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Estimating permeability of 3D micro-CT images by
physics-informed CNNs based on DNS
Stephan Gärttner1,* , Faruk O. Alpak2, Andreas Meier1,
Nadja Ray1, Florian Frank1,*
1Friedrich-Alexander-Universität Erlangen-Nürnberg,
Department Mathematik, Cauerstraße 11, 91058 Erlangen, Germany
2Shell Technology Center, 3333 Highway 6 South, Houston, TX 77082, USA
September 7, 2021
Keywords: digital rock, neural networks, deep learning, permeability, porous media.
MSC classification: 05C21, 68T07, 76D07, 76M10, 76S05.
Abstract
In recent years, convolutional neural networks (CNNs) have experienced an increasing
interest for their ability to perform fast approximation of effective hydrodynamic parameters
in porous media research and applications. This paper presents a novel methodology for
permeability prediction from micro-CT scans of geological rock samples. The training data
set for CNNs dedicated to permeability prediction consists of permeability labels that are
typically generated by classical lattice Boltzmann methods (LBM) that simulate the flow
through the pore space of the segmented image data. We instead perform direct numerical
simulation (DNS) by solving the stationary Stokes equation in an efficient and distributed-
parallel manner. As such, we circumvent the convergence issues of LBM that frequently are
observed on complex pore geometries, and therefore, improve on the generality and accuracy
of our training data set.
Using the DNS-computed permeabilities, a physics-informed CNN (PhyCNN) is trained
by additionally providing a tailored characteristic quantity of the pore space. More precisely,
by exploiting the connection to flow problems on a graph representation of the pore space,
additional information about confined structures is provided to the network in terms of the
maximum flow value, which is the key innovative component of our workflow. As a result,
unprecedented prediction accuracy and robustness are observed for a variety of sandstone
samples from archetypal rock formations.
1 Introduction
Artificial neural networks can accelerate or replace classical methods for estimating various hydro-
logical and petrophysical properties of artificial and natural rock [1]. In [2], deep neural networks
were successfully used to approximate tomography operators to reconstruct wave velocity models
from seismic data. Likewise, convolutional neural networks (CNNs) have proven their ability to
provide fast predictions of scalar permeability values directly from images of the pore space of
geological specimens [3,4]. This paper aims at contributing to the last-mentioned application of
machine learning techniques by presenting a finite-element-based forward simulation approach
as well as introducing a novelly considered characteristic quantity for physics-informed neural
network (PhyCNN) models.
*Corresponding authors. E-mail address: gaerttner@math.fau.de,frank@math.fau.de.
arXiv:2109.01818v1 [cs.LG] 4 Sep 2021
Specimens of natural rock are typically obtained by microcomputed tomography (µCT) scan-
ning [5,6] or indirectly from 2D colored images [7]. As an important characteristic quantity of
porous media, permeability measures the resistance of a fluid to travel through a considered pore
space. However, in general, the precision and generality of the estimates performed by neural
networks heavily depend on the quality of the underlying training data set as shown in [8]. In
terms of permeability prediction, the labeling process of geological specimens is connected to the
computation of 3D stationary flow fields of a single-phase fluid within the pore space. For per-
meability prediction under regular geometries, a broader set of methods is available as described
and compared in [9].
In the literature, various methods are available to compute the stationary flow on complex
geometries such as pore spaces of porous media, a thorough comparison of which is found in [10].
Most commonly, lattice Boltzmann methods (LBM) are used to tackle the flow problem on
a discrete modeling basis, cf. [11]. In this approach, a transient interacting many-particle sys-
tem is driven to equilibrium state, heavily exploiting the inherent parallelism of the underlying
mathematical structure. For a detailed description of LBM fundamentals and numerics, we refer
to [12]. Yet, since a global equilibrium has to be reached, complex geometries containing thin
channels may cause LBM to converge extremely slowly or even diverge [4,13]. This behavior is
frequently observed with common pore-scale flow LBM implementations that rely on the conven-
tional single-relaxation-time Bhatnagar–Gross–Krook (BGK) scheme [14]. However, the novel
multiple-relaxation-time (MRT) scheme addresses this problem to a great extent (e. g., [15])
at the cost of a moderate additional computational overhead. We further note that classical
bounce-back rules used for implementing boundary conditions tend to develop boundary layers
that might locally dominate fluid behavior in scenarios with thin channels, cf. [16]. Severe lim-
itations arising from simplistic implementations of LBM restrict the training data sets to ones
derived from sphere-packs instead of real CT data in many publications, see for instance [13]. On
the other hand, industrial-grade state-of-the-art LBM solvers are typically proprietary without
public access to the code base for academic research purposes (e. g., [15,17]). Moreover, a com-
prehensive permeability computation benchmarking study [18] demonstrated that not all of the
simulation methods deliver a good compromise of accuracy against computational performance
indicating that there exists a clear need for accurate and computationally efficient alternative
methods for permeability computation.
In practice, LBM simulations are often aborted after a maximal number of iterations in
case one or more convergence criteria (typically linked to the relative changes in the velocity
and/or computed permeability) cannot be met [4]. As such, training sets for neural networks
can be artificially filtered by the numerical characteristics of the forward simulation, potentially
resulting in biased data. To improve the generality and quality of our PhyCNN training sets,
we base our machine learning data set on the stationary Stokes equation by performing direct
numerical simulations (DNS) on the pore geometry [19]. More precisely, our forward simulation
is based on a distributed-parallel Stokes solver utilizing the finite element library MFEM [20]. As
studied in [9] for simple cylindrical obstacles, such DNS approaches (in this case FEM) deliver
results comparable to LBM. However, our implementation successfully alleviates the drawback of
an impractically large number of iterations to obtain the desired accuracy on complex geometries.
As such, our approach allows overcoming prior restrictions in setting up representative training
sets including also confined and complex structures. Moreover, no artificial data augmentation
schemes such as pore space dilation are needed in our approach to increase the number of training
data samples or enhance the porosity range covered.
Likewise, novel strategies are employed to our neural networks. More precisely, we exploit
the concept of PhyCNNs, where the CNN is provided with additional (physics-related) input
quantities to improve the reliability and the accuracy of its predictions [3,21]. As shown in [22],
carefully chosen specific quantities derived from the pore space such as connectivity indices
can deliver reasonable approximation quality for permeability estimation. As illustrated in [2],
2
Data acquisition
•Extract 1003-voxel
subsamples.
