Estimating permeability of 3D micro-CT images by physics-informed CNNs based on DNS

Abstract

In recent years, convolutional neural networks (CNNs) have experienced an increasing interest in their ability to perform a 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 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. The robustness of this approach is reflected by very high prediction accuracy, which is observed for a variety of sandstone samples from archetypal rock formations.

Full text

https://doi.org/10.1007/s10596-022-10184-0
ORIGINAL PAPER
Estimating permeability of 3D micro-CT images
by physics-informed CNNs based on DNS
Stephan G¨
arttner1·Faruk O. Alpak2·Andreas Meier1·Nadja Ray1·Florian Frank1,3
Received: 18 April 2022 / Accepted: 29 November 2022
©The Author(s) 2023, corrected publication 2023
Abstract
In recent years, convolutional neural networks (CNNs) have experienced an increasing interest in their ability to perform
a 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 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. The robustness of this approach is reflected by very high prediction accuracy, which is observed
for a variety of sandstone samples from archetypal rock formations.
Keywords Digital rock ·Neural networks ·Deep learning ·Permeability ·Porous media
Mathematics Subject Classification (2010) 05C21 ·68T07 ·76D07 ·76M10 ·76S05
1 Introduction
Artificial neural networks can accelerate or replace classical
methods for estimating various hydrological and petrophys-
ical properties of artificial and natural rock [50]. In [6], deep
neural networks were successfully used to approximate
tomography operators to reconstruct wave velocity models
from seismic data. Moreover, convolutional neural networks
(CNNs) were used in [30] to determine effective electrical
parameters in a homogenized model for electric conductivity.
Nadja Ray
ray@math.fau.de
Florian Frank
frank@math.fau.de
Extended author information available on the last page of the article.
Likewise, CNNs have proven their ability to provide fast
predictions of scalar permeability values directly from
images of the pore space of geological specimens [23,35,
53]. Finally, CNNs were successfully used to replace stan-
dard solution schemes to predict effective permeabilities in
2D multiscale flow simulations [18]. This paper aims at
contributing to the deployment of machine learning tech-
niques in effective permeability prediction 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.
We use the term PhyCNN throughout this paper to under-
line the incorporation of an additional physically motivated
input parameter to the network. This approach is to be care-
fully distinguished from implementing physical laws into
the loss-function as performed in [11] to estimate flow
velocity fields.
Specimens of natural rock are typically obtained by
microcomputed tomography (μCT) scanning [10,52]or
/ Published online: 31 January 2023
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indirectly from 2D colored images [5]. As an important
characteristic quantity of porous media, permeability
measures the ability 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, i. e., the accuracy of the forward simulations in our case,
asshownin[56]. 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 permeability prediction
under regular geometries, a broader set of methods is
available as described and compared in [49].
In the literature, various methods are available to com-
pute the stationary flow on complex geometries as con-
stituted by the pore space of porous media, a thorough
comparison of which is found in [54]. Most commonly,
lattice Boltzmann methods (LBM) are used to solve the
flow problem on a discrete modeling basis, cf. [27]. In this
approach, a transient interacting many-particle system 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 [25]. Yet, since a global equilibrium has to be
reached, complex geometries containing thin channels may
cause LBM to converge slowly or even diverge [23,37],
especially in the case of very constricted and therefore
typically under-resolved pore morphologies. This behav-
ior is frequently observed with common pore-scale flow
LBM implementations that rely on the conventional single-
relaxation-time Bhatnagar–Gross–Krook (BGK) scheme
[8]. However, the novel multiple-relaxation-time (MRT)
scheme addresses this problem to a great extent (e. g.,
[3]) at the cost of a moderate additional computational
overhead. Having stated that, residual discretization errors
stemming from the explicit nature of time integration in
LBM schemes still remain in the numerical solution. Espe-
cially for under-resolved pore morphologies, these errors
may be amplified causing divergence of the numerical flow
solution. We further note that classical bounce-back rules
used for implementing boundary conditions tend to develop
oscillations that might locally dominate fluid behavior in
scenarios with thin channels, cf. [55]. Severe limitations
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
[37]. 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.,
[3,47]). Moreover, a comprehensive permeability compu-
tation benchmarking study [39] 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 [23]. 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 [42]. More precisely, our
forward simulation is based on a distributed-parallel Stokes
solver utilizing the finite element library MFEM [4].
As studied in [49] for simple cylindrical obstacles, such
DNS approaches (in this case FEM) deliver permeability
values comparable to the ones from LBM simulation.
