Stochastic Inversion Research Articles

Nonstationary Prestack Linear Bayesian Stochastic Inversion

Nonstationary Prestack Linear Bayesian Stochastic Inversion

Stochastic inversion for equivalent hydraulic fracture characterization using low-frequency distributed acoustic sensing data

Distributed fiber-optic sensing, a game-changing technology, has been successfully applied in monitoring hydraulic fracturing treatments. The cross-well fiber-optic strain/strain-rate responses are directly related to the fracture-propagation process and dynamic fracture geometry. Extracting the insightful information regarding hydraulic fractures requires robust interpretation workflows. In this paper, we aim to develop a stochastic inversion approach to characterize the dynamic equivalent fracture geometry and quantify the associated uncertainties using cross-well low-frequency distributed acoustic sensing (LF-DAS) measurements. A fast analytical forward solver based on the 3D displacement discontinuity method is developed to model the strain. The objective function is the sum-of-squares error between the modeled strain and the measured strain. We focus on an equivalent fracture geometry with a predefined length centered around the monitoring well, which is a reasonable assumption considering the spatial sensitivity of LF-DAS. Therefore, the model parameters reduce to fracture width and height of the equivalent fracture. We adopted the delayed rejection adaptive metropolis (DRAM) algorithm, an efficient Markov-chain Monte Carlo (MCMC)-based method, to perform the optimization. The efficiency and accuracy of the algorithm are tested using a synthetic case that has three monitoring wells at different locations. The mean values of fracture width and height are both close to the true values with negligible standard deviations for all the three monitoring wells. The algorithm performance is not sensitive to the initial guess and the predefined length needs to be large enough. Then, a field case with a single cluster from the HFTS-2 project is presented. The fracture width increases from about 0.5 mm to about 0.9 mm during the pumping stage and gradually decreases during the shut-in period. The fracture height changes during the pumping stage and stabilizes at about 150 m during the shut-in period, the trend of which is consistent with the LF-DAS signals from the offset vertical monitoring well. To the best of our knowledge, this is the first algorithm that can efficiently quantify the fracture width and height simultaneously. The results could provide novel insights on the characteristics of fracture geometry and significantly improve the understanding of complex fracture propagation processes in subsurface reservoirs.

Stochastic inversion of geophysical data by a conditional variational autoencoder

Recovering geologically realistic physical property models by geophysical inversion is a long-standing challenge. Generative neural networks offer a promising path to meet this challenge because they can produce spatially complex models that exhibit the characteristics of a set of training models, even when those characteristics are difficult to quantify. In the context of geophysical inversion, these characteristics may include faults, layers, and sharp contacts between rock units. Here, we develop a framework for incorporating prior geologic knowledge into geophysical inversions using conditional variational autoencoders (CVAEs). We train a CVAE to reconstruct training density models while honoring relative gravity data. Once trained, the decoder network of the CVAE inverts gravity data. The inputs to the decoder are observed gravity data and a set of latent variables that are sampled from a standard normal distribution. The decoder maps from the observed data and latent variables to density models such that the resulting models are consistent with the training models and the input data. Consequently, the inversion fits the observed data and incorporates the information embedded in the training models. The decoder can produce many inverted models instantaneously, sampling an approximation to the posterior model distribution efficiently. We find that the latent variables correspond to independent interpretable ways in which the model can vary while still honoring the observed data. We draw a connection to linear inverse theory, positing that the latent variables are analogous to the principal components of the local posterior model covariance.

Integrated seismic inversion for clastic reservoir characterization: Case of the upper Silurian reservoir, Tunisian Ghadames Basin

