which is why this will be the first point to be addressed in our future work.

####  8.5.5 Training and Identifying Best Regressors

Training and testing ML regressors followed the same procedure outlined in
Section 8.5.3. The only difference was the fact that we are now aiming to
predict the CN associated with each sample, as opposed to its discrete label.
Given that the degree of clogging is a non-discrete latent variable, it can be
argued that using regressors to predict a CN might actually be more useful
than binary classification. However, it is important to consider that the CN
is an abstract metric; it is not obvious how this number’s magnitude reflects
actual physical processes. For this reason we make no value-judgement as to
which approach might be more useful and instead present the results of both
types of predictors.

The results of our training, testing, and optimization procedure are shown in
Table 8.3, where we evaluate regressor performance by quantifying the
$\mathrm{R^{2}}$ and Mean Absolute Error (MAE) obtained from plotting a
“predicted CN” vs “actual CN” curve (Figure 8.6).

Table 8.3:  Clogging prediction performance of the chosen ML regressors. The
highlighted section represents the regressor with the best performance.

Table 8.3 clearly shows that decision tree-based ML models are, once again,
the best predictors for clogging mechanisms in porous media, closely followed
by neural networks (MLPs). A description of the best optimized model (Random
Forest) can be found in Section 8.6 .

Figure 8.6:  Optimized Random Forest regression results.

####  8.5.6 Regression Feature Analysis

Just as we did for classifiers, we now turn to evaluate feature importance for
our best regression algorithm (Random Forest) through a permutation feature
importance algorithm. The results shown in Table 8.4 are surprising,
especially when compared to the ones presented in Section 8.5.4. The top two
features follow the same pattern we saw before, where $d/D$ and $F_{nd}$ are
once again the most significant features. However, this time, the ratio
between particle size and grain size ($d/G$) takes the third spot, followed by
the particle-wall attraction ($PW_{nd}$), the particle-geometric thickness
ratio ($d/T$), and the porosity ($\phi$). The previously-unseen correlation
between $d/G$ might be explained by the fact that we used the symmetry of the
particle velocity distribution to construct the CN. Larger grains/obstacles
produce larger deviations in the particles’ velocity than their smaller
counterparts, effectively changing the CN without actually signifying the
presence of clogging in the porous medium. A similar thing can be said about
the effect of porosity on the CN, where less porous systems may exhibit more
velocity fluctuations than more porous ones.

Table 8.4:  Tabulated relative feature importance for the top 6 regression
features. The rightmost column identifies the score metrics of the model if it
is trained and tested exclusively with the top “n” features.

This analysis then begs the question of whether the importance of $d/G$ and
$\phi$ is actually predictive of clogging mechanisms or just an artifact of
our simulation characterization procedure. We believe that the answer is the
later, as the marginal increase in the prediction accuracy of these two
feature seen in Table 8.5.4 is very small compared to their counterparts
(about 0.01 each). Furthermore, if we train and test our model without $d/G$
and $\phi$ we only reduce the overall $\mathrm{R^{2}}$ score by 0.02
($\mathrm{R^{2}}=0.90$), while also replicating the original feature rankings
obtained by the classifiers. This implies that even though $d/G$ and $\phi$
can be seen as significant by our permutation feature ranking algorithm, most
of their effects on the CN can be captured by other features.

###  8.6 Conclusions

In this chapter, we combined direct numerical simulations with machine
learning approaches to predict clogging in heterogeneous porous media. In
order to obtain the necessary data for training said algorithms we developed a
computational workflow designed to simulate particle flow through randomly-
generated porous media over thousands of different experimental and parametric
conditions. One particular challenge that arose was the fact that clogging is
a continuous process, not a discrete one. For this reason, we also developed
the concept of a “Clogging Number” in order to properly characterize the level
of clogging in our computational simulations. Furthermore, in order to
standardize our results we developed 11 non-dimensional training features that
can be measured a-priori from any system involving the flow of solid particles
through a static obstacle field.

After training, testing, and optimizing several classifiers and regressors we
concluded that the best performing classifier was a Multi-Layer Perceptron
with a logistic activation function, 175 hidden layers and an L2
regularization term of 0.05. This neural network was able to achieve 0.96 and
0.94 labeling accuracy and precision, respectively. In turn, the best
regressor was a Random Forest regressor with 100 trees, a Mean Squared Error
split criteria, minimum sample split of 2, and no maximum depth. This
regressor achieved an $\mathrm{R^{2}}$ value of 0.92 when used to predict the
samples’ CN.

Feature importance analysis of both Machine Learning approaches showed that
the most predictive features for clogging in porous media are, in order of
importance: the ratio between particle size and pore size, the flux of
particles through the porous medium, particle-particle/wall attraction forces,
and fluid viscosity. They also showed that fluid velocity and grain
heterogeneity did not play a significant role in determining whether a system
clogs or not, a point that we attributed to the fact that we did not probe the
11-dimensional parameter space where these features are relevant.

