Multiple-point Geostatistical Method Research Articles

Hierarchical Multiple-Point Geostatistical Modeling Method and Application Based on Braided River Reservoir Architecture

The Lower Shihezi Formation of the Daniudi gas field in the Ordos Basin is a typical reservoir of a braided river system in an alluvial plain, characterized by extensive braided river development, parallel development from the near source to the center of the basin, and frequent interweaving and cut stacking, as well as a complex deposition process that has seen frequent river channel changes. The braided river belt, braided channel, channel bar inside the river, and interlayer within the channel bar constitute a hierarchical and complicated architectural feature, which poses a great challenge to accurately characterize this type of reservoir for modeling. We proposed a hierarchical, level-by-level embedding, and progressive multiple-point geostatistical modeling strategy that is refined layer by layer according to a 5–3 level architectural unit hierarchy, with the modeling results of each level providing constraints for the next level modeling. The hierarchical geological model based on the combination of qualitative architectural anatomy and quantitative pre-architecture unit scale is critical in guiding the efficient development of the remaining gas in the braided river reservoir in Daniudi.

Reservoir Modeling of Braided River Reservoirs Based on Geological Knowledge Database: A Case Study of P1x Formation of the Daniudi Gas Field, Ordos Basin, China

Abstract The sandy braided river depositional system developed in the Lower Shihezi Formation of the Daniudi gas field, Ordos Basin. It has the characteristics of frequent migration and oscillation of braided channels and large well spacing, making it challenging to portray the braided river sand body in this area, bringing uncertainty to the 3D geological modeling of the reservoir. This study takes the primary gas reservoir H1 member as an example. It establishes a quantitative geological knowledge database for the reservoir by statistically fitting the correlation equations between the braided channel and channel bar in the planes and profiles to reduce the uncertainty of reservoir modeling. This study combines the multisource and multiscale information from modern sedimentation, field outcrops, and tank simulation experiments of the braided river. From the data, the distribution intervals for the thickness and width of the braided channel sand body in the H1 member are 1–22 m and 7–320 m, respectively, and the thickness, width, and length distribution intervals of the sand body in the channel bar are 3–30 m, 80–1500 m, and 240–4200 m, respectively. A 3D training image is established using the object-based simulation method based on the H1 member’s well data and combining the quantitative parameters of various microfacies in the geological knowledge database. The multiple-point geostatistical modeling method is applied to establish a sedimentary microfacies model. The model’s uncertainty is reduced through multi-information fusion constraint modeling, providing a reliable basis for guiding the prediction of the remaining gas in the Daniudi gas field.

MULTIPLE-POINT GEOSTATISTICS TO DERIVE MISSING SURFACE DISPLACEMENT VALUES OF A GLACIER INFERRED FROM DINSAR

Abstract. Glacier displacements play a vital role in the monitoring and understanding of glacier dynamics. Glacier displacement fields are typically retrieved from pre- and post-event SAR images using DInSAR. The glacier displacement map produced by DInSAR contains missing values due to decorrelation of the SAR images. This study demonstrates the utility of direct sampling—a well-established multiple-point geostatistics method—for deriving those missing values. Univariate and bivariate implementations of direct sampling are employed. In the univariate implementation, missing values are derived in single displacement map, whereas in bivariate implementation gaps in two displacement maps are filled simultaneously. Evaluation is carried out by artificially generated missing values on the displacement map of different shapes and sizes at different locations with known values. Imposed missing values are then reconstructed and compared with the original values. Reconstruction results of both implementations were compared with ordinary kriging using qualitative and quantitative measures. The study shows that with an increase in the size of such discontinuities, ordinary kriging predictions deteriorate significantly, whereas only slight decrease in reconstruction accuracy is observed for direct sampling. The results of both implementations are similar with the univariate implementation performing slightly better over bivariate implementation because the information from ancillary data is only partly complementary for bivariate reconstructions. Direct sampling performed better than ordinary kriging with accuracy below the DInSAR detection limit. This study concludes that multiple-point geostatistics is an effective method for deriving missing values in DInSAR derived displacement maps. Direct sampling based reconstruction is fast and straightforward to implement.