•Remove
disconnected pore
space.
•Eliminate
impermeable
subsamples.
Data labeling
Perform forward
simulation to deter-
mine permeability
labels kcmp .
Network training
•Compute physics
input.
•Setup training and
validation data set.
•Optimize network
parameters via
SGD.
Network vali-
dation against
•unseen data
samples of same
sandstone type,
•data samples of
different sandstone
type,
•artificially distorted
data.
Figure 1: Overall workflow in flow chart representation.
also training a network solely on previously extracted features from the raw data may lead to
satisfactory prediction quality.
The outstanding performance of our methodology is achieved by considering the maximum
flow value, a graph-network derived quantity being highly correlated to the target permeability
value and simple to compute. As such, we solve maximum flow problems on a graph repre-
sentation of the pore space based on [23]. Thus, we approximate the Stokes flow through the
pore space by an abstract flow through a graph. By using the scalar quantity of maximum flow
as a second input to our neural network, we additionally provide our CNN with information
that reflects possible thin channel-like structures. As discussed in [24], graph representations
have been shown highly capable of characterizing the pore-space connectivity in fractured rock
and allow for a convenient way to deduce topological quantities of interest. We demonstrate
that involving the maximum flow value in our newly designed PhyCNN in combination with the
DNS-based forward simulation approach, our methodology delivers superior prediction accuracy
and robustness compared to what is found in the literature.
The paper is organized as follows: In Section 2, we describe the sampling and preprocessing
procedure of sandstone specimens including the forward simulation. Subsequently, Section 3
is dedicated to the network architecture used in our study. Finally, we validate the training
performance of our PhyCNN on different types of sandstone in Section 4.
2 Methodology and data preparation
In this section, we describe the workflow and methodology by which we acquire the data set
necessary to train and validate a CNN using a supervised learning approach. The preprocessing
includes the selection and preparation of a set of pore-space geometries in form of voxel sets and
the labeling with their computed permeability value kcmp. Our complete workflow is presented
in Figure 1. We note that all steps except for the data labeling procedure (green box) are
implemented in Matlab 2021a [25].
2.1 Sampling and preprocessing
The training procedure of our PhyCNN is based on a segmented X-ray µCT scan of a Bentheimer
sandstone sample, see Figure 2, with experimentally measured porosity φexp = 22.64% and
permeability kexp = 386 mD, provided by [26,27]. Further characteristic quantities for this
sandstone as well as two other sandstone types used below for validation purposes are listed
in Table 1. Bentheimer sandstone is known to exhibit a broad range in pore volume distribution
and high pore connectivity in comparison to other types of sandstone [28]. As such, this sample
is expected to contain a representative collection of geometrical and topological properties of
the pore space in natural rocks. The data set used in this paper is a 1000×1000×1000 binary
voxel image, in which each voxel either belongs to the pore space (“fluid voxels”) or the rock
3
matrix (“solid voxels”). The voxel edge length is 2.25 µmyielding an overall cube side length of
2.25 mm.
Figure 2: 100022D slices and 100033D pore-space µCT image of all samples considered in this study, illustrating charac-
teristic pore features for Bentheimer, Berea, and Castlegate sandstone.
type φexp φcmp kexp [mD] MCD [µm]Acmp [mm]
Bentheimer 22.64% 26.72% 386 30.0 355
Berea 18.96% 21.67% 121 22.3 284
Castlegate 26.54% 24.67% 269 24.7 335
Table 1: Characteristics quantities of sandstone samples used in this paper: experimentally determined porosity, φexp
(±0.5%), porosity computed from the µCT scan, φcmp, experimentally determined permeability, kexp (±10%), mean capil-
lary diameter, MCD, and interior surface area, Acmp, computed via the Matlab function isosurface. First four quantities
are provided in [27].
In the first step, we extract subsamples of 100×100×100 voxels from the Bentheimer sample.
Henceforth, we use the term ‘subsample’ to refer to segmented µCT-scan pieces of this specific
size. For subsample extraction, we make use of the sliding frame technique. This approach is
commonly used to further exploit a given data set beyond the partition into disjoint subsets [29].
More precisely, we sweep a 100×100×100 voxel frame along the coordinate axes of the original
1000-voxel cube and displace it by steps of 50 voxels (half a subsample size) resulting in a total of
193= 6859 subsamples. Although data are sampled redundantly, the set of obtained subsamples
can be regarded independently when training neural networks [4]. Furthermore, by rotating
the original 100×100×100 voxel subsamples by 90◦around the yand zaxis, the number of
extractable data is increased further by a factor of three. As such, artificial data augmentation
techniques like shrinkage or inflation of the pore space from the µCT image are not required
here for obtaining a sufficient and representative amount of training data. Even though we can
produce 3·193=20 577 subsamples by the method above, we select only the first 10 000 ones,
since this number is sufficient to train our PhyCNN. In particular, this number of available data
samples exceeds that of similar studies, cf. [4,13].
4
Second, fluid voxels belonging to disconnected pore space with respect to the xdirection
possibly occurring within the subsamples are turned into solid voxels. As disconnected pores do
not contributed to Stokes flow being driven from the inhomogeneous boundary conditions (1c)
placed on opposing sides of the subsample, this procedure maintains permeability properties
while facilitating their calculation. To this end, a simple graph walking algorithm is exploited to
identify connected subdomains of the pore space. Starting from a random fluid node, neighboring
nodes within the fluid domain are successively added until the scheme converges. A thorough
description of this algorithm is found in [30].
By projecting the voxels of the connected pore space onto the xaxis, we conclude whether
each subsample has a nontrivial permeability. Subsamples with zero permeability are excluded
from the later workflow (the maximum flow value is exactly zero if and only if the permeability is
zero—therefore no training on such data samples is necessary, cf. Section 3.2.2). By encoding the
simplified pore geometries in 1-bit raw format, memory consumption is 125 kB per subsample.
Accordingly, our whole library of data samples allocates 1.25 GB of disk space and is therefore
still manageable on standard personal computers.
We note that a subsample of size 225 µmis too small to be a representative elementary
volume (REV) for most sandstone types, cf. [27,28]. As such, computed effective properties
of one subsample cannot be expected to represent the whole segmented porous medium out of
which it was extracted. On the other hand, the obtained training data set for our PhyCNN is
therefore expected to be highly diverse, i. e. to contain highly permeable as well as narrow and
confined pore geometries. Consequently, this setup is well suited to underline the robustness and
prediction quality of our proposed methodology. In the conclusions Section 5, we will point out
how the results presented in this paper could be used to approach REV-scale samples.
