WGMwgm QEqe EPep PMSpms BECbec DEde

[1] [2] [3] [1]This document is the results of research funded by the EPSRC
additive manufacturing grant. [2]H.P.M.(First Author) contributed equally to
this work with J.M. (Last Author) [3]H.P.M, J.M., and G.M.M. conceptualised
this study, K.M.H and H.P.M chose the samples, H.P.M. and K.S. did the imaging
and analysis, J.M. and H.M. built the software and ran the models, All authors
contributed to writing the paper.

[type=editor, auid=000,bioid=1, orcid=0000-0002-1445-6354]

[1]

1]organization=Institute of GeoEnergy Engineering, addressline=Heriot-Watt
University, city=Edinburgh, postcode=EH14 4AS, country=United Kingdom

2]organization=Institute for Sustainable Built Environment,
addressline=Heriot-Watt University, city=Edinburgh, postcode=EH14 4AS,
country=United Kingdom

3]organization=Narro Associates, addressline=Orchard Brae House, 30
Queensferry Rd, city=Edinburgh, postcode=EH4 2HS, country=United Kingdom

4]organization=Society for the Protection of Ancient Buildings, addressline=37
Spital Square, city=London, postcode=E1 6DY, country=United Kingdom

[cor1]Corresponding author

# Multi-scale flow, permeability, and heat transport in low-carbon and
traditional building materials

Hannah P. Menke<EMAIL_ADDRESS>[  Katherine M. Hood  Kamaljit Singh
Gabriela M. Medero  Julien Maes  [  [  [

###### Abstract

Permeability and heat transport through building materials ultimately dictates
their insulatory performance over a buildings service lifetime. However,
characterisation of building materials is challenging because porous building
materials are heterogeneous and their macroscopic physical properties (e.g.
permeability, thermal, and mechanical behaviour) depend on their micro scale
characteristics, i.e. the local distribution, material’s fabric, and features
of the solid components and the connectivity of the spaces between them.
Large-scale testing can measure these macro-scale properties, but often does
not give insight into the underlying micro structural properties that
ultimately leads to optimisation. Thus, a knowledge of the 3D structure is
therefore required to assist in the development and implementation process for
new materials. Experiments combining X-ray microtomography with numerical
modelling are an accepted method of studying pore scale processes and have
been used extensively in the oil and gas industry to study highly complex
reservoir rocks. However, despite the obvious similarities in structure and
application, these techniques have not yet been widely adopted by the building
and construction industry.

An experimental investigation was performed on the pore structure of several
building materials, including conventional, historic, and innovative, using
X-ray tomography and direct numerical simulation. Six samples were imaged at
between a 4 and 15 $\mu$m resolution inside a micro-CT scanner. The porosity
and connectivity were extracted with the grain, throat, and pore size
distributions using image analysis. The permeability, velocity, and thermal
conductivity were then investigated using GeoChemFoam, our highly-versatile
and open source numerical solver. It was found that each material had a
unique, heterogeneous and sometimes multi-scale structure that had a large
impact on the permeability and thermal conductivity. Furthermore, it was found
that the method of including sub resolution porosity directly effected these
bulk property calculations for both parameters, especially in the materials
with high structural heterogeneity. This is the first multi-scale study of
structure, flow and heat transport on building materials and this workflow
could easily be adapted to understand and improve designs in other industries
that use porous materials such as fuel cells and batteries technology,
lightweight materials and insulation, and semiconductors.

###### keywords:

multi-scale synthetic building materials concrete pore-scale flow and
transport micro-CT direct numerical simulation GeoChemFoam heat transfer
permeability thermal conductivity

First multi-scale study of both low carbon and traditional building materials.

Micro-CT imaging and image analysis of building materials.

Darcy-Brinkman-Stokes permeability, flow, and heat transport.

Conjugate thermal conductivity.

##  1 Introduction

The Intergovernmental Panel on Climate Change (IPCC) has stated that reducing
carbon emissions from the built environment, which account for almost 40
percent of the total carbon emissions globally, will make a meaningful impact
on global climate change[27, 17]. The global building stock continues to rise
annually and strategies are being developed to mitigate the associated
increases in emissions and achieve net zero carbon, largely by reducing
building emissions [40]. Around 25 percent of embedded carbon in construction
is generated in the building material production, transport, and construction
stages [14]. One method of reducing carbon during the construction process is
through the use of recycled or circularly produced materials that use less
carbon and energy and are resource efficient to produce than traditional
building materials [12, 34, 24]. However, as these building materials are
assimilated into the construction industry, their properties must be measured
both for optimisation during the material development process and for accurate
estimation during building design and construction.

Characterisation of building materials is challenging for several reasons.
Porous building materials are heterogeneous materials and their macroscopic
physical properties (e.g. permeability, thermal, and mechanical) depend on
their micro scale characteristics, i.e. the local distribution and features of
the solid components and the connectivity of the spaces between them. For
example, the porosity and permeability of building materials are ultimately
related to the underlying structure and arrangement of the grains (solid
particles), pores, and binding agent. Thus, two concretes made of the same
materials, but with different sizes of aggregate can have vastly different
thermal, transport, and mechanical properties [11]. Porous building materials
also often contain a binding agent such as cement, which is itself porous, and
has different material properties than the aggregate. The average pore throat
size of the cement is often also on a length-scale orders of magnitude lower
than the pores between the solid grains which makes it difficult to estimate
macro-scale properties of the building material prior to testing. Large-scale
testing can measure these macro-scale properties, but often does not give
insight into the underlying micro structural properties that ultimately leads
to optimisation and thus a knowledge of the 3D structure and particle
arrangement is required to assist in the development, optimisation, and
implementation process for new low carbon building materials.

Experiments combining X-ray microtomography ($\mu$-CT) with numerical
modelling are now an accepted method of studying pore scale processes and have
been used extensively in the oil and gas and carbon storage industries to
study highly complex reservoir rocks [7, 28]. Pore-scale imaging experiments
coupled with simulation are an increasingly important tool used in industry
prediction of geological and petrophysical properties including porosity and
connectivity [25], mineralogical heterogeneity [18], and relative permeability
[33, 1], and thermal conductivity [21, 19]. In addition, new numerical
techniques such as Darcy-Brinkman-Stokes (DBS) have been implemented to tackle
the multi-scale nature of rocks, where processes are modelled explicitly in
the pores using Stokes, and implicitly as an averaged volume in the
microporous regions [26, 13, 38]. However, despite the obvious similarities in
structure and application, these techniques have not yet been widely adopted
by the building materials and construction industry.