Deltaic petroleum systems are often complex and heterogeneous. Delineation of sandstone bodies in such complex depositional settings constitutes a crucial challenge in reservoir exploration. However, reservoir characterization based on conventional well-logs and stratigraphic interpretation in complex settings often leads to uncertainties. In this work, we present an integrated quantitative interpretation to reveal heterogeneity and the reservoir properties in a complex deltaic environment. The study benefits from integrating borehole lithology logs, electrical well logging, and a 3D post-stack seismic cube. To demonstrate the practical application and effectiveness of our proposed approach, seismic inversion has been applied to the Acacus Formation in the Ghadames Basin of southern Tunisia. We estimate the acoustic impedance volume from 3D seismic data using a stochastic inversion algorithm. This algorithm enables the generation of multiple realizations, accounting for uncertainties and providing a more comprehensive understanding of subsurface impedance. Then, the porosity model is computed by establishing a functional relationship between acoustic impedance and porosity. The analysis of the relationship between acoustic impedance and different rock properties helped to differentiate lithological units of sandstone and claystone. This lithology classification is used to estimate the lithofacies model. Finally, the obtained porosity and lithofacies models are validated with well data. The crucial correlation between facies distribution and porosity allows for accurate mapping of individual thin sandstone bodies and their property distribution beyond well control. The lithofacies model shows significant sandstone bodies in the Acacus A and C units that constitute good reservoirs. The porosity model confirms this, as these sandstone facies exhibit high porosity (20%). The proposed approach has proven to be powerful in reservoir delineation, characterization, and exploration, and can be applied in similar geological setting basins, and frameworks.

Seismic fluid identification method based on the joint PP‐ and SH–SH‐wave stochastic inversion

AbstractPre‐stack seismic inversion is an effective method for elastic parameter inversion using seismic data, which facilitates seismic fluid identification. However, pure PP‐wave inversion has issues of strong multi‐solution and limited prediction accuracy. Therefore, we propose a seismic fluid identification approach based on the joint PP‐ and SH–SH‐wave stochastic inversion. First, the linearized SH–SH‐wave amplitude variation with offset approximation parameterized by shear modulus and density is derived. Numerical simulations demonstrate that the SH–SH‐wave amplitude variation with offset approximation has a good accuracy. Reflection coefficient contribution analysis indicates that the new formulation has better parameter sensitivity to shear modulus and density than the PP wave amplitude variation with offset approximation derived by Russell, which helps one to improve the inversion of shear modulus and density. On this basis, we construct a joint inversion equation of PP and SH–SH waves for a Russell fluid indicator, a shear modulus and density and present a novel joint stochastic inversion method based on the ensemble smoother with multiple data assimilation. Stanford VI‐E model tests reveal that the Russell fluid indicator factor, shear modulus and density obtained from the joint PP‐ and SH–SH‐wave inversion have higher identification accuracy and smaller relative errors than those from pure PP‐wave inversion. Furthermore, field data tests indicate that this method has practical applicability in seismic fluid identification.

Stochastic seismic inversion and Bayesian facies classification applied to porosity modeling and igneous rock identification

We apply stochastic seismic inversion and Bayesian facies classification for porosity modeling and igneous rock identification in the presalt interval of the Santos Basin. This integration of seismic and well-derived information enhances reservoir characterization. Stochastic inversion and Bayesian classification are powerful tools because they permit addressing the uncertainties in the model. We used the ES-MDA algorithm to achieve the realizations equivalent to the percentiles P10, P50, and P90 of acoustic impedance, a novel method for acoustic inversion in presalt. The facies were divided into five: reservoir 1, reservoir 2, tight carbonates, clayey rocks, and igneous rocks. To deal with the overlaps in acoustic impedance values of facies, we included geological information using a priori probability, indicating that structural highs are reservoir-dominated. To illustrate our approach, we conducted porosity modeling using facies-related rock-physics models for rock-physics inversion in an area with a well drilled in a coquina bank and evaluated the thickness and extension of an igneous intrusion near the carbonate-salt interface. The modeled porosity and the classified seismic facies are in good agreement with the ones observed in the wells. Notably, the coquinas bank presents an improvement in the porosity towards the top. The a priori probability model was crucial for limiting the clayey rocks to the structural lows. In Well B, the hit rate of the igneous rock in the three scenarios is higher than 60%, showing an excellent thickness-prediction capability.

