Elsevier

Computers & Fluids

Volume 201, 15 April 2020, 104475
Computers & Fluids

A data-driven surrogate to image-based flow simulations in porous media

https://doi.org/10.1016/j.compfluid.2020.104475Get rights and content

Highlights

  • The objective is to develop a data-driven surrogate to numerical flow simulations.

  • Two-dimensional LB simulation runs are used to train and to predict the solutions.

  • Convolutional neural networks is used for predicting the fluid dynamics.

  • The developed model can capture the dynamics of the problem at a much lower cost.

Abstract

The objective for this work is to develop a data-driven surrogate to high-fidelity numerical flow simulations using digital images of porous media. The proposed model can capture the pixel-scale velocity vectors in a large verity of digital porous media created by random two-dimensional (2D) circle packs. To develop the model, images of the 2D media (binary images of solid grains and void spaces) along with their corresponding velocity vectors at the pixel level computed using lattice Boltzmann simulation runs are used to train and to predict the solutions with a high accuracy in much less computational time. The velocity vector predictions made by the surrogate models are used to compute the permeability tensor for samples that have not been used in the training. The results show high accuracy in the prediction of both velocity vectors and permeability tensors. The proposed methodology harness the enormous amount of generated data from high-fidelity flow simulations to decode the often under-utilized patterns in simulations and to accurately predict solutions to new cases. The developed model can truly capture the physics of the problem and enhance the prediction capabilities of the simulations at a much lower cost. These predictive models, in essence, do not spatially reduce the order of the problem. They, however, possess the same numerical resolutions as their Lattice Boltzmann simulations equivalents do with the great advantage that their solutions can be achieved by a significant reduction in computational costs (speed and memory).

Introduction

Darcy’s principles [7] describe the fluid flow of single-phase fluids in porous media at low Reynolds numbers, which is of significant importance in earth sciences, hydrology, and petroleum engineering. According to the Darcy equation, pressure gradients are linearly proportional to the fluid rate; the proportionality constant is permeability, which is merely a function of pore space topology of porous media irrespective of the fluid type. In numerical flow simulators for porous media, permeability values are obtained based on the data collected from the field and experiments. An accurate quantification of permeability is difficult due to the variations in pore space morphology characteristics. Permeability has been obtained from experiments and also from analytical and empirical expressions that relate permeability to some attributes of the porous media, such as porosity and pore size distribution. The analytical expressions are, however, only approximations for ideal cases while the empirical expressions have utility only in media similar to scenarios for which they were obtained and thus, are inaccurate when applied to a wide range of other media. Experimental approaches are generally preferred when it is not possible to account for all relevant physics by an equation or model; however, they tend to be time consuming and expensive. Furthermore, they do not capture the effect of pore space morphology characteristics on the flow field and thus, on permeability.

For certain properties, such as permeability, hydraulic tortuosity, and inertial factors of the porous media, high-fidelity numerical simulations using digital images have become a credible alternative, enabled by improvements of imaging techniques, numerical methods, and computing power [38], [51]. Appealing aspects of this approach include the ability to probe pore-scale physics at a level not possible with traditional experiments and the ability to perform an endless set of numerical tests without degrading or altering the sample. There are considerations that can limit this digital approach including whether the imaging technique can resolve all relevant characteristic scales in the pore space and whether numerical algorithms can accurately model the physical processes. Higher resolution, however, mandates higher computation power. In high-fidelity numerical simulation models, the expensive computational costs, the intensive memory requirements, and the poor scaling performances have traditionally prevented their applications beyond toy or small-scale problems, even using the modern high-performance computing systems.

In image-based pore-scale modeling, the domain is discretized into nodes, voxels, or volume elements, and the resulting grid is used to numerically approximate the relevant partial differential equations for flow, namely computational fluid dynamics (CFD). There is a group of numerical modeling techniques that can utilize the voxel data from X-ray tomography or similar methods as the numerical grid. This gridding approach has become widely used in porous media studies in conjunction with the lattice Boltzmann (LB) simulations and has been proved to be highly effective for simulating fluid flow through porous media [51].