The result of this investigation is a generalized clogging prediction
algorithm that can accurately predict clogging in heterogeneous porous media,
a tool which we hope can help improve the design process of engineered porous
systems. This is the first time that anyone has probed such a large parametric
phase space and/or applied Machine Learning to try and describe this difficult
problem, to the best of our knowledge. However, additional work would be
required the evaluate the ability of our models to predict clogging in most
physical systems. To do so we would need to fine-tune our clogging
characterization algorithm and expand our parameter space into different
length scales, different particle shapes, different grain geometries, and a
larger range of fluid velocities and simulation times. Lastly, and arguably
the most important step, we would need to use actual experimental data to test
(or maybe even train) these models. We hope that this investigation spurns
further work into integrating Machine Learning approaches with numerical
simulations in order to probe and characterize stochastic physical phenomena.
[75mm] In science one tries to tell people, in such a way as to be understood
by everyone, something that no one ever knew before. But in poetry, it’s the
exact opposite. Paul Dirac

##  Chapter 9 Conclusions

In this dissertation, we derived, implemented, benchmarked, and showcased a
novel CFD approach for simulation of multiscale multiphase flow within and
around deformable porous media. This micro-continuum modeling framework is
based on elementary physics and was rigorously derived in Chapter 2 through
the method of volume averaging and asymptotic matching. The result is a set of
partial differential equations that approximate the multiphase Volume of Fluid
equations in solid-free regions and multiphase Biot Theory in porous regions.
These equations are valid in every simulated grid cell within a multiphase
porous system, regardless of content, which obviates the need to define
different meshes, domains, or complex boundary conditions within the
simulation. The solver’s numeric and algorithmic development were presented in
Chapter 3. The equations were implemented into hybridPorousInterFoam and
hybridBiotInterFoam, two open-source packages accessible here for free to any
interested party.

Throughout this thesis we show that the Multiphase DBB model can be readily
used to model a large variety of systems, from multiphase flow in static
porous media, to elastic systems under compression, to viscosity- or
capillarity-dominated fracturing systems, all the way up to multiscale wave
propagation in poroelastic coastal barriers (Chapters 4 to 7). In particular,
we used this model to investigate and obtain parametric relationships for: A)
the permeability-clay content relationship in sedimentary rocks (Chapter 4),
and B) the crossover between fluid drainage and fracturing as a function of
wettability, solid deformability, and hydrodynamics (Chapter 7).

We note, however, that the solver presented here cannot be liberally applied
to any porous system, as it comes with the following inherent limitations.
First, closure of the model’s system of equations requires appropriate
constitutive and parametric relations that describe fluid pressure,
permeability, capillarity, and rheology within volume averaged porous regions.
Therefore, the assumptions present in each of these models should be carefully
considered. Second, volume averaging imposes important length scale
restrictions in order to conform to the scale separation hypothesis: the pore
sizes within the averaging volume must be substantially smaller than the
chosen REV, and the REV must be substantially smaller than the macroscopic
length scale. Third, as implemented here, the multiphase DBB framework only
represents continuum-level elastic or plastic solid mechanics that can be
described from an Eulerian frame of reference. As such, it cannot be used to
model large elastic deformations or phenomena originating from sub-REV
heterogeneities such as fluidization or granular mechanics (Meng et al.,,
2020), except insofar as they are captured in an averaged manner at the REV
scale. Fourth, the use of the CSF as a representation of capillary forces
within solid-free regions enforces mass conservation, but it creates a diffuse
fluid-fluid interface that may generate spurious and parasitic currents.

Finally, although the modeling framework developed here opens up significant
new possibilities in the simulation of coupled fluid-solid mechanics, it also
creates a need for the development of constitutive relations describing the
coupling between multiphase flow and poromechanics. Of particular importance
is the formulation of saturation and deformation-dependent solid rheological
models (both plastic and elastic), as well as the rigorous derivation of the
interfacial condition between solid-free and deformable porous regions. In
this study we proposed a suitable approximation for said boundary condition
based on our single-field formulation, the implementation of a wettability
boundary condition, and the previous work done by Neale & Nader, (1974) and
Zampogna et al., (2019). However, the accuracy and validity of such an
approximation is still an open question, one that is at the frontier of our
modeling and characterization capabilities (Qin et al.,, 2020). The derivation
and implementation of said boundary condition, along with the addition of
erosion and chemical reactions into this modeling framework, will be the focus
of subsequent investigations.

We hope that the results developed of this thesis will inspire more work in
this direction, as suitable and accurate CFD models will be key pieces in our
path to meet the world’s steadily increasing water and energy demand.

