characterization of tight gas reservoirs with …

The Pennsylvania State UniversityRESERVOIR VOLUME: AN ARTIFICIAL INTELLIGENCE APPLICATION
A Thesis in
December 2020
The thesis of Yiming Zhang was reviewed and approved by the following:
Turgay Ertekin Professor of Petroleum and Natural Gas Engineering George E. Trimble Chair in Earth and Mineral Sciences Thesis Advisor
Zuleima T. Karpyn Professor of Petroleum and Natural Gas Engineering Quentin E. and Louise L. Wood Faculty Fellow in Petroleum and Natural Gas Engineering
Hamid Emami-Meybodi Assistant Professor of Petroleum and Natural Gas Engineering
Sanjay Srinivasan Head, John and Willie Leone Family Department of Energy and Mineral Engineering
iii
Abstract
Stimulated reservoir volume (SRV) characterization has been a practical concept
developed by researchers in the petroleum industry and utilized recently for the purposes
of quickly evaluating the effective well drainage area and analyzing the performance of
multistage stimulated horizontal wells in tight formations. In this study, SRV modeling of
tight gas reservoirs is focused for characterization purposes.
Characterization of reservoir properties is an exhaustive and time-consuming
process. For a dual porosity type of reservoir, the characterization of the reservoir is often
much more descriptive than a typical conventional reservoir, thus making the history
matching process much more complex and difficult to solve by hand. The recent
applications of artificial neural network (ANN) technology in petroleum engineering
applications has attracted increasingly more attention.
In this study, two ANN based tool are designed and developed. The first ANN model
is the forward-looking ANN model, which predicts the production profiles based on the
reservoir characteristic parameters from horizontal wells in tight gas sands. The forward-
looking model predicts coefficients of the hyperbolic decline curve with cumulative
production error of less than 3%. The second ANN model is the inverse ANN model, which
focuses on characterization of tight gas reservoir with known reservoir parameters, design
parameters, and production profile. The inverse model is capable of predicting
characteristic parameters with cumulative production error of less than 8% by the
iv numerical model itself, or cumulative production error of less than 13% in conjunction
with the forward-looking model.
Chapter 2 Literature Review ........................................................................................ 3
2.1 Unconventional Gas Reservoirs, Tight Gas Sand ...................................................... 3 2.2 Stimulated Reservoir Volume (SRV) Concept ........................................................... 6 2.3 Artificial Neural Networks (ANN) ............................................................................. 9
Chapter 3 Problem Statement ...................................................................................... 13
Chapter 4 Generation of Training and Testing Sets..................................................... 14
Chapter 5 ANN Model Development ........................................................................... 23
5.1 Forward-Looking ANN Model ................................................................................... 23 5.2 Inverse ANN Model .................................................................................................... 27
Chapter 6 Results & Discussions ................................................................................. 31
6.1 Forward-Looking ANN Model ................................................................................... 31 6.2 Inverse ANN Model .................................................................................................... 50
Chapter 7 Summary and Conclusions .......................................................................... 77
References .................................................................................................................... 82
Appendix B Training and Testing Data Sets ............................................................... 89
Appendix B.1 Testing Data Sets ...................................................................................... 89 Appendix B.2 Data Sets for all Cases .............................................................................. 91
vi
List of Figures
Figure 2.1. a) Conventional sandstone, b) Tight gas sandstone (Naik, 2003) ............. 5
Figure 2.2. Types of fracture growth (Warpinski, 2008) ............................................. 7
Figure 2.3. 2D view of the representative stimulated reservoir and the induced fracture network it contains (Xu, 2009) ................................................................ 8
Figure 2.4. Cotton Valley case study: map view of the induced hydraulic fracture system as mapped using borehole-based microseismic monitoring (Xu, 2009). .................................................................................................................... 9
Figure 2.5. Resource triangle for natural gas (Holditch, 2006) ................................... 10
Figure 4.1. Sample 2D Top View for Reservoir Model with SRV Zone ..................... 16
Figure 4.2. Sample Top View of Fracture Permeability of the Reservoir ................... 17
Figure 4.3. Sample Top View of Fracture Porosity of the Reservoir .......................... 18
Figure 4.4. Sample Top View of Fracture Spacing of the Reservoir ........................... 19
Figure 4.5. Sample curve fitting comparisons. ............................................................ 21
Figure 4.6. Generation of Training and Testing Sets Flow Chart ................................ 22
Figure 5.1. Architecture of Forward-Looking ANN Model Network .......................... 26
Figure 5.2. Architecture of Inverse ANN Model Network ........................................... 30
Figure 6.1. Coefficient qi Prediction Results ............................................................... 34
Figure 6.2. Coefficient Di Prediction Results .............................................................. 34
Figure 6.3. Coefficient b Prediction Results ................................................................ 35
Figure 6.4. Hinton Diagram – Forward-Looking Model ............................................. 37
Figure 6.5. Sample Forward-Looking Model Prediction Case – Best case ................. 39
Figure 6.6. Sample Forward-Looking Model Prediction Case – Intermediate case .... 40
Figure 6.7. Sample Forward-Looking Model Prediction Case – Worst case .............. 41
Figure 6.8. Sample Forward-Looking Model Prediction Case – qi prediction error ... 43
vii Figure 6.9. Sample Forward-Looking Model Prediction Case – qi prediction error ... 44
Figure 6.10. Sample Forward-Looking Model Prediction Case – Di prediction error ....................................................................................................................... 45
Figure 6.11. Sample Forward-Looking Model Prediction Case – Di prediction error ....................................................................................................................... 46
Figure 6.12. Sample Forward-Looking Model Prediction Case – b prediction error .. 48
Figure 6.13. Sample Forward-Looking Model Prediction Case – b prediction error .. 49
Figure 6.14. SRV Zone Major Axis Length Predictions .............................................. 51
Figure 6.15. SRV Zone Minor Axis Length Predictions .............................................. 52
Figure 6.16. SRV Zone Fracture Permeability Predictions .......................................... 53
Figure 6.17. SRV Zone Fracture Porosity Predictions ................................................. 54
Figure 6.18. SRV Zone Fracture Spacing Predictions .................................................. 54
Figure 6.19. Matrix Porosity Predictions ..................................................................... 55
Figure 6.20. Natural Fracture Permeability Predictions .............................................. 56
Figure 6.21. Natural Fracture Porosity Predictions ..................................................... 56
Figure 6.22. Matrix Permeability Predictions .............................................................. 57
Figure 6.23. Natural Fracture Spacing Predictions ...................................................... 58
Figure 6.24. Hinton Diagram for Inverse ANN Model ................................................ 60
Figure 6.25. Method 1 Validation Error Distribution .................................................. 62
Figure 6.28. Sample Inverse Model Comparison – Best Case .................................... 66
Figure 6.29. Sample Inverse Model Comparison – Best Case .................................... 67
Figure 6.30. Sample Inverse Model Comparison – Intermediate Case ....................... 69
Figure 6.31. Sample Inverse Model Comparison – Intermediate Case ....................... 70
viii Figure 6.32. Sample Inverse Model Comparison – Worst Case .................................. 72
Figure 6.33. Sample Inverse Model Comparison – Worst Case .................................. 73
Figure 6.34. Drainage Area versus Predicted Cumulative Production Error .............. 75
Figure A.1. Inverse ANN Model Graphic User Interface............................................. 86
Figure A.2. Forward-Looking ANN Model Graphic User Interface ............................ 88
ix
Table 4.2. Hyperbolic Curve Fitting ............................................................................ 20
Table 5.1. Forward-Looking ANN Model Inputs and Outputs .................................... 24
Table 5.2. Functional Links and Eigenvalues for Forward-Looking Model ............... 25
Table 5.3. Input and Output Parameters for the Inverse ANN Model .......................... 27
Table 5.4. Functional Links for the Inverse ANN Model ............................................. 28
Table 6.1. Forward-Looking Model Prediction Errors ................................................ 33
Table 6.2. Prediction Errors, Inverse ANN Model ....................................................... 50
Table 6.3. Inverse Model Cumulative Production Error.............................................. 61
x
ACKNOWLEDGEMENTS
First and foremost, I would like to express my sincere gratitude to my advisor, Dr. Ertekin,
for this opportunity to pursue my Masters study here in Penn State University as a Petroleum
Engineer. I am grateful to have Dr. Ertekin as my advisor; his patience, wisdom, and support have
helped me throughout my Masters progression. His guidance is a beacon of light; academically, the
vast amount of his knowledge and experiences provide me with insightful suggestions; his great
personality also deeply influenced me as how to approach different situations in a professional
manner and most importantly in how to develop as a better human being. I would also like to Dr.
Zuleima Karpyn and Dr. Hamid Emami-Meybodi for taking their time and their contributions as
committee members for my thesis.
I would also like to express my gratitude to my friend and my mentor, Qian Sun, for his
support and encouragement throughout the development of my thesis work. I thank Qian for his
help in introducing me to the key concepts in reservoir modeling, artificial neural network, and
techniques to help the development of my models. I also would like to thank Miao Zhang for her
support and help during my college life.
At last, I would like to thank my mother, Wenxia Zhang, for all of her love and support,
especially after I made the decision to pursue my study up here in the state of Pennsylvania. Her
experience as a reservoir engineer has helped me tremendously because I have the opportunity to
see closely of what it is like to be a petroleum engineer.
