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The application of multi-stage fractured horizontal well (MSFHW) due to its costly operation necessitates optimization of associated fracture parameters to ensure its economic success. In comparison to significant number of studies dedicated to use of MSFHWs for shale gas reservoirs, there are only few researches available for oil systems. This study explores the optimum criteria for a number of important fracture parameters in low-permeability heavy-oil systems. For this purpose, a response surface methodology (RSM) was employed to examine the simultaneous effect of four fracture parameters, including the number of fracture stages, fracture length, fracture width and fracture conductivity, on well productivity. The evaluations were conducted on two homogeneous and heterogeneous permeability systems. The optimization of fracture parameters was also performed on an economic basis by utilizing the net present value (NPV) concept. Useful charts were also generated providing practical insights into the individual and combinational effects of fracture parameters on well performance. The results from this study demonstrated that the fracture conductivity and the number of fracture stages were, respectively, the first two important parameters controlling the well productivity for rock systems with higher permeability. However, when rock texture became tighter, the number, and to a lesser extent the length, of fractures exhibited more evident role on production improvement, especially in the case of heterogeneous reservoirs. The results also underlined the significance of economic considerations, in particular, when determining the optimum fracture length and number of fracture stages. Keywords Multi-stage fractured well · Heavy-oil reservoir · Fracture conductivity · Response surface methodology · NPV · Low-permeability rock Abbreviations N Cumulativ e production for non-fractured well pnf C F racture conductivity (mD-ft) (bbl) C F ixed cost ($) NPV Net present value ($) fix C Leak off volume coefficient N N umber of stages l s C Total cost ($) P Acid price ($/bbl) T a h F racture height (ft) P Oil price ($/bbl) f o IF Improvement factor (%) V Acid volume (bbl) K F racture permeability (mD) W F racture width (in) f f N Cumulative production for fractured well (bbl) X F racture length (ft) pf f ΔN Cumulative production increment after n years p,n (bbl) * Armin Shirbazo ΔR Revenue after n years ($) arminshirbazo@aut.ac.ir Jalal Fahimpour fahimpour@aut.ac.ir Introduction Babak Aminshahidy Aminshahidy@aut.ac.ir Carbonates, because of their low-permeability rock textures, 1 are good candidates for acid fracturing jobs. In this method, Department of Petroleum Engineering, Amirkabir University acid under high pressures is injected into the target zone of Technology (Tehran Polytechnic), Tehran, Iran Vol.:(0123456789) 1 3 Journal of Petroleum Exploration and Production Technology during which the surface of the rock minerals is dissolved transverse fractures is dependent on increase in fracture unevenly. As a result, a non-uniform pattern is etched on the conductivity and width. Rahman et al. (2014) investigated rock surface, which keeps the fracture tip open constantly, the optimization of fracture configuration under geome- even after the pressure is released. Horizontal well drilling chanical considerations. They concluded that elongating and hydraulic fracturing are two recently used IOR methods the fracture is more beneficial than increasing the fracture utilized to improve the well productivity owing to increased conductivity. well-to-reservoir contact. Obviously, there is no question Soliman and Grieser (2010) studied the effect of frac- left why combining these two techniques, i.e. multi-stage ture spacing and timing of fracturing. They showed that hydraulic fracturing of horizontal wells (MSFHW), has the optimum fracture spacing is influenced by stress inter - gained significant attention for development of hard-to- ference between fractures and fluid-flow conditions. Yang produce resources. However, as the MSFHW technology is et al. (2017) made a study on optimization of the number of a complex and costly process, optimization of the fracture fracture stages and fracture spacing. They concluded that by parameters is a crucial task to ensure an economic operation earning a better description of reservoir heterogeneity, it is and satisfactory well performance. not anymore imperative to space fractures evenly; instead, There are various methods developed for modeling the they can be placed in intervals with higher density of natural horizontal wells with multiple transverse fractures (Deng fractures. et al., 2014; MoradiDowlatabad and Jamiolahmady 2018; Most of the previous works on optimization of fracture Raghavan and Joshi 1993; Valko and Amini 2007; Wang design for multi-stage fractured wells focus mainly on sin- et al. 2010; Zhao et al. 2016). These models can be used gle fracture parameters and their individual effect on well to predict the performance of a fractured well Wang et al. performance. This work presents a full analysis of the simul- (2016) combined response surface methodology with a taneous effect of multiple fracture parameters governing the flow-stress-damage model to estimate the stimulated res- performance of MSFHWs and the level of importance of ervoir area as a function of several parameters such as each parameter for various reservoir