Parameter Space Reduction and Sensitivity Analysis...
Transcript of Parameter Space Reduction and Sensitivity Analysis...
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Parameter Space Reduction and Sensitivity Analysis in
Complex Thermal Subsurface Production Processes
Jacob H. Bauman* and Milind D. Deo
Chemical Engineering Department, University of Utah, 50 S. Central Campus Drive Room 3290,
Salt Lake City, Utah 84112
[email protected], [email protected]
Abstract
As conventional resources for liquid fuels in the world become scarcer and less secure, there is a
need to develop other feasible resources. Oil shale is a massive resource local to the United States for
potential liquid fuel production. In situ oil shale processing strategies are attractive for reduced
environmental impact (in comparison to surface production operations) and provide access to resources
inaccessible to mining. The efficiency of feasible and economical development is greatly enhanced
with predictive power that is both efficient and accurate. However, modeling thermal subsurface
processes is a complex problem involving many simultaneously occurring physical phenomena. In this
study an oil reservoir simulator capable of representing thermal processes was used to explore the
impact and interplay of various pertinent parameters to an in situ oil shale processing strategy. A
statistical methodology was developed using designed factorial experiments (simulations) to expose
probable dominating parameters, including synergistic or diminutive interactions between parameters.
An empirical regression model, or response surface, was built from the simulated data. Monte Carlo
simulations were used to characterize the response surface and to estimate the uncertainty in predicted
oil recovery results due to the explored parameters.
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KEYWORDS: oil shale, thermal reservoir simulation, uncertainty analysis, experimental design
Introduction
The world is facing several interesting energy challenges. Conventional liquid fuel resources are
becoming scarcer. Carbon dioxide emissions associated with global warming demand attention and
technological solutions. Secure energy sources to supply increasing global demand will also be
necessary to address challenges moving forward. Understanding complex thermal and reactive
subsurface processes will facilitate technological development including production of oil from oil shale
and oil sands, thermal treatment of underground coal, carbon dioxide sequestration, geothermal energy
production, and so on.
Oil shale processing technology development is attractive because of the massive resources within the
United States. Resource estimates in the Green River formation located in the United States range from
1.5 trillion to 1.8 trillion barrels original oil in place from relatively rich shales exceeding 15 gallons per
ton1, 2
. Two major processing strategies exist for converting oil shale to oil: ex situ and in situ. Ex situ
strategies include mining organic rich shale followed by crushing and pyrolysis heating. Various ex situ
pyrolysis heating strategies exist. In situ processing strategies attempt to convert the organic matter to
oil underground by some form of heating, and then producing that oil in production wells. Since heat
input is required in both types of processes, heating efficiency is crucial for any successful strategy.
In situ thermal processing is a complex process that requires understanding of multiple phenomena at
multiple scales. The thermal transformation of organic matter (kerogen in oil shale) to useful fuels
(liquids and gases) requires an understanding of the parent composition, the transformation pathways
and detailed understanding of the products. The organic matter coexists with inorganic and mineral
matter, and the heat and mass transfer at the grain scale affect process effectiveness.
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Description of the important physical phenomena and related properties
Heat transfer through reservoir scale systems is not trivial. A wide array of heating strategies
including resistive heating wells3, fracture injection with conductive material
4, underground rubblization
followed by well heating5, or radio frequency heating
6 have been proposed and developed. Effective
heating of the reservoir is crucial to the efficiency of any in situ thermal processing strategy. Modeling
of various strategies requires fundamental understanding of the physics including: thermal conductivities
and heat capacities of inhomogeneous reservoir materials, convection, phase changes, heats of reactions,
and heat losses due to inefficiencies or losses to reservoir boundaries. Estimating the significance of
these various modes of heat transfer could simplify such models since physical field data are sparse and
expensive.
As complex organic material like kerogen in oil shale is heated it is converted into lighter oil and gas
products. A variety of kinetic transformation mechanisms have been reported7-9
. Time and temperature
histories of these complex organic materials can have significant implications on product distributions
of literally thousands of components with their associated properties. Typically these components are
represented with relatively few pseudo components. The complexity of the reaction network to
represent these components is a major consideration10
. Kinetic parameters such as activation energy,
frequency factor, stoichiometry, etc. depend greatly on the complexity of the component representation.
