# Data-driven methods for flow and transport in porous media: a review

Guang Yang  Ran Xu  Yusong Tian  Songyuan Guo  Jingyi Wu  Institute of
Refrigeration and Cryogenics, Shanghai Jiao Tong University, 800 Dongchuan
Road, 200240 Shanghai, China  Xu Chu∗ Department of Engineering, University of
Exeter, United Kingdom  Cluster of Excellence SimTech, University of
Stuttgart, Pfaffenwaldring 5a, 70569 Stuttgart, Germany

###### Abstract

This review examined the current advancements in data-driven methods for
analyzing flow and transport in porous media, which has various applications
in energy, chemical engineering, environmental science, and beyond. Although
there has been progress in recent years, the challenges of current
experimental and high-fidelity numerical simulations, such as high
computational costs and difficulties in accurately representing complex,
heterogeneous structures, can still potentially be addressed by state-of-the-
art data-driven methods. We analyzed the synergistic potential of these
methods, addressed their limitations, and suggested how they can be
effectively integrated to improve both the fidelity and efficiency of current
research. A discussion on future research directions in this field was
conducted, emphasizing the need for collaborative efforts that combine domain
expertise in physics and advanced computationald and data-driven
methodologies.

###### Contents

  1. 1 Introduction
  2. 2 Image-based approach
    1. 2.1 Data augmentation
    2. 2.2 Image segmentation
    3. 2.3 Image reconstruction
    4. 2.4 Super resolution
  3. 3 Data-driven modeling and prediction
    1. 3.1 Macroscopic modeling and prediction
    2. 3.2 Pore-scale prediction
    3. 3.3 Heat transfer
  4. 4 Physics-informed machine learning
    1. 4.1 Physics-informed neural network (PINN)
    2. 4.2 Hybrid data-driven and physics-informed approaches
    3. 4.3 Machine learning augmented numerical solver
  5. 5 Conclusion and perspective

##  1 Introduction

Flow and transport in porous media is a fundamental subject in various
scientific and engineering disciplines, including hydrogeology, petroleum
engineering, environmental engineering, and chemical engineering. Porous
media, characterized by their internal structure of interconnected void
spaces, facilitate the movement of fluids and the transport of solutes, gases,
and particulates [1, 2, 3]. Understanding these processes is crucial for
addressing challenges related to groundwater contamination, oil recovery, CO2
sequestration, and soil remediation. Porous media are typically classified
based on their porosity, permeability, and the nature of their solid matrix.
The porous structure can range from well-ordered, such as in synthetic
materials in industries, to highly heterogeneous, as seen in natural soils and
fractured rock formations. The complexity of these structures necessitates a
multi-scale approach to study the flow and transport phenomena, encompassing
pore-scale interactions to field-scale processes.

The transport of solutes and particles within porous media is influenced by
advection, diffusion, and dispersion mechanisms. Understanding these
mechanisms is essential for predicting the fate and transport of contaminants,
optimizing enhanced oil recovery techniques, and designing efficient
filtration systems. Heat transfer in porous media is another critical aspect
that intersects with flow and transport processes. The interplay between
thermal and mass transport can significantly influence the behavior of
multiphase systems, such as in geothermal reservoirs and thermal energy
storage systems.

Advancements in experimental techniques and numerical simulations have greatly
enhanced our ability to investigate and predict flow and transport in porous
media. X-ray computed tomography (CT) [4], nuclear magnetic resonance (NMR),
and microfluidics [5] are some of the tools that provide detailed insights
into pore-scale processes. The rapid advancement of high-performance
computation (HPC) including heterogeneous HPC enables high-fidelity single-
[6, 7, 8, 9, 10, 11, 12, 13, 14] and multi-phase simulations of different
kinds [15, 16, 17, 18]. Computational approaches, including pore-resolved
methods [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], pore network modeling
[30], and volume-averaged approach [19], enable the simulation of complex
interactions within porous structures. The difficulties of current
experimental and high-fidelity numerical simulations, such as high
computational costs and challenges in accurately representing complex,
heterogeneous structures, can potentially be addressed by state-of-the-art
data-driven methods.

InputOutputHidden layerHidden layer Figure 1: A three-layer multi-layer-
perceptron (MLP) neural network

The use of artificial intelligence (AI) has seen a significant increase across
various complex applications such as self-driving cars, drug discovery,
weather prediction [31], and advanced language processing models like GPT
[32]. Over the past decade, the widespread adoption of ML has been facilitated
by the availability of open-source libraries, a robust community, an abundance
of data from both physical and numerical experiments, and easy access to HPC.
Neural network structures range from simple multi-layer perceptrons (MLP)
(Fig.1) to advanced architectures such as transformers. Flow modeling has
historically relied on first principles, such as conservation laws, which have
been the foundation of research for centuries. However, when dealing with
multi-scale, multi-phase and multi-physical problems – utilizing scale-
resolving simulations based on the Navier-Stokes equations becomes
computationally infeasible due to the immense computational demands. As an
alternative, approximations of these equations, commonly employed in
turbulence modeling, for specific flow configurations are often utilized.
Nevertheless, these methods entail substantial costs and time investments,
particularly when applied to iterative optimization and real-time control
purposes. To address these challenges, substantial efforts have been dedicated
to developing accurate and efficient reduced-order models that capture
essential flow mechanisms while significantly reducing computational expenses.
In recent years, ML has emerged as a promising approach for achieving
dimensionality reduction and developing reduced-order models in fluid
mechanics, such as proper orthogonal decomposition (POD), dynamic mode
decomposition (DMD), and deep autoencoder (DAE) [33, 34]. ML provides a
concise framework that complements and extends existing methodologies. By
harnessing data-driven techniques, ML empowers researchers to gain novel
insights into complex flow behaviors while maintaining a manageable
computational burden.

The integration of ML into fluid mechanics represents a transformative
development, enhancing both modeling and control of fluid flows [35, 36, 37,
38, 39, 40, 41]. This includes the deployment of ML across various aspects of
fluid dynamics, including enhancements to Reynolds-Averaged Navier-Stokes
(RANS) simulations [42], improvements in Large Eddy Simulation (LES) [43, 44],
innovations in flow control, and the burgeoning field of Physics-Informed
Neural Networks (PINN) [45]. These advancements are particularly crucial in
applications involving complex porous media, where traditional methods often
fall short. Machine learning augments RANS models, which are pivotal for
engineering applications involving turbulent flows. By integrating ML, these
models gain enhanced predictive accuracy [46]. Techniques such as supervised
learning can train models on discrepancies between traditional RANS outputs
and experimental data, thereby tuning the turbulence models to reflect actual
flow conditions more accurately. LES is critical for capturing complex fluid
behaviors by resolving large-scale flow structures while modeling smaller,
more turbulent scales. ML, particularly deep learning models like
Convolutional Neural Networks (CNNs), enhances LES by predicting subgrid-scale
stresses directly from the resolved scales. This application not only improves
the accuracy of LES but also makes it computationally less demanding, which is
vital for simulations involving intricate geometries like those found in
porous media.