cases. A response sur- dip angle, stress difference and injection rate. They con- face methodology (RSM) has been used for this purpose. cluded that injection rate is a positive factor increasing Fracture length, fracture width, fracture conductivity and the stimulated area since at higher rates less leakage into number of fracture stages are four main parameters con- the formation discontinuities happens. Tang et al. (2018) sidered for two homogenous and heterogonous reservoirs. investigated the interactive effect of heterogeneity with The net present value (NPV) has also been used to find the some factors like in-situ stress gradient and stress shadow. optimum fracture design criteria from an economic point Their results underlined the significance of considering of view. A description of the reservoir models, the range of formation heterogeneity in designing the fractures. fracture parameters and the methodology used for analyzing Hareland and Rampersad (1994) developed a pseudo- the results are described in the following sections. three-dimensional hydraulic fracturing model to optimize the controllable fracturing parameters such as pump rate, fluid rheology, proppant schedule and fracture length. Their results demonstrated that NPV does not necessarily Description of reservoir model increase with increasing the fracture length. Jabbari and Benson (2013) performed a case study for the optimization A single-well sector model of a carbonate reservoir located in the south-west of Iran was used for the main parts of this of hydraulic fracturing design in a low-permeability shale reservoir. Their results demonstrated the necessity of creat- study. In addition, three other permeability models described later were imposed on the selected sector model to further ing longer fractures in low-permeability rocks. Jahandideh and Jafarpour (2016) studied the impact of heterogeneity in investigate the sensitivity of the MSFHW performance to rock permeability. The oil bearing layer has a thickness of shale brittleness and fracability on fracturing process and emphasized that the optimal number, location and length 250 ft, where a horizontal well, 3000 ft long, is penetrated through its middle. The no-flow-boundary reservoir has an of fractures are dependant on rock fracability distribution. Romero et al. (2003) used a direct boundary element initial pressure of about 4700 psi and contains a relatively heavy-oil fluid with an API of 22.4 and bubble point pres- method to calculate the well productivity index of frac- tured wells. They found that the optimum range for frac- sure of 1663 psi. The single well model produces under a constant bottom-hole pressure of 2000 psi. ture conductivity and its geometry is significantly affected by the fracture face skin. Besler et al. (2007) studied the The Eclipse commercial software was used to simulate the reservoir model (Eclipse Simulation Software 2010). fracture performance in horizontal wells and concluded that while longitudinal fractures exhibit low-conductiv- An explicit design was implemented to model the fracture geometry by ascribing fracture properties including fracture ity requirements, increase of productivity in wells with 1 3 Journal of Petroleum Exploration and Production Technology Fig. 1 Illustration of numerical reservoir sector model corresponding gridding scheme Fig. 2 A cross-section view of the sector model and the corresponding horizontal well section length, width and permeability to the corresponding grid Table 1 Dimension of the X direction 9000 ft numerical sector reservoir blocks around the wellbore. The fractures were considered Y direction 5000 ft model to be transverse to the horizontal well and propagated across Z direction 250 ft the full reservoir thickness. To accurately model the flow behavior around the fracture face, local grid refinement (LGR) technique was employed as shown by Figs. 1 and 2. (Case 3 in Table 2), which is known as the base case here- The simulation model in Cartesian system consists of 177 after. The average horizontal permeability of this sector × 35 × 5 grid blocks, respectively, in x, y and z directions. model is 4.7 mD. In addition, to complement this work and The final dimensions of the sector reservoir model are listed to examine the effect of fracture parameters with respect to in Table 1. rock permeability, three other permeability models were also As mentioned earlier, the main focus of this study is on created. These three cases, as listed in Table 2, include two the performance of MSFHW and level of importance of frac- homogenous models with permeabilities of 5 mD (Case 1) ture parameters in a low-permeability heavy-oil reservoir 1 3 Journal of Petroleum Exploration and Production Technology Table 2 Petrophysical Case number Type K , K (md) K (md) Porosity (%) x y z properties of four reservoir models used in this study Case 1 Homogenous 5 1.5 20 Case 2 Homogenous 0.5 0.15 20 Case 3 (base case) Heterogeneous 4.7 (average) 1.47 (average) 9.20 Case 4 Heterogeneous 0.47 (average) 0.15 (average) 6.02 and 0.5 mD (Case 2) and a heterogonous reservoir with an 2018). Accordingly, a response surface methodology (RSM) average permeability of 0.47 mD (Case 4). It should be noted was employed to scrutinize the effect of fracture parameters that properties of these three cases, except their permeability on well productivity. An evaluation exercise was also carried and porosity, are similar to Case 3 (base case). The perme- out