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Physics Inspired Machine Learning for Solving Fluid Flow in Porous Media: A
Novel Computational Algorithm for Reservoir Simulation
Conference Paper · October 2021
DOI: 10.2118/203917-MS
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SPE-203917-MS
Chico Sambo and Yin Feng, University of Louisiana at Lafayatte, LA, 70504, USA
Copyright 2021, Society of Petroleum Engineers
This paper was prepared for presentation at the SPE Reservoir Simulation Conference, available on-demand, 26 October 2021 – 25 January 2022. The official
proceedings were published online 19 October 2021.
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of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect
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Abstract
The Physics Inspired Machine Learning (PIML) is emerging as a viable numerical method to solve partial
differential equations (PDEs). Recently, the method has been successfully tested and validated to find
solutions to both linear and non-linear PDEs. To our knowledge, no prior studies have examined the PIML
method in terms of their reliability and capability to handle reservoir engineering boundary conditions,
fractures, source and sink terms. Here we explored the potential of PIML for modelling 2D single phase,
incompressible, and steady state fluid flow in porous media.
The main idea of PIML approaches is to encode the underlying physical law (governing equations,
boundary, source and sink constraints) into the deep neural network as prior information.
The capability of the PIML method in handling reservoir engineering boundary including no-flow,
constant pressure, and mixed reservoir boundary conditions is investigated. The results show that the PIML
performs well, giving good results comparable to analytical solution. Further, we examined the potential of
PIML approach in handling fluxes (sink and source terms). Our results demonstrate that the PIML fail to
provide acceptable prediction for no-flow boundary conditions. However, it provides acceptable predictions
for constant pressure boundary conditions.
We also assessed the capability of the PIML method in handling fractures. The results indicate that the
PIML can provide accurate predictions for parallel fractures subjected to no-flow boundary. However, in
complex fractures scenario its accuracy is limited to constant pressure boundary conditions. We also found
that mixed and adaptive activation functions improve the performance of PIML for modeling complex
fractures and fluxes.
Keywords: Physics informed machine learning, deep neural networks, Partial differential equations,
activation functions, fractures, production and injection wells, boundary conditions
Introduction
In the last 20 years, machine learning methodologies, especially deep learning methods have yield
transformative and impressive results in diverse applications; to name a few speech recognition (Lee et al.,
2019; Vishal and Aggarwal, 2019), machine vision (Bernal et al., 2019; Aslam et al., 2019; Wang et al.,
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Physics Inspired Machine Learning for Solving Fluid Flow in Porous Media:
A Novel Computational Algorithm for Reservoir Simulation
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Governing Equation
Here we consider the standard governing equation for incompressible, steady state, homogenous system,
and single-phase fluid flow in porous media. It is derived assuming that the fluid moves into a representative
elementary volume. The fundamental law of mass conservation states that the accumulation of mass inside
of given volume must be equal to net flux over its boundaries. This principle can be written mathematically
as follows:
(1)
where
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2020), medical science (Yang et al., 2016), petroleum problems (Wood and Choubineh, 2019; Mahdiani
et al., 2020; Jin et al., 2014; Dobrescu et al., 2016; Sambo et al., 2018a; Sambo et al., 2018b) and image
recognition (Goodfellow et al., 2016).
These remarkable successes achieved in abovementioned and other areas can be attributed to the
explosive growth of data, computing resources, and advances in optimization algorithms (Raissi et al.,
2017a). However, in petroleum engineering applications the cost of acquiring large amount of data can be
prohibitive, and we are often faced with the challenges of making predictions to making decisions with
small or limited data. In this small data scenario, machine learning algorithms can be used as physics-based
algorithms to solve physical laws written in form of partial differential equations (PDE's).
To solve a PDE via deep learning methods, a key ingredient is to constrain the neural network to minimize
the PDE residual and boundary points, and several methods have been proposed to accomplish this. Among
these methods, some are limited to particular types of problems, for instance image-like input domain (Khoo
et al., 2017, Long et al., 2018; Zhu et al., 2019) or parabolic PDE's (Beck et al., 2017; Han et al., 2018).
