Gradual Deformation Method Research Articles

An improved stochastic inversion method for 3D elastic impedance under the prior constraints of random medium parameters

Three-dimensional (3D) elastic impedance inversion is widely used in oil and gas exploration with its excellent identification of reservoirs and noise immunity. However, modifying the prior information directly in inversion will undoubtedly destroy the spatial autocorrelation structure of the elastic impedance. Maintaining the elastic impedance spatial autocorrelation structure in seismic inversion is of great significance for guiding the reservoir identification and prediction. The 3D prior information construction method is proposed under the constraint of random medium parameters. This method establishes an intrinsic connection between random medium parameters and elastic impedance based on seismic data and well data, which provides a theoretical basis for simulating prior information satisfying the spatial correlation of the subsurface medium. To ensure the invariance of the spatial structure term in the inversion process, the random term and the spatial structure term in the simulation process are separated by the gradual deformation method (GDM), and the priori constrained objective function is further constructed to reduce the inversion uncertainty. Numerical example shows that the relative error of the proposed simulation method is smaller than that of conventional sequential Gaussian simulation, and the details are more accurately depicted based on the priori inversion results in this study. In addition, the elastic impedance equation based on the p-wave modulus is derived by combining with the reservoir sensitive parameters in field data. The field examples document that this method can effectively perform high-resolution geophysical prediction of underground thin-layer reservoirs.

Hybrid Inversion of Reservoir Parameters Based on Cosimulation and the Gradual Deformation Method

A new hybrid inversion methodology for petrophysical properties (porosity, water saturation, and volume of clay) based on partially stacked seismic angle gathers was presented. The innovation of this method is that it combines the advantages of high efficiency of analytical and semianalytical multistep petrophysical estimation with the advantages of spatial continuity and high resolution of geostatistical direct inversion of reservoir parameters. A trace-by-trace linearized amplitude versus offset (AVO) inversion and semianalytical petrophysical estimation approach could efficiently achieve a preliminary estimation of reservoir parameters under some assumptions and approximations. These laterally unconstrained preliminary estimation results are able to provide effective information for finer-resolution geostatistical simulation and improve the computational efficiency of stochastic optimization. The 3-D fast Fourier transform moving average (FFT-MA) geostatistical cosimulation and the stochastic optimization by gradual deformation method (GDM) improve the resolution and spatial continuity of the preliminary estimation results. At the same time, stochastic inversion overcomes the assumptions in the analytical inversion algorithm that the parameters need to conform to the logarithmic Gaussian distribution and the linear approximation of the Zoeppritz equation. The method was tested on a synthetic case and a real case. The results show that compared with the analytical and semianalytical methods, this hybrid inversion can obtain results with finer resolution and better spatial continuity, and the computational efficiency is significantly improved compared with the geostatistical direct inversion of reservoir parameters. Uncertainty evaluation can be obtained by performing statistical analysis on the multiple realizations of stochastic inversions.

Stochastic correlated hydraulic conductivity tensor calibration using gradual deformation

Quasi-point hydraulic properties (K) measured locally under laboratory or field conditions need to be upscaled to block-scale K-tensors for use in flow simulators. The upscaled model also needs to be calibrated to hydraulic head observations. The calibration must preserve spatial covariance, cross-covariance and non-linear relations between tensor components. We apply a new upscaling method that allows to compute and model the covariance between block K-tensor components. We use a gradual deformation method for calibration of simulated K-tensor fields to measured head data. Our method incorporates a new bivariate transform that preserves the non-linear relations between K-tensor components. The ensemble of calibrated realizations allows quantification of uncertainty of groundwater flow models. A comparison with PEST on a test case defining capture zones for water supply wells shows that our method calibrates better to measured heads than PEST, provides more realistic K-tensors and results in larger capture zones.

