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Ensemble-based optimization of hydraulically fractured horizontal well placement in shale gas reservoir through Hough transform parameterization

Ensemble-based optimization of hydraulically fractured horizontal well placement in shale gas... Shale gas reservoirs have been successfully developed due to the advancement of the horizontal well drilling and multistage hydraulic fracturing techniques. However, the optimization design of the horizontal well drilling, hydraulic fracturing, and operational schedule is a challenging problem. An ensemble-based optimization method (EnOpt) is proposed here to optimize the design of the hydraulically fractured horizontal well in the shale gas reservoir. The objective is to maximize the net present value (NPV) which requires a simulation model to predict the cumulative shale gas production. To accurately describe the geometry of the hydraulic fractures, the embedded discrete fracture modeling method (EDFM) is used to construct the shale gas simulation model. The effects of gas absorption, Knudsen diffusion, natural and hydraulic fractures, and gas–water two phase flow are considered in the shale gas production system. To improve the parameter continuity and Gaussianity required by the EnOpt method, the Hough transformation parameterization is used to characterize the horizontal well. The results show that the proposed method can effectively optimize the design parameters of the hydraulically fractured horizontal well, and the NPV can be improved greatly after optimization so that the design parameters can approach to their optimal values. Keywords Shale gas · Ensemble optimization · Embedded discrete fracture model · Hough transformation 1 Introduction Shale gas development has received great attention in recent years because of the increasing demand for natural gas resources and its huge reserves worldwide (Dong et al. 2012, Edited by Yan-Hua Sun 2014). The matrix permeability of the shale gas reservoir is ultra-low due to the nanoscale pore size (Javadpour et al., * Liang Xue 2007; Liu et al. 2019a). With the advancement of horizontal xueliang@cup.edu.cn well drilling and hydraulic fracturing techniques, the natural gas resources in shale reservoir can be successfully unlocked State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum (Beijing), in a cost-effective manner. Beijing 102249, China Shale gas is considered as one of the unconventional Department of Oil-Gas Field Development Engineering, energy resources, and many special physical mechanisms College of Petroleum Engineering, China University are associated with the gas flow in shale reservoirs. The of Petroleum (Beijing), Beijing 102249, China natural gas in the shale gas reservoir can mainly exist in two State Key Laboratory of Shale Oil and Gas Enrichment forms, i.e., free gas and adsorbed gas (Curtis 2002). The Mechanisms and Effective Development, Beijing 100083, adsorbed gas will be desorbed into the matrix pores from China the organic matters when the reservoir pressure decreases Sinopec Petroleum Exploration and Production Research during the production process. In the shale matrix, the non- Institute, Beijing 100083, China Darcy flow can occur due to the nanoscale pore size, and School of Water Conservancy, North China University the Knudsen diffusion has been used to account for the of Water Resources and Electric Power, Zhengzhou 450046, effect of the molecular scale gas flow (Javadpour 2009; Xue Henan, China Vol.:(0123456789) 1 3 840 Petroleum Science (2021) 18:839–851 et al. 2019; Zhang et al. 2020). In the fracture system of the engineering. Brouwer and Jansen (2004) used a systematic shale gas reservoir, the natural fractures and the hydraulic dynamic optimization approach to control the valve setting fractures together form a complex fracture network, and the in the injector and producers for a water flooding project in shale gas will flow through these fractures with various sizes the heterogeneous reservoir. Sarma and Chen (2008) and into the horizontal well (Cipolla et al. 2010; Clarkson et al. Zhang et al. (2010) converted the discrete control variables 2011). The multistage hydraulic fracturing will create mul- to continuous ones and used the adjoints and gradient- tiple transverse fractures along the lateral direction of the based optimization method to determine the optimal the horizontal well in relatively intact reservoirs, or generate well locations. Volkov and Bellout (2018) approximated the complex fracture networks connected to the horizontal well well placement optimization gradient by using finite differ - in fractured reservoirs (Li et al. 2016, 2017; Li and Zhang ence approximations of augmented Lagrangian derivatives 2018; Liu et  al. 2019b). To improve the ultimate recov- with adjoint formulation to improve efficiency. Al Dossary ery of the shale gas in a cost-effective manner, the design and Nasrabadi (2016) proposed an imperialist competitive of the hydraulically fractured horizontal well needs to be algorithm (ICA) to optimize the well location. Hamida optimized. et  al. (2017) modified the traditional genetic algorithm Fractures in the hydraulically fractured shale gas reser- with a similarity operator to optimize the well placement. voir increase the contact surface between shale