Active flow control techniques, which aim to manipulate the flow to achieve
desired outcomes (such as lift increase or drag reduction), are significantly
enhanced by ML [47, 48]. Reinforcement learning, a type of ML, is effectively
used to optimize control strategies dynamically. For example, controllers can
adjust the operation of actuators in real-time based on feedback from flow
sensors, thereby optimizing performance continuously under varying conditions.
ML also revolutionizes passive flow control, which uses fixed devices to alter
flow characteristics without mechanical adjustments. Using algorithms like
genetic algorithms or deep reinforcement learning, designers can simulate
numerous configurations of devices such as vortex generators or dimples to
ascertain the most effective setup. This method drastically reduces
experimental trial and error by predicting performance impacts through
simulation, leading to optimized designs with enhanced aerodynamic properties.

Causal analysis, another critical application, involves understanding the
cause-and-effect relationships within fluid flow systems [49, 50, 51, 52,
liu2023large, 53]. Techniques such as Granger causality and transfer entropy
[54, 55, 24] are used to identify causal links between different flow
variables, providing deeper insights into the underlying dynamics of the
system. This information is crucial for developing predictive models and
designing effective interventions in fluid mechanics applications.

In conclusion, the incorporation of data-driven methods in the study of flow
and transport in porous media represents a potential paradigm shift in the
field. These approaches offer unprecedented potential for unraveling the
complexities of fluid mechanics, enabling more accurate predictions, efficient
computations, and innovative solutions to longstanding challenges. By bridging
the gap between traditional computational/experimental techniques and modern
data science, this review aims to highlight the transformative impact of data-
driven methods and inspire further research and application.

##  2 Image-based approach

###  2.1 Data augmentation

The unprecedented volumes of original data from experiments and field
measurements are important for machine-learning to extract information and
train models. Generally, the four common porous material imaging techniques
belong to time-consuming tasks which included X-ray tomography [56], magnetic
resonance [57], and neutron and positron imaging X-ray tomography [58].
Besides the experimental data, the large number of original data can be
acquired from the data augmentation to learn robust relationships between the
inputs and outputs. The data augmentation refers to methods for constructing
and introducing unobserved data of latent variables which is applicable to the
research of flow and transport in the porous media by data-driven methods
[59].

In the research of porous media, the data augmentation techniques can
generally be categorized as supervised data augmentation and unsupervised data
augmentation. Under supervised data augmentation, predefined data manipulation
methods are usually used to augment the data based on existing data. The basic
data manipulation includes image cropping, mixing, and geometric transforms.
Alqahtani et al. [60] rotated the input images and flipped along horizontal
and vertical axes to artificially increase the number of training instances.
Varfolomeev et al. [61] cropped two-dimensional images using a kernel size of
$496\times 496$ pixels and a stride of 248 pixels to carry out the data
augmentation. Graczyk et al. [62] exploit the property that a horizontal flip
of the input image does not change the porosity and diffusion coefficient of
the porous media. Liu et al. [63] extended 2-D data augmentation techniques to
a 3-D sub-sampling process by regularly or arbitrarily rotating the image to
improve the robustness of the system. Guan [64] obtained the dataset from
micro-X-ray tomography scales of a Betheimer sandstone. The image data was
downsampled to $256^{3}$ voxels to yield 36864 training images with extraction
every 16 voxels. Rabbani et al. [65] inspired by mixing image method and
adopted a weighted interpolation method to binary two textures which is
exhibited in Fig. 2. Ge et al. [66] made a random crop from a 3-D rock image
with a size variation between 50$\%$ to 200$\%$ of the original input size as
data augmentation. Then, they resized the image with tri-linear interpolation
for up-sampling and porosity averaging for down-sampling.

Figure 2: Texture interpolation by weighted averaging of the normalized
distance maps, (a and h) original gray maps, (b and i) binarized geometries,
(c and j) normalized distance maps of the solid space, (e) equally–weighted
average of the distance maps, (d, g, and k) three realizations made by
changing the threshold level that controls the porosity [65]. Reproduced with
permission from Elsevier.

The unsupervised data augmentation learns the distribution of the existing
data and randomly generates the images that are consistent with the
distribution of the training dataset. The representative data augmentation
method is GAN which is discussed in Chapter 2.3 to create artificial image
data from the Generator(G) model for data augmentation.

###  2.2 Image segmentation

After data augmentation, images need to be analyzed by image segmentation. The
image segmentation refers to the division of an image into a number of regions
based on features such as colour and greyscale. The porous material images
generally contain two or more phases, each of which needs to be detected and
separated to analyze and understand the flow and transport characteristics in
porous media. The image segmentation separates the different phases, minerals,
or fluids for further detailed investigation about the bulk measurement [67]
and the CFD modeling [21]. The image segmentation method in the research of
porous media can be categorized as image segmentation based on the classic
segmentation method and deep learning.

The classic image segmentation method used the predefined thresholding level
and interface characteristics to segment original images. Armstrong et al.
[68] accomplished image segmentation using indicator kriging-based
thresholding to characterize the pore morphology and analyze residual oil blob
mobilization. The voxel values below the lower limit are labeled as one phase
and voxel values above the upper limit are labeled as the second phase. Zhang
et al. [69] used the greyscale for quantification and determined a certain
thresholding level for image segmentation. Then they analyzed the
uncertainties and errors by comparison with image segmentation based on grey
scale and experimental results. They concluded that the spatial resolution of
the image could affect image segmentation results. Schlüter et al. [70]
analyzed every histogram-based global thresholding method which is presented
in Fig. 3a. Based on the analysis of the global thresholding method, they also
introduced locally adaptive segmentation methods which are used to smooth
phase boundaries and avoid noise errors. They concluded that the major cause
of poor segmentation results is image blur and proposed locally adaptive
segmentation methods that could remove the image noise and image artifacts.
Brown et al. [71] represented the three phases in model porous media by the
phases contact lines instead of grey scale values. They used the watershed
algorithm which can be relatively effective in avoiding mis-classification
errors at phase boundaries due to partial volume effects [72] and presented in
Fig. 3b. The classic image segmentation method contains many limitations
including low spatial resolution of the raw images [71], shadows and artifacts
at the inter-phase boundary, and the reconstruction noise [73]. Considering
the limitations and difficulties of classic image segmentation and the
development of computility and deep learning models, the deep learning has
been widely used in image segmentation [74].

Recently, image segmentation based on deep learning has become more popular in
porous material studies and it has been used to distinguish different phases.
Andrew [75] found that the image segmentation method included classic and
machine learning have reasonable agreement and similar accuracy in the images
with low-level noise. However, by adding Gaussian blur noise to the input
images, classic segmentation misclassified the voxels up to 50$\%$. Berg et
al. [76] demonstrated the performance of the classic image segmentation method
compared to deep learning through Waikato Environment for Knowledge Analysis
(WEKA). The classic segmentation technique is vulnerable to radial artifacts
that generates a plausible segmentation. Marques et al. [77] employed two deep
learning methods Seg-Net and U-Net for segmenting and classifying thin rock
sections into pores and backgrounds. Varfolomeev et al. [61] estimated
performance of 2-D SegNet, 2-D U-net, 3-D U-net for processing micro-CT image
of rock samples in image segmentation. They also introduced the application of
edge-preserving filtering before the application of deep learning for
segmentation and found that pre-processing of the grey scale images can affect
segmentation results. Da Wang et al. [78] investigated U-Net, Seg-Net, and
Res-Net in image segmentation and found that 3-D U-shaped ResNet performs
better among the others in mineral segmentation accuracy and simulation
permeability. Mahdaviara et al. [79] examined the performance of 2-D and 3-D
U-Net deep learning models for multi-phase segmentation of unfiltered X-ray
tomograms of gas diffusion layers with different percentages of hydrophobic
polytrafluoroethylene and presented in Fig. 3c. Wang et al. [80] applied a
random forest-based machine learning algorithm and UNet++ deep learning for
image segmentation of 3-D scanned shale images from micro-CT. Siavashi et al.
[81] used autoencoders for semantic image segmentation, which was defined as
the categorization of pixels within an image into semantic categories. Four
deep autoencoders, SegNet, UNet, ResNet and UResNet were trained on X-ray
images of a brine decane two-phase experiment and presented in Fig. 3d. After
comparison, they concluded that the UResNet algorithm, upon adequate training,
contained the best proficiency in segmenting multi-phase flow images.