For example, in one model from Braun and Burnham the activation energy for fractions of various
representations of type II kerogen vary from 47 kcal/mol to 54 kcal/mol11
. The stoichiometry of
reactants and products depends on the molecular weights and elemental representations of the lumped
species in order to conserve mass and elemental balances. For further cracking reactions of lumped
components, the kinetic parameters are dependent on molecular weights, aromaticity, thermodynamics,
etc. of all species represented by the representative component. Significant variation in organic
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(kerogen) and inorganic material within and between resources has implications on the appropriate
kinetic representation of thermal processes as well.
Permeability dynamics (initial permeability and its evolution as the process unfolds) are crucial to a
successful in situ oil shale strategy. Oil shale resources are typically characterized with very low initial
permeability. As solid kerogen is heated and converted to liquid and gaseous products pore space is
created. Permeability increase is possible. Experimental studies have shown expansion followed by
subsidence of oil shale rock as it is heated12
. Decomposition of inorganic rock at high heats would have
permeability implications, including possible microfracturing of the rock, for some heating strategies.
Another possible major contributor to permeability dynamics is coke plugging of permeability pathways.
Relative permeability correlations are used in reservoir simulation to account for multiphase Darcy
flow in permeable rock. The relative permeability models are based on experimental data or are
empirically constructed, but the accuracy in relative permeability representation in complex reservoirs
may be crucial. Thermal effects on phase viscosities and dynamic capillary effects with changing rock
mechanics add complexity to relative permeability representation.
Significant interplay of parameters within and between these physical phenomena can exist. For
example, high temperatures required for oil shale pyrolysis have significant impact on organic material
composition, phase and flow behavior, and possibly geomechanics. Although the parameters are
supplied to governing equations and theoretical models in a simulator, the impact each parameter has on
the final recovery of oil predicted is not easily determined or available. This information potentially
would supply researchers and modelers the parameters that are of the greatest importance, and therefore
need the greatest attention for accurate prediction of such processes.
Simulator Description
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The principle balance equations needed to solve thermal reservoir problems are species and energy
balance equations.
The flow term is typically calculated with Darcy’s law.
STARS from Computer Modeling Group is capable of performing four phase multi-component
thermal reservoir simulations13
. Equilibrium calculations are K-value based. The simulations in this
study were run with STARS. In these simulations, vertical heating wells surround a producer in a seven
point pattern. The heating wells supply only heat without injecting any fluids. Only a triangular wedge
Figure 1: Visualization of simulated section with dimensions.
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with fractions of two heaters and a fraction of one producer is discretized and calculated as shown in
Figure 114
. The initial dimensions of the wedge were 53 feet between heaters and 50 feet thickness of
the reservoir. The temperature at the heating wells was raised quickly to approximately 800oC and then
controlled at about 650oC in order to supply sufficient heat for adequate heat transfer throughout the
reservoir, but also maintaining reasonable temperatures near the heating well.
Geological data from the mahogany zone in the Uinta Basin well U059 was used to estimate the
richness of the layers in the reservoir. The richness of the layers varies from 12.5 – 25 wt% of
hydrocarbon material in the oil shale15
. All of the hydrocarbons were assumed to be kerogen, and this
kerogen was assumed to occupy 90-95% of the pore space, the rest of the pore space being occupied
with 99% gas saturation and 1% water saturation. The initial kerogen, water, and gas volumes in the
pore space will vary between resources or sections of a resource. This could have important
implications on the heat transfer dynamics dependent on the initial water mass and volume. Kerogen
was specified with a constant solid density, so rich layers were assigned higher porosities and lean layers
were assigned lower porosities. Porosity is defined here as total pore space occupied by kerogen, water,
and gas. Fluid porosity refers to the pore space occupied by liquids and gases. As solid kerogen is
converted to liquids and gases, fluid porosity increases while total pore space (as defined in this study)
remains unchanged. Green River oil shale has been characterized with low initial fluid porosity16
,
though this can vary between resources. The simulations studied in this paper have relatively low initial
fluid porosities, and are quite dry. Very little initial water is present in these simulations. Horizontal
permeability varied from 0.1 md at the bottom of the reservoir to 1 md at the top and vertical
permeability varied from 0.05 md at the bottom of the reservoir to 0.5 md at the top. Initial permeability
is typically low for oil shale, but permeability models where permeability increases as kerogen is
converted to fluids would make permeability dependent on initial kerogen richness as the process
unfolds. It is likely that a permeability creation/destruction model will be necessary to model this
process due to solid kerogen conversion to fluids, volume expansion of the rock, and coke plugging of
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pores. Increased permeability would allow fluids to flow more efficiently through the reservoir. If
significant permeable pathways develop, residence time of products is reduced which has compositional
implications as liquid oils could further transform to gases and coke. Also, with increased permeability
there is greater opportunity for convective heat transfer for improving heating efficiency. Typical
permeability creation/destruction models relate permeability to fluid porosity. In this particular study,
these dynamic effects were not considered.
