Geostatistical Inversion Method Research Articles

GeosPIn: A MATLAB library for Geostatistical Petrophysical Inversion

Petrophysical inversion of seismic data is one of the key components of seismic reservoir characterization. The goal of petrophysical inversion is to estimate petrophysical properties of reservoir rocks, such as porosity, volumes of minerals or lithologies, and water and hydrocarbon saturations from seismic data. This process can be performed by combining seismic AVO modeling and rock physics relations with deterministic or probabilistic inverse theory methods, either in a two-step approach based on seismic AVO inversion and rock physics inverse mapping or in a single loop step that includes both. We present a comprehensive MATLAB library named Geostatistical Petrophysical Inversion (GeosPIn). The novelty of the library lies in a single-loop geostatistical inversion method based on the Probability Field Simulation (PFS) and the Ensemble Smoother Multiple Data Assimilation (ES-MDA), and its implementation for applications to 3D datasets. The library also includes a Bayesian petrophysical inversion based on the two-step approach for comparison. We illustrate the main algorithms, explain the structure of the source codes, and demonstrate the library applications through 1D and 3D examples.

A facies‐constrained geostatistical seismic inversion method based on multi‐scale sparse representation

AbstractGeostatistical seismic inversion is an important method for establishing high‐resolution reservoir parameter models. There is no accurate representation method for reservoir structural features, and prior information about structural features cannot be incorporated into geostatistical inversion. Based on the assumption of the sparsity of stratigraphic sedimentary features, the same type of structural feature is used to represent the sedimentary pattern of reservoirs within the same facies. Different sparse representation patterns are used to represent the differences in sedimentary patterns between facies. Although changes in depositional environment might result in the multi‐scale characteristics of geological structures for varying sedimentary rhythms, this paper proposes a facies‐constrained geostatistical inversion method based on multi‐scale sparse representation to better accommodate such situation. Using the method of sparse representation combined with wavelet transform, the multi‐scale sedimentary structural features of reservoirs are learned from well‐logging data. Seismic facies and multi‐scale features are used as prior information for geostatistical inversion. Further, the likelihood function is constructed using seismic data to obtain the posterior probability distribution of reservoir parameters. Finally, the accurate inversion result is obtained by using multi‐scale sparse representation as a constraint in the posterior probability distribution of reservoir parameters. Compared with conventional geostatistical methods, this algorithm can better match the structural features of reservoir parameters with varying geological conditions. Field data tests have shown the effectiveness of this method in improving the accuracy and resolution of reservoir parameter structural features.

Inversion géostatistique par itérations de la tomographie de résistivité électrique

AbstractElectrical resistivity tomography (ERT) is a geophysical method used to create an image of the subsurface due to its sensitivity to porosity, water saturation, and pore fluid salinity. This geophysical method has been widely applied in the investigation of mineral and groundwater resources, as well as in archaeological, environmental, and engineering studies. The prediction of subsurface properties, such as electrical conductivity, from measured ERT data requires solving a challenging geophysical inversion problem. This work proposes an iterative geostatistical resistivity inversion method using stochastic sequential simulation and co-simulation as stochastic model perturbation and update techniques. Electrical resistivity models are generated conditioned to a target histogram, often retrieved from available resistivity borehole data, and assuming a spatial continuity pattern described by a variogram model. From the electrical resistivity models, a finite-volume approximation of Poisson’s equation is used to compute synthetic ERT data. The misfit between predicted and observed data drives the convergence of an iterative procedure and conditions the co-simulation of new models in the subsequent iterations. This methodology is applied to a two-dimensional synthetic case, and a set of two-dimensional profiles obtained from an ERT survey carried out in southern Portugal. In both application examples, the final models predict ERT data that match the observed ones while reproducing borehole data and imposed variogram models. The results obtained in both data sets are compared against a commercial deterministic ERT inversion methodology, showing the ability of the proposed method to model small-scale variability and assess spatial uncertainty.

