Stochastic Realization Research Articles

Stochastic multiscale flux basis for Stokes-Darcy flows

Three algorithms are developed for uncertainty quantification in modeling coupled Stokes and Darcy flows. The porous media may consist of multiple regions with different properties. The permeability is modeled as a non-stationary stochastic variable, with its log represented as a sum of local Karhunen-Loève (KL) expansions. The problem is approximated by stochastic collocation on either tensor-product or sparse grids, coupled with a multiscale mortar mixed finite element method for the spatial discretization. A non-overlapping domain decomposition algorithm reduces the global problem to a coarse scale mortar interface problem, which is solved by an iterative solver, for each stochastic realization. In the traditional domain decomposition implementation, each subdomain solves a local Dirichlet or Neumann problem in every interface iteration. To reduce this cost, two additional algorithms based on deterministic or stochastic multiscale flux basis are introduced. The basis consists of the local flux (or velocity trace) responses from each mortar degree of freedom. It is computed for each subdomain independently before the interface iteration begins. The use of the multiscale flux basis avoids the need for subdomain solves on each iteration. The deterministic basis is computed at each stochastic collocation and used only at this realization. The stochastic basis is formed by further looping over all local realizations of a subdomain's KL region before the stochastic collocation begins. It is reused over multiple realizations. Numerical tests are presented to illustrate the performance of the three algorithms, with the stochastic multiscale flux basis showing significant savings in computational cost compared to the other two algorithms.

Convergence and stability of a micro–macro acceleration method: Linear slow–fast stochastic differential equations with additive noise

We analyze the convergence and stability of a micro–macro acceleration algorithm for Monte Carlo simulations of linear stiff stochastic differential equations with a time-scale separation between the fast evolution of the individual stochastic realizations and some slow macroscopic state variables of the process. The micro–macro acceleration method performs a short simulation of a large ensemble of individual fast paths, before extrapolating the macroscopic state variables of interest over a larger time step. After extrapolation, the method constructs a new probability distribution that is consistent with the extrapolated macroscopic state variables, while minimizing Kullback–Leibler divergence with respect to the distribution available at the end of the Monte Carlo simulation. In the current work, we study the convergence and stability of this method on linear stochastic differential equations with additive noise, when only extrapolating the mean of the slow component. For this case, we prove convergence to the microscopic dynamics when the initial distribution is Gaussian and present a stability result for non-Gaussian initial laws.

Portfolio-based airline fleet planning under stochastic demand

Airlines operate their fleet of aircraft over a relatively long time horizon during which the realized stochastic demand has the potential to profoundly impact the airlines’ financial performance. This makes the investment in a fleet of aircraft a highly capital-intensive long-term commitment, associated with inherent risks. We propose an innovative three-step airline fleet planning methodology with the primary objective of identifying fleets that are robust to stochastic demand realizations. The methodology presents two main innovation aspects. The first one is the use of the mean reverting Ornstein–Uhlenbeck process to model the long-term travel demand, which is then combined with discrete-time Markov chain transitions to generate demand scenarios. The second innovative aspect is the adoption of a portfolio-based fleet planning perspective that allows for an explicit comparison of different fleets, in size and composition. Ultimately, the methodology yields for each fleet in the portfolio a distribution of net present values of operating profit across the planning horizon and a list of key financial and operational metrics per year. The robustest fleet can be selected based on the operating profit generating capability across different realizations of stochastic demand. An illustrative case study is presented as a proof of concept. The case study is used to demonstrate the type of results obtained and to discuss the usefulness of the methodology proposed.

Enhancing images of shale formations by a hybrid stochastic and deep learning algorithm

Accounting for the morphology of shale formations, which represent highly heterogeneous porous media, is a difficult problem. Although two- or three-dimensional images of such formations may be obtained and analyzed, they either do not capture the nanoscale features of the porous media, or they are too small to be an accurate representative of the media, or both. Increasing the resolution of such images is also costly. While high-resolution images may be used to train a deep-learning network in order to increase the quality of low-resolution images, an important obstacle is the lack of a large number of images for the training, as the accuracy of the network’s predictions depends on the extent of the training data. Generating a large number of high-resolution images by experimental means is, however, very time consuming and costly, hence limiting the application of deep-learning algorithms to such an important class of problems. To address the issue we propose a novel hybrid algorithm by which a stochastic reconstruction method is used to generate a large number of plausible images of a shale formation, using very few input images at very low cost, and then train a deep-learning convolutional network by the stochastic realizations. We refer to the method as hybrid stochastic deep-learning (HSDL) algorithm. The results indicate promising improvement in the quality of the images, the accuracy of which is confirmed by visual, as well as quantitative comparison between several of their statistical properties. The results are also compared with those obtained by the regular deep learning algorithm without using an enriched and large dataset for training, as well as with those generated by bicubic interpolation.