A multiple reaction scheme was used to estimate kerogen decomposition to products. All
hydrocarbons were lumped into seven representative components: kerogen, heavy oil (HO), light oil
(LO), gas, methane (CH4), char, and coke. The reaction scheme was adapted from a previous study17
and is similar to other kerogen decomposition models18
, though the reaction scheme in this study is
relatively simple. These representative components are lumped together based on molecular weight.
Many more representative components lumped according to other physical characteristics such as
density, viscosity, aromaticity, solubility, and so on could also be represented depending on the desired
complexity in the reaction scheme. The ability to develop a more complex set of lumped components
also depends on the available data from experiments where these physical properties of interest are
analyzed. However, increased complexity can greatly increase computational cost. A simplified
reaction scheme assumes the most important physical properties for predicting reservoir behavior
depend on the molecular weight of all the components in a real system.
Reaction 1 Kerogen -> HO + LO + gas + CH4 + char
Reaction 2 HO -> LO + gas + CH4 + char
Reaction 3 LO -> gas + CH4 + char
Reaction 4 gas -> CH4 + char
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Reaction 5 char -> CH4 + gas + coke
It should also be noted that this reaction scheme does not include mineral reactions. Carbonate
minerals present in oil shale resources in the Green River formation can also decompose at the high
temperatures encountered. These mineral transformations would have implications for CO2 emissions,
and also would have a relationship with porosity and permeability dynamics. Studies for carbonate
mineral decomposition have been done to determine the mineral reactions that could be involved19
.
Experimental Design
Factorial experimental designs give experimenters and analysts efficient tools to understand the
impact parameters have on a response in a process. Unlike “one at a time” experiments, factorial
designs allow the researcher to estimate interactions between parameters with fewer experiments. These
factorial designs allow researchers to evaluate many factors together. Experimental design methods
primarily were developed for quality assurance purposes, but have been used in a wide variety of
applications20, 21
. These experimental design tools have also been applied to various oil reservoir
studies22
. A common experimental design is the 2k full factorial design. These designs test k factors at
two levels for each factor, high levels and low levels. Each combination of high and low values of each
factor is called a run. Full factorial designs require 2k runs to test every possible combination of high
levels and low levels for k factors. When the number of factors is excessive or runs are expensive,
fractional factorial designs are used for efficiency. In fractional factorial designs runs are selectively
eliminated with the assumption that higher order interactions are much less significant than individual
factors without interactions. These designs are represented as 2k-p
fractional factorial designs. Fewer
runs are required, but information about the significance of higher order interactions is confounded with
information about individual parameters.
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Experimental design and analysis methods are useful for comparing the sensitivity of a response due
to variable input parameters, including their possibly significant interactions. The parameters of
particular interest in this study are: molecular weight of kerogen, activation energy for kerogen cracking
reaction 1, activation energy for heavy oil cracking reaction 2, activation energy for light oil cracking
reaction 3, activation energy for gas cracking reaction 4, distributed representation for activation energy
for kerogen cracking reaction 1, relative permeability representation, and reaction enthalpy. Each of
these parameters is required for calculating the mass, energy, and momentum balances solved by the
simulator. Activation energies are required for calculating the reaction rate term in the mass balance
equation. Relative permeability is used to calculate the flow term with Darcy’s law. Reaction enthalpy
is incorporated in the energy balance equation. Ranges for each of these parameters were estimated
from various literature data, inherent uncertainty, or are estimated to explore sensitivities.
The molecular structure of kerogen is largely unknown. The molecular weight of kerogen has been
reported in ranges from about 3,00023
to 27,00024
. The stoichiometry in the chemical reactions and the
initial concentration of kerogen are dependent on the choice for molecular weight of kerogen to
conserve volume and mass for all simulation runs. Consequently, when the molecular weight of
kerogen is changed between simulation runs, the stoichiometry of the reactions and the initial molar
concentration of kerogen in the pore space must also be adjusted for mass and volume consistency. The
range of molecular weight, and associated stoichiometry for reactions 1 and 2 are shown in Table I. The
values in Table I demonstrate that stoichiometry for such a kinetic mechanism depends on the properties
(molecular weight and H/C ratio) of the pseudo components and is therefore “non-unique.” Mass and
elemental balances are important in these representations.
