Multipoint Statistics Research Articles

Training Image Optimization Method Based on Convolutional Neural Network and Its Application in Discrete Fracture Network Model Selection

Abstract Training image (TI) is important for multipoint statistics simulation method (MPS), since it captures the spatial geological pattern of target reservoir to be modeled. Generally, one optimal TI is selected before applying MPS by evaluating the similarities between many TIs and the well interpretations of target reservoir. In this paper, we propose a new training image optimization approach based on the convolutional neural network (CNN). First, candidate TIs were randomly sampled several times to obtain the sample dataset. Then, the CNN was used to conduct transfer learning for all samples, and finally, the optimal TI of the conditioning well data is selected through the trained CNN model. By taking advantage of the strong learning ability of CNN in image feature recognition, the proposed method can automatically identify differences in spatial features between the conditioning well data and the samples of the training image. Hence, it effectively resolves the difficulty of spatial matching between discrete datapoints and grid structures. We demonstrated the applicability of our model via 2D and 3D training image selection examples. The proposed methods effectively selected the appropriate TI, and then the pretreatment techniques for improving the accuracy of continuous TI selection were achieved. Moreover, the proposed method was successfully applied to training image selection of a discrete fracture network model. Finally, sensitivity analysis was carried out to show that sufficient conditioning data volume can reduce the uncertainty of the optimization results. By comparing with the improved MDevD method, the advantages of the new method are verified in terms of efficiency and reliability.

Use of geostatistical modeling to improve the understanding of permeability upscaling in isotropic and anisotropic burrowed reservoirs

Coarse materials filling burrows in reservoirs can provide permeable pathways through an otherwise impermeable host rock matrix —thus enhancing fluid flow properties. However, understanding permeability anisotropy and permeability-upscaling under such conditions represents a challenge in the exploration and development of reservoirs. Herein, we examined 75 hypothetical models that represent diverse types and amounts of bioturbation. The aim is to understand the impacts of burrows on petrophysical characteristics, including upscaling of permeability where the matrix is isotropic. These models were simulated in a high-resolution 1-m3 grid (8 × 106 cells) using multipoint statistics (MPS) modeling techniques to systematically vary burrow morphology and bioturbation intensity in the same host rock matrix. The modeled burrow morphologies (vertical, horizontal, and boxwork morphologies) and bioturbation intensity―expressed as burrow percentage (BP, ranging from 2% to 50%)― show burrow connectivity trends increasing with increasing BP. In each trend, there is a burrow-morphology-dependent critical BP at which burrows develop connected permeability pathways. Accordingly, three unique patterns of burrow connectivity were identified among the 75 models. These include when burrows are: 1) just isolated volumes (isotropic system); 2) create unidirectional flow (anisotropic system); and 3) create omnidirectional flow (isotropic system). These connectivity patterns determine whether the permeability of the burrow-related reservoir is isotropic or anisotropic. Thus, identifying each of these connectivity patterns in burrowed reservoirs would help determine the appropriate averaging method to upscale permeability. The results of this study bridge an essential gap in reservoir modeling and simulation of hydrocarbon reservoirs and aquifers with burrowed strata.

One Step at a Time: The Origins of Sequential Simulation and Beyond

In the mid-1980s, still in his young 40s, André Journel was already recognized as one of the giants of geostatistics. Many of the contributions from his new research program at Stanford University had centered around the indicator methods that he developed: indicator kriging and multiple indicator kriging. But when his second crop of graduate students arrived at Stanford, indicator methods still lacked an approach to conditional simulation that was not tainted by what André called the ‘Gaussian disease’; early indicator simulations went through the tortuous path of converting all indicators to Gaussian variables, running a turning bands simulation, and truncating the resulting multi-Gaussian realizations. When he conceived of sequential indicator simulation (SIS), even André likely did not recognize the generality of an approach to simulation that tackled the simulation task one step at a time. The early enthusiasm for SIS was its ability, in its multiple-indicator form, to cure the Gaussian disease and to build realizations in which spatial continuity did not deteriorate in the extreme values. Much of Stanford’s work in the 1980s focused on petroleum geostatistics, where extreme values (the high-permeability fracture zones and the low-permeability shale barriers) have much stronger anisotropy, and much longer ranges of correlation in the maximum continuity direction, than mid-range values. With multi-Gaussian simulations necessarily imparting weaker continuity to the extremes, SIS was an important breakthrough. The generality of the sequential approach was soon recognized, first through its analogy with multi-variate unconditional simulation achieved using the lower triangular matrix of an LU decomposition of the covariance matrix as the multiplier of random normal deviates. Modifying LU simulation so that it became conditional gave rise to sequential Gaussian simulation (SGS), an algorithm that shared much in common with SIS. With nagging implementation details like the sequential path and the search neighborhood being common to both methods, improvements in either SIS or SGS often became improvements to the other. Almost half of the contributors to this Special Issue became students of André in the classes of 1984–1988, and several are the pioneers of SIS and SGS. Others who studied later with André explored and developed the first multipoint statistics simulation procedures, which are based on the same concept that underlies sequential simulation. Among his many significant intellectual accomplishments, one of the cornerstones of André Journel’s legacy was sequential simulation, built one step at a time.