Stochastic curvature of enclosed semiflexible polymers.

The conformational states of a semiflexible polymer enclosed in a compact domain of typical size a are studied as stochastic realizations of paths defined by the Frenet equations under the assumption that stochastic "curvature" satisfies a white noise fluctuation theorem. This approach allows us to derive the Hermans-Ullman equation, where we exploit a multipolar decomposition that allows us to show that the positional probability density function is well described by a telegrapher's equation whenever 2a/ℓ_{p}>1, where ℓ_{p} is the persistence length. We also develop a Monte Carlo algorithm for use in computer simulations in order to study the conformational states in a compact domain. In addition, the case of a semiflexible polymer enclosed in a square domain of side a is presented as an explicit example of the formulated theory and algorithm. In this case, we show the existence of a polymer shape transition similar to the one found by Spakowitz and Wang [Phys. Rev. Lett. 91, 166102 (2003)PRLTAO0031-900710.1103/PhysRevLett.91.166102] where in this case the critical persistence length is ℓ_{p}^{*}≃a/8 such that the mean-square end-to-end distance exhibits an oscillating behavior for values ℓ_{p}>ℓ_{p}^{*}, whereas for ℓ_{p}<ℓ_{p}^{*} it behaves monotonically increasing.

Identification of dominant gas transport frequencies during barometric pumping of fractured rock

We demonstrate that although barometric pressures are complicated signals comprised of numerous frequencies, it is a subset of these frequencies that drive the overwhelming majority of gas transport in fractured rock. Using an inverse numerical analysis, we demonstrate that a single barometric component with seasonally modulated amplitude approximates gas transport due to a measured barometric signal. If past barometric tendencies are expected to continue at a location, the identification of this frequency can facilitate accurate long term predictions of barometrically induced gas transport negating the need to consider stochastic realizations of future barometric variations. Additionally, we perform an analytical analysis that indicates that there is a set of barometric frequencies, consistent with the inverse numerical analysis, with high production efficiency. Based on the corroborating inverse numerical and analytical analyses, we conclude that there is a set of dominant gas transport frequencies in barometric records.

Pit Optimization on Clustered Realizations: Identifying Functional Scenarios

Scenario-level uncertainty is a representation of large-scale differences encountered when considering ultimate mining pits optimized over various subsets of stochastic realizations. Accounting for scenario-level uncertainty allows mine plans to be assessed against several probabilistic subsurface interpretations. Scenario-level uncertainty is found through clustering stochastic geostatistical realizations and optimizing final pits across the realizations in each cluster. Where these pits are geometrically different, plans for each scenario differ. Knowing these differences allows the value of each planning alternative to be assessed based on the value and relative probability of each scenario. These results also facilitate the use of formal decision-making frameworks for planning, leading to optimization that better represents the full statistical characterization of the deposit. An example utilizing this approach evaluates both infrastructure placement and drilling program requirements. In both cases, scenario-level uncertainty improves initial plans by limiting opportunity costs and identifying additional potential.

State-Space Realizations and Optimal Smoothing for Gaussian Generalized Reciprocal Processes

This technical note derives stochastic realization and optimal smoothing algorithms for a class of Gaussian generalized reciprocal processes (GGRP). The note exploits the interplay between reciprocal processes and Markov bridges which underpin the GGRP model. A forward–backward algorithm for stochastic realization of GGRP is described. The form on the inverse covariance matrix for the GGRP is used, via Cholesky factorization, to derive a procedure for optimal (MMSE) smoothing of GGRP observed in noise. The note demonstrates that the associated smoothing error is also a GGRP with known covariance which may be used to assess the performance of smoothing as a function of the model parameters. A numerical example is provided to illustrate the performance of the MMSE smoother compared to those derived from compatible Markov and reciprocal model-based algorithms.

