Stochastic Permeability Field Research Articles

Model‐Reduced Adjoint‐Based Inversion Using Deep‐Learning: Example of Geological Carbon Sequestration Modeling

AbstractAs an extension of our previous research on deep‐learning‐based adjoint‐state approach (Xiao, Deng, & Wang, 2021, https://doi.org/10.1029/2020wr027400), we present a two‐level hybrid neural network architecture to efficiently derive a model‐reduced adjoint‐based optimization workflow with a large‐scale inverse modeling example in geological carbon‐storage applications. The first level employs a nonlinear dimensionality reduction technique, that is, deep convolutional autoencoder, owing to its high scalability and the availability of high‐performance deep‐learning libraries for flexible implementation. Analogous to the reduced‐order linear model, the second level emulates a neural‐network‐powered linear transition unit based on the reduced subspace. This hybrid framework provides a readily available model‐reduced adjoint derivation with negligible computational cost and computer storage. Once the gradient is obtained in reduced space, it is correspondingly projected back to full space for the inverse modeling. The performance of the method is verified by estimating the space‐dependent stochastic permeability field using two geological carbon storage (GCS) process‐based multiphase flow models with an increasing complexity. The experimental results indicate that the surrogate model can accurately characterize the spatial‐temporal evolution of the CO2 saturation fields using a relatively small number of training data. Although the proposed inversion workflow achieves comparable results similar to the conventional finite‐difference method, it is faster than high‐fidelity model simulations in several orders of magnitude; hence, it will be a beneficial simulation and modeling tool for engineers to manage the uncertainty of GCS processing.

Deep Convolutional Encoder‐Decoder Networks for Uncertainty Quantification of Dynamic Multiphase Flow in Heterogeneous Media

AbstractSurrogate strategies are used widely for uncertainty quantification of groundwater models in order to improve computational efficiency. However, their application to dynamic multiphase flow problems is hindered by the curse of dimensionality, the saturation discontinuity due to capillarity effects, and the time dependence of the multi‐output responses. In this paper, we propose a deep convolutional encoder‐decoder neural network methodology to tackle these issues. The surrogate modeling task is transformed to an image‐to‐image regression strategy. This approach extracts high‐level coarse features from the high‐dimensional input permeability images using an encoder and then refines the coarse features to provide the output pressure/saturation images through a decoder. A training strategy combining a regression loss and a segmentation loss is proposed in order to better approximate the discontinuous saturation field. To characterize the high‐dimensional time‐dependent outputs of the dynamic system, time is treated as an additional input to the network that is trained using pairs of input realizations and of the corresponding system outputs at a limited number of time instances. The proposed method is evaluated using a geological carbon storage process‐based multiphase flow model with a 2,500‐dimensional stochastic permeability field. With a relatively small number of training data, the surrogate model is capable of accurately characterizing the spatiotemporal evolution of the pressure and discontinuous CO2 saturation fields and can be used efficiently to compute the statistics of the system responses.

Gaussian process emulators for quantifying uncertainty in [formula omitted] spreading predictions in heterogeneous media

We explore the use of Gaussian process emulators (GPE) in the numerical simulation of CO2 injection into a deep heterogeneous aquifer. The model domain is a two-dimensional, log-normally distributed stochastic permeability field. We first estimate the cumulative distribution functions (CDFs) of the CO2 breakthrough time and the total CO2 mass using a computationally expensive Monte Carlo (MC) simulation. We then show that we can accurately reproduce these CDF estimates with a GPE, using only a small fraction of the computational cost required by traditional MC simulation. In order to build a GPE that can predict the simulator output from a permeability field consisting of 1000s of values, we use a truncated Karhunen-Loève (K-L) expansion of the permeability field, which enables the application of the Bayesian functional regression approach. We perform a cross-validation exercise to give an insight of the optimization of the experiment design for selected scenarios: we find that it is sufficient to use 100s values for the size of training set and that it is adequate to use as few as 15 K-L components. Our work demonstrates that GPE with truncated K-L expansion can be effectively applied to uncertainty analysis associated with modelling of multiphase flow and transport processes in heterogeneous media.