matrix and Jesmani et al. (2015) used the derivative-free particle swarm fractured horizontal well and form the primary flow path optimization algorithm to optimize the well location with for shale gas flow. Reservoir numerical simulation and data- well design constrains. Tukur et al. (2019) implemented the driven methods can forecast the shale gas production and aid genetic algorithm and simulated annealing methods to opti- the optimization of the fractured horizontal well (Yu et al. mize the well placement. In the optimization of a hydrauli- 2018; Xue et al. 2020, 2021). To take the fractured flow cally fractured horizontal well in a shale reservoir, Ma et al. into consideration, multi-continuum models, such as the dual (2013) optimized the hydraulic fracture placement with the porosity model (Warren and Root 1963; Kazemi 1969; Ros- gradient-based finite difference method (FD), discrete simul- tami et al. 2020) and multi-porosity model (Wu and Pruess taneous perturbation stochastic approximation (DSPSA), and 1988), have been used for shale gas reservoir simulation. genetic algorithm (GA). Yu and Sepehrnoori (2013) used the Zhang et al. (2009) conducted the sensitivity analysis of the response surface method to conduct the optimization of the hydraulic fracture parameters in the shale gas reservoir by multiple hydraulically fractured horizontal wells in shale upscaling the discrete natural fractures to dual porosity sys- gas reservoirs. Wilson and Durlofsky (2013) introduced tem. Cipolla et al. (2010) simulated the shale gas production a direct search algorithm to optimize the shale gas field by using the dual porosity model with the consideration of development by combining with reduced physics model to gas adsorption effect. Rubin (2010) used local grid refine- improve computational efficiency. Rammay and Awotunde ment (LGR) method with dual porosity model to improve (2016) used differential evolution algorithm to optimize the the characterization of the hydraulic fractures. Essentially, hydraulic fracturing parameters and horizontal well length. the multi-continuum model represents the fractures by using Zhang and Sheng (2020) optimized the fractured horizontal an averaged property in the grid block. The discrete frac- well in the shale gas reservoir by considering the effect the ture modeling method can characterize the fracture geometry stimulated reservoir volume. The ensemble optimization explicitly (Karimi-Fard et al. 2004; Gong et al. 2008), but method (EnOpt) is powerful optimization method proposed the unstructured grid has to be used and the computational by Chen et al. (2009). It is a stochastic gradient optimiza- burden can be too heavy to be implemented in the field- tion method, which approximates the gradient through the scale application. The embedded discrete fracture modeling ensemble computation, and thus it is capable of integrating method (EDFM) was proposed by Li and Lee (2008) and with any simulator. In addition, compared to other gradient- improved by Moinfar et al. (2014), which can represent the based optimization method (e.g., finite difference method geometry of each fracture explicitly but with structure grid and adjoint method) and gradient-free optimization method system, and this method has been applied to simulate the (e.g., genetic algorithm and particle swarm method), it is shale gas production (Dai et al. 2017; Yu et al. 2018). Due more efficient when the dimension of the control variables to the uncertainty associated with the geological condition is high. Leeuwenburgh et al. (2010) applied this method to including the conceptual model uncertainty (Xue and Zhang optimize the settings of inflow control valve in a water flood- 2014; Xue et al. 2015), the production data can be used to ing project and compared its performance with the adjoint reduce the predictive uncertainty and obtain more accurate method. However, this method has not been fully investi- prediction results (Dachanuwattana et al. 2018; Kang et al. gated in the optimization problem of hydraulically fractured 2020; Xue et al. 2020). horizontal well in shale gas reservoir. Many optimization methods have been investigated In this research, we propose to use ensemble optimi- to deal with the well placement problem in reservoir zation method to maximize the net present value (NPV) 1 3 Petroleum Science (2021) 18:839–851 841 in the shale gas development project. The EDFM is used f f f f k k g rg to simulate the shale gas production. The optimization f well mf (3) ∇ ⋅ ∇p − q + q = g g g parameters are the location of horizontal well drilled in a shale reservoir with considering the effect of the natu - f f where k is the absolute permeability of fractures, k is the ral fractures, the optimal hydraulic fracturing parameters rg (i.e., number of stages, length of the hydraulic fractures, relative permeability of shale gas in fractures, p is the gas well and conductivity of hydraulic fractures) and the opera- pressure in fractures, q is the mass flow rate of shale gas mf tional parameter (i.e., bottom hole pressure). Usually, to well, q is the mass flow rate of shale gas from matrix to these parameters are optimized independently step by f f fractures,  is the fracture porosity, S is the gas saturation step. However, in our research, all these parameters are in fractures, and  