Journal of Natural Gas Science and Engineering 36 (2016) 71e78
Contents lists available at ScienceDirect
Journal of Natural Gas Science and Engineering
journal homepage: www.elsevier.com/locate/jngse
Stochastic optimization of hydraulic fracture and horizontal well
parameters in shale gas reservoirs
Muzammil Hussain Rammay*, Abeeb A. Awotunde
King Fahd University of Petroleum and Minerals, Saudi Arabia
a r t i c l e i n f o
a b s t r a c t
Article history:
Received 29 December 2015
Received in revised form
2 October 2016
Accepted 5 October 2016
Available online 6 October 2016
Multistage hydraulic fracturing is one of the most important techniques in the successful exploitation
and development of shale gas reservoirs. Most unconventional reservoirs rely on multistage hydraulic
fracturing for commercial success. This success is accomplished by drilling horizontal well of appropriate
length and creating transverse hydraulic fractures in stages across the well. Shale-gas reservoir's contact
with the horizontal well bore is improved by using optimal well length, optimal fracture conductivity,
optimum fracture length and optimum number of fracture stages. Because multistage fracturing of long
horizontal wells increase the cost of field development, the economics of field development can be
improved by using global optimization algorithms to estimate the optimum values of these operational
parameters. In shale gas reservoirs, hydraulic fracture parameters such as fracture half-length, amount of
proppant and fracture spacing should be optimized to maximize the net present value (NPV).
In this work, differential evolution (DE) is implemented on a more realistic LGR-based shale gas
reservoir simulation model. The objective is to maximize the net present value by optimizing hydraulic
fracturing parameters and horizontal well length. Results obtained indicate that significant increase in
NPV can be realized by using the optimization algorithm to estimate the operational parameters of
hydraulic fracturing process in shale gas reservoirs.
© 2016 Elsevier B.V. All rights reserved.
Keywords:
Fracture length
Fracture conductivity
Stochastic optimization
Multi stage hydraulic fracture
Net present value
Fracture spacing
1. Introduction
There is also considerable potential for shale gas exploitation
globally. According to EIA (EIA, 2011), the total gas resources of the
world are estimated to be about 22,600 TCF of which 40% is now
contributed by shale plays. Dong et al. (2014) conducted a probabilistic assessment of world recoverable shale gas resources and
conclude that the amount of shale-gas OGIP worldwide was 34,000
(P90) to 73,000 (P10) Tcf, with total recoverable reserves (TRR) of
4000 (P90) to 24,000 (P10) Tcf. Recently, shale reservoirs have
become technically and economically recoverable. The development of Barnett shale is considered a trendsetter in the shale gas
industry. The two most important factors for this extraordinary
success are horizontal drilling and advancement in fracturing
technology. In 2000, shale gas was 1% of domestic gas production in
the United States but has risen to 32% in 2013 (adapted from “US
EIA, Annual Energy Outlook, 2012 Early Release”).
Multistage hydraulic fracturing has become a critical
* Corresponding author.
E-mail address: [email protected] (M.H. Rammay).
http://dx.doi.org/10.1016/j.jngse.2016.10.002
1875-5100/© 2016 Elsevier B.V. All rights reserved.
component in the successful development of shale gas reservoirs. A
primary goal in unconventional reservoirs is to contact as much
rock as possible with a fracture or a network of fractures of
appropriate conductivities. This objective is typically accomplished
by drilling horizontal wells and placing multiple transverse fractures along the lateral. Reservoir contact is optimized by defining
the lateral length, the number of stages to be placed in the lateral,
the fracture isolation technique and job size.
Shale gas reservoir simulation modeling is different from the
conventional reservoir simulation approaches due to the gas
desorption effect and presence of natural fractures in shale. Zhang
et al. (2009) performed upscaling of properties from the discrete
fracture network model to dual porosity system in order to incorporate natural fractures effects in shale gas reservoirs. Cipolla et al.