There have been several recent studies that have combined imaging and
modelling experimental and modelling workflows to study materials used in the
building industry. Bentz et al. [4] showed the applicability of pore scale
analysis to porous building materials by imaging a lime clinker brick and a
hard-burned clay brick using ($\mu$-CT) and confirmed good agreement with the
Katz-Thompson relationship of permeability, diffusivity, and pore-size.
However, their imaging resolutions were insufficient to accurately compute
these properties with numerical simulations. Nunes et al. [29] used nuclear
magnetic resonance imaging to investigate the influence of pore structure on
moisture transport in lime-plaster-brick systems and found that linseed oil
application hindered the drying of the brick. Chung et al.[9] evaluated the
performance of glass beads on thermal conductivity in insulating concrete and
confirmed the results using numerical modelling. Still, they used limited
image analysis and did not measure the connectivity of the sample, instead
relying on probability functions to describe the structural arrangement of
beads and voids. In addition, while these studies have shown the potential for
modern pore-scale analysis, none of them have incorporated multi-scale
modeling techniques to improve accuracy and computational expense.

Several studies have also attempted to model heat transport in building
materials. Bicer et al. [6] and Guo et al. [16] built purely mathematical
models that did not include any simulations or pore scale information. Zhang
et al. [43] built a theoreticial mesoscale model for thermal conductivity of
concrete that was based on bulk properties and did not consider multi-scale
heterogeneity. Zhu et al. [44] built a multi-scale theoretical model for
concrete and bench marked the results against simulations and experiments.
However, the simulations were only 2D and did not explicitly model pores (only
pores completely filled with aggregate) meaning that the heat conductivity
along connected pathways of grains or empty pores could not be accounted for
explicitly. Panerai et al. [31] investigated thermal conductivity in a range
of fibrous structures using microCT imaging and 3D modeling, but did not
investigate the effect of the model used for effective thermal conductivity of
the microporous fibres. Xiao et al [41] used numerical modelling to
investigate heat conduction of random porous fibres but the simulations were
limited to 2D. Shen et al. [35] studied the thermal conductivity of various
hybrid steel-fine polypropylene and fiber-reinforced concrete with theoretical
models, experiments, and 2D simulations, finding that the touching solid
components acted as a thermal bridge. Yet, 2D numerical models limited the
complexity of these investigations. Yang et al., [42] did numerical
investigations into the effect of fractures on thermal conductivity. However,
their fractures were simple and straight and their digital rock was not
derived from a real 3D sample. Siegert et al [36] were the first to point out
the importance of understanding harmonic vs arithmetic averaging on heat
transfer in porous media structures, showing vast difference in effective
properties depending on which method was used. Nevertheless, their
investigations were limited to meshing of simple, relatively homogeneous
samples that did not contain multi-scale porosity. However, as of yet, no one
has used multi-scale modelling to compare the permeability and thermal
conductivity of different building materials with varying structural
complexity and materiel components.

The objective of this study is to showcase the potential of combined pore-
scale imaging and multi-scale modelling in use-cases relevant to the building
industry. (1) First, six samples of building materials are imaged inside a
micro-CT scanner at a 4-15 micron resolution. (2) The images are then analysed
for pore, throat, and grain size distributions, and the connectivity of the
pore-space is assessed both with and without including the microporosity. (3)
Permeability is then calculated on the images using the Darcy-Brinkman-Stokes
method, both included and excluding the microporous phase. (4) Finally, the
thermal conductivity inside the materials is estimated. As the microporosity
is unresolved in our micro-CT images, its structure is unknown and its thermal
conductivity needs to be modelled by correlations. Here, both harmonic and
arithmetic averaging of the conductivity of the fluid and solid inside the
microporous phase is used and the results compared.

##  2 Sample Selection

Seven samples were selected for this study, five traditional building
materials from various time periods, and a low carbon brick (KBriq) made from
recycled construction and demolition waste and a proprietary binder by Kenoteq
[15]. Five traditional materials were chosen: a fired clay brick from a 1930’s
building, a brick sample (approximately 1900’s) drilled from the Mackintosh
Building at the Glasgow School of Art (GSA), modern aerated concrete, a wooden
beam from a 1930’s building, and Bentheimer sandstone. These materials have a
range of flow and heat transport properties with heterogeneous pore structures
that make them ideal benchmark cases for modern multi-scale numerical solvers.

##  3 Imaging and Analysis Methods

Cylindrical cores with a diameter of 9-mm were drilled from each of the
samples, except the aerated concrete where a larger piece was extracted to
ensure a representative volume due to the large distribution in grain sizes,
and Bentheimer for which a 4 $\mu$m resolution image of which was downloaded
from https://www.digitalrocksportal.org/. The samples were imaged in three
dimensions inside an EasyTom ($\mu$-CT) scanner from Rx Solutions at 100keV
and 10W at between a 6 and 15 $\mu$m resolution.

The raw projections were then reconstructed (Fig. 1A) and post-processed using
the image processing modules in Avizo 2020.3.1 (www.thermofischer.com). The
high-resolution micron-scale images were filtered using the non-local means
filter [8] (Fig. 1B) and segmented using the watershed segmentation algorithm
[5] into three phases (pore, microporosity, and solid grain) (Fig. 1C). The
pores, grains, and throats were then each separated into individual components
using the separate objects module that uses the watershed technique on the
Euclidean distance map of the phase (Fig. 1D) and independently analysed for
size distributions (Fig. 1E). The pore space of each image was then analysed
for connectivity in the Z direction and both connected and unconnected pores
were identified (Fig. 1F). This connectivity analysis was done twice for each
image, once for only the macropores, and again for the macropores and
micropores together, giving an estimation of the connectivity of flow pathways
both including and excluding the sub resolution porosity. The image analysis
results are summarised in Table 1. This connectivity analysis is including the
wooden beam where the cell walls themselves were assumed to be microporous
[39] and thus no solid phase was identified in any segmentation.

Figure 1:  The raw image of KBriq (A), filtered image (B), segmentation (C)
with the primary porosity rendered in light blue, microporosity in dark blue
and solid grains in red shown for the KBriq sample. Grains are separated and
randomly colored (D), and colored by size (E) with small grains in yellow
(0-100$\mu$m equivalent diameter), medium sized grains in blue
(100-200$\mu$m), large grains in red (200-500$\mu$m), and extra large grains
in green (500+$\mu$m). Connected (blue) and unconnected (red) pores (A).

The average porosity of the microporosity was estimated by using the segmented
microporosity as a mask for the grey-scale image. The histogram was then
plotted for the microporous phase only. The histogram values were then
normalised using the grey scale values used in watershed segmentation where
the maximum grey-scale value of the macroporous phase was assigned to a
porosity of 1 and the minimum grey-scale value of the solid phase was assigned
to be a porosity of 0. A linear relationship between grey-scale value and
porosity was assumed for the microporous phase and the mean value of the
porosity histogram was used in the DBS numerical simulations as the porosity
of the microporous phase.