A workflow for modelling fluvial meander-belt successions: Combining forward stratigraphic modelling and multi-point geostatistics

A new workflow has been developed for modelling reservoir successions that comprise fluvial meander-belt deposits, based on algorithms that employ multi-point statistics (MPS). A library of training images – from which MPS modelling algorithms can borrow geological patterns for modelling fluvial meandering systems – has been built. The training images incorporate sedimentary architectures relating to point-bar deposits accumulated by fluvial meander-bend expansion and translation, as observed in high-sinuosity river systems and their preserved deposits in the geological record. The training-image library has been developed using a forward stratigraphic modelling software (PB-SAND) that simulates fluvial meander-bend evolution and resulting point-bar facies organization, and which has been constrained using sedimentological data from geological analogues. The training images are applied to two widely employed MPS modelling algorithms: SNESIM and DEESSE. Solutions to common issues encountered in MPS modelling workflows have been established through optimization of modelling settings for SNESIM and DEESSE. Auxiliary variables are used to simulate common facies trends. Application of the training-image library through the developed workflows for SNESIM and DEESSE has been tested; the sensitivity of unconditional simulation results to input parameters has been analysed to define modelling recipes, consisting of sets of appropriate modelling parameters for use with each training image and modelling algorithm. The creation of fluvial reservoir models that are geologically realistic using MPS algorithms remains challenging, but the proposed approach holds promise.

A note on precision-preserving compression of scientific data

Abstract. Lossy compression of scientific data arrays is a powerful tool to save network bandwidth and storage space. Properly applied lossy compression can reduce the size of a dataset by orders of magnitude while keeping all essential information, whereas a wrong choice of lossy compression parameters leads to the loss of valuable data. An important class of lossy compression methods is so-called precision-preserving compression, which guarantees that a certain precision of each number will be kept. The paper considers statistical properties of several precision-preserving compression methods implemented in NetCDF Operators (NCO), a popular tool for handling and transformation of numerical data in NetCDF format. We compare artifacts resulting from the use of precision-preserving compression of floating-point data arrays. In particular, we show that a popular Bit Grooming algorithm (default in NCO until recently) has suboptimal accuracy and produces substantial artifacts in multipoint statistics. We suggest a simple implementation of two algorithms that are free from these artifacts and have double the precision. One of them can be used to rectify the data already processed with Bit Grooming. We compare precision trimming for relative and absolute precision to a popular linear packing (LP) method and find out that LP has no advantage over precision trimming at a given maximum absolute error. We give examples when LP leads to an unconstrained error in the integral characteristic of a field or leads to unphysical values. We analyze compression efficiency as a function of target precision for two synthetic datasets and discuss precision needed in several atmospheric fields. Mantissa rounding has been contributed to NCO mainstream as a replacement for Bit Grooming. The Appendix contains code samples implementing precision trimming in Python3 and Fortran 95.