(2010) presented more rigorous method for shale gas simulation
modeling by using dual permeability grid and use of Langmuir
isotherm for adsorption effect. Rubin (2010) discussed dual
permeability grid and non Darcy flow incorporation in to the shale
gas reservoir simulation models. The result of this investigation is a
technique which uses small logarithmically spaced dual permeability (DK) LGR grids within the stimulated reservoir volume (SRV)
72
M.H. Rammay, A.A. Awotunde / Journal of Natural Gas Science and Engineering 36 (2016) 71e78
coupled to standard DK grids outside of the SRV. Yu and
Sephernoori (2013a) employed numerical reservoir simulation
techniques to shale gas reservoirs in a manner similar to Cipolla
et al. (2010) and Rubin (2010). The authors validated their model
with field production data from Barnett shale. Yu and Sephernoori
(2014a) performed shale gas simulation modeling with the
consideration of gas desorption and geomechanics effects for Barnett shale and Marcellus shale. The authors studied the effect of gas
desorption on gas recovery with laboratory data of Langmuir
isotherm. Yu and Sepehrnoori (2014b) proposed a reservoir simulation approach to design and optimize unconventional gas production. In their paper, the authors presented a framework to
obtain the optimal gas-production scenario by optimizing the
operational parameters.
Different approaches have been presented to model and optimize hydraulic fracturing in shale gas reservoirs. Zhang et al. (2009)
performed shale gas simulation and sensitivity studies of hydraulic
fracture parameters on cumulative gas production. The authors
observed that, among the hydraulic fracture parameters considered, the fracture half-length had the largest effect on cumulative
gas production. Singh et al. (2011) conducted sensitivity studies of
shale oil production for different hydraulic fracture parameters
such as fracture half length, fracture conductivity and fracture
spacing. Logarithmically-spaced locally-refined grids were used to
capture the transient flow in the shale oil production from the
stimulated reservoir volume. The authors observed that closer
fracture spacings led to higher initial oil production rates and
higher ultimate oil recovery. The authors thus concluded that
longer fractures caused larger stimulated reservoir volume (SRV)
and led to higher cumulative oil production per well.
Holt (2011) estimated the optimal placement of hydraulic fractures stages along a horizontal wellbore by using gradient-based
optimization algorithms. The author used three gradient-based
optimization algorithms (ensemble-based optimization, simultaneous perturbation stochastic approximation and finite-difference
estimation of gradient), but recommended particle swarm optimization or genetic algorithm as a viable alternative to the
gradient-based algorithms for optimization of hydraulic fracture
spacing. Yang and Economides (2012a) stated that man-made
proppants (such as ceramics) should be applied in high closure
stress environments that are often encountered in deep reservoirs.
The authors showed that there is an optimum proppant number
corresponding to maximum NPV for various values of reservoir
permeability. They considered different proppant types and their
associated costs, fracturing fluid costs and pumping charges and
applied a unified fracturing design optimization technique to estimate the optimum mass of proppant that gives the highest NPV. Jin
et al. (2013) used numerical reservoir simulation study to develop
simple correlations that quantify the required fracture spacing
necessary to optimize recovery factors in unconventional shale oil
reservoirs.
Ma et al. (2013a) used the gradient-based finite difference
method (FDM), discrete simultaneous perturbation stochastic
approximation (DSPSA), and genetic algorithm (GA) to estimate
optimal hydraulic fracture placement. The authors concluded that
DSPSA and GA are more efficient than the gradient-based method.
Yu and Sephernoori (2013b) conducted sensitivity studies using
numerical modeling of multistage hydraulic fractures in combination with economic analysis. The authors built response surface
model to optimize multiple horizontal well placements. This integrated approach contributed to obtaining the optimal drainage area
around the wells. Ma et al. (2013b) presented the optimization of
horizontal well placement and hydraulic fracture stages using
simultaneous perturbation stochastic approximation (SPSA) and
covariance matrix adaptation evolution strategy (CMA-ES). The
authors compared the results from SPSA optimization with those
from CMA-ES and found that CMA-ES achieved higher NPVs but
required more computational time than SPSA. Yu et al. (2014)
studied five complex and irregular hydraulic fracture patterns.
Each pattern has different fracture half-lengths. Their results
showed that there is a significant difference in gas recovery between different patterns. Gu et al. (2014) developed a correlation
between cumulative gas production and fracture conductivity using sensitivity study. For a fixed propped length, the authors
observed a critical conductivity beyond which the production is
insensitive to the conductivity. The results showed that this critical
conductivity increases with propped length and decreases with
production time. The authors further showed that critical conductivity is negatively correlated with flowing bottomhole pressure.