LB simulations have been applied to flow simulations of realistic porous media to compute permeability [9], [18], [45] with the advantage of being flexible in the specification of variables on the complex boundaries in terms of simple particle bounce back and reflection. This flexibility has opened up the potential in its use for modeling and simulating flow in complex media, such as porous rocks. Challenges for applying LB to real problems include finite-size effects and relaxation time dependence of no-flow boundaries. In image-based simulations, the accuracy of the calculated macroscopic properties depends on the spatial resolution of the rock image [11], [28]. However, there is always a trade-off between image resolution and computational power. Furthermore, in all digital samples, there is a resolution threshold, below which certain flow characteristics, such as re-circulation, are not resolved [25].

An extensive research has been performed to study LB modeling of fluid flow in the porous media [1], [2], [10], [27], [28], [29], [32]. Additionally, Pan et al. [30], [31] and Stewart et al. [43] studied the effect of sphere size, spatial discretization, and fluid viscosity (relaxation parameter) on the computed permeability of random-sphere packs and Maier et al. [26] investigated flow of single-phase fluid through a column of glass beads. Takbiri Borujeni [48] studied the applicability of lb simulations in porous media for a wide range of Re and verified the results against experimental and other CFD methods. Their computed permeability tensor and non-Darcy factors were validated experimental flow measurements. They also showed that for Re < 1 permeability is not a function of topology of porous media (not a function of pressure gradient and flow velocity), i.e., can be described by the Darcy equation.

The main advantage of pore-scale flow simulations is that explicit influence of each impacting factor can be studied by isolating the effect of other parameters. Attempts of this tabulation of all these impacts have not been manageable yet since such a multi-dimensional parametric study requires comprehensive efforts and time. In this respect, this work aspires to change the status quo and make a transformative leap by combining pore-scale modeling with physics-based ML [34], [35], [36], [49], [50] to develop surrogate models, which can be used to determine the flow fields at very little additional cost. It is also important to note that, using a trained and validated data-driven surrogate model will give us a luxury of performing pore-scale flow simulations, in which computational expenses are not restrictive.

Recently, there have been numerous studies of the application of ML in CFD, most of which are limited to building interpretable reduced-order models (ROMs) [14], [15], [47], [54]. In ROMs, where the number of variables is reduced to simplify the governing equations and the relationships between inputs and outputs, some details are inevitably overlooked. On the other hand, the widespread success of ML-based predictive modeling in other disciplines, such as autonomous cars, suggests a great opportunity to advances in the state-of-the-art by combining conventional CFD simulation techniques with predictive capabilities of data-driven surrogate models to truly capture the physics of the problem and enhance prediction capabilities of the simulations at a much lower cost. They, however, possess the same numerical resolutions as their CFD equivalents do with the great advantage that their solutions can be achieved by a significant reduction in computational costs (speed and memory). Essentially, the predictive models learn the nature of communications among grid cells and decode the spatial correlations between them (auto- and cross-correlations) in the entire computational domain and can accurately predict solutions to completely new sets of simulation runs, from beginning to end.

Recently, Convolutional Neural Networks (CNNs) with hierarchical feature learning capability has outperformed the state of the art in many computer vision tasks, including image classification [41], segmentation [24], and synthesis [13]. Despite in classification tasks, where the network predicts a single class label for an input image, in many visual tasks, the desired output could be a class label, or a continuous value, assigned to each pixel of the input image [59].

Ciresan et al. [6] predicted the class label of each pixel by training a network in a sliding-window fashion which takes a patch around each pixel. This network, then, is able to localize and also is more robust to overfitting the training data, i.e., generated patches, is much larger than the number of training images. However, this framework is quite slow due to the separate processing of each patch, which results in a lot of redundancy on overlapping patches. Moreover, such networks should deal with the trade-off between the localization and context. Large patches need many pooling layers that can reduce the localization performance, while small patches only incorporate little context information in the final decision. More recent studies [24], [39] proposed to fuse the fine to coarse features from multiple layers in different depth. This enables the network to achieve an accurate localization while having a large receptive field (context) at the same time. In the work performed by Ronneberger et al. [37], the authors introduced U-Net which employed contracting path in its Auto-Encoder architecture to capture context and enable precise localization. Furthermore, training a very deep neural network is quite a challenging task. More specifically, it is hard for a deep network to find an optimal solution compared to shallower counterparts. One of the main issues in training a deep network is the vanishing gradient problem, making it difficult to tune the parameters of the early layers in the network [12]. In the past couple of years, multiple training strategies have been proposed to train a deep neural network effectively, including deep supervision in hidden layers [23], initialization scheme [12], and batch normalization [17]. He et al. [16] introduced residual connections in which they employ additive merging of signals to improve the training speed, and gradient flow through the networks.

