3.2 Time-lapse geophysical data response

Primary Subsurface Decision Variable (
): Existence of CO2/brine plume in shallow aquifer due to
leakage from deep C02 storage reservoir via an abandoned wellbore:
o

Subsurface Earth models (z(s)): 5 geologic parameters varied: correlation coefficient in x and z
direction, permeability of sand and clay facies (geologic units), and sand volume fraction

Decision alternatives (a): Given that this groundwater is used for irrigation, which crop should be
planted: a more profitable crop (corn) or more salt-tolerant (wheat)?

Information Considered to make decision variable interpretation (
): electrical resistivity
o
3 different survey acquisition geometries (as location of leak is deemed unknown)
o
2 different acquisition times: time=0 years and time=50 years since start of CO2 injection
Geologic sequestration of CO2 is a possible mitigation technique to address global climate change
(Benson & Surles, 2006). Possible subsurface receptors of this CO2 are depleted gas and oil fields
(Jenkins et al., 2012). Two important unknowns when monitoring CO2 sequestration operations will be:
did a leak occur and if it did, where did the leak occur? This example considers the possibility of CO2 and
accompanying brine leaking from a deep, depleted oil field into a shallow aquifer that is used as a source
of irrigation water. The considered conduit of leakage is an abandoned wellbore that connects the deep
reservoir to the surface; Watson & Bachu (2009) estimate that 4.5% of abandoned wellbores pose a risk
of leakage. Therefore, the prior uncertainty of a leak occurring is
. For all
details of this work, please see Trainor-Guitton et al., (2013).
Southwest Kansas has depleted oil fields that could be potential geologic storage sites of CO2. The High
Plains Aquifer lies above these oilfields and is extremely vital to the agricultural economy. The largest
economic consequence of changing groundwater chemistry due to a combined CO2 and brine intrusion
would be to the irrigated crops, of which corn is the most profitable and ubiquitous. Corn is sensitive to
the salinity of the irrigation water as demonstrated in Table 3, which shows the total dissolved solids
(TDS) concentrations in mg/L concentration that result in crop yield reductions of 10%, 25%, and 50%
(Ayers, 1977). Table 3 also shows the TDS levels that will result in the same crop yield reductions for
wheat. These concentrations are much higher than for corn, illustrating that wheat is much more
tolerant to saline water. Figure 5 demonstrates the decision tree of this example. The value outcomes
are based on revenues and costs for 100 acres of each crop (Dumler et al., 2011; Dumler & Shoup,
2011).
Table 3: Crop Yield Reduction as function of Salinity (modified from Ayers(1977))
Crop Yield
Reduction
0%
10%
25%
50%
Corn
700TDS
1088TDS
1600TDS
2500TDS
Wheat
2560 TDS
3130 TDS
5120 TDS
6960TDS
Figure 5: Decision tree for Example 1: principal uncertainty is if a plume of 2,000 mg/L exists in agricultural aquifer due to
CO2/brine leakage. Value outcomes based on 100 acres of each crop (Dumler, Brien, & Martin, K., 2011; Dumler&Shoup,
2011)
To represent the range of possible contamination, an uncertainty quantification (UQ) analysis provided a
dataset of 714, 3D groundwater samples which assume an abandoned well is a source of leakage. The
UQ study is based on a section of the High Plains Aquifer in southwest Kansas and was performed to see
how 9 different subsurface and leakage parameters would determine the existence and/or extent of
contamination to the shallow aquifer (Mansoor et al., 2012; Sun et al., 2013). See Table 4 for the ranges
sampled for the 5 subsurface heterogeneity parameters and the other hydrogeologic parameters varied
(scaling of leakage rates, brine concentration, etc.). On each sample, groundwater fate and transport
was simulated for 100 years, assuming 50 years of CO2 injection into the deeper reservoir.
Table 4: Parameter ranges
Parameter
Wellbore Leakage Model
Sand volume fraction
0.35 to 0.65
Correlation length in X-direction
200 to 2500 m
Correlation length in Z-direction
0.50 to 25.00 m
van Genuchten α in clay (related to the
inverse of the air entry suction)
-5.50 to -4.14 m-1
Permeability in sand
10-12 to 10-9 m2
Permeability in clay
10-18 to 10-15 m2
CO2 leakage flux scaling parameter
0 to 1.0
Brine leakage flux scaling parameter
0 to 1.0
Brine concentration
0.3-3moles/kg
Our decision variable
indicates whether a plume exists (i=1) or does not exist (i=2), where a plume is
defined as the area or volume of the aquifer that exceeds the maximum contamination level (mcl) of
TDS (EPA, 2010). Here, plumes of 2,000 mg/L or higher results in an economic loss when planting corn,
and therefore is the threshold considered to define the decision variable
