of 0.0156 and 0.008

The lithological characteristics of the sandstones are analyzed, and both isotropic and anisotropic properties, including bulk rock geochemistry, thermophysical, hydraulic and elastic rock properties, are measured on the cubes' faces. In addition, the intrinsic gas permeability under an infinite pressure gradient, the effective porosity, the elemental composition, the thermal conductivity, and the thermal diffusivity together with the P-wave and S-wave velocity are measured on 108 rock cylinders taken from the inside of the cubes representative for each Cartesian direction in order to account for anisotropic patterns.

The measurements are used to interpolate the full 3D field of each property. Prior to interpolation, the discrete measurements are statistically analyzed for correlation and formal relationships. Interpolations are conducted using deterministic and geostatistical methods, including the inverse distance weighting (IDW) and simple kriging (SK) interpolation. The models are evaluated through cross validation, and the observed spatial patterns are categorized. The interpolation results providing the lowest cross validation error are statistically analyzed again and compared with the aforementioned statistical patterns. Eventually, the geological processes, which produced the observed patterns, are interpreted and discussed with the help of qualitative thin section and environmental scanning electron microscope (ESEM) analyses.

The research outputs of this study lie between the scale of a core plug measurement and a wireline log and/or pumping test

In order to cover multiple varieties of sedimentary lithofacies types, a quarry in Obersulzbach (Rhineland-Palatinate, Germany) in the Saar–Nahe basin was selected for the investigations (Fig.

Two rock cubes of

Lateral faces of OSB1_c displayed in the form of an open cube (from left to right:

First, a local metric coordinate system was defined, where each edge of the cube represents an axis in the Cartesian coordinate system in order to reference each measurement to a point in space. The sampling points were set in a raster of

After the extraction, the rock cylinders were ovendried at 105

The permeability was measured with the Hassler cell permeameter, which is described in

The experimental semivariogram represents the cumulative dissimilarity of a discrete set of point pairs

Spatial inter- and extrapolation can be generated with deterministic and geostatistical techniques. All interpolations are based on the assumption that a point

For deterministic interpolation, the

For geostatistical interpolation, simple kriging (SK) is used. Kriging in general is a popular technique for interpolating geological properties in space

Cross validation can be used to assess the quality of a model. During cross validation,

Anisotropy describes the dependence of a physical property on a direction. Rock properties such as stiffness, permeability or thermal conductivity are anisotropic in most cases. Hence, measurements of those properties might show differing magnitudes in different directions if the medium is polar anisotropic. The intrinsic permeability, for example, provides typical ranges for the ratio between the vertical (

In the following, we will provide an exemplary description of the anisotropy of elasticity, and we will provide measures for anisotropy quantification under the simplifying assumption of transverse isotropy. The elastic modulus tensor can be expressed as a fourth rank tensor as follows:

In order to quantify the linear statistical relationship between two independent variables

Regression aims at finding a fitting function between samples of two or more random variables. For curvilinear regression, a function of a degree

The spatial dependence of the discrete values is evaluated through experimental semivariograms. The semivariograms are generated for the single rock faces, where measurements are available, and for the plug measurements. The empirical semivariogram is fitted to a variogram model, which is then used for the geostatistical interpolation. Interpolation analyses are performed as IDW and SK realizations (Fig.

We decided to waive sequential simulation as large amounts of the cubes' volumes are covered by rock samples. Thus, we do not expect a relevant kriging variance. With this in mind, the simulations are assumed to capture most of the total variance from the measurements themselves. The interpolation results that provide the lowest cross validation error are used for statistical analyses in order to derive correlations and regression functions between the scalar fields. Eventually, significant correlations are compared with the noninterpolated data sets. Both the spatial modeling and the statistical analyses are performed with the open source software called Geological Reservoir Virtualization

The sandstones belong to a clinothem strata deposited in a fluvial-dominated lacustrine delta. More specifically, the architectural element represents a distributary mouth bar formed by rapid sandstone deposition in sheet-like bodies, as described in

Petrographic classification after

The average grain size in both cubes ranges from fine to very coarse sand (200–1400

The original rigid detrital components consist of 50 %–60 % quartz, 20 %–30 % strongly altered feldspar and 10 %–25 % lithic fragments. Mica grains are often bent between more rigid grains. The rock matrix accounts for approximately 10 %–20 % and is built up by detrital grains, coated by iron oxides, ductile, autochthonous pelite grains and fine-grained quartz. According to the geochemical analyses, the rocks can be classified as lithic arenites to arkoses or wackes (Fig.

Thin section analysis (Fig.

In order to provide full comparability, the following section will provide an overview of the measurements derived from the rock cylinder analyses. For each property, 79 rock samples from OSB1_c and 29 from OSB2_c were investigated. An overview of the rock properties' ranges is provided in the box and whisker charts shown in Fig.

Box and whisker charts showing the empirical distribution of the rock properties measured in the rock cylinders taken from the rock cubes. Outliers were detected according to Tukey's method

The local variability of OSB1_c is significantly higher than that of OSB2_c. The intrinsic permeability of OSB1_c provides a coefficient of variation of 0.3 and a Dykstra–Parsons coefficient of 0.4, while measurements from OSB2_c show values of 0.2 for the coefficient of variation and 0.18 for the Dykstra–Parsons coefficient, respectively. According to the classification provided by

The range of values in OSB1_c for each property is greater than the range of those in OSB2_c. OSB1_c provides lower values in P-wave and S-wave velocity, thermal conductivity and mass fraction of

Statistically significant linear correlations (Fig.