2.2 Forward simulation
To apply a supervised learning approach as outlined in Section 3, each subsample within the
data set derived by the methods of Section 2.1 needs to be labeled with a computed permeability
kcmp that we use as the reference value.
To this end, in Section 2.2.1, we perform flow simulations on the pore space. More precisely,
for each of the 10 000 subsamples, a stationary flow field along the xdirection is computed by
solving the Stokes equations on the union of fluid voxels Ωfor the fluid velocity uand pressure p.
The discretization uses arbitrary order (stable) Taylor–Hood or reduced-order stabilized Taylor–
Hood mixed finite elements.
In Section 2.2.2, the (absolute, scalar) permeability is computed by averaging the pressure
gradient and velocity field across the subsample in xdirection, cf. Section 2.1. Apparently, as the
data set contains yand z-rotated versions of each subsample, this relates to the determination
of the permeability with respect to all three main axes. By accurate bookkeeping, the diagonal
permeability tensor of each initial subsample is retrievable.
Finally, in Section 2.2.3, we justify our choice of discretization parameters, i.e., mesh refine-
ment level and finite element spaces used to produce the permeability values kcmp to train and
validate our PhyCNN.
2.2.1 Computation of the flow field
We consider the stationary Stokes equation for a Newtonian fluid in the nondimensionalized
form,
−1
Re∆u+∇p=0in Ω, (1a)
∇·u= 0 in Ω, (1b)
where Ω⊂(0,1)3is a domain that consists of the union of fluid voxels of a considered 100×100×100
voxel subsample (i.e. the subsample is inscribed into the unit cube). In (1), u=u(x, y, z)denotes
5
Subsample 0 Subsample 9213
pressure
ph∈[−1.5E0,1.4E0] ph∈[−1.5E0,1.8E0]
velocity magnitude
|uh| ∈ [0,7.5E−3] |uh| ∈ [0,1.9E−3]
Figure 3: Examples of geometries used for the training process, with pressure fields (top) and velocity magnitudes (bottom).
Subsample 0 (left) exhibits moderate pressure gradients due to the wide and highly conductive channels in the pore space.
Contrarily, in Subsample 9213 (right) Stokes pressure is dominated by thin structures leading to an ill-conditioned problem,
i. e., small changes in the diameter of narrow pores have drastic impact on the permeability. Thin pore throats are indicated
by black arrows in the pressure plot.
the (dimensionless) fluid velocity, p=p(x, y, z)the (dimensionless) pressure and Re :=ρ UcLc/µ
the Reynolds number of the system with characteristic length Lc[m], characteristic veloc-
ity Uc[m s−1], fluid density ρ[kg m−3], and fluid viscosity µ[Pa s]. Note that the perme-
ability kcmp [m2]is invariant with respect to Re (see below). The data set is constructed in
such a way that there are connected fluid voxels (i.e. sharing a common face) reaching from
the x=0 plane to the x=1 plane, cf. Figure 3, since impermeable subsamples were excluded from
the workflow in the ‘data acquisition’ step, cf. Figure 1and Section 2.1. The flow field is driven
by a pressure gradient in xdirection induced by the boundary condition
1
Re∇u−pIn=exon ΓN,(1c)
6
where ΓN:=(x, y, z)∈∂Ω |x∈ {0,1}and exdenoting the unit vector in xdirection. On the
remaining boundary ΓD:=∂Ω \ΓN, no-slip boundary conditions are prescribed,
u=0on ΓD.(1d)
The weak formulation of (1) is discretized by generalized Taylor–Hood pairs of spaces,
Q3
`+1/Q`, where Q`denotes the local space of polynomials of degree at most `in each vari-
able x, y, z, cf. [31,32]. For `= 0, the pressure space Q0=P0is discontinuous and consists
of elementwise constants. The respective “reduced Taylor–Hood pair” Q3
1/P0requires stabiliza-
tion (see below) due to a lack of discrete inf-sup-stability [31]. Let φi=φi(x, y , z) : Ω→R3,
i= 1, . . . , n and ψi=ψi(x, y, z) : Ω→R,i= 1, . . . , m denote the basis functions for the global
discrete spaces for velocity and pressure, respectively. The discrete velocity uh=uh(x, y, z)and
discrete pressure ph=ph(x, y, z)then have the representation
uh=
n
X
i=1
[xu]iφi, ph=
m
X
i=1
[xp]iψi(2)
with degree-of-freedom vectors xu∈Rn,xp∈Rm, which are unique solutions of the linear
system
A BT
B−Cxu
xp=bu
0⇐⇒:Ax=b(3)
with right-hand side bu∈Rn,[bu]i:=RΓNex·φi. The sparse blocks in Aare the vector-Laplacian
matrix A∈Rn,n and the divergence matrix B∈Rm,n,
[A]i,j :=ZΩ
∇φi:∇φj,[B]k,j :=−ZΩ
ψk∇·φj,
and Cis a stabilization matrix that is required only for lowest order `= 0 to guarantee the full
rank of A. We choose Cas in [31], (3.84), in which case, Ccan be interpreted as a pressure-
Laplacian discretized by cell-centered finite volumes [33]. For ` > 0,Cis set to zero. In either
case, Ais symmetric and indefinite with npositive and mnegative eigenvalues.
In order to solve the saddle point system (3) efficiently, a preconditioned MINRES method is
applied, as it is the best choice of Krylov subspace methods for symmetric indefinite systems [34].
The chosen precondition operator for Ain (3) is the symmetric and positively definite block-
diagonal matrix
P:= diag (A,W)
with Wbeing the pressure-mass matrix [Wk,l ]:=RΩψkψl. The action of the inverse P−1in each
Krylov iteration is approximated block-wise by one V-cycle of an algebraic multigrid method
(Boomer AMG from the Hypre library [35]). Since Wis spectrally equivalent to the (negative)
Schur complement BA−1BTof A(for ` > 0), and due to the utilization of AMG, the number
of Krylov iterations required to reach a given relative tolerance is bounded independently of the
mesh size [36–38] (however, it highly depends on the geometry of the domain). MINRES with
preconditioning as described above belongs to the state-of-the art Stokes solvers in the high-
performance computing context [39]. In Section 2.2.3, we discuss appropriate choices regarding
mesh size and discretization order.
