In this section we present the comparison between the experimental and
numerical results for the squeeze flow of glycerol and PVP solution. The
comparison is realized by uncertainty propagation, using the forward Monte
Carlo sampling method, and Bayesian inference, using MCMC.

We used Bayesian inference on the constitutive model to obtain a distribution
for the rheological parameters (see Section LABEL:sec:uncertainties). Based on
the simple shear measurements and constitutive model prediction, we expect
that the squeeze flow behavior of glycerol can be described by the Newtonian
squeeze flow model. For PVP, a prediction for the initial shear rate using the
Newtonian model and zero-shear viscosity, yields maximum shear rates in the
order of 102 1/s. Since this is in the shear-thinning regime
(fig:rheodataPVP), we use the truncated power law model to predict the squeeze
flow behavior of PVP solution.

For both of the fluids, we start by using the method of uncertainty
propagation to compare the numerical model to the experimental data, keeping
the parameter distributions fixed. Thereafter, we use Bayesian inference to
update the parameters, and thus the model response, using the experimental
data from the squeeze flow. Finally, we will compare both methods for both
fluids.

##  0.1 Newtonian fluid

The results of the experimental cases 1 to 4 are visualized in
fig:UP_Gly_NLP01. From case 1 to 3 we have an increase in applied force (see
tab:ExpCaseGly and tab:PU_Fapp). In fig:UP_Gly_NLP01 we observe that an
increase in applied force leads to a faster growth of the radius. The applied
force in case 4 is similar to the applied force in case 2 and the volume is
increased (see tab:ExpCaseGly and tab:PU_fluidvolume). We observe that an
increase in fluid volume leads to a slight increase in initial radius.
Analyzing the experimental data, we see that the uncertainty increases as time
proceeds.

### Uncertainty propagation

To use the method of uncertainty propagation, we assign a distribution to the
model input parameters. In Section LABEL:sec:uncertainties, we have discussed
experiments to obtain the uncertainty in the input parameters $F$, $V$,
$R_{0}$ and $\gamma$ and have obtained the distributions for these parameters.
The curvature $\alpha$ has been defined in two limiting cases: $\alpha=0$ and
$\alpha=1$. Using Bayesian inference on the simple shear measurements, we have
retrieved a distribution for the rheological parameters. Every distribution is
to be propagated through the squeeze flow model to obtain the distribution of
the radius over time. We have used 10,000 samples from each distribution of
the model parameters to retrieve 10,000 samples for the radius per point in
time.

We consider the Newtonian model prediction including the two limiting cases of
the Laplace pressure ($\alpha=0$ and $\alpha=1$) in fig:UP_Gly_NLP01. In the
case of $\alpha=0$ (no Laplace pressure), we observe that the model prediction
increasingly deviates from the experimental data as time proceeds. Moreover,
the difference between the model prediction without Laplace pressure and the
experimental data decreases with increased applied force. These observations
can be explained by the capillary number $\mathrm{Ca}=\eta\dot{h}/\gamma$. In
the squeeze flow setting, the radius increases with time meaning that
$\dot{h}$ will decrease and thus also Ca. The capillary number decreases from
$\sim$ $10^{-2}$ at $t=0$ s to $\sim$ $10^{-7}$ at around $t=350$ s in case 1.
A low capillary number indicates that the influence of the surface tension
increases. This explains why the model prediction without Laplace pressure
describes the short-term flow behavior well in comparison to the long-term
flow behavior. It also explains why there is an increasing similarity between
the experimental data and model prediction for an increasing applied force.

Figure 1: Comparison between the experimental data of glycerol and the model
prediction using uncertainty propagation. The subscript ‘m’ denotes the model
prediction and the subscript ‘exp’ denotes the experimental data. The model
prediction including the Laplace pressure is the case where we have the
maximum possible curvature. The two extreme cases encompass the experimental
data for all cases (1 to 4).

In fig:UP_Gly_NLP01, we can see that for $\alpha=1$ (maximum Laplace pressure)
for cases 1 to 4, denoted in yellow, the model response clearly underestimates
the experimental data. Based on the observation in fig:UP_Gly_NLP01, the
curvature should be smaller than the maximum curvature to describe the
experimental data. Because we are uncertain about the value of $\alpha$
throughout the experiment, we have a relatively wide range of possible model
predictions. In order to quantify the curvature we use the method of Bayesian
inference.

### Bayesian inference

In the framework of Bayesian inference, we use the experimental squeeze flow
data to define the likelihood and use the parametric uncertainties as priors.
The likelihood function is defined in a similarly way as the one used in
Bayesian inference on the rheological measurements (see Section
LABEL:sec:uncertainties), where we use the experimental data to determine the
noise in the likelihood function. However, in this case the noise differs per
time step instead of per shear rate.The procedure of Bayesian inference is
conducted using the MCMC method described in Section LABEL:sec:uncertainties.

The prior distribution of the model input parameters is similar to the
distribution used in the method of uncertainty propagation. For these
parameters, we choose normal distributions. The parameters $\eta$, $R_{0}$ and
$V$ should physically be non-negative. Therefore, we use equation
eq:logtransform to obtain a lognormal distribution. We use the transformation
explained in rmk:logtransform to obtain the distribution of the true parameter
values. We define $\alpha$ in between zero and one. Based on the observations
made in fig:UP_Gly_NLP01, we expect that $\alpha$ is closer to zero than to
one. Therefore, we assign a beta distribution to $\alpha$, of which the
probability density function is defined as

$f(\alpha;a,b)=\frac{1}{B(a,b)}\alpha^{a-1}(1-\alpha)^{b-1}\qquad\alpha\in[0,1],$
(1)

where $a$ and $b$ are the shape parameters valued $a=3$ and $b=8$,
respectively and $B(a,b)$ is a normalization constant.

We evaluate the Newtonian squeeze flow model using twelve walkers/chains and
10,000 samples per walker. The burn-in period per walker is $2\times\tau_{f}$
and the thinning is $\tau_{f}$/2, similar to the formulation as described in
Section LABEL:sec:uncertainties. The autocorrelation time per model parameter
for each of the cases 1 to 4 is given in tab:autocor_Gly. The effective sample
size per case is given in tab:Neff_Gly.

Table 1: Autocorrelation times per model parameter and case using twelve
walkers and 10,000 samples per walker.