Hydraulic fracturing modeling using a discrete fracture network in the Barnett Shale

Hydraulic fracturing has been a successful stimulation method for enhancing recovery in very low permeability shale gas reservoirs. Proper modeling of multistage hydraulic fracturing in shale reservoirs requires an accurate representation of the pre-existing natural fractures in the shale. The objective of this study is to utilize a 3D discrete fracture network model to simulate hydraulic fracturing in five horizontal wells A-E in the Barnett Shale. Each horizontal well had 10 stages of hydraulic fracturing stimulation. Only the well C has comprehensive logging data including an FMI (Formation Microresistivity Imager) log of the well. We characterize stochastic realizations pre-existing natural fracture network in the reservoir integrating different types of data from limited wellbore data to the geologic setting. Two fracture sets clustered using Fisher distribution after correcting for the observation biases. Fractures size constrained by induced microseismicity magnitude happened during hydraulic fracturing while fracture intensity correlated to clay contents (Gamma Ray) and distance from the fault crossing the well C. We have modeled the hydraulic fracturing in the five wells and predict that how pre-existing fractures induced to slip and contribute to permeability enhancement of the reservoir.

Stochastic prediction of fractured caprock by history matching pressure monitoring data

In this study, a novel method to stochastically characterize the effective permeability of a fractured caprock (typically fine-grained sedimentary rock or natural mudstones) is presented. The method requires pressure monitoring data from above and below the caprock, which are used in combination with other readily-available data from laboratory and outcrop observations in a history-matching process that maintains the distribution of underlying fracture attributes to characterize the spatial variation in permeability of the caprock. A “best-fitting” geological model of caprock, representing a most-likely scenario, is selected from an ensemble of stochastic realizations generated using optimized parameters. The method is demonstrated using a representative, moderately permeable, heterogeneous caprock, as a reference “truth” to compare accuracy of caprock predicted using pressures from injection and post-injection phases. The accuracy of “best-fitting” caprock is assessed by studying its sensitivity to four variables, which are caprock thickness, leakage intensity, time length of available pressure history, and injection rate.Improved characterization of effective caprock permeability allows more confident prediction of flux through the sealing interval, over time. That information is useful in quantifying potential impacts of fluid migration, evaluating monitoring designs, and designing mitigation alternatives, should they be needed. It has value both during active injection and in building confidence that a site can be safely closed after injection is complete.

Investigating the impact of distributed energy resources on market power of strategic utility corporation

Market participants may employ potential market power improperly in energy trading. On the other hand, integrations of distributed energy resources (DER) are highly complex since it entails the optimal coordination of a diverse portfolio of DER under multiple sources of uncertainty. A large number of possible stochastic realisations that arise can lead to complex operational models that become problematic in real-time market environments. Although recent works have explored the impacts of DER on numerous aspects of grid operation and planning, its role in imperfect competitive energy markets has not been investigated nonetheless. This article proposes the theoretical and quantitative analysis of the withholding strategies for the utility corporations with the integration of DERs for the first time and the corresponding market power effects on utility corporation's profits and market prices. The quantitative demonstration is supported by a bi-level model with the optimal company profit for the upper level and the market clearing for the lower level. This bi-level problem can be solved directly when a single-level problem is obtained with a mathematical program with equilibrium constraints (MPEC). Numerical studies are implemented on a wholesale market with the day-ahead horizon and hourly resolution.

Stochastic Lattice Modelling of the Force-Displacement and Cracking Behaviour of the Steel Reinforced Tie

The stochastic modelling of the microcracking and the force-displacement behaviour of the tensile steel reinforced tie using the lattice model is presented in the current article. The three-dimension problem of the modelling of the tie is reduced to the two-dimensional so as the main stiffness parameters of the concrete and the reinforcement of the two-dimensional model would be the same as for the three-dimensional. The concrete and steel obey the Hook law. All elastic constants, as well as dimensions of the tie, were assumed as the deterministic quantities except for the critical concrete tensile strains which were treated as a two-dimensional stationary uncorrelated truncated Gaussian random field. The discrete element approach and the explicit integration scheme have been used for the modelling. The estimations of the main parameters of the force-displacement behaviour stochastic process and other statistical indexes were obtained using 72 realization of the force-displacement behaviour of a chosen model. Extra two stochastic realizations of the two different models, as well as three deterministic models, were modelled to compare stochastic and deterministic behaviour of the force-displacement behaviour. The analysis showed that the force-displacement behaviour of the tie under tensile force cannot be treated as a Gaussian stochastic process when the p value is 0.05 at the small displacements and within the interval when the cracking of the concrete is very intensive. However, at the bigger displacements, when the cracking becomes less intensive, the tensile force can be treated as a Gaussian random variable.

