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Numerous studies have demonstrated the capability of supervised deep learning techniques for predicting geological features of interest from seismic sections, including features that are difficult to identify using traditional interpretation methods. However, the successful application of these techniques in practice has been limited by the difficulty of obtaining a large training data set where the seismic data and corresponding ground truth labels are well-defined. Manually creating large amounts of labels requires a heavy workload, and the uncertainty of the interpretation and labeling process decreases the model’s ability for making accurate predictions. Using the chalk-flint sequence scenario onshore Denmark as an example, we have developed a novel workflow for predicting subresolution thin layers from seismic sections. It entails generating large quantities of synthetic training data with high-quality labels using stochastic geological modeling, training a convolutional neural network based on the synthetic data set, and applying it to real seismic data. This is, to our knowledge, the first example of using deep learning to predict subresolution thin layers from seismic data based on geostatistically generated training images. It is shown that a neural network trained on synthetic data can predict a realistic number of subresolution flint layers from the real seismic data that have been collected from the Stevns region in Denmark, which has value for the understanding of the overall geological characteristics of succession and engineering applications such as construction site evaluation.

Seismic facies classification aims to predict a facies model, or a set of facies models, from measured seismic data. We focus on stochastic classification methods to estimate the probability distribution of facies conditioned on seismic data. Bayesian classification methods based on analytical solutions are generally applied to seismically inverted elastic attributes or petrophysical properties rather than measured seismic data, and they do not include spatial correlation models in the prior distribution. Instead, iterative stochastic methods, such as Monte Carlo acceptance-rejection sampling and approximate Bayesian computation, can be directly applied to seismic data and can be implemented with complex prior models; however, their convergence is generally slow because it requires the simulation of a large number of facies models from the prior distribution. We develop an efficient implementation of a Markov chain Monte Carlo (MCMC) approach that adopts complex prior models, such as multiple-point statistics simulations based on a training image, to generate geologically realistic facies realizations. The novelty of the approach is that the proposal distribution of the proposed MCMC method is based on a gradual deformation algorithm, in which the proposal distribution is expressed as a linear combination of the indicator variables of the previously accepted model and a random prior simulation, according to a uniform random weight. The synthetic examples indicate accurate facies predictions with a success rate of approximately 80% for well-log facies classification based on elastic attributes and approximately 60% for seismic facies classification based on seismic data. The inversion is compared to an MCMC method with prior models sampled from a first-order Markov chain and Bayesian facies classification.

Shear wave velocity information is valuable in many aspects of seismic exploration and characterization of reservoirs. However, shear wave logs are not always available in the interval of interest due to cost and time-saving purposes. In this study, we present a tailored supervised learning approach to estimate shear wave velocity from well-log measurements in the Lower Cretaceous succession of the Valdemar and Boje fields in the Danish North Sea. Our objective is to investigate the performance of four supervised learning regression models (linear, random forest, support vector and multi-layer perceptron). A limited well-log data set from six wells is used for training and testing the supervised learning models. A set of well data containing normalized gamma ray, compressional wave velocity, neutron porosity and medium resistivity logs gave reasonable shear wave velocity estimates in the test wells with root-mean-square error scores within the range of other published studies. Based on limited input data and complex geology, the multi-layer perceptron was the most successful model in predicting the reservoir sections of the test wells. However, all models lacked stability in the overburden zones. Lastly, re-training the multi-layer perceptron on the six wells to predict missing shear wave velocity in a nearby well showed promising results for further reservoir characterization. The obtained results can yield useful input into, for example, seismic pre-stack inversion, amplitude versus offset analysis and rock physics analysis.

Probabilistic formulations of inverse problems are most often based on Bayes Rule, which is considered a powerful tool for integration of data information and prior information about potential solutions. However, since its introduction it has become apparent that the Bayesian inference paradigm presents a number of difficulties, especially in the phase where the problem is mathematically formulated. Perhaps the most notable difficulty arises because Bayes Theorem is usually formulated as a relation between probability densities on continuous manifolds. This creates an acute crisis because of a problem described by the French mathematician Joseph Bertrand (1889), and later investigated by Kolmogorov and Borel. According to Kolmogorov's (1933/1956) investigations, conditioning of a probability density is underdetermined: In different parameterizations (reference frames), conditional probability densities express different probability distributions. Surprisingly, this problem is persistently neglected in the scientific literature, not least in applications of Bayesian inversion. We will explore this problem and show that it is a serious threat to the objectivity and quality of Bayesian computations including Bayesian inversion, computation of Bayes Factors, and trans-dimensional inversion. Another difficulty in Bayesian Inference methods derives from the fact that data uncertainties, and prior information on the unknown parameters, are often unknown or poorly known. Because they are required in the calculations, statisticians have inventedhierarchical methods to compute parameters (known as hyper-parameters) controlling these uncertainties. However, since both the data uncertainties and the prior information on the unknowns are supposed to be known 'a priori', but are calculated 'a posteriori', this creates another crisis, namely a violation of causality. We will take a close look at the consequences of this mixing of 'prior' and 'posterior', and show how it potentially jeopardizes the validity of Bayesian computations.