dissolved solids at the 50 year time step for each sample
. The concentration of total
is evaluated to determine if a plume exists
or not:
(15)
The samples of the UQ study represent 4.5% of the total event space since the well acts as a leakage
source from the deeper CO2 storage reservoir in all S=714 simulations. Not all samples resulted in a
plume of 2,000 mg/L. No plume is expected when no leak has occurred. Thus, the probabilities of a
plume or no plume occurring (branches in Figure 5) is calculated by:
(16)
(17)
For the case of 2,000 mg/L,
and
. Using
these uncertainties along with the value outcomes of Figure 5 in Equations 4 and 5,
and
is
. Therefore, the value of perfect information (VOIperfect)
(see Table 5).
Table 5: Value of Imperfect Information (VOIimperfect) for 3 Different Electrode Locations
Electrode
Location
(Middle
Electrode
Distance
from
Leaking
Well)
Vprior =$32,744 VOIperfect = $420
Vimperfect(mcl=2000)
VOIimperfect(mcl=2000)
Surface
(400m)
$32,530
$200
Surface &
Borehole
(400m &
210m)
$32,560
$240
Surface
(1500m)
$32,340
$20
2D Electrical Resistivity
From the vantage point of the farmers, knowledge of the existence of a high salinity plume in the
groundwater would be useful before the choice of crop is determined. A remote-sensing technique
would be preferable to well sampling, as the wells are expensive and only sample one lateral position. A
remote-sensing device could detect the plume before it reaches the well. Two factors make this
situation favorable for electrical resistivity: the target (the plume) is electrically conductive and the
aquifer depth is 240m (relatively shallow). Dissolved CO2 will result in dissociated hydrogen ions, which
will increase the ionic conductivity of the groundwater (Carroll et al., 2009). Brine will also increase the
ionic conductivity with dissociated sodium and chloride ions.
An electrical resistivity survey uses surface and/or borehole electrodes to induce electrical current into
the subsurface in order to estimate the electrical resistivity of the underlying media (Daily et al., 1992;
Ramirez et al., 2003). Two electrode pairs alternate between acting as the current electrodes and
potential electrodes which results in a large number of measurements. The bulk resistivity
of a
homogeneous half-space can be calculated using a form of Ohm’s Law
(18)
since the locations of the four electrodes (r) are known and the amount of current input into the ground
( ) and the voltage difference (V) between the two potential electrodes are measured. Each
measurement, known as a transfer resistance (V/I), will have a different depth of investigation due to
the separation of four electrodes used in this measurement.
Different geometries and depths of plumes will influence the reliability of electrical resistivity to
accurately detect the existence of a leak. A significant benefit of utilizing the UQ study samples is that
one can capture a comprehensive set of conditions where false positives and false negatives may occur.
This is accomplished by simulating the electrical resistivity response on all the 714 UQ samples. First,
the geochemical variables must be translated into electrical resistivity (or its inverse: conductivity).
To calculate the ionic conductivity of the pore water
for every simulation sample
, we follow
Hearst et al. (2000). Rock physics relationships are used to estimate the bulk rock electrical resistivity
using the porosity of the media and the calculated pore water conductivity (Archie, 1942; de Lima &
Sharma, 1990). Therefore, we compute and include the bulk electrical resistivity
associated transfer resistances for time=50 years as attributes of vector
(Ω-m) and the
(Equation 1). Figure 6
displays two properties associated with sample 275: total dissolved solids (equivalent to mg/L
concentration) at time = 50 years and the factor change in electrical resistivity from time = 50 years
relative to time=0 years.
Figure 6: Sample 275. Longest mcl = 500 mg/L plume (of 714 simulations). No plume at mcl = 2000 mg/L. Top panel shows
total dissolved solids (TDS). Bottom panel shows resistivity ratio (time=50 years/ time=0)
Recall that the location of the leak is considered to be unknown. Figure 7 demonstrates the three
possible electrical resistivity survey scenarios that are considered to test the sensitivity of the technique
at different distances to the plume. The first (demonstrated in red) is a 1,200 m line of surface-based of
electrodes, where the westernmost electrode is 200 m upstream of the leaking well. The line consists of
120 electrodes. The length of 1,200 m was chosen based on the depth of the aquifer, as it will provide
the necessary depth of investigation. The second survey geometry includes 11 borehole electrodes (in
addition to the 120 surface electrodes) located 250 m downstream of the leak. The last survey scenario