Figure 3illustrates Stokes velocity and pressure fields for two exemplary subsamples exhibit-
ing qualitatively highly different pore-space geometries such as wide pore throats and narrow
channels. We will emphasize the impact of highly and merely permeable rock samples on the
behavior of DNS-based permeability computations and PhyCNN predictions throughout the pa-
per.
7
2.2.2 Permeability estimation
We deduce the permeability value kcmp of interest from the previously calculated Stokes velocity u
and pressure p(we suppress the discretization index hin this section). In the Stokes equations (1),
the inflow and outflow boundaries are defined as
Γin :=(x, y, z)∈∂Ω |x= 0, Γout :=(x, y, z)∈∂Ω |x= 1
and thus disjoint subsets of ΓN. From the solution (u, p)of (1), we compute the approximated
permeability kcmp [m2]by the classical Darcy law, which reads in nondimensionalized form,
Q=−Da Re (Pout −Pin),(4a)
with (dimensionless) volume flow rate Q, and (dimensionless) area-averaged inflow and outflow
pressures, Pin,Pout, given by [40]
Q:=Z
Γout
u·n, Pin :=1
|Γin|Z
Γin
p , Pout :=1
|Γout|Z
Γout
p .
From (4a), kcmp is derived from the Darcy number
Da :=kcmp
L2
c
,(4b)
where Lcis the characteristic length, in our case, the edge length of a 1003-voxel subsample,
i. e., Lc= 225 µm. We choose Re equal to one, since Re does not influence the permeability
value kcmp (a rescaling of uby Re−1implies a rescaling of Qby Re−1and therefore, Re cancels
out in (4a)).
Formula (4a) determines an approximation of the scalar permeability in xdirection by area
averaging. If this approach was applied to all three principal directions, it yielded a diagonal
permeability tensor. In [41], a volume-averaged approach is proposed that is capable of deter-
mining the full permeability tensor. Application to xdirection only, yields a column vector,
whose entries are in general non-trivial. Since we want to train our PhyCNN with one scalar
permeability value only, we decided to utilize the area-averaging approach in this study.
2.2.3 Choice of discretization parameters
In this section, we investigate the influence of mesh refinement and choice of polynomial order
on the solution quality by comparing the computed permeability kcmp obtained from (4a) to the
analytical value kana. In order to have analytical results available, we restrict our considerations
to viscous flow through rectangular channels Ωa,b. This is supposed to constitute a sufficient
benchmark, since the approximation quality of the computed permeability kcmp is dominated by
the discretization error in narrow pore throats due to high local gradients.
For the Stokes equations (1), consider a rectangular channel
Ωa,b := (0,1) ×1
2−a
2,1
2+a
2×1
2−b
2,1
2+b
2⊂R3
of width a∈0,1
2and height b∈0,1
2. An analytical expression for its permeability kana
is [42]
kana :=K∞·min(a, b)3·max(a, b)
12 ,where
Kj:= 1 −
j
X
n=1
1
(2n−1)5·192
π5·min(a, b)
max(a, b)tanh (2n−1)π
2
max(a, b)
min(a, b),
8
denoting the limit of Kjfor j→ ∞ by K∞. By application of the triangular inequality,
0<min(a, b)≤max(a, b), and |tanh(x)|<1∀x∈R, we obtain the following approximation
error bound:
|K∞−Kj| ≤
∞
X
n=j+1
1
(2n−1)5·192
π5.
Piecewise application of Jensen’s inequality to the convex function (2x−1)−5finally yields the
estimate:
|K∞−Kj| ≤ 192
π5
∞
Z
j+0.5
1
(2x−1)5dx=192
128 ·1
π5j4≈0.0049 j−4.
As such, we obtain at least six significant digits using K10 for the calculation of the analytical
reference permeability kana.
order `ref. level kcmp
|kcmp −kana|
kana
m(DOF uh)n(DOF ph)
0 0 8.44E−8 8.82E−2 8 484 1 800
0 1 9.06E−8 2.22E−2 54 873 14 400
0 2 9.22E−8 4.62E−3 390 975 115 200
1 0 9.25E−8 1.20E−3 54 873 2 828
1 1 9.26E−8 4.06E−4 390 975 18 291
Table 2: Computed permeabilities kcmp of a channel with 6×3voxel rectangular cross section for different polynomial
orders `of the pressure space (cf. Section 2.2.1) and mesh refinement levels. As in (2), mand nare the numbers of DOF
for velocity uhand pressure ph, respectively. The relative tolerance of MINRES is set to 1.0E−6, which has shown to be
sufficient for three significant digits in the permeability value using a small scouting test set.
Table 2lists the relative error for the computed permeability kcmp obtained from (4a)
on Ω0.06,0.03 using different polynomial orders `and global mesh refinement levels. As expected,
both refinement and higher-order discrete spaces consistently reduced the error with respect to
the analytical solution. The best cost-to-approximation-quality ratio is achieved by Q3
2/Q1el-
ements (`= 1) on the original grid. In particular, this choice poses a significant improvement
in accuracy over the use of lowest-order spaces Q3
1/Q0(`= 0) with two global mesh refinement
levels.
The advantage of higher order discretizations presented above for narrow rectangular chan-
nels seems to generalize to our actual geological samples, which include confined structures. For
Subsample 0, cf. Figure 3, where the main flow channels are wide with respect to the voxel
resolution, kcmp is hardly affected by increasing the polynomial order pfrom zero to one (4%
deviation). On the other hand, Subsample 9213 (Figure 3) exhibiting narrow and branchy struc-
tures experienced a significant relative increase in computed permeability of 43%. Therefore, we
approve the increased computational complexity of lowest order classical Taylor–Hood elements
(`= 1) for permeability labeling in our forward simulations, cf. Section 2.2.1.
This completes the methodology description for data acquisition and yields a training set
suitable to train our neural network for permeability prediction. We conclusively note that the
solver converged for every subsample in the database particularly maintaining the generality of
our training set.
2.3 Evaluation metrics
In our study, we use the following well-known metrics to characterize and compare different
measures of a data set statistically.