Geostatistical inversion method based on seismic waveform similarity

Geostatistical inversion method based on seismic waveform similarity

Updating Local Anisotropies with Template Matching During Geostatistical Seismic Inversion

Iterative geostatistical seismic inversion methods are widely used to predict petro-elastic rock properties from seismic reflection data. When the model perturbation technique uses two-point geostatistics, these methods struggle to reproduce complex and nonstationary geological environments such as faults, folds and highly variable depositional systems. These limitations are often due to the use of a global variogram model to express the expected spatial continuity pattern of the property of interest. In complex geological environments a global variogram model might be unable to detect local heterogeneities and rapid variations of lithology, and result in nonrealistic geological models. Local heterogeneities might be predicted from the data using seismic attribute analysis, which can be imposed during geostatistical seismic inversion as local anisotropy models. In these approaches, the information about the local spatial continuity patterns is fixed and will guide and condition the entire inversion procedure, which can lead to errors and uncertainty in areas where this approach is not appropriate due to high local uncertainty of geological features, given the poor signal-to-noise ratio of the data or the presence of important geological features below the seismic resolution. This work proposes an iterative geostatistical seismic inversion method which iteratively updates the local spatial continuity models based on the trace-by-trace misfit between observed and predicted seismic data. The update of the local spatial continuity models aims at surpassing the limitations of the seismic inversion methods that use a fixed a priori variogram model. The method is successfully illustrated in a challenging two-dimensional synthetic data set and in a real case application. The results demonstrate the benefit of updating iteratively the imposed local spatial continuity patterns based on the data misfit. The inverted models are capable of better predicting the location of faults and reproducing the continuity of sinuous channels.

Iterative geostatistical seismic inversion with rock-physics constraints for permeability prediction

Accurate prediction of the spatial distribution of subsurface permeability is a fundamental task in reservoir characterization and monitoring studies for hydrocarbon production and CO2geologic storage. Predicting permeability over large areas is challenging, due to their high variability and spatial anisotropy. Common approaches for modeling permeability generally involve deterministic calculations from porosity using precalibrated rock-physics models (RPMs) or geostatistical cosimulation methods that reproduce observed experimental porosity-permeability relationships. Instead, we have predicted permeability from seismic data using an iterative geostatistical seismic inversion method that combines the advantages of rock-physics and geostatistical modeling methods. First, we simulate facies through 1D vertical Markov chain simulations. Then, permeability, porosity, and acoustic impedance are sequentially generated and conditioned to the previously simulated facies model. An RPM is used to evaluate the misfit between the permeability predictions obtained from geostatistical cosimulation at the well locations and well-log values computed from the acoustic impedance. The residuals of the misfit function are used as conditioning constraints in the stochastic update of the models in the subsequent iteration. The outcome of our methodology is a set of multiple geostatistical realizations of facies, permeability, porosity, and acoustic impedance conditioned to seismic data and constrained by an RPM. We first illustrate the method on a synthetic 1D example and compare it to a traditional geostatistical inversion approach. We then apply our inversion to a 3D real data set to assess the methodology performance with scarce conditioning data and in the presence of noise.

Comparaison des méthodes d’inversion géostatistiques et d’apprentissage profond pour la caractérisation de l’architecture de la zone source DNAPL

Comparaison des méthodes d’inversion géostatistiques et d’apprentissage profond pour la caractérisation de l’architecture de la zone source DNAPL

The Application of Stochastic Seismic Inversion Method of Geostatistical Inversion in Putaohua X Area

Stochastic seismic inversion is an important part of geostatistical inversion method, and is the combination of seismic inversion and sequence simulation. In the process of random seismic inversion, firstly, the histogram distribution of random variables is counted, the variation function of random variables is calculated, and the variation range of random variables in different directions is determined. Then, for each implementation, the seismic traces are synthesized at the seismic channel position according to the calculated reflection coefficient, and the error between the synthetic traces and the original seismic traces is compared, and the inversion results are output after the required accuracy is reached. Geostatistical inversion can improve the vertical and horizontal resolution of the reservoir and accurately describe the spatial distribution of the reservoir. Taking the Heidimiao oil layer in the west slope area of Daqing Placanticline as an example, this paper expounds the application of geostatistical inversion in the work area as a whole. Combined with the results of seismic attribute analysis and well logging curve preprocessing, the inversion results formed by accurate time-depth calibration and other processes of reservoir sensitivity parameter analysis provide a basis for optimizing the target well location in the work area.

