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Stochastic optimization of hydraulic fracture and horizontal well parameters in shale gas reservoirs
Accepted Manuscript Stochastic optimization of hydraulic fracture and horizontal well parameters in shale gas reservoirs Muzammil Hussain Rammay, Abeeb A. Awotunde PII:
S1875-5100(16)30712-0
DOI:
10.1016/j.jngse.2016.10.002
Reference:
JNGSE 1848
To appear in:
Journal of Natural Gas Science and Engineering
Received Date: 29 December 2015 Revised Date:
2 October 2016
Accepted Date: 5 October 2016
Please cite this article as: Rammay, M.H., Awotunde, A.A., Stochastic optimization of hydraulic fracture and horizontal well parameters in shale gas reservoirs, Journal of Natural Gas Science & Engineering (2016), doi: 10.1016/j.jngse.2016.10.002. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Stochastic Optimization of Hydraulic Fracture and Horizontal Well Parameters in Shale Gas Reservoirs Muzammil Hussain Rammay and Abeeb A. Awotunde, King Fahd University of Petroleum and Minerals Corresponding Author's Email: [email protected]
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Abstract 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.
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Keywords Fracture length; Fracture conductivity; Stochastic optimization; Multi stage hydraulic fracture; Net present value; Fracture spacing
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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 4,000 (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 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. 1
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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) coupled to standard DK grids outside of the SRV. Yu and Sepehrnoori (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 Sepehrnoori (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 (2014c) 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 2
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hydraulic fracture placement. The authors concluded that DSPSA and GA are more efficient than the gradient-based method. Yu and Sepehrnoori (2013) 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. (2014b) 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 3
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−∇p =
µ
kf
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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 non-Darcy flow, which does not occur within the shale itself. The non-Darcy flow is modeled using the Forchheimer modification to Darcy’s law as shown in Eq. 1.
ν + βρν 2 ,
(1)
where ν is velocity, µ is viscosity, k f is fracture permeability, ρ is phase density and β is the
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non-Darcy turbulent flow factor. β was used in the Forchheimer correction and was determined using a correlation proposed by Evans and Civan (1994) as shown in Eq. 2. 1.485*109 , (2) β= k f 1.021 where the unit of k is md and unit of β is ft-1. The correlation for β 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 β and used in the numerical model accounting for non-Darcy flow in hydraulic fractures. 2.1 Langmuir Isotherm for Desorbed or Adsorbed Gas in Shale Gas Simulation Modeling
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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. V p (3) Gs = L p + pL 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,
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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). 4
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2.2 Shale gas reservoir simulation model calibration/history matching to real field data
Table 1: Basic Shale Gas Reservoir Information Input Parameters Initial reservoir pressure Depth of the reservoir Reservoir temperature Initial gas saturation
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The shale gas reservoir used in this study is of dimension 3000 ft × 1500 ft × 300 ft . 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 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.
Units psi ft o F fraction
Matrix permeability (from history matching) Matrix porosity φ (from history matching) Shale compressibility, c Viscosity of gas, µ Bottom-hole pressure (BHP) Production time period (total production time) Fracture height
0.00001 0.04 10-6 0.02 1500 10 300
md fraction psi-1 cp psi years ft
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
2120 60 30 3 x Gas 20 96 650 100 1 2.58 (161.02)
ft Grids Grids Grids direction
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Values 3800 7000 180 0.7
Stages scf/ton psi ft md-ft g/cm3 (lb/ft3)
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Figure 1: Top view of Shale gas reservoir model with hydraulic fractures of different lengths
Figure 2: History Matched Results of Shale gas reservoir simulation model with Real Field data 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. N fsj N ts N wells N wells (Qgas ,k ∆tk Rgas − Qw,k ∆tk Rwater ) (4) NPV = ∑ ∑ − FC + C + C ∑ fsij tk well j ∑ k =1 j =1 j = 1 i = 1 365 (1 + d ) 6
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In Eq. 4, N ts is the total number of time steps, N wells is the total number of wells, N fsj 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, ∆tk 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
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discount rate, FC is the fixed cost, Cwell j is the cost of well j and C fsij is the cost of fracturing
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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).
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Table 2: Economic parameters used to calculate the cost of hydraulic fracturing Parameters Value Pumping charges, $ (fixed cost) 100,000 Mob/Demob, $ (fixed cost) 70,000 Proppant (brown sand) cost, $/lbm 0.159 Proppant (white sand) cost, $/lbm 0.182 Proppant (ceramic) cost, $/lbm 0.6 Fracturing fluids (40 lb/1000gal X-linked gel) cost, $/gal 0.37 Gas price, $/Mscf 4 Discount rate 0.1
V f = 2w f x f h f ,
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The cost of multistage hydraulic fracturing depends on the volume of each fracture stage. The hydraulic fracture volume, V f , is calculated from Eq. 5. (5)
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where w f is the width of hydraulic fracture, x f is the half length of hydraulic fracture and h f is the height of hydraulic fracture. The 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. Vf (6) C ff = 7.48052 R ff η f In Eq. 6, V f is the volume of hydraulic fracture at a single stage, η f is the efficiency of the fracturing fluid, R ff is the unit cost of fracturing fluid in dollar per unit gallon and C ff 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, C p , as shown in Eq. 7.