Figure 3: Examples of porous material image segmentation using different
methods. (a) Global thresholding and locally adaptive segmentation methods
[70]. Reproduced with permission from John Wiley and Sons, (b) Watershed
method[71]. Reproduced with permission from John Wiley and Sons. (c) 2-D U-Net
and 3-D U-Net [79]. Reproduced with permission from Elsevier. (d) UResNet[81].
Reproduced with permission from Elsevier.

###  2.3 Image reconstruction

To acquire the morphological, topological, and flow properties of porous
media, we need not only the division of an image into a number of regions
through image segmentation, but also reliable tomography images of porous
media. The experimental images are generally 2-D image data which contains
low-resolution cells. The image reconstruction technique and the super
resolution technique should be introduced. The image reconstruction technique
provides a 2-D/3-D complete structure realistic-look image from the
experimental image data for predicting effective thermal conductivity,
permeability, and interfacial heat transfer coefficient [82, 83]. The super
resolution technique is presented in the Chapter 2.4.

The image can be reconstructed through classic reconstruction methods, which
included stochastic reconstruction methods [84, 85, 86], (non) ballistic
procedures [87, 88, 89], process-based reconstruction methods [90] and
discrete-element methods [91, 92]. The classic reconstruction methods are
mostly time-consuming and in some cases overestimate or underestimate the
dynamical properties [93]. The deep learning methods show more realistic image
reconstruction [94].

Furthermore, GAN as an unsupervised deep learning methodology, has been
designed to perform fast sampling from the learned probability density
representation allow full parallel generation, and become the dominant model
of image reconstruction, making the generator(G) appropriate to generate large
volumetric 3-D images of porous media. Mosser et al. [95] developed a Deep
Convolutional GAN(DCGAN) to generate the 3-D images which has been presented
in Fig. 4a. The G and D networks are both fully CNNs. G tries to generate 3-D
images, while D tries to distinguish between fake and real images’ probability
density. D tries to label any received sample correctly, meanwhile, G learns
how to create images to fool D. In the process of the adversarial battle
between G and D, the visual inspection, different morphological, statistical,
and dynamical parameters are evaluated the reconstruction performance [96,
97]. In some cases, the training dataset of images is incomplete which is
difficult to reconstruct. Feng et al. [98] proposed the conditional GAN(CGAN)
to recover the full 2-D images and generate the 3-D image through stacking in
Z-direction which has been presented in Fig. 4b. Three-step sampling(TSS) is a
powerful layer-by-layer reconstruction method. After TSS, the sampling images
are available and reconstructed using CGAN considering the continuity and
variability of adjacent layers in 3-D porous media. The coupled TSS-CGAN
method is 750 times faster than the classic multiple-point statistics
method(MPS). In order to better style and levels of detail in synthetic
images, the style-based GAN is proposed which has been presented in Fig. 4c. D
consists of a down-sampling block and fully connected layers which add more
feature maps to the output. Hajizadeh and Farhadpour [99] applied image
quilting on porous media images created by StyleGAN. Shams et al. [100]
developed a coupled GAN and auto-encoder(AE) networks to reconstruct 3-D
porous media images which have been presented in Fig. 4d. The AE consists of
an encoder, code, and decoder. The encoder extracts features of input images
and compresses them to code. The coupled GAN-AE algorithms provide more
realistic results compared with the Modified Joshi-Quiblier-Adler
reconstruction(MJQA) [83, 101]. In addition, the resolution of generated
images of the GAN-AE model is four times the resolution of the original coarse
representation. Because of the GAN model’s low reconstruction efficiency, Feng
et al. [102] proposed a general end-to-end deep learning-based 3D
reconstruction framework called the state-of-the-art GAN-based method (Bicycle
GAN) which has been presented in Fig. 4e. The mapping between a 2-D slice and
its 3-D structure is learned by the neural network. This method can achieve
3.6$\times$ $10^{4}$ speedup factor compared with the classical method. Zhang
et al. [103] proposed a deep learning algorithm based on GAN and CNN named
LGCNN. In particular, the size of constructed porous media is far larger than
previous GAN reconstruction algorithms as much as 3-4 orders of magnitude.
Amiri et al. [104] employed the SliceGAN comprised of a 3-D G with 3-D
transpose convolution layers, alongside three 2-D discriminators which has
been presented in Fig. 4f. During training, the G creates 3-D images and the
random slices are fed into the D. This model can capture the heterogeneity and
variability that cannot be captured by the limited field of SEM tomography and
sidestep the 3-D reconstruction constraints. The images produce by Generator
has been extensively used to unsupervised data augmentation in Argilaga [105,
106] and corrales2022wasserstein.

Figure 4: Six GAN models that have been used to reconstruct tomographic images
of geo porous material. (a) DCGAN [95]. Reproduced with CC BY 4.0. (b) CGAN
[98]. Reproduced with permission from Elsevier. (c) StyleGAN [99]. Reproduced
with permission from Springer Nature. (d) GAN-AE [100]. Reproduced with
permission from Elsevier. (e) BicycleGAN [102]. Reproduced with permission
from Elsevier. (f) SliceGAN [104]. Reproduced with CC BY 4.0.

In addition to the deep learning GAN-based model, many studies have utilized
other algorithms commonly used in computer vision. In the research of Zhang et
al. [107], a recurrent neural network (RNN) based model, namely 3D-PMRNN has
been proposed to solve the 2D-TO-3D reconstruction of porous media. This model
requires only one 3D sample at least and reconstructs the 3D structure layer-
by-layer. The result shows that the synthetic layers of 3D-PMRNN show better
diversity than BicycleGAN [102] and the reconstruction efficiency has been
improved compared to the GAN-based model. Papakostas et al. [108] studied six
nature-inspired optimization algorithms and concluded that there are needs for
more descriptive and high-order functions for measuring the material’s phase
distribution to increase the fidelity of the reconstructed microstructure.

###  2.4 Super resolution

The super resolution technique involves the inference of a high-resolution
(HR) image from low-resolution (LR) measurements [109]. The quest for HR
images has been one of the major pursuits in both experimental and numerical
research. The numerical method to simulate the velocity fields within the
porous media is generally DNS method which is highly demanding in terms of
computational time and mesh requirements. Furthermore, the super resolution
technique can improve the flow prediction performance effectively instead of
increasing spatial grid resolution [110]. In the experimental, the porous
media images lack sub-micron structure information which is important to
predict the fluid transportation properties [82, 100] because of the micro-CT
probing limitations. Because of the micro-CT resolution limitations, the super
resolution reconstruction provides robust approaches to improve resolution and
remove noise and corruption based on statistical inference. The super
resolution reconstruction enables a realistic resemblance of real HR porous
media structure images with a desirable field of view [111]. Each of the two
studies is described as follows respectively.