XXXXXXX Case & $F$ $\hat{V}$ $\hat{R}_{0}$ $\hat{\eta_{\mathrm{N}}}$ $\gamma$
$\alpha$
1 79 76 87 82 83 83
2 96 128 158 110 84 127
3 103 92 84 72 96 95
4 108 82 102 72 120 82

Table 2: Effective sample size per case based on the highest autocorrelation
time per case for a sample size $N=120,000$.

XXXXX Case & 1 2 3 4
$N_{\mathrm{final}}$ 2736 1488 2304 1944

In fig:BI_Gly_NLP_ppd, we show the posterior predictive distribution of the
model response $R(t)$ and the experimental data for cases 1 to 4. The width of
the model prediction is comparable to the width of the experimental data. The
initial data point in case 3 shows an increased distribution width for the
posterior predictive distribution as well as the experimental data, because
the noise in the posterior predictive distribution is determined using the
standard deviation of the experimental data, which differs per time step. The
short-term flow behavior predicted using uncertainty propagation shows similar
behavior to the prediction made using Bayesian inference, because the Laplace
pressure’s influence decreases toward the short-term flow behavior.
Furthermore, the similarities in model response determined using uncertainty
propagation and Bayesian inference indicates that the uncertainty in the model
input parameters are well-quantified, meaning that the parametric uncertainty
determined using independent parameter experiments shows a similar amount of
uncertainty as the parametric uncertainty obtained using the squeeze flow
experimental data.

Figure 2: Comparison between the posterior predictive distribution and
experimental data of the squeeze flow using Bayesian inference. The subscript
’m’ corresponds to the model and the subscript ’exp’ to the experimental data.

The posterior of the model parameters and the correlation between them are
visualized in fig:BI_Gly_NLP_corner for case 1. Similar plots for the
remaining cases can be found in the supplementary information. The posterior
per input parameter is visualized by the histogram and the prior distribution
by the green graph. Furthermore, the area in between the dashed black lines
indicates the 95% credibility interval. We observe that for the parameters
$\eta_{\mathrm{N}}$, $F$ and $\gamma$ the posterior looks similar to the prior
distribution, indicating that the model parameters are not updated through the
likelihood function. We also observe that the prior of the fluid volume is not
similar to the posterior. This suggests that the pipetting and subsequent
weighting of the sample yield predicted volumes that are inconsistent with the
squeeze flow model predictions. The posterior of $\alpha$ is not similar to
the prior distribution and is close to zero, suggesting that the influence of
the Laplace pressure is minimally present. We would expect a perfect
correlation between $\alpha$ and $\gamma$ based on the expression for the
Laplace pressure (see equation eq:Laplace and equation eq:curvekappa).
However, because we are much more certain about $\gamma$ in comparison to
$\alpha$, the uncertainty is dominated by $\gamma$, due to which the
parameters do not seem to be correlated. The posteriors in the corner plot are
given as $\hat{\boldsymbol{\theta}}$ for the parameters to which we have
applied the log-transformation. The mean, standard deviation and coefficient
of variation are given in tab:musig_BI_squeezeGly for the actual model
parameters $\boldsymbol{\theta}$. For the model parameters that are difficult
to determine, such as the curvature-related parameter $\alpha$, Bayesian
inference can give an accurate probablistic prediction. The uncertainty in
viscosity and surface tension is expected to be similar for cases 1 to 4. In
LABEL:app:uncertaintytab we show the uncertainty for the remaining cases
(cases 2 to 4), from which it can be concluded that the uncertainty per case
is comparable.

Figure 3: Corner plot showing the correlation between the posterior of the
model parameters for case 1, where we use glycerol. The histogram and dark-red
graphs denote the posterior and prior, respectively. The region in between the
black dashed lines defines the 95% credibility interval. Table 3: Uncertainty
in the Newtonian squeeze flow model parameters $\boldsymbol{\theta}$ for case
1.

XXXX & $\mu$ $\sigma$ CV [%]
$F$ [N] 2.88 5.78$\,\times\,10^{-2}$ 2.01
$V$ [mL] 1.59$\,\times\,10^{-1}$ 2.95$\,\times\,10^{-3}$ 1.86
$R_{0}$ [cm] 7.53$\,\times\,10^{-1}$ 8.27$\,\times\,10^{-3}$ 1.10
$\eta_{\mathrm{N}}$ [Pa$\cdot$s] 8.69$\,\times\,10^{-1}$
1.24$\,\times\,10^{-2}$ 1.43
$\gamma$ [N/m] 4.46$\,\times\,10^{-2}$ 2.49$\,\times\,10^{-3}$ 5.59
$\alpha$ [-] 2.48$\,\times\,10^{-2}$ 1.13$\,\times\,10^{-2}$
5.36$\,\times\,10^{1}$

##  0.2 Generalized Newtonian fluid

We now consider the comparison between the model predication and experimental
data for PVP. The results for the experimental cases 5 to 8 are visualized in
Figures 4 and 5. We observe that cases 5 and 6 show a larger initial radius in
comparison to cases 7 and 8, which can be explained by the increased fluid
volume in cases 5 and 6. Comparing cases 5 and 6, we see that the radius in
the latter case increases faster than in the former, which is related to the
larger applied force in case 6. Similar observations and conclusions are made
between cases 7 and 8, where the latter shows the larger increase in radius.
The uncertainty in experimental data increases over time as shown in Figure 4
and 5, similar to the observations made for the experimental data of glycerol
(see fig:UP_Gly_NLP01).

### Uncertainty propagation

The parametric uncertainties of $F$, $V$, $R_{0}$ and $\gamma$ have been
determined through additional experiments and the uncertainty in the
rheological parameters $\lambda$, $\eta_{0}$, $\eta_{\infty}$ and $n$ have
been determined using Bayesian inference applied on the simple shear
measurements. These uncertainties are propagated through the squeeze flow
model to predict the outward motion of the fluid. We have used 12,240 samples
from each distribution of the model parameters to retrieve 12,240 samples for
the radius per point in time.

We start by comparing the Newtonian squeeze flow model to the experimental
data obtained for PVP solution, including and excluding Laplace pressure. We
choose to implement the viscosity at the zero-shear rate plateau as input for
the Newtonian squeeze flow model. We expect to observe a mismatch between the
data and model prediction, because of the shear thinning flow behavior at high
shear rates (see fig:rheodataPVP).