SUMMARY Markov chain Monte Carlo (MCMC) sampling of solutions to large-scale inverse problems is, by many, regarded as being unfeasible due to the large number of model parameters. This statement, however, is only true if arbitrary, local proposal distributions are used. If we instead use a global proposal, informed by the physics of the problem, we may dramatically improve the performance of MCMC and even solve highly nonlinear inverse problems with vast model spaces. We illustrate this by a seismic full-waveform inverse problem in the acoustic approximation, involving close to 106 parameters. The improved performance is mainly seen as a dramatic shortening of the burn-in time (the time used to reach at least local equilibrium), but also the algorithm’s ability to explore high-probability regions (through more accepted perturbations) is potentially better. The sampling distribution of the algorithm asymptotically converges to the posterior probability distribution, but as with all other inverse methods used to solve highly nonlinear inverse problems we have no guarantee that we have seen all high-probability solutions in a finite number of iterations. On the other hand, with the proposed method it is possible to sample more high-probability solutions in a shorter time, without sacrificing asymptotic convergence. This may be a practical advantage for problems with many parameters and computer-intensive forward calculations.

PurposeWe present a probabilistic approach to medical image analysis that requires, and makes use of, explicit prior information provided by a medical expert. Depending on the choice of prior model the method can be used for image enhancement, analysis, and segmentation.MethodsThe methodology is based on a probabilistic approach to medical image analysis, that allows integration of 1) arbitrarily complex prior information (for which realizations can be generated), 2) information about a convolution operator of the imaging system, and 3) information about the noise in the reconstructed image into a posterior probability density. The method was demonstrated on positron emission tomography (PET) images obtained from a phantom and a patient with lung cancer. The likelihood model (multivariate log-normal) and the convolution operator were derived from phantom data. Two examples of prior information were used to show the potential of the method. The extended Metropolis-Hastings algorithm, a Markov chain Monte Carlo method, was used to generate realizations of the posterior distribution of the tracer activity concentration.ResultsA set of realizations from the posterior was used as the base of a quantitative PET image analysis. The mean and variance of activity concentrations were computed, as well as the probability of high tracer uptake and statistics on the size and activity concentration of high uptake regions. For both phantom and in vivo images, the estimated images of mean activity concentrations appeared to have reduced noise levels, and a sharper outline of high activity regions, as compared to the original PET. The estimated variance of activity concentrations was high at the edges of high activity regions.ConclusionsThe methodology provides a probabilistic approach for medical image analysis that explicitly takes into account medical expert knowledge as prior information. The presented first results indicate the potential of the method to improve the detection of small lesions. The methodology allows for a probabilistic measure of the size and activity level of high uptake regions, with possible long-term perspectives for early detection of cancer, as well as treatment, planning, and follow-up.

Numerous studies have demonstrated the capability of supervised deep learning techniques for predicting geological features of interest from seismic sections, including features that are difficult to identify using traditional interpretation methods. However, successful application of these techniques in practice has been limited by the difficulty of obtaining large training dataset where seismic data and corresponding ground truth labels are well defined. Manually creating large amounts of labels requires a heavy workload, and the uncertainty of the interpretation and labeling process decreases the model’s ability for making accurate predictions. Using the chalk-flint sequence scenario onshore Denmark as an example, we present a novel workflow of generating large quantities of synthetic training data with high-quality labels using stochastic geological modelling, and we investigate the capability of a synthetic data-trained convolutional neural network for predicting sub-resolution thin layers from seismic sections. It is shown that a neural network trained on synthetic data can predict a realistic number of sub-resolution flint layers from real seismic data that have been collected from the Stevns region in Denmark, which has value for understanding of overall geological characteristics of the succession and engineering applications such as construction site evaluation.