Geostatistical seismic inversion constrained by reservoir sedimentary structural features

Geostatistical inversion is a technique widely used in seismic inversion. It builds the posterior distribution by incorporating prior information. Abundant prior information for geostatistical inversion is provided by the spatial variability of well log represented by variograms. However, a variogram can only describe the spatial variability of lithologic parameters at a single point. Describing the structural relationship of rock deposition in the vertical direction (the sedimentary rhythm) requires a continuous curve, rather than one single point. We have developed a geostatistical seismic inversion method constrained by sedimentary structural features to enhance the consistency between inversion results with the sedimentary features of a reservoir. This method first uses sparse representation to learn the sedimentary structural features of a reservoir. Then, the structural features are taken as prior information of geostatistical inversion to improve the accuracy of inversion results. The novelty of our method is that the element describing the spatial variability of model parameters is extended from a single point to a sedimentary structural block. We derive an analytical expression of posterior distribution of model parameters with constraints of sedimentary structural features and realize a data-driven geostatistical seismic inversion. Experiments on synthetic and field data indicate promising results consistent with the sedimentary features of reservoir.

A Multi-Point Geostatistical Seismic Inversion Method Based on Local Probability Updating of Lithofacies

In order to solve the problem that elastic parameter constraints are not taken into account in local lithofacies updating in multi-point geostatistical inversion, a new multi-point geostatistical inversion method with local facies updating under seismic elastic constraints is proposed. The main improvement of the method is that the probability of multi-point facies modeling is combined with the facies probability reflected by the optimal elastic parameters retained from the previous inversion to predict and update the current lithofacies model. Constrained by the current lithofacies model, the elastic parameters were obtained via direct sampling based on the statistical relationship between the lithofacies and the elastic parameters. Forward simulation records were generated via convolution and were compared with the actual seismic records to obtain the optimal lithofacies and elastic parameters. The inversion method adopts the internal and external double cycle iteration mechanism, and the internal cycle updates and inverts the local lithofacies. The outer cycle determines whether the correlation between the entire seismic record and the actual seismic record meets the given conditions, and the cycle iterates until the given conditions are met in order to achieve seismic inversion prediction. The theoretical model of the Stanford Center for Reservoir Forecasting and the practical model of the Xinchang gas field in western China were used to test the new method. The results show that the correlation between the synthetic seismic records and the actual seismic records is the best, and the lithofacies matching degree of the inversion is the highest. The results of the conventional multi-point geostatistical inversion are the next best, and the results of the two-point geostatistical inversion are the worst. The results show that the reservoir parameters obtained using the local probability updating of lithofacies method are closer to the actual reservoir parameters. This method is worth popularizing in practical exploration and development.

Study of Geostatistical Inversion in the Lithologic Distribution and Velocity Modeling of Thick Igneous Rock in the FY Area, Northern Tarim Basin, China

Abstract —In the northern Tarim Basin, a large number of thick igneous rocks are encountered in the drilling process in the Permian. Their lithology and velocity are very strongly, which has a great influence on migration imaging of the “beaded” areas. It is very important to conduct the fine lithology identification and high-precision velocity modeling of the igneous rocks for the exploration and development of the reservoirs. A geostatistical inversion method to obtain the igneous-rock lithologic distribution pattern and velocity modeling in the FY area of the northern Tarim Basin is introduced in this paper. The results show that the application of the geostatistical inversion method greatly improves the resolution of lithology identification. This helps us further understand the Permian igneous rocks distribution in the FY area. Comparison between the seismic facies classification maps of the FY study area shows that the obtained velocity model can reflect the lateral distribution of igneous rocks well. At the same time, the velocity model can reflect the variation of igneous rocks velocity in detail and has a high precision. The average velocity error of the wells participating in the inversion is less than 2%, and the minimum average velocity error is 0.23%. Finally, the velocity model is applied to seismic data processing, and the processing results indicate that it can help to improve seismic migration imaging. The study demonstrates that the geostatistical inversion method can provide a high-precision velocity model for formation pressure prediction and seismic data processing and interpretation, ultimately guiding the exploration and development of oil.