Figure 4: Newtonian model prediction compared to the experimental squeeze flow
data of PVP using uncertainty propagation, where the subscript ’m’ corresponds
to the model and the subscript ’exp’ to the experimental data.

In fig:UP_PVP_NLP01, we show the Newtonian model prediction and experimental
data for PVP. There is no significant distinction between the model prediction
including and excluding the Laplace pressure. Furthermore, we see that
initially the experimental data is underestimated by the model prediction.
This can be explained by the constant viscosity at the zero shear rate
plateau. Due to the high initial shear rates, the viscosity is lower than the
viscosity at the zero-rate plateau, meaning that the evolution rate of the
fluid front is higher. We expect that the model prediction improves as we
incorporate shear thinning flow behavior in the model through the truncated
power law model.

Figure 5: Truncated power law model prediction compared to the experimental
squeeze flow data of PVP using uncertainty propagation. The subscript ’m’
denotes to the model and the subscript ’exp’ to the experimental data

The truncated power law model prediction for $\alpha=0$ and $\alpha=1$ is
visualized in fig:UP_PVP_TPLLP01. We observe that for the initial stages the
model prediction has improved using the truncated power law model instead of
the Newtonian model. As time proceeds, the model prediction describes the
experimental data similarly as the Newtonian model prediction, which can be
explained by the decrease in shear rate. Around $t=1\,\mathrm{s}$ for cases 5
to 8, the model overestimates the experimental data, independent of the
inclusion of the Laplace pressure.

We can use Bayesian inference to update the rheological parameters using the
rheological data as prior knowledge. In this way, we can investigate whether
the rheological parameters as determined in simple flow are representative for
the squeeze flow. Furthermore, we will use this method to investigate the
uncertainty in curvature of the interface at the fluid front. We will not
apply Bayesian inference on the Newtonian model separately, since it is a
special case (_i.e._ , specific parameter setting) of the truncated power law
model.

### Bayesian inference

We apply the framework of Bayesian inference, using MCMC, on the squeeze flow
model with the truncated power law model incorporated. The prior distribution
of the model input parameters are similar to the ones used in uncertainty
propagation. The rheological parameters, $\eta_{0}$, $\eta_{\infty}$, $n$ and
$\lambda_{\mathrm{cr}}$, should be non-negative and therefore are assigned a
lognormal distribution (rmk:logtransform). The surface tension and applied
force have a normal distribution and we use a lognormal distribution like the
one used for the rheological parameters for the fluid volume and initial
radius. The prior distribution of the curvature is a Beta-distribution, given
by equation eq:betadist, where $a=3$, $b=8$.

We use 27 walkers/chains, 10,000 samples per walker, a burn-in period of
$2\times\tau_{f}$ and $\tau_{f}$/2 for the thinning to apply Bayesian
inference on the squeeze flow model. The autocorrelation time for cases 5 to 8
is provided in tab:autocor_PVP per model parameter. The effective sample size
is given in tab:Neff_PVP for cases 5 to 8.

Table 4: Autocorrelation times per model parameter and case using 27 walkers
and 10,000 samples per walker.

XXXXXXXXXX Case & $F$ $\hat{V}$ $\hat{R}_{0}$ $\hat{\eta}_{0}$
$\hat{\eta}_{\infty}$ $n$ $\lambda_{\mathrm{cr}}$ $\gamma$ $\alpha$
5 111 95 111 97 113 120 101 105 111
6 113 98 115 94 101 111 97 104 130
7 114 124 135 127 127 132 137 129 170
8 96 95 110 107 115 106 107 115 136

Table 5: Effective sample size $N_{\mathrm{final}}$ per case based on the
highest autocorrelation time per case, for sample size $N=270,000$.

XXXXX Case & 5 6 7 8
$N_{\mathrm{final}}$ 4374 4023 3267 3915

In fig:BI_PVP_TPLLP01, the experimental cases 5 to 8 and model predictions
using the Bayesian inference framework are given. The mismatch between the
experimental data and model prediction around $t=1\,\mathrm{s}$ is
significantly decreased in the model prediction using Bayesian inference,
meaning that the data available through the likelihood improved the model
prediction.

Figure 6: Truncated power law model prediction compared to the experimental
squeeze flow data of PVP using Bayesian inference. The subscript ’m’
corresponds to the model and the subscript ’exp’ to the experimental data

To get a closer look at the posterior distribution for the individual
parameters, we analyze the two-dimensional posterior marginals provided in
fig:BI_PVP_TPLLP_corner. The posterior distributions are given by the
histograms on the far right image of every row, including the prior
distribution given in red. We observe that the posterior distribution
corresponds to the prior distribution for the applied force $F$ and
rheological parameters $\hat{\eta}_{0}$ and $\hat{\eta}_{\infty}$ and
reasonably corresponds to the rheological parameters $\hat{n}$ and
$\hat{\lambda}_{\mathrm{cr}}$. The posterior distribution of $\gamma$ is
similar to the prior distribution and the prior distribution does not
correspond to the posterior distribution of $\alpha$. Note that $\alpha$ is
shifted toward zero in comparison to the prior distribution, but not as strong
as in the glycerol case (fig:BI_Gly_NLP_corner). The posterior of the initial
radius corresponds to the prior distribution, where the posterior distribution
is more narrow than the prior distribution and the mean remains approximately
the same. Using Bayesian inference led to a decrease in uncertainty in the
initial radius. A significant mismatch can be observed in the prior and
posterior distribution of the fluid volume. The mismatch in prior and
posterior distribution of the parameters underlies the improved prediction
using Bayesian inference around $t=1\,\mathrm{s}$ in fig:BI_PVP_TPLLP01 in
comparison to fig:UP_PVP_TPLLP01 for cases 5 to 8.

In analyzing the correlation between the parameters, we investigate the two-
dimensional posterior marginals provided in fig:BI_PVP_TPLLP_corner. We
observe that nearly every posterior marginal does not show a significant
correlation between parameters. The posterior marginal between the applied
force $F$ and fluid volume $V$ is elliptic, which could indicate a correlation
between the parameters or a difference in level of informativeness in the
prior. For the parameters to which we have applied the log-transformation, the
posteriors are given by $\hat{\boldsymbol{\theta}}$ in the corner plot. The
statistical moments are given in tab:musig_BI_squeezePVP for the actual model
parameters $\boldsymbol{\theta}$. The uncertainty in the rheological
parameters and surface tension is expected to be similar for cases 5 to 8. In
LABEL:app:uncertaintytab we show the uncertainty in the case independent
parameters ($\eta_{0}$, $\eta_{\infty}$, $n$, $\lambda_{\mathrm{cr}}$ and
$\gamma$) for the remaining cases (6 to 8), from which it can be concluded
that the uncertainty per case is comparable as expected.