1
Introduction
In the past decade, the petroleum industry has seen tremendous strides in the
exploration and production of unconventional resources and related technologies. An
unconventional reservoir is defined as a reservoir where advanced technology is required
in order to produce the resources at economical flow rates. Examples of unconventional
gas are tight gas, coalbed methane, shale gas, and gas hydrate. This study focuses on tight
gas formations.
In the 1970s, United States defined “tight gas” as the producible natural gas from
reservoirs with permeability values less than 0.1 millidarcy. Tight gas reservoirs are found
throughout the world and are typically deposited in sandstone or carbonate rocks (Holditch,
2006). Stimulated reservoir volume (SRV) has been a practical concept developed by
researchers in the field for the purposes of quickly evaluating the effective well drainage
area and analyzing the actual performance of multistage horizontal wells in tight
formations.
performance. Reservoir modeling and simulation can be costly and time-consuming
especially for complex formations. Artificial neural network (ANN) technology has been
widely used in science and engineering and can be applicable in studies involving
forecasting. There have been many studies in the oil and gas applications utilizing the ANN
2
tools. They can be applied in almost every situation if the predictor variables and the
predicted variables are mathematically related.
The present study represents the development of two ANN models with SRV in tight
gas sands. Computer Modeling Group’s (CMG®1) Implicit – Explicit (IMEXTM2) black-oil
simulator is used to generate production profile of the reservoirs, with the reservoir
parameters randomly generated by the Matrix Laboratory (MATLAB3). MATLAB neural
network extension toolbox is utilized in the training of the ANNs. Two ANN models are
developed in this study. The first ANN model is a forward-looking model that forecasts the
hyperbolic decline production profile for the tight gas reservoir with SRV. The second ANN
model is an inverse ANN model, which predicts the reservoir parameters and characterizes
the tight gas reservoir using the production data.
Chapter 2 reviews the information about tight gas reservoirs, stimulated reservoir
volume model, and ANN applications for oil and gas reservoirs. Chapter 3 states the
problem explored in this study. Chapter 4 describes how the training and testing sets for
the ANN models are generated. Chapter 5 elaborates on the set up of the two ANN models
studied in this thesis. Chapter 6 discusses the results and observations obtained from the
ANN models and finally, Chapter 7 summarizes and provides the conclusions reached.
1 CMG®: Computer Modeling Group, A numerical reservoir simulator by Computer Modeling Group Ltd. 2 IMEXTM: IMplicit – EXplicit 3 MATLAB: MATrix LABoratory, A tool for numerical computation by The MathWorks™ Inc.
3
Chapter 2
Literature Review
This chapter briefly describes tight gas sand class of unconventional gas reservoirs,
stimulated reservoir volume zone, and artificial neural network knowledge applications in
petroleum and natural gas engineering.
2.1 Unconventional Gas Reservoirs, Tight Gas Sand
Although there are different perceptions on the definition of an unconventional gas
system globally, economic values have defined the differences between conventional and
unconventional resources in the United States since 1970s. Natural gas resources at the
time such as coalbed methane, shale gas, gas hydrates, and tight gas were challenging to
produce at economic flow rates and were considered as unconventional by most
exploration geologists. Unconventional gas reservoirs are typically difficult to produce
technically without any stimulations due to their relatively low permeability whereas
conventional reservoirs will produce without significant stimulation treatments or any
other special recovery process. Unconventional reservoirs usually have large volume of
hydrocarbons in place as compared to conventional reservoirs. Conventional and
unconventional gas resources can also be distinguished based on their geological features:
conventional resources are buoyancy-driven deposits whereas unconventional resources
are generally not buoyancy-driven as the latter are commonly independent of structural and
stratigraphic traps (Law, 2002).
4
In the 1970s, the U.S. government defined tight gas reservoirs as having less than
0.1 millidarcy in terms of expected value of permeability to gas flow. Some “ultra-tight”
gas reservoirs may even have in-situ permeability values down to 0.0000001 millidarcy.
Reservoir parameters such as effective porosity, capillary pressure, fluid viscosity, and
fluid saturation are also critical as they affect the effective permeability of the reservoir.
Rock parameters are also important as they are controlled by depositional settings. Tight
gas resources typically are deposited in sandstone or carbonate matrices. In the United
States, tight gas sandstones are generally “clean sandstones deposited in high-energy
depositional settings whose intergranular pores have been largely occluded by authigenic
cements (mainly quartz and calcite)” (Dutton, 1993). To illustrate the comparisons between
a conventional sandstone and a tight gas sandstone, Figure 2.1 shows a thin section view
of each, injected with blue epoxy as the blue areas represent the pore space. For Figure
2.1a, the conventional sandstone contains interconnected pore space, as gas is able to flow
easily from the rock. Whereas the tight gas sandstone in Figure 2.1b displays irregularly
distributed pore space and contains much less effective pores than the conventional
sandstone. The narrow capillaries in the tight gas sandstone poorly connect to the pore
space resulting in very low permeability, as gas flow is difficult to produce without any
types of stimulations.
5
(a)
(b)
Figure 2.1. a) Conventional sandstone, b) Tight gas sandstone (Naik, 2003)
One of the most common depositional settings for tight gas reservoirs is the basin-
centered gas system, as the deep basin (>15,000ft) typically consists “an abnormally-
pressured, gas-saturated accumulation in low-permeability reservoirs lacking a down-dip
water contact” (Law, 2002). Contrary to what many explorationists believe, tight reservoirs
6
can also be found in various ages and types of depositional settings, such as tectonic
settings dominated by extensional, compressional, or wrench faulting and folding. There
are also tight reservoirs resulted from late burial diagenesis of the sandstones.
2.2 Stimulated Reservoir Volume (SRV) Concept
Stimulated Reservoir Volume (SRV) is a practical and useful engineering concept
for the purposes of quickly evaluating the effective well drainage area and analyzing the
actual performance of multistage horizontal wells in tight formations (Zhao, 2012).
Different researchers calculate the volume in slightly different ways as it can also be
referred to as stimulated reservoir area (SRA) or effective stimulated volume (ESV), but the
overall concept is very similar.
SRV is typically applied as a correlation parameter for well performance and used
as a proxy for fracture geometry. For naturally fractured tight gas reservoirs, horizontal
wells can be drilled but these wells need to be stimulated. The concept of the SRV from
microseismic-event distributions can be applied to reservoirs where complex fracture
networks are created (Mayerhofer, 2010). Figure 2.2 illustrates the different types of
fracture growth, from a simple fracture to a complex fracture network.
7
Figure 2.2. Types of fracture growth (Warpinski, 2008)
The SRV number is often used semi-quantitatively as larger SRV number will lead
to larger expected cumulative production. The size of the created fracture network can be
approximated as the 3-D volume of the microseismic event cloud (Mayerhofer, 2008).
However, SRV is not the only driver of well performance. In addition, production values
also depend on the volume influenced by hydraulic treatment, spacing of the fractures,
conductivity of the fractures (both natural and hydraulic fractures), permeability and
porosity of hydraulic fractures (Zimmer, 2011).
Elliptical shaped zone can be utilized to represent the SRV (SRA for 2D) geometry
in terms of reservoir modeling. Xu et. al. (2009) represented hydraulically fractured
8
network in tight gas formations with horizontally expanding ellipse as shown in Figure
2.3. They also presented a tight gas case study from Cotton Valley using borehole-based
microseismic monitoring illustrated in Figure 2.4 as an elliptical shape with the fracture
height, the major axis, and the minor axis of values h≈120m, a≈400m, and b≈60m,
respectively. Kulga (2014) implemented SRV concept to simulate the behavior of an
equivalent discrete fracture network model in his examination of shale gas reservoirs. His
models are categorized into low-rate, mid-rate, and high-rate based on the numbers of SRV
grid in 17x17 reservoir grid systems. The SRV grid blocks for each case are mimicking an
elliptical shape for the stimulated zone, with 17 grid blocks for the low-rate cases, 39 grid
blocks for the mid-rate cases, and 49 grid blocks for the high cases. The SRV reservoir
modeling presented in this paper uses a similar approach as Kulga used in his study and is
explained in detail in Chapter 4.
Figure 2.3. 2D view of the representative stimulated reservoir and the induced fracture network it contains (Xu, 2009)
9
Figure 2.4. Cotton Valley case study: map view of the induced hydraulic fracture system as mapped using borehole-based microseismic monitoring (Xu, 2009).
2.3 Artificial Neural Networks (ANN)
Unconventional resources have seen tremendous growth contributing to the ever-
increasing worldwide oil and gas demand over the past decade. In 2016, United States
withdrew and produced 32.65 trillion cubic feet of natural gas (EIA, 2017). It is estimated
that 20% of the gas production in the U.S. currently comes from tight gas reservoirs (EIA,
2017). Figure 2.5 illustrates the principle of a gas-resource triangle, as the reservoirs are
lower grade with decreasing permeability as it goes into deeper into the triangle. The low
quality resources of natural gas require improved technology and adequate gas prices
before they can produce at economic flow rates. However, the size of these reservoirs can
be quite large compared with the high-quality reservoirs, but the analysis of these
unconventional reservoirs are also much more complex than the conventional reservoirs.
10
With the recent technological advances, more and more unconventional reservoirs are put
into play and produce at economic flow rates.
Figure 2.5. Resource triangle for natural gas (Holditch, 2006)
Artificial neural networks (ANN) models proved to be an efficient tool for analyzing
huge data sets with techniques such as history matching, production forecasting, and
characterization. Bansal (2013) utilizes ANN to map the existing complex relationships
between seismic data, well logs, completion parameters, and production characteristics.