Conditional multiple-point geostatistical simulation for unevenly distributed sample data

To expand the applicability/versatility of multiple-point geostatistical (MPS) methods to the unevenly distributed sample data acquired from geological and environmental surveys, this paper presents a conditional MPS-based simulation method which considers the distribution characteristics of sample data adequately. In this work, we mainly focus on the improvement of two key steps in MPS methods, i.e. the selection of simulation paths and the construction of data events, aiming at mitigating the adverse effects of unevenly distributed conditioning data. First, a simulation path sensitive to the distribution density of informed samples is adopted to ensure that each simulation of the unknown nodes in a simulation grid is done from the location with the highest density of informed nodes around. Second, a stable data event is obtained by evenly extracting several informed nodes closest to the current node from each subarea. This improvement avoids the excessive concentration of the nodes in a data event, so that the nodes in an obtained data event are more evenly distributed around the current node. The two improvements are embedded into a widely used MPS method, the direct sampling. Several 2D and 3D synthetic experiments with categorical or continuous variables are used to test the proposed method. The results demonstrate their applicability in characterizing heterogeneous phenomena when faced with unevenly distributed conditioning data.

Using input-adaptive dictionaries trained by the method of optimal directions to estimate the permeability model of a reservoir

The thirst for perfect characterization of petroleum reservoirs is yet to be quenched. Sparse transform is an image compression method the full potential of which is still not employed by the geoscientists in the reservoir characterization workflow. Sparse transform is the state of representing an image model based on a weighted linear combination of a sparse set of columns from a matrix, called dictionary. The required dictionary in the sparse transform scheme is obtained by learning over a training set of model images using dictionary learning algorithms. Meanwhile, the sparse coding algorithms are used to select the sparse proper subset of columns from the trained dictionary and specify appropriate weight to each column in the linear combination. To enhance the results, the property of internal structure and generality for the implicit dictionaries are integrated with the property of adaptability for the explicit dictionaries to regularize and enhance the effectiveness of the input-adaptive dictionary in reconstructing the approximate model.This phenomenon can be employed in the geosciences to estimate the reservoir model if the dictionary is trained over a set of geologic models of a delighted property. The geologic models should be diverse enough to represent the possible variety of the desired property and the limited number of columns from the dictionary should be properly selected such that it adequately captures the variety inherent within the geologic models. In this study, the variety of the geologic models are provided by manipulating two training images by FilterSim multiple-point geostatistical method and generating a large set of realizations. The training images are extracted from a spectral decomposition profile to represent a stationary fracture system and a nonstationary delta system model in an Iranian petroleum reservoir. The dictionary learning algorithm is an alternation of dictionary updating and sparse coding steps. In this paper, the method of optimal directions is used as dictionary learning algorithm and the least angle regression with shrinkage algorithm is used as the sparse coding method.The resultant approximate model achieved by the sparse transform compression scheme is comparable to the reference models both in terms of physical structures and fluid flow characteristics. For the current study, the approximate fracture system model is shown to be superior to 86.16% of the model images in the corresponding population space in terms of fluid flow characteristics. In the case of the delta system, the approximate model is superior to 99.17% of the model images in the population space. Further experiments indicate similar deductions using other methods for dictionary learning, sparse coding, and multiple-point geostatistical modeling.

Bridging iterative Ensemble Smoother and multiple-point geostatistics for better flow and transport modeling

Inverse method can be used to fill the gap between huge amount of data from sensors and complex groundwater model. The iterative Ensemble Smoother (iES) is one of the most efficient algorithms applied to groundwater modeling for data assimilation. However, the iES only works for multi-Gaussian fields, because two-point statistics are used to estimate the co-relation between state variables and parameters. In curvilinear geometries, such as sinuous channels in fluvial deposits, the distribution of hydraulic conductivity is non-multiGaussian. Multiple-Point Geostatistics (MPG) method has gained popularity for modeling curvilinear structures by conditioning on directly measured data, such as conductivities. This paper is aimed at bridging the iES and MPG method via pilot points to deal with inverse problem for further conditioning on indirect data, such as piezometric head, in non-Gaussian fields. As a result, the better flow and transport modeling will be achieved because both data and the concept model (e.g., geological structures) are honored after data assimilation. To do that, the iES is used to update conductivities at pilot points by assimilating indirect data, then the updated values at pilot points together with measured conductivities will be used as hard data to model hydraulic conductivity field via Direct Sampling, an MPG method. A synthetic example was used to demonstrate the methodology in terms of characterization of conductivity and flow and transport predictions. The results show that this new approach can not only assimilate dynamic data into groundwater flow model but also preserve curvilinear structures.