Figure 7: Corner plot showing the correlation between the posterior of every
model parameter for case 5, where we use PVP solution. The histogram and dark-
red graphs denote the posterior and prior, respectively. The region in between
the black dashed lines defines the 95% credibility interval. Table 6:
Uncertainty in the truncated power law squeeze flow model parameters
$\boldsymbol{\theta}$ for case 5.

XXXX & $\mu$ $\sigma$ CV [%]
$F$ [N] 2.80 1.00$\,\times\,10^{-1}$ 3.60
$V$ [mL] 5.03$\,\times\,10^{-1}$ 1.68$\,\times\,10^{-2}$ 2.99
$R_{0}$ [cm] 1.18 3.88$\,\times\,10^{-2}$ 3.30
$\eta_{0}$ [Pa$\cdot$s] 3.20$\,\times\,10^{1}$ 1.32$\,\times\,10^{-1}$
4.13$\,\times\,10^{-1}$
$\eta_{\infty}$ [Pa$\cdot$s] 1.02$\,\times\,10^{-3}$ 3.04$\,\times\,10^{-4}$
2.97$\,\times\,10^{1}$
$n$ [-] 7.73$\,\times\,10^{-1}$ 4.87$\,\times\,10^{-3}$
6.30$\,\times\,10^{-1}$
$\lambda_{\mathrm{cr}}$ [s] 7.00$\,\times\,10^{-1}$ 3.28$\,\times\,10^{-2}$
4.68
$\gamma$ [N/m] 6.59$\,\times\,10^{-2}$ 1.97$\,\times\,10^{-3}$ 2.99
$\alpha$ [-] 1.24$\,\times\,10^{-1}$ 6.47$\,\times\,10^{-2}$
5.22$\,\times\,10^{1}$

### Model-based results

In this section we analyze model-based results for the PVP solution, _i.e._ ,
results acquired for non-observable quantities. We analyze two model-based
results at $t=1.0\,\times\,10^{-2}$ s, $t=1.0$ s, $t=10$ s and $t=30$ s: the
local viscosity regimes and the velocity profile. The results are shown for
case 5. Using the data available in the supplementary information, the results
can be obtained for cases 6 to 8 as well.

First, we discuss the local viscosity regimes. In Section LABEL:sec:model, we
have discussed the three regimes in the truncated power law model: Newtonian
at zero-shear rate plateau, power law at intermediate shear rates and
Newtonian at infinite shear rates. Using the squeeze flow model, we can obtain
the interfaces $w_{1}(t,r)$, $w_{2}(t,r)$ and $h(t)$. In addition, we can
obtain the uncertainty in the position of the interfaces. In fig:w1w2ht531, we
show each of the interfaces with the uncertainty. Note that the domain is half
of the height of the fluid layer. In this case, we do not reach a high enough
shear rate to reach the third regime, where $\eta=\eta_{\infty}$, therefore,
$w_{2}(r,t)=h(t)$, due to which we cannot distinguish these two regimes in the
figure. In time, we observe a decrease in height and increase in radius as
expected. By comparing the local viscosity regime for $t=1.0\,\times\,10^{-2}$
s to $t=30$ s we observe that the area covered by the Newtonian regime, where
$\eta=\eta_{0}$, increases, whereas the power law regime decreases. This shows
that the shear thinning behavior decreases through time. The uncertainty in
$w_{1}(r,t)$ is not constant in the radial domain. In time, the level of
uncertainty in $h(t)$ decreases.

Figure 8: Uncertainty in the local viscosity regimes of case 5 for various
points in time. The error bars cover the $\mu\pm 2\sigma$ region. Due to the
relatively small uncertainty in $r$ in comparison to the uncertainty in
$w_{1}(r,t)$, $w_{2}(r,t)$ and $h(t)$, we plot the mean of the distribution of
$r$.

In fig:velproft531 we show the velocity profile at $r=R$ for
$t=1.0\,\times\,10^{-2}$ s, $t=1.0$ s, $t=10$ s and $t=30$ s. Note that we
show the full height of the fluid layer. We observe a rapid decrease in
velocity between $t=1.0\,\times\,10^{-2}$ s and $t=1.0$ s, after which the
velocity slightly decreases. Furthermore, the height decreases over time in
the squeeze flow and we show that the height decreases the fastest in the
initial stages. Near the walls, the uncertainty is equal to zero because we
assume a no-slip condition. Toward the middle of the velocity profile, the
level of uncertainty increases. We expect the profile to change with time,
because the shear thinning behavior is only present during the initial stages.
By comparing the model prediction of the velocity for the four different
points in time, we do not directly observe the changes in the form of the
velocity profile, which we attribute to the logarithmic scale and/or the
decrease in height. The level of uncertainty decreases in time, especially
from $t=1.0\,\times\,10^{-2}$ s to $t=1.0$ s.

Figure 9: Uncertainty in the velocity profile of case 5 for various points in
time. The error bars cover the $\mu\pm 2\sigma$ region. Due to the relatively
small uncertainty in $r$ in comparison to the uncertainty in $v_{r}(r,z,t)$,
we plot the mean of the distribution of $R$. Furthermore, the uncertainty in
$h$ is relatively small in comparison to the uncertainty in $v_{r}(r,z,t)$ due
to which we plot the mean of the distribution of $h$.


ResUNet++ assisted porosity estimation

# Machine learning-based porosity estimation from
spectral decomposed seismic data

Honggeun Jo11footnotemark: 1 Yongchae Cho22footnotemark: 2 Michael
Pyrcz33footnotemark: 3 Hewei Tang44footnotemark: 4 Pengcheng Fu44footnotemark:
4

###### Abstract

##  1 Abstract

Estimating porosity models via seismic data is challenging due to the signal
noise and insufficient resolution of seismic data. Although impedance
inversion is often used by combining with well logs, several hurdles remain to
retrieve sub-seismic scale porosity. As an alternative, we propose a machine
learning-based workflow to convert seismic data to porosity models. A
ResUNet++ based workflow is designed to take three seismic data in different
frequencies (i.e., decomposed seismic data) and estimate their corresponding
porosity model. The workflow is successfully demonstrated in the 3D
channelized reservoir to estimate the porosity model with more than 0.9 in R2
score for training and validating data. Moreover, the application is extended
for a stress test by adding signal noise to the seismic data, and the workflow
results show a robust estimation even with 5% of noise. Another two ResUNet++
are trained to take either the lowest or highest resolution seismic data only
to estimate the porosity model, but they show under- and over-fitting results,
supporting the importance of using decomposed seismic data in porosity
estimation.