Stochastic petrophysical inversion is a method used to predict reservoir properties from seismic data. Recent advances in stochastic optimization allow generating multiple realizations of rock and fluid properties conditioned on seismic data. To match the measured data and represent the uncertainty of the model variables, many realizations are generally required. Stochastic sampling and optimization of spatially correlated models are computationally demanding. Monte Carlo methods allow quantifying the uncertainty of the model variables but are impractical for high-dimensional models with spatially correlated variables. We have developed a Bayesian approach based on an efficient implementation of the Markov chain Monte Carlo (MCMC) method for the inversion of seismic data for the prediction of reservoir properties. Our Bayesian approach includes an explicit vertical correlation model in the proposal distribution. It is applied trace by trace, and the lateral continuity model is imposed by using the previously simulated values at the adjacent traces as conditioning data for simulating the initial model at the current trace. The methodology is first presented for a 1D problem to test the vertical correlation, and it is extended to 2D problems by including the lateral correlation and comparing two novel implementations based on sequential sampling. Our method is applied to synthetic data to estimate the posterior distribution of the petrophysical properties conditioned on the measured seismic data. The results are compared with an MCMC implementation without lateral correlation and demonstrate the advantage of integrating a spatial correlation model.

An optimization problem is proposed for inferring physical properties of polycrystals given ultrasonic (elastic) wave velocity measurements, made across multiple sample orientations. The feasibility of the method is demonstrated by inferring both the effective grain elastic parameters and the grain c -axis orientation distribution function (ODF) of ice-core samples from Priestley glacier, Antarctica. The method relies on expanding the ODF in terms of a spherical harmonic series, which allows for a non-parametric estimation of the sample ODF. Moreover, any linear combination of the Voigt (strain) and Reuss (stress) homogenization scheme is allowed, although for glacier ice, the exact choice is found to matter little for bulk elastic behaviour, and thus for inferred physical properties, too. Finally, the accuracy of the inferred grain elastic parameters is discussed, including the well-posedness and shortcomings of the inverse problem, relevant for future adoptions in glaciology, geology and elsewhere.

Mapping landforms on the Moon is of great interest and importance for future human settlements and resources exploration. One of the first steps is to map the topography in great detail and resolution. However, data from the Lunar Orbiter Laser Altimeter (LOLA) provide low-resolution elevation maps in comparison to the size of detailed geological features. To improve resolution, we developed a new method to upscale topographic maps to a higher resolution using images from the Lunar Reconnaissance Orbiter Camera (LROC). Our method exploits the relation between topographic gradients and degrees of shading of incoming sunlight. In contrast to earlier published methods, our approach is based on probabilistic, linear inverse theory, and its computational efficiency is very high due to its formulation through the Sylvester Equation. The method operates on multiple images and incorporates albedo variations. A further advantage of the method is that we avoid/reduce the use of arbitrary tuning parameters through a probabilistic formulation where all weighting of data and model parameters is based on prior information about data uncertainties and reasonable bounds on the model. Our results increase the resolution of the topography from -60 m per pixel to 0.9 m per pixel, bringing it to the same pixel resolution as the optical images from LROC, allowing in some cases detection of craters as small as -3 m of diameter. We estimate uncertainties of the topographic model due to noise in the images, and in the low-resolution (LOLA) model.

The mapping of faults provides essential information on many aspects of seismic exploration, characterisation of reservoirs for compartmentalisation and cap-rock integrity. However, manual interpretation of faults from seismic data is time-consuming and challenging due to limited resolution and seismic noise. In this study, we apply a convolutional neural network trained on synthetic seismic data with planar fault shapes to improve fault mapping in the Lower and Upper Cretaceous sections of the Valdemar Field in the Danish North Sea. Our objective is to evaluate the performance of the neural network model on post-stack seismic data from the Valdemar Field. Comparison with variance and ant-tracking attributes and a manual fault interpretation shows that the neural network predicts faults with more details that may improve the overall geological and tectonic understanding of the study area and add information on potential compartmentalisation that was previously overlooked. However, the neural network is sensitive to seismic noise, which can distort the fault predictions. Therefore, the proposed model should be treated as an additional fault interpretation tool. Nonetheless, the method represents a state-of-the-art fault mapping tool that can be useful for hydrocarbon exploration and CO2 storage site evaluations.