Some recent studies adopted the minimization of corresponding energy functional using variational form of
PDE (E and Yu., 2018). However, not all PDE's can be derived from a known functional; therefore, Galerkin
type projections have also been considered (Meade and Fernandez, 1994). Alternatively, one could use the
PDE in its strong form directly as presented in the literature works (Lagaris et al., 1998; Lagaris et al.,
2000; Berg and Nystrom, 2018; Raissi et al., 2017; Sirignano and Spiliopoulos,2018). This strong form uses
automatic differentiation to a known and differentiable activation function, thus, avoiding truncation errors
and numerical quadrature errors of variational forms. This strong form is widely known as Physics-informed
neural networks (Samaniegoa et al., 2019; Raissi et al., 2017a; Raissi et al., 2017b; Raissi et al., 2019).
Although Physics-informed neural networks have been successfully tested and validated to find solutions
to both linear and non-linear PDE's (Dwivedi et al., 2019; Samaniegoa et al., 2019; Raissi et al., 2017a;
Chengping et al., 2020; Raissi et al., 2017b; Raissi et al., 2019). However, the problems dealt with are not
in general problems directly related to reservoir engineering applications and the literature lacks detailed
investigation of PIML for solving fluid flow in porous media with reservoir engineering constraints.
Thus, it is extremely important to understand the effectiveness of PIML method for reservoir engineering
applications. Here, we investigated the application of physics inspired machine learning method for the
simulation of steady state and incompressible fluid flow in 2D systems. We examined the performance of
the PIML method for a system containing (a) reservoir engineering boundary conditions, (b) fluxes (source/
sink), and (c) fractures. The objective is to assess how well this method performs for reservoir engineering
related problems.
The remaining part of the paper proceed as follows: Section II describes the governing equation
for incompressible single phase fluid flow in 2D homogenous system. Section III introduces PIML
methodology. Section IV presents a set of numerical examples to demonstrate the effectiveness and the
capability of PIML method. Finally, we conclude the paper in section V.
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ρ is the fluid density, ϕ stands for formation porosity, is the macroscopic Darcy velocity, denotes
the normal at the boundary (∂Ω) of the computational domain(Ω), and q denotes fluid sources and sinks,
i.e., outflow and inflow of fluids per volume at certain locations. Applying Gauss’ theorem, Eq. (1) can be
written on the alternative integral form as:
This equation is valid for particular volumes that are infinitesimally small in domain (Ω), and hence it
follows that the macroscopic behavior of the single-phase fluid must satisfy the continuity equation.
(3)
Eq. (3) contains more unknowns than number of equations and to derive a closed mathematical model.
Therefore, some constitutive equations can be introduced. In this regard, the rock compressibility is
introduced, which describes the relationship between the porosity (ϕ) and the pressure(p).
(4)
It worth mentioned that changes take place under constant temperature and pressure, respectively. Since
∂V is constant for a fixed number of particles, hence, Eq. (4) can be writen in the equivalent form as follows:
(5)
In many subsurface systems, the density changes slowly so that heat conduction keeps the temperature
constant, in which case Eq. (5) simplifies to:
(6)
The macroscopic fluid velocity is defined as volume per area occupied by fluid per time. This usually
defined by Darcy law shown as follows:
(7)
Introducing Darcy's law, the compressibility of fluid and rock in Eq. (3), we obtain the following parabolic
equation for the fluid pressure.
(8)
Assuming that the fluid and rock are both incompressible, meaning cr and cf are independent of p so that
(ct = 0), (8) simplifies to the following equation:
(9)
If we introduce the fluid potential,Φ = p−gρz, Eq. (9) can be recognized as the (generalized) Poisson's
equation ∇. k ∇ϕ = ∓q or as the Laplace equation ∇. k ∇ϕ=∓q if there are no volumetric fluid sources or
sinks. Assuming that the formation is horizontal and homegenous, incompressible fluid, and isotropic, the
permeability, k, is constant(assunmed to be equal to 1). The problem shown in Eq. (9) can be transformed
into:
(10)
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(2)
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The PDE shown in Eq. (10) is the subject of this work, and its solution represent pressure distribution
within the Ω. Depending on the experiment, Eq. (10) can be solved either analytically or numerically. Those
solutions are then used to validate the results from PIML method.