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Table I: High and low molecular weights and associated stoichiometry for reactions 1 and 2.
Ranges for appropriate activation energies have been reported18
, and can vary significantly depending
on experimental methods, and analysis of results. These ranges for activation energy are shown in Table
II. Studies have reported that activation energy for kerogen pyrolysis is most appropriately modeled
with some distribution18, 25
, but it is uncertain how much impact different representations of activation
energy have on the simulation results at large scales. A normal distribution with 5 kJ/mol standard
deviation is shown in Figure 2. Distribution of activation energies for kerogen pyrolysis is a complex
function, but is sometimes represented by the normal distribution26
. A perfect activation energy
distribution cannot be represented exactly in STARS, so discrete quantities, determined by integrating
under the distribution curve, represent kerogen reacting with a specified activation energy according to
the distribution.
Table II: Range of activation energies.
Reaction Low Activation
Energy (kJ/mol)
High Activation
Energy (kJ/mol)
1. Kerogen Cracking 195 225
2. Heavy Oil Cracking 208 260
3. Light Oil Cracking 208 260/233
4. Gas Cracking 235 270
Kerogen Heavy Oil Light Oil Gas Methane char coke
MW (+) 20000.55 424.49 152.03 52.01 16.04 12.60 14.55
MW (-) 2974.84 424.61 151.99 51.95 16.04 12.55 14.55
Formula (+) C1479H2220 C31.75H42.82 C11.19H17.51 C3.35H11.63 CH4 CH0.55 C1.19H0.32
Formula (-) C220H330 C31.76H42.81 C11.19H17.50 C3.35H11.62 CH4 CH0.53 C1.19H0.32
Stoic rxn 1 (+) -1 37.29 13.86 25.03 17.06 38.71 0
Stoic rxn 1 (-) -1 5.55 2.06 3.72 2.54 5.8 0
Stoic rxn 2 (+) -1 2.18 0.06 0.03 7.13 0
Stoic rxn 2 (-) -1 2.18 0.06 0.03 7.13 0
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Figure 2: Normal distribution of activation energies for reaction 1.
Relative permeability representations are often approximated in simulation, but such approximations
may have significant implications. The range of relative permeability curves are shown in Figure 3, the
low level being more linear and the high level being curved. The shape of the relative permeability
curves depends on the resource, the wetting characteristics of the rock, and the constituents present in
the pore space. Finally, heat of reaction could play an important role in the heat transfer efficiency
depending on the characteristics of the associated reactions. Efficient heat transfer through an oil shale
reservoir is crucial to any successful operation. Heat of reaction for oil shale pyrolysis has been
reported27
, but it is not certain how much heat is “lost” to reaction compared to heat required to raise the
rock temperature.
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Figure 3: Oil/water and liquid/gas relative permeability curves.
Results and Discussion
The initial experimental design was a 27-4
fractional factorial design. The eight run design for the
initial 7 factors (excluding (8) heat of reaction) is shown in Figure 4. Each row represents a simulation
Run X1 X2 X3 X4 X5 X6 X7
1 - - - + + + -
2 + - - - - + +
3 - + - - + - +
4 + + - + - - -
5 - - + + - - +
6 + - + - + - -
7 - + + - - + -
8 + + + + + + +
Figure 4: Eight run fractional factorial screening experimental design.
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run at the parameter levels specified in the run. The un-coded levels for these parameters are shown in
Table III. Details about each of these parameters have been described. The numerical performance for
each of these runs was not equal. Some of the runs had excessive time step cuts due to rapid changes in
gas saturation. The response chosen for these runs was simulation time in order to pinpoint the possible
causes of these time step reductions. Figure 5 is a Pareto chart displaying the impact each of these
parameters have on the simulation time. Activation energy for reaction 3, or factor X4, had the greatest
impact on the simulation time. After investigation, it appeared that simulation time increased
significantly when the activation energy for reaction 3 was greater than the activation energy for reaction
4. This could be due to the combination of rapid gas creation coupled with high gas mobility causing
rapid gas saturation changes. The high value for factor X4 was lowered to 233 kJ/mol as shown in
Table II, and no major differences in simulation time were observed in subsequent runs.
Table III: Un-coded parameters for screening design.