The developed ANN models are used to forecast oil, water, and gas cumulative production
for a two-year period. The presented study utilizes the expert systems to characterize tight
oil reservoirs and the results are further extended to identify potential infill drilling
locations. The ANN models also allow the entire reservoir to be analyzed in a time efficient
manner and can be applied for different tight reservoirs around the world.
11
Kulga (2010) developed an ANN model to predict monthly production profile for
hydraulically fractured horizontal wells in tight gas sand reservoirs. The ANN model can
accommodate several production regimes for any given data sets in the range of reservoir
parameters and predicts production data with an error of less than ±10%.
Siripatrachai (2012) implemented two ANN models to characterize hydraulic
fractured horizontal wells completed in shale gas reservoirs. The “Gas Production
Prediction ANN” model predicts the production profile of the shale gas reservoirs with error
margins of less than ±10%. The “Equivalency ANN” model converts equivalent hydraulic
fracture representations from transverse hydraulic fracture representation and crushed zone
representation. The second model characterizes the representations with cumulative gas
production error within ±15%.
Implementations of ANN models are also widely used in other petroleum industry
related applications as well. Ramgulam (2007) developed an inverse ANN model to assist
and improve the history matching process for a reservoir that produced oil, gas, and water
over ten years of production. The model was also applied for a field case in a reservoir of
the Tahoe Field and predicted close matched properties to the oil and gas production from
three producers for the case study.
Artun (2008) developed two ANN models for design optimization of cyclic pressure
pulsing in a depleted, naturally fractured reservoir. The forward model is designed to
predict the corresponding performance indicators with given design parameters; the inverse
model is designed to predict the corresponding design parameters with the given set of
desired performance characteristics. The models are tested with a single-well reservoir
model of the Big Andy Field and the networks are able to accurately predict the
12
performance indicators including the peak rate, time to reach the peak rate, cycle flow rates,
incremental oil production, and gas-oil-ratio.
Sun (2013) developed expert network systems to determine the optimal design of
carbon dioxide injection pattern for deep saline formation. Three ANN models were
developed to conduct the case study for MT. Simon sandstone formation, including end-
point forward-looking solution network, injection efficiency profile network, and injection
well bottom-hole pressure profile network.
The ANN modules developed in this study are for characterization of tight gas
reservoir models and are explained in detail in Chapter 5. The selected ranges for reservoir
parameters used for ANN trainings are based on tight gas reservoir characteristics.
Chapter 3
Problem Statement
Reservoir simulation and the reservoir engineering techniques often play crucial
roles for finding the optimum scenario for the characterization of a field. However, this
process can be inefficient especially in terms of time cost and the possibility of missing the
optimal solutions due to the lack of rock or fluid properties or the uncertainties associated
with the availability of the field data. In this study, ANN technology is implemented to
overcome these challenges.
Before developing the ANN models to capture the characterizations of the
reservoirs, the first part of this study focuses on the development of the modeled reservoirs
utilizing Monte Carlo Simulation protocol for the parameters in order to reduce the
uncertainties originating from various reservoir properties. These models have one
horizontal wellbore with an elliptical stimulated reservoir volume zone placed around the
wellbore to simulate the complex fracture of a tight gas reservoir. After running these
models through a commercial modeling software, the results are then used as training and
validation sets for the ANN models.
Two ANN models are developed in this study; the first model is a forward-looking
ANN model to predict the coefficients for the hyperbolic decline curves characterizing the
production profiles. The second model is an inverse ANN model to characterize the
reservoir properties.
Chapter 4
Generation of Training and Testing Sets
Training and testing sets for ANN are generated by using the numerical model
simulator for the following reservoir conditions:
• 2 dimensional representation of the flow domain
• Homogeneous, isotropic square reservoir with non-uniform grid
distribution based on the dimensions of the elliptical zone’s major and
minor axes
• Gas reservoir, no oil saturation or water saturation (So=0, Sw=0)
• One horizontal well in the center of the Y-direction
• Production with bottom hole pressure specified (psf)
• SRV zone along the horizontal wellbore in an elliptical shape
• Isothermal reservoir with 275°F (Ti)
In addition, these tight gas reservoirs can be modeled with dual-porosity, single-
permeability system. The following constraints are also established to ensure logical
parameter combinations for the reservoir modeling (other constraints not listed are already
established by the ranges of the parameters).
• Horizontal wellbore length (LHW) ≤ square root of drainage area (A) * 0.6
• SRV zone minor axis length (LMinor) < 1000 ft.
• Natural fracture permeability (kf) > matrix permeability (km)
15
The combinations of reservoir parameters are generated through the computer
programming software using the random numbers generator from the continuous uniform
distribution (unifrnd.m) and the ranges for each parameter are shown in Table 4.1. The
parameters must be generated through unifrnd.m in order to obtain applicable and
functional ANNs, which can handle a wide range of inputs. The distributions of the
generated parameter are listed in Appendix A. The generated combinations of the
parameters also provide the necessary inputs for the numerical model.
Table 4.1. Range of Reservoir Parameters
Parameter Minimum Value
Maximum Value Unit
Drainage Area (A) 100 550 acre Thickness (h) 100 300 ft
Matrix Permeability (km) 0.000001 0.0001 md Matrix Porosity (m) 0.05 0.15 % Initial Pressure (pi) 3,000 8,000 psi
Horizontal Wellbore Length (LHW) 2,000 5,000 ft SRV Major Axis Length (LMajor) LHW+100 1.4*LHW ft SRV Minor Axis Length (LMinor) 0.25*LMajor 0.4*LMinor ft Natural fracture permeability (kf) 0.000102 0.001099 md
Natural fracture porosity (f) 0.005 0.02 % Fracture Permeability of the SRV Zone (kSRV) 0.5 5 md
Fracture Porosity of the SRV Zone (SRV) 1.2*f 1.5*f % Specific Gravity (γg) 0.6 0.9
Bottom Hole Pressure (psf) 14.7 14.7+0.5*Pi psia Natural fracture spacing (yf) 50 200 ft
Fracture Spacing of the SRV Zone (ySRV) 1 50 ft
The implementation process for the SRV zones of the reservoirs are indicated by
the dimensions of the LMajor, LMinor, and LHW. The midpoint of the horizontal well is placed
in the middle of the drainage area for all reservoirs. The area of the SRV zone is represented
16
by a grid system to characterize an elliptically shaped area. The grid system design is based
on the dimensions of the major and minor axes of the ellipse. The grid along the horizontal
well will have 21 blocks. There are seven total rows for the SRV zone; each sets of
symmetric rows about the wellbore will have two less grid blocks each time as the SRV
zone advances away from the horizontal well; the elliptical shape is symmetrical about the
minor axis as well. An example of the reservoir is displayed in Figure 4.1 with the red grid
blocks indicating the SRV zone. Overall, each reservoir will have a dimension of 23x9, but
the dimensions of the blocks will vary based on the reservoir’s drainage area, horizontal
wellbore length, major axis length, and minor axis length of the SRV zone.
Figure 4.1. Sample 2D Top View for Reservoir Model with SRV Zone
A sample top view of a reservoir model (Reservoir Case 0001) is utilized to
demonstrate the differences between the natural fracture properties and the fracture
properties of the SRV zone. For fracture permeability properties, the kf values are assigned
L Major
L Major
17
to the blue colored grid blocks in the reservoir as illustrated in Figure 4.2 to represent the
natural fracture permeability. The kSRV values are assigned to red colored grid blocks in the
reservoir to represent the fracture permeability of the SRV zone. In this example, the natural
fracture has a permeability value of 0.00019 md as indicated by the blue colored blocks
and the fracture of the SRV zone has a permeability value of 1.79 as indicated by the red
colored blocks.
Figure 4.2. Sample Top View of Fracture Permeability of the Reservoir
For fracture porosity, the f values are assigned to the blue colored grid blocks in
the reservoir to represent the natural fracture porosity as demonstrated in Figure 4.3. The
SRV values are assigned to the red colored grid blocks in the reservoir to represent the
fracture porosity of the SRV zone. In this example, the natural fracture has a porosity value
RESERVOIR CASE 0001 PROPERTIES 23 x 9 grid system A = 467 acres h = 249 ft p
i = 3769.07 psi
18
of 1.7% as indicated by the blue colored blocks and the fracture of the SRV zone has a
porosity value of 2.2% as indicated by the red colored blocks.
Figure 4.3. Sample Top View of Fracture Porosity of the Reservoir
For fracture spacing properties, the yf values are assigned to the red colored grid
blocks in the reservoir to represent the natural fracture spacing as demonstrated in Figure
4.4. The ySRV values are assigned to the blue colored grid blocks in the reservoir to represent
the fracture spacing in the SRV zone. In this example, the natural fracture network has a
fracture spacing value of 190 ft. as indicated by the blue colored blocks and the fracture
network of the SRV zone has a fracture spacing value of 45 ft. as indicated by the red
colored blocks.
i = 3769.07 psi
19
Figure 4.4. Sample Top View of Fracture Spacing of the Reservoir
The reservoir models are studied using the numerical model with the input files
generated through an automated process using the parameter data sets. The total simulation
period is five years and the total number of data sets is 2,000. After running through each
of the data sets, daily production data were extracted for each reservoir and a hyperbolic
decline curve fitting process was implemented using Eq. 4.1 for each data set to obtain the
coefficients, qi, Di, and b for the ANN models.