shifts the 1,200 m line of surface electrodes downstream (away from the leak) by 1,100 m. The distance
of the closest electrode to the leaky well is 900 m (no overlap as in the first survey scenario). This
distance was chosen based on the average well spacing of the area of interest of the High Plains aquifer,
such that the position of the electrode line is equidistance between the hypothetical leaking well and a
neighboring well.
Figure 7: Three electrode configurations. (1) Red: 120 surface electrodes straddling leaking well. (2) Red and purple:
straddling electrodes plus 11 borehole electrodes200 m downstream of leaky well. (3) Green: surface electrodes 900 m
downstream of leaky well
Since the electrode lines only span the x-axis of the aquifer grid, 2D slices of the 3D aquifer simulation
results are extracted, representing the plane made by the x-axis and z-axis. This profile captures the
effect that regional flow will have on any pH/TDS plume.
Leak Diagnostic
Electrical resistance measurements were computed for all
represents forward geophysical calculation).
samples of
at 0 and 50 years (function
(19)
3% random Gaussian noise was added to each of the transfer resistance predictions to simulate realistic
field measurements (LaBrecque & Daily, 2008; LaBrecque et al., 1996).
Typically, these transfer resistances would be used in geophysical inversion (
) to estimate the
resistivity structure of the earth; however, geophysical inversion can be computationally intensive for
one set of field measurements. Thus, it is computationally expensive to perform inversion for all
resistivity profiles. Considering the decision variable, the goal is to use the transfer resistance
measurements to determine if a leak has occurred and resulted in a plume, not to know the spatial
structure of the plume; we follow the approach of Daily et al. (2004) and borrow their mean of
logarithmic ratios (MLR) as a leak diagnostic metric
(20)
wheren is the number of measurements taken on each sample, rktime is the kth transfer resistance at year
= time, and rk0 is the transfer resistance at year = 0. A MLR is calculated for each aquifer realization s,
and this is used as an interpretation of electrical resistivity detecting aconductive plume, represented by
. The index j represents the J discrete bins of the MLR values.
The 714 calculated MLR’s only represent leak events (estimated as 4.5% of the event space). Therefore,
15,153 “synthetic” MLR’s (representing non-leak events) are generated by utilizing the statistics of the
MLR’s calculated from non-plume simulations (mcl<500 mg/L). These synthetic MLR’s include the effects
of noise added to the transfer resistances.
The data reliability of the electrical resistivity technique can be calculated by comparing MLR’s (
the plume existence (red) or non-existence (green)
) to
(21)
The reliability in the top plot in Figure 8 is for the surface electrode configuration straddling the leaky
well. The information posterior
(22)
is the bottom plot of Figure 8. In general, MLR’s < 0.001 indicate that no plume (green) is present while
MRS’s > 0.002 are indicative of a plume (red). The range between 0.001 and 0.002 demonstrate
ambiguity in the electrical resistivity message: the probability of a plume existing is comparable to it not
existing. This exhibits the imperfectness of the electrical resistivity measurement.
Figure 8: (a) Data reliability and (b) information pre-posterior for mcl = 2000 mg/L at time = 50 years
There is an obvious deviation around MLR = 0.002. This is due to sample 275, which has the longest
plume at threshold mcl ≥ 500 mg/L (for all 714 samples), but no plume for mcl ≥ 2000 mg/L (see Figure
6). The electrodes have a high sensitivity to the resistivity change of the long, lower concentration
plume, but sample 275 does not contain concentrations greater than 1,000 mg/L at 50 years. Therefore,
sample 275 represents a situation where the message from the information will not uniquely identify if a
plume of 2,000 mg/L exists.
The information posterior is used to calculate the value with imperfect information (Equation 8). The
same process is repeated for the two other electrode configurations. The three values of imperfect
information (VOIimperfect) for the three configurations are summarized in Table 5. As expected, all are less
than VOIperfect : the VOIimperfect range from 5-55% of the VOIperfect value. Also, the configuration that
includes borehole electrodes has the highest value and the configuration farthest from the leak
(downstream surface electrodes) has the lowest. This is expected given that the strength of the
electrical resistivity signal will increase when electrodes are placed in-situ (e.g. the borehole) and the
signal will drop with increasing distance from the target (Equation 12).
Summary: CO2 Leak Example