on the cost of fracturing operation to find the optimum ability map and its distribution in each layer of Case 3 are fracture parameters from an economic perspective. depicted by Figs. 3 and 4, respectively. More details of the properties of the reservoir model used in this study can be Selected range for fracture parameters found in Table 3. Al-Ameri and Gamadi (2019) in their study demonstrated that it is possible to have a full combination of fracture Methodology length, fracture width, and fracture conductivity by control- ling the acid properties and the post-flush stage. Controlling Four fracture parameters including fracture length, fracture the operational parameters such as pumping rate, injection width, fracture conductivity and the number of fractur- rate, acid type, injection time and fluid loss control will help ing stages were selected for this study. The range of these the engineers to adjust the fracture parameters and hence parameters will be discussed in the next sections. In total, to maximize the fracture conductivity near the wellbore 600 simulations were conducted for each reservoir model. (Aljawad et al. 2019). The MINITAB software was then used to analyze the results According to studies performed on hydraulic fracturing of the numerical simulations performed (Minitab Software and real practices carried out on fracturing operations in Fig. 3 Permeability maps in each geologic layer of the base case (case 3) 1 3 Journal of Petroleum Exploration and Production Technology Fig. 4 The log-normal scale histograms of rock permeability distributed in each five layers of the base case (case 3) Table 3 Properties of the sector reservoir model under study Under low temperatures, the fracture propagates to short dis- tances due to inadequate stimulation. At high temperatures Formation Carbonate also the etched fracture length is limited because of the high Temperature 203 (°F) reaction rate. All these parameters are controlled by various Bubble point pressure 1663 (psi) additives such as surfactants, retarders and emulsified acids Reservoir pressure 4700 (psi) (Dehghani et al. 2019). Williams and Nierode (1972) stud- Bottom hole pressure 2000 (psi) ied the acid penetration distance and stated that the fracture Formation volume factor 1.18 (bbl/STB) geometry is influenced by formation temperature, acid injec- Fluid viscosity 3.62 (cp) tion rate, acid concentration, and rock type. GOR 330 (scf/STB) Creating a desired fracture length is achieved by design- Specific water gravity 1.15 ing the desired acid fluid system in many cases. Acid fluid Oil API gravity 22.4 loss during acid-fracturing treatments significantly limits the Specific gas gravity 0.82 effective fracture lengths obtained in carbonate reservoirs. The fluid-loss control systems like a gelled-acid system (White et al. 1992) are effective on controlling the fracture the oil and gas industry, the appropriate range for fracture length. Mukherjee and Cudney (1993) introduced a new gel- based acid that eliminates wormhole growth and increases parameters were selected for the purpose of this study as follows. the length of the fracture. The viscosity of the acid is tempo- rarily increased from 30 to 1000 cP after the acid is injected into the formation, preventing acid leakage and leading to Fracture length create more extended fracture length with the same volume of injected acid. Emulsified acids are capable of creating longer fracture Zhou et al. (2007) developed leak-off control acid length without deterioration of fracture conductivity (Nav- (LCA) systems to reduce acid leak-off. The initial viscos- arrete et al. 1998). Alghamadi (2006) expressed that the ity of LCA designed at a low value .The viscosity of LCA effectiveness of a hydraulic fracturing operation is strongly increased impressively when the value of pH decreases to a dependent on the effective fracture length and conductivity. 1 3 Journal of Petroleum Exploration and Production Technology specific value, and this can reduce the leak-off and enhance (2015), in his experimental work found out that not only acid penetration. Rahman (2010) investigated the effect of the fracture permeability, but also the cross-section area for injected acid rheology represented by the power-law expo- flow or fracture width, are crucial parameters affecting the nent, injected rate and initial oil flow rate and interaction flow capacity of the fracture. Acid fracture conductivity can between them on fracture length and etched fracture width. be controlled by choosing proper operational parameters, He showed that fracture length decreases and etched fracture although reaching a definite value is not assured. It has been width increases by increasing power-law exponent. demonstrated that extremely high contact times and high de Rozieres et al. (1994) investigated the diffusion coef- operation temperature do not necessarily guarantee higher ficient to predict the fracture length. They concluded appro- fracture conductivity (Melendez Castillo 2007). priate design for acid fluid by a suitable combination of dif- Acid properties such as viscosity, concentration and type ferent acid type and diffusion coefficient, can significantly ae ff ct the fracture conductivity. Also, the post-u fl sh stage has affect the fracture length and subsequently, the production an essential role on the magnitude of fracture conductivity of the fractured well. (Al-Ameri and Gamadi 2019). Anderson and Fredrickson Specific