##  4 Numerical Methods

For each image, a $400^{3}$ voxel sub-volume was extracted from each image and
used in all numerical simulations. Permeability was calculated using the DBS
approach, both including and excluding the microporous phase. Steady state
heat transfer (thermal conductivity) was then solved with a constant value for
the heat conductivity of the solid $\kappa_{s}=1.4$ $kW/m/K$ and heat
conductivity of the fluid $\kappa_{f}$ ranging from 0.0006 to 666 $kW/m/K$.
All simulations used modules in GeoChemFoam 5.0 [22, 20, 23], our numerical
toolbox based on OpenFOAM® [30].The full code can be downloaded free of charge
from http://github.com/geochemfoamhttps://github.com/GeoChemFoam.

###  4.1 Calculation of Permeability

Flow in the images was solved using the simpleDBSfoam module using the DBS
approach [38] in which one equation is used to model the flow within the fully
resolved macropores (i.e. voxel porosity equal to 1.0) and the micropores
(i.e. voxel porosity lower than 1.0).

$\mu\nabla^{2}\mathbf{u}-\nabla\mathbf{p}-\mathbf{\mu}k^{-1}\cdot\mathbf{u}=0,$
(1)

$\nabla\cdot\mathbf{u}=0,$ (2)

where $\mathbf{u}$ [m.s-1] is the fluid velocity, $\mathbf{p}$ [Pa] is the
pressure gradient, $\mu$ [kg.m1.s-1] is the fluid viscosity and $k$ [m2] is
the local permeability, i.e. the permeability of the computational cell. This
coefficient is assigned to a large value of 1013 m2 in the pores and to a very
small value of 10-21 m2 in the solid phase in order to obtain a no-flow no-
slip condition at the fluid-solid interface.

The permeability of each microporous voxel is assigned using the Kozeny-Carmen
relationship, which assumes that the microporosity consists of an even packing
of equally-sized elliptical grains. As there is no information about the size
of the grains in the microporous phase _a priori_ , the grain size of this
phase was taken from literature values of typical grain size distributions for
each material. The permeability of the microporous phase was estimated using
the equation

$K_{\mu}=\frac{h^{2}}{180}\frac{\phi_{\mu}^{3}}{(1-\phi_{\mu})^{2}},$ (3)

where $\phi_{\mu}$ is the porosity of the microporous phase and $h$ is the
average grain radius. Additional information on Darcy-Brinkman-Stokes
modelling methods can be found in [26].

To calculate the overall permeability $K$ of the sample, a pressure drop
$\Delta P$ is applied between the left and the right boundaries, and the
velocity field is calculated. The permeability is then obtained as

$K=-\frac{U_{D}L}{\mu\Delta P}$ (4)

where $L$ is the length of the sample and $U_{D}$ (m/s) is the Darcy velocity,
defined as

$\displaystyle U_{D}=\frac{Q}{A},$ (5)

where $A$ (m2) is the cross-sectional area of the domain and $Q$ is the flow
rate (m3/s).

###  4.2 Calculation of effective heat conductivity

Heat transport was solved using a simplified temperature equation,

$0=\nabla\cdot\overline{\kappa}\nabla T$ (6)

where $\overline{\kappa}$ is the local heat conductivity, i.e. the heat
conductivity of the computational cell. This coefficient must include the
contribution of the pores, micropores and solid present inside the
computational cell. Inside the solid phase, $\overline{\kappa}=\kappa_{s}$,
and inside the pores, $\overline{\kappa}=\kappa_{f}$. In the microporous
phase, $\overline{\kappa}$ will depend on the heat conductivities of the fluid
and solid, of the porosity $\phi_{\mu}$, and of the underlying structure of
the pores inside the microporous phase. Calculating this coefficient
accurately would require a high resolution image of this underlying structure.
In the absence of such image, the local heat conductivity must be modelled. In
this work, the results are compared using two different models:

$\mathbf{\kappa}_{\mu(harmonic)}=\frac{\mathbf{\kappa}_{f}\mathbf{\kappa}_{s}}{((1-\mathbf{\phi}_{\mu})\mathbf{\kappa}_{f}+\mathbf{\kappa}_{s}\mathbf{\phi}_{\mu})}$
(7)

$\mathbf{\kappa}_{\mu(arithmetic)}=(1-\mathbf{\phi}_{\mu})\mathbf{\kappa}_{s}+\mathbf{\phi}_{\mu}\mathbf{\kappa}_{f}$
(8)

Additional information on the $heatTransportFoam$ solver numerical methods in
GeoChemFoam can be found in [21].

Upscaling heat transport requires calculating the effective heat conductivity
coefficient $\kappa_{eff}$ [$kW/m/K$] between the fluid and the solid, defined
by [36]. The effective heat conductivity regroups the contribution of fluid,
solid and microporous phase into one single-field coefficient. To calculate
it, a temperature drop $\Delta T$ is established between the left and right
boundaries, the temperature field inside the domain is calculated and the
effective heat conductivity coefficient is obtained as,

$\kappa_{eff}=\frac{Q_{h}L}{A\Delta T},$ (9)

where $Q_{h}$ is the heat flow rate, calculated as the integrate of the heat
flux $\mathbf{J}=-\overline{\kappa}\nabla T$ across the cross secitonal area
A.

##  5 Results and Discussion

###  5.1 Image Analysis

The images were first segmented into three phases: macro pore, micro pore, and
solid grain as show in Fig. 2. The volume fraction of each phase is shown in
Table 1. The connectivity of each phase of porosity was then assessed with
just the macro porosity and then the connected macro+micro porosity.

In this analysis the aerated concrete shows similar volume fractions to the
KBriqs with 0.18-0.20 macro porosity, 0.18-0.38 micro porosity, and 0.42-0.64
solid grain. This high macroporosity volume fraction led to a high
connectivity of macroporosity of 0.99 for all three samples. The GSA Brick and
Fired Clay Brick each have large volume fractions of microporosity of 0.89
with little microporosity of 0.06-0.10 and little solid grain (0.01-0.05). The
lack of macroporosity led to zero connectivity in the macroporosity in the
Fired Clay Brick. However, in the GSA Brick, the porosity was present
predominantly as long, thin fractures, leading to a relatively high connected
volume fraction of the macroporosity of 0.3. In the wooden beam the porosity
was present in laminated layers of pores with low porosity regions between.
However, the connectivity was high regardless with a connected macroporosity
of 0.97. Bentheimer did not contain microporosity but was almost entirely
connected due to its strong homogeneity and large grains.