Physics Inspired Machine Learning Algorithm
Figure 1—General PIML workflow for governing PDE. The right side, neural
networks is informed with while in the left side is uninformed neural networks.
Neural networks are known to be universal function approximator (Hornik et al., 1989). They can
approximate a continuous function to any acceptable level of precision. Based on the approach introduced
by Raissi et al., 2019, the solution p(x,y) to the given governing partial differential equation can be
approximated by DNN parametrised by set of trainable parameters (i.e., weight and biases). This simply
means that PDE solution can be represented as nested compositional functions such as:
(11)
x represents input vectors, 2D space coordinates such as x = (x,y), the symbol θ ensemble all trainable
parameters such as:
(12)
Here, σ is activation function and nl represents the number of hidden layers. Now let us define vi(x) =
σ(wix +bi) for a given i = 1,…..,nl on other hand vi(x) = (wix +bi) for i = nl+1. As a result, we can write
the governing equation (PDE) as follows:
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Mathematically, deep neural networks are choice of nested compositional functions. The deep neural
networks employed here is based on feedforward neural network (FNN), also known as multilayer
perceptron (MLP). It uses linear transformation to the inputs and output layers while applying nonlinear
transformations to the hidden layers recursively. FNN is adopted in our modelling process amongst many
other types of neural networks available in the literature, such as the convolutional neural networks (CNN)
and the recurrent neural networks (RNN). This is because the FNN is good enough for most PDE's and easier
to train for deep neural networks (DNNs). The detailed methodology used in the work is shown in Fig. 1.
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(13)
The governing equation can be written in terms of neural networks parameters as follows:
(14)
The PDE in its general form:
Subject to the Dirichlet and Neumann BC's:
(15)
Here, Ω is the computational domain limited with the boundary (∂Ω), where ∂Ω = ∂DΩ∪∂NΩ and
∂DΩ∩∂NΩ ≡ ∅, n is the unit vector normal to the boundary (∂Ω).
To learn the PDE with DNN, we approximate the p(x) with another trial solution denoted as
, this
is in line with the principle that says the neural networks are universal approximator [
]. The
main objective in PIML is to train the DNN by optimizing the parameters (θ). We convert the problem
into optimization problem and the objective function is defined as mean square error (MSE). The MSE
illustrated here is in its general case where the system contains the combination of Neumann, injection well,
production well, and Dirichlet conditions.
where:
(16)
The terms in Eq. (16) such as MSEb, MSEfu, MSEs0/i represent mean root square for boundary data,
colocation points and sink and source. The first term (MSEb) force Dirichlet and Neumann boundary, the
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We consider a single phase and incompressible fluid system that is characterized by p(x). In some
examples presented in this work, some boundary data were assumed (Dirichlet boundary conditions) while
on other cases boundary data are unavailable (Neumann boundary conditions). We also presented numerical
examples where fractures and fluxes terms are included in the system. Therefore, we have constructed
unique loss functions for each problem. However, the loss function provided in this section is for general
governing equation:
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second term (MSEfu) enforce the PDE over colocation points over the domain that are generated in uniform
or nonuniform fashion. The last term (MSEs0/i) is used to enforce fluxes (source/sink). Where, Nint, Nneu, and
(17)
The process of minimizing the error is carried out with the help of optimization algorithm such as Adam
(Kingma and Ba, 2014).