From the literature, the optimization of only some of the hydraulic fracture parameters has been proposed. For example, optimization of fracture spacing using global optimization algorithm
was proposed by Ma et al. (2013b) while the optimization of the
horizontal well length using gradient-based optimizer was proposed by Holt (2011). To our knowledge, the simultaneous optimization of hydraulic fracture conductivity, fracture length, fracture
spacing and horizontal well has not been studied. Thus, we propose
the joint optimization of these variables and studied the
improvement in NPV attainable by the optimization of these
operational parameters using Differential Evolution. Differential
Evolution (DE) is a global optimization algorithm introduced by
Storn and Price (1995). DE has been shown to be one of the most
efficient optimization algorithms to solve problems in reservoir
engineering (Awotunde, 2015; Sibaweihi et al., 2015) Further details on the formulation and implementation of DE can be obtained
from Price et al., 2005.
We used reservoir model calibrated to real data and deployed
local grid refinement around the horizontal well and fracture to
ensure accurate simulation results. Results from these studies
showed that optimization of the hydraulic fracture parameters can
significantly increase the net present value of investment in a shale
gas reservoir project.
2. Shale gas reservoir simulation model
Reservoir simulation is preferred to analytical models or decline
curve analysis to predict and evaluate shale gas production because
simulations can model hydraulic fractures with interference of
production between fractures, non-Darcy flow in hydraulic fractures, varying fracture lengths and fracture conductivities of multi
stage hydraulic fractures and decrease in fracture conductivity due
to increase in effective stress or decrease in pore pressure around
hydraulic fractures. Thus, in this study, a commercial reservoir
simulator was used to model multi hydraulic fracture stages placed
across a horizontal well in a shale gas reservoir for gas production
forecasting and prediction. For realistic modeling of multi-fracture
stages from horizontal well in shale gas reservoir logarithmically
spaced local grid refinement was used in grid cells (global cells) of
simulation model. Logarithmically spaced local grid refinement is
able to accurately model multistage hydraulic fracture and long
transient behavior of gas from shale matrix to hydraulic fractures.
In the global grid cells, the hydraulic fracture was explicitly
modeled using logarithmically spaced local grid refinement and the
hydraulic fracture local grid cells is 1 ft wide. Furthermore, rest of
the logarithmically spaced local grids described the shale reservoir
matrix.
In addition, a dual porosity grid is used to allow simultaneous
matrix to natural fracture flow. The turbulent gas flow due to high
gas flow rate in propped hydraulic fractures is modeled as nonDarcy flow, which does not occur within the shale itself. The non-
M.H. Rammay, A.A. Awotunde / Journal of Natural Gas Science and Engineering 36 (2016) 71e78
Darcy flow is modeled using the Forchheimer modification to
Darcy's law as shown in Eq. (1).
Vp ¼
m
kf
n þ brn2 ;
(1)
where nis velocity, m is viscosity, kf is fracture permeability, r is
phase density and b is the non-Darcy turbulent flow factor.b was
used in the Forchheimer correction and was determined using a
correlation proposed by Evans and Civan (1994) as shown in Eq. (2).
b¼
1:485*109
;
k1:021
f
(2)
where the unit of k is md and unit of b is ft1. The correlation for b
was obtained using over 180 data points including those for
propped fractures and it was found to match the data very well
with the correlation coefficient of 0.974. This equation was used to
calculate b and used in the numerical model accounting for nonDarcy flow in hydraulic fractures.
installed in offset wells (Grieser et al., 2009; Yu and Sephernoori,
2013a). The horizontal well with its multiple fractures is shown in
Fig. 1. Fig. 2 shows the history matching results. A reasonable
match between the simulated data and the actual field data was
obtained. The history matching result (Fig. 2) shows that the shale
gas reservoir simulation model is well-calibrated so that the given
shale gas reservoir model can be used for production forecasting
and optimization. Pertinent reservoir information is shown on
Table 1.