Potential overlooked bioturbated reservoir zones in the shallow marine strata of the Hanifa Formation in central Saudi Arabia

A comprehensive study including sedimentological, ichnological, petrographic and petrophysical investigations and geostatistical modeling of the late Kimmeridgian bioturbated strata of the upper Hanifa Formation suggests that the grain-dominated burrow fills of these strata enhance the stratum porosity and permeability by creating permeable pathways in the otherwise tight mud-dominated unit. The bioturbation patterns in these strata reveal a boxwork pattern organized as a horizontal network and vertical shafts. These shafts exhibit circular to elliptical cross-sections with the long axis ranging from 0.39 to 1.65 cm that intersects at angles of 60°, 90°, and 120°. This pattern suggests the domination of Thalassinoides ichnotaxa in the studied strata. The Thalassinoides, with a bioturbation intensity ranging from 10% to 40%, contain passive fills that are characterized by coarse-grained (the majority of the grain sizes is > 0.5 mm) carbonates with a packstone texture consisting of skeletal, coated and peloidal grains. These passive fills have a porosity (a mean (μ) of 7% ± 2%) and permeability (μ of 82 ± 98 mD) substantially higher than the matrix porosity (μ of 1.5% ± 0.5%) and permeability (μ of 0.6 ± 0.6 mD). Multipoint statistics (MPS) models suggest that the Thalassinoides in these strata generated connected porous and permeable pathways even at a low bioturbation intensity (~10%). The MPS models also indicate that the bulk permeability of these strata increases with the bioturbation intensity and reaching up to 204 mD for models with a 40% bioturbation intensity. The similarity of these results to those published for well-studied ichnofossil-hosted tight gas reservoirs suggests that the mud-dominated units of the upper Hanifa Formation could form potential flow zones. Interestingly, however, these strata are overlooked as subsurface reservoirs, probably because of the limitations of downhole logging tools and the lack of integrated, comprehensive investigations. This study provides an illustrative example of shallow marine strata in which mud-dominated sediments could be flow units because of the impact of burrows. Hence, understanding the bioturbation trends in these strata could be beneficial for the prediction of the reservoir quality of similar strata elsewhere.

Vug and fracture characterization and gas production prediction by fractals: Carbonate reservoir of the Longwangmiao Formation in the Moxi-Gaoshiti area, Sichuan Basin

A comprehensive knowledge of the development and connectivity of fractures and vugs in carbonate reservoirs plays a key role in reservoir evaluation, ultimately affecting the gas prediction of this kind of heterogeneous reservoir. The carbonate reservoirs with fractures and vugs that are well developed in the Longwangmiao Formation, Sichuan Basin are selected as a research target, with the fractal dimension calculated from the full-bore formation microimager (FMI) image proposed to characterize the fractures and vugs. For this purpose, the multipoint statistics algorithm is first used to reconstruct a high-resolution FMI image of the full borehole wall. And then, the maximum class-variance method (the Otsu method) realizes the automatic threshold segmentation of the FMI image and acquisition of the binary image, which accurately characterizes the fractures and vugs. Finally, the fractal dimension is calculated by the box dimension algorithm, with its small value difference enlarged to obtain a new fractal parameter ([Formula: see text]). The fractal dimensions for four different kinds of reservoirs, including eight subdivided models of vugs and fractures, show that the fractal dimension can characterize the development and the connectivity of fractures and vugs comprehensively. That is, the more developed that the fractures and vugs are, the better the connectivity will be, and simultaneously the smaller that the values of the fractal dimensions are. The fractal dimension is first applied to the gas production prediction by means of constructing a new parameter ([Formula: see text]) defined as a multiple of the effective thickness ([Formula: see text]), porosity (Por), and fractal dimension ([Formula: see text]). The field examples illustrate that the fractal dimensions can effectively characterize the fractures and vugs in the heterogeneous carbonate reservoir and predict its gas production. In summary, the fractals expand the characterization method for the vugs and fractures in carbonate reservoirs and extend its new application in gas production prediction.

Effect of the intermittency dynamics on single and multipoint statistics of turbulent boundary layers