##  2 Introduction

Integrating seismic data into subsurface modeling can help capture possible
geological trends in inter-well regions Pyrcz and Deutsch, (2014). In a
conventional workflow as illustrated in Figure 1, (1) seismic data are first
converted to a 3D acoustic impedance field, (2) the correlation between
acoustic impedance and porosity is inferred from well logs (e.g., sonic log,
density log, and neutron log), and (3) porosity models are constructed through
co-kriging using the acoustic impedance field as secondary data, which relies
on the correlation built in Step (2).

Figure 1: The (a) prevailing workflow versus (b) machine learning-solution
solution for porosity model estimation. This study focused on developing the
ML-based model to estimate porosity from seismic data.

This method has several drawbacks. First, building the acoustic impedance
model from seismic data often incurs significant computational cost, as it
requires full-waveform inversion with corresponding seismic depth imaging Cho
et al., (2018, 2020). Although employing a model-based approach can be an
alternative to reduce the computing cost, this method requires manual seismic
horizon picking, a very labor-intensive task susceptible to subjective
interpretations Li, (1994); Cho et al., (2020). Second, this workflow relies
on an overly simplistic impedance-porosity correlation established in Step
(2). Typically, the correlation is assumed to be linear, and co-kriging (or
co-simulation) only represents the relationship with a linear correlation
coefficient Pyrcz and Deutsch, (2014). However, as shown in Figure 2, the
actual relationship between acoustic impedance and porosity is complex and
variant. Due to the poor representation of the impedance-porosity
relationship, the typically large spatial connectivity of acoustic impedance
often leads to overestimated porosity connectivity. In other words, sub-
seismic scale heterogeneity, which is important for reservoir simulation, is
not naturally honored in this workflow. Finally, most of the previous methods
are not capable take advantage of spectrally decomposed seismic data. When
converting seismic data to acoustic impedance, only seismic data from the
highest frequency is often used, not considering the relationship among
different frequency band widths.

Figure 2: An example of the relationship between porosity and acoustic
impedance. The data is acquired from the Great Australian Bright field and the
multiple well logs (e.g., sonic, density, and porosity logs) are used to
collect 2,803 data points Huuse and Feary, (2005). The plot shows strong
heteroscedasticity, as well as non-linear relationship between porosity and
acoustic impedance, making it challenge to integrate acoustic impedance by
simply implementing a linear correlation.

We hypothesize that a fully automated workflow can be built in a machine
learning (ML) framework to overcome the above issues. Instead of converting
seismic data to an acoustic impedance field, a properly trained ML model can
convert seismic data to porosity models directly (Figure 1). The suggested ML-
based workflow is composed of (1) stochastically generating a set of porosity
models conditioned on well data without the use of secondary data, (2)
generating spectrally decomposed seismic data from the porosity realizations
using a petrophysical forward model, (3) using the data from the two above
steps as “labeled data” and “features”, respectively, to train an ML model
that predicts porosity fields from seismic data, and (4) feed the actual
seismic data to the trained ML model to predict the porosity field.

Out of many possible ML algorithms, we propose to apply ResUNet++ Jha et al.,
(2019), originally inspired by U-net Ronneberger et al., (2015) and ResNet He
et al., 2015a  neural network structures. Although these algorithms were
originally developed for image processing, such as medical image segmentation,
in recent studies, the applications are expanded to a variety of different
scopes, such as fault detection from seismic sections Li et al., (2019); Yang
et al., (2020), enhancement of time-lapse seismic data Jun and Cho, (2021),
surrogate reservoir flow simulator Yan et al., (2020); Maldonado-Cruz and
Pyrcz, (2021), and proxy computational fluid dynamics in digital rocks Santos
et al., (2020, 2021). In applying this class of ML models, we analogize 3D
physical fields (e.g., velocity, porosity fields, etc.) to 3D images. In other
words, “grid cells” carrying physical values are treated as “pixels” or
“voxels”. ResUNet++’s image-to-image architecture is naturally suited for our
objective as both the input, spectrally decomposed seismic data, and the
output, porosity fields, attached to the same grid cells. In “training” the
neural network, we are essentially establishing correlations between fields of
two types of physical quantities. Inherent spatial characteristics, such as
heterogeneity of porosity in shorter length scales and large spatial
connectivity of acoustic impedance, are naturally “learned” by the neural
network and reflected in the results, thereby overcoming a drawback of the
conventional workflow. Because the training data are conditioned on well data,
the trained ML model cannot be readily used to predict porosity fields at a
different site. However, the conditioning of the models effectively reduce the
uncertainty space, so the training data set does not need to be very large.

In this paper, we propose an automated, ML-based porosity model estimation
from the decomposed seismic data. In Theory & Method, we explain (1)
geostatistics for generating multiple porosity realizations, (2) geophysical
forward model for converting the reservoir models to seismic responses (e.g.,
low-, mid-, and high-resolution seismic data), and (3) structure of ResUNet++
to estimate the porosity model from the decomposed seismic data. In Numerical
Examples, the suggested workflow is demonstrated in 3D channelized reservoir
models and tested its robustness by (1) expanding applications to different
geological scenarios and (2) adding signal noises, commonly observed in
acquiring seismic data. Moreover, we perform a comparative study between using
multiple-frequency seismic data and single-frequency seismic data in porosity
estimation to reveal advantage in using decomposed seismic data. In
Conclusion, we summarized observations in demonstrations and conclude our
work.

##  3 Theory & Method

###  3.1 Geostatistics

The reference model (Figure 3) is originally designed by Bosshart et al.,
(2018); Tang et al., (2020, 2021) for geologic carbon storage. The reservoir
model has an aerial extent of 40.96 km2 and the total thickness of 320 m,
discretized into 64$\times$64$\times$32 grids. Therefore, each grid block has
100$\times$100$\times$10 m3 dimensions. The first three top layers are the cap
rock (e.g., shale), and the remaining layers consist of two rock facies (e.g.,
shaly sandstone and sandstone), as shown in Figures 3 and 3. Sandstone has a
higher porosity than shaly sandstone, and shale has the least porosity.