= (1+∗∗)1 ⁄ (4.1)
The first five days of production are omitted for the hyperbolic curve fitting, as the
infinitely large production at the initial stage would cause difficulties in obtaining a well-
fitted decline curve. Table 4.2 shows the average values of the coefficient of regression,
R2, with values closer to 1.0000 indicating a good fit. The qi coefficients are representations
i = 3769.07 psi
20
of the initial productions, while the decline rates and curvatures of the production profiles
are captured by the Di and b coefficients. Other curve fitting techniques are also tested,
however, their results are not presented in this study due to their poor R2 values. Holditch
(2006) also stated, “Hydraulically fractured tight gas wells do not decline exponentially.
In almost all cases, they will decline hyperbolically”. Table 4.2 also shows the coefficient
range for qi, Di, and b.
Table 4.2. Hyperbolic Curve Fitting
Decline Curve R2 Coefficient Ranges
Hyperbolic 0.9992 2.5*107 < qi < 5.2*109
0.015 < Di < 1.0 0.06 < b < 0.67
Due to the reservoir properties being generated randomly and also have wide
ranges, after obtaining the coefficients from the curve fitting process, the results need to be
carefully examined to avoid and eliminate any non-feasible data sets. In the 2,000 data sets
generated for this study, none of them was needed for omission as they all produced
feasible production data.
A best-fitted case example of the curve fitting is demonstrated in Figure 4.5a in
comparison with the original gas production profile produced by the numerical model. A
worst fitted case example of the curve fitting is also demonstrated in Figure 4.5b. For both
cases, the curve fitting production curves are represented by the dash lines, whereas the
original production profiles are represented by the solid lines. Overall, the hyperbolic curve
fitting is able to capture the production profile quite well, with an average R2 value of
0.9992 out of the 2,000 cases.
21
(a)
(b)
Figure 4.5. Sample curve fitting comparison for (a) Case 0014, (b) Case 0233.
0 200 400 600 800 1000 1200 1400 1600 1800
Time (day)
Numerical Model
Curve Fitting
0 200 400 600 800 1000 1200 1400 1600 1800
Time (day)
Numerical Model
Curve Fitting
Reservoir Case 0014 PROPERTIES 23 x 9 grid system A = 318 acres h = 200 ft p
i = 5411.40 psi
2 = 1.0000
Reservoir Case 0233 PROPERTIES 23 x 9 grid system A = 317 acres h = 214 ft p
i = 3400.93 psi
22
To summarize the process of generating the training and testing sets for the ANN
models, a flow chart is illustrated in Figure 4.6.
Figure 4.6. Generation of Training and Testing Sets Flow Chart
Chapter 5
ANN Model Development
This chapter discusses the development of the forward-looking ANN model and the
inverse ANN model. Using the data sets described from Chapter 4, the ANN extension
toolbox is utilized to train both of the ANN networks.
5.1 Forward-Looking ANN Model
The inputs and outputs along with the functional links are listed in Table 5.1 for
the forward-looking ANN model. Inputs for this model include drainage area (A), thickness
(h), matrix permeability (km), matrix porosity (m), initial pressure (pi), natural fracture
permeability (kf), natural fracture porosity (f), natural fracture spacing (yf), specific gravity
(γg), fracture permeability of the SRV zone (kSRV), fracture porosity of the SRV zone (SRV),
fracture spacing of the SRV zone (ySRV), SRV major axis length (LMajor), SRV minor axis
length (LMinor), horizontal wellbore length (LHW), and bottom hole pressure (psf).
The outputs for this model are the coefficients for the hyperbolic decline curve
characterizing the gas production curve, qi, Di, and b. Table 5.1 displays the input
parameter list and the output parameter lists for the model. Since the magnitude for the qi
term is relatively large as it captures the initial production of the decline curve, taking the
logarithm scale of the term improves the performance of the model. The Di term on the
other hand has a much smaller magnitude; however, taking the logarithm scale of the term
24
will result in negative values. By adjusting the logarithm scales of the term into positive
values, the model seem to improve in terms of predicting performance of the Di term.
Table 5.1. Forward-Looking ANN Model Inputs and Outputs
Input Output 1 A 1 log(qi) 2 h 2 (-)log(Di) 3 km 3 b 4 m 5 pi 6 LHW 7 LMajor 8 LMinor
9 kf 10 f 11 SRV
12 kSRV 13 γg 14 psf 15 yf 16 ySRV
Functional links are added to output layers for the model as they provide
mathematical relationships between the parameters. They can also be incorporated as input
parameters as long as the functional links do not consist any of the output parameters; in
this model, input functional links did not have a significant impact for the performance of
the predictions. These additional relationships improve the performance of the model
significantly as oppose to a model without any functional links. The functional links for
25
the forward model are listed in Table 5.2. One of the most important output functional
links are the logarithmic values of the cumulative gas production, qT, in terms of improving
the performance of the model. Another important functional link is the second functional
link as the decline curve qi coefficients should be as close as possible to the initial
production, q0. Eigenvalues from 2x2 matrices are also introduced in the functional links
and their respective matrices are listed in Table 5.2.
Table 5.2. Functional Links and Eigenvalues for Forward-Looking Model
Output Functional Links Eigenvalues 1 log(qT) 1
log(qi) b
3 b/kSRV 2 log(qi) log(q0)
4 log(qi)*ySRV LMajor LHW
5 log(qi)*kSRV 6 (pi - psf)*km/log(qi) 7 log(qi)*m 8 log(qi)*yf*kf 9 sqrt[log(qi)*LMajor] 10 (-)log(Di)/ySRV 11 (-)log(Di)/(-)log(SRV) 12 yf/(-)log(Di) 13 b*ySRV/yf 14 b*h/LMinor 15 qi/A 16 log(qT)/log(qi) 17 Eigen 1(1) 18 log(Eigen 2(2))
The initial structure for the ANN model started as a three-layer feed-forward
backpropagation network (newff) as this type of supervised learning is more applicable in
26
static types of problems. Each of the hidden layers consisted 30 neurons initially with
tansig activation functions. This initial structure setting is optimized locally through
parallel training (Sun, 2015) based on 500 trials with five shuffles for each of the trials.
The range for the hidden layers is one to three and the range for the number of neurons for
each layer is 15 to 80. The final optimized architecture shown in Figure 5.1 consists of
three hidden layers of 37, 60, 38 neurons and tansig, tansig, logsig as the activation
function, respectively. The results from the final architecture achieved a ~3% average error
for each of the outputs and are discussed in detail in Chapter 6.1.
Figure 5.1. Architecture of Forward-Looking ANN Model Network
27
5.2 Inverse ANN Model
The inputs and outputs for the inverse ANN model are listed in Table 5.3. The
inverse model is designed to characterize the reservoir, especially the characteristics of the
SRV zone. The input includes the essentially known properties as production already has
taken place. The same logic was followed from the previous model in terms of adjusting
the parameters. For example, the large values of initial production (q0) and cumulative gas
production (qT) were converted to logarithmic scales to have a much smaller value range.
The matrix permeability (km) and natural fracture permeability (kf) were also converted to
positive logarithmic scales due to the small magnitude of the parameter values.
Table 5.3. Input and Output Parameters for the Inverse ANN Model
Input Output
28
For the inverse model, functional links are included for both the inputs and outputs.
The inverse model is more complex and difficult to predict, especially as the number of
predicting output parameters increases. A large numbers of functional links and
eigenvalues were tested through iteratively and the final set is listed in Table 5.4 including
parameters such as area of the ellipse (ASRV).
Table 5.4. Functional Links for the Inverse ANN Model
Output Functional Links Output Functional Links 1 Eigen 1 (1) 20 LMajor/LHW 2 Eigen 1 (2) 21 yf/ySRV 3 Eigen 2 (1) 22 kSRV*ySRV 4 Eigen 4 (2) 23 h/ySRV 5 Eigen 5 (2) 24 log(qT)/ySRV 6 log(ASRV) 25 log(q0)/ySRV 7 log(A/ASRV) 26 Eigen 6 (1) 8 kSRV/log(qi) 27 Eigen 6 (2) 9 log(qT)/kSRV 28 Eigen 6 (3) 10 kSRV/b 11 ySRV/log(qi) 12 ySRV/γg Input Functional Links 13 yf/log(qT) 1 Eigen 3 (1) 14 yf*b 2 Eigen 3 (2) 15 log(qT)/log(q0) 16 kf*h 17 log(kSRV/kf) 18 kf/km 19 LMinor/LMajor
29

Eigen 2 represents the eigenvalues of the matrix − log()

The development for the design of architecture follows the same method as the
forward-looking ANN model. The final structure shown in Figure 5.2 consists of two
hidden layers with 37, 53 neurons and tansig, logsig as the activation functions,
respectively. The performance for this architecture achieved the target tolerance level and
the results are discussed in Chapter 6.2.
30
31
Chapter 6
Results & Discussions
This chapter discusses the results obtained from both the forward-looking ANN
model for performance prediction and the inverse ANN model for the characterization of
the reservoir.
6.1 Forward-Looking ANN Model
Errors for the forward-looking ANN model are analyzed through four different
methods. The errors of the hyperbolic curve fitting coefficients are analyzed first. As
described previously in Chapter 5.1, qi and Di are both represented in logarithmic scales
with the latter also converted to positive values to improve the performance of the model.