Figure 2:  The raw (A, E, I, M, Q, U) , 3-phase segmentation (B,F,J,N,R,V,Y),
connectivity of the macro porosity (C,G,K,O,S,W,Z), and the connectivity of
the micro+macro porosity (D,H,L,P,T,X) of each sample. In the 3-phase
segmentation, macro pores are light blue, micro pores are dark blue, and solid
grains are red. In the connectivity of the macro porosity and macro+micro
porosity, unconnected pores are red and connected pores are blue. Bentheimer
does not have microporosity and the image was obtained pre-segmented.

The images were then analysed for pore, grain, and throat size distributions
(Fig. 3). Aerated concrete had the largest grains with large numbers of grains
even at the larger radii (Fig. 3A). The grain size distribution of the KBriq
are very similar to Aerated Concrete, with a peak in grain radius of around
250 microns. Bentheimer had the smallest variance in grain size with a peak at
250 microns. The Clay Fired and GSA bricks had a small number of isolated
grains that were all under 500 microns. No grains were extracted from the
wooden beam as all of its solid components were microporous.

When comparing pore size (Fig. 3B), Aerated Concrete had the largest pores
with an even distribution stretching from 10-1500 microns. The other samples
all had peaks between 100 and 200 microns, except the GSA brick which had a
long tail indicative of the large cracks that can be seen throughout the
sample. It is interesting to note that the Wooden Beam shows a peak much
larger than would be expected for its small pores, which is due to the long
pores stretching end to end of the sample and the calculation of pore
equivalent diameter rather than maximal ball radius [10].

The throat size distributions are shown in Fig. 3C. Wooden beam and Bentheimer
have the smallest pore throats with a peak around 20 microns and a very narrow
distribution. Aerated concrete has a very wide distribution with a peak around
100 microns. The rest of the samples peak around 50 microns and have a small
tail at larger throat sizes indicating some heterogeneity within the samples.

Sample | resolution |  macro $\mathbf{\phi}$ |  micro $\mathbf{\phi}$ | solid grain |  connected $\mathbf{\phi}$ |  connected $\mathbf{\phi}$ |  micro $\mathbf{\phi}$ | $h$ | $\mathbf{K_{\mu}}$
---|---|---|---|---|---|---|---|---|---
|  $\mathbf{\mu}$m | vol Frac | vol Frac | vol Frac | macro | macro+micro | avg porosity |  $\mathbf{\mu}$m |  m2
Aerated Concrete | 15 | 0.20 | 0.38 | 0.42 | 0.99 | 0.99 | 0.41 |  5 [3] |  2.75 x $10^{-14}$
Fired Clay Brick | 7 | 0.06 | 0.48 | 0.43 | 0.00 | 1.00 | 0.27 |  6.5 [2] |  1.10 x $10^{-14}$
GSA Brick | 7 | 0.06 | 0.89 | 0.05 | 0.33 | 1.00 | 0.34 |  6.5 [2] |  2.12 x $10^{-14}$
Wooden Beam | 7 | 0.34 | 0.66 | 0.0 | 0.97 | 1.0 | 0.72 | N/A |  2.70 x $10^{-20}$ [32]**
KBriq | 8 | 0.18 | 0.18 | 0.64 | 0.955 | 0.996 | 0.35 | 5* |  1.41 x $10^{-14}$
Bentheimer | 4 | 0.26 | 0 | 0.74 | 0.99 | N/A | N/A | N/A | N/A
Table 1: Porosity and properties of the samples measured using image analysis
and estimated from literature values.*The KBriq samples were estimated to have
a microporous grain size $h$ of 5 $\mathbf{\mu}$m due to its composition of
construction and demolition inert waste. **The permeability of the microporous
phase of wood is analogous to the permeability of a cell wall.  Figure 3:  The
grain size distributions (A), pore size distributions (B), and throat size
distributions (C) of the samples. Note that the grain size distribution of the
wooden beam is excluded because it does not contain grains.

###  5.2 Numerical Simulation: Permeability with Darcy-Brinkman-Stokes

The permeability $K$ was calculated on a $400^{3}$ voxel subvolume of the
images both including and excluding the microporous phase using the DBS solver
in GeoChemFoam. When excluding the microporosity both the solid and
microporous phases were given a porosity of 0.0001 and a permeability of
$10^{-20}$. When including the microporosity the average porosity of the
microporosty was used along with the Kozeny-Carmen calculated $K_{micro}$,
with the solid values remaining 0.0001 and $10^{-20}$ $m^{2}$. All calculated
$\mathbf{\phi}$, $K$, $L$, and $U{{}_{D}}$ for each scenario are shown in
Table 2. The streamlines for each scenario and the pds of velocity are shown
in Fig.4.

It is observed in Aerated Concrete (Fig.4. A, B, C) that the macro porous
regions are highly connected with fast flow in many of the pores resulting in
a moderate peak in the velocity PDF at high velocities. This peak is reduced
and flattened with the inclusion of microporosity indicating that the binder
is contributing to flow in some regions. The permeability of Aerated concrete
was the highest of all the samples at 5.4 x $10^{-11}$ $m^{2}$ , which is a
reflection of the large pores and throat sizes, and increases less then 1
percent with the addition of microporosity. The Clay Fired Brick (Fig.4. D, E)
is not connected in the macro pores, however, when microporosity is included a
low permeability of 5.9 x $10^{-14}$ $m^{2}$ is calculated and it is observed
that flow is slow through the clay and then increases when an unconnected pore
or grain is encountered. Nevertheless the difference between the minimum and
maximum velocities is not more than one order of magnitude and thus only one
peak is seen in the PDF. The GSA Brick (Fig.4.F, G, H) shows only a single
connected path along the fracture with a permeability of 2 x $10^{-14}$
$m^{2}$.. The fraction of the sample volume containing high velocities is so
low that the high velocities barely register on the PDF. When microporosity is
included, flow is fast in much of the sample as the micropores connect
previously unconnected fracture networks and this is reflected in the
permeability increase of 3.5x and the PDF with a strong peak at high
velocities and a tail leading to lower velocities.

The Wooden Beam (Fig.4. I, J, K) has pores that stretch end to end in the
direction of flow, resulting in a flow field that is analogous to a bundle of
tubes with a permeability of 5.9 x $10^{-13}$ $m^{2}$. For the Stokes case
this results in several small peaks and troughs that depend on the various
pore radii, with the few pores that are slightly bigger than the others
becoming the preferential flow paths. When microporosity is added some of the
smaller tubes become more connected and another peak is seen in the mid-range
velocities and a small 1.2x increase in permeability. Bentheimer (Fig.4. L,M )
does not contain sub resolution porosity and is well connected resulting in
only a single peak at high velocities and a permeability of 3.8 x $10^{-12}$
$m^{2}$.