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(7)
(8)
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0.012 Fcd 0.541 0 ≤ Fcd < 1000md − ft Cp = 1.084 1000 ≤ Fcd ≤ 10000md − ft 0.479( Fcd / 1000) The mass of proppant is calculated from (Economides 1992) Eq. 8. M p = C p (2 x f h f ) and the cost of proppant at a single stage is calculated from Eq. 9. C prop = M p R p so that the total cost of single fracturing stage i is given in Eq. 10. C fsi = C prop ,i + C ff ,i
(9) (10)
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The costs of a horizontal well, Cwell , used in this work are taken from Yu and Sepehrnoori (2013b) and shown for different well lengths on Table 3.
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Table 3: Costs of a horizontal well for different well lengths (Yu and Sepehrnoori, 2013b) Horizontal Well Length (ft) Cost ($) 1,000 2,000,000 2,000 2,100,000 3,000 2,200,000 4,000 2,300,000
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This cost of horizontal well length is given in Eq. 11. (11) Cwell = 100 Lwell + 19*105 , 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 , x fi ) subject to all known
constraints.
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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, 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). Figure 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.
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(a) (b) Figure 3: Total Cumulative Gas Production and NPV versus Fracture Conductivity at Fracture Spacing 50 to 200 ft for Fracture length = 800 ft and 5 Fracture stages
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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 Figure 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). Figures 4a-c show the NPV versus number of function evaluations. Figure 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 NPVs. This is expected because more operational parameters were estimated in Case 3, as compared with Cases 1 and 2.
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(a)
(c) (d) Figure 4: Comparison of NPVs attained by different cases (a) Best realization (b) Median realization (c) Worst Realizations (d) Final optimized NPV in all realizations
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Figure 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 7 to 10 have the low fracture conductivities. Figure 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. Figure 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. 10
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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.
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Figure 5: Optimal fracture lengths and fracture conductivities from the best realization of Case 1
Figure 6: Optimal fracture lengths, conductivities, spacings and number of fracture stages \00 the best realization of Case 2
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Figure 7: Optimal Fracture Lengths, Conductivities, Spacings, Number of Fracture Stages and Well Length from the best realization of Case 3
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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, stress-sensitive 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 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.
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Nomenclature Cwell = Cost of well, $ C fs = Cost of fracturing stage, $ C ff = Total cost of fracturing fluid of single stage, $
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C p = Proppant concentration, lb/ft2
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C prop = Cost of Proppant,
Cwell = Cost of Horizontal well, $ DE = Differential Evolution d = Discount rate FC = Fixed cost, $ Fcd = Fracture conductivity, md-ft Gs = Gas content, scf/ton h f = Height of the fracture, ft
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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.
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i = Index of first fracturing stage to total number of fracturing stages j = Index of first well to total number of wells k = Index of first time step to total number of time steps k f = Fracture permeability, md
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Lwell = Horizontal well length, ft M = Thousand (103) MM = Million (106) M p = Mass of Proppant, lb NPV = Net present value, $ N ts = Total number of time steps N wells = Total number of wells N fs = Total number of fracturing stages pL = Langmuir pressure, psi p = Pressure, psi Qgas = Producing rate of gas, Mscf/day Qw = Producing rate of water, bbl/day Rgas = Unit Price of the gas, $/Mscf
Rwater = Unit Cost of water disposal, $/bbl R ff = Unit cost of fracturing fluid, $/gallon 13
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R p = Unit price of Proppant, $/lb
∆tk = Time step at index k, days V ff = Volume of fracturing fluid, ft3
VL = Langmuir Volume, scf/ton w f = Width of hydraulic fracture stage, inch (ft) x f = Half length of hydraulic fracture stage, ft
η f = Efficiency of fracturing fluid
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ρ B = Bulk density of shale, g/cm3 (lb/ft3) ν = Velocity, m/s (ft/s) µ = Viscosity, cp ρ = Phase density, g/cm3 (lb/ft3) β = Non-Darcy Flow beta factor, ft-1
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V f = Volume of hydraulic fracture, ft3
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Highlights:
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Shale gas reservoir simulation model was built and calibrated by field production data. A realistic LGR-based simulation model and appropriate mathematical models of cost functions were used. Differential evolution algorithm was implemented to estimate optimal hydraulic fracture parameters and horizontal well length.
