Journal of Technology: Fall 2022, additional articles

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Fracture Permeability Estimation Under

Complex Physics: A Data-Driven Model Using

Machine Learning

Xupeng He, Marwah M. AlSinan, Dr. Hyung T. Kwak and Dr. Hussein Hoteit

Abstract /	The permeability of fractures, including natural and hydraulic, are essential parameters for the modeling
	of fluid flow in conventional and unconventional fractured reservoirs. Traditional analytical (Cubic Law-
	based) models used to estimate fracture permeability show unstable performance when dealing with
	different complexities of fracture cases. This work presents a data-driven, physics included model based
	on machine learning as an alternative to traditional methods.
		The workflow for the development of the data-driven model includes four steps.
	1.	Identify uncertain parameters and perform Latin Hypercube Sampling (LHS). We first identify the
		uncertain parameters that affect the fracture permeability. We then generate training samples using
		LHS.
	2.	Perform training simulations and collect inputs and outputs. In this step, high-resolution simulations
		with parallel computing for the Navier-Stokes (NS) equations are run for each of the training samples.
		We then collect the inputs and outputs from the simulations.
	3.	Construct an optimized data-driven surrogate model. A data-driven model based on machine learn-
		ing is then built to model the nonlinear mapping between the inputs and outputs collected from Step
		2. Herein, artificial neural network (ANN) coupling with the Bayesian optimization algorithm is
		implemented to obtain the optimized surrogate model.
	4.	Validate the proposed data-driven model. In this step, we conduct blind validation on the proposed
		model with high fidelity simulations.
		We further test the developed surrogate model with newly generated fracture cases with a broad range
	of roughness and tortuosity under different Reynolds numbers. We then compare its performance to the
	reference NS equation solutions. Results show that the developed data-driven model delivers a good
	accuracy, exceeding 90% for all training, validation, and test samples. This work introduces an integrat-
	ed workflow for developing a data-driven, physics included model using machine learning to estimate
	fracture permeability under complex physics, e.g., inertial effect.
		To our knowledge, this technique is introduced for the first time for the upscaling of rock fractures.
	The proposed model offers an efficient and accurate alternative to the traditional upscaling methods that
	can be readily implemented in reservoir characterization and modeling workflows.

Introduction

Equivalent continuum and discrete fracture models are generally used for modeling fluid flow in fractured reservoirs at field-scale. The proper application of these models requires an accurate assessment of the permeability of rock fractures as input. The permeability of a rock fracture is a complex function of various static parameters, such as mean aperture, roughness, and contact areas, all of which are subjected to the dynamic stress acting on the fracture walls1. Herein, we investigate the relationship between fracture permeability and static parameters.

The full physics Navier-Stokes (NS) equations provide the most accurate approach for estimating the permeability of rock fractures2. Its expensive computational cost makes it infeasible for real applications3. In addition, experimental measurements are even more time-consuming.

Typically, discrete rock fractures have been idealized as two smooth, parallel plates with constant distance4. Based on this assumption, the NS equations lead to the well-known Cubic Law (CL). The CL has been widely used in various disciplines, including hydrology and petroleum, due to its simplicity and efficiency; however, the CL generally overshoots the hydraulic properties of rock fractures5, as rock fractures are formed by two rough surfaces with variable apertures. Subsequently, various CL-based models have been developed in the literature for improving the accuracy of classical CL. These models could be categorized into two kinds: (1) by modifying the definition of the aperture used in the CL, such as arithmetic mean5, geometric mean6, and harmonic mean7, and (2) by incorporating correction factors for fracture roughness6, 8-12, flow tortuosity5, and combined effect of

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roughness and tortuosity3, 13.

These models, however, show unstable performance when dealing with different complexities of fracture cases. In addition, all these models assume the laminar flow regimen and don't include complex physics within rock fractures, i.e., ignoring the inertial effect. To our knowl- edge, only one empirical model developed by Xiong et al. (2011)14 was found in the literature to account for the inertial effect. This empirical model is obtained by fitting with a limited amount of data, which makes its broad applicability questionable. All these traditional models discussed here exhibit different levels of limitations, which inspires us to develop a big data-driven, physics included model for estimating the permeability of rock fractures.

Recent advances in machine learning have revolutionized many industries. They show that provided with a high-quality data set, a well-designed network structure, and proper hyperparameters, this technology is competitive to traditional models in terms of accuracy and efficiency. As a result, it inspires various geoscience and petroleum engineering applications.

Examples include multicomponent flash calculation15, equivalent continuum model construction from discrete fracture characterization16, fracture network recognition from outcrops17, and well data history analysis18. These four applications correspond to four neural networks: artificial neural network (ANN), convolutional neural network, U-Net, and long short-term memory, which are designed for value-to-value,image-to-value,image-to-im- age, and time series problems, respectively.