Finally, the KBriq (Fig.4. Q,R,S ) a small peak at high velocities with Stokes
flow and a similar permeability of 4 x $10^{-12}$ $m^{2}$. The peak in the PDF
is then shifted to lower velocities and widened when microporosity is included
due to connecting some stagnant pores that then contribute to over all flow,
however, these connected change the permeability by less than 1 percent, which
is attributed to the already high permeability and relatively homogeneous
arrangement of pores and grains.

It is therefore concluded that if the macropores are well connected, there are
only minor effects on the flow field and permeability. However, if the
macropores are poorly connected or disconnected then microporosity must be
included in the flow calculations to solve for permeability.

Sample | $K_{Stokes}$ | $K_{DBS}$ | $\mathbf{\phi}_{Stokes}$ | $\mathbf{\phi}_{DBS}$ | $L_{Stokes}$ | $L_{DBS}$ | $U_{D(Stokes)}$ | $U_{D(DBS)}$
---|---|---|---|---|---|---|---|---
| $[m^{2}]$ | $[m^{2}]$ | [-] | [-] | [m] | [m] | $[m.s^{-1}]$ | $[m.s^{-1}]$
Aerated Concrete |  5.386 x $10^{-11}$ |  5.626 x $10^{-11}$ | 0.210 | 0.371 |  4.535 x $10^{-5}$ |  3.483 x $10^{-5}$ |  2.634 x $10^{-3}$ |  2.751 x $10^{-3}$
Fired Clay Brick | 0 (unconnected) |  5.92 x $10^{-14}$ | N/A | 0.365 | N/A |  1.319 x $10^{-6}$ | N/A |  1.215 x $10^{-1}$
GSA Brick |  2.022 x $10^{-14}$ |  7.13 x $10^{-14}$ | 0.039 | 0.359 |  2.033 x $10^{-6}$ |  1.26 x $10^{-6}$ |  1.20 x $10^{-2}$ |  4.230 x $10^{-2}$
Wooden Beam |  5.866 x $10^{-13}$ |  6.856 x $10^{-13}$ | 0.353 | 0.819 |  3.64 x $10^{-6}$ |  2.589 x $10^{-6}$ |  2.103 x $10^{-3}$ |  2.460 x $10^{-3}$
KBriq |  4.033 x $10^{-12}$ |  4.125 x $10^{-12}$ | 0.179 | 0.245 |  1.33 x $10^{-5}$ |  1.161 x $10^{-5}$ |  6.239x $10^{-3}$ |  6.382 x $10^{-3}$
Bentheimer |  3.750 x $10^{-12}$ | N/A | 0.226 | N/A |  1.150 x $10^{-5}$ | N/A |  5.80 x $10^{-3}$ | N/A
Table 2: Permeability and flow properties of the samples as calculated using
GeoChemFoam’s Darcy-Brinkman-Stokes Flow Solver (simpleDBSFoam) in 4003
subvolumes.  Figure 4:  The velocity streamlines of each sample both excluding
(Stokes) [A,F,I,L,N] and including (DBS) microporosity [B,D,G,J,O]. The
probability density functions of $U/U_{avg}$ for each simulation are shown in
column 3 [C,E,H,K,M,P].

###  5.3 Numerical Simulations: Steady State Heat Transfer

As the true thermal conductivity of a microporous solid is dependent on a
structure that can not be known in sub resolution region without imaging at
another scale, harmonic or arithmetic averaging of that region is a reasonable
approximation. However, it has yet to be investigated as to if there is a
difference in overall calculated $\mathbf{\kappa}_{eff}$ when each method is
used and indeed which method produces more accurate results under which
circumstances. For each sample two solver configurations were then used: (1)
$\mathbf{\kappa}_{eff}$ calculated with arithmetic averaging of
$\mathbf{\kappa}_{s}$ and $\mathbf{\kappa}_{f}$ for the microporous solid
phase and (2) $\mathbf{\kappa}_{eff}$ calculated with harmonic averaging
$\mathbf{\kappa}_{s}$ and $\mathbf{\kappa}_{f}$ for the microporous phase.
Input $\mathbf{\kappa}_{s}$ values for the simulations are shown in Table 3.
Heat Transfer was then simulated on the samples for fluids ranging from
$\mathbf{\kappa}_{f}$ = 0.0006 to $\mathbf{\kappa}_{f}$ = 666. Fig. 6 shows
the J flux plotted for each of the samples for the highest and lowest values
of $\mathbf{\kappa}_{f}$. The results of each simulation shown in Fig. 5 are
then compared to the theoretical value of $\mathbf{\theta}_{eff}$,
corresponding to the theoretical limit as calculated with the same method:

$\mathbf{\theta}_{eff(harmonic)}=\frac{\mathbf{\kappa}_{f}\mathbf{\kappa}_{s}}{((1-\mathbf{\phi}_{total})\mathbf{\kappa}_{f}+\mathbf{\kappa}_{s}\mathbf{\phi}_{total})}$
(10)
$\mathbf{\theta}_{eff(arithmatic)}=(1-\mathbf{\phi}_{total})\mathbf{\kappa}_{s}+\mathbf{\phi}_{total}\mathbf{\kappa}_{f}$
(11)

It is important to note that while in typical systems the pore fluid will be
dominated by air or water with a low $\mathbf{\kappa}_{f}$, in extreme weather
events such as floods this may not always be the case with penetration of mud
or contaminant laden fluids, possibly even metal particulates. Thus we have
included a large range of $\mathbf{\kappa}_{f}$ values to span a range of
possible pore fluids.

Figure 5:  The $\mathbf{\kappa}_{eff}$ and $\mathbf{\theta}_{eff}$ normalised
to $\mathbf{\kappa}_{s}$ for each of the building materials using both
harmonic and arithmetic averaging of the components of the solid phase.
Figure 6:  The heat flux J of the simulations for $\mathbf{\kappa}_{f}$ =
0.0006 (columns 1 and 3) and $\mathbf{\kappa}_{f}$ = 666 (columns 2 and 4)
using $\mathbf{\kappa}_{\mathbf{\mu}(arithmetic)}$ and
$\mathbf{\kappa}_{\mathbf{\mu}(harmonic)}$ for all of the samples. For visual
illustration purposes, each simulation is scaled based on linear scaling for
arithmetic cases and logarithmic scaling in harmonic cases with the same
maximum value where red is high J and blue is low J.