Stochastic Facies Inversion with Prior Sampling by Conditional Generative Adversarial Networks Based on Training Image

Stochastic Facies Inversion with Prior Sampling by Conditional Generative Adversarial Networks Based on Training Image

Reservoir characterization in coastal-continental environment of the Jurassic complex based on seismic data inversion

A modern seismic interpretation of coastal-continental deposits in the Jurassic complex of one of the fields in the north of the West Siberian Oil and Gas Province, including simultaneous stochastic inversion of seismic data was performed. The main object of the study was the interval of the group of J2 reservoirs. Petro-elastic modeling was carried out on the project. Modelling includes six wells with geophysical well logs. Feasibility study shows possibility of predicting lithology by stochastic inversion. In addition, optimal parametrization of elastic properties was determined to separate water-saturated and gas-saturated reservoirs from clay deposits. Interpretation of well and seismic data, as well as the results of stochastic inversion, made it possible to identify play zones in target interval. Probability cubes of gas-saturated reservoirs, made it possible to map the lenticular structure of the reservoirs and substantiate hydrodynamic barriers between individual gas plays.

Refinement of the geological model of Jurassic deposits accounting the results of stochastic inversion and facies modeling

The paper presents the results of a comprehensive interpretation showing the effective way to integrate seismic data into a three-dimensional geological model. Stochastic inversion was used to increase the reliability of forecasts of productive thicknesses. A comprehensive interpretation of the geological and geophysical information of the Yu21 formation deposits of the Malyshevskaya formation was carried out, including sedimentological analysis of core data, petroelastic modeling of well logging curves for the purposes of stochastic inversion and stochastic inversion of seismic data. An areal forecast of sedimentation environments (facies) was carried out. The resulting three-dimensional geological model, in more detail, compared to the model without taking into account the spatial seismic forecast, emphasizes the heterogeneity of the distribution of properties in the geological environment, which is especially important when planning production drilling with horizontal wells.

Integrating seismic attributes, post-stack seismic inversion, and static modeling for assessing the Pliocene-Pleistocene hydrocarbon potentialities, Offshore North Sinai Field, Egypt

Optimal gas production from marginal fields vulnerable to tidal energetic effects is exceedingly challenging. Conventional seismic analysis is insufficient for a significant understanding of the geomorphological description, volumetric calculations, and microfacies analysis of the anticipated heterogeneous gas-bearing sands. New insights into the offshore North Sinai (ONS) Field based on quantitative and geostatistical interpretations integrated with physical and geometrical attribute analyses are boosted for reconstructing diverse-scale reservoir internal architecture.Deterministic band-limited inversion using acoustic impedance logs contributed to the initial knowledge of facies migration during tidal-dominated delta progradation and revealed four 2nd-order sequence boundaries dependent on P-velocity variations with depth as a function of rock elasticity, low impedance gas content of 2500–3000 kPa s/m and overburden compressional factors. Gradual boundaries confining reservoir bases due to spatiotemporal facies discrepancies triggered the implementation of stochastic algorism as a model norm for tracing thin-bedded and internal heterogeneous reservoirs with no predictable gas-water contacts across subtle dismantled areas. Stochastic inversion with intensive layering below −1 ms along with static property reservoir modeling allowed the evaluation of both fine and coarse-scale reservoirs discovered as extensions of injector-controlled stacked mouth bars, fan-lobes, and distributary channels crosscutting tidal mudflats of delta top facies associations within a system of spatially migrated clinoforms to the north. Stream avulsion-induced and accommodation space inconsistencies across syn-sedimentary faults caused the shifting of the Late Plio-Pleistocene clinoform depocenter northward and deltaic facies variation inside each individual sand system, not eustatic sea level fall.After calculating the biogenic gas reserves occupying each reservoir, the present strategy offers an additional seven broad gas prospects within four dismantled blocks, raising the ONS initial gas in place to 967.95 Bcf. In addition, it depicts five gas shoals emplaced at shallow depths, posing considerable drilling hazards. This paper workflow is the first of its kind in the Neogene section of the OSN Basin.