9
First, we define the standard deviation σby
σ=v
u
u
t
1
N−1
N
X
i=1
(ti−¯
t)2
for a number Nof real-valued data tiand arithmetic mean value ¯
t, measuring the expected
deviation from ¯
t. In order to quantify the correlation between different measures of a data set,
we further introduce the coefficient of determination R2
R2= 1 −
N
P
i=1
(ti−yi)2
N
P
i=1
(ti−¯
t)2
for targets tiand corresponding predictions yi, encoding the share of variance in the targets that
is covered by the predictions.
Finally, we define the mean-squared error MSE by
MSE =
N
X
i=1
(ti−yi)2
N.
3 Machine learning model
In this section, we motivate and present our suggested neural network architecture for perme-
ability prediction. Therefore, we provide a brief introduction to the class of convolutional neural
networks in Section 3.1 as the basis for our specific network architecture. For further reading, we
refer to [43]. In Section 3.2, this model is augmented by an additional physical input quantity,
which is derived in Section 3.2.2 in detail. For simplicity, we refer to the 1003-voxel geologi-
cal subsamples derived in Section 2.1 as (data) samples in the context of supervised machine
learning.
3.1 Convolutional neural networks
In the following, we discuss the building blocks required to set up an artificial neural network
tailored to the task of permeability predictions from 3D binary images of pore-space geometries.
In a broader sense, we start with the more general concept of feed-forward neural networks.
As indicated by the name, feed-forward neural networks (FFNs) process their input data
by successively propagating the input information through a finite number of layers. Each of
these layers i∈ {0, . . . , L}consists of Nineurons, where the indices 0and Lcorrespond to the
network’s input and output layer, respectively. Typically, the action of a layer 1≤i≤Lon
input data xi−1∈RNi−1is given as an affine-linear transformation, followed by application of
a nonlinearity (activation function) σi:
xi=σi(Wixi−1+bi).
We refer to Wi∈RNi,Ni−1as the weight matrices and to bi∈RNias the biases. Their combined
entries are the learnable parameters of the network.
Convolutional neural networks (CNNs) constitute a subclass of FFNs exhibiting a dedicated
internal structure specialized to the interpretation of image data. Furthermore, CNNs naturally
incorporate translational invariance with respect to the absolute spatial position of features
within the input data. In the following, we give a short introduction to this kind of networks.
For a broader overview of common network designs and layer types, we refer to the respective
literature, e. g. [43]. A list of the different layer types used in our study is given in Table 3.
10
Typically, CNNs consist of several convolutional (conv) lower layers (filter layers) including
pooling operations, followed by dense neuron layers. As indicated by the name, convolutional
layers perform the discrete analog of a convolution, where the convolution kernel is a learnable
parameter of the network. As such, they possess a highly structured weight matrix Wand are
therefore superior to fully connected layers in terms of computational complexity, especially for
highly localized kernels (sparsity). This restriction is justified in our application scenario, since
most (low-level) geometry features are based on highly localized correlations among neighboring
voxels.
MaxPooling (MP) layers are used to reduce the resolution of the output image. Presenting
a natural bottleneck, the most significant pieces of information are selected and passed to the
subsequent neuron layers. Application of several different filters to the same input within each
layer of the network extracts increasingly complex characteristic features of the pore geometry.
Finally, this information is interpreted by the subsequent dense neural layers.
Convergence of the training procedure is improved by exploiting additional batch normaliza-
tion (BN) layers, calibrating the mean and spread of the provided inputs. Throughout the network,
we apply LeakyReLU nonlinearities in an elementwise manner. For a parameter α∈(0,1), the
LeakyReLU(α) function is defined by
LeakyReLU(α;x):= max(0, x)−αmax(0,−x).
Due to the slope of this nonlinearity being bounded away from zero, we circumvent the dying
neuron problem occurring in deep neural networks with ReLU nonlinearities [44], which arise
formally by setting α= 0. Furthermore, backpropagation during the learning phase is facilitated.
Our implementation is conducted using the Deep Learning Toolbox in Matlab R2021a [25].
3.2 Physics-informed convolutional neural networks
In the following, we use the building blocks for general CNNs as introduced in Section 3.1 and
derive a network architecture specifically tailored to our application in permeability prediction.
Therefore, we first give an introduction to PhyCNNs and thereafter discuss the choice of mean-
ingful additional input quantities to the network.
3.2.1 General architecture
As discussed above, standard CNNs are required to extract all important features and information
solely from the input image data to predict the correct output. Due to the complex interplay
of weights and biases within the layers of a CNN, it is not possible to input a priori knowledge
of important features that one would wish the network to take into account. Nevertheless,
it is possible to provide the network with additional input derived from the image data in
a preprocessing step, leading to the field of PhyCNNs. An exemplary schematic representation of
such a network structure is provided in Figure 5. As the additional physical input is typically not
of image type (and hence does not exhibit spatially correlated features), it omits the convolutional
layers and is directly coupled to the upper dense layers. As such, PhyCNNs predict by considering
both the extracted geometry features as well as the provided physical input quantities. Depending
on the quality and quantity of the extracted features as well as the specific application, these
may even provide sufficient information to the network to accurately perform predictions, cf. [2],
underlining the potential of physics-informed strategies.
This approach has been successfully applied to the prediction of permeabilities from pore-
space geometries: In [3], porosity and surface area have been used to increase predictions per-
formance for 2D image data. The specific choice of these parameters is related to the Kozeny–
Carman formula estimating permeability from porosity and surface area. As indicated in [45],
the proposed relation deteriorates in quality for increasingly complex geometries. However, [21]
found porosity to be the most prominent input factor for their CNN among a set of various
11
derived quantities such as coordinate number or mean pore-size. Furthermore, characteristics
obtained by pore network modeling have shown low general accuracy [21].
As indicated by our data set statistics in Figure 4, porosity and surface area are only weakly
correlated with the computed permeability kcmp for our complex 3D setting. For a given sur-
face area Acmp, we observe data samples covering four orders of magnitude in their respective
permeability values, about half the range is observed for fixed porosities φcmp. Hence, drastic
increases in accuracy cannot be expected by regarding these quantities as additional physical
input in our case. Instead, we construct another more specific physical quantity that can be
computed efficiently from the input data using graph algorithms included in Matlab [25]. More
precisely, we solve a surrogate maximum flow problem on a graph representation of the pore
space. In Section 3.2.2, we give a thorough description of this chosen quantity.