Effect of the Galactic Disk and Satellite Galaxies on the Spatial Distribution of Globular Clusters

The distribution anisotropy in systems of globular clusters (GCs) in satellite galaxies of the Milky Way is considered using the inertia tensor. The statistical significance of the results is estimated by repeating this analysis for random catalogs. This method reproduces a known planar structure in the distribution of satellite galaxies; however, the significance of this structure is slightly lower than that for other methods used in the literature. For the GCs at distances 2 < R < 10 kpc, the structure is coplanar with the galactic plane. At large distances, the structure orientation is close to the orientations of the satellite galaxies, i.e., is perpendicular to the Galaxy plane. The probability of stochastic realization of such distribution is 1.7%. The GC distribution anisotropy is also measured without the GCs belonging to the Sagittarius tidal stream. Measurements show that the effect of the stream on the GC perturbation is weak.

Using Memristors for Robust Local Learning of Hardware Restricted Boltzmann Machines

One of the biggest stakes in nanoelectronics today is to meet the needs of Artificial Intelligence by designing hardware neural networks which, by fusing computation and memory, process and learn from data with limited energy. For this purpose, memristive devices are excellent candidates to emulate synapses. A challenge, however, is to map existing learning algorithms onto a chip: for a physical implementation, a learning rule should ideally be tolerant to the typical intrinsic imperfections of such memristive devices, and local. Restricted Boltzmann Machines (RBM), for their local learning rule and inherent tolerance to stochasticity, comply with both of these constraints and constitute a highly attractive algorithm towards achieving memristor-based Deep Learning. On simulation grounds, this work gives insights into designing simple memristive devices programming protocols to train on chip Boltzmann Machines. Among other RBM-based neural networks, we advocate using a Discriminative RBM, with two hardware-oriented adaptations. We propose a pulse width selection scheme based on the sign of two successive weight updates, and show that it removes the constraint to precisely tune the initial programming pulse width as a hyperparameter. We also propose to evaluate the weight update requested by the algorithm across several samples and stochastic realizations. We show that this strategy brings a partial immunity against the most severe memristive device imperfections such as the non-linearity and the stochasticity of the conductance updates, as well as device-to-device variability.

Bayesian Stochastic Inversion (A case study from an Iranian oil field)

We have implemented an estimation procedure whereby wireline data can be extrapolated away from existing well using geostatistical inversion of post-stack 3D seismic data. This procedure works directly in a fine-scale stratigraphic grid, and is conditioned by well and seismic data. It uses a Bayesian framework and a linearized, weak contrast approximation of the Zoeppritz equation to construct a joint log-Gaussian posterior distribution for Pwave impedances. Variograms are also estimated from well-log, seismic data and acoustic inversion results that define the expected degree of lateral smoothness away from the well. A sequential Gaussian Simulation algorithm is applied to sample the posterior PDF and generates multiple, high-resolution realizations of the acoustic impedance which can be utilized to generate stochastic realizations of petrophysical variables that not only honor the well-log data, but most importantly, that fully honor the 3D seismic. Sensitivity analysis was also performed to find the optimum values for S/N ratio, Variogram ranges and prior standard deviation. We had just one well in the area and preferred to keep it to perform quality control of the inversion results. Without constraining with well, geostatistical inversion still can be used to estimate acceptable static reservoir model for the subsequent simulation and planning of in-fill drilling and/or enhanced-oil-recovery operations.

Drywell infiltration and hydraulic properties in heterogeneous soil profiles

Drywells are increasingly used to capture stormwater runoff for surface infiltration and aquifer recharge, but little research has examined the role of ubiquitous subsurface heterogeneity in hydraulic properties on drywell performance. Numerical experiments were therefore conducted using the HYDRUS (2D/3D) software to systematically study the influence of subsurface heterogeneity on drywell infiltration. Subsurface heterogeneity was described deterministically by defining soil layers or lenses, or by generating stochastic realizations of soil hydraulic properties with selected variance (σ) and horizontal (X) and vertical (Z) correlation lengths. The infiltration rate increased when a high permeability layer/lens was located at the bottom of the drywell, and had larger vertical and especially horizontal dimensions. Furthermore, the average cumulative infiltration (I) for 100 stochastic realizations of a given subsurface heterogeneity increased with σ and X, but decreased with Z. This indicates that the presence of many highly permeable, laterally extending lenses provides a larger surface area for enhanced infiltration than the presence of isolated, highly permeable lenses. The ability to inversely determine soil hydraulic properties from numerical drywell infiltration results was also investigated. The hydraulic properties and the lateral extension of a highly permeable lens could be accurately determined for certain idealized situations (e.g., simple layered profiles) using constant head tests. However, variability in soil hydraulic properties could not be accurately determined for systems that exhibited more realistic stochastic heterogeneity. In this case, the heterogeneous profile could be replaced with an equivalent homogeneous profile and values of an effective isotropic saturated conductivity (Ks) and the shape parameter in the soil water retention function (α) could be inversely determined. The average value of Ksfor 100 stochastic realizations showed a similar dependency to I on σ, X, and Z. Whereas, the average value of α had large confidence interval for soil heterogeneity parameters and played a secondary role in drywell infiltration. This research provides valuable insight on the selection of site, design, installation, and long-term performance of a drywell.