A quantitative seismic interpretation study is presented for the Lower Cretaceous Tuxen reservoir in the Valdemar Field, which is associated with heterogeneous and complex geology. Our objective is to better outline the reservoir quality variations of the Tuxen reservoir across the Valdemar Field. Seismic pre-stack data and well logs from two appraisal wells form the basis of this study. The workflow used includes seismic and rock physics forward modelling, attribute analysis, a coloured inversion, and a Bayesian pre-stack inversion for litho-fluid classification. Based on log data, the rock physics properties of the Tuxen interval reveal that the seismic signal is more governed by porosity than water-saturation changes at near-offset (or small angle). The coloured and Bayesian inversion results were generally consistent with well-log observations at the reservoir level and conformed to interpreted horizons. Although the available data have some limitations and the geological setting is complex, the results implied more promising reservoir quality in some areas than others. Hence, the results may offer useful information for delineating the best reservoir zones for further field development and selecting appropriate production strategies.

The estimation of crustal structure and thickness is essential in understanding the formation and evolution of terrestrial planets. Initial planetary missions with seismic instrumentation on board face the additional challenge of dealing with seismic activity levels that are only poorly constrained a priori. For example, the lack of plate tectonics on Mars leads to low seismicity, which could, in turn, hinder the application of many terrestrial data analysis techniques. Here we propose using a joint inversion of receiver functions and apparent incidence angles, which contain information on absolute S‐wave velocities of the subsurface. Since receiver function inversions suffer from a velocity depth trade‐off, we in addition exploit a simple relation that defines apparent S‐wave velocity as a function of observed apparent P‐wave incidence angles to constrain the parameter space. We then use the Neighborhood Algorithm for the inversion of a suitable joint objective function. The resulting ensemble of models is then used to derive uncertainty estimates for each model parameter. In preparation for the analysis of data from the InSight mission, we show the application of our proposed method on Mars synthetics and sparse terrestrial data sets from different geological settings using both single and multiple events. We use information‐theoretic statistical tests as model selection criteria and discuss their relevance and implications in a seismological framework., Key Points: We propose the joint inversion of receiver functions and apparent S‐wave velocity curves to estimate crustal thickness. Using the Neighborhood Algorithm, we show how a full uncertainty estimate can be computed from an ensemble solution. The method is applied to Martian synthetics and terrestrial data sets comprising single and multiple events., IMPRS, Emeritus group, DLR German Space Agency, http://www.orfeus-eu.org/data/eida/, http://instaseis.ethz.ch/marssynthetics/

Mapping landforms on the Moon is of great interest and importance for future human settlements and resources exploration. One of the first steps is to map the topography and investigate their shape...

Isothermal titration calorimetry (ITC) is a widely used method to determine binding affinities and thermodynamics in ligand-receptor interactions, but it also has the capability of providing detailed information on much more complex events. However, the lack of available methods to analyze ITC data is limiting the use of the technique in such multifaceted cases. Here, we present the software ANISPROU. Through a semi-empirical approach that allows for extraction of quantitative information from complex ITC data, ANISPROU solves an inverse problem where three parameters describing a set of predefined functions must be found. In analogy to strategies adopted in other scientific fields, such as geophysics, imaging, and many others, it employs an optimization algorithm which minimizes the difference between calculated and experimental data. In contrast to the existing methods, ANISPROU provides automated and objective analysis of ITC data on sodium dodecyl sulfate (SDS)-induced protein unfolding, and in addition, more information can be extracted from the data. Here, data series on SDS-mediated protein unfolding is analyzed, and binding isotherms and thermodynamic information on the unfolding events are extracted. The obtained binding isotherms as well as the enthalpy of different events are similar to those obtained using the existing manual methods, but our methodology ensures a more robust result, as the entire data set is used instead of single data points. We foresee that ANISPROU will be useful in other cases with complex enthalpograms, for example, in cases with coupled interactions in biomolecular, polymeric, and amphiphilic systems including cases where both structural changes and interactions occur simultaneously.

When hydrocarbon reservoirs are used as a CO2 storage facility, an accurate uncertainty analysis and risk assessment is essential. An integration of information from geological knowledge, geological modelling, well log data, and geophysical data provides the basis for this analysis. Modelling the time development of stress/strain changes in the overburden provides prior knowledge about fault and fracture probability in the reservoir, which in turn is used in seismic inversion to constrain models of faulting and fracturing. One main problem in solving large scale seismic inverse problems is high computational cost and inefficiency. We use a newly introduced methodology -Informed-proposal Monte Carlo (IPMC) -to deal with this problem, and to carry out a conceptual study based on real data from the Danish North Sea. The result outlines a methodology for evaluating the risk of having sub-seismic faulting in the overburden that potentially compromises the CO2 storage of the reservoir.