equipment can be used to provide the desired (1989) in their dynamic acid etching tests, demonstrated that fracture length during acid fracturing operations. For exam- higher fracture conductivity is achieved at higher tempera- ple, diverters are commonly used along the stimulated well- tures. Mou et al. (2010) studied the relationship between bore to create more uniform stimulation and to enhance acid etching pattern, conductivities and mineralogy distributions penetration radius (Aljawad et al. 2019). In summary, creat- and permeability and explored the formation specification ing an appropriate fracture length is related to a number of that leads to deep and narrow channels. As the result of their factors, including the volume of injectable fluid, flow rate study, the rougher the surface the higher the conductivity and fluid injection pressure, the type and viscosity of the is, which is because of wider fractures and deeper channels injected fluid, and the amount of fluid that leaks into the for - that are formed. mation during the fracturing process (Economides and Nolte Pournik et al. (2013) examined the effect of acid spending 2000). Many of these factors can be controlled by choosing on fracture conductivity and fracture length. In his study, an appropriate design and selecting a suitable fluid system. unspent acid generated a large amount of etching and par- According to the real data collected from the fracturing tially acid system created more conductive etching pattern. operations in different fields, to cover a reasonable range, Also, fluid selection plays a crucial role on fracture con- five fracture half-length (X ) values from 50 to 400 ft was ductivity. Favorable fracture conductivity can be achieved used in this work. by appropriate design of acid fluid. Cash et al. (2016) per - formed acid conductivity experiments on samples containing Fracture width a high amount of calcite. He concluded that a concentration of 15 wt% HCl has better-sustained conductivity compared Fracture aperture or width (W ) is one of the chief parame- with 28 wt% HCl. Also, the conductivity can be adjusted if ters affecting the fracture performance. For propped fractur - the optimal design for acid injection time and acid concen- ing treatments, desired fracture width can easily be reached tration is performed. Pournik et al. (2007) used a different by appropriate proppant pack selection. However, in acid acid system for fracturing carbonate formations. Acid vis- fracturing, initial fracture width is a function of the rate of cosified with surfactant, emulsified acid and acid viscosified acid injection, acid type, acid pumping time and acid vol- with polymer used in their studies. There were significant ume, acid contact time, acid temperature and etching pat- differences in the fracture conductivity created between tern (Penaloza 2013; Pournik 2008; Suleimenova 2015). three acid systems. Their results showed that at different It has also been shown that the fracture width increases contact times and temperatures, conductivities from hundred with decreasing the number of fracture stages (Al-Ameri to tens of thousands can be achieved. and Gamadi 2019). An appropriate fracture width can be Zhang et al. (2020) performed a series of experiments on achieved by controlling these factors. reservoir cores to study the effect of clean acid and gelled For the reservoir model used in this study, four explicit acid on carbonate formations. Rough etching created by values for fracture width ranging from 0.09 to 0.36 inches clean acid, and channel etching created by gelled acid was were selected. observed. Accordingly, higher fracture conductivity and longer fracture length was obtained as a result of gelled acid injection. Fracture conductivity In this study, five different fracture permeabilities ranging from 1 to 50 Darcy were used according to the published Fracture conductivity (C ) is the product of fracture width data in the experimental and field studies, and subsequently (W ) and fracture permeability (K ) (Eq. 1). Suleimenova f f 1 3 Journal of Petroleum Exploration and Production Technology fracture conductivities between 7.5 and 1500 mD-ft were code was developed (MATLAB Software 2017b). It could obtained. automatically create the corresponding Eclipse data file to each MSFHW scenario, run the model and export the results. To C = K × w . f f f (1) evaluate the fracture performance, an improvement factor (IF) was defined, expressing the percentage gain in cumulative oil production of the fractured well compared to the non-fractured Number of fracture stages well. The corresponding formula is shown by Eq. 2. The horizontal well section of the sector model has a length N − N pf pnf IF = , (2) of 3000 ft and according to this, a maximum number of pnf 9 fracture stages (N ) were envisaged for it. More fracture stages, due to geomechanical considerations, such as stress where N and N is the cumulative oil production of frac- pf pnf shadowing phenomena, seemed not to be a reasonable sce- tured and non-fractured wells, respectively. nario for this system. Based on this, different cases with To gain a better picture of the individual and combi- fracture stages of 1, 3, 5, 7 and 9 were simulated. The frac- national effects of fracture parameters on the well perfor - tures were spaced evenly with equal spacing between them mance, practical charts were also generated. The results will along the wellbore. To make the comparison