Two-level uncertainty assessment in stochastic seismic inversion based on the gradual deformation method

Geostatistical seismic inversion can combine seismic data, well data, and spatial continuity of the property of interest to obtain high-resolution reservoir models and evaluate uncertainties. Some workflows estimate global geostatistical parameters, such as correlation length, and keep them fixed in all simulations and inversions. This can introduce biases due to the sparsity of available well data and underestimate the uncertainty of inversion. A better approach is to incorporate the uncertainty in these global parameters. Lateral correlation length is one of the most difficult parameters to estimate. We have developed a seismic inversion method based on local gradual deformation method, which incorporates the uncertainty of lateral correlation length and provides a two-level uncertainty evaluation. We first estimate a uniform prior distribution of lateral correlation length from well data and additional geologic expert knowledge. After using fast Fourier transform (FFT) moving average simulations and local gradual deformation optimization, we obtain multiple realizations from which we could extract the lateral correlation lengths and calculate their posterior distribution. The FFT moving average method generates reservoir models by a convolution between a filter operator and a random noise field. The filter operator does not change during inversion, and the correlation structure of the random noise field could be changed by the local gradual deformation method to match the seismic data. A synthetic model test shows that the correlation lengths and the global probability distribution of the inverted results tend to the true geostatistical characteristics. The posterior distribution of the lateral correlation length narrows after inversion. Compared with conventional geostatistical seismic inversion techniques, uncertainties in the results increase because we incorporate the uncertainty in the global parameters. A real case also demonstrated that by modifying the random noise field locally, thin layers in a thick formation are well restored, even if they are not interpreted in advance.

Three-dimensional stochastic assimilation of gravity data in Lalor volcanogenic massive sulphide, Manitoba, Canada

We propose a new numerical workflow based on stochastic data integration where we merge a conceptual geological model, drillhole geophysical and geological logs, and surface geophysical data to compute a unified numerical model of a volcanogenic massive sulphide (VMS) deposit. The first step of the workflow consists in building a three-dimensional (3D) numerical conceptual model of the geology. This conceptual model, as well as geological logs, is then used to generate multiple equiprobable scenarios of the geology by means of multiple-point simulation (MPS). The MPS method studies high-order statistics in the space of a numerical conceptual model, making it possible to reproduce complex geological structures. We then use conventional conditional sequential Gaussian simulation, which is a method based on a node-by-node sequential process, to stochastically populate the geological grid with densities. For this purpose we use available density logs to simulate multiple equiprobable spatial distributions of the density at high spatial resolution within each geological unit separately. The stochastic high-resolution density models are iteratively combined by the gradual deformation method to minimize the difference between measured Bouguer anomaly data and the data computed on the combined realizations of density. Application of the proposed method to the Lalor deposit, a VMS deposit in Manitoba, Canada, produces a density model that honours the geology of the deposit and the Bouguer anomaly data. This unified model has the advantage to include all the available information (geological and density logs and surface geophysics) at scales appropriate for mining applications.

Calibration of categorical simulations by evolutionary gradual deformation method

Methods to simulate facies (or categorical) fields are numerous. However, calibration of simulated facies fields to large-scale or dynamic data still remains an important challenge due to the discrete nature of the fields, the non-linearity of the response with respect to the facies fields, and the non-derivability of the objective function used in calibration. A new gradual deformation method (GDM) is presented and tested for the calibration of facies realizations obtained by patch-multipoint simulation (MPS). The proposed method borrows ideas from pluriGaussian simulation, evolutionary algorithms, and GDM. Various test cases are considered: proportion maps, section of seismic amplitudes, inlet to outlet travel time along the shortest path, and water-cut curves obtained with a flow simulator. Both conditional/unconditional MPS simulations and 2D/3D problems are considered. In all studied test cases, the new GDM approach has provided excellent calibration to the target variables. The method is general as it can be used in conjunction with any facies simulator.