Numerical results and Discussion
In all examples presented in this research, we employed the fully connected neural network consisting
in 5 hidden layers with 50 neurons per hidden layer. The single and fixed activation functions (i.e.,
hyperbolic tangent, rectified linear unit) are used in all hidden layers for all numerical examples in reservoir
engineering boundary conditions section. On the other hand, the mixed and adaptive activation functions
are employed on numerical examples related to fractures and fluxes. The Xavier initialization method is
adopted to initialize weights (Glorot and Bengio, 2010). The mean square error is optimized with an Adam
optimization algorithm (Kingma and Ba, 2014). For all numerical examples except mentioned, we use
uniformly distributed points on ∂Ω and Ω. The computational domain (Ω) is given by (x, y) ∈ [0, 1] × [0,
1]. The PIML method is implemented in Tensorflow v2.0.
Reservoir Engineering Boundary Conditions (REBC's)
Although the physics informed/inspired machine learning has attracted the interest of many researchers.
However, its effectiveness when subjected to REBC's, have received little attention. Therefore, in this
section, we investigate the capability and effectiveness of PIML method exposed to widely known REBC's
namely, (i) No-flow boundary, (ii) constant pressure boundary, and(iii) mix boundary conditions. The
investigation of other BCs with little application to petroleum industry is beyond the scope of this work.
For each example presented here, either an analytical solution or a conventional numerical algorithm is
available to verify the correctness of the model and quantify the relative error.
Constant Pressure boundary. In reservoir engineering applications, it is common that parts of the reservoir
may be in communication with a larger aquifer system that provides external pressure support. Also,
reservoir with strong gas cap fall in this category. These scenarios tend to maintain reservoir pressure as
constant at boundaries. These types of reservoir boundaries are named constant pressure boundaries and
they can be modeled in terms of Dirichlet boundary condition. The single phase, homogenous system,
incompressible fluid flow equation along with constant pressure at boundaries is given by:
(18)
The neural networks parameters can be learned by minimizing the mean squared error loss:
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Ndir represent the number of points (xi),(
), in the interior domain, on the Dirichlet
boundary and the Neumann boundary. Moreover, adir and aneu represent penalty terms for the Dirichlet and
Neumann boundaries.
Now the DNN is trained with the main goal of minimizing the objective function (MSE). This is achieved
by optimizing the trainable parameters θ using the following equation.
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7
where:
(19)
(20)
Fig. 2b illustrates predicted pressure values
. These predicted pressure values are obtained with 5000
iterations and the boundary scaling factor(α = 10). We consider a data set comprising of 200 boundary
points and inner colocation points of 800. The fixed tangent hyperbolic activation function was adequate for
this example. The exact (true) pressure values are shown in Fig. 2a. These exact pressures p(x) values are
generated with the analytical solution of the PDE (16) derived based on method of separation of variable.
Fig. 2c presents the absolute error which is difference between the exact and the neural network prediction
results. The results indicate smaller values of absolute error (−0.005≤ε≤ 0.04). Although the PIML method
obtained good results highlighted with smaller (ε). Howver, it is worth highlighting some boundary effects
experienced in the left and right-side boundary (see Fig. 2c). We believe that those effects can be eliminated
by finding the most optimum scaling factor. The accuracy of the model is quantified through average relative
error. The resulting relative L2-norm between predicted and exact solution is 2.44×10−2. The computed
relative is very small, this suggest that the PIML method produces accurate predictions of the given PDE
subjected to constant pressure boundary. The PIML predictions are also consistent with lower Laplace values
(∇⋅∇p(x)≈0) presented in Fig. 2d. However, we observe that not always zero Laplace translate to better
predictions, there are cases where zero Laplace led to poor predictions or vice versa.
Figure 2—(a) Illustrate pressure predicted from PIML model for strong aquifer. (b) True solution estimated
from analytical solution. The absolute error and Laplace distribution are shown in (c) and (d), respectively.