A two-stage adaptive stochastic collocation method on nested sparse grids for multiphase flow in randomly heterogeneous porous media

A new computational method is proposed for efficient uncertainty quantification of multiphase flow in porous media with stochastic permeability. For pressure estimation, it combines the dimension-adaptive stochastic collocation method on Smolyak sparse grids and the Kronrod–Patterson–Hermite nested quadrature formulas. For saturation estimation, an additional stage is developed, in which the pressure and velocity samples are first generated by the sparse grid interpolation and then substituted into the transport equation to solve for the saturation samples, to address the low regularity problem of the saturation. Numerical examples are presented for multiphase flow with stochastic permeability fields to demonstrate accuracy and efficiency of the proposed two-stage adaptive stochastic collocation method on nested sparse grids.

Stochastic Galerkin Formulations for $$\hbox {CO}_2$$ Transport in Aquifers: Numerical Solutions with Uncertain Material Properties

Abstract A stochastic Galerkin formulation for the transport of $$\hbox {CO}_2$$ CO 2 in a tilted aquifer with uncertain heterogeneous properties is presented. We consider a simplified physics model assuming capillary pressure to be negligible compared to hydrostatic and viscous pressure. The flow is dominated by buoyancy and capillary trapping. We assume a stochastic permeability field and a stochastic model for the uncertain relative permeabilities. We prove that the proposed stochastic Galerkin formulation results in a hyperbolic system of equations, and we devise a numerical method that captures the expected solution discontinuities. The shock-capturing solver for the flux function is combined with an adaptive quadrature method for discontinuous isosurfaces that is used to compute the discontinuous stochastic accumulation coefficient. The stochastic solver is validated against Monte Carlo sampling of an analytical solution for the deterministic problem. The sharp features of the statistics of the solution are accurately captured by the numerical solver. The polynomial chaos framework admits low-cost post-processing of the output to obtain statistics of interest. By construction of an accurate polynomial chaos surrogate model of the output, fast sampling admits calculation of risk for leakage and failure probabilities.

Anisotropic Dispersion and Upscaling for Miscible Displacement

Summary Dispersion is the irreversible mixing that occurs during miscible displacements. Dispersion can reduce local displacement efficiency by lessening solvent peak concentration or increase volumetric sweep efficiency by spreading of the injected solvent to more of the reservoir. Dispersion is therefore an important parameter in predicting and simulating miscible displacements. The difficulty of simulating miscible displacement and understanding the effect of dispersion is also compounded by numerical dispersion, which increases the apparent dispersion in finite-difference simulation models. This paper presents an approach to estimate the total longitudinal and transverse dispersion in large-scale media by use of continuous solvent injection in a medium of finite thickness. The simulations are based on the experimental arrangement of Blackwell (1962) to estimate transverse dispersion, with experiments consisting of coinjecting two miscible fluids into different sections of the medium at similar rates. This model arrangement, coupled with analytical solutions for the 2D convection/dispersion equation for a continuously injected solvent, allows us to determine longitudinal and transverse dispersivity simultaneously for the flow medium. In this manner, we investigate the effects of stochastic permeability distributions and other scaling groups affecting first-contact-miscible simulations on dispersion. Sensitivity analysis of dispersion in stochastic permeability fields confirms that both longitudinal and transverse dispersion are scale dependent. Results also show that the effect of increasing autocorrelation of cell permeability in the longitudinal direction (parallel to flow) is to increase longitudinal dispersion, as solvent travels through more continuous layers, while reducing transverse dispersion. Increasing autocorrelation in the transverse direction reduces dispersion in both directions. This reduction is caused by equilibration of solvent concentrations in continuous sections of the reservoir, resulting in reduced solute fingering and channeling. Finally, we developed a simple procedure to use the estimated dispersivities to determine a priori the maximum gridblock size that will maintain an equivalent level of dispersion between fine-scale models and upscaled coarse models. Large gridblock sizes can be used for highly heterogeneous and layered reservoir models. Nonuniform coarsening (upscaling) methods were also recommended and validated for reservoir models with sets of sequential but different permeability distributions. The procedure was extended to multicontact miscible simulations. The sweep and recovery from upscaled multicontact miscible simulations were comparable with those of fine-scale models.