Prediction of Marine Thin Shale Gas Reservoir with Seismic Phase-Controlled Nonlinear Stochastic Inversion

Due to the complicated interference sources and low signal-to-noise ratio of seismic records, conventional seismic inversion methods are difficult to accurately identify shale gas reservoirs with a thickness of less than 10 m. This presents great challenges to shale gas exploration and development in China. Seismic phase-controlled nonlinear stochastic inversion (SPCNSI) is related to the heterogeneity of underground media. With the constraints of the stratigraphic sequence or seismic facies models, the minimum value between the seismic model and seismic record can be solved through iterative processes. Based on the solved acoustic velocity in formation, the constraints for SPCNSI can be formed with the matching relationship between target layers and different sequences in three-dimensional space. The prediction resolution of an unconventional reservoir can be effectively improved by combining logging, geological and seismic information. The method is suitable for predicting thin shale gas reservoirs in complex geological structures. In this study, SPCNSI was developed to predict the thin-layer marine shale gas reservoir in the southeast of Chongqing; the horizontal and vertical distribution characteristics were proven in the Longmaxi Formation and the Wufeng Formation; thin layers with a thickness of less than 7 m were also discovered. According to the results, three sets of potential development areas were determined by the inversion method, and the accuracy and reliability of the method were verified by logging and productivity testing.

Estimation of eruption source parameters for the 2021 La Soufrière eruption (St Vincent): implications for quantification of eruption magnitude on volcanic islands

Abstract Eruption source parameters (ESPs) used to characterize explosive eruptions are estimated from tephra deposit data using different models (statistical or numerical) and inversion approaches. The ESPs thus derived are subject to substantial uncertainties when the bulk of the tephra deposit, including information about its full spatial extent and spatial variation in grain-size distribution is missing due to geographical and environmental conditions. We use an advection–diffusion model coupled with a Bayesian inversion and uncertainty quantification algorithm to investigate how ESPs can be robustly estimated given such conditions. The 2021 eruption of La Soufrière volcano (St Vincent and the Grenadines) is our case study. An inversion is conducted for the first two explosive phases of this eruption (U1 and U2). We estimate: an erupted mass of 3.3 × 10 10 ± 1 × 10 10 kg for U1 and 3.1 × 10 10 ± 1.9 × 10 9 kg for U2, with an average particles release height of c. 13.5 km a.s.l. ±0.5 km for both phases. Given the efficiency of the proposed approach and the plausibility of the stochastic inversion results, we recommend this procedure for estimating ESPs for explosive eruptions for which the bulk of the deposit is missing or is inaccessible.

The influence of different back analysis conditions on the stochastic back analysis of arch dam material Parameters: A case study

It is very important to back analyse arch dam material parameters through monitoring data. The selection of the monitoring points, prior distribution of the material parameters and likelihood function have a significant impact on the stochastic back analysis of arch dam material parameters based on Bayesian inference. In this paper, the sensitivity analysis method is used, and the sensitivity of the comprehensive elastic modulus of dam concrete and bedrock for arch dam deformation is analysed. Through high-sensitivity monitoring points and all monitoring points in the arch dam, the influence of the monitoring point selection on the stochastic back analysis of the arch dam material parameters is studied. Considering the prior distribution of the material parameters and likelihood function forms, the influence of different prior distributions and likelihood function forms on the stochastic back analysis of the arch dam material parameters is studied based on high sensitivity measuring points. The example have shown that the sensitivity analysis can be used for optimizing the selection of monitoring points. Different parameter prior distributions and likelihood function forms have certain influences on the stochastic back analysis of the arch dam material parameters. This paper provides a reference for selecting the stochastic inversion conditions for the arch dam material parameters.

Bayesian ensemble machine learning-assisted deterministic and stochastic groundwater DNAPL source inversion with a homotopy-based progressive search mechanism

A hyperheuristic homotopy algorithm (HH-HA) and a homotopy-based swarm intelligence algorithm for parallel stochastic search are proposed to improve the search ergodicity and stability of deterministic and stochastic inversion approaches, respectively, for the identification of dense nonaqueous phase liquid (DNAPL) source characteristics and contaminant transport parameters in groundwater. Meanwhile, a Bayesian ensemble machine learning (BEML) method for numerical simulation surrogate modeling is proposed to enhance the learning and generalization capability, while significantly reducing the computational cost of repetitive CPU-demanding likelihood evaluations in inversion iterations. Two cases are presented for the verification and application of the proposed inverse modeling methods. The results show that the BEML surrogate model is powerful in approximating the numerical model outputs, reducing the mean relative errors of the contaminant concentration and pressure head predictions to 1.18% and 1.29%, respectively. The homotopy-based progressive search mechanism is highly effective in achieving more accurate identification values for deterministic inversion and a more reliable posterior distribution for stochastic inversion. The HH-HA shows stable performance, approaching the global optimum in wide areas and reducing the mean relative error of identification from 11.95% to 4.11%. The homotopy-based parallel stochastic search lines adequately cover the search space tracking the change of homotopy parameter, and the mean relative error of point estimation is reduced to 2.03%. Though the deterministic inversion approach can provide intuitive identification values, the reliability of the inversion outputs cannot be quantified. In contrast, the proposed stochastic inversion system is of great significance for providing decision makers with comprehensive reference information.