3D spatial characterization of sand body for uranium reservoirs based on geostatistical resistivity inversion

The concern over climate change and a transition away from fossil fuels has reinvigorated interest in using nuclear energy in producing electricity. Uranium is the principal fuel used in nuclear power plants, and mineral deposits containing this element are of strategic importance. The successful development of sandstone uranium deposits benefits from 3D geophysical characterization of sand bodies in uranium reservoirs. In view of this, we have evaluated a method based on 3D geostatistical resistivity inversion. First, we analyze the application of that method to the problem at hand and introduce a workflow for analyzing the data. Second, through petrophysical sensitivity analysis, we identify the logging parameters that can characterize sandstone in this context and geostatistical inversion to estimate those parameters. We then use 3D geostatistical resistivity inversion to map a sandstone uranium reservoir and find it to provide accurate results. Finally, the 3D data of inversion are applied to 3D spatial characterization of a sand body in the Sifangtai uranium reservoir inverting a field data set in the western of Songliao Basin in China. We believe this geostatistical inversion method to be useful in determining the location of drilling wells for exploration and development of sandstone uranium deposits.

Geostatistical Seismic Inversion with Self-Updating of Local Probability Distributions

Three-dimensional subsurface elastic models inverted from seismic reflection data are the basis of the geo-modeling workflow. These models are often used to predict the spatial distribution of reservoir rock properties such as porosity, volume of minerals and fluid saturations. Stochastic seismic inversion methods are important modeling tools, as they allow one to infer high-resolution subsurface models and assess uncertainties related to the spatial distribution of the inverted petro-elastic properties. Within this framework, iterative geostatistical seismic inversion methods use stochastic sequential simulation and co-simulation as a model generation and perturbation technique based on the mismatch between synthetic and real seismic data. This work proposes an alternative approach of iterative geostatistical seismic inversion based on the concept of self-updating of local probability distributions of the elastic property of interest to be inverted. The model perturbation is conditioned by local probability distribution functions, which are iteratively updated based on the data misfit at previous iterations. This approach allows for better exploration of the model parameter space, avoiding local fast convergence at early steps of the inversion, and wider exploration of the model parameter space. The method is applied to a two-dimensional nonstationary synthetic dataset and to a three-dimensional real case example with a blind well test.

Direct Geostatistical Seismic Amplitude Versus Angle Inversion for Shale Rock Properties

The purpose of seismic reservoir characterization is to predict the spatial distribution of the subsurface rock properties from a set of direct and indirect measurements. Commonly, rock property volumes are obtained in a two-steps approach. First, the elastic properties are inverted from seismic reflection data and then used to compute rock properties volumes by applying a calibrated rock physics models. Such an approach is not only time consuming but may also lead to biased results as the uncertainties related to seismic inversion may not be propagated through the entire geomodeling workflow. Here, we propose an iterative geostatistical shale rock physics seismic AVA inversion method to invert seismic reflection data directly for shale rock properties. The workflow consists of three main steps starting from shale properties model generation of brittleness, total organic carbon (TOC), and porosity using stochastic sequential simulation and cosimulation and calculation of a volume of shale. In the following step, elastic property volumes are calculated based on a calibrated shale rock physics model using the self-consistent approximation. The elastic models are then used for the calculation of the synthetic seismic data. In the final step, the misfit between synthetic and real data is calculated and used as part of the stochastic update of the model parameter space. The whole process is repeated until a minimum misfit between the observed and synthetic seismic is achieved. The proposed method is successfully tested on an onshore Lower Paleozoic shale gas reservoir in Northern Poland, and the predicted shale rock properties agree with those observed at a blind-well location.