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DOI:
10.1016/j.jngse.2016.10.002
Reference:
JNGSE 1848
To appear in:
Journal of Natural Gas Science and Engineering
Received Date: 29 December 2015 Revised Date:
2 October 2016
Accepted Date: 5 October 2016
Please cite this article as: Rammay, M.H., Awotunde, A.A., Stochastic optimization of hydraulic fracture and horizontal well parameters in shale gas reservoirs, Journal of Natural Gas Science & Engineering (2016), doi: 10.1016/j.jngse.2016.10.002. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Stochastic Optimization of Hydraulic Fracture and Horizontal Well Parameters in Shale Gas Reservoirs Muzammil Hussain Rammay and Abeeb A. Awotunde, King Fahd University of Petroleum and Minerals Corresponding Author's Email: [email protected]
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Abstract 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.
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Keywords Fracture length; Fracture conductivity; Stochastic optimization; Multi stage hydraulic fracture; Net present value; Fracture spacing
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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 4,000 (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 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. 1
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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) coupled to standard DK grids outside of the SRV. Yu and Sepehrnoori (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 Sepehrnoori (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 (2014c) 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 2
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hydraulic fracture placement. The authors concluded that DSPSA and GA are more efficient than the gradient-based method. Yu and Sepehrnoori (2013) 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. (2014b) 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 3
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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 non-Darcy flow, which does not occur within the shale itself. The non-Darcy flow is modeled using the Forchheimer modification to Darcy’s law as shown in Eq. 1.
ν + βρν 2 ,
(1)
where ν is velocity, µ is viscosity, k f is fracture permeability, ρ is phase density and β is the
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non-Darcy turbulent flow factor. β was used in the Forchheimer correction and was determined using a correlation proposed by Evans and Civan (1994) as shown in Eq. 2. 1.485*109 , (2) β= k f 1.021 where the unit of k is md and unit of β is ft-1. The correlation for β 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 β and used in the numerical model accounting for non-Darcy flow in hydraulic fractures. 2.1 Langmuir Isotherm for Desorbed or Adsorbed Gas in Shale Gas Simulation Modeling
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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. V p (3) Gs = L p + pL 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,
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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). 4
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2.2 Shale gas reservoir simulation model calibration/history matching to real field data
Table 1: Basic Shale Gas Reservoir Information Input Parameters Initial reservoir pressure Depth of the reservoir Reservoir temperature Initial gas saturation
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The shale gas reservoir used in this study is of dimension 3000 ft × 1500 ft × 300 ft . 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 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.
Units psi ft o F fraction
Matrix permeability (from history matching) Matrix porosity φ (from history matching) Shale compressibility, c Viscosity of gas, µ Bottom-hole pressure (BHP) Production time period (total production time) Fracture height
0.00001 0.04 10-6 0.02 1500 10 300
md fraction psi-1 cp psi years ft
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
2120 60 30 3 x Gas 20 96 650 100 1 2.58 (161.02)
ft Grids Grids Grids direction
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Values 3800 7000 180 0.7
Stages scf/ton psi ft md-ft g/cm3 (lb/ft3)
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Figure 1: Top view of Shale gas reservoir model with hydraulic fractures of different lengths
Figure 2: History Matched Results of Shale gas reservoir simulation model with Real Field data 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. N fsj N ts N wells N wells (Qgas ,k ∆tk Rgas − Qw,k ∆tk Rwater ) (4) NPV = ∑ ∑ − FC + C + C ∑ fsij tk well j ∑ k =1 j =1 j = 1 i = 1 365 (1 + d ) 6
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In Eq. 4, N ts is the total number of time steps, N wells is the total number of wells, N fsj 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, ∆tk 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
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discount rate, FC is the fixed cost, Cwell j is the cost of well j and C fsij is the cost of fracturing
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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).
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Table 2: Economic parameters used to calculate the cost of hydraulic fracturing Parameters Value Pumping charges, $ (fixed cost) 100,000 Mob/Demob, $ (fixed cost) 70,000 Proppant (brown sand) cost, $/lbm 0.159 Proppant (white sand) cost, $/lbm 0.182 Proppant (ceramic) cost, $/lbm 0.6 Fracturing fluids (40 lb/1000gal X-linked gel) cost, $/gal 0.37 Gas price, $/Mscf 4 Discount rate 0.1
V f = 2w f x f h f ,
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The cost of multistage hydraulic fracturing depends on the volume of each fracture stage. The hydraulic fracture volume, V f , is calculated from Eq. 5. (5)
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where w f is the width of hydraulic fracture, x f is the half length of hydraulic fracture and h f is the height of hydraulic fracture. The fracture volume of each stage is then used to calculate amount/volume of fracturing fluid and mass of proppant. Cost of fracturing fluid is given in Eq. 6. Vf (6) C ff = 7.48052 R ff η f In Eq. 6, V f is the volume of hydraulic fracture at a single stage, η f is the efficiency of the fracturing fluid, R ff is the unit cost of fracturing fluid in dollar per unit gallon and C ff is the total cost of fracturing fluid at a single stage. Efficiency of fracturing fluid is defined as the ratio of the volume of the fracture created to the volume of fracturing fluid. In this work, the efficiency of the fracturing fluid is taken as 0.5. We have used two sets of data: one set from Economides and Nolte (1989) and the other set from Zhang et al. (2013) to form a relation between the fracture conductivity, Fcd , and proppant concentration, C p , as shown in Eq. 7.