between dif- be discussed in the next section. ferent cases meaningful, the location of the fractures was not changed with increasing number of stages. Only in the Ranking effective parameters on well performance case of 7 stages, because of asymmetrical fracture numbers, two fracturing schemes were considered. That is, in Type-I, Having generated a large set of simulation data (a total of fractures were concentrated toward the toe and hill of the 2400 simulation cases for 4 reservoir models), the response horizontal section, while in Type-II, they were concentrated surface methodology (RSM) was employed to explore the toward the middle of the wellbore as shown in Fig. 5. relationship between the fracture parameters and the calcu- Table 4 summarizes the full range of all fracture parameters lated IF. The level of impact of parameters on well perfor- used in this study. The simulation results demonstrated that mance was ranked according to the fitted model. Type-I of seven-stage fractured well performed slightly better than Type-II for all simulation cases, although the observed difference was insignificant. Therefore, only the results of the Table 4 The values of fracture parameters used in simulation model Type-I has been presented and discussed in this work. Parameter Values Analysis of simulation results Fracture half length 50 100 200 300 400 (ft) A full factorial experimental design was used to generate a full Fracture width (in) 0.09 0.18 0.27 0.36 – combination of all fracture parameters. Based on this, a total fracture Permeability 1000 10,000 20,000 40,000 50,000 (mD) of 600 fracturing scenarios were simulated for each reservoir Number of stages 1 3 5 7 (two types) 9 model. To facilitate the simulation runs, a MATLAB computer Fig. 5 Two types of seven-stage hydraulic fracture allocation in horizontal well (Top view) 1 3 Journal of Petroleum Exploration and Production Technology RSM is a useful technique to find the correlation between and N is the number of fracture stages. More information the interpretive variable and response variable. RSM and details about MSFHW cost can be found in Table 5. employs different functions such as linear and full quadratic V = 2.x .w .h .c , a f f f L (6) polynomial regression, to create the relationship between the output and input parameters (Al-Mudhafar and Sepehr- where x is the fracture half-length, w is fracture width, h is f f f noori 2018; Fedutenko et al. 2012; Zubarev 2009). The full fracture height and c is leak-off volume coefficient. quadratic regression model with interaction terms, used in The leak-off volume coefficient (c ) in above formula- this study, has the following form: tion is considered to ensure enough volume of fracturing n n n n fluid is injected into the formation to reach ideal frac- f = 𝛽 + 𝛽 x + 𝛽 x x + 𝛽 x , ture geometry. The c coefficient is a function of various (x) 0 i i ij i j ii (3) i=1 i=1 j=1j≻i i=1 parameters such as porosity, permeability, pressure drop across the mud cake and fluid and rock properties. Con- where β , β , β and β are equation coefficients and x are 0 i ij ii i,j sidering that leak-off volume of 1.4–3.3 times the actual input variables. fracture volume has been practiced during the common In this work, the input variables were fracture length, fracturing operations (Economides et al. 2002), an aver- fracture conductivity and the number of fracture stages and age value of 2 was assumed for c coefficient in this study. the improvement factor (IF) was considered as the output It should be mentioned that in multi-stage fracturing variable. According to this, the general form of the fitted operations, it is usually assumed that addition of each proxy model is as follows: fracture stage adds up a fixed cost of 10% to the operation cost. The cost of operation is obtained by summing up the F = + X + Ns + C + X (x) 0 1 f 2 3 f 4 values of items 2–4 in Table 5 that equals to 300,000$ for 2 2 + Ns + C + X Ns 5 6 7 f the case under study. + X C + NsC . The following sequence was considered to calculate the 8 f f 9 f (4) corresponding NPV for each fracturing scenario: It should be noted that the fracture width was not used in the model as a single variable, since its effect on well 1. The required volume of acid is calculated based on frac- production has been considered together with fracture per- ture dimensions. meability by introducing the fracture conductivity into the 2. The fixed cost of operation is calculated. (assumed model. 300,000 $ in this study). 3. The total cost of operation is calculated using Eq. 5. Economic analysis 4. The increase in cumulative oil production from frac- tured, compared to non-fractured, well is calculated by Considering that MSFHW is a costly operation, optimi- employing the simulation results (Eq. 7): zation of fracture parameters from an economic perspec- f nf ΔN = N − N , p,n (7) p,n p,n tive is a crucial part of this job. In other words, each of the MSFHW parameters can restrict the treatment design where ΔN is the increase in cumulative production p,n with respect to their operation costs. For example, fractur- f nf after n years, N and N are the cumulative oil produc- p,n p,n ing cost increases considerably with increasing number of tion from fractured and non-fractured well after n years. fracture stages (Rahman et al. 2014). The net present value 5. The revenue after acid fracturing treatment is calculated (NPV) is a simple but very useful measure that can be using Eq. 8. used