Iterative geostatistical seismic inversion incorporating local anisotropies

Geostatistical seismic inversion methods use stochastic sequential simulation as the model generation and perturbation technique. These stochastic simulation methods use a global variogram model to express the expected spatial continuity pattern of the subsurface elastic properties of interest. The conditioning to a single variogram model is not suitable for complex and non-stationary geological environments, resulting in poor inverted models unable to reproduce non-stationary features such as channels, folds, and faults. The proposed method uses a stochastic sequential simulation and co-simulation method able to cope with spatially varying information using local and independent variogram models. The information about the dip, azimuth, and ranges of the local variogram model is inferred directly from the observed data. First, local dip and azimuth structural volumes are computed from seismic attribute analysis. Then, local variogram models are fitted along the directions estimated from the previous step. This information is used as steering data during the inversion, acting as proxy of the true subsurface geological complexities. Application examples in synthetic and real datasets with complex geometries show the impact of using local anisotropy models in both the reproduction of the original seismic data and the reliability of the inverted models. The resulting inverted models show enhanced consistency, where small-to-large scale discontinuities and complex geometries are better reproduced, allowing reducing the spatial uncertainty associated with the subsurface properties. This work represents a step forward in integrating geological consistency into geostatistical seismic inversion, surpassing the limitation of using a single variogram model to reproduce complex geological patterns.

Geostatistical electrical resistivity tomography using random mixing

With modern data acquisition technology, Electrical Resistivity Tomography (ERT) surveys can be deployed easily and run cost-efficiently over large areas and over long periods. Inversion of the electric potential data to image the subsurface resistivity distribution is a key step in the workflow. In contrast to deterministic inversion commonly used in ERT, geostatistical inversion methods construct a variety of possible realizations of resistivity distributions and allow for a better interpretation of results and a quantification of the uncertainty. In this paper we present a geostatistical approach for the inversion of ERT survey data based on Random Mixing. The method of Random Mixing had been developed and tested for inverse groundwater modelling and is applied here to ERT as an example of geophysical inversion for the first time. Random Mixing is based on the idea of constructing realizations of the unknown electric conductivity field by combining random fields representing the spatial correlation of conductivity such that direct measurements of conductivity if available and indirect dipole measurement of electric potentials at electrodes are matched. Implementation is based on the three-dimensional finite element method to solve the forward model where properties in a core region are updated through the Random Mixing iteration. To reduce computational costs for the evaluation of the misfit cost function one-dimensional interpolation is applied. The results of Random Mixing for a synthetic example and a field survey inversion are discussed and compared with deterministic inversion results.

The impact of a priori elastic models into iterative geostatistical seismic inversion

In reservoir modelling and characterization, the integration of geological information within seismic inversion is critical to efficiently obtain accurate inverted models. The existing knowledge about the subsurface geology should be gathered and used as conditioning data to generate robust a priori models. Usually, these models explain the expected background trend of the subsurface property of interest. One of the main differences between deterministic and iterative geostatistical seismic inversion methods is how the a priori knowledge about the spatial continuity pattern of the subsurface elastic property to be inferred, is used as constraint for the inversion. Deterministic approaches use explicit a priori models, often low-frequency models, to represent the geological background of the inverted properties. On the other hand, iterative global geostatistical seismic inversion methods do not include explicitly a priori models, since the spatial continuity pattern of the elastic property to be inverted is described exclusively by a variogram model. This work explores the impact of using explicit a priori models when integrated into iterative geostatistical seismic inversion methods. Three a priori models were built with different techniques and incorporated into the inversion process using two distinct methodologies: the first directly constrains the generation of elastic models, using stochastic sequential simulation with simple kriging with locally varying means; alternatively, these a priori models are also incorporated as part of a multi-objective function, accounting for both data and model deviations. All scenarios are applied to a synthetic case study, and those with the best results are then applied to a three-dimensional real dataset. The results of this work illustrate the drawbacks and advantages of using explicit a priori models in iterative geostatistical seismic inversion, highlighting the impact on the reproduction of channels under complex and heterogeneous siliciclastic geological environments and the need of reliable a priori models for accurate predictions about the subsurface.