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 net-
works. 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 [46,53]. As shown in [44], carefully cho-
sen specific quantities derived from the pore space such as
connectivity indices can deliver reasonable approximation
quality for permeability estimation. As illustrated in [6],
also training a network solely on previously extracted fea-
tures 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 representation
of the pore space based on [12]. 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 [48],
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Fig. 1 Overall workflow in flow chart representation
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 3is 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 Fig. 1. We note that all
steps except for the data labeling procedure (green box) are
implemented in Matlab 2021a [31].
2.1 Sampling and preprocessing
The training procedure of our PhyCNN is based on a seg-
mented X-ray μCT scan of a Bentheimer sandstone sample,
Fig. 2 100022D slices and 100033D pore-space μCT image of all samples considered in this study, illustrating characteristic pore features for
Bentheimer, Berea, and Castlegate sandstone
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see Fig. 2, with experimentally measured porosity φexp =
22.64 % and permeability kexp =386 mD, provided by
[32,33]. Further characteristic quantities for this sand-
stone 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 vol-
ume distribution and high pore connectivity in comparison
to other types of sandstone [20]. As such, this sample is
expected to contain a representative collection of geometri-
cal and topological properties of the pore space in natural
rocks. The data set used in this paper is derived from
a 1000×1000×1000 binary voxel image, in which each
voxel either belongs to the pore space (“fluid voxels”) or
the rock matrix (“solid voxels”). The voxel edge length
is 2.25 μm yielding an overall cube side length of 2.25 mm.
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 [43]. 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 [23].
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 erosion
and dilation 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 using natural
Tab le 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 capillary diameter, MCD, and
interior surface area, Acmp, computed via the Matlab function
isosurface
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
First four quantities are provided in [32]
rock sample for training, cf. [23,37]. However, using
artificially generated pore-space geometries, even larger
data sets including 90 000 data point are available in the
literature, cf. [35]. More precisely, these data are obtained
from stochastic computer models, evading the necessity of
expensive imaging and segmentation of real rock.
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 [24].
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 only 1.25 GB of
disk space. Therefore, it is manageable on standard personal
computers and file read/write access is reasonably cheap.
We note that a subsample of size 225 μmistoosmall
to be a representative elementary volume (REV) for most
sandstone types, cf. [20,32]. 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. Moreover, hierarchical neural
networks have proven to be a powerful tool to leverage the
performance of CNNs predicting permeability to samples of
REV-size (in the order of 5003voxels), cf. [38]. Therefore,
we believe that our proposed methodology also benefits
existing REV-scale models.
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
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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 u
and pressure p. The discretization uses arbitrary order
(stable) Taylor–Hood or reduced-order stabilized Taylor–
Hood mixed finite elements. In this paper, we consider voxel
meshes only, yet the usage of unstructured grids by using
obvious respective discrete spaces is straight forward.
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 refinement level and
finite element spaces used to produce the permeability
values kcmp to train and validate our PhyCNN.
We note that—despite the finite resolution of the
μCT images—the voxel scans are henceforth considered to
represent the actual (ground-truth) pore geometry. Hence,
errors arising from imaging are neglected as they pose
an independent part of the total methodology that is beyond
the scope of this paper.
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=0inΩ, (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 Eq. 1,u=u(x,y,z) denotes the (dimensionless)
fluid velocity, p=p(x,y, z) the (dimensionless) pressure,
and Re :=ρU
cLc/μ the Reynolds number of the
system with characteristic length Lc[m], characteristic
velocity Uc[ms
−1], fluid density ρ[kg m−3], and fluid
viscosity μ[Pa s]. Note that the permeability kcmp [m2]is
Fig. 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
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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−pIn=exon ΓN,(1c)
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 Eq. 1is 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 variable x,y,z, cf. [13,14]. For =0,
the pressure space Q0=P0is discontinuous and consists
of elementwise constants. The respective “reduced Taylor–
Hood pair” Q3
1/P0requires stabilization (see below) due
to a lack of discrete inf-sup-stability [13]. 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