Stochastic seismic inversion based on an improved local gradual deformation method

A new stochastic seismic inversion method based on the local gradual deformation method is proposed, which can incorporate seismic data, well data, geology and their spatial correlations into the inversion process. Geological information, such as sedimentary facies and structures, could provide significant a priori information to constrain an inversion and arrive at reasonable solutions. The local a priori conditional cumulative distributions at each node of model to be inverted are first established by indicator cokriging, which integrates well data as hard data and geological information as soft data. Probability field simulation is used to simulate different realizations consistent with the spatial correlations and local conditional cumulative distributions. The corresponding probability field is generated by the fast Fourier transform moving average method. Then, optimization is performed to match the seismic data via an improved local gradual deformation method. Two improved strategies are proposed to be suitable for seismic inversion. The first strategy is that we select and update local areas of bad fitting between synthetic seismic data and real seismic data. The second one is that we divide each seismic trace into several parts and obtain the optimal parameters for each part individually. The applications to a synthetic example and a real case study demonstrate that our approach can effectively find fine-scale acoustic impedance models and provide uncertainty estimations.

Nonlinear inversion based on Metropolis sampling algorithm

Nonlinear inversion based on Metropolis sampling algorithm is formulated in the Bayesian framework. As one kind of Monte Carl non-linear inversions, it can effectively integrate high frequency information of well logging data, and obtain inversion results with a higher resolution. Firstly, we get the priori information through fast Fourier transform moving average (FFT-MA) and gradual deformation method (GDM). Second, we structure likelihood function. Then we apply Metropolis algorithm in order to obtain an exhaustive characterization of the posteriori probability density. FFT-MA is a kind of efficient simulation method. Combined with GDM, it can constantly modify reservoir model and keep the spatial structure unchanged until it matches the observed seismic data. According to the model trial and real data processing, we can conclude that nonlinear inversion based on Metropolis sampling algorithm provide reasonable elastic parameter information, especially it improves the resolution of P-wave velocity. Even when the signal noise ratio (SNR) is relatively low, it can still show reasonable elastic parameter information, which proves the effectiveness of the proposed method. The inversion resolution of P-wave and S-wave impedances is higher than elastic parameters inversion if we do not consider the noise. ©, 2015, Science Press. All right reserved.

Simultaneous inversion of petrophysical parameters based on geostatistical a priori information

The high-resolution nonlinear simultaneous inversion of petrophysical parameters is based on Bayesian statistics and combines petrophysics with geostatistical a priori information. We used the fast Fourier transform-moving average (FFT-MA) and gradual deformation method (GDM) to obtain a reasonable variogram by using structural analysis and geostatistical a priori information of petrophysical parameters. Subsequently, we constructed the likelihood function according to the statistical petrophysical model. Finally, we used the Metropolis algorithm to sample the posteriori probability density and complete the inversion of the petrophysical parameters. We used the proposed method to process data from an oil field in China and found good match between inversion and real data with high-resolution. In addition, the direct inversion of petrophysical parameters avoids the error accumulation and decreases the uncertainty, and increases the computational efficiency.

A generalization of the local gradual deformation method using domain parameterization

Reservoir model needs to be constrained by various data, including dynamic production data. Reservoir heterogeneities are usually described using geostatistical approaches. Constraining geological/geostatistical model realizations by dynamic data is generally performed through history matching, which is a complex inversion process and requires a parameterization of the geostatistical realizations for model updating. However, the parameterization techniques are still not very efficient and need to be improved.In recent years, the local gradual deformation method has been widely used to parameterize geostatistical realizations. The domain deformation technique has also been developed to improve the history matching efficiency. Both methods can smoothly modify model realizations while conserving spatial geostatistical properties. The first one consists in locally combining two or more realizations while the second one allows the optimization process to change the model realization via the variation of the shape of geometrical domains. In this paper, we generalize the local gradual deformation method by adding the possibility to change the geometry of local zones through the domain deformation. This generalization provides a greater flexibility in the definition of the local domains for the local gradual deformation method. In addition, we propose a new way to initialize the realization which guarantees a good initial point for the optimization and potentially improves the efficiency of history matching.