to evaluate the profitability of a project with time. In this study, considering the small size of the sector model and short production life of the reservoir, we employed a Table 5 Cost of different parts No. Service type Cost 1-year scenario for this economic exercise. of acid fracturing scenario used The total cost of the MSFHW job can be calculated in this study 1 Consumable 1000 $/ materials bbl according to Eq. 5. (acid and C = N × V × p + C + 0.1 ×(Ns − 1)× C , additives) (5) T s a a Fix Fix 2 Operation 200,000 $ where C is the total cost of fracturing job, V is the required T a 3 Transport and 85,000 $ volume of injected acid for two wings that is given in Eq. 6, equipment and staff P is the acid price per unit volume, C is a fixed cost a Fix 4 Crew 15,000 $ related to transportation, crew, equipment and facilities costs 1 3 Journal of Petroleum Exploration and Production Technology when all fracture parameters are at their maximum level, ΔR = P ×ΔN , n o p,n (8) IFs are about 100% and 600% for Case 1 with K = 5mD where P is the price of oil assuming 70 $/bbl at time (Fig. 6) and Case 2 with k = 0.5 mD (Fig. 7), respec- of this study. tively. Similarly, in case of heterogeneous reservoirs, IFs 6. NPV is calculated using Eq. 9. are about 30% and 130% for Case 3 (Fig. 8) and Case 4 (Fig. 9), respectively. In addition, it is noted that the multi- ΔR stage fractured well has considerably better performance NPV = − C , (9) (1 + i) in homogenous reservoirs in comparison with their analog n=1 heterogeneous ones, despite the average reservoir perme- where m is the number of years evaluation is performed, ability is almost similar in both cases. For instance, com- and “i” is the currency escalation rate, assumed 10% in pare Fig. 6 (Case 1) with Fig. 8 (Case 3) and Fig. 7 (Case this work. 2) with Fig. 9 (Case 4). This, in turn, demonstrates that the distribution of rock permeability within the reservoir, especially in the regions around the fractured zones, is an Results and discussion important parameter affecting the fracture performance. Analyzing the performance of fractured wells under dif- Eec ff t of individual fracture parameters ferent levels, it is observed that in all four reservoir cases the effect of any individual parameter on fracture perfor - The effect of each individual fracture parameter on well mance is negligible when all other parameters are at their performance was studied by analyzing the trend of IF with minimum level (blue curves in Fig. 6, 7, 8, 9). As noted, increasing the magnitude of that parameter, whilst the other the positive effect of an individual parameter becomes evi- parameters were kept at their minimum, medium and maxi- dent when other fracture parameters are at their medium mum levels according to the ranges given in Table 4. The level (orange curves). More importantly, this positive effect curves corresponding to four selected reservoir models are becomes notably more significant when other parameters depicted through Figs. 6, 7, 8, 9. are at their maximum level (green curves). In other words, Comparison of IFs between four different reservoir there is a monotonic increase in IF as the magnitude of an models shows that, first of all, the performance of frac- individual fracture parameter increases when other param- tured well is more remarkable in rocks with lower per- eters are at their medium and—in particular—maximum meability. For instance, in homogenous reservoir models, levels. For example, for the real sector model, Case 3, it Fig. 6 Effect of fracture parameters on production improvement after 1 year for case 1 1 3 Journal of Petroleum Exploration and Production Technology Fig. 7 Effect of fracture parameters on production improvement after 1 year for case 2 Fig. 8 Effect of fracture parameters on production improvement after 1 year for case 3 is observed that when X increases from its minimum (50 another example, in Case 4, increasing the number of frac- ft) to maximum (400 ft) magnitude, IF increases from 4 to ture stages from 1 to 9, results in increase in production gain 8% and from 10 to 30%, for the cases with parameters on from 5 to 50% when other parameters are at their medium their medium and maximum levels, respectively (Fig. 8). As level and from 15 to 130% when they are at their maximum 1 3 Journal of Petroleum Exploration and Production Technology Fig. 9 Effect of fracture parameters on production improvement after 1 year for case 4 level (Fig. 9). The previous observations demonstrate that Level of importance of fracture parameters the positive impact of an individual fracture parameter becomes more pronounced with increasing level of other To find the level of importance of each fracture parameter parameters. Furthermore, there is typically a certain range on well performance, the RSM technique was employed. for fracture parameters below which the fracturing process Following this, a full quadratic proxy model was fitted to becomes inefficient. For instance, for the cases considered the available simulation results (IFs) for each reservoir in this study, each fracture parameter showed minimal effect case. Fracture length, fracture conductivity and the number on well performance when all other fracture parameters were of fracture stages were selected as the model variable. It at their minimum level. should be noted that the fracture conductivity accounts for It is also interesting to note that in all four reservoir cases the effects of fracture