i=1
[xu]iφi,p
h=
m

i=1
[xp]iψi(2)
with degree-of-freedom vectors xu∈Rn,xp∈Rm,which
are unique solutions of the linear system
AB
T
B−Cxu
xp=bu
0⇐⇒ : Ax=b(3)
with right-hand side bu∈Rn,[bu]i:=ΓNex·φi.The
sparse blocks in Aare the vector-Laplacian matrix A∈Rn,n
and the divergence matrix B∈Rm,n,
[A]i,j :=Ω
∇φi:∇φj,[B]k,j :=−
Ω
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 [13], (3.84), in which case, Ccan
be interpreted as a pressure-Laplacian discretized by cell-
centered finite volumes [16]. 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 [51]. The chosen precondition operator
for Ain Eq. 3is the symmetric and positively definite
block-diagonal matrix
P:=diag (A,W)
with Wbeing the pressure-mass matrix [Wk,l]:=Ωψkψl.
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
[15]). 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 [7,34,36](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 [19]. In Section 2.2.3,wediscuss
appropriate choices regarding mesh size and discretization
order.
Figure 3illustrates Stokes velocity and pressure fields for
two exemplary subsamples exhibiting 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 paper.
2.2.2 Permeability estimation
We deduce the permeability value kcmp of interest from the
previously calculated Stokes velocity uand pressure p(we
suppress the discretization index hin this section). In the
Stokes Eq. 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 Eq. 1, we compute the approximated permeabil-
ity 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 (dimension-
less) area-averaged inflow and outflow pressures, Pin,Pout ,
givenby[28]
Q:=
Γout
u·n,P
in :=1
|Γin|
Γin
p, P
out :=1
|Γout|
Γout
p.
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From Eq. 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 Eq. 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 [22],
a volume-averaged approach is proposed that is capable
of determining 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 refine-
ment and choice of polynomial order on the solution quality
by comparing the computed permeability kcmp obtained
from Eq. 4a to the analytical value kana.Inordertohave
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 Eq. 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)and height b∈(0,1). An analytical
expression for its permeability kana is [9]
kana :=K∞·min(a, b)3·max(a, b)
12 ,where
Kj:=1−
j

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) ,
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|≤
∞

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
∞

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 permeabil-
ity kana.
Tabl e 2lists the relative error for the computed
permeability kcmp obtained from Eq. 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/Q1elements (=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.
Tab le 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)andmesh
refinement levels
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
As in Eq. 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
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The advantage of higher order discretizations presented
above for narrow rectangular channels 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 p
from zero to one (4% deviation). On the other hand,
Subsample 9213 (Fig. 3) exhibiting narrow and branchy
structures 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.
We emphasize that our scheme does not require mesh
refinement, which is typically required in LBM simulations
of natural rock.
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.
First, we define the standard deviation σby
σ=