The instantaneous velocity field of turbulent flows is traditionally described in a statistical sense as a sum of an ensemble or temporal mean velocity at a specific location and a fluctuating term. While the Reynolds decomposition is well-established for fully turbulent flows, its applicability for flows that show intermittent behaviour is questionable. As intermittency is an integral part of turbulent boundary layer and turbulent jet and wake flows, an analysis of the effect of the intermittency on the statistical quantities is of fundamental interest. In this work, the effect of intermittency on single and multipoint statistics is studied for an experimentally obtained turbulent boundary layer data set at high Reynolds numbers by means of three different decomposition approaches. The analysis clearly shows that the Reynolds decomposition is only valid for single point statistics in the inner fully turbulent region of wall-bounded flows. In the wake region of a turbulent boundary layer, the Reynolds decomposition overestimates the turbulent kinetic energy due to the intermittency of the turbulent/non-turbulent interface. It is shown herein that these artefacts can be reduced or even completely avoided by applying either the boundary layer height or zonal based decomposition approach. In this study, the limitations of the different methods are analysed in detail and the approaches are evaluated comparatively by investigating statistical quantities such as turbulence intensity, anisotropy of fluctuations and characteristic length scales of coherent motions. The results make it possible to better understand and interpret the characteristics of the turbulent statistics of intermittent flows.

Quasi‐Online Groundwater Model Optimization Under Constraints of Geological Consistency Based on Iterative Importance Sampling

AbstractThe increasing use of wireless sensor networks and remote sensing permits real‐time access to environmental observations. Data assimilation frameworks tap into such data streams to autonomously update and gradually improve numerical models. In hydrogeology, such methods are relevant in areas of long‐term interest in water quality and quantity, for example, in drinking water production. Unfortunately, accurate hydrogeological predictions often demand a degree of geological realism, which is difficult to reconcile with the operational limitations of many data assimilation frameworks. Alluvial aquifers, for example, are sometimes characterized by paleo‐channels of unknown extent and properties, which may act as preferential flow paths. Gradually optimizing such fields in real‐time or quasi‐real‐time settings is a formidable task. Besides subsurface properties, ill‐specified model forcings are a further source of predictive bias, which an optimizer could learn to compensate. In this study, we explore the use of a quasi‐online optimizer based on the iterative batch importance sampling framework for a groundwater model of a field site near Valdobbiadene, Italy. This site is characterized by the presence of paleo‐channels and heavily exploited for drinking water production and irrigation. We use Markov chain Monte Carlo steps to explore new parameterizations while maintaining consistency between states and parameters as well as conformance to a multipoint statistics training image. We also optimize a preprocessor designed to compensate for potential bias in the model forcing. We achieve promising and geologically consistent quasi‐real‐time optimization, albeit at the loss of parameter uncertainty.

Depositional conditioning of three dimensional training images: Improving the reproduction and representation of architectural elements in sand-dominated fluvial reservoir models

Fluvial deposits create significant hydrocarbon reservoirs, although their characterisation can be difficult due to their differing scales of heterogeneity. Whilst numerical modelling methods have advanced to statistically honour fluvial input datasets, geologically realistic features are often lost, impacting hydrocarbon recovery predictions. Two dimensional training images are often used to dictate what heterogeneity is inputted into multi-point statistics based reservoir models. In this study, a three dimensional training image is built, based upon depositional conditions derived from outcrop and modern satellite imagery data of a fluvial system. The aims of this study are to: identify the heterogeneity within the modern and outcrop data and to replicate it in a three dimensional training image; to model such heterogeneity using object-based, sequential indicator simulation and multi-point statistics; and to qualitatively and quantitatively (through static net-connectivity testing) analyse the reproducibility and geological realism of the generated reservoir models. Digital photogrammetric data from Tuscher Canyon, Utah, of the Lower Castlegate Sandstone and satellite imagery from the Jamuna River, northern India, are used to depositionally condition a three dimensional training image. This training image was then used to generate the multi-point statistics models, which were then tested against more traditional object-based and sequential indicator simulation reservoir models. Results indicated that object-based models realistically reproduced heterogeneous architectural elements, however, the connectivity of net-reservoir elements were unrealistically shaped and over-connected. The sequential indicator simulated models produced unrealistic heterogeneous architectural elements and overestimated the connectivity of net-reservoir elements. The multi-point statistical models realistically reproduced heterogeneous architectural elements geometries and the connectivity of net-reservoir elements. Study implications suggest that, based upon limited data, depositional conditioning can generate three dimensional training images to produce reservoir models that are both geologically realistic and reproducible. • Heterogeneity within the Lower Castlegate of Tuscher Canyon, Utah is characterised. • MPS, SIS and OBM models of fluvial reservoirs are compared. • MPS is found to produce the most realistic and reproducible results.

Measuring vorticity vector from the spinning of micro-sized mirror-encapsulated spherical particles in the flow.