##  Appendix A Relative Permeability and Capillary Pressure Models

###  A.1 Relative Permeability Models

The two relative permeability models used in this paper and implemented in the
accompanying code depend on the definition of an effective saturation in order
to account for the presence of irreducible saturations within a porous medium,

${\alpha}_{w,eff}\ \ =\
\frac{{\alpha}_{w}-{\alpha}_{w,irr}}{1-{\alpha}_{w,irr}-{\alpha}_{w,irr}\ }$

Here, ${\alpha}_{w,eff}$ is the wetting fluid’s effective saturation, which is
the wetting fluid’s saturation normalized by each fluid’s irreducible
saturation ${\alpha}_{i,irr}.\ $The Brooks & Corey, (1964) model relates each
phase’s relative permeability to saturation through the following expressions:

$k_{r,n}=\ {\left(1-{\alpha}_{w,eff}\right)}^{m}$

$k_{r,w}\ ={\left({\alpha}_{w,eff}\right)}^{m}$

where $m$ is a non-dimensional coefficient that controls how sensitive the
relative permeability is with respect to saturation. The van Genuchten, (1980)
model calculates relative permeabilities in the following way:

$k_{r,n}\ =\ {\left(1-{\alpha}_{w,eff}\
\right)}^{\frac{1}{2}}{\left({\left(1-{\alpha}_{w,eff}\right)}^{\frac{1}{m}}\right)}^{2m}$

$k_{r,w}={\left({\alpha}_{w,eff}\right)}^{\frac{1}{2}}\
{\left(1-{\left(1-{\left({\alpha}_{w,eff}\right)}^{\frac{1}{m}}\right)}^{m}\right)}^{2}$

In this case, $m$ controls how wetting (or non-wetting) the porous medium is
to a given wetting (or non-wetting) fluid. High values of $m$ indicate high
relative permeabilities for the non-wetting fluid, while low values of $m$
indicate very low relative permeabilities for the same fluid.

###  A.2 Capillary Pressure Models

The implemented capillary pressure models also depend on the definition of an
effective wetting-fluid saturation ${\alpha}_{w,pc}$,

${\alpha}_{w,pc}\ =\
\frac{{\alpha}_{w}-{\alpha}_{pc,irr}}{{\alpha}_{pc,max}-{\alpha}_{pc,irr}}\ $

Here, ${\alpha}_{pc,max}$ is the maximum saturation of the wetting fluid and
${\alpha}_{pc,irr}$ is its irreducible saturation. The Brooks & Corey, (1964)
model uses the following expression to calculate the capillary pressures
within a porous medium:

$p_{c}\ =\ p_{c,0}{\left({\alpha}_{w,pc}\right)}^{-\beta}$

where $p_{c,0}$ is the entry capillary pressure, and $\beta$ is a parameter
depending on the pore size distribution. Conversely, the van Genuchten, (1980)
model calculates the capillary pressure with the following relation:

$p_{c}\
={p_{c,0}\left({\left({\alpha}_{w,pc}\right)}^{-\frac{1}{m}}-1\right)}^{1-m}\
$

##  Appendix B Solid Rheology Models

###  B.1 Hershel-Bulkley Plasticity

A Bingham plastic is a material that deforms only once it is under a
sufficiently high stress. After this yield stress is reached, it will deform
viscously and irreversibly. The Herschel-Bulkley rheological model combines
the properties of a Bingham plastic with a power-law viscosity model, such
that said plastic can be shear thinning or shear thickening during
deformation. In OpenFOAM® this model is implemented as follows:

$\boldsymbol{\sigma}={\mu}^{eff}_{s}\left(\ \nabla{\boldsymbol{U}}_{s}+{\
\left(\nabla{\boldsymbol{U}}_{s}\right)}^{T}-\frac{2}{3}\nabla\cdot\left({\boldsymbol{U}}_{s}\mathrm{\
}\boldsymbol{I}\right)\right)\mathrm{\ }$

where ${\mu}^{eff}_{s}$ is the effective solid plastic viscosity, which is
then modeled through a power law expression:

${\mu}^{eff}_{s}={\mathrm{min}\left({\mu}^{0}_{s}\mathrm{\ ,\ \ \
}\frac{\tau}{\eta}+{\mu}_{s}{\eta}^{n-1}\right)\ }$

where$\ {\mu}^{0}_{s}$ is the limiting viscosity (set to a very large value),
$\tau\ $is the yield stress, ${\mu}_{s}$ is the viscosity of the solid once
the yield stress is overcome, $n$ is the flow index ($n=1$ for constant
viscosity), and $\eta$ is the shear rate.

###  B.2 Quemada Rheology Model

The Quemada rheology model (Quemada,, 1977; Spearman,, 2017) is a simple model
that accounts for the fact that the average yield stress and effective
viscosity of a plastic are functions of its solid fraction. These two
quantities are large at high solid fractions and small at low solid fractions,
as described by the following equations:

$\tau={\tau}_{0}{\left(\frac{\left({{\phi}_{s}}/{{\phi}^{max}_{s}}\right)}{\left(1-{{\phi}_{s}}/{{\phi}^{max}_{s}}\boldsymbol{\
}\right)}\right)}^{D}$

${\mu}_{s}=\frac{{\mu}_{0}}{{\left(1-\frac{{\phi}_{s}}{{\phi}^{max}_{s}}\right)}^{2}}$

here, ${\phi}^{max}_{s}$ is the maximum solid fraction possible (perfect
incompressible packing),$\ {\tau}_{0}$ is the yield stress at
${\phi}_{s}={\phi}^{max}_{s}/2$ ,$\ {\mu}_{0}$ is the viscosity of the fluid
where the solid would be suspended at low solid fractions (high fluid
fractions), and $D$ is a scaling parameter based on the solid’s fractal
dimension.