1
N−1
N

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

i=1
(ti−yi)2
N

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

i=1
(ti−yi)2
N.
3 Machine learning model
In this section, we motivate and present our suggested neural
network architecture for permeability 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 [2].
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 geological
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 0 and 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 sub-
class 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. [2]. A list of the different layer types used in
our study is given in Table 3.
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
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Tab le 3 Layer structure of our
PhyCNN Block Layers Learnables
input1 image input 100×100×100 −
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
concat1 concat(dense1, input2(physics input)) −
dense2 dense(32) – LeakyReLU(0.1) 4 161
output regression(1) −
The physical scalar input2 is expanded to dimension 64×1 and concatenated with the output of ‘dense1’
using a depth concatenation layer concat1. 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
concat(in1,in2): concatenation layer
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 normalization (BN) layers, cal-
ibrating the mean and spread of the provided inputs.
Throughout the network, we apply LeakyReLU nonlinear-
ities 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
[29], 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 [31].
3.2 Physics-informed convolutional neural networks
In the following, we use the building blocks for gen-
eral 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 intro-
duction to PhyCNNs and thereafter discuss the choice of
meaningful 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. 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
Fig. 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. To achieve
the desired increase in prediction accuracy and robustness,
the additional data must be well-chosen and complement
the insights gained by the CNN from the pure image data.
Depending on the quality and quantity of the extracted
features as well as the specific application, these alone
may even provide sufficient information to the network
to accurately perform predictions, cf. [6], underlining the
potential of physics-informed strategies.
This approach has been successfully applied to the
prediction of permeabilities from pore-space geometries:
In [37], the Euclidean distance map, maximum inscribed
sphere radii, and time of flight maps are inserted as inputs
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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
Fig. 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 (R2=
0.055), computed porosity φcmp (R2=0.547), specific surface area
Aspec (R2=0.455), maximum flow on graph representation fmax
(R2=0.869). The given R2values refer to a linear regression model
to determine the overall fluid field. In [53], porosity
and surface area have been used to increase predictions
performance for 2D image data. Similar observations have
been made on 3D data where the additional consideration
of porosity and tortuosity significantly reduced the number
of outliers, especially when training on small data sets, cf.
[45]. The specific choice of these parameters is related to
the Kozeny–Carman formula estimating permeability from
porosity and surface area / tortuosity. As indicated in [41],
the proposed relation deteriorates in quality for increasingly
complex geometries. However, [46] found porosity to be
the most prominent input factor for their CNN among
a set of various derived quantities such as coordinate
number or mean pore size. Furthermore, characteristics
obtained by pore network modeling have shown low general
accuracy [46].
As indicated by our data set statistics in Fig. 4, porosity
and surface area are only weakly correlated with the
computed permeability kcmp for our complex 3D setting.
For a given surface 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 [31]. 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 that
specifically encodes information about the sample that is
difficult to access by standard CNNs.
Our final network architecture is presented schematically
in Fig. 5using the hyper-parameters of Table 3and
Fig. 5. The structure is based on the findings of [23],
Fig. 5 Schematic representation of physics-informed convolutional
neural networks (PhyCNNs). The image input is processed by convo-
lutional layers, condensing extracted features in cubes of size 23.The
scalar valued physical input circumvents the image convolution lay-
ers and is directly fed into the subsequent dense layers. As such,
the interpretation of the given data performed in the network’s upper
dense layers relies on both information sources. Numbers at the lay-
ers denote their dimension (number of neurons). Note that the cubic
convolutional kernels and their dimension are illustrated in 2D for
clarity. Graphics produced using NN-SVG [26]
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where the hyper-parameters of a purely convolutional neural
network for permeability prediction have been optimized
using a grid search algorithm. Since this optimization
is specific to the data set considered, slight manual
adjustments further increased accuracy on our data. More
specifically, 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. As a result, the network’s upper dense layers
are able to perform predictions from the extracted image
data as well as the additional scalar input. For details on the
parameters used within the network’s training procedure,
see Section 4.1. A pretrained version of our network is
available as part of the toolbox RTSPHEM [17] on GitHub.
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 N
correspond to distribution nodes, edges Eto connections by
pipes, and the weights ωencode the maximal capacity of
each pipe. For any two nodes n1= 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 [12]. 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 Fig. 6
for 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. Fig. 6. As such, we can approximate the
permeability determination problem by a maximum flow
problem through Gbetween the nodes nin,n
out.
For the samples considered in this paper, the compu-
tational effort for calculating fmax using Matlab’s built-in
routine maxflow with about one second per sample on
a single CPU core, also cf. Section 4, 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. [12], 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. Fig. 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 Fig. 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.
We further note that for moderate tortuosity, fmax is
approximately proportional to the volume of the channel
Fig. 6 Left: Derivation of the graph representation from a pore space
illustrated by a 2D 4×4 pixel 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
8×8 pixel setup exhibiting fmax of three w.r.t. the horizontal axis and
zero w.r.t. the vertical axis
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needed to transport the maximum flow through the sample.
As such, by exclusively regarding the subgraph of G