Multiscale Parameterization of Petrophysical Properties for Efficient History-Matching

The prediction of fluid flows within hydrocarbon reservoirs requires the characterization of petrophysical properties. Such characterization is performed on the basis of geostatistics and history-matching; in short, a reservoir model is first randomly drawn, and then sequentially adjusted until it reproduces the available dynamic data. Two main concerns typical of the problem under consideration are the heterogeneity of rocks occurring at all scales and the use of data of distinct resolution levels. Therefore, referring to sequential Gaussian simulation, this paper proposes a new stochastic simulation method able to handle several scales for both continuous or discrete random fields. This method adds flexibility to history-matching as it boils down to the multiscale parameterization of reservoir models. In other words, reservoir models can be updated at either coarse or fine scales, or both. Parameterization adapts to the available data; the coarser the scale targeted, the smaller the number of unknown parameters, and the more efficient the history-matching process. This paper focuses on the use of variational optimization techniques driven by the gradual deformation method to vary reservoir models. Other data assimilation methods and perturbation processes could have been envisioned as well. Last, a numerical application case is presented in order to highlight the advantages of the proposed method for conditioning permeability models to dynamic data. For simplicity, we focus on two-scale processes. The coarse scale describes the variations in the trend while the fine scale characterizes local variations around the trend. The relationships between data resolution and parameterization are investigated.

Calibrating transmissivities from piezometric heads with the gradual deformation method: An application to the Culebra Dolomite unit at the Waste Isolation Pilot Plant (WIPP), New Mexico, USA

Summary The gradual deformation method (GDM) is a geostatistical parameterization technique introduced in reservoir engineering for compelling reservoir models to respect production history. It makes it possible to adjust the spatial distribution of the hydraulic properties populating the reservoir model so that the flow responses simulated for the modified model closely replicate the production data collected on the field. The work presented in this paper investigates the potential of the GDM to solve inverse problems in hydrogeology, or subsurface hydrology, and produce heterogeneous transmissivity fields conditional to both transmissivity and piezometric head measurements within a stochastic context. It focuses on the calibration of the Culebra Dolomite unit, which lies approximately 450 m above the repository horizon of the Waste Isolation Pilot Plant (WIPP) in southeastern New Mexico, USA. Results show that the proposed version of the GDM, with a local deformation in each sub-domain neighboring a well, is efficient and competitive compared to other geostatistical-based inverse methods such as the pilot point method.

Updating multipoint simulations using the ensemble Kalman filter

In the last two decades, the multipoint simulation (MPS) method has been developed and increasingly used for building complex geological facies models that are conditioned to geological and geophysical data. In the meantime, the ensemble Kalman filter (EnKF) approach has been developed and recognized as a promising way for assimilating dynamic production data into reservoir models. So far, the EnKF approach is proven efficient for updating continuous model parameters that have a linear statistical relation with the flow responses. It remains challenging to extend the EnKF approach to updating complex geological facies models generated by MPS, while preserving their geological and statistical consistency.In this paper, we introduce a new method for parameterizing geostatistical reservoir models generated by MPS. It is mathematically proven that updating these parameters during a history matching process does not compromise the hard data conditioning and the geological and statistical consistency of the reservoir model defined by the training image and other information including global facies proportions, trend maps etc. This method is an alternative to the gradual deformation method but has an enlarged search space for covering possible solutions. Based on the above parameterization, we present two algorithms of using EnKF approach to update multipoint simulations to dynamic data. We also present results of using the above methodology to condition a sector model of a fluvial reservoir to dynamic data.

History Matching of Production and 4D Seismic Data: Application to the Girassol Field, Offshore Angola