In Bentheimer (Fig. 5. E ), it is observed that all simulations the
$\mathbf{\kappa}_{eff}$ tracks the $\mathbf{\theta}_{eff(arithmatic)}$. This
is because there is no porosity in the solid and thus fluid inside the solid
phase to modify the $\mathbf{\kappa}_{\mathbf{\mu}}$ of the solid. In
addition, both grains and pores are well connected so the heat transfer can be
modelled conceptually as a combination of two stacked phases as is analogous
to arithmetic averaging.

For Aerated Concrete good agreement is observed between
$\mathbf{\kappa}_{eff}$ and $\mathbf{\theta}_{eff}$ when $\mathbf{\kappa}_{f}$
¡ $\mathbf{\kappa}_{s}$ . However, when $\mathbf{\kappa}_{f}$ ¿
$\mathbf{\kappa}_{s}$ both simulations $\mathbf{\kappa}_{eff}$ track the
$\mathbf{\theta}_{eff(arithmatic)}$ . This is because the fluid pathways are
large and well connected with little tortuosity, so when $\mathbf{\kappa}_{f}$
is high, those pathways can be used for heat transport and the porous medium
can be effectively modelled arithmetically. However, when
$\mathbf{\kappa}_{f}$ is low, most of the heat is transported through the
grains and then the simulations follow the exact law in the solver.
Additionally, using the harmonic function pulls the
$\mathbf{\kappa}_{\mathbf{\mu}}$ towards the smallest of $\mathbf{\kappa}_{f}$
and $\mathbf{\kappa}_{s}$ . The solid inside the microporous phase thus
behaves like a barrier.

For the GSA Brick good agreement is observed between $\mathbf{\theta}_{eff}$
and $\mathbf{\kappa}_{eff}$ at all $\mathbf{\kappa}_{f}$ except for a small
deviation at high $\mathbf{\kappa}_{f}$ in the harmonic case when the
fractures act as a heat conduit and some pull towards
$\mathbf{\theta}_{eff(arithmatic)}$ is seen. However, the fractures are only a
small portion of the sample and thus this deviation is only small. The Clay
Fired Brick shows very similar behavior where vugs and cracks act as faster
heat conduits. However, there are enough of them to impede heat flow in the
solid when $\mathbf{\kappa}_{f}$ ¡ $\mathbf{\kappa}_{s}$ and thussome
deviation of $\mathbf{\kappa}_{eff(harmonic)}$ from
$\mathbf{\theta}_{eff(harmonic)}$ is observed in these cases.

In the Wooden Beam , $\mathbf{\theta}_{eff}$ and $\mathbf{\kappa}_{eff}$ are
in agreement when $\mathbf{\kappa}_{f}$ ¡ $\mathbf{\kappa}_{s}$. However, when
$\mathbf{\kappa}_{f}$ is high, the harmonic simulations track
$\mathbf{\theta}_{eff(arithmatic)}$. This is attributed to the long and thin
pores that act as direct conduits. Yet, these pores do not stretch completely
end to end, so the fluid has to pass through some solid and in the harmonic
case this results in much lower $\mathbf{\kappa}_{eff}$ for the solid phase.

For the KBriq a similar trend to Aerated concrete when $\mathbf{\kappa}_{f}$ ¿
$\mathbf{\kappa}_{s}$ is observed. Nevertheless, there is a smaller influence
of the microprosity when $\mathbf{\kappa}_{f}$ ¡ $\mathbf{\kappa}_{s}$, which
is attributed to a lower overall porosity and thus greater ability of the
solid phase to transmit heat through the grains.

It is thus concluded that in many cases knowledge of both the macro and micro
structures are required when choosing how to model heat transfer. In cases
where there is no microporosity and the solid and grains are well-connected,
the arithmetic function should be used. Still, when microporosity is present,
the structure of the macropores must be considered. The macro pores themselves
can act as either barriers or conduits to heat flow depending on their
arrangement and the ratio between $\mathbf{\kappa}_{f}$ and
$\mathbf{\kappa}_{s}$. Furthermore, use of arithmetic or harmonic averaging of
the microporous phase can change the resulting $\mathbf{\kappa}_{eff}$ by
several orders of magnitude in some cases. It is thus posited that some
knowledge of the underlying nanostructure within the microporous phase must be
incorporated in order to choose an accurate solver method.

Sample | $\mathbf{\phi}$ | $\mathbf{\kappa}_{s}$ | $\mathbf{\gamma}_{s}$ | $\mathbf{\rho}_{s}$ | U (arithmetic) | U (harmonic) | t (arithmetic) | t (harmonic)
---|---|---|---|---|---|---|---|---
| $[-]$ | $kW/m/K$ | $kJ/kg/K$ | $kg/m^{3}$ | $W/m^{2}/K$ | $W/m^{2}/K$ | hr | hr
Aerated Concrete | 0.372 | 1.4 | 0.96 | 2400 | 0.14 | 0.29 | 5.67 | 11.78
Fired Clay Brick | 0.365 | 1.0 | 1.05 | 1362 | 0.15 | 0.57 | 3.90 | 14.36
GSA Brick | 0.359 | 1.0 | 1.05 | 1362 | 0.16 | 0.62 | 3.97 | 15.85
Wooden Beam | 0.819 | 0.2 | 2.00 | 600 | 1.18 | 1.44 | 7.16 | 8.75
KBriq | 0.179 | 1.4 | 0.96 | 2400 | 0.11 | 0.15 | 5.83 | 7.90
Bentheimer | 0.226 | 3.3 | 0.92 | 2650 | 0.07 | 0.08 | 3.54 | 4.17
Table 3: Thermal and porosity properties of the samples used in the heat
transfer simulations. As the makeup of a KBriq is proprietary, the same values
for density and specific heat capacity as Aerated Concrete were used. The
calculated $\mathbf{U_{eff}}$ for air is shown for both arithmetic and
harmonic as well as the time for heat to diffuse through a 0.1 m wall.

###  5.4 Upscaling to real systems

It is then possible to extrapolate these results to a real system by
considering a wall of 0.1m thickness $\mathbf{L}$. Here the $\mathrm{U-value}$
of the system $\mathrm{[Wm^{-2}K^{-1}]}$ can be approximated as
$\mathbf{L}/\kappa_{eff}$. The time $\mathbf{t}$ for heat to diffuse through
the wall is equal to $\mathbf{L^{2}/D}$ where $\mathbf{L}$ is the wall
thickness and $\mathbf{D}$ =
$\mathbf{\kappa}_{eff}/[\rho_{s}(1-\theta)\gamma_{s}+\rho_{f}\theta\gamma_{f}]$
where $\mathbf{\gamma}$ is the specific heat capacity in
$\mathrm{[kJkg^{-1}K^{-1}]}$. A summary table of the calculate times using
both the Keff from arithmetic and harmonic averaging and resultant U values
and diffusion time are shown in Table 3. Here it is shown that, depending on
geometry and the presence of microporosity the difference in calculated U
value and heat diffusion time can vary between almost no change in the case of
Bentheimer (with no microporosity), to almost a 5-fold change in the cases of
the Fired Clay and GSA Bricks, where microporosity dominates. This again
illustrates the importance of understanding the microporous structure for
effective upscaling of these parameters.