###  B.3 Linear Elasticity

A linear elastic solid assumes that a solid exhibits very small reversible
deformations under stress. Linear elasticity is described by the following
relation:

$\boldsymbol{\sigma}={\mu}_{s}\nabla{\boldsymbol{u}}_{\boldsymbol{s}}+{{\mu}_{s}\left(\nabla{\boldsymbol{u}}_{\boldsymbol{s}}\right)}^{T}+{\lambda}_{s}tr\left(\nabla{\boldsymbol{u}}_{\boldsymbol{s}}\right)I$

where $u_{s}\ $is the solid displacement vector (not to be confused with solid
velocity $U_{s}=\frac{\partial u_{s}}{\partial t}$), and ${\mu}_{s}$ and
${\lambda}_{s}$ are the Lamé coefficients. The implementation of linear
elasticity in OpenFOAM® follows the procedure outlined in Jasak & Weller,
(2000).

##  Appendix C Fracturing Instabilities

The following figures demonstrate how different fracturing patterns can result
from different solid fraction initializations. Here we set up two sets of four
identical experiments. In the first set, the only difference between cases is
the value of the standard deviation of their respective normally-distributed
solid fraction field (all centered at ${\phi}_{s}=0.64$). These experiments
follow the same simulation setup used for the fracturing case shown in Figure
6.4K.

Figure C.1: Effects of the solid fraction field’s standard deviation on
fracturing.

In the second set of experiments we simulated the base case presented in
Figure 6.9 with different solid fraction profiles picked from the same normal
distribution ${\phi}_{s}=0.6\ \pm 0.05$.

Figure C.2: Effects of different solid fraction field initializations on
fracturing.

Figures C.1 and C.2 clearly show that the shape of the created fractures is
dependent on the initial solid fraction distribution.

##  Appendix D Additional Derivation Steps

###  D.1 Skipped Steps in the Derivation of the Fluid Continuity Equation

We start with a basic continuity equation for fluid phase $i$,

$\frac{\partial{\rho}_{i}}{\partial
t}+\nabla\cdot\left({\rho}_{i}\boldsymbol{U}_{i}\right)=0$ (D.1)

where ${\rho}_{i}$ and $\boldsymbol{U}_{i}$ are the fluid’s density and
velocity, respectively. Applying the averaging operators for a volume
containing a solid phase ($s$) and an additional fluid phase ($j$) we get

$\overline{\frac{\partial{\rho}_{i}}{\partial
t}}+\overline{\nabla\cdot\left({\rho}_{i}\boldsymbol{U}_{i}\right)}=0$ (D.2)

Expanding the averaged functions

$\frac{\partial\overline{{\rho}_{i}}}{\partial
t}-\frac{1}{V}\int_{A_{i,s}}{{\rho}_{i}{\boldsymbol{U}}_{s}\cdot{\boldsymbol{n}}_{i,s}dA}-\frac{1}{V}\int_{A_{i,j}}{{\rho}_{i}{\boldsymbol{U}}_{i}\cdot{\boldsymbol{n}}_{i,j}dA}\\\
+\nabla\cdot\left(\overline{{\rho}_{i}{\boldsymbol{U}}_{i}}\right)+\frac{1}{V}\int_{A_{i,s}}{{\rho}_{i}{\boldsymbol{U}}_{i}\cdot{\boldsymbol{n}}_{i,s}dA}+\frac{1}{V}\int_{A_{i,j}}{{\rho}_{i}{\boldsymbol{U}}_{i}\cdot{\boldsymbol{n}}_{i,j}dA}=0$
(D.3)

Canceling like terms we then obtain

$\frac{\partial\overline{{\rho}_{i}}}{\partial
t}+\nabla\cdot\left(\overline{{\rho}_{i}{\boldsymbol{U}}_{i}}\right)+\frac{1}{V}\int_{A_{i,s}}{{\rho}_{i}{(\boldsymbol{U}}_{i}-{\boldsymbol{U}}_{s})\cdot{\boldsymbol{n}}_{i,s}dA}=0$
(D.4)

Now, given that the fluid and solid velocities are equal at the fluid-solid
interface (${\boldsymbol{U}}_{i}={\boldsymbol{U}}_{s}$ at $A_{i,s}$) we can
cancel the terms within the integral. This results in Equation 2.17 presented
in the main text.