Image-based velocity estimation of rock using Convolutional Neural Networks

Digital images of rock samples have been using extensively in Digital Rock Physics (DRP) to evaluate physical parameters of rock such as permeability, P- and S-wave velocities and formation factor. The parameters are numerically computed by simulation of the corresponding physical processes through segmented image of rock, which provide a direct and accurate evaluation of rock properties. However, recent advances in machine learning and Convolutional Neural Networks (CNN) allow using images as input. Such networks, however, require a considerable number of images as input. In this paper, CNNs are used to estimate the P- and S-wave velocities from images of rock medium. To deal with lack of input data, a hybrid pattern- and pixel-based simulation (HYPPS) is used as an efficient data augmentation method to increase the training data set. For each input image, 10 stochastic realizations are produced. Compare to the case wherein the stochastic models are not used, the new results from the enhanced network indicate a sharp improvement in the estimations such that R2 is increased to 0.94. Furthermore, the newly developed CNN network, unlike the one with the small data set (R2=0.75), manifests no over/underestimation. The estimated properties, in comparison with the computational results, indicate that CNNs perform outstandingly in predicting the physical parameters of rock without conducting any time-demanding forward modeling if enough input data are provided.

Stochastic Dual Dynamic Programming for Operation of DER Aggregators Under Multi-Dimensional Uncertainty

The operation of aggregators of distributed energy resources (DER) is highly complex, since it entails the optimal coordination of a diverse portfolio of DER under multiple sources of uncertainty. The large number of possible stochastic realizations that arise can lead to complex operational models that become problematic in real-time market environments. Previous stochastic programming approaches resort to two-stage uncertainty models and scenario reduction techniques to preserve the tractability of the problem. However, two-stage models cannot fully capture the evolution of uncertain processes and the a priori scenario selection can lead to suboptimal decisions. In this context, this paper develops a novel stochastic dual dynamic programming approach which does not require discretization of either the state space or the uncertain variables and can be efficiently applied to a multi-stage uncertainty model. Temporal dependencies of the uncertain variables as well as dependencies among different uncertain variables can be captured through the integration of any linear multidimensional stochastic model, and it is showcased for a p-order vector autoregressive model. The proposed approach is compared against a traditional scenario-tree-based approach through a Monte-Carlo validation process, and is demonstrated to achieve a better trade-off between solution efficiency and computational effort.

Realization and identification algorithm for stochastic LPV state-space models with exogenous inputs

In this paper, we present a realization and an identification algorithm for stochastic Linear Parameter-Varying State-Space Affine (LPV-SSA) representations. The proposed realization algorithm combines the deterministic LPV input output to LPV state-space realization scheme based on correlation analysis with a stochastic covariance realization algorithm. Based on this realization algorithm, a computationally efficient and statistically consistent identification algorithm is proposed to estimate the LPV model matrices, which are computed from the empirical covariance matrices of outputs, inputs and scheduling signal observations. The effectiveness of the proposed algorithm is shown via a numerical case study.

Prediction of groundwater impacts from coal seam gas extraction in the Surat Basin

The Queensland regulatory framework recognises that the impacts of groundwater extraction activities can overlap in areas of concentrated development. Such areas of overlapping impacts can be declared ‘cumulative management areas’ (CMAs). When a CMA is established, the Office of Groundwater Impact Assessment (OGIA) becomes responsible for carrying out a cumulative impact assessment and preparing an Underground Water Impact Report (UWIR). The Surat CMA was declared in March 2011, and since this time, two iterations of the UWIR have been published in 2012 and 2016, underpinned by a gradually evolving regional groundwater flow model. This case study presentation will examine a number novel features of the regional groundwater flow model, resulting from OGIA’s ongoing research and development program, including the development of an adapted version of the MODFLOW-USG groundwater flow modelling code and approximation of coal desaturation and dual-phase flow effects using a modified van Genuchten function. The presentation will also look at the simulation of CSG extraction using a ‘descending drain’ methodology that recognises the gas-filled nature of CSG wells and generating up-scaled properties of highly heterogenous sedimentary material by first generating stochastic realisations of fragments of these layers, and then using ‘numerical permeameters’ to determine both the expected value and stochasticity of these properties, at the regional scale.