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The accuracy of PIML method was assessed through analytical solution written as following:
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Mixed Reservoir Boundary Conditions. Combinations of these conditions are used when studying parts
of a reservoir (e.g., sector models). There are also cases, for instance when describing groundwater systems
or CO2 sequestration in saline aquifers, where (parts of) the boundaries are open, or the system contains a
background flow. The governing equation under these conditions can be reduced in form of:
(21)
(22)
Subjected to the constant pressure conditions (Dirichlet BC's)
(23)
Therefore, the loss function can be written as follows:
MSE = MSEfu + MSED + MSEN
where:
(24)
The accuracy of PIML method was assessed through analytical solution written as following:
(25)
Our goal in this example remains the same as in previous ones but this time we focus on mixed
reservoir boundary conditions. The pressure solutions obtained from PIML tests are presented in Fig. 3b.
These predicted pressure values are obtained with 8000 iterations and the boundary scaling factor (α = 5).
The boundary and inner training datasets arrangement is similar to the previous example. The fixed relu
activation function was adequate for this example. The analytical solutions are shown in Fig. 3a. We also
present the difference between true and predicted pressure distribution as indicated Fig. 3c. The results
indicate that the absolute error is very insignificant (−0.0028≤ε≤0.0014). The computed relative L2 norm
between predicted and actual solution is 4.17×10−2. The Laplace values (see Fig. 3d) are close to zero
(∇⋅∇p(x)≈0). Therefore, it is evident that the PIML method provides an accurate estimate of p(x) with mixed
reservoir boundary conditions.
Reservoirs with No-Flow Boundary Conditions. In reservoir engineering applications, one is often
interested in describing closed flow systems that have no fluid flow across its external boundaries. This is
a natural assumption when studying full reservoirs that have trapped and contained petroleum fluids for
million of years (Knut-Andreas Lie, 2019). Mathematically, no-flow conditions across external boundaries
are modeled by specifying homogeneous Neumann conditions (
). In this section, we
consider pressure equation given as follows:
(26)
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Subjected to the no-flow boundary conditions (Neumann BC's)
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Subjected to the No-flow boundary (Neumann BC's)
(27)
With no-flow boundary conditions, the solution of PDE above is a constant pressure (initial reservoir
pressure distribution). Here, we assume that the initial reservoir pressure is uniformly distributed (p = 2).
Thus, the analytical solution is written as follows:
(28)
The loss function (MSE) for no-flow boundary system has slightly different form compared to other
previous examples. This is because the solution of pure no-flow boundary tends to converge to different
constants for every single experiment. In order to avoid this issue, we have added a new term in loss function
based on our analytical solution. The addition of this term ensures that the neural networks learning process
not only minimize the residual of PDE and boundary points but also it converges to a constant uniform
pressure. Thus, the loss function takes the form:
where:
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Figure 3—(a) Illustrate pressure predicted from PIML model for mixed boundary conditions. (b) Exact solution
estimated from analytical solution. The absolute error and Laplace distribution are shown in (c) and (d), respectively.
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(29)
Figure 4—(a) Illustrate pressure predicted from PIML model. (b) True solution estimated from
analytical solution. The absolute error and Laplace distribution are shown in (c) and (d), respectively.
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The number of hidden layers, activation function, number of nodes and many other neural networks
parameters remain the same from mixed boundary case (αneu = 7). Unlike, in the previous numerical example,
here we deal with pure no-flow boundary case. Fig. 4a shows the exact solution p(x) field generated with
analytical solution p(x)=2,whereas the predicted pressure values are shown in Fig. 4b. It is important to note,
although the visual observation indicate significant color contrast between those two Figures. However,
the numerical values of those two pressure field maps are very close to each other. Likewise, in previous
numerical examples we present 2D plots shown in Fig. 4c which represent absolute error. The results indicate
that the absolute error is very insignificant(−0.00002≤ε≤0.0000216). Another interesting point that worth
noting is close to zero Laplace values (∇⋅∇p(x)≈0) over the computational domain which is indicated in the
Fig. 4d. The computed relative L2 norm between predicted and actual solution is 4.17×10−2. This highlights
the capability of this approach in handling no-flow boundary problems.