A multiscale preconditioner for stochastic mortar mixed finite elements

The aim of this paper is to introduce a new approach to efficiently solve sequences of problems that typically arise when modeling flow in stochastic porous media. The governing equations are based on Darcy’s law with a stochastic permeability field. Starting from a specified covariance relationship, the log permeability is decomposed using a truncated Karhunen–Loève expansion. Multiscale mortar mixed finite element approximations are used in the spatial domain and a nonintrusive sampling method is used in the stochastic dimensions. A multiscale mortar flux basis is computed for a single permeability, called a training permeability, that captures the main characteristics of the porous media, and is used as a preconditioner for each stochastic realization. We prove that the condition number of the preconditioned interface operator is independent of the subdomain mesh size and the mortar mesh size. Computational results confirm that our approach provides an efficient means to quantify the uncertainty for stochastic flow in porous media.

Pilot Block Method Methodology to Calibrate Stochastic Permeability Fields to Dynamic Data

In the present paper, a new geostatistical parameterization technique is introduced for solving inverse problems, either in groundwater hydrology or petroleum engineering. The purpose of this is to characterize permeability at the field scale from the available dynamic data, that is, data depending on fluid displacements. Thus, a permeability model is built, which yields numerical flow answers similar to the data collected. This problem is often defined as an objective function to be minimized. We are especially focused on the possibility to locally change the permeability model, so as to further reduce the objective function. This concern is of interest when dealing with 4D-seismic data. The calibration phase consists of selecting sub-domains or pilot blocks and of varying their log-permeability averages. The permeability model is then constrained to these fictitious block-data through simple cokriging. In addition, we estimate the prior probability density function relative to the pilot block values and incorporate this prior information into the objective function. Therefore, variations in block values are governed by the optimizer while accounting for nearby point and block-data. Pilot block based optimizations provide permeability models respecting point-data at their locations, spatial variability models inferred from point-data and dynamic data in a least squares sense. A synthetic example is presented to demonstrate the applicability of the proposed matching methodology.

Mixed Multiscale Finite Element Methods Using Limited Global Information

In this paper, we present a mixed multiscale finite element method using limited global information. We consider a general case where multiple global information is given such that the solution depends smoothly on these global fields. The global fields typically contain small-scale (local or global) information required for achieving a convergence with respect to the coarse mesh size. We present a rigorous analysis and show that the proposed mixed multiscale finite element methods converge. Some preliminary numerical results are shown. We study a parameter dependent permeability field (a simplified case for general stochastic permeability fields). As for spatial heterogeneities, channelized permeability fields with strong nonlocal effects are considered. Using a few global fields corresponding to realizations of permeability fields, we show that one can achieve high accuracy in numerical simulations.

Mixed Multiscale Finite Element Methods for Stochastic Porous Media Flows

In this paper, we propose a stochastic mixed multiscale finite element method. The proposed method solves the stochastic porous media flow equation on the coarse grid using a set of precomputed basis functions. The precomputed basis functions are constructed based on selected realizations of the stochastic permeability field, and furthermore the solution is projected onto the finite-dimensional space spanned by these basis functions. We employ multiscale methods using limited global information since the permeability fields do not have apparent scale separation. The proposed approach does not require any interpolation in stochastic space and can easily be coupled with interpolation-based approaches to predict the solution on the coarse grid. Numerical results are presented for permeability fields with normal and exponential variograms.

Integrating Multiphase Production History in Stochastic Reservoir Characterization

Summary This paper focuses on integrating field-production history into reservoir characterization through stochastic inverse modeling. A key element of our approach is a three-dimensional (3D) streamline simulator that is orders of magnitude faster than traditional numerical simulators and thus allows for rapid inversion of multiphase production data. Stochastic permeability fields, conditioned to field-production history, are then generated with simulated annealing. We also explore the spatial resolution associated with estimates of reservoir permeability variations derived by use of field-production history. On the basis of techniques from geophysical inverse theory, we address such issues as data sensitivity, spatial resolution, averaging kernels, and uncertainties associated with our estimates of reservoir permeability. The proposed inversion technique has been applied to synthetic as well as field cases. The synthetic example involves a sensitivity analysis of multiphase production history in heterogeneous five-spot and nine-spot patterns. The field example consists of production history from a five-spot pattern in the north Robertson unit, a low-permeability carbonate reservoir in West Texas. Watercut history at the producers are used to estimate permeability variations in a two-layer (matrix-fracture) model of the reservoir. All computations were performed on a 125 MHz Pentium with an average run time of about 4 wall-clock hours, indicating the feasibility of our approach.