Eq. 6.1.1 is used to calculate the errors for the qi coefficients, as they need to be converted
back from logarithmic scales first. Eq. 6.1.2 is used to calculate the errors for Di
coefficients, as they need to be converted to negative values then back from logarithmic
scales. For the b coefficients, Eq. 6.1.3 is used to calculate the errors since they did not
require any conversions.
10− × 100% (6.1.2)
= |−|
32
For the coefficients error equations, qi, Di, and b are the actual coefficients of the
hyperbolic decline curve; qiANN, DiANN, and bANN are the values predicted by the ANN model.
These equations provide the absolute errors between the actual and predicted values for
each of the testing cases.
The last method analyzes the errors of cumulative gas production, qT, for each case
using the hyperbolic decline equation through the actual and predicted coefficients. The
comparison of the cumulative production will provide a better understanding of the model
and its performance. The calculation for errors of cumulative gas production are described
using Eq. 6.1.4.
0 −∫

0
× 100% (6.1.4)
The target tolerance for this forward model is to obtain an average error per testing
case of 5%. Initially, with the total cases of 1,000 it was not possible to achieve the intended
error tolerance. After adding an additional 1,000 scenarios, the forward model is able to
predict the coefficients with an average error within the targeted threshold. After
optimizing the structure of the forward model through parallel training, the errors for the
forward model prediction using the selected testing cases are summarized in Table 6.1.
33
Table 6.1. Forward-Looking Model Prediction Errors
Average Error Minimum Error Maximum Error qi 3.05% 0.17% 14.56% Di 3.12% 0.12% 28.70% b 3.33% 0.01% 24.34% qT 2.97% 0.04% 14.00%
The proposed forward-looking ANN model is tested with a trained and tested with
a total of 2,000 data sets. Out of the 2,000 cases, 1,900 are used as training cases, 50 are
used as validation cases, and 50 are used as testing cases. For the 50 testing cases, the
coefficient prediction results are illustrated in Figure 6.1 for qi, Figure 6.2 for Di, and
Figure 6.3 for b. The “original” represents the coefficients from the original reservoir
fitted using the hyperbolic decline curve and the “ANN” represents the coefficients
predicted from the forward-looking model. The results are sorted from the lowest to highest
in terms of values for each coefficient as this type of demonstrations provide a clearer view
of how the ANN model is predicting. Overall, the predictions for the forward-looking ANN
model achieved well within the target tolerance range, with the average error of 3.05% for
qi, 3.12% for Di, 3.33% for b coefficients.
34
0
200000000
400000000
600000000
800000000
q i
Case Number
qi Results
D i
Case Number
Di Results
Figure 6.3. Coefficient b Prediction Results
A Hinton diagram (Sun, 2017) is constructed in Figure 6.4 to demonstrate the
weights for each of the input parameters relative to each of the forecasting coefficients.
The green color block represents a positive weight relationship and the red color block
represents a negative weight relationship for the parameters; the magnitude of the weight
is described by the size of the blocks with larger blocks representing a larger weight
magnitude and vice versa.
The fracture permeability of the SRV zone parameter has the largest weight in terms
of magnitude for all three coefficients, which agrees with the setup of the reservoir, since
most of the initial production will come from the SRV zone as dedicated by this parameter.
The initial pressure parameter has large magnitude of influences toward the qi and Di
coefficients as these two coefficients control the initial point and initial decline rate of the
0
0.1
0.2
0.3
0.4
0.5
0.6
b
Original
ANN
36
decline curve. Whereas the sandface pressure parameter has a large magnitude of influence
towards the b coefficient as this coefficient controls the degree of curvature of the line.
Both of these pressure parameter influences make valid physical senses and further validate
the setup of the reservoirs, as initial reservoir pressure would influence heavily to the initial
production stage and the sandface pressure of the wells would influence heavily toward the
rest of the production stage. The fracture spacing of the SRV zone parameter, surprisingly,
has the largest positive influence to the coefficients in this setup. As one of the more
difficult parameter to predict in the inverse model, this parameter is relatively less sensitive
to the overall production of the reservoirs.
Overall, the Hinton diagram for the forward-looking model provides insight details
regarding the relationship between the inputs and outputs. This diagram is also particularly
helpful for setting up the functional links for the inverse model as it provides a better
understanding on how each of the parameters relates to the forecasting of hyperbolic
decline curve coefficients.
Predicted gas production profile and cumulative production curves can be
generated through the predicted coefficients using the hyperbolic equation. The proposed
forward-looking model has an average error of 2.97% for the cumulative production
calculated using Eq. 6.1.4, which is well within the targeted error tolerance for the 50
testing cases. Based on the cumulative production error lists, a best case (Figure 6.5), an
intermediate case (Figure 6.6), and a worst case (Figure 6.7) are selected as sample cases
demonstrated below with gas production profile (a) and cumulative gas production (b). The
log(q i ) -log(D i ) b
A
h
k f
38
blue dash lines represent the original hyperbolic curve fittings and the red solid lines
represent the forward-looking model predictions.
For the best-case scenario, the cumulative production error is 0.04% and has a qi
error lower than the average error at 2.67% and below average errors for Di, 1.55%, and b,
1.99% as well. The intermediate sample case has a cumulative production error of 2.90%
with rather low Di and b errors of 0.47% and 0.39%, respectively. However, the
intermediate sample case has qi error at around the average predicted qi error while the
cumulative production error is also around the cumulative production average error. The
cumulative production error for the worst case scenario is 14.00% as it also has the highest
qi error of 14.56% as Figure 6.7b clearly demonstrates during the initial production stage
for this reservoir. This case also has a slightly higher than average Di error at 4.26% and a
relatively high b error at 24.34%.
(a)
Time(days)
q=qi/(1+b*Di*t) (1/b)
Forward ANN
Reservoir Case 1797 ERRORS qi = 2.67% Di = 1.55% b = 1.99% Cum. Production = 0.04% PROPERTIES A = 266 acres h = 240 ft pi = 4287.48 psi km = 0.000004 md m = 5.2% kf = 0.00064 md f = 1.0% γg = 0.66 yf = 126 ft kSRV = 1.07 md SRV = 1.5% LHW = 2043 ft psf = 1446 psi LMajor = 2205 ft LMinor = 682 ft ySRV = 44 ft
39
(b)
(a)
Time(days)
q=qi/(1+b*Di*t) (1/b)
Forward ANN
Time(days)
q=qi/(1+b*Di*t) (1/b)
Forward ANN
Reservoir Case 0891 ERRORS qi = 3.08% Di = 0.47% b = 0.39% Cum. Production = 2.90% PROPERTIES A = 350 acres h = 176 ft pi = 6954 psi km = 0.00006 md m = 7.4% kf = 0.00049 md f = 1.9% γg = 0.74 yf = 178 ft kSRV = 0.70 md SRV = 2.3% LHW = 2343 ft psf = 1554 psi LMajor = 2846 ft LMinor = 727 ft ySRV = 40 ft
40
(b)
(a)
Time(days)
q=qi/(1+b*Di*t) (1/b)
Forward ANN
Time(days)
q=qi/(1+b*Di*t) (1/b)
Forward ANN
Reservoir Case 0545 ERRORS qi = 14.56% Di = 4.26% b = 24.34% Cum. Production = 14.00% PROPERTIES A = 120 acres h = 193 ft pi = 7808 psi km = 0.00005 md m = 6.5% kf = 0.00038 md f = 1.8% γg = 0.85 yf = 97 ft kSRV = 4.56 md SRV = 2.2% LHW = 1372 ft psf = 2170 psi LMajor = 1664 ft LMinor = 530 ft ySRV = 5 ft
41
(b)
Figure 6.7. Sample Forward-Looking Model Prediction Case – Worst case
To further investigate into the relationships between each of the predicted
coefficients errors and the cumulative production errors, two testing cases, case 0477 and
case 0545, are selected. Individual coefficient difference disparity is studied for each case
and the set up for this investigation is listed as the following:
• To study the qi error effect, the original hyperbolic curve fitting values for Di and b
in conjunction with the ANN predicted qi value are used to generate the production
profile and cumulative production.
• To study the Di error effect, the original hyperbolic curve fitting values for qi and b
in conjunction with the ANN predicted Di value are used to generate the production
Time(days)
4.2
4.4
4.6
4.8
5
5.2
5.4
5.6
5.8
q=qi/(1+b*Di*t) (1/b)
Forward ANN
42
• To study the b error effect, the original hyperbolic curve fitting values for qi and Di
in conjunction with the ANN predicted b value are used to generate the production
The effect of qi error in relation to cumulative production error is demonstrated in
Figure 6.8 for testing case 0477 and Figure 6.9 for testing case 0545. The two selected
cases demonstrated an almost one to one relationship, as 6.88% in qi error resulted in 6.88%
in cumulative production error for the first case and 14.56% qi error led to 14.56%
cumulative production error for the second case. This came out to be expected as the qi
term has a relative large magnitude and controls the initial production stage of the
production profile, which would contribute greatly to the overall production curve.