We demonstrate a nonintrusive technique that is capable of measuring all three-components of vorticity following small tracer particles in the flow. The vorticity is measured by resolving the instantaneous spin of the microsized spherical hydrogel particles, in which small mirrors are encapsulated. The hydrogel particles have the same density and refractive index as the working fluid-water. The trajectory of the light reflected by the spinning mirror, recorded by a single camera, is sufficient to determine the 3D rotation of the hydrogel particle, and hence the vorticity vector of the flow at the position of the particle. Compared to more conventional methods that measure vorticity by resolving velocity gradients, this technique has much higher spatial resolution. We describe the principle of the measurement, the optical setup to eliminate the effect of particle translation, the calibration procedure, and the analysis of measurement uncertainty. We validate the technique by measurements in a Taylor-Couette flow. Our technique can be used to obtain the multipoint statistics of vorticity in turbulence.

Multilevel Strategies and Geological Parameterizations for History Matching Complex Reservoir Models

Summary In this study, we explore using multilevel derivative-free optimization (DFO) for history matching, with model properties described using principal-component-analysis (PCA) -based parameterization techniques. The parameterizations applied in this work are optimization-based PCA (O-PCA) and convolutional-neural-network (CNN) -based PCA (CNN-PCA). The latter, which derives from recent developments in deep learning, is able to accurately represent models characterized by multipoint spatial statistics. Mesh adaptive direct search (MADS), a pattern-search method that parallelizes naturally, is applied for the optimizations required to generate posterior (history-matched) models. The use of PCA-based parameterization considerably reduces the number of variables that must be determined during history matching (because the dimension of the parameterization is much smaller than the number of gridblocks in the model), but the optimization problem can still be computationally demanding. The multilevel strategy introduced here addresses this issue by reducing the number of simulations that must be performed at each MADS iteration. Specifically, the PCA coefficients (which are the optimization variables after parameterization) are determined in groups, at multiple levels, rather than all at once. Numerical results are presented for 2D cases, involving channelized systems (with binary and bimodal permeability distributions) and a deltaic-fan system using O-PCA and CNN-PCA parameterizations. O-PCA is effective when sufficient conditioning (hard) data are available, but it can lead to geomodels that are inconsistent with the training image when these data are scarce or nonexistent. CNN-PCA, by contrast, can provide accurate geomodels that contain realistic features even in the absence of hard data. History-matching results demonstrate that substantial uncertainty reduction is achieved in all cases considered, and that the multilevel strategy is effective in reducing the number of simulations required. It is important to note that the parameterizations discussed here can be used with a wide range of history-matching procedures (including ensemble methods), and that other derivative-free optimization methods can be readily applied within the multilevel framework.

Dynamic pore network modelling of real gas transport in shale nanopore structure

Gas transport in shale nanopores is controlled by the complex transport mechanisms and pore structure characteristics. So far much work has been done on the single pore and regular structure based pore network but little is known on real gas transport behavior on a realistic shale pore structure model. In this work, a dynamic pore network model is proposed to describe single component methane transport in nano-scale porous media. Gas transport behavior takes transport mechanisms of slip flow, transition flow, surface diffusion and ad/desorption into account. Real gas effect under high pressure and temperature is considered when calculating gas properties. A three dimensional pore network model is built from a three dimensional image that is reconstructed using multi-point statistics from a small organic-rich area on a two dimensional shale SEM image. This pore network model is used to analyze dynamic gas transport and pressure drop transmission process. Our simulated results reveal that there exists a time period that gas permeability on the pore network is influenced by the pressure drop transmission process and pore structure that flows on. When pressure drop reaches outlet, gas transport turns steady state. At steady state period gas permeability becomes constant and is influenced by the effective stress, pressure, temperature and shale rock property.