width and permeability. The corre - studied here (Figs. 6, 7, 8, 9), increasing the fracture conduc- sponding fitted proxy models are presented in Appendix B. tivity has the most positive effect on improving the well pro- The values of the coefficient of determination of the fitted ductivity compared to other parameters including fracture regression proxy models for each reservoir case are listed length, fracture width and number of fracture stages. This in Table 6. The estimated R values verify the accuracy is realized by noting that the slope of corresponding line to and reliability of the fitted models. The pareto charts cor - the conductivity parameter in each reservoir case has the responding to each reservoir case have also been generated largest slope compared to other parameters. In other words, and shown by Figs. 10, 11, 12, 13. when fracture conductivity increases from its minimum to As Fig. 10 shows, in homogenous reservoir with k = 5 maximum value, IF increases to a greater extent compared mD, the fracture conductivity and in the second place, the to what observed for other fracture parameters. number of fracture stages are the most important param- More useful charts were also prepared from the simu- eters on enhancing the well productivity. The interaction lation results for different reservoir cases, demonstrating between these two parameters also in the third place domi- the interaction effect of fracture parameters on well perfor - nates the fracture performance. It is interesting to note mance. The results are presented in Appendix A. that in this case, the impact of fracture length compared to other parameters is considerably less important. In the second reservoir model, Case 2, which in com- parison to Case 1 has a tighter rock with one order of 1 3 Journal of Petroleum Exploration and Production Technology Table 6 Coefficient of determination values for all reservoir models 2 2 2 R (%) R (adjusted) (%) R (pre- diction) (%) Case 1 98.62 98.60 98.54 Case 2 95.25 95.20 95.00 Case 3 96.94 96.89 96.75 Case 4 93.12 93.01 92.74 Fig. 12 The level of importance of fracture parameters in case 3 Fig. 10 The level of importance of fracture parameters in Case 1 Fig. 13 The level of importance of fracture parameters in case 4 Investigating the importance of fracture parameters in the heterogeneous sector model, Case 3 (Fig. 12), demonstrates more or less the same trend as observed for the homogeneous Case 1. That is, the fracture conductivity and number of frac- ture stages are first two important parameters, respectively, that influence the well performance. An obvious difference between two cases is the more pronounced effect of frac- ture length in Case 3 compared to Case 1. The importance of X becomes even more important than that of interaction Fig. 11 The level of importance of fracture parameters in case 2 between C and N , observed in Case 1. This behavior can be f s attributed to the distribution of various rock permeabilities in Case 3, compared to a single value used in Case 1. magnitude, the number of fracture stages (N ) dominates The results from the heterogeneous tight reservoir model the well performance in the first place (Fig. 11). The frac- (Case 4), depicted by Fig. 13, demonstrate that similar to ture conductivity and then the fracture length are the next Case 2 (low-permeability homogeneous reservoir), the num- two subsequent parameters that almost with the same ber of fracture stages is the dominant parameter that controls level of importance control the fracture performance. The the performance of fractured well. This is in contrast to Cases observations demonstrate that in tighter reservoir systems 1 and 3 (reservoirs with higher permeabilities), where the increasing the number of fracture stages becomes signifi- fracture conductivity was the dominating parameter. Also, cantly more important when higher well productivity is the fracture length and facture conductivity are in the second required. Furthermore, fracture length also plays more and third place, respectively, affecting the well performance crucial role in rocks with poor flow characteristics. 1 3 Journal of Petroleum Exploration and Production Technology in Case 4. The observations put more emphasis that as rock for some cases lie in an obviously different range. For permeability decreases, increasing the number of fracture instance, the optimum fracture length for the Case 3 is 50 stages rather than fracture conductivity become more impor- ft and optimum fracture width and conductivity for Case tant and profitable to expedite the well production rate. 4 are 0.09 in and 375 md-ft, respectively (Table 7). Such observations place emphasis on the significance of care - Optimization under economic considerations ful economic analysis within reasonable evaluation time intervals, where an optimum fracture design is demanded. In the previous sections, the performance of fractured well To provide a better picture of importance of economic was investigated from a technical point of view. In other analysis for fracture designs, the well production IFs corre- words, maximizing the well productivity by changing the sponding to technical (maximum) and economic (optimum) fracture parameters was considered as the main object of fracture designs have been compared in Table 8. It is inter- fracturing job. As discussed earlier, due to excessive cost of esting to note that, especially for the heterogeneous reser- multi-stage