3. Economic model, objective and cost functions
The objective of optimization in shale gas reservoir is to increase
gas recovery or the net present value of investment. The net present
value of a project involving with horizontal wells and multi stage
hydraulic fracturing is given in Eq. (4).
NPV ¼
Nts N
wells
X
X
Langmuir isotherm is used to model gas desorption and
adsorption phenomena in shale gas simulation models. It is one of
the widely used methods to include gas desorption/adsorption effects in shale gas reservoirs. The amount of gas adsorbed on a solid
surface is given by the Langmuir equation (Langmuir, 1918) as
shown in Eq. (3).
Gs ¼
VL p
:
p þ pL
(3)
It is used for characterizing physical desorption processes as a
function of pressure at constant temperature. In Eq. (3), Gs is the gas
content in scf/ton, VL is the Langmuir volume in scf/ton, pL is the
Langmuir pressure in psi, and p is pressure in psi. The bulk density
of shale is needed to convert the typical gas content in scf/ft3 to scf/
ton.
Langmuir pressure and Langmuir volume are the two most
important parameters for gas desorption or adsorption phenomena. Langmuir volume is defined as the gas volume adsorbed or
desorbed at infinite pressure. Langmuir volume also represents the
maximum storage capacity of shale rock for gas. Langmuir pressure
is defined as the pressure corresponding to one-half the Langmuir
volume. A high Langmuir pressure causes large adsorbed gas to be
released at the same reservoir pressure. The Langmuir isotherm is
often determined in laboratory using core samples. The simulator
used in this work for shale gas simulation, has the capability to take
Langmuir isotherm into the consideration if the reservoir is defined
as coal. We have used Barnett shale gas Langmuir isotherm data.
The data consists of Langmuir pressure of 650 psi and Langmuir
volume of 96 scf/ton (Mengal and Wattenbarger, 2011).
Qgas;k Dtk Rgas Qw;k Dtk Rwater
tk
k¼1 j¼1
0
2.1. Langmuir isotherm for desorbed or adsorbed gas in shale gas
simulation modeling
73
@FC þ
N
wells
X
j¼1
0
ð1 þ dÞ365
@Cwell þ
j
Nfsj
X
11
(4)
Cfsij AA
i¼1
In Eq. (4), Nts is the total number of time steps, Nwells is the total
number of wells, Nfsj is total number of fracturing stages in well j, i is
the index of the fracturing stage in well j, Qgas,k is the producing rate
of gas at time step k, Dtk is the time step at index k, Rgas is the price
of the gas, Qw,k is the water production rate at time step k, Rwater is
the cost of water disposal, d is the discount rate, FC is the fixed cost,
Cwellj is the cost of well j and Cfsij is the cost of fracturing stage i in
well j. The first term in Eq. (4) represents revenue less water
disposal cost discounted to the beginning of the project while the
second term is the combined costs of drilling the horizontal wells
and placing multistage hydraulic fracturing in the wells. We have
used only one horizontal well and no water production is considered. Table 2 shows the economic parameters used to compute the
cost of hydraulic fracturing (Yang and Economides, 2012b).
The cost of multistage hydraulic fracturing depends on the
volume of each fracture stage. The hydraulic fracture volume,Vf, is
calculated from Eq. (5).
2.2. Shale gas reservoir simulation model calibration/history
matching to real field data
The shale gas reservoir used in this study is of dimension
3000ft1500ft300ft. Production dataset from Barnett shale was
used to perform history matching in order to validate the hydraulic fracture model (Grieser et al., 2009) adopted. In this
simulation model, the fracture half-lengths differ from one
another and predictions of various values of fracture half-lengths
were provided by the fracture maps obtained by using geophones
Fig. 1. Top view of Shale gas reservoir model with hydraulic fractures of different
lengths.
74
M.H. Rammay, A.A. Awotunde / Journal of Natural Gas Science and Engineering 36 (2016) 71e78
fracture volume of each stage is then used to calculate amount/
volume of fracturing fluid and mass of proppant. Cost of fracturing
fluid is given in Eq. (6).