Figure 3: Porosity and rock facies models (cross- and inline-sections) in
different vintages: (a) fence diagram of porosity, (b) horizontal and cross-
sectional views of porosity, (c) fence diagram of rock facies, and (d)
horizontal and cross-sectional views of rock facies. The black dash-lines show
the relative location of each cross- and inline-section. We applied a vertical
exaggerations ($\times$100) for displaying purpose.

As ML models require a relatively large number of stochastic realizations as
the training data, we generate ensemble of the reservoir models from the
reference model following the workflow in Figure 4. First, we extract the hard
data (porosity and rock types) along the well trajectories (Figure 3) from the
reference model. As the reservoir model consists of more than two rock types,
we applied normal-score transformation (NST) Pyrcz and Deutsch, (2014); Jo et
al., (2017) to porosity to make all rock types have the same scale (i.e., zero
means and unity standard deviation). Then, Sequential Gaussian Simulation
(SGS) and Sequential Indicator Simulation (SIS) are applied to generate NST
porosity and rock facies models, respectively Pyrcz and Deutsch, (2014). The
variogram of NST porosity is assumed to be isotropic with a horizontal
connectivity range of 250 m and a vertical range of 10 m. The variogram of
rock facies is assumed to be anisotropic with the 3 km and 1.5 km ranges in
easting- and northing-directions. Detailed parameters of the variograms are
summarized in Table 1. After generating a rock facies realization, we rescale
NST porosity to original porosity following their corresponding rock type.

Figure 4: The workflow to generate seismic data from the porosity and facies models. Table 1: Variogram model parameters of porosity and rock facies. Porosity has isotropic variogram, while rock facies has anisotropic variogram whose azimuth is 0∘ (i.e., easting). Property | Model type |  Major range |  Minor range |  Vertical range | Azimuth | Sill | Nugget
---|---|---|---|---|---|---|---
Porosity (SGS) | Exponential | 250 m | 250 m | 10 m |  $-$ (Isotropic) | 1 | 0
Rock facies (SIS) | Exponential | 3,000 m | 1,500 m | 10 m |  0∘ (Eastward) | 1 | 0
|  |  |  |  |  |  |

Following the geostatistic workflow, we generate 100 realizations of porosity
and rock facies models, and GeostatsPy Pyrcz et al., (2021) is employed for
SGS and SIS in this study. The first and second rows of Figure 5 presents
porosity and rock facies models. Moreover, the petrophysical model conversion
is applied to generate the corresponding post-stack seismic data of
realizations, as visualized in the third, fourth, and fifth rows of Figure 5.
The three seismic data and porosity models are input and output of ResUnet++,
respectively.

Figure 5: Examples of realizations.

We perform a statistical analysis of one realization in Figure 5. Figure 6
presents the relationships among porosity and three seismic data with
different frequencies. Each column and row indicate different properties. Even
though there is a visible porosity difference between shaly sandstone and
sandstone, the difference becomes negligible in seismic data histograms,
indicating seismic responses are insensitive to porosity. This is because the
porosity contrast with adjacent grid blocks is more sensitive to seismic
responses. Therefore, straightforward correlations between the porosity and
seismic data are not available to predict porosity models, rationalizing our
hypothesis of applying an ML workflow to consider these geometric features in
predicting porosity models.

Figure 6: Scatter plot matrix between porosity and seismic data with different
frequencies. Blue, orange, and green points indicate shaly sandstone,
sandstone, and shale, respectively. Diagonal plots are the histograms of
properties, and the rest of the plots are scatter plots between two
corresponding properties in the row and column.

###  3.2 Petrophysical model conversion to seismic data

There are a number of methods to infer the velocity from the porosity models
Wyllie et al., (1956); Todd and Simmons, (1972); Han et al., (1986). In this
research, we applied Eberhart-Phillips’ method Eberhart-Phillips et al.,
(1989), in which the empirical equation is widely used for the sandstone
environment. Eberhart-Phillips et al., (1989) determined the empirical
relationship (equation 1) among multiple earth parameters: velocity, effective
pressure, porosity, and clay contents in sandstone core samples, which can be
expressed as

$\begin{split}V_{\text{p}}^{*}&=5.77-6.94\phi-1.73\sqrt{\gamma}+0.446~{}(P_{\text{e}}-e^{-16.7P_{\text{e}}}),\\\
V_{\text{s}}&=3.70-4.94\phi-1.57\sqrt{\gamma}+0.361~{}(P_{\text{e}}-e^{-16.7P_{\text{e}}}),\end{split}$
(1)

where $\phi$ and $\gamma$ denote porosity and clay contents, respectively.
$P_{\text{e}}$ means effective pressure in kbar unit, and we assume that the
effective pressure is linearly proportional to the depth. $V_{\text{p}}^{*}$
and $V_{\text{s}}$ are matrix velocity of P- and S-wave. Eberhart-Phillips et
al., (1989) applied corrections to the matrix P-velocity as follows:

$\frac{1}{V_{\text{p}}}=\frac{1-\phi}{V_{\text{p}}^{*}}+\frac{\phi_{\text{g}}}{V_{\text{g}}}+\frac{\phi_{\text{w}}}{V_{\text{w}}}+\frac{\phi_{\text{o}}}{V_{\text{o}}},$
(2)

where $V_{\text{p}}$ is the bulk P-velocity includes the influence of fluid
components. The subscript o, g, and w represent oil, gas, and water,
respectively. The pore space is assumed to be fully filled up with fluids
($\phi=\phi_{\text{o}}+\phi_{\text{g}}+\phi_{\text{w}}$), and the constant
fluid velocity is applied as follows: $V_{\text{o}}=1300$m/s,
$V_{\text{g}}=4800$m/s, $V_{\text{w}}=1500$m/s. We obtain the bulk density
volume by taking average of the matrix, and fluid velocity is calculated by
considering the clay contents ratio. We used the Gardner’s relationship
Gardner et al., (1974) to make synthetic density model.