0 0.1 0.2 0.3 0.4
10-4
10-2
100
102
0 0.2 0.4 0.6 0.8
10-4
10-2
100
102
0 20406080
10-4
10-2
100
102
01234
10-4
10-2
100
102
Figure 4: Correlation of different physical measures of the underlying geometry with the computed permeability values
kcmp for Bentheimer data samples. From left to right: Computed surface area Acmp , computed porosity φcmp , specific
surface area Aspec, maximum flow on graph representation fmax.
Our final network architecture is presented in Table 3and Figure 5. The structure is based on
the findings of [4], where the hyper-parameters of a purely convolutional neural network have been
optimized using a grid search algorithm. For our study, we decrease the bottleneck constituted
by the final convolutional layer and improve convergence by adding batch normalization layers
and relaxed cut-offs as nonlinearities (LeakyReLU). Furthermore, the maximum flow value fmax
is treated as an additional scalar physical input quantity which is duplicated to a vector of
length 64 and subsequently concatenated with the first dense layer of the network. By doing
so, we balance both inputs to the second dense layer which results in a more uniform weight
distribution.
convolution section dense section
100
100
100
5
520
20
20
4
4
4
64 32
1
2
2
2
5
5
2
2
1
64
image input physics input
Figure 5: Schematic representation of physics-informed convolutional neural networks (PhyCNNs). The scalar valued
physical input circumvents the image convolution layers and is directly fed into the subsequent dense layers. Graphics
produced using NN-SVG [46].
12
block layers learnables
input1 image input 100×100×100 —
input2 physics input 1×1×1 —
conv1 conv(32,5) – BN –LeakyReLU(0.1) – MP(5,5) 4 096
conv2 conv(64,5) – BN –LeakyReLU(0.1) – MP(5,5) 256 192
conv3 conv(128,3) – BN –LeakyReLU(0.1) – MP(2,2) 221 568
dense1 dense(64) – LeakyReLU(0.1) 65 728
dense2 dense(32) – LeakyReLU(0.1) 4 161
output regression(1) —
Table 3: Layer structure of our PhyCNN. The physical scalar input2 is expanded to dimension 64×1and concatenated
with the output of ‘dense1’ using a depth concatenation layer. Nomenclature:
conv(N,K): convolutional layer with Nchannels and K×Kkernel size;
BN: batch normalization layer;
MP(N,P): maxPooling Layer, size Nstride P;
dense(N): dense layer with Nneurons;
regression(N):regression layer with Nneurons;
LeakyReLU(α): leaky rectified linear unit, slope αon negative inputs.
3.2.2 Maximum flow problems on graphs
In the field of optimization, maximum flow problems aim at identifying the maximum flow
rate a network of pipes is able to sustain. Let the network be described by an undirected graph
G(N, E , ω), where the nodes Ncorrespond to distribution nodes, edges Eto connections by pipes,
and the weights ωencode the maximal capacity of each pipe. For any two nodes n16=n2∈N
(source and sink), we can determine the maximum flow between those nodes allowed by the
network. For a thorough introduction to graph algorithms and maximum flow problems, we
refer to [23]. In this sense, we aim at approximating relevant properties of the physical flow
between two opposite sides of a sample by solving flow problems on a suitable graph network.
To do so, we consider the pore space of our segmented image data (stemming from geological
specimens) to be a graph by identifying each voxel as a node and each neighboring relation of
voxels via a common face as an edge. This procedure is illustrated exemplarily in Figure 6for
a 2D structure. For simplicity, all edge weights ωiare assumed to be one. Note that this graph
representation still comprises multiple standard characteristics of the pore space. For example,
porosity translates to the number of nodes in the graph divided by the total number of voxels; the
number of surface elements is approximately determined by 6 card(N)−2 card(E). Moreover, we
add a further node nin connected to each voxel of the sample’s inflow face, as well as nout being
connected to each voxel of the sample’s outflow face, cf. Figure 6. As such, we can approximate
the permeability determination problem by a maximum flow problem through Gbetween the
nodes nin, nout.
For the samples considered in this paper, the computational effort for calculating fmax using
Matlab’s built-in routine maxflow is reasonably small compared to the solution of stationary
Stokes equations as described in Section 4.3. By the min-cut max-flow theorem, cf. [23], the
maximum flow problem allows for another interesting interpretation. In the setup introduced
within this section, fmax corresponds to the minimal number of edges that need to be deleted
from Gto obtain a disconnected graph such that inflow and outflow faces of the sample are
contained in different connected components, cf. Figure 6. Hence, we obtain information about
the connectivity of the pore space with respect to the direction of interest. More precisely, fmax
classifies structures containing thin channels regarding their restricting effects on fluid flow. This
behavior is comprehensible using the geometries illustrated in Figure 3: Exhibiting fmax of 1230,
the permeability of Subsample 0 is governed by wide channels. On the other hand, exhibiting
fmax of 55, narrow pore throats hamper the fluid flow of Subsample 9213.
13
nin nout
Figure 6: Left: Derivation of the graph representation from a pore space illustrated by a 2D 4×4pixel image. Nodes are
depicted as circles, edges as lines. Gray voxels refer to the solid matrix. The maximum flow value fmax of the depicted
graph is one. Right: More complex 8×8pixel setup exhibiting fmax of three w.r.t. the horizontal axis and zero w.r.t. the
vertical axis.
We further note that for moderate tortuosity, fmax is approximately proportional to the
volume of the channel needed to transport the maximum flow through the sample. As such, by
exclusively regarding the subgraph of Gactively used by the maximum flow, we cut off parts
of Gthat do not effectively contribute to fluid flow. Hence, the determination of fmax can
be understood as approximating the porosity with respect to the pore space participating in
fluid transport. That way, we naturally improve the permeability estimation based on classical
porosities.
Plotting fmax against the calculated permeabilities kcmp, we conclude an almost linear rela-
tionship between both quantities in the logarithmic scale. Using mean-square linear regression,
we obtain the approximate functional relation
kcmp ∼50625 ·(fmax)1.407 ·10−8.183 [D],(5)
cf. Figure 4. On the displayed logarithmic scale, this regression yields an approximation quality
of R2= 0.8692, cf. Section 2.3. Apparently, the derived quantity is strongly correlated to the
target permeability kcmp. Therefore, the permeability estimation using formula (5) is considered
a suitable initial guess, which the CNN is able to improve by relating to features extracted from
the pore geometry.