is a coefficient to account for non-Darcy optimized simultaneously so that the most optimal shale flow. gas development plan can be designed. When optimizing The water flow equation in the shale matrix is: the horizontal well location, the coordinates of the well are m m m m discrete variables. Here, we further propose to use Hough k k   S w rw w m mf ∇ ⋅ ∇p − q = (4) transformation-based parameterization to transform the w w Cartesian space into Hough space. This transformation can improve the Gaussianity of the parameter so that it where  is the water density, k is the apparent permeability can be better integrated with EnOpt method. of water in the shale matrix, k is the relative permeability rw of water in the shale matrix,  is the water viscosity, p is w w mf the water pressure in the shale matrix, q is the mass flow rate from matrix to fracture, and S is the water saturation. 2 Methodology The water flow equation in the fracture is: f f f f k k   S 2.1 Shale gas flow simulation w rw w f well mf ∇ ⋅ ∇p − q + q = (5) w w w The EDFM is used to solve the simulation of shale gas f well production. The detailed equations to characterize the mul- where p is the pressure of water phase in fractures and q w w tiscale shale gas flow have been presented in Xue et al. is the mass flow rate of water to well. (2020), and the design of the simulation software was In the EDFM, the mass flow rate from matrix to fractures, mf introduced in Li et al. (2015). The brief governing equa- q , and two fracture segments connections in the same tions of the gas–water two phase flow model of shale gas matrix block or in two adjacent matrix blocks can be com- production through EDFM method are introduced here. puted through the non-neighboring connection (NNC) The gas flow equation in shale matrix is: (Moinfar et al. 2014): m m m m S  m f nnc m f k k N N g nnc nnc k p − p g a rg  k k p − p  rg g g rg m s mf j g g (1) mf nnc nnc nnc ∇ ⋅ ∇p + q − q = q = q = A = T g c g g g,c j nnc j g g g j=1 j j=1 (6) where  is the density of the shale gas, k is the apparent g a where p is the gas pressure in fractures, N is the number nnc permeability of shale matrix, k is the relative permeability rg of the non-neighboring connection grids, T is the NNC nnc of gas in the shale matrix, is the gas viscosity,  is the nnc transmissibility factor, A is the contact area of fracture and nnc matrix porosity, S is the gas saturation in the shale matrix, g matrix, k is the harmonic mean of matrix and fracture nnc q is the desorption/adsorption mass flow rate, which can be permeability, and d is the characteristics distance between characterized by Langmuir isothermal adsorption model, matrix block and fracture plane. mf and q is the mass flow rate from matrix to fractures. Together with the NNC between matrix and facture in To account for the Knudsen diffusion, the apparent per - EDFM mentioned above, two more NNCs need to be con- meability k can be evaluated by (Tang et al. 2005): sidered, i.e., the two fracture segments connections in the same matrix block and two adjacent matrix blocks. The m m 2 k = k 1 + 8C Kn + 16C Kn (2) 1 2 a ∞ NNC transmissibility, T , can be written as: nnc nnc nnc where k is the absolute permeability, C and C are con- T T 1 2 k A ∞ 1 2 T = = (7) nnc stant coefficients, and Kn is the Knudsen number. nnc d T + T 1 2 The shale gas flow equation in fractures is: 1 3 842 Petroleum Science (2021) 18:839–851 with where  is the updated control variable,  is the control k+1 k variable before updating,  is the coefficient determining the k w L f1 f1 int updating step size,  is the prior covariance matrix of the T = , xx f1 control variables, and  is the sensitivity of NPV function k w L f2 f2 int g(x) to the control variables. T = The prior covariance matrix of the control variables f2 xx is: where L is the length of the fracture intersection, k is the int f fracture permeability, w is the fracture aperture, and d is the f f (11) xx N − 1 normal distance between the center of the fracture and the fractures intersection. The developed EDFM model has been where N is the ensemble size. validated against the commercial software CMG to show its The control variable matrix  can be expressed as: accuracy (Dai et al. 2017). x − x … x − x ⎛ ⎞ 1,1 1 N ,1 N x x 2.2 Ensemble optimization method ⎜ ⎟ = ⋮⋱ ⋮ (12) ⎜ ⎟ x − x … x − x ⎝ ⎠ 1,N 1 N N N e x e x The ensemble optimization method was proposed by Chen 1 N et al. (2009), and it has been applied in several production where x = x is the ensemble mean of the control i ij j=1 optimization problems (Chen and Oliver 2010; Fonseca et al. variables. 2014; Tueros et al. 2018). A vector of the control variable is The product term   can be approximated by xx defined in the EnOpt method, which contains all the variables that need to be optimized. The control variable vector can be (13) xx xg defined as: where  is the covariance between X  and G. It can be xg = x , x , x , … , x (8) 1 2 3 N expressed as: where N is the number of control variables. 