A leak diagnostic (MLR) was introduced to assess if the electrical resistivity technique can
achieve a holistic measure of how well ER data will indicate the existence of plume.

o
Simulated electrical response on all 714 UQ samples.
o
Avoided geophysical inversion (computationally expensive for 714 datasets).
Imperfectness of electrical resistivity was captured with the MLR diagnostic.
o
The electrical resistivity response to larger but lower concentration plumes can be
equivalent to the response of higher concentrated plumes. This creates false positives
and affects the information reliability.
o
Ambiguity in the message at MLR’s between 0.001 and 0.002: similar probabilities are
assigned for plume and no plume.

Decision alternative is not spatial.
o
A possible decision could be where (if at all) the aquifer requires remediation due to the
CO2/brine leak. However, after conferring with local agriculturists regarding how they
currently address issues of the salinity of irrigation water, crop rotation was chosen as
the decision alternative.
3.3 Equivalent Collocated Geophysical Inversions and Geologic Observations
Aquifer vulnerability: removal of contaminate sources at critical locations

Primary Subsurface Decision Variable (
): Surface-locations that act as entry points into aquifer
o


Subsurface Earth models (z(s)): Depositional system: buried glacial valley
o
18 training images: capturing different dimensions of buried valleys
o
10 realizations each for S=180 subsurface models
Decision alternatives (a): Which farms should be compensated to not use fertilizers?
o