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0.012 Fcd 0.541 0 ≤ Fcd < 1000md − ft Cp = 1.084 1000 ≤ Fcd ≤ 10000md − ft 0.479( Fcd / 1000) The mass of proppant is calculated from (Economides 1992) Eq. 8. M p = C p (2 x f h f ) and the cost of proppant at a single stage is calculated from Eq. 9. C prop = M p R p so that the total cost of single fracturing stage i is given in Eq. 10. C fsi = C prop ,i + C ff ,i
(9) (10)
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Table 3: Costs of a horizontal well for different well lengths (Yu and Sepehrnoori, 2013b) Horizontal Well Length (ft) Cost ($) 1,000 2,000,000 2,000 2,100,000 3,000 2,200,000 4,000 2,300,000
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This cost of horizontal well length is given in Eq. 11. (11) Cwell = 100 Lwell + 19*105 , 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.
{
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The optimization problem is thus stated as max NPV ( Lwell , Nfs, Fcdi , x fi ) subject to all known
constraints.
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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, 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). Figure 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.
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(a) (b) Figure 3: Total Cumulative Gas Production and NPV versus Fracture Conductivity at Fracture Spacing 50 to 200 ft for Fracture length = 800 ft and 5 Fracture stages
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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 Figure 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). Figures 4a-c show the NPV versus number of function evaluations. Figure 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 NPVs. This is expected because more operational parameters were estimated in Case 3, as compared with Cases 1 and 2.
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(c) (d) Figure 4: Comparison of NPVs attained by different cases (a) Best realization (b) Median realization (c) Worst Realizations (d) Final optimized NPV in all realizations
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Figure 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 7 to 10 have the low fracture conductivities. Figure 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. Figure 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. 10
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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.
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Figure 5: Optimal fracture lengths and fracture conductivities from the best realization of Case 1
Figure 6: Optimal fracture lengths, conductivities, spacings and number of fracture stages \00 the best realization of Case 2
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Figure 7: Optimal Fracture Lengths, Conductivities, Spacings, Number of Fracture Stages and Well Length from the best realization of Case 3
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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, stress-sensitive 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 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.
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Nomenclature Cwell = Cost of well, $ C fs = Cost of fracturing stage, $ C ff = Total cost of fracturing fluid of single stage, $
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C p = Proppant concentration, lb/ft2
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C prop = Cost of Proppant,
Cwell = Cost of Horizontal well, $ DE = Differential Evolution d = Discount rate FC = Fixed cost, $ Fcd = Fracture conductivity, md-ft Gs = Gas content, scf/ton h f = Height of the fracture, ft
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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.
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i = Index of first fracturing stage to total number of fracturing stages j = Index of first well to total number of wells k = Index of first time step to total number of time steps k f = Fracture permeability, md
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Lwell = Horizontal well length, ft M = Thousand (103) MM = Million (106) M p = Mass of Proppant, lb NPV = Net present value, $ N ts = Total number of time steps N wells = Total number of wells N fs = Total number of fracturing stages pL = Langmuir pressure, psi p = Pressure, psi Qgas = Producing rate of gas, Mscf/day Qw = Producing rate of water, bbl/day Rgas = Unit Price of the gas, $/Mscf
Rwater = Unit Cost of water disposal, $/bbl R ff = Unit cost of fracturing fluid, $/gallon 13
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R p = Unit price of Proppant, $/lb
∆tk = Time step at index k, days V ff = Volume of fracturing fluid, ft3
VL = Langmuir Volume, scf/ton w f = Width of hydraulic fracture stage, inch (ft) x f = Half length of hydraulic fracture stage, ft
η f = Efficiency of fracturing fluid
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ρ B = Bulk density of shale, g/cm3 (lb/ft3) ν = Velocity, m/s (ft/s) µ = Viscosity, cp ρ = Phase density, g/cm3 (lb/ft3) β = Non-Darcy Flow beta factor, ft-1
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V f = Volume of hydraulic fracture, ft3
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Highlights:
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Shale gas reservoir simulation model was built and calibrated by field production data. A realistic LGR-based simulation model and appropriate mathematical models of cost functions were used. Differential evolution algorithm was implemented to estimate optimal hydraulic fracture parameters and horizontal well length.
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