In the study of flow fields in the porous media, the DNS can generally be
implemented using the lattice Boltzmann method, finite volume method, or
finite element method. In order to capture turbulent flow within porous media
precisely, the refined mesh is necessary. The ML is capable of learning the
complex relationship between experiments and a cost-effective prediction in
the simulation of coarse mesh. The CNN and U-net models are commonly used for
flow field prediction, but the prediction accuracy in complex porous
structures is low especially the pore flow with small dimensions [112, 113,
114]. The super resolution techniques have been introduced and developed to
reconstruct fine-scale turbulent flow from coarse velocities [115, 116].
Although super resolution has been used in many flow field reconstruction
research[117, 118, 119], the flow field in the porous media has not been
extensively studied due to the complexities involved in the structure. Zhou et
al. [110] developed the U-net, a successive encoder-decoder network that used
the geometry and coarse velocities as the model input with the super-
resolution technique. The super-resolution can improve the prediction accuracy
and regularize the ill-posedness of this learning problem which has been
presented in Fig. 5a. Pawar et al. [120] employed vanilla SRGAN and Gg-SRGAN
to generate HR concentration profiles which was presented in Fig. 5b. The
results show that the difference between the images reconstructed from Gg-
SRGAN and the ground truth solutions generated by the finite element method is
low. Doloi et al. [121] proposed using super-resolution techniques to
reconstruct fine-scale oil saturation map of an upscaled-reservoir model from
a low-resolution simulation.

Figure 5: Super resolution examples of porous material reconstruction
including the flow fields. (a) The prediction of velocity by UNet+SR [110].
Reproduced with permission from APS. (b) The prediction of concentration by
Gg-SRGAN [120]. Reproduced with permission from Elsevier. (c) The porous
structure by SRCNNR [122]. Reproduced with permission from John Wiley and
Sons. (d) SRGAN[123]. Reproduced with permission from Springer Nature. (e)
HSDL[124]. Reproduced with permission from Elsevier. (f) EAST[125]. Reproduced
with permission from Elsevier.

In the study of porous media structure, super resolution techniques have shown
promise as a way to enhance images of real porous media structures. The super
resolution can improve the quality of the LR images by reducing the difference
error between LR and HR images [126] or finding the mapping relationship
between LR and HR images [111]. Wang et al. [127] used the LR and HR 3-D
micro-CT images as input. The LR images can be generated from the cubic spline
interpolation by HR images. A CNN model was trained to reconstruct the 2-D
layers and stacked those 2-D images to build the 3-D HR images. The developed
SRCNN can be used to improve the resolution of 3-D porous media images by the
corresponding HR 2-D images. They found that the SRCNN model would retain the
notable noises in the LR images. In other words, the SRCNN reconstructed
images have been shown low perceptual quality and even lost some certain
texture features loses. Wang et al. [122] further introduced the Super
Resolution GAN(SRGAN) to improve their previous work(SRCNN) to be usable
various images. The DeepRock-SR dataset [128] was used for training the SRGAN
network. The SRGAN network can better reconstruct the grains, pores, edges,
and HR-SR images due to significant improvement in perceptual and textural
features which have been presented in Fig. 5c. Janssens et al. [129] used
U-Net generator to reconstruct HR images and compared the porosity and
permeability in reconstruction results with experimental results to validate
the super resolution model. In particular, two-phase displacement images were
used to train U-Net GAN, and the image resolution is improved. Zhang et al.
[123] proposed a 3-D reconstruction method SRCSGAN based on ConSinGAN [130]
and residual networks for the SR reconstruction of porous media. They also
compared the SNESIM [131], MSPGAN [107], and ConSinGAN and found that SRCSGAN
has been proven its accuracy in the SR reconstruction by sandstone and shale
samples. Meanwhile, SRCSGAN can save and reuse the training model after the
first training which has shown its high efficiency in multiple reconstructions
of porous media and presented in Fig. 5d. Kamrava et al. [124] generated an
augmented dataset of 2-D porous media CT images by cross correlation-based
simulation(CCSIM). The hybrid stochastic deep-learning algorithm was proposed.
The result confirmed that the HSDL model enabled a better estimation of the
physical properties of complex porous media and presented in Fig. 5e. Zhao et
al. [132] consumed that the result of super resolution was determined by not
only voxel size but also the porous media feature such as pore throat. They
used the ratio of pore throat size and voxel size to group training set and
registered micro-CT as input instead of synthetically down-sampled images. The
combination of those two techniques can produce images with better sharpness.
Xing et al. [125] introduced self-attention and channel attention mechanisms
to undergo structural optimization and proposed a novel Efficient Attention
Super Resolution Transformer(EAST) model. The self-supervised fine-tuning
approach can enhance the model robustness and resist noise and blur
interference and the result is presented in Fig. 5f.

##  3 Data-driven modeling and prediction

The processes of flow and heat transfer in porous media are generally
characterized by multi-scale. The mass and momentum transport phenomena within
porous media are highly complex, as the fluid dynamics evolve across scales
that differ by several orders of magnitude in both space and time [133].
Consequently, the study of flow and heat transfer in porous media has been a
challenging problem in the scientific community. In this section, we focus on
the prediction of the typical physical fields of flow and heat transfer within
porous media based on the data-driven approach. Specifically, the flow
dynamics problems are divided into macroscopic and pore-scale views. The
relationship between macroscopic flow and pore-scale flow is illustrated in
Fig. 6.

Figure 6: Relationship between macroscopic flow and pore-scale flow [134].
Reproduced with permission from John Wiley and Sons.

###  3.1 Macroscopic modeling and prediction

Based on the continuum hypothesis, continuum-scale models are frequently used
to describe macroscopic modeling and prediction. In these models, a porous
medium is typically treated as a homogeneous and isotropic region, neglecting
the pore-scale heterogeneity. Macroscopic conservation equations are solved,
which are obtained based on volumetric averaging theories for a representative
elementary volume (REV) in which phenomenological parameters are introduced to
implicitly account for the microstructures of porous media [135]. These
phenomenological parameters include porosity, pore size, specific surface
area, tortuosity, permeability and relative permeability, effective
diffusivity/thermal conductivity, effective reaction rate, etc. For instance,
single-phase fluid flow is often described using the Darcy, Brinkman-extended
Darcy, or Forchheimer-extended Darcy equations [136, 137], wherein
permeability quantifies the flow capacity of porous media. Continuum-scale
models are widely adopted for predicting large-scale transport processes of
engineering interest. However, since the average length of a computational
element in the continuum-scale models is usually much larger than the typical
pore size of a porous medium, the microscale heterogeneity of porous media is
neglected in the continuum-scale models. In macroscopic modeling, porous media
are characterized by structural macroparameters including the porosity and the
permeability. Permeability describes the ease with which a fluid can traverse
a medium, and it is a critical transport property for 3D structures. This is
of significant interest to the research community, particularly because
analytical solutions are generally unavailable except for the most simplified,
ideal geometries. Continuum-scale models are promising in terms of
computational efficiency and user-friendliness. However, these models require
additional input parameters from experiments or empirical equations to solve
the momentum and energy equations. Moreover, these models do not facilitate
flow visualization. Despite numerous efforts to develop relationships that
model the permeability based on the structural characteristics of the domain,
a universal relationship remains elusive [138, 139].