Inversion of ERT-3D data using PSO and weighting functions

In the last decade, a new data acquisition technique has emerged in 3D electrical resistivity tomography studies, where the target to study is beneath a structure such as a building and it is not possible to use the traditional technique. This is a very common scenario when investigating urban areas or archaeological sites. The technique, called Perimeter Data Acquisition (PDA) in this paper, allows the acquisition of irregularly distributed data beneath structures. Although electrical anomalies can be detected, the solution does not correctly determine their vertical location because of the irregular distribution of the data and the lack of data towards the central portion of the survey region limited by the electrodes.The methodology presented here aims to improve the calculation of the vertical location of the anomalous bodies in the subsoil using weighting functions in conjunction with a stochastic inversion method called Particle Swarm Optimization (PSO). Three weighting functions defined from the theoretical depth of the arrays and the resistivity standard deviation were tested by using synthetic models and comparing the solutions between them and with non-weighted solutions. Finally, the algorithm and weighting functions were applied on a case study to compute the vertical location of a cavern beneath a Mayan pyramid.The results indicate that when using the PDA technique, it is necessary to employ a weighting function to obtain an acceptable and reasonable solution. It is found that the weighting function defined from the resistivity standard deviation allowed to determine with the least error the vertical characteristics of the anomalous bodies.

Fracture network characterization with deep generative model based stochastic inversion

Fracture network characterization with deep generative model based stochastic inversion

Hydrogeological modeling of the Roussillon coastal aquifer (France): stochastic inversion and analysis of future stresses

Global climate change-induced stresses on coastal water resources include water use restrictions, saline intrusions, and permanently modifying or damaging regional resources. Groundwater in coastal regions is often the only freshwater resource available, so an in-depth understanding of the aquifer, and the aquifer’s response to climate change, is essential for decision-makers. In this study, we focus on the coastal aquifer of Roussillon (southern France) by developing and investigating a steady-state groundwater flow model (MODFLOW 6) and calibrated with PEST++ on a Python interface (FloPy and PyEmu). Model input and boundary conditions are constrained by various scenarios of climate projections by 2080, with model results predicting the aquifer’s response (and associated uncertainty) to these external forcings. Using simple assumptions of intrusion estimates, model results highlight both strong climatic and anthropogenic impacts on the water table. These include aquifer drawdowns reaching several meters locally, and the seawater interface advancing locally several hundred meters inland and rising by several meters. Intrusions of this magnitude risk endangering exploited water wells and their sustainability. Our results demonstrate the critical importance of properly characterizing the geology and its heterogeneity for understanding aquifers at risk because poor predictions may lead to inappropriate decisions, putting critical resources at risk, particularly in coastal environments.

Stochastic inversion combining seismic data, facies properties, and advanced multiple-point geostatistics

We proposed a novel seismic inversion approach that integrates the physical properties of litho-facies, and geophysical data, within a multiple-point geostatistical framework to reduce the uncertainty in predictions of litho-facies spatial arrangement away from wells or control points. The litho-facies groups (rock-type) in the well locations are defined and conditioned to the distribution of elastic properties, including the values of P-wave velocity (Vp) and facies density (ρ) within the borehole. A conceptual geological model (training image) is utilized within a wavelet-based multiple-point geostatistical simulation (WAVESIM) algorithm to generate litho-facies realizations. In our inversion algorithm, the forward model is created by implementing the bivariate Kernel density estimation (KDE) technique of the litho-facies properties (Vp and ρ) that are distributed within the well locations. We utilize an iterative inversion framework, where a particular number of elastic properties (Vp and ρ) for each WAVESIM realization are drawn. For each generated set of the WAVESIM realizations, we simulate reflectivity series by KDE that are convolved with the seismic wavelet to create synthetic seismograms. Then, using a normalized, cross-correlation function we select the realization that provides the best-match between synthetic seismogram and the input seismic data. Our inversion technique is successfully applied to synthetic and field datasets. Our results demonstrate the efficiency of our inversion to characterize highly heterogeneous reservoirs in a reasonable computational time and control the connectivity between the channels.