1 (14) xg The NPV is used as the objective function during the opti- N − 1 mization process here, and the formula is written as: And the NPV vector G can be expressed as: Q(i)P g(x)= − w P − 2hf hf P (9)  = g x − g, g x − g, … , g x )− g (15) L hw xf stage hf 1 2 N (1 + r) i=1 � � where g = g x is the ensemble mean of the NPV where i is the time step index, N is the total production time, t N l=1 Q(i) is the cumulative shale gas production within the given values. time step which can be computed from the above EDFM- In the EnOpt method, we can substitute Eq. (13) in Eq. based shale gas production model, P is the price of shale (10) and use  as the filtering matrix. Then the optimiza- xx gas which is set as 3 CNY/m , r is the discount rate which is tion parameters are updated by set as 6%, w is the length of horizontal well, P is the drill- L hw ing cost per unit well length which is set as 30,000 CNY/m, = x + k+1 k xx xg (16) hf is the half-length of the hydraulic fractures, hf is the k xf stage stage number of the hydraulic fractures, and P is the well hf complete cost per unit fracture length, which is set as 20,000 CNY/m. All these parameter setting values are inferred from 2.3 Hough transform parameterization a laboratory report to describe the shale gas development in for horizontal well Zhaotong shale gas field in China. During the optimization process, the NPV is maximized by optimizing the control Hough transformation was proposed by Hough (1962), which variables. was used to detect lines. Duda and Hart (1972) extended In each optimization step, the control variable is updated Hough transformation to detect any arbitrary objects in the through image analysis. Hough transform converts the parameters from the Cartesian coordinate space to the Hough space. Hough =  + space is basically an accumulator space, and it uses the vot- k+1 k xx (10) ing method in the accumulation space to find the local maxi- mum value to conduct feature detection. The accumulator to 1 3 Petroleum Science (2021) 18:839–851 843 transform x–y Cartesian coordinate system space into ρ–θ 0 if t < a 𝛿 (u − a)du = (18) polar coordinate system of Hough space can be expressed as: 1 if t > a −∞ +∞ +∞ As shown in Fig.  1, the straight line y =−x + 5 in the A(, )= I(x, y)( − x cos  − y sin )dxdy −∞ −∞ Cartesian coordinate system can be transformed into a point � � � (17) in the Hough space π∕4, 5 2 2 . where A(, ) is recording the how many sinusoidal curves The accumulator in the Hough space provides the infor- actually pass through the (, ) point in the Hough space, mation on the total number of sinusoidal curves that pass I(x, y) is the point in the Cartesian coordinate space. (⋅) is through the point (, ) , and all the grid blocks along the the Dirac delta function, and it can be defined as: line segment, characterized by (, ) in Hough space, will contribute to the accumulator. The Hough transform has the capability to transform any line to the parameter set that y ρ can take a better continuity than the discrete point in the Cartesian coordinate (Lu and Zhang 2015; Yao et al. 2018); therefore, it can be used to represent a line segment in the 4 Cartesian coordinate space. By using this method, the hori- y = -x + 5 zontal well is not characterized by its endpoint coordinates x , y , z and x , y , z , but it is represented by a parameter 1 1 1 2 2 2 set (as shown in Fig. 2): (, , D, L, ,  , ) (19) x 01 (a) Cartesian coordinate space (b) Hough space where  is the vertical line distance between the origin to line segment in the x–y projection plane;  is the angle between Fig. 1 Transformation of a straight line in Cartesian space to Hough x-axis and the vertical line in the x–y projection plane; D is space A″′ B″′ A″ B″ Fig. 2 Parameterization of a horizontal well in the Hough space 1 3 B′ A′ L 844 Petroleum Science (2021) 18:839–851 the distance between the vertical line and the center point of e the line segment in the x–y projection plane; L is the length North e of the line segment in the x–y projection plane;  is the angle between x-axis and the line segment in the x–z projection plane; and  is the angle between z-axis and the line segment in the y–z projection plane;  is the angle between y-axis and the line segment in the x–y projection plane. The horizontal well drilled in the shale gas reservoir, which can be regarded as the line segment. When defining West East the horizontal well with the Hough transformation-based α parameterization method, the horizontal well is within the x–y plane in the coordinate system, and  = 0 and  = 0 in this case. In addition, it can be seen from the triangu- lar relationship that  =  . Therefore, the horizontal well South can be represented by the parameter set in the optimization problem: (, , D, L) (20) Fig. 3 Characterization of the fracture plane (Song et al. 2019) With consideration of the shale matrix permeability, the 2.4 Equivalent permeability conversion of natural final permeability tensor can be computed by: fractures K + K K K ⎛ ⎞ eN11 m eN12 eN13 In the shale gas reservoir, there may exist a large set of natu- ⎜ ⎟ K K + K K (23) eN21 eN22 m eN23 ral fractures, which results in the shale gas reservoir with ⎜ ⎟ ⎝ K K K + K ⎠ severe heterogeneous permeability distribution (Khanal eN31 eN32 eN33 m and Weijermars 2019). The hydraulic fractures can be rep- resented explicitly with its full geometrical properties in the EDFM method. However, the number of the natural fractures is very large, which is infeasible to obtain the information on 3 Results and discussion each natural fracture. Even we can know the exact distribu- tion of the natural fractures, the simulation model can be