For clarification, the terminology used in the remainder of the paper is the following. The term input is used to denote the binary (zeros and ones) images of porous media, where 0 denotes the void spaces and 1 denotes the solid grains. The term pixel and numerical grids are used interchangeably due to the fact that the numerical method, LB, use image pixels as the numerical grid. The term output refers to the velocity vectors computed at each pixel of each input using LB simulations.

Applications of ML have gained lots of popularity in the past few years throughout various industries. The application of ML in CFD has gained considerable interest recently, mostly to build ROMs. However, in such applications of ML in CFD, it is inevitable to overlook some details. On the other hand, predictive ML techniques suggest a greater opportunity, when the conventional CFD simulation techniques are combined with predictive capabilities of data-driven models. Such approaches can truly capture the physics of the problem and enhance the prediction capabilities of the simulations at a much lower cost.

Unlike the automotive industry, the application of Artificial Intelligence (AI) in CFD has been limited to interpretable models from data [21], [40], [53], and predictive models are yet to be employed. The widespread success of predictive modeling in complex problems suggests a great opportunity to advances in the state-of-the-art by combining conventional CFD simulation techniques with ML predictive modeling to truly capture the physics of the problem and enhance prediction capabilities of the simulations at a much lower cost. This can be achieved by developing physically interpretable spatio-temporal simulations of complex CFD problems and introducing a significant reduction in computational cost (speed and memory).

Section snippets

Lattice Boltzmann Mmethod

The Boltzmann equation isfαt+eα.fα=Ωα,where fα(x,t) is the fraction of fluid particles that have traveled in the α-direction in the phase space directions, eα is the particle velocity in the α-direction, and Ωα is the collision operator [3]. The LB simulation method is a discrete form of the continuous Boltzmann equation in which time and space are discretized with velocity limited to a finite set of admissible directions in which the particles can travel [5], [46]. The basic LB algorithm

Methodology

In LB simulations, the solutions, u(s,x(s)), are obtained at spatial locations s, where the pixels of the binary input image x(s) and S={s1,,sns} are the index set for the spatial grid locations, sSRds(ds=1,2,3) are the spatial locations. The simulations can be considered as a mapping of x{0,1}Rdxns to its corresponding solution uURduns,η:{0,1}U,where u=η(x). The purpose for building the surrogate model is to develop a new mapping function, u^=F(x,θ), to be trained using a limited

Deep convolutional neural network

Neural networks and specially CNNs, are known for being a powerful tool with the ability to process high dimensional data and vast data sets. The universal approximation theorem indicates that NNs can approximate any arbitrary functions on compact subspaces. NNs comprise a set of vector-valued functions known as layers of neurons. Each layer learns a linear transformation of the input vector, x(s), through its matrix of weights, θ, and vector of biases, b. A non-linear activation function, F,

Results

To develop the model, velocity values for the entire output set are normalized between zero and one (the minimum value of the velocity values is transformed linearly into zero, the maximum value is transformed into one, and every other value is transformed into a decimal between 0 and 1). All the simulation cases are divided into two sections; the first section with 12.8% of the data is used to train the model while the remaining data are used as test data. For the training, only x-direction

Performance of surrogate models trained with less data

In this section, the performance of the approach presented is evaluated using the models trained with fewer number of data samples. X-direction velocity profile for flow in the x-direction is plotted along a vertical line is depicted in Fig. 15. It can be seen that as the number of the training data increases, the velocity profiles tend to become closer to the LB simulation results (shown by the dashed red line in the right figure in Fig. 15). It should be pointed out that even for the smallest

Conclusions

A data-driven surrogate to high-fidelity numerical flow simulations is presented by employing a deep convolutional neural network based on contracting paths and residual blocks. The network consists of only convolutional layers and can take any arbitrary-sized image as input and generate an output of a similar size. The developed model captures the flow fields at the grid level for samples that had not been used in the development of the model. Permeability tensor for the samples of porous

Author contribution

All authors have contributed sufficiently to the research work to be included as authors. To the best of our knowledge, no conflict of interest, financial or other, exists.

Declaration of Competing Interest

To the best of our knowledge, no conflict of interest, financial or other, exists.

Acknowledgments

The authors gratefully acknowledge the support of NVIDIA Corporation with the donation of the Titan Xp GPU used for this research.

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