Seasonally Resolved Excess Urban Methane Emissions from the Baltimore/Washington, DC Metropolitan Region.

Urban areas are increasingly recognized as an important source of methane (CH4), but we have limited seasonally resolved observations of these regions. In this study, we quantify seasonal and annual urban CH4 emissions over the Baltimore, Maryland, and Washington, DC metropolitan regions. We use CH4 atmospheric observations from four tall tower stations and a Lagrangian particle dispersion model to simulate CH4 concentrations at these stations. We directly compare these simulations with observations and use a geostatistical inversion method to determine optimal emissions to match our observations. We use observations spanning four seasons and employ an ensemble approach considering multiple meteorological representations, emission inventories, and upwind CH4 values. Forward simulations in winter, spring, and fall underestimate observed atmospheric CH4 while in summer, simulations overestimate observations because of excess modeled wetland emissions. With ensemble geostatistical inversions, the optimized annual emissions in DC/Baltimore are 39 ± 9 Gg/month (1 δ), 2.0 ± 0.4 times higher than the ensemble mean of bottom-up emission inventories. We find a modest seasonal variability of urban CH4 emissions not captured in current inventories, with optimized summer emissions ∼41% lower than winter, broadly consistent with expectations if emissions are dominated by fugitive natural gas sources that correlate with natural gas usage.

High-resolution water-saturation prediction using geostatistical inversion and neural network methods

One of the main challenges that we face is the accurate prediction of pore-fluid properties with the highest possible resolution. The seismic resolution is the most limiting factor, especially in our case, in which the main reservoirs are deepwater turbidite channels and their thin beds typically fall below the seismic tuning thickness. Therefore, we designed a new workflow that combines the geostatistical inversion and the neural network analysis with the aim of predicting a 3D high-resolution water saturation (sampled every 1 ms), overcoming the limitation of seismic detectability and providing better reservoir characterization. The power of the geostatistical inversion is that it provides multiple model realizations, and each realization honors the well data (statistical information and logs) and the seismic data. These realizations are more reliable and high-resolution versions of the elastic parameters. On the other hand, the main advantage of the neural network is that it establishes a stable nonlinear link between the input seismic and inversion results and the target water saturation. The available data set for this study includes three angle stacks and seven wells from Scarab field, offshore Nile Delta. The resulted high-resolution saturation volume was tested using blind-well analysis and revisit post the drilling of a new well later on. It gave spectacular results in both cases. The normalized correlations between the predicted saturation volume and the real saturation logs are 0.87 and 0.89, respectively. The results prove the validity of the workflow to accurately predict water saturation with a higher resolution than ever before.

Multimodel Bayesian analysis of groundwater data worth

AbstractWe explore the way in which uncertain descriptions of aquifer heterogeneity and groundwater flow impact one's ability to assess the worth of collecting additional data. We do so on the basis of Maximum Likelihood Bayesian Model Averaging (MLBMA) by accounting jointly for uncertainties in geostatistical and flow model structures and parameter (hydraulic conductivity) as well as system state (hydraulic head) estimates, given uncertain measurements of one or both variables. Previous description of our approach was limited to geostatistical models based solely on hydraulic conductivity data. Here we implement the approach on a synthetic example of steady state flow in a two‐dimensional random log hydraulic conductivity field with and without recharge by embedding an inverse stochastic moment solution of groundwater flow in MLBMA. A moment‐equations‐based geostatistical inversion method is utilized to circumvent the need for computationally expensive numerical Monte Carlo simulations. The approach is compatible with either deterministic or stochastic flow models and consistent with modern statistical methods of parameter estimation, admitting but not requiring prior information about the parameters. It allows but does not require approximating lead predictive statistical moments of system states by linearization while updating model posterior probabilities and parameter estimates on the basis of potential new data both before and after such data are actually collected.