##  6 Conclusions

A novel workflow for upscaling permeability, flow, and thermal conductivity in
a range of porous building materials is presented. Six building material were
imaged with micro-CT and analyzed for microporosity and connectivity. The flow
and permeability solved on the images both with and without microporosity.
Finally, the effective thermal conductivity was solved using both harmonic and
arithmetic averaging of the microporous phase and the U value and diffusive
time was calculated for a 0.1m thick wall.

A strong dependence is found on the connectivity of the pore structure for
both permeability and thermal conductivity. When the pores were well connected
the microporous phase did not effect permeability appreciably. However, in
cases where the microporosity connects otherwise disconnected macropores, it
must be included to get an accurate measure of permeability and flow. Indeed,
in some cases where the macropores are wholly disconnected, permeability
cannot be computed without including the microporous phase.

Thermal conductivity was also effected by local heterogeneity and the
arrangement of pores, where macro pores act as either heat conduits or
barriers depending on the ratio of thermal conductivity between the
microporous solid and liquid phases. When the macropores are not well-
connected, the choice of arithmetic or harmonic averaging of thermal
conductivity in the sub-resolution porosity can change the effective thermal
conductivity by order of magnitude. Furthermore, this in turn can change the
calculated U-values and heat diffusion timescale of the bulk materials by up
to half an order of magnitude. It is thus imperative that the nano structure
is investigated to inform this decision and will be a target for future work.

This is the first study to combine micro-CT imaging and multi-scale modelling
of flow and heat transfer that has applications in the sustainable building
industry. These techniques open up the possibility of using this workflow to
streamline the design custom materials for optimum permeability and insulation
properties for individual use cases. In addition, this workflow could easily
be adapted to understand and improve designs in other industries that use
porous materials such as fuel cells and batteries technology, lightweight
materials and insulation, and semiconductors as well as multi-scale structures
from the natural world such as termites nests[37].

##  7 Acknowledgements

This work was generously funded by the EPSRC project EP/P031307/1. H.P.M. and
K.M.H. would also like to thank the Glasgow School of Art for permission to
use their samples.