Grade and Tonnage Uncertainty Analysis of an African Copper Deposit Using Multiple-Point Geostatistics and Sequential Gaussian Simulation

Spatial uncertainty analysis is a complex and difficult task for orebody estimation in the mining industry. Conventional models (kriging and its variants) with variogram-based statistics fail to capture the spatial complexity of an orebody. Due to this, the grade and tonnage are incorrectly estimated resulting in inaccurate mine plans, which lead to costly financial decision. Multiple-point geostatistical simulation model can overcome the limitations of the conventional two-point spatial models. In this study, a multiple-point geostatistical method, namely SNESIM, was applied to generate multiple equiprobable orebody models for a copper deposit in Africa, and it helped to analyze the uncertainty of ore tonnage of the deposit. The grade uncertainty was evaluated by sequential Gaussian simulation within each equiprobable orebody models. The results were validated by reproducing the marginal distribution and two- and three-point statistics. The results show that deviations of volume of the simulated orebody models vary from − 3 to 5% compared to the training image. The grade simulation results demonstrated that the average grades from the different simulation are varied from 3.77 to 4.92% and average grade 4.33%. The results also show that the volume and grade uncertainty model overestimates the orebody volume as compared to the conventional orebody. This study demonstrates that incorporating grade and volume uncertainty leads to significant changes in resource estimates.

A Fast Approximation for Seismic Inverse Modeling: Adaptive Spatial Resampling

Seismic inverse modeling, which transforms appropriately processed geophysical data into the physical properties of the Earth, is an essential process for reservoir characterization. This paper proposes a work flow based on a Markov chain Monte Carlo method consistent with geology, well-logs, seismic data, and rock-physics information. It uses direct sampling as a multiple-point geostatistical method for generating realizations from the prior distribution, and Metropolis sampling with adaptive spatial resampling to perform an approximate sampling from the posterior distribution, conditioned to the geophysical data. Because it can assess important uncertainties, sampling is a more general approach than just finding the most likely model. However, since rejection sampling requires a large number of evaluations for generating the posterior distribution, it is inefficient and not suitable for reservoir modeling. Metropolis sampling is able to perform an equivalent sampling by forming a Markov chain. The iterative spatial resampling algorithm perturbs realizations of a spatially dependent variable, while preserving its spatial structure by conditioning to subset points. However, in most practical applications, when the subset conditioning points are selected at random, it can get stuck for a very long time in a non-optimal local minimum. In this paper it is demonstrated that adaptive subset sampling improves the efficiency of iterative spatial resampling. Depending on the acceptance/rejection criteria, it is possible to obtain a chain of geostatistical realizations aimed at characterizing the posterior distribution with Metropolis sampling. The validity and applicability of the proposed method are illustrated by results for seismic lithofacies inversion on the Stanford VI synthetic test sets.

LSHSIM: A Locality Sensitive Hashing based method for multiple-point geostatistics

Reservoir modeling is a very important task that permits the representation of a geological region of interest, so as to generate a considerable number of possible scenarios. Since its inception, many methodologies have been proposed and, in the last two decades, multiple-point geostatistics (MPS) has been the dominant one. This methodology is strongly based on the concept of training image (TI) and the use of its characteristics, which are called patterns. In this paper, we propose a new MPS method that combines the application of a technique called Locality Sensitive Hashing (LSH), which permits to accelerate the search for patterns similar to a target one, with a Run-Length Encoding (RLE) compression technique that speeds up the calculation of the Hamming similarity. Experiments with both categorical and continuous images show that LSHSIM is computationally efficient and produce good quality realizations. In particular, for categorical data, the results suggest that LSHSIM is faster than MS-CCSIM, one of the state-of-the-art methods.

Gap-Filling of Landsat 7 Imagery Using the Direct Sampling Method

The failure of the Scan Line Corrector (SLC) on Landsat 7 imposed systematic data gaps on retrieved imagery and removed the capacity to provide spatially continuous fields. While a number of methods have been developed to fill these gaps, most of the proposed techniques are only applicable over relatively homogeneous areas. When they are applied to heterogeneous landscapes, retrieving image features and elements can become challenging. Here we present a gap-filling approach that is based on the adoption of the Direct Sampling multiple-point geostatistical method. The method employs a conditional stochastic resampling of known areas in a training image to simulate unknown locations. The approach is assessed across a range of both homogeneous and heterogeneous regions. Simulation results show that for homogeneous areas, satisfactory results can be obtained by simply adopting non-gap locations in the target image as baseline training data. For heterogeneous landscapes, bivariate simulations using an auxiliary variable acquired at a different date provides more accurate results than univariate simulations, especially as land cover complexity increases. Apart from recovering spatially continuous fields, one of the key advantages of the Direct Sampling is the relatively straightforward implementation process that relies on relatively few parameters.