too In this research, the EnOpt method based on Hough trans- time-consuming to be used in practice. Here, we establish form is used to optimize the economic benefits of shale gas a method to convert the discrete natural fractures to their reservoir produced by fractured horizontal wells. The loca- equivalent permeability. Let us define the azimuth angle of tion parameters of fractured horizontal wells are transformed the fracture is  , the dip angle of the fracture is  , and per- into Hough space, and the EnOpt algorithm is used for the meability parallel to the fracture is K (as shown in Fig. 3). integrated optimization of the design parameters. The Hough The permeability tensor of each fractures can be com- transformation-based parameterization can be suitable to puted by (Song et al. 2019): improve the continuity of the discrete design parameters, 2 2 2 2 ⎛ cos  ⋅ cos  + sin  sin  ⋅ cos  ⋅ sin  cos  ⋅ sin  ⋅ cos  ⎞ i i i i i i i i i 2 2 2 2 ⎜ ⎟ (21) =  sin  ⋅ cos  ⋅ sin  cos  ⋅ sin  + cos  − cos  ⋅ sin  ⋅ sin ei i i i i i i i i i i ⎜ ⎟ cos  ⋅ sin  ⋅ cos − cos  ⋅ sin  ⋅ sin  sin ⎝ ⎠ i i i i i i i When a number of N fractures exist in a single grid block, such as the location or central point coordinate of the hori- the permeability tensor can be expressed by: zontal well, and can be helpful to the linearization of the 2 2 2 2 N N cos  ⋅ cos  + sin  sin  ⋅ cos  ⋅ sin  cos  ⋅ sin  ⋅ cos ⎛ ⎞ i i i i i i i i i � � 2 2 2 2 ⎜ ⎟ (22) =  =  sin  ⋅ cos  ⋅ sin  cos  ⋅ sin  + cos  − cos  ⋅ sin  ⋅ sin eN ei i i i i i i i i i i ⎜ ⎟ i=1 i=1 cos  ⋅ sin  ⋅ cos  − cos  ⋅ sin  ⋅ sin  sin ⎝ ⎠ i i i i i i i 1 3 Fracture Petroleum Science (2021) 18:839–851 845 nonlinear parameter in the set optimization algorithm. The distribution. For example, the natural fracture apertures transformed parameters can improve the Gaussian assump- follow a Gaussian distribution (characterized by the mean tion of the EnOpt process and thus provide a more reason- and variance in the generation function), the lengths of the able optimization performance. natural fractures follow a uniform distribution (character- ized by the minimal and maximal values in the generation 3.1 Synthetic model construction of the shale gas function), and the azimuth follows a Gaussian distribution. reservoir The random discrete natural fracture model is generated by using the parameter setting listed in Table  2. The central −8 To demonstrate the workflow and performance of the pro- point density of the natural fractures are set as 4.0 × 10 / posed EDFM-EnOpt method, a synthetic model of shale gas m . According to the relationship of the generated fracture reservoir simulation is constructed here. In the synthetic endpoints coordinate data and the computational grid, the model, a hydraulically fractured horizontal well is located fracture endpoint coordinate information is converted into in a 3D shale gas reservoir. The shale gas simulation model computational grid property, i.e., the equivalent grid block is constructed using EDFM method. During the simulation permeability. Then, the shale gas reservoir geological model process, it considers the special gas flow mechanisms in the with natural fractures can be constructed as shown in Fig. 4. shale gas reservoir. The simulation model is constructed For the hydraulic fractures, the number of hydraulic frac- based on gas–water two-phase flow model. In the shale tures is small and the fracture conductivity is large. They matrix flow, it considers the adsorption/desorption of the directly connect with the horizontal wellbore and the shale shale gas on the organic content in the shale reservoir and matrix, which form the primary flow path for shale gas and the Knudsen diffusion effect of the shale gas flow caused by has a great impact on the flow field. Therefore, the hydraulic the nanoscale pore structure. In the fracture flow, it consid- fractures should be accurately described by considering their ers the gas flow in natural fractures by upscaling the natural explicit fracture geometry. In this paper, the hydraulic frac- fractures using equivalent permeability method and the gas tures are established by EDFM, and the hydraulic fractures flow in hydraulic fractures by explicitly taking the fracture are transverse fractures under the reservoir condition, that is, geometry into account. the hydraulic fractures are rectangular plates perpendicular The parameter values used in the synthetic model to char- to the horizontal stratum. The parameter values are listed in acterize the shale gas reservoir properties are summarized Table 3, which are used as the initial design of the hydraulic in Table 1. These data are collected from a laboratory report fracturing project before optimization. to study the Zhaotong shale gas field in China and represent Once combining the properties of shale matrix, natural the state of the shale gas reservoir before the well drilling fractures and hydraulic fractures, the reservoir model for and gas production. the shale gas production simulation can be generated, as A large number of natural fractures can be developed shown in Fig. 5. in the shale gas reservoir. It is hard to obtain the fracture geometry for each natural fracture, but the statistical dis- 3.2 Optimization results of the hydraulically tribution of the natural fractures can be obtained by geo- fractured