## References

  * Armstrong et al. [2016] Armstrong, R.T., McClure, J.E., Berrill, M.A., Rücker, M., Schlüter, S., Berg, S., 2016\.  Beyond darcy’s law: The role of phase topology and ganglion dynamics for two-fluid flow.  Physical Review E 94, 043113\.
  * Baspinar et al. [2010] Baspinar, M.S., Demir, I., Orhan, M., 2010.  Utilization potential of silica fume in fired clay bricks.  Waste management & research 28, 149–157.
  * Bentz et al. [1999] Bentz, D.P., Garboczi, E.J., Haecker, C.J., Jensen, O.M., 1999\.  Effects of cement particle size distribution on performance properties of portland cement-based materials.  Cement and concrete research 29, 1663–1671.
  * Bentz et al. [2000] Bentz, D.P., Quenard, D., Kunzel, H.M., Baruchel, J., Peyrin, F., Martys, N.S., Garboczi, E., 2000.  Microstructure and transport properties of porous building materials. ii: Three-dimensional x-ray tomographic studies.  Materials and Structures 33, 147–153.
  * Beucher and Meyer [1993] Beucher, S., Meyer, F., 1993\.  The morphological approach to segmentation: the watershed transformation.  Mathematical morphology in image processing 34, 433–481.
  * Bicer and Devecioglu [2023] Bicer, A., Devecioglu, A.G., 2023\.  Modelling for determining the thermal conductivity of porous solid materials.  Magazine of Concrete Research , 1–8.
  * Blunt et al. [2013] Blunt, M.J., Bijeljic, B., Dong, H., Gharbi, O., Iglauer, S., Mostaghimi, P., Paluszny, A., Pentland, C., 2013\.  Pore-scale imaging and modelling.  Advances in Water resources 51, 197–216.
  * Buades et al. [2011] Buades, A., Coll, B., Morel, J.M., 2011.  Non-local means denoising.  Image Processing On Line 1, 208–212.
  * Chung et al. [2016] Chung, S.Y., Han, T.S., Kim, S.Y., Kim, J.H.J., Youm, K.S., Lim, J.H., 2016\.  Evaluation of effect of glass beads on thermal conductivity of insulating concrete using micro ct images and probability functions.  Cement and Concrete Composites 65, 150–162.
  * Dong and Blunt [2009] Dong, H., Blunt, M.J., 2009\.  Pore-network extraction from micro-computerized-tomography images.  Physical review E 80, 036307\.
  * Dos Santos [2003] Dos Santos, W.N., 2003.  Effect of moisture and porosity on the thermal properties of a conventional refractory concrete.  Journal of the European Ceramic Society 23, 745–755.
  * Etxeberria et al. [2007] Etxeberria, M., Marí, A.R., Vázquez, E., 2007.  Recycled aggregate concrete as structural material.  Materials and structures 40, 529–541.
  * Faris et al. [2020] Faris, A., Maes, J., Menke, H., 2020.  An investigation into the upscaling of mineral dissolution from the pore to the core scale, in: ECMOR XVII, European Association of Geoscientists & Engineers. pp. 1–15.
  * Gan et al. [2017] Gan, V.J., Cheng, J.C., Lo, I.M., Chan, C.M., 2017\.  Developing a co2-e accounting method for quantification and analysis of embodied carbon in high-rise buildings.  Journal of cleaner production 141, 825–836.
  * Grose [2022] Grose, T.K., 2022.  Grown to last.  ASEE Prism 31, 28–31.
  * Guo et al. [2011] Guo, L., Guo, L., Zhong, L., Zhu, Y., 2011.  Thermal conductivity and heat transfer coefficient of concrete.  Journal of Wuhan University of Technology-Mater. Sci. Ed. 26, 791–796.
  * IEA [2019] IEA, U., 2019.  Global status report for buildings and construction 2019, in: NA, IEA Paris, France. p. NA.
  * Lai et al. [2015] Lai, P., Moulton, K., Krevor, S., 2015.  Pore-scale heterogeneity in the mineral distribution and reactive surface area of porous rocks.  Chemical Geology 411, 260–273.
  * Liu and Wu [2016] Liu, Z., Wu, H., 2016.  Pore-scale study on flow and heat transfer in 3d reconstructed porous media using micro-tomography images.  Applied Thermal Engineering 100, 602–610.
  * Maes and Menke [2020] Maes, J., Menke, H.P., 2020\.  A bespoke openfoam toolbox for multiphysics flow simulations in pore structures, in: Proceedings of the 17th International Conference on Flow Dynamics (ICFD2020), pp. 1–15.
  * Maes and Menke [2021a] Maes, J., Menke, H.P., 2021a.  Geochemfoam: Direct modelling of flow and heat transfer in micro-ct images of porous media.  arXiv preprint arXiv:2110.03311 .
  * Maes and Menke [2021b] Maes, J., Menke, H.P., 2021b.  GeoChemFoam: Direct Modelling of Multiphase Reactive Transport in Real Pore Geometries with Equilibrium Reactions.  Transport in porous media doi:10.1007/s11242-021-01661-8.
  * Maes and Menke [2021c] Maes, J., Menke, H.P., 2021c.  Geochemfoam: Operator splitting based time-stepping for efficient volume-of-fluid simulation of capillary-dominated two-phase flow.  arXiv:2105.10576 doi:arXiv:2105.10576.
  * Medero and Chapman [2020] Medero, G., Chapman, S., 2020\.  Construction units in form of bricks, blocks or tiles made from recyclable materials and by-products, methods of making the construction units and their use.  US Patent 10,669,205.
  * Menke et al. [2019] Menke, H., Gao, Y., Linden, S., Andrew, M., 2019\.  Using nano-xrm and high-contrast imaging to inform micro-porosity permeability during stokes-brinkman single and two-phase flow simulations on micro-ct images.  EarthArXiv .
  * Menke et al. [2021] Menke, H.P., Maes, J., Geiger, S., 2021.  Upscaling the porosity–permeability relationship of a microporous carbonate for darcy-scale flow with machine learning.  Scientific reports 11, 1–10.
  * Metz et al. [2005] Metz, B., Davidson, O., De Coninck, H., Loos, M., Meyer, L., 2005.  IPCC special report on carbon dioxide capture and storage.  Cambridge: Cambridge University Press.
  * Noiriel [2015] Noiriel, C., 2015.  Resolving time-dependent evolution of pore-scale structure, permeability and reactivity using x-ray microtomography.  Reviews in Mineralogy and Geochemistry 80, 247–285.
  * Nunes et al. [2017] Nunes, C., Pel, L., Kuneckỳ, J., Slížková, Z., 2017.  The influence of the pore structure on the moisture transport in lime plaster-brick systems as studied by nmr.  Construction and Building Materials 142, 395–409.
  * OpenCFD [2016] OpenCFD, 2016.  OpenFOAM, the open source cfd toolbox, User Guide.  OpenCFD Ltd.
  * Panerai et al. [2017] Panerai, F., Ferguson, J.C., Lachaud, J., Martin, A., Gasch, M.J., Mansour, N.N., 2017\.  Micro-tomography based analysis of thermal conductivity, diffusivity and oxidation behavior of rigid and flexible fibrous insulators.  International Journal of Heat and Mass Transfer 108, 801–811.
  * Petty and Palin [1983] Petty, J., Palin, M.A., 1983\.  Permeability to water of the fibre cell wall material of two hardwoods.  Journal of experimental botany 34, 688–693.
  * Reynolds et al. [2017] Reynolds, C.A., Menke, H., Andrew, M., Blunt, M.J., Krevor, S., 2017.  Dynamic fluid connectivity during steady-state multiphase flow in a sandstone.  Proceedings of the National Academy of Sciences 114, 8187–8192.
  * Robayo-Salazar et al. [2017] Robayo-Salazar, R.A., Rivera, J.F., de Gutiérrez, R.M., 2017.  Alkali-activated building materials made with recycled construction and demolition wastes.  Construction and building materials 149, 130–138.
  * Shen et al. [2023] Shen, L., Di Luzio, G., Cao, M., Ren, Q., Ren, X., Jiang, M., Zhu, D., Yao, X., 2023.  Insights and theoretical model of thermal conductivity of thermally damaged hybrid steel-fine polypropylene fiber-reinforced concrete.  Cement and Concrete Composites 138, 105001.
  * Siegert et al. [2021] Siegert, M., Gurris, M., Saenger, E.H., 2021.  Validation suite for numerical solvers calculating effective thermal conductivity in porous media.  Journal of Applied Geophysics 189, 104323.
  * Singh et al. [2019] Singh, K., Muljadi, B.P., Raeini, A.Q., Jost, C., Vandeginste, V., Blunt, M.J., Theraulaz, G., Degond, P., 2019\.  The architectural design of smart ventilation and drainage systems in termite nests.  Science Advances 5, eaat8520.
  * Soulaine et al. [2016] Soulaine, C., Gjetvaj, F., Garing, C., Roman, S., Russian, A., Gouze, P., Tchelepi, H.A., 2016.  The impact of sub-resolution porosity of x-ray microtomography images on the permeability.  Transport in porous media 113, 227–243.
  * Stamm [2002] Stamm, A.J., 2002.  Density of wood substance, adsorption by wood, and permeability of wood.  The Journal of Physical Chemistry 33, 398–414.
  * Twinn et al. [2019] Twinn, R., Desai, K., Box, P., 2019.  Net zero carbon buildings: a framework definition.  NA .
  * Xiao et al. [2023] Xiao, T., Zhang, Q., Yang, X., Hooman, K., Li, G., 2023\.  Influence of solder condition on effective thermal conductivity of two-directional random fibres: Pore-scale simulation.  International Journal of Heat and Mass Transfer 202, 123715.
  * Yang et al. [2019] Yang, H., Zhang, L., Liu, R., Wen, X., Yang, Y., Zhang, L., Zhang, K., Askari, R., 2019.  Thermal conduction simulation based on reconstructed digital rocks with respect to fractures.  Energies 12, 2768\.
  * Zhang et al. [2015] Zhang, W., Min, H., Gu, X., Xi, Y., Xing, Y., 2015\.  Mesoscale model for thermal conductivity of concrete.  Construction and Building Materials 98, 8–16.
  * Zhu et al. [2023] Zhu, J., Wang, Y., Xiao, R., Yang, J., 2023.  Multiscale theoretical model of thermal conductivity of concrete and the mesoscale simulation of its temperature field.  Journal of Materials in Civil Engineering 35, 04022359.