Parametric generation of conditional geological realizations using generative neural networks

Deep learning techniques are increasingly being considered for geological applications where—much like in computer vision—the challenges are characterized by high-dimensional spatial data dominated by multipoint statistics. In particular, a novel technique called generative adversarial networks has been recently studied for geological parametrization and synthesis, obtaining very impressive results that are at least qualitatively competitive with previous methods. The method obtains a neural network parametrization of the geology—so-called a generator—that is capable of reproducing very complex geological patterns with dimensionality reduction of several orders of magnitude. Subsequent works have addressed the conditioning task, i.e., using the generator to generate realizations honoring spatial observations (hard data). The current approaches, however, do not provide a parametrization of the conditional generation process. In this work, we propose a method to obtain a parametrization for direct generation of conditional realizations. The main idea is to simply extend the existing generator network by stacking a second inference network that learns to perform the conditioning. This inference network is a neural network trained to sample a posterior distribution derived using a Bayesian formulation of the conditioning task. The resulting extended neural network thus provides the conditional parametrization. Our method is assessed on a benchmark image of binary channelized subsurface, obtaining very promising results for a wide variety of conditioning configurations.

Multiscale modeling of CO2-induced carbonate dissolution: From core to meter scale

Reactive transport models can be used to predict the long-term storage capacity of CO2 in carbonate formations if the parameters that couple physical and chemical interactions can be calibrated from centimeters (laboratory-scale) to kilometers (field-scale). However, calibration across length scales is challenging in highly reactive and heterogeneous carbonate reservoirs. In this study we translated simulations of CO2-induced carbonate dissolution and transport processes of centimeter-sized cores to meter-sized rock blocks. The meter-sized rocks represent an intermediate scale between laboratory experiments and field demonstrations.We conducted brute-force, meter-sized simulations maintaining the same grid resolution used to successfully model rock heterogeneities and physical behaviors observed in laboratory experiments. To do this, we applied multi-point statistics to generate rock heterogeneity that honors characterization data, and adopted the reaction and transport parameters (e.g., reaction rate kinetics, porosity-permeability relationship) from models calibrated by core-flood experiments.We used the high-resolution simulation data sets to understand the interplay between carbonate reactivity, flow velocity, and rock heterogeneity on the development and consequences of dissolution fronts at a meter scale, and in particular how parameters that couple chemistry and transport change at variable length scales. The modeling results reveal scale dependence of porosity-permeability relationship and carbonate reactivity. We found that upscaling reactive transport processes from centimeters to meters requires: (1) Effective representation of initial rock heterogeneity and related preferential flow field. In more permeable rocks, several dissolution fingers or wormholes form and grow along the flow direction, competing for reactive solution and leading to fast breakthrough. In less permeable rocks, a single dissolution channel forms with a dramatic increase in lateral branching due to the resistance from low-permeability regions along the flow path. (2) Higher exponential values for porosity-permeability power-law correlations. The value of the power parameter n increases by about two times as we scale the dissolution process from centimeters to meters. (3) Lower carbonate reaction rates. Carbonate reaction rates obtained for meter-sized models are about ten times slower than those for the smaller core-flood experiments. We also used the high-resolution meter-sized simulations to upscale carbonate dissolution by coarsening the representative elementary volumes (or grids). The calibrated reactive transport parameters are consistent with those derived from scaling analysis.The upscaling analysis developed in this study improves our understanding of carbonate dissolution (especially wormhole development) and associated reactive transport across time and length scales, providing a useful basis for establishing a direct relationship between core-scale and field-scale models. Additional work is needed to see if relationships developed from single-phase reactive transport processes shown in this work are directly applicable to in-situ reservoir multiphase flow conditions arising from geological storage of CO2.

On the symmetry properties of a random passive scalar with and without boundaries, and their connection between hot and cold states

We consider the evolution of a decaying passive scalar in the presence of a gaussian white noise fluctuating linear shear flow known as the Majda Model. We focus on deterministic initial data and establish the symmetry properties of the evolving point wise probability measure for the random scalar. We identify, for both point line source initial data, regions in the x-y plane outside of which the PDF skewness is sign definite for all time, while inside these regions we observe multiple sign changes corresponding to exchanges in symmetry between hot and cold leaning states using exact representation formula for the PDF at the origin, and away from the origin, using numerical evaluation of the exact available Mehler kernels for the scalars statistical moments. A new, rapidly convergent Monte-Carlo method is developed, dubbed Direct Monte-Carlo (DMC), using the random Green's functions allowing for the fast construction of the PDF for single point statistics, and multi-point statistics natural for full Monte-Carlo simulations of the underlying stochastic differential equations (FMC). This new method demonstrates the full evolution of the PDF from short times, to its long time, limiting and collapsing universal distribution at arbitrary points in the plane. Further, this method provides a strong benchmark for FMC and we document numbers of field realization criteria for the FMC to faithfully compute this complete dynamics. Armed with this benchmark, we apply the FMC to a channel with a no-flux boundary condition enforced on a channel and observe a dramatically different long time state resulting from the wall. In particular, the channel case collapsing invariant measure has negative skewness, with random states heavily leaning heavily towards the hot state, in stark contrast to free space, where the limiting skewness is positive, with its states leaning heavily towards the cold state.