actively 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 relationship 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. Fig. 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. Furthermore, 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
computational 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 (Fig. 7), we disregarded all data samples ranging
outside the interval of [50 mD, 50D]. 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 [21].
Using a standard mean-squared-error (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) optimizer with momen-
tum 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.
4.2Predictionquality
The network’s predictive performance is illustrated in Fig. 8
via 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 Fig. 8show a low
number of outliers, underlining the stability of our approach.
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 [32,33], 200 additional
subsamples were extracted from each Berea and Castlegate
Fig. 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=8319mD
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Fig. 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
10-1 10010110 2
10-1
100
101
102a)
10-2 10-1 10010 1102
10-2
10-1
100
101
102b)
10-3 10-2 10-1 10 0101
10-3
10-2
10-1
100
101c)
10-2 10-1 10010 1102
10-2
10-1
100
101
102d)
μCT scan, see Table 1, using the same methods as
described in Section 2. Achieving R2values 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. Figs. 2,8and 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 of Bentheimer. However, Table 4
marks a very high standard deviation for the permeability
Tab le 4 Overview prediction
quality Measure Bentheimer (train.) Bentheimer (val.) Berea Castlegate
Data set mean kcmp [mD] 3417.7 3628.8 660.9 1661.0
σ(k
cmp)[mD] 4948.7 5233.2 657.8 2442.2
PhyCNN(fmax)R292.89% 88.20% 89.43% 94.55%
R2log 96.33% 93.22% 94.13% 94.51%
PhyCNN(φ-σ)R285.65% 73.55% 72.81% 74.86%
R2log 91.27% 79.57% 58.81% 75.12%
plain CNN R278.92% 73.11% 64.54% 62.66%
R2log 84.48% 77.97% 58.36% 74.12%
plain fmax R257.78% 57.67% 42.82% 56.46%
R2log 85.15% 85.03% 82.65% 81.68%
In the first block, mean permeability and standard deviation σ(k
cmp)are listed for the respective data sets.
Subsequently, R2values in natural and logarithmic scale for the training and validation Bentheimer data as
well as for Berea and Castlegate sandstone are presented as achieved by our PhyCNN, i. e., the maxflow-
informed CNN (PhyCNN(fmax)), the porosity/surface-area-informed CNN (PhyCNN(φ-σ)), and a plain
CNN without additional inputs. Finally, the additional input quantity fmax is considered in a power-law
(linear fitting applied to logarithmic variables) regression model (plain fmax)
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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.
Moreover, Table 4lists the prediction quality indices
of other related CNNs for comparison. To clearly and
consistently separate the different approaches investigated,
we refer to the PhyCNN described in Section 3.1 more
precisely as PhyCNN(fmax) within this comparison. In
case of the PhyCNN(φ-σ), the general structure depicted
in Fig. 5is maintained while replacing the maxflow
input variable by the porosity φand surface area σ,and
retraining the network, cf. Fig. 4. As the data of Table 4
show, the PhyCNN(φ-σ) cannot reach the accuracy of our
PhyCNN(fmax). Moreover, robustness appears significantly
lower since accuracy on other sandstone types than
the training data decreases rapidly. Especially for Berea
sandstone, the results are poor, which might be a direct
consequence of the significantly different pore-space
characteristics in comparison to Bentheimer, cf. Table 1.
Furthermore, we also compare to the analogous network
without any additional input quantities (plain CNN),
resulting in slightly worse predictions than the PhyCNN(φ-
σ). Finally, we note that both of these CNNs provide
less accurate predictions on validation samples than fmax
alone (plain fmax) being fitted to the Bentheimer training
data set using a power law approach and measured
with respect to R2-log. As such, we conclude that fmax
already contains highly relevant information regarding the
samples’ permeability delivering more accurate predictions
in the logarithmic measure than standard CNN approaches.
However, our PhyCNN(fmax) improves the results of pure
fmax by extracting further information directly from the
pore-space, cf. Fig. 5.
Finally, we investigate the generalization limits of
our PhyCNN by subjecting it to an artificially distorted
data set covering a challengingly wide permeability and
porosity range. To achieve that, we applied a level-set-
based algorithm already used in [18] to erode or dilate
the pore space of Subsample 0 displayed in Fig. 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 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 Fig. 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, 50D]
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 Fig. 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 Eq. 1may
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 [40].
Fig. 9 Testing generalization ability on artificially distorted data sam-
ples. PhyCNN prediction performance on eroded and dilated pore
spaces exhibiting a large range of porosities φcmp. For each manipu-
lated geometry, Stokes simulation data kcmp (red) and CNN prediction
kprd (blue) are compared. Furthermore, we provide relative predic-
tion 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
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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 generation 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
[1], 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 −6 in the norm ·
P−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. [23].
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 Phy-
CNN’s training process terminated 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. Accord-
ingly, 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) implementations
while successfully addressing convergence challenges that
typically appear on complex 3D geometries including
poorly resolved subdomains.
As a result of computations with the proposed numeri-
cally robust forward simulation algorithm, an unbiased (by
computational artifacts) data set was constructed to train
a convolutional neural network for accelerated permeability
predictions. An easily-computable graph-based characteris-
tic 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. Moreover, similar pre-
diction 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 under-
line the robustness of our artificial intelligence augmented
permeability estimation approach.
In a one-on-one comparison before accounting for
the training-investment-related computational 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
approach 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.
Finally, we note that the refined methodology presented
in this paper may generalize to larger sample sizes
approaching the REV-scale. Recent publications suggest
hierarchical multiscale neural networks [38] 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.
Notation
Symbols
Acmp Computed interior surface area of μCT scan [mm2].
259Computational Geosciences (2023) 27:245–262
Content courtesy of Springer Nature, terms of use apply. Rights reserved.