As our suggested workflow is demonstrated to estimate the porosity models from
the given post-stack seismic data, we generated synthetic seismic data by
providing the porosity and rock facies models (the first column in Figure 3)
as an input. A schematic sketch of the workflow for seismic data generation is
displayed in Figure 7. We infer the velocity and density information from the
porosity and rock facies models, then calculate acoustic impedance models to
convolve with the source wavelet. To demonstrate the capability of the
proposed method for resolving sub-seismic scale geology, we generated three
different seismic volumes with different frequency bandwidth: 10 Hz, 20 Hz,
and 30 Hz, which we label high-, mid-, and low-resolution seismic data in
Figure 7.

Figure 7: The process to convert porosity and facies model to seismic data in
multi-scales (10 Hz, 20 Hz, and 30 Hz).

###  3.3 ResUNet++ for porosity model estimation

With a basis of convolutional neural network and ResUnet structure He et al.,
2015a , ResUnet++ includes new advanced techniques, such as the squeeze &
excite block Hu et al., (2018), atrous spatial pyramid pooling (ASPP) Chen et
al., (2017), and the attention block Vaswani et al., (2017), which enhance the
stability and performance of the model. From the workflow for 2D image
segmentation, we expand it for 3D image prediction (e.g., 3D seismic data and
porosity models), and the architecture of neural networks is described in
Figure 8. The detailed structure of ResUnet++ is available via the author’s
repositories: https://github.com/whghdrms.

Figure 8: The block diagram of the proposed ResUnet++ architecture (modified
from Jha et al., (2019))

In this study, input and output of our ResUnet++ are seismic data (three
different resolutions) and porosity model. The ResUnet++ consists of two main
steps: encoding and decoding, which are connected by a bridging block (i.e.,
ASPP). Through the encoding stem, the dimensions of feature maps decrease
while the number of channels increases to extract main geometric features in
seismic data. After passing the ASPP block, the feature maps go through the
decoding stem, where their dimension is expanding back to the original
reservoir size (i.e., 64$\times$64$\times$32). In restoring the original
dimension, the encoded seismic information is transformed to the corresponding
porosity model.

Typical neural network operations (e.g., convolution, batching normalization,
activation functions, up-sampling, and pooling) are visualized in white and
blue blocks in Figure 8. Jun and Cho, (2021) extensively explained the
mathematical backgrounds of these operations. The yellow blocks indicate
element-wise addition and multiplication, and the purple blocks denote
concatenating feature maps over channels. Especially, the addition and
concatenation are for skip connections (or shortcuts) that adjusts the
complexity of the neural network flexibly He et al., 2015a ; Ronneberger et
al., (2015). The green blocks are the squeeze & excite block and attention
block whose detailed sequence are visualize in the right side of Figure 8. The
key idea of these two blocks is to emphasize regional information of feature
maps to improve the performance of the neural network model Vaswani et al.,
(2017); Hu et al., (2018), which are widely used for recent natural language
processing studies, such as Wang et al., (2018); Young et al., (2018). The red
blocks indicate the ASPP, a technique for resampling features in multi-scales
(He et al., 2015b, ).

The seismic data and porosity models are re-scaled through Min-max
normalization to have the range between -1 and 1, as described in the
following equations:

$\begin{split}{\phi_{\text{normalized}}}=\frac{\phi-\phi_{\min}}{\phi_{\max}-\phi_{\min}}\times
2-1,\\\ {S_{\text{normalized}}}=\frac{S-S_{\min}}{S_{\max}-S_{\min}}\times
2-1,\end{split}$ (3)

where $\phi$ and $S$ are porosity and seismic data, and subscript normalized,
$\min$, and $\max$ represent normalized (or re-scaled), minimum, and maximum
values, respectively.

The mean squared error (MSE) of the estimated porosity is used as a loss
function ($L$) in the proposed ResUnet++ and can be expressed as follows:

${L}=\frac{1}{N_{\text{batch}}\times
N_{\text{grid}}}\sum_{n=1}^{N_{\text{batch}}}\sum_{i,j,k=1}^{N_{\text{grid}}}(\hat{\phi}_{n,i,j,k}-\phi_{n,i,j,k})^{2}$
(4)

where $N_{\text{batch}}$ and $N_{\text{grid}}$ are the number of training data
in each training batch and the number of grid cells in each porosity model.
Moreover, the subscript $n$, $i$, $j$, and $k$ indicate the $n$-th training
data and values at the ($i$, $j$, and $k$)-th grid cells in x-, y-, and
z-directions, respectively. $\hat{\phi}$ and $\phi$ are estimated and true
porosity.

##  4 Numerical examples

###  4.1 Training and validation data

We demonstrate the ResUnet++ assisted porosity estimation workflow with the
ensemble of 3D porosity models and seismic data (Figure 5). The seismic data
from three different frequencies (e.g., 10 Hz, 20 Hz, and 30 Hz) are used as
inputs, and the porosity models are the output of ResUnet++. Out of 100
realizations, 70 realizations are used to train the neural networks, and the
rest are retained and used for validation. Min-max normalization is a
prerequisite for implementing the neural network model to remove the impact of
different scales of properties (e.g., porosity and seismic data).

###  4.2 Porosity estimation via ResUNet++

We train ResUnet++ for 1,000 epoch with a batch size of 25, requiring about 5
hours on a desktop computer with an environment of CPU Intel E5-1650 3.60 GHz
and GPU NVIDIA Quadro M6000 24 GB. The stop criterion of training ResUnet++ is
when the loss is stabilized and shows no decrease further in both training and
validating data. For deployment, the trained ResUnet++ predicts a porosity
model in less than a second.

As shown in the upper plot of Figure 9, a significant discrepancy between
losses of training and validating data are observed until 300 epochs, but both
of them decrease simultaneously between 400 and 1,000 epochs, indicating a
stable training process. Moreover, no over-fitting is observed with the given
neural network structure, as the difference between training and validating
loss is minimal in the end. As a secondary metric, we also compute mean
absolute error (MAE), and MAE shows a similar pattern as MSE (lower plot of
Figure 9).

Figure 9: Loss curves of the ResUnet++ over 1,000 epochs

Figure 10 presents the demonstration of porosity estimation from training
data. Out of 70 training data, four realizations are visualized in each row of
Figure 10. In the figure, the first three columns show seismic data in
different resolutions, and the fourth column shows the estimate of the trained
ResUnet++. The fifth and sixth columns depict the truth model and error
between the estimate and truth, respectively. Even though the detailed
heterogeneity in estimated porosity models is not identical to the truth
models, the global trend is well preserved in the estimated porosity model.
Furthermore, errors are close to zero and stochastically distributed in space,
indicating there is no systemic bias in the trained ResUnet++.