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Modelling Fluxes with PIML
In this example, we consider our previous problem with new constraints (source and sink). These are well
known in reservoir engineering applications. The fluid flow equation with sink and source can be written
as follows:
(30)
The source and sink are represented as q+ (injection intensity) and q− (production intensity). The location
of all five wells are shown as follows:
(31)
(32)
(33)
(34)
(35)
Although several studies in the literature (e.i. Raissi et al., 2017a; Raissi et al., 2017b; Raissi et al.,
2019) have generated the training datasets using random method such as Latin hypercube sampling strategy
(Stein,1987). However, these studies have not substantiated their motivations of adopting the random
approach instead of uniform input data generation. To fill this gap, we investigated the impact of these two
methods of training data generation. Training datasets generated using random and uniform method are
shown in Figure 5.a and Figure 5.b, respectively. It turns out that no matter which method is used to generate
the training datasets, their impact on the performance of the PIML method is inconsequential. As illustrated
in figures, we have enforced the training points around the wells. This was achieved by adding more training
data sets in circular shape with different radius (rw = 0.1, r1 = 0.15, r2 = 0.2, r3 = 0.25, r4 = 0.3) and the rw
represents the wellbore radius. Around 100 training datasets were allocated for each radius (r1 = r2 = r3 =
r4 = 100). It noteworthy that the literature lacks a scientific method to obtain optimum radius and number
of training points. Therefore, the trial-and-error method has been used in this study. We have observed that
adding more training datasets around source and sink has no benefit on the performance of PIML method.
In our experiments, a five-spot injection pattern has been used to test the performance of PIML method.
It consists of four production wells located at the corners of square domain {(0.1, 0.1);(0.1, 0.9);(0.9, 0.9);
(0.9, 0.1)} and the injection well located in the center{(0.5, 0.5)}. The highest-pressure values can be found
around the injection well (p = +1) while the lowest pressure values can be found at the production well (p =
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Subjected to Neumann and Dirichlet BC's:
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Figure 5—Uniformely distributed training dasets(a) and Randomly distributed training datsets (b)
The results of five spot pattern of PIML has not matched perfectly with those from FEM. PIML method
experience a wellbore related problem which we defined in this study as "wellbore extension phenomena".
The PIML tends to disturb larger radius around the wellbore compared to FEM (See Fig. 6a and Fig. 6b).
One of the possible reasons for this small discrepancy, could be related with the way how we define the loss
functions. Overall, our findings on application of PIML for the simulation of single fluid flow in porous
media with source and source at least hint the potential of this method. Our future research will investigate
this issue in more details.
The universal approximation theorem states that neural networks have properties of universal
approximator (Hornik et al., 1989; Cybenko, 1989), which means they can approximate any continuous
function. However, the neural networks failed to provide acceptable solution for no flow boundary
conditions. This could be because the PIML requires some data on boundary to avoid sink and source
interference. We also believe that if the problem is unsteady the no flow boundary condition has no inference
on the performance of PIML because there will be one more constraint such as initial conditions. This is not
the first attempt that shows that the neural networks fail to provide solution when applied to solve petroleum
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−1). Unlike previous section, in this example wells are present in the domain of interest (qsc≠0). The PIML
model consisting of 5 hidden layers with 50 neurons in each layer, mixed activation functions with Tanh in
the first two hidden layers and Relu for the rest of hidden layers was employed to compute results illustrated
in Fig. 6b. The training points are uniformly distributed on the boundary (Nb = 800) and computational
domain(Nint = 1,000). Here we also compared the results from PIML with those from FEM method shown in
Fig. 6a. In order to numerically compare the results, two lines (x = 0.51 and y = 0.3) were drawn from both
2D plots and the data was extracted. The extracted data was then used to create plots shown in Fig. 6c and
Fig. 6d. Numerical experiments indicated that fixed activation function was inadequate for more problems
containing complex fractures and well patterns. Our observation is in line with previous research (Jagtap
et al.,2019). Adaptive functions provide better results compared to fixed activation functions because they
accelerate the convergence.
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related problem. In fact, previous study (Fuks and Tchelepi, 2020) also pointed out similar issue when they
used PIML to solve fluid flow equation.