Mapping landforms on the Moon is of great interest and importance for future human settlements and resources exploration. One of the first steps is to map their topography and investigate their sha...

Inversion of seismic data using information from horizontal wells is often hampered by cumulative well-location errors. These errors can have a significant influence on the final subsurface model derived from the data. To achieve a proper data integration and arrive at correct uncertainty estimates, we formulate the problem in a fully probabilistic framework and present a numerical approach for improving subsurface imaging using uncertain well-log data and their uncertain locations as well as uncertain seismic data. The result is improved model error quantification in the seismic inversion process.

Outcrops are valuable for analogous subsurface reservoirs in supplying knowledge of fine-scale spatial heterogeneity pattern and stratification types, which are difficult to obtain from subsurface reservoir cores, well logs or seismic data. For petrophysical properties in a domain where the variations are relatively continuous and not dominated by abrupt contrasts, the spatial heterogeneity pattern can be characterized by a semivariogram model. The outcrop information therefore has the potential to constrain the semivariogram for subsurface reservoir modelling, even though it represents different locations and depths, and the petrophysical properties may differ in magnitude or variance. However, the use of outcrop-derived spatial correlation information for petrophysical property modelling in practice has been challenged by the scale difference between the small support volume of the property measurements from outcrops and the typically much larger grid cells used in reservoir models. With an example of modelling the porosity of an outcrop chalk unit in eastern Denmark, this paper illustrates how the fine-scale spatial correlation information obtained from the sampling of outcrops can be transferred to coarser-scale models of analogue rocks. The workflow can be applied to subsurface reservoirs and ultimately improves the representation of geological patterns in reservoir models.

We developed a method for quantification of uncertainties on spatial data acquired at points with uncertain location coordinates. We applied it to well-log data from wells with uncertain trajectori...

Inverse problems where relationships are linear arise in many fields of science and engineering and, consequently, algorithms for solving them are widespread. However, when the size of the problem increases, the computational challenge becomes huge, hence, unless some simplifying assumptions are yielded, it becomes impossible to solve the problem. Our study addresses these issues for large linear(izable) inverse problems when the uncertainties can be modeled as Gaussian and the forward relationship and covariance matrices can be expressed in terms of Kronecker products. Under these conditions, we illustrate an algorithm capable of addressing very large problems with very limited storage requirements and much faster than the traditional approach. The result is a complete characterization of the posterior distribution in a probabilistic sense in terms of mean and covariance. We extend this method also to nonlinear problems where a Gauss-Newton algorithm is employed. Applications to reflection seismology, magnetic anomaly inversion and image restoration are presented.

Since InSight (the Interior Exploration using Geodesy and Heat Transport) landed 26 months ago and deployed an ultra sensitive broadband seismometer(SEIS) on the surface of Mars, around 500 seismic events of diverse variety have been detected, making it possible to directly analyze the subsurface properties of Mars for the very first time. One of the primary goals of the mission is to retrieve the crustal structure below the landing site. Current estimates differ by more than 100% for the average crustal thickness. Since data from orbital gravity measurementsprovide information on relative variations of crustal thickness but not absolute values, this landing site measurement could serve as a tie point to retrieve global crustal structure models. To do so, we propose using a joint inversion of receiver functions and apparent incidence angles, which contain information on absolute S-wave velocities of the subsurface. Since receiver function inversions suffer from a velocity depth trade-off, we in addition exploit a simple relation which defines apparent S-wave velocity as a function of observed apparent P-wave incidence angles to constrain the parameter space. Finally we use the Neighbourhood Algorithm for the inversion of a suitable joint objective function. The resulting ensemble of models is then used to derive the full uncertainty estimates for each model parameter. Before its application on data from InSight mission, we successfully tested the method on Mars synthetics and terrestrial data from various geological settings using both single and multiple events. Using the same method, we have previously been able to constrain the S-wave velocity and depth for the first inter-crustal layer of Mars between 1.7 to 2.1 km/s and 8 to 11 km, respectively. Here we present the results of applying this technique on our selected data set from the InSight mission. Results show that the data can be explained equally well by models with 2 or 3 crustal layers with constant velocities. Due to the limited data set it is difficult to resolve the ambiguity of this bi-modal solution. We therefore investigate information theoretic statistical tests as a model selection criteria and discuss their relevance and implications in seismological framework.