Matrix visualization of the Pearson correlation coefficient derived from the plug measurements. Statistically significant correlations with a

The spatial dependence of the discrete measurements is estimated using experimental semivariograms. Therefore, the geochemical representatives

Empirical semivariograms of the mass fraction of

The spatial distributions of the rock properties are interpolated with Shepard's inverse distance weighting (IDW) and simple kriging (SK). Both realizations of a single scalar field provide comparable patterns, which is due to the high sampling density. The interpolation errors are also located in similar ranges; however, IDW seems to be more sensitive to outliers, resulting in much higher interpolation errors with regard to properties like P-wave velocity or mass fraction of

The rock properties exhibit a multitude of spatial patterns. Here, discrete, layered and homogeneous patterns, both connected and disconnected to primary sedimentary structures, could be observed in the interpolations.

RMSE and MAE for the interpolation results of IDW and SK for OSB1_c.

A bedding-connected pattern is exhibited in the intrinsic permeability and

The bedding structures in OSB1_c are well reflected by the spatial pattern of the interpolated intrinsic permeability gradually increasing from low values, between 0.7 and 2

Spatial distribution of the intrinsic permeability modeled with a simple kriging interpolation. The histogram shows a bimodality of the distribution split up into the basal beds and uppermost beds.

The spatial distribution of the mass fraction of

Spatial distribution of the mass fraction of

Other scalar fields are decoupled from depositional bounding surfaces. For instance, the geochemical mass fractions of

Spatial distribution of the mass fraction of

Spatial distribution of the mass fraction of

Higher fractions of

The overall aim of this study was to quantify the 3D interdependencies of thermophysical, hydraulic, elastic and geochemical scalar fields in sandstone media at the lithofacies scale and to identify the controlling factors for the property distributions. With a high-resolution study at the lithofacies scale, statistical and spatial interrelationships between characteristic physicochemical fields could be discovered and traced back to depositional and diagenetic processes.

Recent multiscale modeling approaches without the use of local constraints show that the prediction of permeability and porosity in siliciclastic systems is still challenging

Although statistically significant correlations may imply a natural relationship between physicochemical properties, this relationship could also be based on random processes requiring causality to be verified. Weak correlations were found between the effective porosity and the intrinsic permeability, which are usually positively correlated

The linear correlation analysis revealed a significant negative relationship between hydraulic and geochemical properties that fits to a polynomial regression (Fig.

Geochemical analyses, in contrast to petrographic ones, limit the interpretations of geological processes as mineral phases can only be assumed and not determined for certain. A high mass fraction of

Regression analysis of the relationship between intrinsic permeability and mass fraction of

A significant correlation between

Significant nonintuitive relationships between the physical and geochemical scalar fields at the lithofacies scale have been revealed with a deductive approach of spatial field modeling and statistical data analysis. All in all, the following conclusions can be drawn from this study:

As specific properties such as the mass fraction of

This study demonstrates that the observation of bedding structures does not necessarily indicate a stronger polar anisotropy compared to macroscopically unstructured lithologies. Here, the microscopic characteristics, like the amount of secondary porosity, might play a more important role in the attenuation of physical waves than the bounding surfaces.

It could be shown that hydraulic properties are dependent on the intergranular matrix and cement amount, which are in turn controlled by depositional processes and eogenetic precipitates. Those findings are not new (see

We demonstrate that the strength of statistical correlation can be preserved in spatial interpolations as long as the sampling density is sufficient. If the sampling density is too low, a statistical correlation might be inadvertently feigned.

As shown in this study, the local geological variability should not be underestimated as an uncertainty factor in spatial predictions and upscaling procedures. In fact, the local geological variability of physicochemical properties might nearly cover the variability being present in an entire formation. Therefore, a high-resolution analysis of physicochemical rock properties can assist in assessing the uncertainty of field-scale property models which is induced by the local geological variability at the lithofacies scale.

GeoReVi is an open source software for Windows systems available at

The investigated rock samples are available at the Institute of Applied Geosciences, TU Darmstadt, and can be requested from linsel@geo.tu-darmstadt.de. Also, the samples are registered in the System for Earth Sample Registration (SESAR;

AL conceptualized and prepared the paper. AL and SW conducted the laboratory and field measurements. JH contributed to the conceptualization of the study. MH was the overall supervisor of the study.

The authors declare that they have no conflict of interest.

The authors are grateful for the permission to work in the sandstone quarry of Konrad Müller GmbH in Obersulzbach, Germany. Also, we would like to thank Reimund Rosmann and Institut IWAR (Technische Universität Darmstadt, Germany) for the preparation of the rock cubes. We are extremely thankful to Mattia Pizzati and Giacomo Medici for their time and effort in putting together constructive reviews. Adrian Linsel has received financial support from the Friedrich-Ebert-Stiftung, Germany, which is gratefully acknowledged.

This research has been supported by the Friedrich-Ebert-Stiftung, Germany.

This paper was edited by Kei Ogata and reviewed by Giacomo Medici and Mattia Pizzati.