Information Considered to interpret decision variable (
): existing time-domain electromagnetics
collocated with drillers logs (subjective lithology information)
This example is inspired by a Northern European situation, where the populace relies solely on its
groundwater resources for drinking water. The aquifers have been compromised by surface-sourced
contaminants due to agricultural activities. Contamination will continue to be a threat until the crops or
farms that are located at entry points into the aquifer are identified and the source of contamination
removed. The key uncertainty (surface locations that allow contaminants to enter the aquifer) is related
to the geologic depositional system. For this location, it is buried glacial valleys, which present
overlapping and cross-cutting, sinuous features (Figure 9). A reasonable assumption is that the buried
valley facies are filled with sand and represent high volume aquifers; sand facies are therefore assigned
a high permeability value. Conversely, the non-valley or background facies are assumed to be aquitards
and are assigned a low permeability value. The details of this example can be found in Trainor-Guitton et
al. (2013).
Figure 9: Plane-view of network of buried valleys; darker to lighter representing older to younger buried valley generations
(Jørgensen&Sandersen, 2006)
For this example, 18 training images are generated to represent the model uncertainty of the buried
valley length, width and thickness dimensions. Figure 10a demonstrates two of these training images.
Ten Earth models are generated for each training image (examples shown in Figure 10b). As seen in
Figure 9, the network of connected buried valleys is complex; ‘‘significant parts of the recharge area may
therefore lie at relatively large distances from the valley [which represents the deep aquifer]’’
(Sandersen & Jørgensen, 2003). Thus, contamination can be transported kilometers from its surficial
entry point into a deep aquifer. Therefore, the dynamic response or transport of surface-borne
contaminants into the aquifer is the decision variable
.
Figure 10: ) Two examples of training images used to represent patterns of glacial, buried valleys b) Examples of models or
samples generated from these training images
Aquifer vulnerability, the decision variable, is determined by placing a tracer on the surface of all Earth
models
, simulating the groundwater flow and transport (
), and tracking the volume of aquifer
affected by entry at each surface location .
(23)
Figure 11 is an example vulnerability map resulting from one individual model via flow simulation. The
magnitude of vulnerability reflects the volume of aquifer affected from entering at that surface location.
Figure 11: One vulnerability map assessed from one individual model via flow simulations. Vulnerability defined as volume
of aquifer (m3) affected by entry at particular surface location.
Equivalentto Equation 3, the function
predicts the outcome of alternative on the decision variable
of aquifer vulnerability:
(24)
For this case, the decision is made independently at each possible location of a farm (represented by )
and the alternatives are to compensate or not. Four possible outcomes are possible at each surface
location depending on that location’s vulnerability
(25)
The first two outcomes represent the cost (negative value) of compensating a farmer to stop using
chemicals to avoid contamination of the aquifer. The third cost (negative value) is twice as much,
representing the environmental consequences of not compensating a farm located on an effective entry
point to the aquifer. The last cost outcome has a zero cost as no compensation or contamination occurs.
Note that we have chosen to threshold the aquifer vulnerability to two levels: 0 or positive. Other levels
of vulnerability (i.e.
100) could be considered to be more appropriate for particular situations.
The Vprior expression with the possible farm locations is
(26)
.
This prior value includes a sum over the locations
since it is considered that this decision is made
independently at each farm. There are L=7,776 farm locations on the surface of the models for this
example. Please see the original manuscript for another approach that does not assume independence
for each decision location.
Calibrated TEM and drillers logs
No information source directly or indirectly measures aquifer vulnerability. Time-domain
electromagnetic (TEM) data is currently being collected to assist in the decision of determining which
farms are at vulnerable locations. The time-domain electromagnetic data is collected to infer lithology
types (valley or non-valley). Although lithology is not equivalent to vulnerability, the lithology
information can be used to “condition” models, and therefore constrain the possible aquifer
vulnerability calculation.
This example takes advantage of the fact that TEM information has already been collected in proximity
(within 10 meters) to drillers logs: subjective observations of lithology. Therefore, one can quantify of
how well TEM can differentiate between valley (sand) and non-valley (clay) lithologies by comparing the
collocated datasets. Figure 12 plots the occurrence of electrical resisitivities (from inverting the TEM
measurements) and the collocated drillers log observations for sand and clay. In general, sand does have
electrical resistivities above 60 ohm-m and clay below 40 ohm-m. However, there is still uncertainty as
both lithology types have occurrence of high and low electrical resistivities.
Figure 12: Likelihood of electrical resistivity given lithology values from 7 collocated TEM and drillers log a) Electrical
Resistivity of Sand (valley). b) Electrical Resisitivity of Clay (non-valley)
Both electrical resistivity and lithology are properties contained within vector
. Figure 12 represents
the relationship
(27)
I will call this the data likelihood, not the reliability, since it does not include the decision variable
outright.
Soft Probability
However, the data reliability is necessary to estimate the posterior value or value with information.
Thus, a connection must be made from the data likelihood in Equation 25 to the decision variable. This
is achieved by sampling Howard’s (1966) formulation of the value with information:
(28)
Here d represents the data and
, unlike all previous equations, represents a generic probability density
function. In the previous examples, many datasets (d) were explicitly generated (simulating the physics
of measurement) to represent the possible variability expected from the information source. Here, the
data likelihood of Equation 25 (from the collocated datasets seen in Figure 12), the prior models