Pattern-Based History Matching Maintains Consistency for Complex-Facies Reservoirs

This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper SPE 190128, “Pattern-Based History Matching for Reservoirs With Complex Geologic Facies,” by A. Golmohammadi, M.R. Khaninezhad, and B. Jafarpour, SPE, University of Southern California, prepared for the 2018 SPE Western Regional Meeting, Garden Grove, California, USA, 22–27 April. The paper has not been peer reviewed. A challenging problem of automated history-matching work flows is ensuring that, after applying updates to previous models, the resulting history-matched models remain consistent geologically. This is particularly challenging in formations with complex connectivity patterns. In this work, the authors introduce a novel machine-learning approach with the aim of preserving the main connectivity patterns of previous reservoir models during history matching of complex geologic formations. Introduction The authors introduce a machine-learning algorithm to incorporate discrete connectivity patterns in history matching of complex-geologic-facies models. This is achieved by splitting the introduced history-matching optimization problem into two iterative subproblems: a continuous approximation of the solution that is obtained by solving a regularized least-squares inversion (while maintaining the expected connectivity of the patterns) followed by a machine-learning-based mapping of the continuous solution to the discrete feasible set defined by previous models. The second step involves a machine-learning approach that uses offline training to implement the mapping. The offline learning process uses the k-nearest neighbor (k-NN) algorithm to construct local pattern (feature) vectors and compare them with the feature vectors in the training data set. For each spatial template, the feature vectors with the smallest distance in the learning data set are selected and their corresponding label vectors (i.e., multivariate discrete patterns) are identified and stored. Once all local patterns are scanned and processed using a defined template size, an aggregation step is applied on the overlapping templates to incorporate the multipoint statistics patterns collectively in assigning discrete labels to each gridblock. Methodology History Matching With Facies-Feasibility Constraint. An indirect method is developed for solving the constrained minimization problem by defining a two-step alternating-directions method. In this approach, the first step uses a standard gradient-based method to find an approximate continuous solution to the problem and the second step maps the resulting continuous solution onto the feasible set while ensuring that the updated solution remains close to the continuous solution from the first step. These two steps are repeated until no further improvement in the data match is obtained. A machine-learning algorithm is used to implement the mapping in the second step. Mapping Onto the Feasible Set. The authors use a supervised-learning approach to implement the mapping in Step 2 of the solution approach. The k-NN method is implemented in this paper as a simple classification technique in which the decision about the label of a feature vector is based on a distance measure defined between the feature vector and the learning data set. Using a predefined distance measure, the algorithm selects feature vectors with minimum distances from the training data set and assigns the most-frequent label corresponding to these feature vectors as the output label.

A Deep-Learning-Based Geological Parameterization for History Matching Complex Models

A new low-dimensional parameterization based on principal component analysis (PCA) and convolutional neural networks (CNN) is developed to represent complex geological models. The CNN-PCA method is inspired by recent developments in computer vision using deep learning. CNN-PCA can be viewed as a generalization of an existing optimization-based PCA (O-PCA) method. Both CNN-PCA and O-PCA entail post-processing a PCA model to better honor complex geological features. In CNN-PCA, rather than use a histogram-based regularization as in O-PCA, a new regularization involving a set of metrics for multipoint statistics is introduced. The metrics are based on summary statistics of the nonlinear filter responses of geological models to a pre-trained deep CNN. In addition, in the CNN-PCA formulation presented here, a convolutional neural network is trained as an explicit transform function that can post-process PCA models quickly. CNN-PCA is shown to provide both unconditional and conditional realizations that honor the geological features present in reference SGeMS geostatistical realizations for a binary channelized system. Flow statistics obtained through simulation of random CNN-PCA models closely match results for random SGeMS models for a demanding case in which O-PCA models lead to significant discrepancies. Results for history matching are also presented. In this assessment CNN-PCA is applied with derivative-free optimization, and a subspace randomized maximum likelihood method is used to provide multiple posterior models. Data assimilation and significant uncertainty reduction are achieved for existing wells, and physically reasonable predictions are also obtained for new wells. Finally, the CNN-PCA method is extended to a more complex non-stationary bimodal deltaic fan system, and is shown to provide high-quality realizations for this challenging example.