Factor Physical Parameter Low Level (-) High Level (+)
X1 MW/Stoic/Concentration 3000 20000
X2 Eact Reaction 1 195 225
X3 Eact Reaction 2 208 260
X4 Eact Reaction 3 208 260
X5 Eact Reaction 4 235 270
X6 Eact distribution Rxn 1 Without With
X7 Relative permeability Linear Curved
Figure 5: Pareto chart for parameter effects on simulation time.
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Using the lowered value for factor X4, runs in the fractional factorial design were completed with
ultimate recovery of oil as the output response. None of the simulations produced acceptable amounts
of oil. Upon inspection it was found that oil generated from kerogen had inadequate mobility in lower
temperature zones far from the heaters to flow to the producer. As a result, oil components had large
residence times in the reservoir, and eventually converted further to gas and residual components. This
result gives insight into the design of such a process, specifically the spacing needed between wells for
successful operation. If heating wells are drilled too far from producing wells the residence time of the
oil in hot zones of the reservoir will be excessive, and these oils will convert to gasses or residual solids
significantly reducing or even prohibiting production of oil. However, capital and operating costs
increase with the number of wells drilled. Well spacing is a crucial design consideration for this process
since excessive residence time of products in the reservoir and the cost of drilling wells are competing
considerations for optimal process design.
The initial dimensions of the simulated domain were changed to resolve this issue of excessive
residence time of the oil. Reducing the spacing between these wells assures the whole reservoir was at a
high enough temperature for adequate oil mobility. Figure 6 shows the modified dimension of the
simulated wedge, changing the distance between heaters from 53 ft to 26.5 ft. The same fractional
factorial design was used with ultimate recovery of oil as the response. The normal probability plot in
Figure 7 illustrates the results of the runs. Normal probability plots, like Pareto charts, are useful
visualizing the significance of the effects for each factor. Dominating effects will appear as outlier
points on a normal probability plot. The points of the effects on this plot in Figure 7 are linear without
outliers indicating that there is no evidence from these runs that any factors are dominant or
insignificant. With the 27-4
fractional factorial design used, single factor effects are confounded with
pair interaction effects and higher order interactions. Additional runs are necessary to isolate the effects
of individual parameters from confounding with the effects of higher order interactions.
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Figure 6: Aerial view of simulated wedge. Distance between heating wells was reduced to 26.5 ft.
Figure 7: Normal probability plot of the effects on ultimate recovery of oil from 27-4
fractional factorial design.
Further runs were done with a sixteen run fractional factorial design for 6 to 8 factors. All 8 factors,
including heat of reaction were tested with this design. The design used is shown in Figure 8 where
factors E1 – E7 represent possible interactions between parameters, but individual factors are isolated
from possible confounding with interactions.
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Run Mean X1 X2 X3 X4 X5 X6 X7 X8 E1 E2 E3 E4 E5 E6 E7
1 + - - - - - - - - + + + + + + +
2 + + - - - + - + + - - - + - + +
3 + - + - - + + - + - + + - - + -
4 + + + - - - + + - + - - - + + -
5 + - - + - + + + - + - + - - - +
6 + + - + - - + - + - + - - + - +
7 + - + + - - - + + - - + + + - -
8 + + + + - + - - - + + - + - - -
9 + - - - + - + + + + + - + - - -
10 + + - - + + + - - - - + + + - -
11 + - + - + + - + - - + - - + - +
12 + + + - + - - - + + - + - - - +
13 + - - + + + - - + + - - - + + -
14 + + - + + - - + - - + + - - + -
15 + - + + + - + - - - - - + - + +
16 + + + + + + + + + + + + + + + +
Figure 8: Experimental design for 6-8 factors without confounding of individual parameters.
The results from these runs are displayed in a Pareto chart in Figure 9. It appears that the most
significant factors are X2, X4, X6, and X7 along with higher order interactions between parameters,
likely between these most significant factors. These factors are activation energy for reaction 1,
activation energy for reaction 3, activation energy distribution representation for kerogen conversion,
and relative permeability representation. It appears that activation energy for reaction 4 and heat of
reaction have the least impact on ultimate recovery of oil.
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Figure 9: Pareto chart from 16 run fractional factorial design for 8 factors.
The results from these runs can be used in a 24 full factorial design without any additional runs. The
data were regressed with the polynomial model shown in Equation 1, where β0 = the intercept (global
mean), β = single and higher order interaction linear coefficients, and x = input variables. This
polynomial model forms a multivariate surface called a response surface. The effects are calculated by
taking the difference of the averages of the responses at high and at low levels of each factor, and for
interactions between factors, and the coefficients β are half of those effects. The coefficients are
summarized in Table IV.