Figure 10: Porosity estimation for training data

Figure 11 displays porosity estimation from four validating data. Same as the
demonstration of training data, the estimated porosity models successfully
capture global trends. Especially, even though the third validating data shows
connected high porosity regions in the northing direction, which is not
observed commonly in training data, the ResUnet++ successfully predicts the
porosity model that follows such trend. The error maps also show near zero in
the entire extent, as shown in the last columns of Figure 11.

Figure 11: Porosity estimation for validating data

We visualize the truth versus prediction plots for training and validating
data, as shown in the first two rows of Figure 12. For both training and
validating data set, most of the points along with 1:1 lines with over 0.9
coefficient of determination (i.e., R2 score). Besides, no significant bias
(e.g., systemic under- or over-estimating) is observed in the plots.

Figure 12: Truth versus prediction plots of ResUnet++

###  4.3 Demonstration of robustness

We perform the stress test to the trained ResUNet++ to verify the
expandability of applications to more practical cases, such as estimating
porosity models for different geological scenarios or adding signal noise to
seismic data. In addition, we conduct a comparative study using single
frequency seismic data instead of three frequencies.

####  4.3.1 Application to different geological scenarios

One of the sailent feature of the reference model and ensemble is that the
major connectivity of rock facies is along with the easting direction,
creating a larger high porosity connection along with the easting direction as
shown in Figures 3 and 5. We alter the major connectivity direction from
easting to northing, which is the most significant change we can make, and
generate another set of realizations as visualized in Figure 13. Even though
the ResUNet++ has never seen this new configuration in training, the trained
ResUNet++ shows reliable prediction performance in Figure 13. Most of the high
porosity regions are well captured in the estimated porosity model, same as
the truth models, and error maps have negligible values.

Figure 13: Porosity estimation with different geological scenarios

The third row of Figure 12 shows the truth versus prediction of the ensemble
with major connectivity directions. Same as training and validating data, most
points are located along the 1:1 line, and R2 scores are over 0.86 in all
cases. The degree of misfits between prediction and truth becomes slightly
larger than the initial ensemble, but they are within an acceptable range.
Therefore, with a properly trained ResUNet++, we can expand the applications
to seismic data from different geological scenarios, such as different major
directions and lengths.

####  4.3.2 Resilience against signal noise

After generating another set of realizations and calculating seismic data, we
add 5% signal noise to the seismic data (i.e., 5% signal-to-noise ratio). Even
though, none of these ensemble are used in training ResUNet++, Figure 14 shows
that the trained ResUNet++ successfully predict high and low permeability
regions. The error maps show larger values than the demonstration of training
and validating data set, but they are still close to zero without any systemic
bias. The fourth row of Figure 12 presents the truth versus prediction of the
ensemble with 5% signal noise. The degree of misfits becomes larger than in
previous cases, but R2 scores are still as high as 0.79.

Figure 14: Porosity estimation with 5% signal noise

###  4.4 Comparative study with single-frequency seismic data

We perform a comparative study using single-channel seismic data. Instead of
three different channels (e.g., low-, mid-, and high-resolution from 10 Hz, 20
Hz, and 30 Hz frequencies), we redesigned the first layer of ResUNet++ to take
a single frequency seismic data. We train the models by only taking either the
lowest and highest resolution seismic data. Figure 15 shows the loss curves of
training and validating data.

Figure 15: Loss curves of the ML-based models with (a) low- and (b) high-
resolution seismic data only.

The model that only takes the lowest resolution seismic data results under-
fitting as the losses of training and validating data do not effectively
decrease until 1,000 epochs (Figure 15). For example, the loss of training
data is twice larger than the model with three different seismic data.
Moreover, the loss of validating data becomes three times as large as the
original model.

On the other hand, the model that only takes the highest resolution seismic
data shows over-fitting. Figure 15 displays that the discrepancy between
training and validating losses is significantly large at the end of the
training, indicating the unreliable predictive performance of ResUNet++ out of
training data. Under- and over-fitting results with single seismic data show
the importance of using multi-resolution seismic data to estimate porosity.

##  5 Conclusion

An ResUNet++-based model is suggested to estimate porosity from three post-
stack seismic volumes with different frequencies and demonstrated in the 3D
channelized reservoirs. The suggested workflow directly takes the spectral
decomposed seismic data and estimates the porosity model in less than one
second. Moreover, we expand the application of the workflow to different
geological scenarios and noisy seismic data, proving the robustness of the
workflow. Even with Seismic data from different major connectivity
orientations, the suggested method shows reliable prediction performance. The
suggested workflow also stable prediction outcomes up to 5% signal noise in
seismic data, which is crucial for practical implementation for real data in
future studies. Our work leads to the following three main conclusions:

First, by reversing the sequence of porosity estimation, the suggested
workflow can shorten computational cost and alleviate subjectivity in
interpretations in converting the acoustic impedance model from seismic data.
The suggested workflow generates stochastic porosity realizations and converts
them to post-stack seismic volume, which is more straightforward than
converting seismic data to acoustic impedance. Moreover, our results suggest
that 100 realizations are enough to train ResUNet++ in estimating porosity
models of 64$\times$64$\times$32 grid, computationally available in typical
desktop computers.

Second, the ResUNet++-based model can linearity assumption between porosity
and acoustic impedance, mitigating possible artifacts such as a poor
representation of the impedance-porosity relationship and overestimation of
porosity connectivity. Instead of relying on bi-variate correlation (e.g., co-
kriging), the ResUNet++-based model enables to use spacial-correlations
between seismic data and porosity in estimating porosity.

Third, using multiple spectral decomposed seismic data helps stabilize the
training process of ResUNet++. With the comparative study between single-
frequency and multiple-frequency seismic data, our results support the
importance of employing multiple seismic data with different resolutions to
estimate porosity. Moreover, whereas conventional methods have no
straightforward way to integrate multi-spectral decomposed seismic data, the
ResUNet++-based model can easily take a set of seismic data with various
frequencies thanks to the flexibility of neural networks.

## Data availability statement

The synthetic model and the raw/processed data required to reproduce these
findings cannot be shared since authors do not have legal right to release the
data.

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