Time-lapse seismic provides a source of valuable information about the evolution in space and time of the distribution of hydrocarbons inside reservoirs. Seismic monitoring improves our understanding of production mechanisms and makes it possible to optimize the recovery of hydrocarbons. Although 4D seismic data are increasingly used by oil companies, they are often qualitative, due to the lack of suitable interpretation techniques.Recent modeling experiments have shown that the integration of 4D seismic data for updating reservoir flow models is feasible. However, methodologies based on sequential interpretation of 4D seismic data, trial and error processes and fluid flow simulation tests require a great effort from integrated teams. The development of assisted history matching techniques is a significant improvement towards a quantitative use of 4D seismic data in reservoir modeling.This paper proposes an innovative methodology based on advanced history matching solutions to constrain 3D stochastic reservoir models to both production history and 4D seismic attributes. In this approach, geostatistical modeling, upscaling, fluid flow simulation, downscaling and petro-elastic modeling are integrated into the same history matching workflow. Simulated production history and 4D seismic attributes are compared to real data using an objective function, which is minimized with a new optimization algorithm based on response surface fitting. The gradual deformation method is used to constrain the facies realization, globally or locally, which populates the geological model at the fine scale. Moreover, a new method is proposed to update facies proportions during the optimization process according to 4D monitoring information.We present here a successful application to the Girassol field. Girassol is a large, complex and faulted turbidite field, located offshore Angola. First, a detailed geostatistical geological model was built to describe reservoir heterogeneity at the fine scale, while respecting 3D base seismic data. Second, the model was constrained to production data and 4D seismic attributes, applying gradual deformation to facies realizations and varying facies proportions. The integration of 4D seismic data led to better production forecasts and improved predictions confirmed by a new seismic survey shot two years after the history matching period. 4D seismic data also contributed to better characterize the spatial distribution of heterogeneities in the field. As a result, the fine scale geological model was improved consistently with respect to the fluid flow simulation model and the observed data. The Girassol study, already presented in (Roggero et al., 2007, 2008), has been updated with recent information and a more detailed presentation concerning the construction of the geological model based on 3D seismic data.

Preselection of Reservoir Models From a Geostatistics-Based Petrophysical Seismic Inversion

Summary A new geostatistics-based seismic-inversion method is introduced in this paper for determining reservoir models consistent with base seismic information. The proposed methodology entails two steps, with only the second being examined in this paper. First, cubes of acoustic velocities or impedances are derived from seismic inversion. Second, these data are incorporated into a matching process to identify reservoir models leading to acoustic responses close to the reference acoustic data. The parameterization of the facies and petrophysical properties populating the reservoir models is based upon the gradual-deformation method (GDM), which relies on geostatistical concepts. This particular feature makes it possible to change the spatial distribution of the property of interest from a few parameters while preserving its spatial variability. The matching process is driven from a global optimization algorithm known as the particle-swarm optimization (PSO). Such a global approach is reasonable for the problem considered because the forward modeling is very fast. A variant of the PSO algorithm is implemented to take advantage of the GDM properties. This approach yields reservoir models that honor the seismic data better than those derived from stochastic simulation only with seismic used as a secondary variable. A numerical experiment is then presented to stress the applicability of the proposed matching methodology: The GDM-based PSO approach is used to identify facies reservoir models and water/oil contact consistent with some reference acoustic P-wave impedances.

Approximate Derivative Computations for the Gradient-Based Optimization Methods in the Local Gradual Deformation for History Matching

Reservoir characterization needs the integration of various data through history matching, especially dynamic information such as production or four-dimensional seismic data. To update geostatistical realizations, the local gradual deformation method can be used. However, history matching is a complex inverse problem, and the computational effort in terms of the number of reservoir simulations required in the optimization procedure increases with the number of matching parameters. History matching large fields with a large number of parameters has been an ongoing challenge in reservoir simulation. This paper presents a new technique to improve history matching with the local gradual deformation method using the gradient-based optimizations. The new approach is based on the approximate derivative calculations using the partial separability of the objective function. The objective function is first split into local components, and only the most influential parameters in each component are used for the derivative computation. A perturbation design is then proposed to simultaneously compute all the derivatives with only a few simulations. This new technique makes history matching using the local gradual deformation method with large numbers of parameters tractable.