Deep learning based models have demonstrated excellent accuracy in predicting
flow dynamics and superior computational efficiency compared to traditional
continuum-scale models [140]. Through extensive data learning, flow
visualization and prediction of heterogeneous macro flow variables are
realized. Tang et al. [141, 142] combined a residual U-Net with convLSTM
networks to predict the temporal evolution of the saturation and pressure
fields in 2-D and 3-D oil production simulations. Their recurrent R-U-Net
model was later applied to CO2 storage with the coupled flow and geomechanics
[143]. Wen et al. [144] developed an R-U-Net-based surrogate model for CO2
plume migration, where injection durations, injection rates, and injection
locations are encoded as channels of input images. Wen et al. [145, 146]
combined U-Net and Fourier neural operator by adding convolutional information
in the Fourier layer, which yields significantly improved cost-accuracy trade-
off. Data-efficient multiphase flow predictions for gas saturation and
pressure were built. Deep-learning-based surrogate models are proposed by Ju
et al. [147] to address the challenge of accurately capturing the impact of
faults on CO2 plume migration in subsurface flow problems. Their approach,
integrating Graph Convolutional Long Short-Term Memory (GConvLSTM) and
MeshGraphNet (MGN), demonstrates the potential of GNN-based methods in
accurately predicting gas saturation and pore pressure evolution in reservoirs
with impermeable faults, highlighting its effectiveness in modeling complex
subsurface flow for CO2 geological storage. Mo et al. [148] developed a DL
surrogate model that integrates an autoregressive model with a convolutional-
NNs-based (CNNs) encoder-decoder network to forecast CO2 plume migration in
random 2-D permeability fields.

###  3.2 Pore-scale prediction

Pore-scale models have been extensively used in the porous media study in
order to obtain the microscopic flow topologies [149, 150]. The pore-scale
models provide distribution details of interested variables at the pore scale,
enable a direct link of the complex transport processes to the realistic
porous structures, and thus can provide a deep understanding of the
relationships between structures, processes, and performance. The role of the
pore-scale modeling as well as its typical sub-processes is shown in Fig. 7.

Figure 7: Implementation processes and roles of pore-scale modeling [151].
Reproduced with permission from Elsevier.

Various computational methods have been developed and applied to the pore
scale to study flow and transport phenomena [152]. Considering the complexity
of the pore geometries, the first step prior to modeling is to characterize
the pore space and structures. Imaging techniques such as X-ray
microtomography (XMT) [153] and magnetic resonance velocimetry (MRV)
measurements [154] as well as the image-based methods in Section 2 have made
it possible to obtain accurate 3-D characterizations of the pore geometry at
high resolution. Pore-scale modeling can then proceed either by simulating
directly on the complex pore geometry, or on a conceptualized pore network
that maintains the same topological structure. The first class of model is
typically referred to as direct numerical simulations. These approaches
include mesh-based method (MBM) [155, 156], lattice Boltzmann method (LBM)
[157, 158, 159] , and smoothed particle hydrodynamics (SPH) [160]. The second
class of model represents the pore space as a network connected by
geometrically simplified pore bodies and pore throats and most commonly takes
the form of pore-network models (PNM) [161]. Both flow and transport processes
can be represented using either of these approaches. Chen et al. [162]
compared the velocity field of different pore-scale flow and solute transport
simulation methods as shown in Fig. 8a. These approaches can be combined with
machine learning to complement each other.

Figure 8: (a) The velocity field predicted by traditional method [163].
Reproduced with permission from Elsevier. (b) Hidden states of the
system—pressure and velocity fields—obtained by using the ML method based on
the data on the concentration field [164]. Reproduced with permission from
AAAS. (c) The electric potential and the streamline plot from the pore-scale
prediction by ML method [165]. Reproduced with CC BY 4.0.

Using machine learning (ML) models as surrogate models to estimate properties
from data is a promising avenue. Current work in pore-scale ML models has
shown promising results [166, 167]. Raissi et al. [164] developed hidden fluid
mechanics (HFM), a physics-informed deep-learning framework capable of
encoding the Navier-Stokes equations into the neural networks while being
agnostic to the geometry or the initial and boundary conditions. Hidden states
of the system —pressure and velocity fields—obtained by using HFM method based
on the data on the concentration field, as shown in Fig. 8b. The high accuracy
that ML models are beginning to demonstrate may allow them to be orders-of-
magnitude faster surrogates for simulation flow of effective properties of
pore-scale materials, or for initializing simulations into a near-converged
state. Wen et al. [166] demonstrate their ML-based scale-bridging framework to
capture adsorption under nanoconfinement where there is no clear separation of
scales. They incorporate atomistic adsorption effects that occur within a
nanoconfined pore as simulated accurately by MD into a continuum LBM that is
capable of simulating larger scales. Nevertheless, the pore-scale ML subspace
remains vastly unexplored, in part due to the complexity of obtaining and
processing data. As examples, current works are often limited to models
trained with small 3D samples [167] or models trained with bigger samples but
with restricted geometry type (spherepacks) [168]. Notably, Prifling et al.
[169] used 90,000 synthetic microstructures from 9 families of distributions
downsampled to 963 to perform 3D machine learning. For porous media, datasets
of natural examples are beginning to be assembled, but of limited scale and
lack flow information [170]. Large, labeled datasets are usually limited to
synthetic examples [169] and/or 2-D flow. A large-scale collection of pore-
scale data is critical for developing advanced ML algorithms that generalize
to unseen samples from many real-life sources. In order to push the state of
the art for ML forward and provide comparisons between ML techniques, it would
be very useful to have a large number of labeled data (the results of
expensive full-physics simulations) to build models that work in the
complexity of real-world pore structures. Santos et al. [165] developed a
diverse dataset that represents challenging, complex porous media in the
context of 3D simulation, empirical functional forms, and machine learning, as
shown in Fig. 8c. This diversity covers dimensions such as porous media
lithology, boundary conditions, geometric resolution, and physical processes
simulated. They introduced the large-scale collection of images, geometric
data, and flow and electrical simulations.

###  3.3 Heat transfer

Data-driven machine learning methods have proven effective in studying heat
conduction in porous media [171]. Wei et al. [172] employed a convolutional
neural network (CNN) to predict the effective thermal conductivity of porous
media, achieving high accuracy and negligible time required for predicting. By
constructing a convolutional encoder-decoder neural network, Wang et al. [173]
also demonstrated that the temperature field in porous media can be accurately
predicted through data-driven analysis. As an image-to-image regression model
widely used for image denoising and reconstruction [174], this method yields
speedups of several orders of magnitude compared to numerical calculations
when predicting the temperature field of the input porous media images.
However, a major deficiency of data-driven machine learning methods is that
sufficient labeled data is required to constrain the training process and
achieve high prediction accuracy [173]. Although the labeled training data can
theoretically be obtained using any method with good accuracy, in practice,
they are commonly obtained from numerical calculations that require solving
physics-based partial differential equations. Acquiring substantial amounts of
labeled training data is usually challenging, which hinders the application of
data-driven machine learning methods for efficient prediction of heat
conduction in different porous media [175]. Xu et al. [176] applied physics-
informed neural networks to investigate heat conduction in porous media. The
obtained effective thermal conductivity values for an ensemble of porous media
samples had an average relative error of 2.49%. They also illustrated that
physics-informed neural networks could be easily extended to predict nonlinear
heat conduction in porous media. Hajimirza and Sharadga [177] investigated
structural features that have significant effects on thermal transport in
porous media and identified five physics-based descriptors to characterize the
structural features: shape factor, bottleneck, channel factor, perpendicular
nonuniformity, and dominant paths. The proposed descriptors were incorporated
into machine learning models to predict the effective thermal conductivity of
porous media, and the results were shown to be fairly accurate, as shown in
Fig. 9a.