Since the '60s of the last century, the calculation of the magnetic anomalies caused by 2D uniformly polarized bodies with polygonal cross-section has been mainly performed using the popular algorithm of Talwani and Heirtzler (1962, 1964). Recently, Kravchinsky et al. (2019, https://doi.org/10.1029/2019GL082767) claimed errors in the above algorithm formulation, proposing new corrective formulas and questioning the effectiveness of almost 60 years of magnetic calculations. Here we show that the two approaches are equivalent and Kravchinsky et al.'s formulas simply represent an algebraic variant of those of Talwani and Heirtzler. Moreover, we analyze a large amount of random magnetic scenarios, involving both changing-shape polygons and a realistic geological model, showing a complete agreement among the magnetic responses of the two discussed algorithms and the one proposed by Won and Bevis (1987, https://doi.org/10.1190/1.1442298). We release the source code of the algorithms in Julia and Python languages.

Summary Fractures are important for reservoir characterization and development. In this paper, we present a new method of Bayesian linearized inversion to invert aspect ratio and fracture density for a quantitative description of fractured reservoirs. The first-order Hudson’s model is linearized based on the Taylor series expansion. Moreover, the model parameters and data error are assumed to be Gaussian distributed. Then the posterior distribution of model parameters can be explicitly expressed and computed analytically. The proposed approach is demonstrated on a synthetic study in which a fractured model is built based on a thin-section micrograph. The inverted rock velocities are used as the inputs for the prediction of fracture parameters and at the same time the estimation uncertainty analysis is performed by the proposed Bayesian scheme.

The Hamiltonian Monte Carlo method (HMC) is gaining popularity in the geophysical community to fully address nonlinear inverse problems and related uncertainty quantification. We present here an application of HMC to invert seismic data in the acoustic approximation in the context of reflection seismology. We address a 2-D problem, in the form of a vertical cross section where both source and receivers are located near the surface of the model. To solve the forward problem we utilise the finite-difference method with PML absorbing boundary conditions. The observed data are represented by a set of shotgathers. The crucial aspect for a successful application of the HMC lies in the capability of performing gradient computations in an efficient manner. To this end, we use the adjont state method to compute the gradient of the misfit functional, which has a computational cost of only about twice that of the forward computation, a very efficient strategy. From the collection of samples characterising the posterior distribution obtained with the HMC, we can derive quantities of interest using statistical analysis and assess uncertainties. We illustrate an application of this methodology on a synthetic test mimicking the setup encountered in exploration problems., EGUsphere

mproved oil recovery from chalk reservoirs by water‐flooding may cause mechanical weakening and change in elasticity. Confined compressive strength testing of chalk from a North Sea reservoir was done in water‐saturated and in oil‐saturated condition. During testing elastic wave velocities were sampled by ultrasonic transducers, so that subsequently Biot's coefficient could be modelled. The porosity declined via an “elastic phase”, a “transitional phase”, an “elastoplastic phase”, and a “strain hardening phase”, but Biot's coefficient indicates that these terms may be partly misleading. In the “elastic phase” porosity and Biot's coefficient decreases, indicating elastoplastic deformation. In the “transitional phase” Biot's coefficient increases as a reflection of breaking contact cement (pore collapse), whereas Biot's coefficient remain stable in the “elastoplastic phase” indicating elastic deformation on the virgin curve. Plastic deformation takes place during phases of creep, where both porosity and Biot's coefficient decrease. Similarly, in the “strain hardening phase”, both porosity and Biot's coefficient decrease as a reflection of elastoplastic deformation. For chalk with 45% – 47% porosity, the “transitional phase” begins at 8 MPa axial stress when water‐saturated and at 12 MPa when oil‐saturated. For chalk with 41% – 43% porosity the corresponding stresses are 16 MPa and 20 MPa. For chalk with 32%–36% porosity the corresponding stresses are 23 MPa and 31 MPa.Chalk samples with irreducible water‐saturation and movable oil were water‐flooded. They yield at stresses close to corresponding oil‐saturated samples, but after flooding show compaction trends not significantly different from the water‐saturated samples. Water‐flooding promotes pore collapse as reflected in an increasing Biot's coefficient. The consequent softening effect on acoustic impedance is small as compared to the effect of increasing fluid density. With respect to 4D seismic, water‐flooding causes distinctly higher acoustic impedance and Poisson's ratio irrespective of compaction.