Cff ¼ 7:48052
Vf
!
hf
(6)
Rff
In Eq. (6), Vf is the volume of hydraulic fracture at a single stage,
hfis the efficiency of the fracturing fluid, Rff is the unit cost of
fracturing fluid in dollar per unit gallon and Cff is the total cost of
fracturing fluid at a single stage. Efficiency of fracturing fluid is
defined as the ratio of the volume of the fracture created to the
volume of fracturing fluid. In this work, the efficiency of the fracturing fluid is taken as 0.5.
We have used two sets of data: one set from Economides and
Nolte (1989) and the other set from Zhang et al. (2013) to form a
relation between the fracture conductivity, Fcd, and proppant
concentration, Cp, as shown in Eq. (7).
Fig. 2. History Matched Results of Shale gas reservoir simulation model with Real Field
data.
Cp ¼
0:012Fcd0:541
0:479ðFcd=1000Þ1:084
0 Fcd < 1000md ft
1000 Fcd 10000md ft
(7)
The mass of proppant is calculated from (Economides, 1992) Eq.
(8).
Table 1
Basic shale gas reservoir information.
Input Parameters
Values
Units
Initial reservoir pressure
Depth of the reservoir
Reservoir temperature
Initial gas saturation
Matrix permeability (from history matching)
Matrix porosity f (from history matching)
Shale compressibility, c
Viscosity of gas, m
Bottom-hole pressure (BHP)
Production time period (total production time)
Fracture height
Horizontal wellbore length
Number of grid block in X direction
Number of grid block in Y direction
Number of grid block in Z direction
Direction of minimum horizontal stress
Phases
Number of multi-fracturing stages
Langmuir volume
Langmuir pressure
Fracture spacing
Fracture conductivity of all fracturing Stages
Bulk density
3800
7000
180
0.7
0.00001
0.04
106
0.02
1500
10
300
2120
60
30
3
x
Gas
20
96
650
100
1
2.58 (161.02)
psi
ft
F
fraction
md
fraction
psi-1
cp
psi
years
ft
ft
Grids
Grids
Grids
direction
Mp ¼ Cp 2xf hf
and the cost of proppant at a single stage is calculated from Eq. (9).
Cprop ¼ Mp Rp
(9)
so that the total cost of single fracturing stage i is given in Eq. (10).
Cfsi ¼ Cprop;i þ Cff ;i
(10)
The costs of a horizontal well, Cwell, used in this work are taken
from Yu and Sephernoori (2013b) and shown for different well
lengths on Table 3.
This cost of horizontal well length is given in Eq. (11).
Stages
scf/ton
psi
ft
md-ft
g/cm3 (lb/ft3)
Vf ¼ 2wf xf hf ;
(8)
(5)
where wf is the width of hydraulic fracture, xf is the half length of
hydraulic fracture and hf is the height of hydraulic fracture. The
Cwell ¼ 100Lwell þ 19*105 ;
(11)
where Lwell is the well length. Thus, the net present value is a
function of the number of fracture stages, the conductivity of each
fracture, the horizontal well length and the fracture half lengths.
The objective is to determine the combination of these parameters
that gives the maximum NPV. The optimization problem is thus
stated as max{NPV(Lwell, Nfs, Fcdi, xfi)} subject to all known
constraints.
4. Sensitivity study and optimization of hydraulic fracturing
parameters
In order to select appropriate ranges or define appropriate
search spaces of the operational parameters in hydraulic fracturing,
Table 2
Economic parameters used to calculate the cost of hydraulic fracturing.
Parameters
Value
Pumping charges, $ (fixed cost)
Mob/Demob, $ (fixed cost)
Proppant (brown sand) cost, $/lbm
Proppant (white sand) cost, $/lbm
Proppant (ceramic) cost, $/lbm
Fracturing fluids (40 lb/1000gal X-linked gel) cost, $/gal
Gas price, $/Mscf
Discount rate
100,000
70,000
0.159
0.182
0.6
0.37
4
0.1
Table 3
Costs of a horizontal well for different well lengths (Yu and
Sephernoori, 2013b).
Horizontal well length (ft)
Cost ($)
1000
2000
3000
4000
2,000,000
2,100,000
2,200,000
2,300,000
M.H. Rammay, A.A. Awotunde / Journal of Natural Gas Science and Engineering 36 (2016) 71e78
sensitivity studies were performed to analyze the effect of hydraulic fracture conductivity and fracture spacing on the cumulative gas production and net present value of investment (Fig. 3).