(a)
Time(days)
q=qi/(1+b*Di*t) (1/b)
Forward ANN
Reservoir Case 0477 ERRORS qi = 6.88% *Di = 0.00% *b = 0.00% Cum. Production = 6.88% PROPERTIES A = 408 acres h = 221 ft pi = 4530 psi km = 0.00002 md m = 5.3% kf = 0.00107 md f = 1.9% γg = 0.78 yf = 168 ft kSRV = 0.74 md SRV = 2.8% LHW = 2528 ft psf = 1234 psi LMajor = 2736 ft LMinor = 870 ft ySRV = 41 ft
* Original hyperbolic coefficient values
Figure 6.8. Sample Forward-Looking Model Prediction Case – qi prediction error
(a)
Time(days)
q=qi/(1+b*Di*t) (1/b)
Forward ANN
Time(days)
q=qi/(1+b*Di*t) (1/b)
Forward ANN
Reservoir Case 0545 ERRORS qi = 14.56% *Di = 0.00% *b = 0.00% Cum. Production = 14.56% PROPERTIES A = 120 acres h = 193 ft pi = 7808 psi km = 0.00005 md m = 6.5% kf = 0.00038 md f = 1.8% γg = 0.85 yf = 97 ft kSRV = 4.56 md SRV = 2.2% LHW = 1372 ft psf = 2170 psi LMajor = 1664 ft LMinor = 530 ft ySRV = 5 ft
Figure 6.9. Sample Forward-Looking Model Prediction Case – qi prediction error
The effect of Di error in relation to cumulative production error is demonstrated in
cases displayed a slightly less effect as compared to the previous qi investigations, as
28.70% in Di error resulted in 21.79% in cumulative production error for the first case and
4.26% Di error led to 3.23% cumulative production error for the second case. The accuracy
of Di term is still significant as it controls the initial decline rate of the curve. It is in the
denominator of the decline curve equation in conjunction with time, and the effect is shown
in both cases as in terms of the initial decline rate of the prediction plots versus the original
reservoir production plots.
Time(days)
q=qi/(1+b*Di*t) (1/b)
Forward ANN
Figure 6.10. Sample Forward-Looking Model Prediction Case – Di prediction error
0 50 100 150 200 250 300
Time(days)
q=qi/(1+b*Di*t) (1/b)
Forward ANN
Time(days)
q=qi/(1+b*Di*t) (1/b)
Forward ANN
Reservoir Case 0477 ERRORS *qi = 0.00% Di = 28.70% *b = 0.00% Cum. Production = 21.79% PROPERTIES A = 408 acres h = 221 ft pi = 4530 psi km = 0.00002 md m = 5.3% kf = 0.00107 md f = 1.9% γg = 0.78 yf = 168 ft kSRV = 0.74 md SRV = 2.8% LHW = 2528 ft psf = 1234 psi LMajor = 2736 ft LMinor = 870 ft ySRV = 41 ft
Figure 6.11. Sample Forward-Looking Model Prediction Case – Di prediction error
0 50 100 150 200 250 300
Time(days)
q=qi/(1+b*Di*t) (1/b)
Forward ANN
Time(days)
q=qi/(1+b*Di*t) (1/b)
Forward ANN
Reservoir Case 0545 ERRORS *qi = 0.00% Di = 4.26% *b = 0.00% Cum. Production = 3.23% PROPERTIES A = 120 acres h = 193 ft pi = 7808 psi km = 0.00005 md m = 6.5% kf = 0.00038 md f = 1.8% γg = 0.85 yf = 97 ft kSRV = 4.56 md SRV = 2.2% LHW = 1372 ft psf = 2170 psi LMajor = 1664 ft LMinor = 530 ft ySRV = 5 ft
47
The effect of b error in relation to cumulative production error is demonstrated in
cases showed different effects as 16.65% in b error resulted in 12.88% in cumulative
production error for the first case and 24.34% Di error led to 2.86% cumulative production
error for the second case. Since the b coefficient is in the power term in the denominator,
the actual value of b will contribute more as demonstrated in the two selected cases. The
predicted value for b is 0.51 for the first case and 0.15 for the second case. The effect of b
coefficient will result in the form of power, as larger value of b will have a much more
impact (Figure 6.12b) whereas smaller value will have less of an impact (Figure 6.13b)
to the cumulative production error.
(a)
Time(days)
q=qi/(1+b*Di*t) (1/b)
Forward ANN
Reservoir Case 0477 ERRORS *qi = 0.00% *Di = 0.00% b = 16.65% Cum. Production = 12.88% PROPERTIES A = 408 acres h = 221 ft pi = 4530 psi km = 0.00002 md m = 5.3% kf = 0.00107 md f = 1.9% γg = 0.78 yf = 168 ft kSRV = 0.74 md SRV = 2.8% LHW = 2528 ft psf = 1234 psi LMajor = 2736 ft LMinor = 870 ft ySRV = 41 ft
Figure 6.12. Sample Forward-Looking Model Prediction Case – b prediction error
(a)
Time(days)
q=qi/(1+b*Di*t) (1/b)
Forward ANN
Time(days)
q=qi/(1+b*Di*t) (1/b)
Forward ANN
Reservoir Case 0545 ERRORS *qi = 0.00% *Di = 0.00% b = 24.34% Cum. Production = 2.86% PROPERTIES A = 120 acres h = 193 ft pi = 7808 psi km = 0.00005 md m = 6.5% kf = 0.00038 md f = 1.8% γg = 0.85 yf = 97 ft kSRV = 4.56 md SRV = 2.2% LHW = 1372 ft psf = 2170 psi LMajor = 1664 ft LMinor = 530 ft ySRV = 5 ft
Figure 6.13. Sample Forward-Looking Model Prediction Case – b prediction error
0 50 100 150 200 250 300
Time(days)
q=qi/(1+b*Di*t) (1/b)
Forward ANN
6.2 Inverse ANN Model
For the inverse model, error analysis is done in three types of equations. For the
parameters kf and km, Eq.6.2.1 is used to calculate the errors between the actual and
predicted values. For the parameters LMajor, LMinor, kSRV, m, f, SRV, yf, and ySRV, Eq.6.2.2
is used to calculate the errors between the actual and predicted values.
= 10−−10− 10−
× 100% (6.2.1)
× 100% (6.2.2)
The proposed inverse ANN model is tested and trained with 2,000 cases which were
also used in the forward-looking ANN model. Out of the 2,000 cases, 1,900 are used as
training cases, 50 are used as validation cases, and 50 are used as testing cases. The errors
for each of the output parameters are summarized in Table 6.2.
Table 6.2. Prediction Errors, Inverse ANN Model
Average Error
Minimum Error
Maximum Error
LMajor 5.82% 0.26% 14.19% LMinor 9.61% 0.46% 28.66% kSRV 13.36% 0.25% 48.40% SRV 24.59% 0.12% 171.11% ySRV 25.03% 0.70% 302.46% m 8.53% 0.15% 35.81% kf 26.03% 1.31% 118.36% f 25.36% 1.25% 177.19% km 158.36% 3.68% 2867% yf 42.24% 0.28% 159.56%
The predictions for the SRV zone parameters have achieved well within reasonable
tolerance as these parameters are particular important for the reservoir models in this study.
51
The inverse model is able to capture the dimensions of the SRV zone quite well as elliptical
major (Figure 6.14) and minor axis (Figure 6.15) lengths resulted the best performances
in terms of prediction accuracy with an average error of 5.82% for LMajor and 9.61% for
LMinor. The “original” represents the parameter values from the original reservoir and the
“ANN” represents the parameter values predicted by the inverse model. The results are
sorted from the lowest to highest in terms of values for each parameter as this type of
demonstrations provide a clearer view of how the ANN model is predicting.
Figure 6.14. SRV Zone Major Axis Length Predictions
0
500
1000
1500
2000
2500
3000
3500
4000
Le ng
th (f
Original
ANN
52
Figure 6.15. SRV Zone Minor Axis Length Predictions
The SRV fracture permeability (Figure 6.16) also predicted well with an average
error of 13.36% even as the scale range for this parameters is relatively small as compared
to the other SRV parameters. This parameter is particularly focused with functional links
as kSRV has major influence on gas production of the reservoirs as noticed through trial and
errors. The priority during the developmental stage of the inverse model is to aim as low
as possible for the kSRV prediction errors.
0
200
400
600
800
1000
1200
Le ng
th (f
Original
ANN
53
Figure 6.16. SRV Zone Fracture Permeability Predictions
For the SRV fracture porosity and SRV fracture spacing, the average errors are in
the 20% range with 24.59% for SRV and 25.03% for ySRV. The inverse model is able to
predict the trends despite having relative higher errors as shown in Figure 6.17 for SRV
and Figure 6.18 for ySRV. The ySRV was quite difficult to predict with the initial SRV fracture
spacing parameter range as it was too small to have any noticeable impact to the overall
production. After tweaking the parameter through trial and errors, the final parameter range
for ySRV was able to have some sensitivities to the production. The errors for SRV can be
explained due to the small magnitude of the parameter and due to the dominance of the
matrix porosity for this type of dual-porosity, single permeability model.
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Pe rm
ea bi
lit y
(m d)
Original
ANN
54
0
0.005
0.01
0.015
0.02
0.025
0.03
Po ro
sit y
Original
ANN
0
10
20
30
40
50
60
Le ng
th (f
Original
ANN
55
The matrix porosity has a dominant effect for the type of reservoir models presented
in this study as compared to the fracture porosity in the matrix and fracture porosity in the
SRV zone. The inverse model is able to predict the matrix porosity exceptionally well as
demonstrated in Figure 6.19. The f only has an average error of 8.53% despite the
parameter having small values of magnitude in the 10-2 range.