SUMMARY Any search or sampling algorithm for solution of inverse problems needs guidance to be efficient. Many algorithms collect and apply information about the problem on the fly, and much improvement has been made in this way. However, as a consequence of the No-Free-Lunch Theorem, the only way we can ensure a significantly better performance of search and sampling algorithms is to build in as much external information about the problem as possible. In the special case of Markov Chain Monte Carlo (MCMC) sampling we review how this is done through the choice of proposal distribution, and we show how this way of adding more information about the problem can be made particularly efficient when based on an approximate physics model of the problem. A highly non-linear inverse scattering problem with a high-dimensional model space serves as an illustration of the gain of efficiency through this approach.

We present a framework to solve geophysical inverse problems using the Hamiltonian Monte Carlo (HMC) method, with a focus on Bayesian tomography. Recent work in the geophysical community has shown the potential for gradient-based Monte Carlo sampling for a wide range of inverse problems across several fields. Many high-dimensional (non-linear) problems in geophysics have readily accessible gradient information which is unused in classical probabilistic inversions. Using HMC is a way to help improve traditional Monte Carlo sampling while increasing the scalability of inference problems, allowing access to uncertainty quantification for problems with many free parameters (>10'000). The result of HMC sampling is a collection of models representing the posterior probability density function, from which not only "best" models can be inferred, but also uncertainties and potentially different plausible scenarios, all compatible with the observed data. However, the amount of tuning parameters required by HMC, as well as the complexity of existing statistical modeling software, has limited the geophysical community in widely adopting a specific tool for performing efficient large-scale Bayesian inference. This work attempts to make a step towards filling that gap by providing an HMC sampler tailored for geophysical inverse problems (by e.g. supplying relevant priors and visualizations) combined with a set of different forward models, ranging from elastic and acoustic wave propagation to magnetic anomaly modeling, traveltimes, etc.. The framework is coded in the didactic but performant languages Julia and Python, with the possibility for the user to combine their own forward models, which are linked to the sampler routines by proper interfaces. In this way, we hope to illustrate the usefulness and potential of HMC in Bayesian inference. Tutorials featuring an array of physical experiments are written with the aim of both showcasing Bayesian inference and successful HMC usage. It additionally includes examples on how to speed up HMC e.g. with automated tuning techniques and GPU computations.

Heat storage in the Danish subsurface is gaining increasing interest for optimizing the use of energy resources, but no deep heat storage facilities have yet been established. As an analogue we study the Gassum Formation in the Stenlille structure that is presently used for gas storage. This allows us to discuss geological and technical characteristics of an aquifer relevant for heat storage in Denmark. We develop a 3D model for a high-temperature aquifer thermal energy storage system using analysis of geological core data, sedimentological description, geophysical data including well logs and seismic lines, as well as a finite difference model to calculate the recovery efficiency, heat storage capacity and thermal breakthrough time. Based on geostatistical methods we made three realisations and found similar results for the three cases. In accordance with results from published simplified models we found a high recovery efficiency of 70% after 4 years and 69% after 20 years, a high heat storage capacity of 1.8×1018 J, and a long thermal breakthrough time of 66–77 years. These results reflect the excellent reservoir properties of the Gassum Formation in Stenlille, characterised by a uniformly layered sand/shale sedimentology, a high average porosity of 25% and a high permeability of 1000 to 10 000 mD of sandstone intervals.

Multiple-point-based geostatistical methods are used to model complex geological structures. However, a training image containing the characteristic patterns of the Earth model has to be provided. If no training image is available, two-point (i.e., covariance-based) geostatistical methods are typically applied instead because these methods provide fewer constraints on the Earth model. This study is motivated by the case where 1D vertical training images are available through borehole logs, whereas little or no information about horizontal dependencies exists. This problem is solved by developing theory that makes it possible to combine information from multiple- and two-point geostatistics for different directions, leading to a mixed-point geostatistical model. An example of combining information from the multiple-point-based SNESIM algorithm and two-point-based SISIM algorithm is provided. The mixed-point geostatistical model is used for conditional sequential simulation based on vertical training images from five borehole logs and a range parameter describing the horizontal dependencies.