A multiscale method for subsurface inverse modeling: Single-phase transient flow

High-resolution geologic models that incorporate observed state data are expected to effectively enhance the reliability of reservoir performance prediction. One of the major challenges faced is how to solve the large-scale inverse modeling problem, i.e., to infer high-resolution models from the given observations of state variables that are related to the model parameters according to some known physical rules, e.g., the flow and transport partial differential equations. There are typically two difficulties, one is the high-dimensional problem and the other is the inverse problem. A multiscale inverse method is presented in this work to attack these problems with the aid of a gradient-based optimization algorithm. In this method, the model responses (i.e., the simulated state data) can be efficiently computed from the high-resolution model using the multiscale finite-volume method. The mismatch between the observations and the multiscale solutions is then used to define a proper objective function, and the fine-scale sensitivity coefficients (i.e., the derivatives of the objective function with respect to each node’s attribute) are computed by a multiscale adjoint method for subsequent optimization. The difficult high-dimensional optimization problem is reduced to a one-dimensional one using the gradient-based gradual deformation method. A synthetic single-phase transient flow example problem is employed to illustrate the proposed method. Results demonstrate that the multiscale framework presented is not only computationally efficient but also can generate geologically consistent models. By preserving spatial structure for inverse modeling, the method presented overcomes the artifacts introduced by the multiscale simulation and may enhance the prediction ability of the inverse-conditional realizations generated.

Uncertainty quantification of reservoir performance predictions using a stochastic optimization algorithm

The simultaneous perturbation stochastic approximation (SPSA) algorithm is modified to obtain a stochastic Gaussian search direction (SGSD) algorithm for automatic history matching. The search direction in the SGSD algorithm is obtained by simultaneously perturbing the reservoir model with unconditional realizations from a Gaussian distribution. This search direction has two nice properties: (1) it is always downhill in the vicinity of the current iterate and (2) the expectation of the stochastic search direction is an approximate quasi-Newton direction with a prior covariance matrix used as the approximate inverse Hessian matrix. For Gaussian reservoir models, we argue and demonstrate that the SGSD algorithm may generate more geologically realistic reservoir description than is obtained with the original SPSA algorithm. It is shown that the SGSD algorithm represents an approximation of the gradual deformation method but its search direction has the desirable properties that are lacking in all gradual deformation methods. The SGSD algorithm is successfully applied to the well-known PUNQ-S3 test case to generate a maximum a posteriori estimate and for uncertainty quantification of reservoir performance predictions using the randomized maximum likelihood method.

Reservoir Characterization from Insufficient Static Data Using Gradual Deformation Method with Ensemble Kalman Filter

Reservoir characterization is critical in order to estimate reserves and optimize oil and gas production. Ensemble Kalman filter characterizes the spatial distribution of reservoir parameters using covariances between static and dynamic data. Ensemble Kalman filter can rapidly provide results reflecting its overall tendency of parameter distribution, but may not characterize them in detail because ensemble Kalman filter does not minimize an objective function. Gradual deformation method is one of the well known non-gradient-based methods, which guarantees a global minimum theoretically. However, if permeability samples available do not represent the whole distribution, it can be hard for gradual deformation method to reduce the objective function sufficiently. In this study, results of ensemble Kalman filter are utilized as input permeability fields in gradual deformation method so that the demerit of each method is compensated by the merit. Although samples from a reservoir have no information on extreme permeability values, the developed gradual deformation method model with ensemble Kalman filter reliably characterizes hidden extreme values.

Efficient Inverse Modeling Using Probabilistic Sensitivity Analysis with Gradual Deformation Algorithm

One of the challenging topics in inverse modeling is numerical efficiency and efficient optimization algorithms have been researched. In optimization algorithms, we need efficient perturbations to find a global minimum. Geostatistical methods combined with perturbation have an important role in generating input parameter fields. However, in cases where the given static sample data are not pertinent considering dynamic data, it is difficult to find a global minimum in the inverse modeling. To overcome this problem, we propose the concept of probabilistic sensitivity analysis. From the sensitivity analysis, cumulative probability distribution functions, which are used as indicators of perturbations, are constructed. Then, they are used for field generations in the gradual deformation method. By enforcing the restart of the optimization process at a local minimum state, the inverse modeling becomes stable and efficient. From the applications using various synthetic reservoirs, the proposed scheme works even in cases where reservoirs have hidden abnormal permeability values, which are not represented by sampling data.