Groundwater Level Mapping Using Multiple-Point Geostatistics

Methods based on two-point geostatistics have been routinely used to interpolate random variables such as groundwater level and concentration and to estimate their values at un-sampled locations. These methods use the observed data to analyze spatial two-point correlations and ignore the higher order moments that may play a key role in the characterization of complex patterns. In this work, a multiple-point geostatistics method is applied to interpolate groundwater level data. To do this, the ensemble simulated groundwater level maps after modeling full physics are used as training images to derive the multiple correlations. An example shows the advantages of using multiple-point geostatistics compared with the traditional two-point geostatistics methods (such as Kriging) for the interpolation of groundwater level data in a complex geological formation.

A multiple-point geostatistical method based on geological vector information

The reproduction of the non-stationary distribution and detailed characteristics of geological bodies is the main difficulty of reservoir modeling. Recently developed multiple-point geostatistics can represent a stationary geological body more effectively than traditional methods. When restricted to a stationary hypothesis, multiple-point geostatistical methods cannot simulate a non-stationary geological body effectively, especially when using non-stationary training images (TIs). According to geologic principles, the non-stationary distribution of geological bodies is controlled by a sedimentary model. Therefore, in this paper, we propose auxiliary variables based on the sedimentary model, namely geological vector information (GVI). GVI can characterize the non-stationary distribution of TIs and simulation domains before sequential simulation, and the precision of data event statistics will be enhanced by the sequential simulation’s data event search area limitations under the guidance of GVI. Consequently, the reproduction of non-stationary geological bodies will be improved. The key features of this method are as follows: (1) obtain TIs and geological vector information for simulated areas restricted by sedimentary models; (2) truncate TIs into a number of sub-TIs using a set of cut-off values such that each sub-TI is stationary and the adjacent sub-TIs have a certain similarity; (3) truncate the simulation domain into a number of sub-regions with the same cut-off values used in TI truncation, so that each sub-region corresponds to a number of sub-TIs; (4) use an improved method to scan the TI or TIs and construct a single search tree to restore replicates of data events located in different sub-TIs; and (5) use an improved conditional probability distribution function to perform sequential simulation. A FORTRAN program is implemented based on the SNESIM.

Stochastic simulation of geological data using isometric mapping and multiple-point geostatistics with data incorporation

Constrained by current hardware equipment and techniques, acquisition of geological data sometimes is difficult or even impossible. Stochastic simulation for geological data is helpful to address this issue, providing multiple possible results of geological data for resource prediction and risk evaluation. Multiple-point geostatistics (MPS) being one of the main branches of stochastic simulation can extract the intrinsic features of patterns from training images (TIs) that provide prior information to limit the under-determined simulated results, and then copy them to the simulated regions. Because the generated models from TIs are not always linear, some MPS methods using linear dimensionality reduction are not suitable to deal with nonlinear models of TIs. A new MPS method named ISOMAPSIM was proposed to resolve this issue, which reduces the dimensionality of patterns from TIs using isometric mapping (ISOMAP) and then classifies these low-dimensional patterns for simulation. Since conditional models including hard data and soft data influence the simulated results greatly, this paper further studies ISOMAPSIM using hard data and soft data to obtain more accurate simulations for geological modeling. Stochastic simulation of geological data is processed respectively under several conditions according to different situations of conditional models. The tests show that the proposed method can reproduce the structural characteristics of TIs under all conditions, but the condition using soft data and hard data together performs best in simulation quality; moreover, the proposed method shows its advantages over other MPS methods that use linear dimensionality reduction.

Multiple-point geostatistical simulation of dykes: application at Sungun porphyry copper system, Iran

Variogram-based methods are not capable of capturing high (>2) order statistics since the variogram measures the relationship between two points at a time only. Multiple-point geostatistics (MPS) has brought new insights into many geological modeling problems. The application of MPS methods has been well documented in realizing complex geological patterns. These methods have often been used in reservoir characterization since their advent in recent decades. The frequent non-linear behaviors of geologic continuity are not limited to reservoirs, but mineral deposits bear complicated formations in many cases. Relying on the power of MPS methods and considering the complexity of geological scenarios in mineral deposits, we have applied MPS in the modeling of mineral deposits. A training image (TI) is produced using geological data from upper horizons of a porphyry copper ore deposit which have been mined out during the previous mining operations. In this study, the SNESIM algorithm has been used. A number of realizations are produced using this multiple-point geostatistical method. Extensive validation steps are performed considering the TI as the reference model. These validations first show that the TI is representative for the domain under study and also illustrates some degrees of similarity between the TI and the realizations. Despite simplifications made to the problem, the application of MPS in mineral deposit modeling still faces many challenges.