horizontal well physical method, such as imaging well logging. Usually, the properties of the natural fractures follow a certain random When the shale gas production simulation model is estab- lished, the EnOpt proposed in this research can be used to optimize the hydraulic fracturing parameters. The objective Table 1 Values of the reservoir properties in the shale reservoir is to maximize the NPV shown in Eq. (9) by optimizing the parameters of well drilling, well completion and reservoir Reservoir properties Value operator simultaneously. Traditionally, the optimization pro- Reservoir dimension, m × m × m 1000 × 580 × 30 cess is conducted in a sequential way where these parameters Buried depth, m 2000 Temperature, °C 85 Rock density, kg/m 2.579 Table 2 Values to generate random discrete natural fracture param- Initial pressure, MPa 40 eter Initial gas saturation, % 85 Natural fracture Generation parameter Distribution Matrix permeability, mD 0.0005 property Matrix porosity, % 6.0 3 −3 −6 Langmuir volume, m /kg 0.0035 Aperture, m N (4.0 × 10, 10 ) Gaussian Langmuir pressure, MPa 3 Azimuth, degree N (45, 400) Gaussian Planned production time, year 10 Length, m U (40, 150) Uniform 1 3 846 Petroleum Science (2021) 18:839–851 Permeability, mD 0.0001 0.0117 0.0184 0.0251 0.0318 0.0385 (a) Discrete natural fractures in shale gas reservoir (b) Converted permeability distribution Fig. 4 Conversion of discrete fracture model to equivalent permeability model realizations can be sampled from any prescribed distribution Table 3 Values of hydraulic fracture properties to avoid Gaussian assumption, but the updating process only Hydraulic fracture property Value uses the first moment (i.e., mean) and second moment (i.e., covariance) in the EnOpt method. If the distribution of the Stage number 11 horizontal well parameter is far from the Gaussian distribu- Half-length of fractures, m 160 tion, the updating or the optimizing performance will be Fracture conductivity, mD m 400 deteriorated step by step. In the EDFM simulation method, the horizontal well is characterized by each discrete grid block in the background matrix grid system. The Hough transformation-based parameterization method here is used to transform the discrete horizontal well grid block to the continuous Hough parameter space. The location of the hori- zontal well can be defined by the angle and radius (i.e.,  and ) in the Hough space. The parameters after Hough trans- formation distribution are more continuous and smoother, and thus the Hough transformation-based parameterization is beneficial to improve the performance of the EnOpt when they are combined together. During the optimization process, the parameters to char- Permeability, mD 0.00010.01170.01840.02510.0318 0.0385 acterize the horizontal well with Hough transformation- based parameterization are updated through the optimiza- tion method. Once the horizontal well parameter updating is Fig. 5 Shale gas reservoir model with fractured horizontal well done and a shale gas simulation process is required to evalu- ate the NPV, the horizontal well parameters in the Hough space are transformed back to its original space by: are optimized one by one independently. This method can- not guarantee to find the optimal design scheme. Due to x =  cos  + D − sin the ensemble feature of our proposed method and the joint � (24) optimization paradigm, the obtained solution is expected to be optimal globally. Before starting the optimization process, we need to pre- y =  sin  − D − cos (25) pare the parameters that will be optimized. The parameters to characterize the horizontal well placement are the param- eters (, , D, L) shown in Eq. (20). The well location is a x � =  cos  + D + sin (26) crucial parameter that controls the well performance of the shale gas development. The EnOpt method requires a Gauss- ian updating during the optimization process. The initial 1 3 Petroleum Science (2021) 18:839–851 847 L iterations. It can be seen that the design of the fractured y =  sin  − D + cos � (27) 2 horizontal well has been changed dramatically. Figure 6a visualizes the initial design of fractured horizontal well with With these two endpoints, the horizontal well can be con- the design parameters shown in the Table 4. Under the ini- structed into the matrix grid system by setting the grid tial condition, the associated cumulative shale gas produc- blocks that intersect with the straight line between these two 7 3 tion is 4.57 × 10 m and the NPV value is 34.84 million endpoints as horizontal well. CNY. Figure 6b shows the optimization results after 4 itera- The parameters to characterize the hydraulic fracturing tions. The parameters (, ,D) characterizing the horizontal design are the number of fracturing stage hf , the half- stage well location have been changed to 315.16 m, 89.01°, and length of the hydraulic fractures hf , and the conductivity xf 562.69 m. Due to the existence of the natural fractures, the of hydraulic fractures hf . The operational parameter con- condc distribution of the natural distribution makes the shale reser- sidered in the shale gas development process is the bottom voir anisotropic and the adjustment of the well location is to hole flowing pressure p . Augmenting all these parameters wf improve the well productivity. The length of the horizontal