Figure 9: (a) Thermal conductivities and typical porous structures [178].
Reproduced with permission from Elsevier. (b) Ray tracing for ray with an
entrance angle of 30/60 deg in a porous domain [177]. Reproduced with
permission from Elsevier. (c) Neural network results with the prediction
performance measures, regression plots for transmissivity, reflectivity,
absorptivity [177]. Reproduced with permission from Elsevier.

The presence of various-sized pores and particles in porous media results in a
complex mechanism of radiative transfer, as rays continually interact with
particle-void domains. To accurately demonstrate the radiative properties of
structures, a large number of light bundles must be traced to completion or
reach the hypothesized conditions. Due to the presence of numerous particles
in porous media, the interaction between light, particles, and voids can lead
to a variety of patterns and trajectories. This necessitates running many
simulations at a high computational cost. ML algorithms have proven to be an
efficient alternative for addressing this challenge by learning the nonlinear
and complex relationship between the inputs of the problem and the outputs,
i.e., the radiative properties. ML can be employed for fast and reliable
prediction of radiative properties through appropriate abstraction of the
geometric configuration and involved physics, in terms of suitable geometrical
and physical features. In recent studies, Petrasch et al. [179] utilized the
two-point correlation function and minimum size of representative elementary
volume to conduct tomography-based analysis of reticulated porous ceramics.
Additionally, other studies have utilized parameters such as tortuosity, chord
length distribution, and other geometrical metrics to determine the structure-
related transport measures of a 3-D microstructure of porous fuel cell [180].
In another study [181], an artificial neural network (ANN) framework was
designed that considered bed height, particle diameter, and emissivity of
monodisperse spherical packed beds as input and achieved close approximations
for radiative properties compared to Monte Carlo ray tracing (MCRT) ground
truth. By utilizing the geometric abstraction, eight geometric features were
employed in an ANN algorithm to estimate the optical properties of
monodisperse and polydisperse circular and square-shaped pores in porous
media. In a more comprehensive study, Hajimirza and Sharadga [177] proposed a
range of geometrical features for the prediction of the radiative properties
of heterogeneous non-overlapping and overlapping circular particles in porous
media (Fig. 9 b and c). A sensitivity analysis was conducted to determine the
significance of the features and concluded that the addition of directional
features could improve the accuracy of the ANN model prediction. Eghtesad and
Hajimirza [182] achieved an enhanced estimation of temperature-dependent
radiative properties of heterogeneous porous media by using the Monte Carlo
ray tracing algorithm for ground truth data generation. This process involves
developing expertly-designed engineered features for geometry abstraction and
implementing artificial neural networks for quick and efficient predictions.

In addition, due to the complexity of heat transfer mechanisms involving
porous media, machine learning methods are also commonly used for model
fitting, including heat transfer coefficients for single-phase and two-phase
processes. For example, Jahanbakhsh et al. [183] developed a generalized
predictive model for analyzing heat flux of thin film evaporation within
hierarchical structures.

##  4 Physics-informed machine learning

Relying solely on massive amounts of data as direct inputs and simply
increasing the depth and width of neural network models to construct complex
mapping relationships is not an optimal modeling strategy. Furthermore, data-
driven approaches usually require a large amount of high-confidence sample
data. However, such methods become impractical when sample data is scarce or
when flow field sample data cannot be obtained. Due to the complexity of
problems, system characteristics often need to be characterized by partial
differential equations (PDEs). Therefore, based on a thorough understanding of
flow physics and model architecture, feature extraction of input data can
effectively balance model accuracy and generalization capability by reducing
input information while ensuring accuracy, simplifying the dimensions and
complexity of network models. This approach enhances model interpretability
and aids in understanding physical systems. In this section, we will focus on
the application of machine learning methods integrated with physical systems
in the context of porous media flow and heat transfer.

###  4.1 Physics-informed neural network (PINN)

The Physical Information Neural Network (PINN) constitutes a novel method
within the domain of machine learning, particularly notable in fluid
mechanics, as it directly engages with PDEs through neural networks.
Originally introduced by Raissi and Karniadakis et al. [45, 184], PINN has
progressively emerged as a new paradigm for solving both forward and inverse
problems in partial differential equations in recent years. Its escalating
prominence stems from its adaptable and versatile nature relative to
conventional numerical techniques. Unlike standard neural networks reliant
solely on data-driven principles, the core effectiveness of PINN resides in
its incorporation of physical priors within the network’s loss function,
encapsulating governing equations alongside initial/boundary conditions,
thereby guiding its learning trajectory. By encapsulating underlying physical
conservation laws as prior knowledge, PINN offers a synergistic amalgamation
of adherence to physical laws and precision in data-driven predictions.
Additionally, PINN’s versatility extends to scenarios marked by sparse
measurement data. Through judicious selection of training samples within
convection fields, PINN facilitates the comprehensive reconstruction of all
pertinent variables within the flow field, thereby enhancing its applicability
in real-world settings beset by data scarcity.

In the investigation of porous media problems, the intrinsic attributes of
PINN render it a universal instrument for confronting the multifaceted
challenges prevalent in porous media. Within PINN-based methodologies, the
loss function chiefly comprises the $L^{2}$ norm of the residual associated
with the governing equation governing fluid flow within porous media [185].
Tartakovsky et al. [186] utilized the PINN method to predict the Darcy law
governing fluid flux in porous media. Hanna et al. [187] developed a residual-
based adaptive PINN using fully connected networks for two-phase Darcy flow
simulation and capturing moving flow fronts. Shokouhi et al. [188]
incorporated a simple set of two-phase flow equations to impose constraints on
two deep learning models used for predicting the simulation response of carbon
dioxide storage sites, comparing data-driven and physics-information-driven
deep learning models. Despite the PINN model requiring almost twice the
training time compared to the data-driven model, it exhibited marginally
superior performance in pressure and gas saturation domains. Almajid and Abu-
Al-Saud [189] implemented PINN by an open-source library (DeepXDE)[190] to
solve the forward and inverse two-phase flow problems of the Buckley-Leverett
PDE equation, which differs from that of Fraces et al. [191] in the
construction of the loss function. The documentation and complete code can be
found at Github [191], where a small amount of observed data is included for
improved prediction accuracy.

For Darcy’s law of fluid flux in porous media, an accurate theoretical model
for closing the system of conservation equations is available. However, It is
not available for more complex systems, including multiphase flows and
transport in porous media [186]. Consequently, to describe unsteady multiphase
flows in such situations, it becomes necessary to linearize the equation
system. Dieva et al. [192] investigated the feasibility of parameterize
function application to solve direct and inverse problems of non-Newtonian
fluid flows in porous media by using the experimental results of core samples
processing in laboratory and field data from a real oil field as examples.