Characterizing heterogeneous permeable media with spatial statistics and tracer data using sequential simulated annealing

Characterizing heterogeneous permeable media using flow and transport data typically requires solution of an inverse problem. Such inverse problems are intensive computationally and may involve iterative procedures requiring many forward simulations of the flow and transport problem. Previous attempts have been limited mostly to flow data such as pressure transient (interference) tests using multiple observation wells. This paper discusses an approach to generating stochastic permeability fields conditioned to geologic data in the form of a vertical variogram derived from cores and logs as well as fluid flow and transport data, such as tracer concentration history, by sequential application of simulated annealing (SA). Thus, the method incorporates elements of geostatistics within the framework of inverse modeling. For tracer-transport calculations, we have used a semianalytic transit-time algorithm which is fast, accurate, and free of numerical dispersion. For steady velocity fields, we introduce a “transit-time function” which demonstrates the relative importance of data from different sources. The approach is illustrated by application to a set of spatial permeability measurements and tracer data from an experiment in the Antolini Sandstone, an eolian outcrop from northern Arizona. The results clearly reveal the importance of tracer data in reproducing the correlated features (channels) of the permeability field and the scale effects of heterogeneity.

Stochastic Reservoir Modeling Using Simulated Annealing and Genetic Algorithms

Summary This paper discusses and compares three different algorithms based on combinatorial optimization schemes for generating stochastic permeability fields. The algorithms are not restricted to generating Gaussian random fields and have the potential to accomplish geologic realism by combining data from many different sources. We have introduced a "heat-bath" algorithm for simulated annealing (SA) as an alternative to the commonly used "Metropolis" algorithm and a new stochastic modeling technique based on the "genetic" algorithm. We applied these algorithms to a set of outcrop and tracer flow data and examined the associated uncertainties in predictions. All three algorithms reproduce the major features of permeability distribution and fluid flow data. For relatively small problems, the Metropolis algorithm is the fastest. For larger problems, the heat-bath algorithm is at least as fast and often faster than the Metropolis algorithm, with significant potential for parallelization. The performance of the genetic algorithm is highly dependent on the choice of population size and probabilities of crossover, update, and mutation.

A Type-Curve Approach to Analyzing Two-Well Tracer Tests

Summary The traditional approach to analyzing field tracer data generally involves a history-matching procedure that is time-consuming and frequently results in a nonunique solution. In this paper, we use theoretical frequency response functions to build type curves of "transfer function" and "phase spectrum" that have a dimensionless heterogeneity index as a parameter to characterize a stochastic permeability field. The transfer function contains information regarding the amplitude-attenuating characteristics of the heterogeneous medium, and the phase spectrum provides information about the spacing between the frequency components of the input and output tracer pulses. In a manner analogous to type-curve matching for well testing, we can analyze field tracer history by comparing the theoretical and field-observed transfer function and phase spectrum. This analysis leads to an estimate of the heterogeneity index and PV. We demonstrate the technique by analyzing field tracer data from the Big Muddy field, WY, and compare the use of parameters obtained in scaling up. Results show very good agreement with the field data.

CO2 Flow Patterns Under Multiphase Flow: Heterogeneous Field-Scale Conditions

SummaryA finite-difference, equation-of-state (EOS), compositional simulator has been used to study CO2 flooding. First, unstable first-contact-miscible (FCM) displacements were simulated with a fine mesh to investigate the transition from gravity override to viscous fingering. Next, a direct comparison was made for FCM and multiple-contact-miscible (MCM) displacements under the same conditions to investigate effects of phase behavior on the growth of viscous fingers. Then, the effects of gravity, physical dispersion, capillary pressure, phase behavior, and heterogeneity were combined and simulated for CO2 flooding on a field scale with stochastic permeability fields.