together, the optimization parameter set is: well, L, has been increased to 676.45 m. The number of the hydraulic fracture stages has been slightly decreased to , , D, L, hf , hf , hf , p (28) stage xf condc wf 10, but the half-length of the fracture has been increased to 184 m and the conductivity of the hydraulic fractures has To initialize the ensembles of the optimization parameter set, 50 initial ensemble realizations are generated by sam- been increased to 427.42 mD m. These parameter changes show that the optimization algorithm tends to find the most pling the parameters in the synthetic shale gas production model with a uniformly distributed random perturbation. economical scenario to design the fractured horizontal well and improve the productivity. The bottom hole flow - The parameter values used in the synthetic model and the optimization process are listed in Table 4. ing pressure value has been decreased to 25.7 MPa. After four optimization iterations, the corresponding cumulative During the optimization process, the EDFM-based shale 7 3 gas production model is used to compute the 10-year cumu- shale gas production is 6.46 × 10  m and the NPV value is 61.52 million CNY. Figure  6c shows the optimization lative shale gas production. The computation is conducted for each ensemble member in the generated initial parameter results after 11 iterations. The parameters (, ,D) charac- terizing the horizontal well location have been changed to set, and the NPV is evaluated with considering the shale gas production revenue and the cost of the well drilling and 353.56 m, 85.79º, and 586.57 m. The number of the hydrau- lic fracture stages remains the same as 10. The length of completion. The covariance between the prior optimization parameters themselves,  , and the covariance between the the horizontal well, and the half-length and the conductivity xx of the hydraulic fractures have been increased to 702.93 m, prior optimization parameters and the NPV values,  , are xg computed by Eqs. (11) and (14), respectively. Then the opti- 214 m, and 522.4 mD m. The bottom hole flowing pressure has been decreased to 21.6 MPa. After 11 optimization itera- mization parameters in the initial ensembles can be updated by Eq. (16). The stopping criterion is that the increment rate tions, the corresponding cumulative shale gas production is 7 3 8.02 × 10  m and the NPV value is 89.69 million CNY. Fig- of the NPV in two consecutive iterations is < 1%. Figure 6 depicts the optimization process of the design ure 6d shows the optimization results after 18 iterations. The parameters (, ,D) characterizing the horizontal well loca- parameter for fractured horizontal well in the shale gas res- ervoir with natural fractures. Here, the iteration step is 33, tion have been changed to 343.86 m, 85.68º, and 602.21 m. The number of the hydraulic fracture stages remains the which means that the optimization process stops after 33 same as 10. The length of the horizontal well, and the half-length and the conductivity of the hydraulic fractures Table 4 Values of design parameters of fractured horizontal well have been increased to 743.02 m, 219 m, and 542.7 mD m. The bottom hole flowing pressure has been decreased to Optimization parameter Value in syn- Parameter perturbation 21.02  MPa. After 18 optimization iterations, the corre- thetic model range in initialization 7 3 sponding cumulative shale gas production is 8.32 × 10  m ρ, m 290 100–600 and the NPV value is 95.87 million CNY. Figure 6e shows θ, ° 90 0–180 the optimization results after 25 iterations. The parameters D, m 500 200–800 (, ,D) characterizing the horizontal well location have L, m 620 250–900 been changed to 338.93 m, 85.18°, and 594.07 m. The length hf 11 5–18 stage of the horizontal well, and the stage number, half-length hf , m 160 80–230 xf and the conductivity of the hydraulic fractures have been hf , mD m 400 100–800 condc increased to 782.7 m, 13, 221 m, and 586.7 mD m. The bot- p , MPa 30 19–35 wf tom hole flowing pressure has been decreased to 19.22 MPa. 1 3 848 Petroleum Science (2021) 18:839–851 (a) Initial optimization (b) 4th optimization (c) 11st optimization (d) 18th optimization (e) 25th optimization (f) 33rd optimization Fig. 6 Optimization process of the hydraulically fractured horizontal well After 25 optimization iterations, the corresponding cumula- half-length and the conductivity of the hydraulic fractures 7 3 tive shale gas production is 9.42 × 10  m and the NPV value have been increased, and the bottom hole flowing pressure is 115.87 million CNY. Figure 6f shows the optimization has been decreased. All the adjustments are conducive to results after 33 iterations. The parameters (, ,D) charac- the improvement of well productivity. In the meanwhile, the terizing the horizontal well location have been changed to associated location adjustment of fractured horizontal well 354.66 m, 78.36º, and 528.15 m. The length of the hori- tends to maximize the connections with the natural fractures zontal well, and the stage number, half-length and the con- so that they can form an effective fracture network. It creates ductivity of the hydraulic fractures have been increased to an advantageous condition for the shale gas to flow from the 817.2 m, 17, 229 m, and 621.4 mD m. The bottom hole matrix to the wellbore. flowing pressure