Multi-point Monin–Obukhov similarity in the convective atmospheric surface layer using matched asymptotic expansions

The multi-point Monin–Obukhov similarity (MMO) was recently proposed (Tong & Nguyen, J. Atmos. Sci., vol. 72, 2015, pp. 4337–4348) to address the issue of incomplete similarity in the framework of the original Monin–Obukhov similarity theory (MOST). MMO hypothesizes the following: (1) The surface-layer turbulence, defined to consist of eddies that are entirely inside the surface layer, has complete similarity, which however can only be represented by multi-point statistics, requiring a horizontal characteristic length scale (absent in MOST). (2) The Obukhov length $L$ is also the characteristic horizontal length scale; therefore, all surface-layer multi-point statistics, non-dimensionalized using the surface-layer parameters, depend only on the height and separations between the points, non-dimensionalized using $L$. However, similar to MOST, MMO was also proposed as a hypothesis based on phenomenology. In this work we derive MMO analytically for the case of the horizontal Fourier transforms of the velocity and potential temperature fluctuations, which are equivalent to the two-point horizontal differences of these variables, using the spectral forms of the Navier–Stokes and the potential temperature equations. We show that, for the large-scale motions (wavenumber $k<1/z$) in a convective surface layer, the solution is uniformly valid with respect to $z$ (i.e. as $z$ decreases from $z>-L$ to $z<-L$), where $z$ is the height from the surface. However, for $z<-L$ the solution is not uniformly valid with respective to $k$ as it increases from $k<-1/L$ to $k>-1/L$, resulting in a singular perturbation problem, which we analyse using the method of matched asymptotic expansions. We show that (1) $-L$ is the characteristic horizontal length scale, and (2) the Fourier transforms satisfy MMO with the non-dimensional wavenumber $-kL$ as the independent similarity variable. Two scaling ranges, the convective range and the dynamic range, discovered for $z\ll -L$ in Tong & Nguyen (2015) are obtained. We derive the leading-order spectral scaling exponents for the two scaling ranges and the corrections to the scaling ranges for finite ratios of the length scales. The analysis also reveals the dominant dynamics in each scaling range. The analytical derivations of the characteristic horizontal length scale ($L$) and the validity of MMO for the case of two-point horizontal separations provide strong support to MMO for general multi-point velocity and temperature differences.

Applying multi-point statistical methods to build the facies model for Oligocene formation, X oil field, Cuu Long basin

Multi-point statistic method (MPS) can overcome the inherent disadvantages of traditional method based on variogram and object modeling, simultaneously allow the modeling progress, and become more flexible and rational. The algorithms based on variogram and gridding geological model are able to control the final result under the collection of samples’ data (well data) and another corresponding data (seismic). Though, these methods have trouble in modeling the shape of geological features. Then, object-modeling method can generate digitized geological features with responsible shapes; conversely, a final result in accordance with an input data is difficult to achieve. Combining the advantages of two mentioned methods, MPS describes the relationship of data in space based on the group of adjacent points or has a certain relationship, it allows the generation of digitized geological features corresponding with responsible shapes, and moreover, it is able to control the final result under a collection of input data (whose nature is still the pixel-based). The Oligocene reservoir, X field, was formed in fluvial/lacustrine and sedimentary mainly deposited in Northwest–Southeast, which is primarily affected by latitude—sub-latitude faults’ system. An Oligocene facies model of X field is built based on MPS, and it will show the geological features more clearly than the existing one. It also shows the remarkable ability on controlling the final result. MPS allows to combine a lot of different data (geology, seismic, outcrop, etc.) with the geological viewpoints which are shown by training image and itself proves the superiority over traditional methods. Duration of each model simulation is approximately 3000 s and huge size (over 15 million cells), and it is better while compared with 1717.8750 s in case of sequential simulation by SISIM method and default properties.