Classification of UXO and non-UXO from magnetic anomaly data: a case study on inversion of drone magnetic data from Rømø, Denmark

SUMMARY A test site containing 24 targets of various disarmed unexploded ordnance (UXO) and non-UXO items were placed on a beach on the island of Rømø (Denmark) in a 600 m × 100 m area. Scalar magnetic anomalies were measured at 3–5 m altitude using an uncrewed aerial vehicle (UAV), towing a bird with a three-sensor triangular configuration to achieve a dense coverage with flight lines of 2 m spacing. The triple-sensor data set is utilized in a probabilistic inversion setup to infer the magnetic moments of the 24 targets. The purpose of the study, is to try and distinguish between different types of ferromagnetic objects (UXO, non-UXO) using magnetic anomaly data. The inversion methodology uses different forward models (prolate spheroids, rectangular prisms) to infer target shape, size and orientation in an attempt to discriminate between UXO and non-UXO items. Stochastic inversions are carried out using different prior assumptions of remanent magnetization strength (10, 50 and 80 per cent) of the induced dipole moment. Among the three levels of remanent magnetization strength in the prior, only some cases of discrimination seem evident for the lowest strength of remanence. One item is correctly classified as a true-negative (i.e. non-UXO) when assuming low remanent magnetization strength (10 per cent of the induced moment). However, at low remanent strength, one false-negative classification emerges, making any discrimination unreliable when assuming such low remanent magnetization. In addition to the discrimination study, different covariance models are utilized to optimize the inversion by addressing correlated errors and noise in the triple-sensor data set. Three covariance models are tested to try and account for spatially correlated noise and potential errors among the three sensors of each overflight. In many cases, the covariance models presented show a potential increase in sampling efficiency and consistency between data and the noise model, suggesting a more robust approach to a noise model in magnetic anomaly inversions. If the noise model is poor, however, it may bias the results by addressing the anomaly signal as noise. The inversions with correlated noise models are compared with inversions using a simple uncorrelated noise model. For several cases of data anomalies, differences between the inversion estimates when using correlated and uncorrelated noise models were evident, indicating that some bias may appear when assuming uncorrelated noise. Due to the general high presence of correlated signals in magnetic survey data, correlated noise models can significantly improve the overall uncertainty estimate of the estimated dipole moment. The study demonstrates, in terms of the 24 targets considered, that discrimination between UXO and non-UXO using magnetics is difficult. However, when using scalar magnetic data of high quality and resolution, the estimated dipole moments are often well resolved and uniquely defined in magnitude and position. This could provide valuable posterior information for future inversion studies by building a library of inferred magnetic moments from targets that have been found and inspected.

An Integrated Workflow for the Probabilistic Estimation of Pressure and Saturation Changes from 4D Seismic Data: Application to the Catcher Fields, Central North Sea

We present 4D seismic inversion to reservoir pressure and saturation changes applied to data from the Catcher fields. The inversion workflow integrates data from reservoir simulation, well logs and production volumes, time-lapse time shifts and angle-stacked 4D seismic amplitudes as well as machine learning and Bayesian inversion methods. It begins with a petro-elastic model and reservoir pressure sensitivity calibration step, using well log data and time-lapse time-shifts. A machine learning inversion is then used to create an initial estimate of the reservoir property changes. This estimate is then used as prior information, in conjunction with reservoir simulation pressure data, for a stochastic Bayesian inversion workflow. We show that the Bayesian inversion benefits from the use of machine learning prior, leading to improvements in the match to the observed 4D seismic signal as well as the injected and produced water volumes. In addition to a most probable solution, stochastic sampling of the Bayesian posterior distribution also produces uncertainty estimates which are valuable in such inversion problems. With this result, we extract multiple equiprobable realisations and define conditional bounds for the pressure and saturation changes, which we recognise as helpful for reservoir management and 4D seismic data assimilation into reservoir simulation models.