4 PhyCNN training and workflow performance
In this section, information concerning our PhyCNN’s training procedure is provided. Further-
more, we evaluate the quality of our resulting predictions on three different types of sandstone
as well as a challenging artificially deformed series of geometries. Finally, we compare the com-
putational efficiency of our Stokes solver and the PhyCNN.
4.1 PhyCNN training
As the data points obtained by the forward computation are quite sparse for extremely high and
low permeability values (Figure 7), we disregarded all data samples ranging outside the interval
of [50 mD,50 D]. As a result, we improve the quality of predictions within the permeability range
where the availability of data is favorable and reliable. Splitting the remaining data set according
to a 90%/10% key, 8 876 samples are used for training the network, another 987 for validation.
In order to accurately account for the labels covering three orders of magnitude, a logarithmic
transformation was applied before training as done in [47]. Using a standard mean-squared-error
14
Figure 7: Computed porosity φcmp and permeability kcmp distribution among the total of 10 000 Bentheimer samples.
Mean porosity is 26.20% with standard deviation σ= 0.067, mean permeability is 3930 mD with σ= 8319 mD.
(MSE) loss in the regression as given in Section 2.3, this results in measuring the error in relative
deviation rather than absolute deviation.
The training was performed over 15 epochs using a stochastic gradient descent (SGD) op-
timizer with momentum 0.9. An almost constant validation loss over the last epochs indicated
convergence of the network. Starting from an initial learning rate of 0.0020, the step size has
been decreased by 60% every four epochs.
4.2 Prediction quality
The network’s predictive performance is illustrated in Figure 8via regression plots. Our PhyCNN
achieved coefficient of determination (R2) values of 96.33% on training data and 93.22% on
validation data, cf. Table 4. As such, the network accomplished very high accuracy with only
weak tendency to overfitting. Furthermore, the plots in Figure 8show a low number of outliers,
underlining the stability of our approach.
Bentheimer (train.) Bentheimer (val.) Berea Castlegate
R292.89% 88.20% 89.43% 94.55%
R2log 96.33% 93.22% 94.13% 94.51%
mean kcmp [mD] 3417.7 3628.8 660.9 1661.0
σ(kcmp)[mD] 4948.7 5233.2 657.8 2442.2
Table 4: Overview prediction quality. R2values in natural and logarithmic scale for the training and validation Bentheimer
data as well as for Berea and Castlegate sandstone. Furthermore, mean permeability and standard deviation σ(kcmp)are
listed for the respective data sets.
In order to further stress the robustness, we additionally validate our PhyCNN on samples
originating from different types of sandstone that were not used for training. Using the data
provided by [26,27], 200 additional subsamples were extracted from each Berea and Castlegate
µCT scan, see Table 1, using the same methods as described in Section 2. Achieving R2val-
ues of 89.43% and 94.13% (logarithmic) on Berea as well as 94.55% and 94.51% (logarithmic)
on Castlegate, the networks proves excellent generalization properties across different types of
pore geometries, cf. Figure 8, Figure 2, and Table 1. As such, the network seems in fact to
perform slightly better on Berea and Castlegate data samples than on the validation data set
15
10-1 10010 1102
10-1
100
101
102a)
10-2 10-1 10 0101102
10-2
10-1
100
101
102b)
10-3 10-2 10 -1 100101
10-3
10-2
10-1
100
101c)
10-2 10-1 10 0101102
10-2
10-1
100
101
102d)
Figure 8: PhyCNN accuracy on validation a) and training data b) originating from Bentheimer sandstone. The related
R2values are 0.8820,0.9289 in the natural scale and 0.9322,0.9633 in the logarithmic scaling. Images c) and d) show our
PhyCNN’s accuracy on the Berea and Castlegate test sets, respectively.
of Bentheimer. However, Table 4marks a very high standard deviation for the permeability
in Bentheimer compared to the other sandstone types. As such, the latter data sets comprise
less heterogeneity facilitating the network’s prediction. We further emphasize that both Berea
and Castlegate validation sets have not been screened to match the trained range [50 mD,50 D].
More precisely, eleven Berea data samples exhibit a permeability kcmp below 50 mD as well as
two Castlegate data samples.
Finally, we investigate the generalization limits of our PhyCNN by subjecting it to an ar-
tificially distorted data set covering a challengingly wide permeability and porosity range. To
achieve that, we applied a level-set-based algorithm already used in [48] to expand or shrink the
pore space of Subsample 0 displayed in Figure 3. More precisely, the pore space is contracted
with a uniform level-set velocity directed perpendicularly to the pore walls until all flow channels
collapse, i. e. the data sample becomes impermeable. The same number of deformation steps
is also used to expand the pore volume of Subsample 0. Each of those steps corresponds to
16
8/30/2021
Dilate.svg
file:///Volumes/NO NAME/Dilate.svg
1/1
Figure 9: Testing generalization ability on artificially distorted data samples. PhyCNN prediction performance on diluted
and reduced pore spaces exhibiting a large range of porosities φcmp . For each manipulated geometry, Stokes simulation
data kcmp (red) and CNN prediction kprd (blue) are compared. Furthermore, we provide relative prediction errors. Markers
referring to the original, not manipulated data sample are increased in size. The permeability range spanned by our
PhyCNN’s training data set is highlighted in gray.
a constriction/expansion of the pore space by a single additional voxel layer on average.
Subsequently, we compute the permeabilities kcmp using the Stokes solver and kprd using
the PhyCNN of the series of pore spaces and compare the results in Figure 9. The data show
an almost perfect match for expanded pore spaces even for porosities up to 70%. Using linear
interpolation, we estimate kcmp to exceed the trained permeability range [50 mD,50 D] beyond
φcmp = 57.33%. Therefore, we conclude that our PhyCNN is capable of properly characterizing
also highly permeable samples. On the other hand, predictions remain reasonably accurate for
strongly confined geometric structures. Three samples in Figure 9show porosities below 9.08%,
which approximately refers to the lower end of the trained permeability range for this example.
These exhibit fmax of 4, 16, and 30, respectively, referring to an almost disconnected pore space.
Since the flow channels narrow in these cases to only very few voxels in diameter, errors from
the µCT scan as well the discretization of Stokes equations (1) may become non-negligible. As
such, these samples may leave the current operating limits of digital rock physics, resulting in
a significant systematic overestimation of permeability compared to lab experiments [49].