Figure 6.19. Matrix Porosity Predictions
The average errors for natural fracture permeability and natural fracture porosity
are also in the 30% range with 26.03% for kf and 25.36% for f. The inverse model is able
to capture the kf values but with less accuracy compared to the kSRV predictions as shown
in Figure 6.20. The larger errors can be explained by the much smaller magnitude of kf
compare to kSRV and the fact that the production has to go through the SRV fracture zones
first before reaching the matrix fractures. The same observations can be made for the f
predictions as explained for SRV with the results displayed in Figure 6.21.
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Po ro
sit y
0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
Pe rm
ea bi
lit y
(m d)
Po ro
sit y
Original
ANN
57
The predictions for the parameters km and yf have the worst performance for the
inverse model with 158.36% for km and 42.24% for yf. The large discrepancy for km can be
explained due to the parameter consists of relatively small values with magnitude as low
as 10-6. Figure 6.22 compares the ANN predictions and the original value for km and it
displays a trend that clearly shows the inverse model has difficulties in predicting this
parameter. Similar difficulties are also present for the parameter yf as it shows in Figure
6.23. Since this is a reservoir dominated by the fracture permeability of the SRV zone and
the natural fracture permeability, km should have minimal impact to the gas production of
the reservoir as expected. Through manual testing, the parameter yf also has minimal impact
to the gas production. Since the product of yf and kf contributes to the production in the
modeling equations and with kf consists of small values, thus explains the insensitivity of
the parameter to the model.
Figure 6.22. Matrix Permeability Predictions
0.00E+00
Pe rm
ea bi
lit y
(m d)
Figure 6.23. Natural Fracture Spacing Predictions
A Hinton diagram (Sun, 2017) is constructed for the inverse model and the final
chart is illustrated in Figure 6.24. The green color block represents a positive weight and
the red color block represents a negative weight for the parameters; the magnitude of the
weight is described by the size of the blocks with larger blocks representing a larger weight
and vice versa. This diagram demonstrates the weights for each of the input parameters
relative to each of the output reservoir parameters. The Hinton diagram serves similar
purpose as described in the previous forward-looking model; it is a quite useful tool
especially in creating and testing of the functional links between the parameters.
For the inverse model, the Hinton Diagram clearly demonstrates the importance of
the decline curve coefficients in predicting the characteristic parameters. The qi coefficient
has the largest influence as expected, as the initial production point is critical in terms of
0
50
100
150
200
250
Le ng
th (f
Original
ANN
59
predicting the parameters especially for the SRV zone. The Di coefficient influences the
initial decline rate whereas b indicates the curvature of the production curve, both of these
coefficients are also critical to the SRV zone and the reservoir characteristics.
The well design parameters, horizontal wellbore length and sandface pressure, have
almost no influences toward the predictions of the characteristic parameters. This
observation makes physical senses and validates the setup of the model as these well design
parameters would not tell much about the characteristics of the SRV zone and the reservoir.
The drainage area has noticeable influence towards the prediction of the
parameters, especially for the SRV fracture permeability, SRV porosity, and SRV fracture
spacing parameters, as opposed to the other well design parameters. This is as expected as
the SRV properties should be closely related to the drainage area once the production comes
online in a given period. The thickness and initial pressure of the reservoir also have
influential impact towards the prediction of the parameters as expected, especially for the
reservoir characteristics such as matrix porosity and natural fracture properties in
geological physical senses.
For the specific gravity parameter, since it is used as a black oil model in the
simulator, further observations cannot be stated except for the parameter does have
influences toward the prediction of the characteristic parameters. However, if a
compositional model were used instead, it would provide much deeper insights toward the
relationship between the gas composition and the characteristic parameters for the inverse
model. One speculation would be the gas composition would have a larger impact in
predicting the reservoir characteristics, but also have noticeable impact in predicting the
SRV characteristics as well.
60
Overall, the Hinton Diagrams are recommended for ANN applications, as they
provide detailed insights regarding the relationship between the inputs and outputs. They
also provides validation towards the setup of the models, especially in problems such as
history matching and characterizations, as the relationship between the inputs and outputs
would explain whether the setup of the models make any physical senses.
Figure 6.24. Hinton Diagram for Inverse ANN Model
k SRV
f y
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After obtaining the predicted results from the inverse model within reasonable
tolerance, the next step is to validate the results by comparing the production rate curve
and the cumulative production curve. Three types of validations will be conducted based
on the algorithm described below. Table 6.3 summarizes the cumulative production error
as compared to the original cases for the three validations.
Table 6.3. Inverse Model Cumulative Production Error
Min (%) Max (%) Mean (%) Cumulative Error Numerical Model (ANN) 0.6545 29.5723 8.1648
Cumulative Error FF(Original) 0.6405 37.8026 13.7809
Cumulative Error FF(Predicted) 0.0485 42.4629 12.4558
The first validation check is to find the testing case numbers based on the index
provided by the inverse model. By combining the predicted parameter values with their
respective input parameters, the next step is to use the updated reservoir parameters and
construct the reservoir model for each testing case through the numerical model. After
simulating through the numerical model, gas production rates and cumulative gas
productions are extracted and compared with the original rates and productions, indicted
by the red dash lines and labeled as “Numerical Model (ANN) Predicted” in the result
figures. A histogram for the cumulative production error distribution is displayed in Figure
6.25 for this method of validation check.
62
The second validation check utilizes the forward-looking ANN model developed
previously in this study. Following the initial procedure as the first validation check, the
reservoir parameters combined with the predicted results are then inputted through the
developed forward-looking model. The forward-looking model predicts the coefficients of
the decline curve using the predicted parameters from the inverse model. Gas production
curves and cumulative gas production curves are then derived using the coefficients, thus
provides a material balance check by comparing with the original and the “Numerical
Model (ANN) Predicted”. This validation is indicated by the purple dash-dot lines and
labeled as “FF (ANN) Predicted” in the results figures. A histogram for the cumulative
production error distribution is displayed in Figure 6.26 for this method.
0 5 10 15 20 25 30
error percentage (%)
63
Figure 6.26. Method 2 Validation Error Distribution
The last validation check also utilizes the forward-looking ANN model developed
previously in this study. However, instead of using the predicted parameters from the
inverse model, gas production curves and cumulative gas production curves are generated
using the coefficients predicted by the original parameters as the input for the forward-
looking model. This method provides another material balance check to the original and
the previous two methods of validation checks. The yellow dot lines indicate this validation
as “FF (Original) Predicted” in the results figures. A histogram for the cumulative
production error distribution is displayed in Figure 6.27 for this method.
0 5 10 15 20 25 30 35 40 45
64
Figure 6.27. Method 3 Validation Error Distribution
After the cumulative errors for each method are obtained, two cases are selected
each for the best cases, intermediate cases, and worst cases for analysis. The case selections
are based on the average cumulative production errors for all three methods. Each selected
case are shown in figures in the same format as previously, with “figure a” for gas
production profile and “figure b” for cumulative gas production. The analysis for each case
will not have a detail look for the parameter of km and yf, since the inverse model cannot
really predict these two parameters as mentioned earlier in the section. The average
parameter errors are calculated based on the parameters LMajor, LMinor, kSRV, and m.
0 5 10 15 20 25 30 35 40
65
For the best case scenarios, case 0574 and case 1402 are selected and shown in
Figure 6.28 and Figure 6.29, respectively. The average cumulative error is 1.33% for case
0574 and 2.59% for case 1402.
For case 0574, LMinor and m are accurately predicted, as the errors for these two
parameters are around 1%. LMajor and kSRV are also well predicted with errors around 12%.
The other parameters are predicted within reasonable range except for ySRV. The average
parameter error is 6.41% for this case and the cumulative production error came out to be
the lowest out of the 50 testing cases. For this case, the low error predictions of LMajor,
LMinor, kSRV, and m really helped the overall predictions of the reservoir’s performance
throughout the production stage.
For case 1402, most of the predicted parameters are around 5% for prediction error,
except for kf, which is still well predicted with 13.50% for prediction error. The overall
predictions of the reservoir’s production agree with the parameter predictions as Figure
6.29 demonstrated excellent predictions for all three methods.
66
(a)
(b)
0 50 100 150 200 250 300 350 400
Time(Day)
Original Case
Time(Day)
Original Case
FF(Original) Predicted
FF(ANN) Predicted
Reservoir Case 0574 ERRORS LMajor = 11.47% LMinor = 1.29% kSRV = 12.11% SRV = 28.69% ySRV = 52.41% km = 45.69% m = 0.77% kf = 24.81% f = 24.00% yf = 28.24% *Avg. Error = 6.41% Cum. ANN = 1.59% Cum. FF(Orig.) = 2.09% Cum. FF(Pred.) = 0.32% Cum. Avg. = 1.33% PROPERTIES A = 402 acres h = 222 ft pi = 6543.35 psi km = 0.00007 md m = 13.6% kf = 0.00087 md f = 1.0% γg = 0.72 yf = 87 ft kSRV = 1.63 md SRV = 1.3% LHW = 2511 ft psf = 875 psi LMajor = 2809 ft LMinor = 960 ft ySRV = 8 ft *Avg. Error is calculated for LMajor, LMinor, kSRV, and Φm
67
(a)
(b)
0 50 100 150 200 250 300 350 400
Time(Day)
Original Case
Time(Day)
Original Case
FF(ANN) Predicted
Reservoir Case 1402 ERRORS LMajor = 3.26% LMinor = 2.82% kSRV = 5.89% SRV = 2.27% ySRV = 5.63% km = 44.29% m = 2.12% kf = 13.50% f = 6.09% yf = 76.93%
*Avg. Error = 3.52% Cum. ANN = 1.83% Cum. FF(Orig.) = 2.59% Cum. FF(Pred.) = 3.34% Cum. Avg. = 2.59% PROPERTIES A = 376 acres h = 127 ft pi = 4920.75 psi km = 0.00006 md m = 14.9% kf = 0.00035 md f = 1.2% γg = 0.71 yf = 68 ft kSRV = 0.80 md SRV = 1.8% LHW = 2427 ft psf = 479 psi LMajor = 2907 ft LMinor = 989 ft ySRV = 26 ft *Avg. Error is calculated for LMajor, LMinor, kSRV, and Φm
68
For the intermediate case scenarios, case 0098 and case 0374 are selected and
shown in Figure 6.30 and Figure 6.31, respectively. The average cumulative error is
4.28% for case 0098 and 8.04% for case 0374.