This study strives to develop a data-driven, physics featuring model based on ANN for estimating fracture permeability, with consideration of static geometric prop- erties, e.g., mean aperture, minimum aperture, rough- ness, tortuosity, etc., and dynamic flow parameters, e.g., Reynolds number. Specifically, a value-to-value model is established to capture the nonlinear relationship between these static geometric and dynamic flow properties as input, and fracture permeability as output. In this study, high-resolution simulations for NS equations based on Latin Hypercube Sampling (LHS) are used to generate the data sets.

Herein, ANN coupled with Bayesian optimization is implemented to obtain the optimized surrogate model with accuracy exceeding 90% for both training and validation sets. We further test the developed surrogate model with newly generated fracture cases with a broad range of roughness and tortuosity under different Reynolds numbers. We also test its performance with the reference solutions from NS equations.

This work introduces an integrated workflow for developing a data-driven, physics included model using machine learning to estimate fracture permeability under complex physics, e.g., inertial effect. To our knowledge, this technique is introduced for the first time. The proposed model offers an efficient and accurate alternative to the traditional upscaling methods that can be readily implemented in reservoir characterization and modeling workflows.

Global Sensitivity Analysis

We can conclude from the Literature Review that the permeability of rock fractures is a complex function of some static geometric properties, e.g., mean aperture, roughness, and tortuosity, and other dynamic flow prop- erties, e.g., Reynolds number. These traditional models, however, ignore the effect of minimum aperture. As the minimum aperture mainly controls the fluid flow within rock fractures for 2D fracture cases, its influence on fracture permeability should be considered.

In this section, global sensitivity analysis is implemented to explore the importance of these static and dynamic parameters on the model response - fracture permea- bility. For the generality of describing 2D rock fractures, we summarized the following five parameters as the model input parameters, including four dimensionless parameters as follows:

1. Mean Aperture: a
2. Relative Roughness: Defined as the ratio of mean to standard deviations of aperture field5:

Relative Roughness = a

σ a1

3. Tortuosity: Defined as the ratio of flow path distance to fracture straight-line distance19:

d flow− path
τ = dstraight −line	2

4. λ: Defined as the ratio of minimum to mean apertures:

λ = amin
a	3

5. Reynolds number: Defined as the ratio of inertial forces to viscous forces20:

Re =	ρQ	=	ρU a
μw	μ	4

where p is the fluid density, Q is the flow rate through rock fractures, µ is the fluid viscosity, w is the fracture width, and U is the characteristic velocity choosing as the averaged mean velocity. The w is set to be 1 m for 2D fracture cases.

We perform global sensitivity analysis based on Sobol or variance-based decomposition to determine the impact of these five parameters on fracture permeability (refer to Saltelli et al. (2010)21 for further mathematic formu- lation). The ranges of these uncertain parameters are collected based on the open-source data set with 7,680 2D fractures3 and the work of Zimmerman et al. (2004)20.

UQLab22 is used to conduct the sensitivity analysis with

1 million realizations, which is proven to be stable. We observe in Fig. 1 that relative roughness shows the most significant influence on the fracture permeability, followed by the ratio of minimum to mean apertures, mean aper- ture, Reynolds number, and tortuosity. Meanwhile, we demonstrate the non-ignorable impact from the ratio of

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Fig. 1 The first-order sensitivity analysis indices using UQLab.

minimum to mean apertures on the permeability of 2D rock fractures. All reviewed models from the literature, however, neglect its influence on fracture permeability, making their applicabilities questionable.

Proposed Workflow

The objective of this section is to develop a data-driven, physics featuring model based on ANN for estimating fracture permeability, with consideration of static geometric properties, e.g., mean aperture, minimum aperture, roughness, tortuosity, etc., and dynamic flow parameters, e.g., Reynolds number.

Figure 2 illustrates the proposed workflow. A detailed description is presented as follows.

STEP 1: Identify Uncertain Parameters and Perform LHS

We first identify the uncertainty parameters that affect the fracture permeability. As previously mentioned, the following five uncertain parameters show a non-ignorable influence on the fracture permeability. We summarized these five significant parameters (heavy hitters) and their corresponding ranges in Table 1.

Herein, we assume uniform distributions and independent relations for all these uncertain parameters. We then generate n realizations of samples, including training and validation, using LHS theory23 to guarantee the space filling manner. Figure 3 shows the LHS from uniform distribution (discrete design) with the different number of realizations.

STEP 2: Perform NS Simulations and Collect Inputs and Outputs from Simulations

In this step, high-fidelity simulations with parallel computing for the NS equations are run for each of the generated samples. These NS equations are solved under the mixed finite element (FE) framework, which is known for approximating the velocity fields accurately24. A detailed explanation of the mixed FE implementation of the NS equations is provided in Appendix A. We then collect the inputs (five uncertain parameters) and outputs (fracture permeabilities) from the simulations.