4.3 Computational performance
In this section, we provide computational performance indicators for the forward simulations as
performed using the method described in Section 2.2 as well as the ones obtained for estimations
using the PhyCNN. Subsequently, we compare the forward simulation run times for the gener-
ation of 10 000 data samples including both permeability computation approaches on the same
compute cluster to estimate the actual speed up. All subsequent specifications of computation
times refer to the wall time.
Each of the 10 000 forward simulations on Bentheimer sandstone was performed with classical
Taylor–Hood elements on voxels (`=1, cf. Section 2.2.3) in parallel on 50 compute nodes of the
Emmy compute cluster at RRZE [50], each being equipped with two Intel®Xeon 2660v2 ‘Ivy
Bridge’ processors (10 physical cores per chip, i. e., 50·2·10=1000 cores total, no hyper-threading)
and 64 GB of RAM. The linear solver (MINRES) converged within 969 iterations on average with
a relative tolerance of 1.0E−6in the norm k·kP−1. Furthermore, meshing, assembly, and solution
were performed within 2.35,0.44, and 33.95 seconds on average per solver call, respectively. As
such, the labeling procedure of all 10 000 data samples was completed within roughly 100 hours.
We conclude that the forward simulation effort of our Stokes approach is roughly comparable to
the LBM-based simulations applied in related publications, cf. [4].
Operating on a graphics cluster featuring two Intel®Xeon E5-2620v3 (6 cores each), two
Nvidia®Geforce Titan X GPUs, and 64 GB of RAM, the PhyCNN’s training process ter-
17
minated within two hours. Using 10 CPU cores and a single graphics chip on this machine,
the permeabilities of the 10 000 elements data set of Bentheimer sandstone is estimated within
1294 seconds by our PhyCNN. More precisely, 867 seconds on CPU were spent on the graph
flow problems related to fmax as described in Section 3.2.2 while 367 seconds were required to
perform the subsequent network inferencing on GPU. Estimating the time required to solve the
related Stokes problem (1) extrapolated from a subset of ten data samples indicates a total time
of approximately 1469 hours. Accordingly, we conclude an acceleration factor of 4087 by using
our PhyCNN on the stated hardware configuration and data set.
From the data presented above, we infer that the time invested in the network’s training
procedure including the calculation of fmax for the complete database is compensated after
about 15 data samples computed beyond the training and validation data sets. This strongly
underlines the capability of our approach to pose a time-saving yet accurate alternative to flow
simulation-based permeability estimators on large data sets.
5 Summary and conclusions
In this work, we demonstrated the feasibility of direct numerical simulation (DNS) for flow
through porous media to generate a library of computed permeability labels from 3D images
acquired using specimens of natural rocks. Our distributed-parallel stationary Stokes solver
achieved computational efficiency comparable to classical lattice Boltzmann method (LBM) ap-
proaches while resolving LBM convergence issues that typically appear on complex 3D geome-
tries.
As a result of computations with the proposed numerically robust forward simulation algo-
rithm, an unbiased (by computational artifacts) data set was constructed to train a convolutional
neural network for accelerated permeability predictions. An easily-computable graph-based char-
acteristic quantity of the pore space, namely the maximum flow value, was introduced to the
neural network as another novel development in our work. As a result, this methodology enabled
our machine learning model to achieve an R2value of 93.22% on the validation data set. More-
over, similar prediction qualities were found for types of sandstone rock that are different from
the training samples, as well as for artificially generated voxel sets. These observations underline
the robustness of our artificial intelligence augmented permeability estimation approach.
In a one-on-one comparison before accounting for the training-investment-related computa-
tional costs, the neural-network-based permeability estimation approach delivered a speed-up in
excess of 4000 fold. On the other hand, computational results indicate that the proposed ap-
proach exceeds the performance of a purely DNS-based workflow beyond approximately 15 data
samples when the training-related computational costs are accounted for. This indicates that the
proposed artificial intelligence augmented permeability estimation workflow is viable for real-life
digital rock physics applications.
Further research is required to extend our refined methodology to geological samples on the
REV scale. Recent publications suggest hierarchical multiscale neural networks [51] to alleviate
the memory requirements of the training and inference procedure. Merging both approaches
holds the potential of leveraging our current results to larger scale geological samples.
Data availability
The CNN code used in this paper is available as part of the porous media numerical toolkit
RTSPHEM [52] on Github.
18
Notation
Symbols
Acmp Computed interior surface area of µCT scan [mm2].
Aspec Specific (w.r.t. material volume) interior surface area [mm−1].
αParameter in LeakyReLU nonlinearity.
fmax Maximum flow value.
kana Analytical permeability (all permeabilities in darcies D, millidarcies mD).
kcmp Computed permeability, cf. (4).
kexp Experimental permeability.
kprd Predicted permeability.
`Polynomial degree along principle axes (Q`is the local space of polynomials of degree
at most `in each variable).
ΩPore space, domain Ω⊂R3of the Stokes equations (1).
pHydrostatic pressure on the pore scale (dimensionless), solution p:Ω→Rof the
Stokes equations (1).
φcmp Computed porosity.
φexp Experimental porosity.
R2R-squared value, coefficient of determination.
Re Reynolds number (definition is irrelevant here, since Re cancels out in the computation
for kcmp, cf. (4)).
σStandard deviation.
uSolenoidal fluid velocity on the pore scale (dimensionless), solution u:Ω→R3of the
Stokes equations (1).
nNode n∈Nin an undirected graph G(N, E, ω), cf. Section 3.2.2.
Abbreviations
CNN Convolutional neural network.
DNS Direct numerical simulation.
DOF Degrees of freedom.
FFN Feed forward network.
LBM Lattice Boltzmann method.
µCT Microcomputed tomography.
MSE Mean squared error.
PhyCNN Physics-informed convolutional neural network.
ReLU Rectified linear unit.
REV Representative elementary volume.
SGD Stochastic gradient descent.
Acknowledgments
S. Gärttner and A. Meier were supported by the DFG Research Training Group 2339 Interfaces,
Complex Structures, and Singular Limits.
N. Ray was supported by the DFG Research Training Group 2339 Interfaces, Complex Structures,
and Singular Limits and the DFG Research Unit 2179 MadSoil.
F. Frank was supported by the Competence Network for Scientific High Performance Computing
in Bavaria (KONWIHR).
We further thank Martin Burger for insightful discussions and Fabian Woller for assisting with
the MFEM implementation.
19
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