Analyzing the effects of geological and parameter uncertainty on prediction of groundwater head and travel time

Abstract. Uncertainty of groundwater model predictions has in the past mostly been related to uncertainty in the hydraulic parameters, whereas uncertainty in the geological structure has not been considered to the same extent. Recent developments in theoretical methods for quantifying geological uncertainty have made it possible to consider this factor in groundwater modeling. In this study we have applied the multiple-point geostatistical method (MPS) integrated in the Stanford Geostatistical Modeling Software (SGeMS) for exploring the impact of geological uncertainty on groundwater flow patterns for a site in Denmark. Realizations from the geostatistical model were used as input to a groundwater model developed from Modular three-dimensional finite-difference ground-water model (MODFLOW) within the Groundwater Modeling System (GMS) modeling environment. The uncertainty analysis was carried out in three scenarios involving simulation of groundwater head distribution and travel time. The first scenario implied 100 stochastic geological models all assigning the same hydraulic parameters for the same geological units. In the second scenario the same 100 geological models were subjected to model optimization, where the hydraulic parameters for each of them were estimated by calibration against observations of hydraulic head and stream discharge. In the third scenario each geological model was run with 216 randomized sets of parameters. The analysis documented that the uncertainty on the conceptual geological model was as significant as the uncertainty related to the embedded hydraulic parameters.

A reservoir skeleton-based multiple point geostatistics method

Traditional stochastic reservoir modeling, including object-based and pixel-based methods, cannot solve the problem of reproducing continuous and curvilinear reservoir objects. The paper first dives into the various stochastic modeling methods and extracts their merits, then proposes the skeleton-based multiple point geostatistics (SMPS) for the fluvial reservoir. The core idea is using the skeletons of reservoir objects to restrict the selection of data patterns. The skeleton-based multiple point geostatistics consists of two steps. First, predicting the channel skeleton (namely, channel centerline) by using the method in object-based modeling. The paper proposes a new method of search window to predict the skeleton. Then forecasting the distributions of reservoir objects using multiple point geostatistics with the restriction of channel skeleton. By the restriction of channel centerline, the selection of data events will be more reasonable and the realization will be achieved more really. The checks by the conceptual model and the real reservoir show that SMPS is much better than Sisim (sequential indicator simulation), Snesim (Single Normal Equation Simulation) and Simpat (simulation with patterns) in building the fluvial reservoir model. This new method will contribute to both the theoretical research of stochastic modeling and the oilfield developments of constructing highly precise reservoir geological models.

A Practical Data Integration Approach to History Matching: Application to a Deepwater Reservoir

Summary This paper presents an innovative methodology to integrate prior geologic information, well-log data, seismic data, and production data into a consistent 3D reservoir model. Furthermore, the method is applied to a real channel reservoir from the African coast. The methodology relies on the probability-perturbation method (PPM). Perturbing probabilities rather than actual petrophysical properties guarantees that the conceptual geologic model is maintained and that any history-matching-related artifacts are avoided. Creating reservoir models that match all types of data are likely to have more prediction power than methods in which some data are not honored. The first part of the paper reviews the details of the PPM, and the next part of this paper describes the additional work that is required to history-match real reservoirs using this method. Then, a geological description of the reservoir case study is provided, and the procedure to build 3D reservoir models that are only conditioned to the static data is covered. Because of the character of the field, the channels are modeled with a multiple-point geostatistical method. The channel locations are perturbed in a manner such that the oil, water, and gas rates from the reservoir more accurately match the rates observed in the field. Two different geologic scenarios are used, and multiple history-matched models are generated for each scenario. The reservoir has been producing for approximately 5 years, but the models are matched only to the first 3 years of production. Afterward, to check predictive power, the matched models are run for the last 1½ years, and the results compare favorably with the field data. Introduction Reservoir models are constructed to better understand reservoir behavior and to better predict reservoir response. Economic decisions are often based on the predictions from reservoir models; therefore, such predictions need to be as accurate as possible. To achieve this goal, the reservoir model should honor all sources of data, including well-log, seismic, geologic information, and dynamic (production rate and pressure) data. Incorporating dynamic data into the reservoir model is generally known as history matching. History matching is difficult because it poses a nonlinear inverse problem in the sense that the relationship between the reservoir model parameters and the dynamic data is highly nonlinear and multiple solutions are avail- able. Therefore, history matching is often done with a trial-and-error method. In real-world applications of history matching, reservoir engineers manually modify an initial model provided by geoscientists until the production data are matched. The initial model is built based on geological and seismic data. While attempts are usually made to honor these other data as much as possible, often the history-matched models are unrealistic from a geological (and geophysical) point of view. For example, permeability is often altered to increase or decrease flow in areas where a mismatch is observed; however, the permeability alterations usually come in the form of box-shaped or pipe-shaped geometries centered around wells or between wells and tend to be devoid of any geologica. considerations. The primary focus lies in obtaining a history match.