Figure 10: Examples of the PINN frameworks of latter versions with various
enhancements and extensions. (a) PI-DeepONets [193]. Reproduced with CC BY
4.0. (b) PPNN [194]. Reproduced with CC BY 4.0.

Building upon the conventional PINN framework, various researchers have
introduced subsequent versions with a plethora of enhancements and extensions.
Examples include fPINN [195], HFM [196], PI-DeepONets [193], PPNN [194], PINN-
SR [197], PeRCNN [198], PINO [199], and some of these frameworks are presented
in Fig. 10. Based on the XPINN model proposed by Jagtap and Karniadakis [200]
as shown in Fig. 11a, Alhubail et al. [201] augmented the parallelization
ability to solve simple fluid flow scenarios in heterogeneous porous media.
Due to the conspicuous discontinuities in hydraulic conductivity, they
employed the XPINN concept to partition the complex domain into smaller
subdomains, deploying multiple neural networks with optimally selected
hyperparameters by enforcing residual continuity conditions on the common
interfaces of adjacent subdomains. Zhang [202] built a physics-informed
convolutional neural network (PICNN) framework grounded on the finite volume
method for simulating and predicting single-phase Darcy flows within
heterogeneous porous media without labeled data, which can be also extended to
simulate two-phase Darcy flows [203] and presented in Fig. 11b, but only
structured grids are allowed. Wang et al. [204, 205, 206, 207] proposed a
theory-guided convolutional neural network (TgCNN) for predicting subsurface
flow problems, forecasting subsurface flow problems, incorporating not only
governing equations but also expert knowledge and engineering control into the
training process. Subsequent iterations of the proposed TgCNN framework extend
to two-phase flow problems in porous media and are employed for efficient
inverse modeling.

Figure 11: Three extended frameworks of PINN for analyzing flow in porous
media. (a) XPINN [200]. Reproduced with CC BY 4.0. (b) PICNN [202]. Reproduced
with permission from Elsevier. (c) PINN embedded with Fourier basis [208].
Reproduced with permission from Elsevier.

As shown in Fig. 11c,Du et al. [208] proposed physics-informed neural networks
enriched by Fourier basis embedding, wherein the input space of spatial-
temporal coordinates undergoes transformation into spectral space, to examine
unsteady carbon dioxide injection processes in porous media. The incorporation
of Fourier features notably enhances the prediction accuracy of long-term
dynamics and representation of high-frequency fingering patterns to address
the spectral bias problem [209], and the proposed approach has enhanced
numerical stability even with low resolutions.

Numerous efforts have been devoted to analyze flow in porous media with
physics-informed neural networks. Recently, employing PINNs to predict
temperature/heat flux fields and effective thermal conductivity of porous
media has gained attention [183]. Xu et al. [176] employed physics-informed
neural networks to investigate heat conduction problems in various porous
media and the framework as shown in Fig. 12a. A good performance was shown in
predicting temperature fields when compared with numerical calculation
results, with an average relative prediction error of 2.49%, which was
presented in Fig. 12. On the other hand, owing to the absence of data labeling
requirements, PINN necessitates considerably less time to develop the proxy
model compared to data-driven neural networks, and also achieves a speedup of
up to 5 orders of magnitude in prediction over numerical calculations.

Figure 12: (a) PINN maps the input porous media to desired thermophysical
characteristics. The neural network comprises two components. Physics-informed
loss functions can be constructed, which constrain the training of the neural
network using the physical governing equations without labeled data. (b-c)
Temperature/heat flux fields of two porous media samples in the test dataset.
The results from numerical calculations, the PINNs, and the errors between
them are shown [176]. Reproduced with permission from Elsevier.

###  4.2 Hybrid data-driven and physics-informed approaches

Physics-Informed Neural Networks (PINNs) have emerged as powerful tools for
modeling physical phenomena governed by empirical laws. However, a notable
limitation of PINNs is their tendency to incorporate a large number of
parameters, often numbering in the millions, to capture the intricacies of
these phenomena. These parameters are typically restricted to non-trainable
filters within the final layer, leading to potential overfitting issues [210],
where the deep learning models excel at fitting training data, but have poor
extrapolation capabilities when applied to unseen data [211]. To address the
limitations of deep learning surrogate models, Wang et al. [212] presents the
PhyFlow-HierCAE model, tailored for predicting structure-dependent pore-fluid
velocity fields in rock formations. By embedding the incompressible Navier-
Stokes continuity equation and the momentum equation upstream of the
Convolutional Autoencoder (CAE), the model gains a fundamental understanding
of fluid dynamics principles as presented in Fig. 13a. Fig. 13b shows CAE
alone cannot predict meaningful flow velocities, both PhyFlow-HierCAE and
PhyFlow-CAE successfully yield flow fields similar to the LBM data, while
PhyFlow-HierCAE being the most quantitatively accurate. This approach exhibits
proficiency in training on limited data and generalizing adeptly to unseen
pore structures. However, conventional CNN-based approaches for training
necessitate labeled data formatted on a Cartesian grid with uniform spacing,
potentially leading to unrealistic representations of geometric features near
pore-grain boundaries.

Figure 13: (a) Input/output images and components of the PhyFlow-HierCAE
Model. W is the size of the sub-images (windows). N is the number of sub-
images. (b) Predictions on the LBM velocity testing data [212]. Reproduced
with permission from Elsevier.

In contrast, Kashefi and Mukerji [213] proposed the Physics-Informed PointNet
(PIPN) to address these issues, which can allow users to represent the
geometry of the pore space and its boundaries smoothly and realistically, and
no data interpolation is needed, to predict steady-state Stokes flow of fluids
within porous media at pore scales [185]. By exclusively utilizing the pore
spaces of porous media data as input, PIPN offers adaptable spatial resolution
across the physical domain, facilitating to achieve optimal resolution with
minimal computational overhead. Moreover, Tartakovsky et al. [214] proposed a
physics-informed ML method, which used conservation laws in addition to data
to train DNNs representing states, space-dependent coefficients, and
constitutive relationships. This method enables the prediction of saturated
flow in heterogeneous porous media with unknown conductivity and unsaturated
flow in homogeneous porous media with an unknown relationship between
unsaturated conductivity and capillary pressure. He et al. [215] extended the
PINN-based parameter estimation methodology of Tartakovsky et al. [214] to
predict the transport problem with sparse measurements of hydraulic
conductivity, hydraulic head, and solute concentration and refer to this
multiphysics-informed neural network approach as MPINN. In which uses the
Darcy and advection-dispersion equations together with data to train the deep
neural network, the governing equations and the associated boundary conditions
are enforced at the “residual” points over the domain. For sparse data, the
MPINN approach significantly improves the accuracy of parameter and state
estimation as compared to standard DNNs trained with data only. In the
subsequent research work, He and Tartakovsky [216] present the PINN method for
solving the coupled advection-dispersion equation (ADE) and Darcy flow
equation with space-dependent hydraulic conductivity and test it for one- and
two-dimensional forward and backward ADEs for a range of Pe. They demonstrate
that assigning larger weights to the residuals of initial and boundary
conditions relative to the partial differential equation residuals in the loss
function is crucial for obtaining accurate solutions.