Fig. 3a shows that the cumulative gas production increased with
increase in hydraulic conductivity up to about 100 md-ft at which
further increment in conductivity did not produce any appreciable
increase in cumulative gas production. The effect of this on NPV is
shown in Fig. 3b, where the NPV initially increased with increasing
conductivity and then declined at values on hydraulic conductivity
greater than 100 md-ft. From Fig. 3a, we also observe that cumulative gas production increased with increase in fracture spacing up
to 150 ft of spacing. Beyond this value, the cumulative gas production remained almost constant. A similar trend is reflected in
the NPV beyond 100 ft where the NPV remained constant.
To study the improvement in estimation of optimal parameters
achieved by the optimization procedure, we tested three optimization cases, starting with a few set of parameters in the first
optimization case. The first case, Case 1, considers only two optimization variables per stage: the fracture conductivity and the
fracture length of each stage. The number of frac stages was set at
20 in this case and the conductivity and fracture length in each frac
stage is uniquely estimated by the optimization algorithm. In Case
2, the fracture conductivities and fracture half lengths of all stages;
and the spacings between the fractures are optimized. Optimizing
the fracture spacing indirectly optimizes the number of frac stages
because closely spaced fractures leads to more fracture stages for
the same horizontal well length. In Case 3, the horizontal well
length is optimized in addition to the other optimization variables
considered in Case 2. Differential evolution (DE) algorithm was
used as the global optimizer to estimate the stated parameters in
each of these cases. The DE was run five times on each case so that
five realizations of the results are obtained for each case. This is
necessary because global optimization is based on random searches
that yield new results every time the optimization is rerun on the
same problem. Each run involves 3000 function evaluations.
The NPVs attained by running the DE five times on each of the
three cases are shown on Fig. 4. For each of the three cases
considered, the five realizations were ordered from best realization
(i.e. the realization with the highest NPV) to the worst realization
(the realization with the lowest NPV). Fig. 4aec shows the NPV
versus number of function evaluations. Fig. 4d shows the highest
values of the NPV from all five realizations plotted against the
iteration index. The figures clearly show that Case 3 yielded the
highest NPVs in all the realizations while Case 1 yielded the lowest
75
NPVs. This is expected because more operational parameters were
estimated in Case 3, as compared with Cases 1 and 2.
Fig. 5 shows the optimal fracture lengths and fracture conductivities for Case 1 obtained from the realization that gave the
highest NPV. The fracture lengths of fracture stages 8, 14 and 15 are
1450, 1400 and 1450 ft respectively. All fracture stages except
Stages 8, 14 and 15 have full fracture lengths of 1500 ft. The color
bar shows the magnitude of fracture conductivity with the lowest
conductivities represented in blue and the highest conductivities
represented in red. In this case, the fracture conductivity is highest
at the first stage, while fracture stages 7e10 have the low fracture
conductivities.
Fig. 6 shows the optimal fracture lengths, fracture conductivities, number of fracture stages and fracture spacing for the best
realization from Case 2. The optimal number of fracture stages in
the horizontal well of length 2000 ft is found to be 32.21 of the 32
frac stages have full fracture length indicating that the ratio of
stages with reduced fracture length to those with full fracture
length is smaller in Case 2 than in Case 1. Fracture conductivities
tend to be higher near the toe of the horizontal well.
Fig. 7 shows the optimal fracture lengths, fracture conductivities, number of fracture stages, fracture spacing and well length
from the best realization of Case 3. The optimal number of fracture
stages and optimal horizontal well length are found to be 36 and
2350 ft, respectively. About half of the fractures have full fracture
length. Also, we observe higher fracture conductivities close to the
toe of the horizontal well than in other parts of the well.
5. Conclusion
We considered the optimization of the operational parameters
of shale gas reservoir. To do this, a shale gas reservoir model
incorporating dual-porosity flow, gas desorption effect, stresssensitive fracture conductivity and local-grid refinements. The
incorporation of these effects helps us achieved a realistic modeling
of flow in hydraulic fractures in the reservoir. This shale gas
simulation model was then calibrated with real field data.