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 dar-
cies 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 Eq. 1.
pHydrostatic pressure on the pore scale (dimension-
less), solution p:Ω→Rof the Stokes Eq. 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 (dimen-
sionless), solution u:Ω→R3of the Stokes
Eq. 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.
Acknowledgements S. G¨
arttner 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.
Funding Open Access funding enabled and organized by Projekt
DEAL.
Data Availability The CNN code used in this paper is available as part
of the porous media numerical toolkit RTSPHEM [17] on GitHub.
Declarations
Conflict of Interests The authors have no relevant financial or non-
financial interests to disclose.
Open Access This article is licensed under a Creative Commons
Attribution 4.0 International License, which permits use, sharing,
adaptation, distribution and reproduction in any medium or format, as
long as you give appropriate credit to the original author(s) and the
source, provide a link to the Creative Commons licence, and indicate
if changes were made. The images or other third party material in this
article are included in the article’s Creative Commons licence, unless
indicated otherwise in a credit line to the material. If material is not
included in the article’s Creative Commons licence and your intended
use is not permitted by statutory regulation or exceeds the permitted
use, you will need to obtain permission directly from the copyright
holder. To view a copy of this licence, visit http://creativecommons.
org/licenses/by/4.0/.
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Publisher’s note Springer Nature remains neutral with regard to
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Affiliations
Stephan G¨
arttner1·Faruk O. Alpak2·Andreas Meier1·Nadja Ray1·Florian Frank1,3
Stephan G¨
arttner
gaerttner@math.fau.de
1Department Mathematik, Friedrich-Alexander-Universit¨
at
Erlangen-N¨urnberg, Cauerstraße 11,
91058 Erlangen, Germany
2Shell Technology Center, 3333 Highway 6 South,
Houston, TX 77082, USA
3Math2Market GmbH, Richard-Wagner-Straße 1,
67655 Kaiserslautern, Germany
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