and Monte Carlo simulation are used to generate the posterior value. The integrals are approximated by
using an arithmetic average calculated through the Monte Carlo sampling. The approximation can be
described in 5 steps (shown in Figure 13):
(1) Use the likelihood function to generate a synthetic electrical resistivity cube
lithology of each prior Earth model
from the
.
(2) For each of the synthetic electrical resistivity cubes
facies. This is known as the ‘‘soft probability’’,
, derive a probability for both lithologic
, for conditioning multiple-point realizations
(Caers, 2005).
(27)
(3) Use that information posterior probability
as a soft probability in a geostatistical simulation
algorithm to generate multiple (w), new Earth models constrained to each of the synthetic data
sets:
.
(4) Create realizations (w) of aquifer vulnerability
function
by applying the dynamic simulation
to the conditioned (interpreted) Earth models.
(5) Calculate the value
aquifer vulnerability:
of the various alternatives
based on the values of posterior
.
Repeating these steps provides multiple value outcomes from which an estimate of the expectation in
Equation 26 can be calculated.
Figure 13: Steps of Monte Carlo simulation to estimate the value with data
For each of our buried valley prior models, one electrical resistivity dataset is generated, for a total of
180 datasets from the likelihood of Figure 12. Figure 13 demonstrates this Step 1 and displays an
example synthetic dataset of electrical resistivity that was generated using one prior model and the cdf
versions of the pdf’s in Figure 12. These electrical resistivity datasets represent the information we could
expect to collect, given our uncertainty of both the subsurface heterogeneity (represented with the
prior models) and the ability of the TEM data to resolve lithology (represented through the data
likelihoods). Step 2 translates this resistivity into a soft probability cube for each lithology.
Realizations, represented by w, are generated that are “conditioned” to the soft probability. By
generating several conditioned Earth models, we can capture the different possible Earth model
interpretations which result from the overlap in the data likelihood, i.e. imperfect information. For each
, two (W = 2) new conditioned Earth models are created
. This can be considered the
minimum of conditioned models that should be generated. More conditioned models will capture the
possible variability due to the imperfect geophysical information message. From the data likelihood in
Figure 12, 360 conditioned models are created, each constrained to their respective synthetic datasets.
Finally, the “conditioned” aquifer vulnerability
is calculated (Equation 23).
The following expression is for the value with imperfect information using the Monte Carlo sampling
method:
(30)
In the case of perfect information
perfectly recovered through the data into
will be equal to
, as all prior models will be
. Therefore, we are assured that the best possible
decision is made given our prior models. Whereas with data that has no information content, the
interpreted or conditioned models will poorly represent the prior Earth model they originated from and
will be quite dissimilar from each other. Therefore decisions made on ‘‘inaccurate interpretations’’ will
lead to lower value outcomes on average. Higher quality data will ultimately lead to higher valued
decision outcomes and consequently, a higher VOIimperfect. If the proposed information, represented
synthetically with
, can constrain the results of the dynamic simulation function and subsequently
the decision variable, then this imperfect information may have value. The degree of ‘‘constraining’’ is
measured by estimating the value of imperfect information: VOIimperfect= Vimperfect - Vprior.
VOIimperfect is calculated for the data likelihood in Figure 12 (which represents the 7 collocated existing
data) and is also calculated for a synthetically generated data likelihood shown in Figure 14. This second
likelihood represents a more imperfect message as the overlap in electrical resistivity between the two
lithology’s is larger. Table 6 contains the VOIperfect andVOIimperfect results; the VOIperfect results are ~17
times larger than that of the imperfect results. And the VOIimperfect using the data likelihood with a better
electrical resistivity separation for the two facies (Figure 12) is slightly higher than the VOIimperfect that
utilizes the data likelihood of Figure 14. Recall that the Vperfect, Vimperfect and Vpriorinclude summations over
the 7,776 possible farm locations, which explains the magnitude for all quantities.
Table 6: VOIimperfect for Aquifer Vulnerability example (summed over all 7,776 possible farm locations; value is unitless)
Vprior=
-$1,088,000
Data Likelihood
Vimperfect
VOIperfect =
$1,024,000
VOIimperfect
From
Collocated
Driller’s Logs
and TEM
(Figure 12)
-$1,030,000
$58,000
Synthetic that
models larger
overlap of
resistivity for
sand and clay
(Figure 14)
-$1,036,000
$52,000
Figure 14: Synthetic likelihood of electrical resistivity given lithology values a) Electrical Resistivity of Sand (valley). b)
Electrical Resistivity of Clay (non-valley)
Summary: Aquifer Vulnerability

Aquifer Vulnerability is key response to determine which farms to compensate.
o
The geologic heterogeneity is of certain pattern: buried glacial valleys. This is
represented with 18 training images.
o
3D groundwater flow and transport is simulated to estimate aquifer vulnerability.
o
Geophysics won’t directly detect this, but can reveal the facies structure that will
influence which surface locations are large entry points into the aquifer.

Actual field data used to describe the data likelihood: 7 TEM (1D inversions) and drillers log
located within 10 meters of each other.
o
Drillers logs are subjective observations of sand and clay facies.
o
Overlap in the electrical resistivity of the two facies represents ambiguity in the
relationship between the geophysical attribute and the key geologic indicator.

Additionally, the two information sources have vastly different scales: TEM
layers are ~50m thick while the observations of the drillers logs are <1 meter.

Monte Carlo sampling is used to estimate the posterior model space and Vimperfect.
o
Conditioned realizations are generated from the information posterior (soft probability).
o
Only 2 realizations were created from each soft probability because flow simulation
must be performed on each realization created (i.e. computational expensive). More
realizations would give a better estimate of Vimperfect.

Decision alternatives are spatial: made at each surface location within the domain of the model