fracturing operations, it is vital to optimize the voir cases (Cases 3 and 4), there is an obvious difference fracture design from an economic perspective. Accordingly, between IFs of optimum and maximum design scenarios. It to find the optimum economic fracture parameters, maxi- is observed that for the real reservoir model (base Case 3), mizing NPV was used as the final criteria to select the best the maximum IFs can reach about 29%, but the economic scenario. The evaluations were performed for all four reser- design restricts this value to about 11%. The observations voir models based on the procedures discussed previously in underline the strict economic analyses required to achieve Sect. 3.4. As mentioned earlier considering the short produc- the most profitable fracture design for field applications. tion life cycle of the reservoir, a 1-year production scheme was used for the NPV calculations. Table 7 shows the results of the economic exercise per- Conclusions formed, where the optimum fracture parameters and the cor- responding maximum NPV for each reservoir model have The work presented in this paper focused on optimization of been determined. fracture parameters in MSFHWs in low-permeability heavy- It should be mentioned that when only the technical oil systems. Two types of homogeneous and heterogeneous considerations were important, i.e. evaluations presented sector reservoir models with permeabilities ranging between in Sect. Effect of Individual Fracture Parameters, the 0.5 and 5 mD were used and about 2400 simulation sce- maximum MSFHW productivity for all four reservoir narios were run. The RSM methodology was then employed models was attained when all fracture parameters were to rank the level of importance of fracture parameters. The at their maximum levels. That is, X = 400 f t, W = 0.36 optimum fracture parameters were also determined by con- f f in, C = 1500 md-ft and N = 9. However, in case of eco- ducting an economic exercise by calculating NPVs corre- f s nomic considerations, the optimum fracture parameters sponding to each fracture design. The main conclusions from this study are as follows: Table 7 The optimum economic fracture parameters and correspond- ing NPVs • For the reservoir cases with average permeability of 5 Case no. Opti- Opti- Optimum Optimum Max mD, fracture conductivity and number of fracture stages, mum Xf mum Wf Cf (md-ft) Ns NPV(M$) were in turn the main effective parameters improving (ft) (in) the well productivity. As rock permeability decreased, 1 100 0.36 1500 9 21.68 i.e. 0.5 mD, the number of fracture stages became the 2 400 0.18 750 9 14.27 dominant parameter controlling the fractured well per- 3 50 0.36 1500 7 2.53 formance. 4 300 0.09 375 7 1.44 • In tight reservoir systems, increasing the fracture length became a beneficial practice to reach higher production rates. However, it should be mentioned that if the frac- Table 8 Comparison of Case no. Maxi- Opti- ture conductivity is low, it is a good practice to keep the maximum and optimum IF mum IF mum IF fracture length shorter to avoid additional costs. (%) (%) • Considering reservoir heterogeneity, especially distribu- tion of rock permeability in the stimulated area around 1 98 73 the wellbore, showed a significant role in determining 2 622 491 the optimum fracture design. For the tight heterogene- 3 29 11 ous sector model, the fracture length had a more posi- 4 131 63 1 3 Journal of Petroleum Exploration and Production Technology tive effect on well productivity compared to fracture conductivity, whereas the later parameter was always the dominating factor over the former one in homogeneous systems. The economic analysis of the fracturing operation played a key role in designing the optimum fracture parame- ters. More specifically determination of maximum frac- ture length and number of fracture stages are crucially affected by profitability of the operation. Open Access This article is licensed under a Creative Commons Attri- bution 4.0 International License, which permits use, sharing, adapta- tion, distribution and reproduction in any medium or format, as long Fig. 15 Main effects plot for case 2 as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativ ecommons .or g/licenses/b y/4.0/. Appendix A The “main effects plot” examine differences between level means (average input values) for various parameters. These graphs are prepared based on analysis of variance (ANOVA). The response means (average output values) for each level of parameter are plotted and connected by a line. In these charts, the value of each parameter increases while Fig. 16 Main effects plot for case 3 the value of the other parameters are kept at average values. The baseline represents the value of the response level in the mean value of all parameters. Also, a reference line (dashed line) is drawn at the overall mean. The reference line repre- sents the value of the response level in the mean value of all parameters. These charts verified our results through Effect of Individual Fracture Parameters and Level of importance of fracture parameters sections (Figs. 14, 15, 16, 17). Fig. 17 Main effects plot for case 4 The following contour plots show the interaction effect of fracture parameters on well performance (IF) for dif- ferent reservoir cases. As the highlighted region becomes darker, the effect of parameters become more favorable and IF increases (Figs. 18, 19, 20, 21). 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