Figure 4 shows the normalized velocity field and streamline profiles obtained from the high-resolution NS simu- lations. We observe in Fig. 4 that an eddy occurs around places with local large asperity of the sharp corner with a high Reynolds number regimen. These eddies exert a significant influence on fluid flow through rock fractures by shrinking the effective flow channel.

The specific procedure of calculating fracture permeability based on NS solutions is detailed as follows.

• Flow rate calculation: Integrating the velocity across

Fig. 2 The workflow for the development of the data-driven, physics featuring model.

Uncertain Parameters	Input Data	Surrogate-Model
(e.g., roughness, tortuosity,
(Heavy-Hitters)
Reynolds number, etc.)
Latin Hypercube Sampling
				Bayesian
				Optimization
Model 1	…	Model ns	Machine Learning
			(ANN)
Numerical Simulations		Blind Validation
Using High-fidelity NSEs
Objective of Interests	Output Data	Test on New Cases
(Fracture Permeability)

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Table 1 Uncertain parameters used in this study and their corresponding ranges.

Uncertain Parameters	Lower Bound	Upper Bound	Distribution
Mean aperture (mm)	1.11	1.29	Uniform
Relative roughness (-)	2.23	6.96	Uniform
Tortuosity (-)	1.01	1.34	Uniform
Ratio of minimum to mean apertures (-)	0.08	0.67	Uniform
Reynolds number (-)	0.1	100	Uniform
(Laminar flow)	(Nonlinear flow)

Fig. 3 The LHS with different numbers of realizations: (a) 50, (b) 100, (c) 300, and (d) 500. For illustration purposes, we take relative roughness vs. tortuosity as an example. As observed, the space filling trend has become more obvious with the increase of realizations.

the fracture outlet leads to flow rate (denoted as QNS).

QNS = ∫w ∫a (uoutlet n)dwda
5
0	0

where w and a are the fracture width and aperture, re- spectively; uoutlet is the velocity at the outlet; and n is the unit vector normal to the outlet. The w is set to be 1 m for 2D fracture cases.

• Fracture permeability calculation: Combined Darcy's law with CL, fracture permeability (denoted as Kf)

is computed as:

⎛ 12Q μ	⎞1/3
ah = ⎜			NS	⎟
⎝		w∇P	⎠
				6
K f =	a2		1/3	⎛ Q	μ⎞2/3	7
	h	= (12)	⎜	∇	⎟
					NS
	12			⎝ w P ⎠
where ah	is the effective hydraulic aperture, and ∇P is
the pressure gradient across the flow direction.

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Fig. 4 The high-resolution NS solutions with normalized velocity field and streamline profiles under high Reynolds number with non-ignorable inertial effect.

STEP 3: Construct Optimized Data-Driven Surrogate Model

This step strives to establish an optimized data-driven surrogate to map the nonlinear relationship between the inputs and outputs collected from step 2. We automate the process of tuning the hyperparameters for ANN using the Bayesian optimization algorithm instead of the manual trial-and-error tuning process. Bayesian optimization attempts to find the global optimum in a minimum number of steps. A detailed description of Bayesian optimization could be found in Brochu et al. (2010)25.

Figure 5 illustrates the structure implemented in this work, including input layers, hidden layers, and output layers. A detailed introduction of ANN could be found in Goodfellow et al. (2016)26.

STEP 4: Validate the Developed Surrogate Model

We conduct blind validation on the optimized data-driven surrogate model. We should pay attention to its performance on the validation samples, as the overfitting cases often occur with good predictions for training samples, yet poor predictions for validation samples. In this study, the optimization process is conducted until the accuracy exceeding 90% for both training and validation samples is reached.

The following parameters are used to assess the performance of the developed ANN model.

1. APE: The average of prediction errors, PE, between the predicted (denoted by K fpredict ) and ground truth (K truthf ) fracture permeability.

PE =	K fpredict	− K truthf	×100%
	K truthf
				8
APE = 1 ∑N	PE	i
	N	i	=			9
	1

2. PPE: The percentage of PE within an acceptable error margin - 10% is the threshold in this study.

10

We further test the developed surrogate model with 500 newly generated fracture cases with a broad range of roughness and tortuosity under different Reynolds numbers. In this study, inputs and outputs collected from the 500 simulations (corresponding to Fig. 3d) are fed to the ANN model for the coupling training validation process.

We summarized the optimum ANN architecture, optimum hyperparameters, and the related model evaluation

Fig. 5 The ANN structure used in this study, consisting of three types of layers: input, hidden, and output layers. The Bayesian optimization is employed to obtain the optimum hyperparameters.

a

aσ a

τ

K f

amin a

Re

Input Hidden Output