Modelling Fractures with PIML
The capability of PIML in handling fractures is assessed in this section. The governing equation remaining
the same from previous section except for fractures cases line sources are allocated in the system. We started
with parallel fractures with no-flow boundary conditions. A fully connected neural network with 5 hidden
layers of 50 neurons each has been employed for both parallel and complex fractures cases. The training
datasets are distributed as boundary points (Nb = 800), fractures points (Nf = 1000), and collocation points
(Nint = 3000).
Four parallel fractures are present in a system with no flow boundary conditions, they are located at
x=0.05,x = 0.3,x = 0.70,x = 0.95 as illustrated in the Fig. 7a. The pressure distribution in fracture tends to
be different from pressure distribution matrix, for this reason we denoted fractures with negative unit (p =
−1.0,x = 0.30,x = 0.95) and positive unit (p = +1,x = 0.05,x = 0.70). The computed pressure distribution
with PIML is shown in Fig. 7b. We compare the prediction results with those from FEM (see Fig. 7a). To
show the results in the same graph, 1D plots are presented based on pressure extracted from selected points.
Since the fractures are all parallel to each other, taking lines perpendicular to fracture seems to be more
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Figure 6—The results of five spot pattern from FEM (a) and PIML (b).
The values of pressure distribution for y = 0.3 (c) and x = 0.51 (d)
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representative. Therefore, different lines are drawn at y = 0.5 and y = 0.70 as shown in Fig. 7c and Fig. 7d,
respectively. We note that for this problem the PIML can predict pressure with high accuracy.
Next, we study the performance of PIML when subjected to more complex fractures. The system contains
four (4) fractures (see Fig. 8a). This numerical example is designed to show the performance of the PIML
method in handling more complex fracture system. The predicted pressure distribution pattern is found to be
similar to those from FEM as illustrated in Fig. 8a and Fig. 8b. We also note that while in previous example
(parallel fractures) the PIML accurately predicted pressured distribution with fixed activation function and
no flow boundary conditions. However, for complex fracture case, we considered the tanh for the first
hidden layer and relu for next 4 hidden layers, adaptive activation function (n = 20,a = 0.1). The results
highlight that increase number of fractures improved the solution because more constraints are present in
the system as illustrated in Fig. 8.c and Fig. 8. d. Overall, the results obtained illustrate that the PIML can
approximate the solution to the PDE with Dirichlet boundary conditions.
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Figure 7—The results of parallel fractues from FEM (a) and PIML
(b). The values of pressure distribution for y = 0.7 (c) and y = 0.5 (d).
SPE-203917-MS
15
Conclusions
In this work, we investigated the capability of the physics inspired machine learning (PIML) method for
the simulation of single-phase, incompressible fluid, steady state fluid flow in porous media. The main
idea of physics inspired machine learning method is to encode the underlying physical law (governing
equations) into the deep neural network as prior information. The performance of the PIML method in
handling reservoir boundary conditions, fluxes term and fractures. The following conclusions are drawn:
1. The capability of the PIML method in handling reservoir engineering boundary conditions including
no-flow, constant pressure, and mixed reservoir boundary conditions is investigated. When comparing
PIML results to those of analytical solution, it must be point out that the PIML performs well, giving
good results comparable to analytical solution.
2. The capability of the PIML method in handling fluxes (sink and source terms) is examined. Our results
demonstrate that the PIML fail to provide acceptable prediction for no-flow boundary conditions.
However, it provides acceptable predictions for constant pressure boundary conditions.
3. We also assessed the capability of the PIML method in handling fractures. The results indicate that the
PIML can provide accurate predictions for parallel fractures subjected to no-flow boundary. However,
in complex fractures scenario its accuracy is limited to constant pressure boundary conditions. We also
found that mixed and adaptive activation functions improve the performance of PIML for modelling
complex fractures and fluxes.
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Figure 8—The results of parallel fractues from FEM (a) and PIML
(b). The values of pressure distribution for x = 0.7 (c) and y = 0.6 (d).
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