$\frac{\partial\overline{{\rho}_{i}}}{\partial
t}+\nabla\cdot\left(\overline{{\rho}_{i}{\boldsymbol{U}}_{i}}\right)=0$ (D.5)

###  D.2 Derivation of the Single-Field Viscous Stress Tensor

Following the main derivation in Section 2.5, the single-field viscous stress
tensor should have the following form:

$\overline{\boldsymbol{S}}={\alpha}_{w}{\mu}_{w}\left(\nabla{\overline{\boldsymbol{U}}}^{w}_{w}+{\left(\nabla{\overline{\boldsymbol{U}}}^{w}_{w}\right)\
}^{T}\right)+{\alpha}_{n}{\mu}_{n}\left(\nabla{\overline{\boldsymbol{U}}}^{n}_{n}+{\left(\nabla{\overline{\boldsymbol{U}}}^{n}_{n}\right)}^{T}\right)$
(D.6)

Additionally, the definition of the single field and relative velocities (see
Sections 2.3 and 2.4) state that

${\overline{\boldsymbol{U}}}^{w}_{w}={\phi}^{-1}_{f}{\boldsymbol{U}}_{f}+{\alpha}_{n}{\boldsymbol{U}}_{r}$
(D.7)
${\overline{\boldsymbol{U}}}^{n}_{n}={\phi}^{-1}_{f}{\boldsymbol{U}}_{f}-{\alpha}_{w}{\boldsymbol{U}}_{r}$
(D.8)

Taking the gradient within the free fluid (where the viscous stress tensor is
dominant), we get,

${\nabla\overline{\boldsymbol{U}}}^{w}_{w}=\nabla{\boldsymbol{U}}_{f}+{\alpha}_{n}\nabla{\boldsymbol{U}}_{r}-{\boldsymbol{U}}_{r}\cdot\nabla{\alpha}_{w}$
(D.9)
$\nabla{\overline{\boldsymbol{U}}}^{n}_{n}=\nabla{\boldsymbol{U}}_{f}-{\alpha}_{w}\nabla{\boldsymbol{U}}_{r}-{\boldsymbol{U}}_{r}\cdot\nabla{\alpha}_{w}$
(D.10)

Inserting these definitions into the first equation and expanding we then
obtain the following equation.

$\overline{\boldsymbol{S}}=({\alpha}_{w}{\mu}_{w}+{\alpha}_{n}{\mu}_{n})\left(\nabla{\boldsymbol{U}}_{f}+{\left(\nabla{\boldsymbol{U}}_{f}\right)}^{T}\right)+{\alpha}_{w}{\alpha}_{n}\left({\mu}_{w}-{\mu}_{n}\right)(\nabla{\boldsymbol{U}}_{r}+\\\
{\left(\nabla{\boldsymbol{U}}_{r}\right)}^{T})-\left({\alpha}_{w}{\mu}_{w}+{\alpha}_{n}{\mu}_{n}\right){\boldsymbol{U}}_{r}\cdot\left(\nabla{\alpha}_{w}+{\left(\nabla{\alpha}_{w}\right)}^{T}\right)$
(D.11)

We can further simplify this expression by defining the single-field fluid
viscosity ${\mu}_{f}={\alpha}_{w}{\mu}_{w}+{\alpha}_{n}{\mu}_{n}$. The result
is a complete expression for the single-field viscous stress tensor.

$\overline{\boldsymbol{S}}={\mu}_{f}\left(\nabla{\boldsymbol{U}}_{f}+{\left(\nabla{\boldsymbol{U}}_{f}\
\right)}^{T}\right)\\\
+{\alpha}_{w}{\alpha}_{n}\left({\mu}_{w}-{\mu}_{n}\right)\left(\nabla{\boldsymbol{U}}_{r}+{\left(\nabla{\boldsymbol{U}}_{r}\right)}^{T}\right)-{\mu}_{f}{\boldsymbol{U}}_{r}\cdot\left(\nabla{\alpha}_{w}+{\left(\nabla{\alpha}_{w}\right)}^{T}\right)$
(D.12)

If ${\boldsymbol{U}}_{r}\ll{\boldsymbol{U}}_{f}$, as argued in Fleckenstein &
Bothe, (2015), this expression reduces to the equation presented in Section
2.5.

$\overline{\boldsymbol{S}}\approx{\mu}_{f}\left(\nabla{\boldsymbol{U}}_{f}+{\left(\nabla{\boldsymbol{U}}_{f}\
\right)}^{T}\right)$ (D.13)

###  D.3 Derivation of the Single-Field Material Derivative

We start with the standard material derivative for a given fluid phase $i$.

$\frac{\partial{\rho}_{i}{\boldsymbol{U}}_{i}}{\partial
t}+\nabla\cdot\left({\rho}_{i}{\boldsymbol{U}}_{i}{\boldsymbol{U}}_{i}\right)$
(D.14)

We then apply the averaging operators for a volume that contains an additional
fluid phase and a solid phase.

$\overline{\frac{\partial{\rho}_{i}{\boldsymbol{U}}_{i}}{\partial
t}}+\overline{\nabla\cdot\left({\rho}_{i}{\boldsymbol{U}}_{i}{\boldsymbol{U}}_{i}\right)}$
(D.15)

Expanding the terms according to the averaging theorems, we obtain:

$\frac{\partial\overline{{\rho}_{i}{\boldsymbol{U}}_{i}}}{\partial
t}-\frac{1}{V}\int_{A_{i,s}}{{\rho}_{i}{\boldsymbol{U}}_{i}{\boldsymbol{U}}_{i,s}\cdot{\boldsymbol{n}}_{i,s}dA}-\frac{1}{V}\int_{A_{i,j}}{{\rho}_{i}{\boldsymbol{U}}_{i}{\boldsymbol{U}}_{i,j}\cdot{\boldsymbol{n}}_{i,j}dA}\\\
+\nabla\cdot\left(\overline{{\rho}_{i}{\boldsymbol{U}}_{i}{\boldsymbol{U}}_{i}}\right)+\frac{1}{V}\int_{A_{i,s}}{{\rho}_{i}{\boldsymbol{U}}_{i}{\boldsymbol{U}}_{i}\cdot{\boldsymbol{n}}_{i,s}dA}\
\ \
+\frac{1}{V}\int_{A_{i,j}}{{\rho}_{i}{\boldsymbol{U}}_{i}{\boldsymbol{U}}_{i}\cdot{\boldsymbol{n}}_{i,j}dA}$
(D.16)

Given that the velocities of two immiscible fluids are equal to each other at
the fluid-fluid interface (i.e.
${\boldsymbol{U}}_{i}={\boldsymbol{U}}_{j}={\boldsymbol{U}}_{i,j}$ at
$A_{i,j})$ and that the fluid and solid velocities are equal at the fluid-
solid interface (i.e.
${\boldsymbol{U}}_{i}={\boldsymbol{U}}_{s}={\boldsymbol{U}}_{i,s}$ at
$A_{i,s})$, we can obtain the following equation (Higuera,, 2015):

$\frac{\partial\overline{{\rho}_{i}{\boldsymbol{U}}_{i}}}{\partial
t}+\nabla\cdot\left(\overline{{\rho}_{i}{\boldsymbol{U}}_{i}{\boldsymbol{U}}_{i}}\right)=0$
(D.17)

We can further simplify the convective term by separating the fluid’s velocity
into its intrinsic average and deviation terms
${\boldsymbol{U}}_{i}={\overline{\boldsymbol{U}}}^{i}_{i}+{\widetilde{\boldsymbol{U}}}_{i}$
(Whitaker,, 2013), which results in

$\nabla\cdot\left(\overline{{\rho}_{i}{\boldsymbol{U}}_{i}{\boldsymbol{U}}_{i}}\right)=\nabla\cdot\left({{\phi}_{f}{\alpha}_{i}\rho}_{i}\overline{{\boldsymbol{U}}_{i}{\boldsymbol{U}}_{i}}\right)=\nabla\cdot\left({\phi}_{f}{\alpha}_{i}{\rho}_{i}{\overline{\boldsymbol{U}}}^{i}_{i}{\overline{\boldsymbol{U}}}^{i}_{i}\right)+\nabla\cdot\left({\rho}_{i}\overline{{\widetilde{\boldsymbol{U}}}_{i}{\widetilde{\boldsymbol{U}}}_{i}}\right)$
(D.18)

For low Re number simulations, we can neglect the deviation term to obtain the
averaged material derivative for a single fluid phase.

$\frac{\partial{\rho}_{i}{\phi}_{f}{\alpha}_{i}{\overline{\boldsymbol{U}}}^{i}_{i}}{\partial
t}+\nabla\cdot\left({\phi}_{f}{\alpha}_{i}{\rho}_{i}{\overline{\boldsymbol{U}}}^{i}_{i}{\overline{\boldsymbol{U}}}^{i}_{i}\right)$
(D.19)

To obtain the equivalent single-field expression we add the material
derivatives of a wetting ($w$) and non-wetting ($n$) fluid,

$\frac{\partial{{\phi}_{f}(\rho}_{w}{\alpha}_{w}{\overline{\boldsymbol{U}}}^{w}_{w}+\
{\rho}_{n}{\alpha}_{n}{\overline{\boldsymbol{U}}}^{n}_{n})}{\partial
t}+\nabla\cdot\left({\phi}_{f}\left({\alpha}_{w}{\rho}_{w}{\overline{\boldsymbol{U}}}^{w}_{w}{\overline{\boldsymbol{U}}}^{w}_{w}+{\alpha}_{n}{\rho}_{n}{\overline{\boldsymbol{U}}}^{n}_{n}{\overline{\boldsymbol{U}}}^{n}_{n}\right)\right)$
(D.20)

From the definitions of the single-field and relative velocities we can write
the following relations:

${\overline{\boldsymbol{U}}}^{w}_{w}={\phi}^{-1}_{f}{\boldsymbol{U}}_{f}+{\alpha}_{n}{\boldsymbol{U}}_{r}$
(D.21)
${\overline{\boldsymbol{U}}}^{n}_{n}={\phi}^{-1}_{f}{\boldsymbol{U}}_{f}-{\alpha}_{w}{\boldsymbol{U}}_{r}$
(D.22)

Based on these definitions, the terms within the time derivative and within
the divergence operator in Eqn. D.20 can be expressed as