For case 0098, the average cumulative error is relatively low; however, the
cumulative error for using the first method falls into the intermediate range with an error
of 7.86%. Most of the parameters are well predicted with errors within 10%, except for kf
(22.4%), SRV (20.29%), and ySRV (32.42%). The average parameter predicting error is
6.97%. However, the reservoir predicting performance using the second and third method
is excellent as the cumulative errors for each of these two methods are only around 2%.
For case 0374, this reservoir is quite similar as compared to case 0098, except for
more than double the initial reservoir pressure than the latter. The production profile shows
the similar production but case 0374 has a much longer depletion period due to its higher
initial pressure. The parameters, LMinor (11.77%), kSRV (17.77%), and m (12.68%) are all
above the average prediction error, respectively, whereas the prediction error for case 0098
is much smaller. These higher errors possibly contribute to the higher cumulative
production errors for this reservoir, especially compared with case 0098 as average
parameter predicting error for case 0374 is 11.16%.
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(a)
(b)
0 50 100 150 200 250 300 350 400
Time(Day)
Original Case
Time(Day)
Original Case
FF(ANN) Predicted
Reservoir Case 0098 ERRORS LMajor = 6.26% LMinor = 9.57% kSRV = 4.83% SRV = 20.29% ySRV = 32.42% km = 530.12% m = 7.22% kf = 22.40% f = 11.54% yf = 37.16% *Avg. Error = 6.97% Cum. ANN = 7.86% Cum. FF(Orig.) = 2.43% Cum. FF(Pred.) = 2.53% Cum. Avg. = 4.28% PROPERTIES A = 311 acres h = 272 ft pi = 3161.25 psi km = 0.000006 md m = 13.7% kf = 0.00063 md f = 1.4% γg = 0.65 yf = 108 ft kSRV = 3.13 md SRV = 2.1% LHW = 2209 ft Psf = 259 psi LMajor = 2721 ft LMinor = 815 ft ySRV = 7 ft *Avg. Error is calculated for LMajor, LMinor, kSRV, and Φm
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(a)
(b)
0 50 100 150 200 250 300 350 400
Time(Day)
Original Case
Time(Day)
Original Case
FF(ANN) Predicted
*Avg. Error = 11.16% Cum. ANN = 11.95% Cum. FF(Orig.) = 5.13% Cum. FF(Pred.) = 7.04% Cum. Avg. = 8.04% PROPERTIES A = 292 acres h = 280 ft pi = 7093.93 psi km = 0.000001 md m = 14.54% kf = 0.00077 md f = 1.5% γg = 0.79 yf = 107 ft kSRV = 1.11 md SRV = 2.1% LHW = 2142 ft psf = 2753 psi LMajor = 2571 ft LMinor = 795 ft ySRV = 30 ft
*Avg. Error is calculated for LMajor, LMinor, kSRV, and Φm
71
For the worst case scenarios, case 0040 and case 0175 are selected and shown in
Figure 6.32 and Figure 6.33, respectively. The average cumulative error is 23.29% for
case 0040 and 29.64% for case 0175.
For case 0040, the errors for parameter LMinor (28.04%), SRV (38.10%), m 29.02%),
and f (45.32%) are all quite large. Even with accurately predicted kSRV (1.87%) and LMajor
(5.58%), the overall reservoir performance prediction has the highest error in terms of
cumulative production using the first method.
For case 0175, the errors for parameter LMinor (27.25%), kSRV (30.75%), and kf
(41.86%) are also quite high as the cumulative production prediction really deviates away
from original reservoir’s production. This particular case also has the highest average
cumulative production error out of the 50 testing cases.
Both of these two cases had high average parameter predicting errors with 16.13%
and 20.72%, respectively, and their overall reservoir performance demonstrated the high
average errors could lead to high cumulative production errors.
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(a)
(b)
0 50 100 150 200 250 300 350 400
Time(Day)
Original Case
Time(Day)
Original Case
FF(ANN) Predicted
Reservoir Case 0040 ERRORS LMajor = 5.58% LMinor = 28.04% kSRV = 1.87% SRV = 38.10% ySRV = 11.44% km = 160.77% m = 29.02% kf = 12.48% f = 45.32% yf = 36.87% *Avg. Error = 16.13% Cum. ANN = 29.57% Cum. FF(Orig.) = 14.87% Cum. FF(Pred.) = 25.44% Cum. Avg. = 23.29% PROPERTIES A = 116 acres h = 266 ft pi = 6734.08 psi km = 0.00002 md m = 5.4% kf = 0.00028 md f = 0.9% γg = 0.81 yf = 177 ft kSRV = 1.52 md SRV = 1.3% LHW = 1346 ft psf = 513 psi LMajor = 1577 ft LMinor = 626 ft ySRV = 16 ft *Avg. Error is calculated for LMajor, LMinor, kSRV, and Φm
73
(a)
(b)
0 50 100 150 200 250 300 350 400
Time(Day)
Original Case
Time(Day)
Original Case
FF(ANN) Predicted
*Avg. Error = 20.72% Cum. ANN = 19.33% Cum. FF(Orig.) = 37.80% Cum. FF(Pred.) = 31.78% Cum. Avg. = 29.64% PROPERTIES A = 119 acres h = 112 ft pi = 4114 psi km = 0.00009 md m = 14.67% kf = 0.00087 md f = 1.8% γg = 0.64 yf = 54.12 ft kSRV = 2.49 md SRV = 2.5% LHW = 1368 ft psf = 1339 psi LMajor = 1368 ft LMinor = 1534 ft ySRV = 25 ft
*Avg. Error is calculated for LMajor, LMinor, kSRV, and Φm
74
To further analyze whether a certain type of reservoir would affect the prediction
error, the data for each testing case are examined along with the errors for each cases. The
data and prediction errors for the testing cases of the inverse model are listed in Appendix
B.1.
One observation from the analysis is that smaller drainage area for the reservoir
seem to result in higher cumulative production error as shown in Figure 6.34. This figure
is plotted with drainage area against the predicted cumulative production error resulted
from the forward model using the predicted parameters. The plot shows most of the larger
drainage area reservoirs (>200 acres) have good results in terms of cumulative production
error. The error bar shifts to the right especially with smaller reservoirs (<200 acres). This
is most likely due to the testing cases having various flow profiles.
For example, for Case 1438, it has a cumulative production error of 42.46% and
has the smallest drainage area out of the testing cases. The reservoir has a lower than
average thickness. It also has matrix permeability, natural fracture permeability, and SRV
fracture permeability values higher than the averages. This reservoir has the lowest initial
pressure and a higher than average sandface pressure. All these parameters indicate the
likelihood of a different flow profile than reservoirs with larger drainage area, thus
resulting the inverse model prediction with the highest cumulative production error even
with most of the parameter predicted quite well.
Another example case is Case 0175, as it also has one of the lowest drainage area
with the second lowest cumulative production error at 31.78%. This case shares similar
trends with Case 1438, as it has values lower than average for thickness and initial pressure
75
and higher than average values for matrix permeability, natural fracture permeability, SRV
fracture permeability, and sandface pressure.
The Case 1399 has the most accurate prediction in terms of cumulative production
error using the forward model at 0.05%, but a rather high cumulative production error using
the numerical model at 20.63%. This reservoir has the largest drainage area with slightly
below average value for SRV fracture permeability.
Another example is Case 0157, which displays similar observations as Case 1399,
but with a cumulative production error of 10.52% using forward model and a cumulative
production error of 16.96% using the numerical model. This reservoir has a relatively small
drainage area compare to Case 1399, but it does have smaller than average permeability
values.
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These analysis further points toward the speculation as smaller drainage area would
shift the cumulative production error bars to the right, especially if different flow profiles
are introduced to the cases. Similar approaches are also performed for the other parameters.
However, they did not display noticeable relationships between these parameters versus
the cumulative production errors.
The next approach is to exam the area of the SRV zone versus the cumulative
production error to analyze the observed relationship. However, the results were not
sufficient to provide any relationships between the area of the SRV zone and the
cumulative production error.
Overall, the inverse model is able to predict the characteristic parameters within
reasonable tolerance. The cumulative production error using the predicted parameters are
also within reasonable error range as the average cumulative production error simulating
through the numerical model using the predicted parameters is 8.16% and more
importantly, the average cumulative production error utilizing the predicted parame

characterization of tight gas reservoirs with …

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