Sensitivity studies of the cumulative gas production and NPV to
hydraulic fracture conductivity and fracture spacing were conducted to determine the search space to impose on the optimization problem. Three optimization cases, in which the number of
hydraulic fracture parameters increased gradually, were then
studied. The first case, Case 1, has the hydraulic fracture conductivities and fracture lengths in all the hydraulic fractures as the
Fig. 3. Total Cumulative Gas Production and NPV versus Fracture Conductivity at Fracture Spacing 50e200 ft for Fracture length ¼ 800 ft and 5 Fracture stages.
76
M.H. Rammay, A.A. Awotunde / Journal of Natural Gas Science and Engineering 36 (2016) 71e78
Fig. 4. Comparison of NPVs attained by different cases (a) Best realization (b) Median realization (c) Worst Realizations (d) Final optimized NPV in all realizations.
optimization variables. Case 2 has the fracture conductivities,
fracture lengths as well as fracture spacings as the optimization
variables. In Case 3, horizontal well length was optimized along
with the other parameters considered in Case 2. Results obtained
show that the optimization of hydraulic fracture parameters can
considerably improve the net present value of investment. Results
also show that high hydraulic fracture conductivities are obtained
in regions close to the toe of the horizontal well.
Fig. 5. Optimal fracture lengths and fracture conductivities from the best realization of
Case 1.
Fig. 6. Optimal fracture lengths, conductivities, spacings and number of fracture stages
\00 the best realization of Case 2.
M.H. Rammay, A.A. Awotunde / Journal of Natural Gas Science and Engineering 36 (2016) 71e78
Rgas
Rwater
Rff
Rp
Dtk
Vff
Vf
VL
wf
xf
hf
rB
n
m
r
b
77
Unit Price of the gas, $/Mscf
Unit Cost of water disposal, $/bbl
Unit cost of fracturing fluid, $/gallon
Unit price of Proppant, $/lb
Time step at index k, days
Volume of fracturing fluid, ft3
Volume of hydraulic fracture, ft3
Langmuir Volume, scf/ton
Width of hydraulic fracture stage, inch (ft)
Half length of hydraulic fracture stage, ft
Efficiency of fracturing fluid
Bulk density of shale, g/cm3 (lb/ft3)
Velocity, m/s (ft/s)
Viscosity, cp
Phase density, g/cm3 (lb/ft3)
Non-Darcy Flow beta factor, ft1
References
Fig. 7. Optimal Fracture Lengths, Conductivities, Spacings, Number of Fracture Stages
and Well Length from the best realization of Case 3.
Acknowledgements
The authors would like to acknowledge the support provided by
Saudi Aramco through the Research Institute at King Fahd University of Petroleum & Minerals (KFUPM) for funding this work
through project no. ES002357 as part of the Trilateral Research
Initiative. The authors would also like to express their gratitude to
Dr. Hasan Al-Yousuf and Dr. Abdullah A. Alshuhail for their careful
review of this work.
Nomenclature
Cwell
Cfs
Cff
Cp
Cprop
Cwell
DE
d
FC
Fcd
Gs
hf
i
j
k
kf
Lwell
M
MM
Mp
NPV
Nts
Nwells
Nfs
pL
p
Qgas
Qw
Cost of well, $
Cost of fracturing stage, $
Total cost of fracturing fluid of single stage, $
Proppant concentration, lb/ft2
Cost of Proppant
Cost of Horizontal well, $
Differential Evolution
Discount rate
Fixed cost, $
Fracture conductivity, md-ft
Gas content, scf/ton
Height of the fracture, ft
Index of first fracturing stage to total number of fracturing
stages
Index of first well to total number of wells
Index of first time step to total number of time steps
Fracture permeability, md
Horizontal well length, ft
Thousand (103)
Million (106)
Mass of Proppant, lb
Net present value, $
Total number of time steps
Total number of wells
Total number of fracturing stages
Langmuir pressure, psi
Pressure, psi
Producing rate of gas, Mscf/day
Producing rate of water, bbl/day
Annual Energy Outlook Early Release, 2012, 2012. US DOE.
Awotunde, A.A., 2015. Estimation of well test parameters using global optimization
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