###  4.3 Machine learning augmented numerical solver

The integration of ML models into CFD solvers poses significant challenges due
to the disparate nature of their programming languages and optimization
domains. While CFD software is typically coded in lower-level languages like
Fortran or C++, optimized for CPU computations, ML research often relies on
higher-level languages such as Python, tailored for GPU computing.
Consequently, for machine learning to gain further widespread application in
computational fluid dynamics, there is an urgent need for a new generation of
algorithms capable of seamlessly integrating with solver frameworks, operable
across CPU, GPU, and TPU platforms. Differentiable solvers are gaining
increasing interest in engineering and physical sciences as they promise to
bridge the previously mentioned gap between computational physics and machine
learning and to ensure the automatic differentiation (AD) of the entire
algorithmic representation of the chosen numerical approximation of the PDE
evolution law [217]. Recent advancements in ML have seen neural networks
trained using a combination of AD and accompanying equations [218, 219], the
interest in differentiable simulators has also been driven by the development
of powerful general-purpose AD libraries, such as PyTorch [220], TensorFlow
[221], and JAX [222], and also numerous specialized libraries, such as JAX-MD
[223], JAX-FEM [224], Phiflow [225]. Bezgin et al. [226] developed JAX-Fluids,
which can be directly embedded into the framework using existing machine
learning models, and uses higher-order numerical methods to compute three-
dimensional compressible two-phase flows. JAX-Fluids supports fully
differentiable algorithms for end-to-end data-driven model optimization,
including both physical and numerical method information, and has recently
been expanded to include parallel computation in version 2.0 [227]. Fraces et
al. [191] presented an approach relying on a series of deep adversarial neural
network architectures with physics-based regularization to simulate two-phase
flow in porous problems (Buckley-Leverett), which is highly scalable to
leverage CPUs, GPUs and other computing architectures. In subsequent work,
they further discussed the limitations of Physics Informed Machine Learning
for Nonlinear Two-Phase Transport in Porous Media and further the
applicability of this method in forward pure hyperbolic settings [228, 229].

Figure 14: (a) Porous medium configuration. (b) Flowchart of the optimization
algorithm using a differentiable Navier-Stokes equation solver. Here F, g, and
h are the loss function and the equality and inequality constraints,
respectively. (c) Visualization of the different optimization cases: (i)
square, (ii) rectangular, (iii) centered square, (iv) off-centered square, (v)
centered rectangular, and (vi) off-centered rectangular. (d) Optimized lattice
parameters for the off-centered rectangular case [230]. Reproduced with
permission from American Physical Society.

To enhance the accessibility, scalability, parallel computing performance, and
multi-physics modeling capabilities of machine learning platforms, Ataei and
Salehipour [231] introduce the XLB library, a Python-based differentiable
Lattice Boltzmann Method library. This library facilitates the straightforward
addition of new boundary conditions and collision models, demonstrating its
effective scalability across many GPU devices. To maximize flow rate in the
direction of applied pressure gradient in porous media of periodically
arranged circular rods, Alhashim and Brenner [230] used JAX-CFD [232] to solve
the two-dimensional periodic velocity field past the array and found that the
system with equal particle number density provided an optimum configuration.
The periodic structure of porous medium configuration as shown in Fig. 14a and
Fig. 14b gives a flowchart of the backward differentiation optimization
algorithm. In this work, they minimized the impact on the computational cost
when increasing the introduction of additional constraints or the number of
design parameters. The black dash-dotted curves in Fig. 14c show a comparison
between the numerical and the theory obtained from the optimization of the
full flow problem.

##  5 Conclusion and perspective

This review has examined the current advancements in data-driven methods for
analyzing flow and transport in porous media. By focusing on three pivotal
methodologies—image-based techniques, data-driven flow modeling, and physics-
informed machine learning, we have highlighted how these approaches are
transforming the field.

Image-based techniques have significantly enhanced the detail and accuracy of
images derived from porous structures through innovative methods such as image
segmentation, image reconstruction, and super-resolution technologies. These
data-driven imaging techniques provide a more precise base for simulation and
analysis, allowing for better characterization of flow dynamics within porous
media. Recent breakthroughs have integrated these techniques into both
traditional and novel modeling frameworks, enhancing the overall fidelity of
simulations. Data-driven flow modeling employs CNNs, U-Nets, GNNs, and other
neural network structures to capture the complex patterns of flow and
transport dynamics through porous media. These tools have shown substantial
promise in offering predictive insights at both microscopic and macroscopic
levels, significantly reducing the cost and time associated with traditional
modeling methods. By leveraging vast datasets from experiments and
simulations, data-driven models can predict critical parameters such as
permeability and porosity with moderate to high accuracy. Physics-informed
machine learning is an emergent field that combines the strengths of data-
driven methods with fundamental physical principles. Techniques such as PINNs
and other hybrid approaches integrate known physical laws directly into the
learning process, enhancing the physical plausibility and accuracy of
predictions. These methods are particularly effective in addressing complex
problems where traditional data is sparse or noisy. Additionally, ML-augmented
numerical solvers seamlessly combine simulation and data-driven technologies.

Throughout this review, we have summarized the potential of state-of-art data-
driven methods, addressed their limitations, and suggested ways they can be
effectively integrated to improve both the fidelity and efficiency of current
research. Looking forward, several key areas offer significant opportunities
for future research and development:

1\. Enhanced data integration: Combining data from multiple sources, such as
laboratory experiments, field measurements, and high-fidelity simulations,
will provide a more comprehensive understanding of porous media systems.
Developing robust techniques for data fusion and assimilation will be
essential for this integration.

2\. Advanced ML algorithms: Continued exploration and development of novel
neural network architectures and training techniques will lead to more
sophisticated models.

3\. Scalable and real-time applications: As these methods mature, developing
scalable algorithms and efficient computational frameworks will be crucial for
deploying these technologies in practical settings. This will enable real-time
monitoring and control, enhancing their applicability in various industrial
and environmental contexts.

4\. Seamless hybridization of ML and simulations: The seamless integration of
ML with CFD through differential fluid dynamics based on automatic
differentiation promises to revolutionize the field. Tools like JAX-based, ML-
augmented numerical solvers enable this integration by embedding ML models
within CFD frameworks. These solvers support fully differentiable algorithms,
allowing for end-to-end optimization of multi-physical simulations. This
hybrid approach can significantly enhance simulation efficiency and accuracy,
unlike the off-line ML in the current stage.

5\. Uncertainty quantification and robustness: Addressing the uncertainties
inherent in porous media systems and ensuring the robustness of data-driven
models against variations in input data will build trust in these methods.
Techniques for uncertainty quantification and model validation will be
necessary to enhance the reliability and applicability of data-driven
approaches.

In conclusion, the incorporation of image-based techniques, data-driven
modeling, and physics-informed machine learning into the study of flow and
transport in porous media represents a transformative advancement in the
field. By continuing to refine these techniques and explore new applications,
researchers can unlock deeper insights and develop innovative solutions to
complex thermal fluidic challenges.

## Acknowledgements

This study is funded by the National Natural Science Foundation of China
(Grant No. 52276013), SFB 1313 from DFG (German Research Foundation, Project
No. 327154368).

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