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Table IV: Summary of calculated effects.
Factors β
Intercept 294.6071
X1 -25.9846
X2 48.71381
X3 -8.88369
X4 -36.7407
X1X2 17.66644
X1X3 -19.9203
X1X4 7.082688
X2X3 14.89081
X2X4 -2.82519
X3X4 1.098062
X1X2X3 -5.35956
X1X2X4 0.575937
X1X3X4 -3.37056
X2X3X4 -0.70869
X1X2X3X4 2.191438
This model fit the experimental output data exactly. Although this is not a theoretical model and may
have little physical significance, insight about the significance of each parameter in the explored ranges
can be garnered. Typically higher order linear interaction effects are assumed to be negligible and can
be used to estimate error22
. Expert opinion and knowledge is advantageous for estimating error, and
elimination of terms in this model perhaps are not justified since this knowledge is unknown22
.
Three random validation simulations within the experimental space were run to estimate the quality of
the response surface, the empirical regression model, compared to a STARS simulation. The difference
between the response surface approximations for ultimate oil recovery and STARS simulation results
ranged from 3% to 15%. The quality of the response surface could be improved at the cost of more
experimental runs, either by reducing the experimental space or by adding additional runs to estimate
curvature due to nonlinearities when parameters are continuous. Alternative experimental designs could
possibly provide more accurate response surfaces with comparable or fewer total runs, however many of
these alternative designs require additional expert knowledge about the problem or unjustified
assumptions.
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Monte Carlo simulations were performed to characterize the response surface. Random values for
each parameter with uniform distributions were chosen for each run. A histogram of 80,000 Monte
Carlo runs is shown in Figure 10. The average value in these runs was 294.7 bbls oil with a standard
deviation of 40.1 bbls oil. A normal distribution with these values is also shown in the figure for
comparison. It appears the Monte Carlo results are slightly skewed to the right of a normal distribution.
This exercise helps to quantify the effects of variations in input parameters on the desired output. The
shape of this distribution could be affected by the response surface itself, the sampling locations for
Monte Carlo simulation, or by the distributions assigned to each of the factors.
Figure 10: Histogram of Monte Carlo calculations of response surface.
Conclusions
Although results for oil shale simulations in this study are calculated with theoretical governing
equations, the interplay within various parameters is not trivial due to competing physical phenomena.
Combinations of parameters that expose possible competing phenomena can have significant numerical
implications. Molecular representations for kerogen with associated stoichiometry, heat of reaction for
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kerogen decomposition, intermediate oil cracking reaction (reaction 2) activation energy, and continuing
gas cracking (reaction 4) reaction activation energy are insignificant in determining the ultimate
recovery of oil at the scale simulated in this paper. Kerogen cracking (reaction 1) activation energy,
relative permeability representation, oil cracking to gas (reaction 3) activation energy, and activation
energy distribution representation have significant impacts on the ultimate recovery of oil in these
simulations. Expert knowledge or similar studies including large scale physical experiments are
important for estimating statistical error for developing validated surrogate models. Otherwise, more
runs are necessary for improving these models quality for approximating simulator results.
The interplay between various flow and kinetic parameters has been explored. Geomechanical, heat
transfer, and equilibrium parameters for example may also play significant roles at certain scales in
production results for such complex reactive transport systems. Parameters from acceptable theoretical
models can also be included in experimental designs to evaluate their impact on results and to include
these parameters in constructing response surface approximations as illustrated in this paper. Response
surfaces can be characterized to quantify risk and uncertainty of simulations according to variation in
input data.
Acknowledgements
The authors would like to acknowledge financial support from the U.S. Department of Energy, National
Energy Technology Laboratory – Grant Number: DE-FE0001243. We would also like to thank
Computer Modeling Group Limited (CMGL), Calgary Canada for providing academic licenses to their
reservoir simulators.
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Nomenclature
Eact – Activation energy
– Phase specific gravity
- Phase enthalpy
- Heat of reaction
- Permeability tensor
– Phase relative permeability
- Phase viscosity
– Number of fluid phases
– Number of phases
- Number of reactions
- Phase pressure
– Phase index
- Porosity
- Source term for energy
- Source term for mass of component i
- Residual mass for component i
- Residual energy
- Reaction rate
- Reaction index
- Phase molar density
- Phase saturation
– Stoichiometric factor of component i in reaction r
– Temperature
– Time
- Phase internal energy
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- Rock internal energy
- Phase velocity
– Mole fraction of component i in phase p
- Height or depth
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