${{\phi}_{f}(\rho}_{w}{\alpha}_{w}{\overline{\boldsymbol{U}}}^{w}_{w}+\
{\rho}_{n}{\alpha}_{n}{\overline{\boldsymbol{U}}}^{n}_{n})=\left({{\alpha}_{w}\rho}_{w}+{{\alpha}_{n}\rho}_{n}\right){\boldsymbol{U}}_{f}+{\phi}_{f}{\alpha}_{w}{\alpha}_{n}{\boldsymbol{U}}_{r}({\rho}_{w}-{\rho}_{n})$
(D.23)

and,

${\phi}_{f}\left({\alpha}_{w}{\rho}_{w}{\overline{\boldsymbol{U}}}^{w}_{w}{\overline{\boldsymbol{U}}}^{w}_{w}+{\alpha}_{n}{\rho}_{n}{\overline{\boldsymbol{U}}}^{n}_{n}{\overline{\boldsymbol{U}}}^{n}_{n}\right)=\\\
{\phi}_{f}{({\alpha}_{w}\rho}_{w}+{\alpha}_{n}{\rho}_{n})\left({\phi}^{-2}_{f}{\boldsymbol{U}}_{f}{\boldsymbol{U}}_{f}+{\boldsymbol{U}}_{r}{\boldsymbol{U}}_{r}\right)+2{\phi}^{-1}_{f}{\alpha}_{w}{\alpha}_{n}{\boldsymbol{U}}_{f}{\boldsymbol{U}}_{r}({\rho}_{w}-{\rho}_{n})$
(D.24)

To simplify things, we now define the single-field density
${\rho}_{f}={\alpha}_{w}{\rho}_{w}+{\alpha}_{n}{\rho}_{n}$, and the viscosity
difference $\Delta\rho={\rho}_{w}-{\rho}_{n}$. Combining the previous
equations we obtain the complete averaged single-field expression for the
material derivative

$\frac{\partial{\rho}_{f}{\boldsymbol{U}}_{f}}{\partial
t}+\frac{\partial{\phi}_{f}{\alpha}_{w}{\alpha}_{n}\Delta\rho{\boldsymbol{U}}_{r}}{\partial
t}+\nabla\cdot\left(\frac{{\rho}_{f}}{{\phi}_{f}}{\boldsymbol{U}}_{f}{\boldsymbol{U}}_{f}\right)+\\\
\nabla\cdot\left({\phi}_{f}{\rho}_{f}{\boldsymbol{U}}_{r}{\boldsymbol{U}}_{r}\right)+\nabla\cdot\left(2{\alpha}_{w}{\alpha}_{n}\Delta\rho{\boldsymbol{U}}_{f}{\boldsymbol{U}}_{r}\right)$
(D.25)

Finally, if ${\boldsymbol{U}}_{r}\ll{\boldsymbol{U}}_{f}$, as argued in
Fleckenstein & Bothe, (2015), this expression reduces to the expression used
in the derivation of Equation 2.40.

$\frac{\partial{\rho}_{f}{\boldsymbol{U}}_{f}}{\partial
t}+\nabla\cdot\left(\frac{{\rho}_{f}}{{\phi}_{f}}{\boldsymbol{U}}_{f}{\boldsymbol{U}}_{f}\right)$
(D.26)

###  D.4 Recovery of Biot Theory from the Fluid and Solid Momentum Equations

We start by adding together the final fluid and solid momentum equations
(Eqns. 2.90 and 2.92) under the assumption of low Reynolds numbers and low
permeability. This is analogous to the steps used to obtain Eqn. 2.84 in
Section 2.10.

$-\nabla\cdot\overline{\boldsymbol{\sigma}}={\phi}_{s}\nabla\cdot{\overline{\boldsymbol{\tau}}}^{s}-{\phi}_{f}\nabla
p+\left({{\phi}_{s}\rho}_{s}+{\phi}_{f}{\rho}_{f}\right)\boldsymbol{g}+{\boldsymbol{F}}_{c,2}$
(D.27)

Now, assuming uniform confining pressure and no swelling pressure (i.e
$\nabla\cdot{\overline{\boldsymbol{\tau}}}^{s}\boldsymbol{=-}\nabla p)$ we
obtain:

$-\nabla\cdot\overline{\boldsymbol{\sigma}}=-{\phi}_{s}\nabla
p-{\phi}_{f}\nabla
p+\left({{\phi}_{s}\rho}_{s}+{\phi}_{f}{\rho}_{f}\right)\boldsymbol{g}+{\boldsymbol{F}}_{c,2}$
(D.28)

We can then set
${\rho}^{*}=({{\phi}_{s}\rho}_{s}+{\phi}_{f}{\rho}_{f}\boldsymbol{)}$ and
input the definition of the capillary force term (Eqn. 2.95), such that

$-\nabla\cdot\overline{\boldsymbol{\sigma}}=-\nabla
p+{\rho}^{*}\boldsymbol{g}-p_{c}\nabla{\alpha}_{w}$ (D.29)

The resulting expression is the momentum conservation equation used in
multiphase Biot Theory (Jha0 & Juanes,, 2014; Kim et al.,, 2013). In said
papers, the corresponding fluid mass conservation equation is

$\frac{\partial m}{\partial t}+\nabla\cdot{\boldsymbol{U}}_{f}=0$ (D.30)

where $m$ is the mass of fluid per control volume. The time derivative is
often expressed in terms of the fluid pressure and volumetric strain
($\epsilon$) by applying the following a pressure-strain relation (Coussy,,
2010; Kim et al.,, 2011):

$\frac{1}{{\rho}_{f}}\left(m-m_{0}\right)=b\boldsymbol{\epsilon}+\frac{1}{M}(p-p_{0})$
(D.31)

Therefore, the continuity equation becomes,

${\rho}_{f}\left(\frac{1}{M}\frac{\partial p}{\partial
t}+b\frac{\partial\epsilon}{\partial
t}\right)+\nabla\cdot{\boldsymbol{U}}_{f}=0$ (D.32)

where $M$ and $b$ are the Biot modulus and coefficient, respectively. In our
modelling framework, however, we note that the initial continuity equation is
equal to the averaged fluid continuity equation presented in the main text by
considering that $m={\phi}_{f}{\rho}_{f}$.

$\frac{\partial{\phi}_{f}{\rho}_{f}}{\partial
t}+\nabla\cdot{\boldsymbol{U}}_{f}=0$ (D.33)

###  D.5 Integral Geometric Relation for Vectors

The following geometric relation holds for vector values within the averaging
integrals; note its similarity to the geometric relation for scalars shown in
the main text (Whitaker, 1986a, )

$\frac{1}{V}\int_{A_{i,s}}{\boldsymbol{I}\cdot{\boldsymbol{n}}_{f,s}dA}\ =\
-\boldsymbol{I}\cdot\nabla{\phi}_{f}$ (D.34)

##  Appendix E Semi-Analytic Solution for the Seismic Stimulation of a
Poroelastic Core

Here we present the analytical solution used to describe the oscillating
elastic system in Section 6.1.2. Given a Biot coefficient of unity and
incompressible fluids, the fractional change in an oscillating poroelastic
core’s fluid content $\Omega\ $as a function of time $t$ is given by (Lo et
al.,, 2012):

$\Omega\left(t\right)=-a_{1}\upsilon+a_{2}{\alpha}_{w}p_{a}+0.5\left(a_{2}p_{0}sin\mathrm{}(\omega
t\right)-a_{2}{\alpha}_{w}p_{a})+\ \sum^{\infty}_{n=1}{A}$ (E.1)

$A={\left(n\pi\right)}^{-2}2{cos\left(n\pi\right)\
}a_{2}p_{0}\left({sin\left(\omega
t\right)+\frac{{\omega}^{2}_{n}{sin\left(\omega t+{\delta}_{n}\right)\
}}{{{{((\omega}^{2}-{\omega}^{2}_{n})}^{2}+D^{2}{\omega}^{2})}^{0.5}}\
}\right)\left(1-{cos\left(n\pi\right)\ }\right)$ (E.2)

where $\upsilon$ is the uniaxial confining pressure, $p_{a}$ is the fixed
pressure at the left boundary, $p_{0}$ is the amplitude of the oscillating
pressure at the right boundary, and $\omega=2\pi f$ is the angular frequency
of the pressure variation. The summation terms ${\omega}_{n}$ and
${sin\left({\delta}_{n}\right)\ }$ are defined as

${\omega}_{n}={\left(\frac{Cn\pi}{Length}\right)}^{2}$ (E.3)

${sin\left({\delta}_{n}\right)\
}=\frac{D\omega}{{{{((\omega}^{2}-{\omega}^{2}_{n})}^{2}+D^{2}{\omega}^{2})}^{0.5}}$
(E.4)

${cos\left({\delta}_{n}\right)\
}=\frac{{\omega}^{2}-{\omega}^{2}_{n}}{{{{((\omega}^{2}-{\omega}^{2}_{n})}^{2}+D^{2}{\omega}^{2})}^{0.5}}$
(E.5)

Furthermore, the dissipation constant $D$, the wave speed $C$, and the
compressibility constants $a_{1}$ and $a_{2}$ are defined as follows

$D=\frac{1}{k_{0}}\
\frac{1}{\frac{T}{{\phi}_{f}}\left(\frac{{{\rho}_{w}M}_{w}}{{\alpha}_{w}}+\frac{{{\rho}_{n}M}_{n}}{{\alpha}_{n}}\right)-\left({{\rho}_{w}M}_{w}+{{\rho}_{n}M}_{n}\right)}$
(E.6)

$C^{2}=\left(K_{b}+\frac{4}{3}G\right)\frac{M}{\frac{T}{{\phi}_{f}}\left(\frac{{{\rho}_{w}M}_{w}}{{\alpha}_{w}}+\frac{{{\rho}_{n}M}_{n}}{{\alpha}_{n}}\right)-\left({{\rho}_{w}M}_{w}+{{\rho}_{n}M}_{n}\right)}$
(E.7)

$a_{1}={\left(3K_{b}\right)}^{-1}$ (E.8)

$a_{2}=K^{-1}_{b}$ (E.9)

where $T=0.5\left(1+{\phi}^{-1}_{f}\right)$ is the tortuosity, $K_{b}$ is the
bulk modulus of the solid matrix, $G$ is the shear modulus of the solid
matrix, and the rest of the variables are defined as in the main manuscript.
The infinite sum in Eqn. E.1 was calculated through a python script, where it
was truncated at the point where the last sum term represented 0.01% of the
previous term.

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