has been decreased to 19.02 MPa. After The objective of the optimization is to maximize the NPV 33 optimization iterations, the corresponding cumulative value through the obtained 10-year shale gas cumulative 7 3 shale gas production is 10.01 × 10  m and the NPV value is production from the simulation model with a given frac- 143.42 million CNY. tured horizontal design. Figure 7 shows the changes of the During the entire optimization process, it can be seen cumulative production of shale gas and the NPV during the that the length of the horizontal well, and the stage number, entire optimization process. Both the shale gas production 1 3 Petroleum Science (2021) 18:839–851 849 10 150 8 120 6 90 4 60 2 30 0 0 05 10 15 20 25 30 05 10 15 20 25 30 Iteration Iteration (a) Cumulative production change (b) NPV value change Fig. 7 Optimization process of the hydraulically fractured horizontal well and the NPV values are increasing in the optimization pro- shale gas reservoir are optimized simultaneously to achieve cess. The cumulative production changes from its initial a global optimal performance. The horizontal well is param- 7 3 7 3 value 4.57 × 10  m to 10.01 × 10  m after optimization, eterized through Hough transform method, which improves and the NPV value changes from its initial value 34.84 mil- the continuity of the parameters and the Gaussianity required lion CNY to 143.42 million CNY after optimization. Since by the ensemble optimization algorithm. The optimization the objective function is NPV, it can be found that the NPV results show that the ensemble optimization algorithm is has been increased to 4.1 times more than its original value effective, and the NPV can be greatly increased to approach after optimization. to its maximal value after optimization. Acknowledgements This work is funded by the National Science and Technology Major Project of China (Grant Nos. 2016ZX05037003-003 4 Conclusions and 2017ZX05032004-002), PetroChina Innovation Foundation (Grant No. 2020D-5007-0203), the National Natural Science Foundation of An ensemble-based optimization method is proposed to cope China (Grant No. 51374222), the Sinopec fundamental perspective research project (Grant No. P18086-5) Joint Funds of the National Nat- with the optimization design problem of the hydraulically ural Science Foundation of China (U19B6003-02-05), and supported fractured horizontal well in the shale gas reservoir. When by Science Foundation of China University of Petroleum, Beijing (Nos. building the simulation model of shale gas reservoir, the 2462018QZDX13 and 2462020YXZZ028). effects of gas–water two phase flow, gas absorption/desorp- tion, and Knudsen diffusion in the shale matrix are consid- Open Access This article is licensed under a Creative Commons Attri- bution 4.0 International License, which permits use, sharing, adapta- ered. The equivalent permeability tensor method is used to tion, distribution and reproduction in any medium or format, as long consider the effect of the large number of developed natural as you give appropriate credit to the original author(s) and the source, fractures, which greatly improves the calculation efficiency. provide a link to the Creative Commons licence, and indicate if changes The embedded discrete fracture (EDFM) is used to explicitly were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated account for the geometry of the hydraulic fractures so that otherwise in a credit line to the material. If material is not included in the fracture flow can be better characterized. Based on the the article’s Creative Commons licence and your intended use is not steepest ascent method, the ensemble optimization method permitted by statutory regulation or exceeds the permitted use, you will is used to approximate the update gradient by covariance, 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/. which greatly reduces the difficulty of obtaining the update gradient. 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Ensemble-based optimization of hydraulically fractured horizontal well placement in shale gas reservoir through Hough transform parameterization

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Abstract

Shale gas reservoirs have been successfully developed due to the advancement of the horizontal well drilling and multistage hydraulic fracturing techniques. However, the optimization design of the horizontal well drilling, hydraulic fracturing, and operational schedule is a challenging problem. An ensemble-based optimization method (EnOpt) is proposed here to optimize the design of the hydraulically fractured horizontal well in the shale gas reservoir. The objective is to maximize the net present value (NPV) which requires a simulation model to predict the cumulative shale gas production. To accurately describe the geometry of the hydraulic fractures, the embedded discrete fracture modeling method (EDFM) is used to construct the shale gas simulation model. The effects of gas absorption, Knudsen diffusion, natural and hydraulic fractures, and gas–water two phase flow are considered in the shale gas production system. To improve the parameter continuity and Gaussianity required by the EnOpt method, the Hough transformation parameterization is used to characterize the horizontal well. The results show that the proposed method can effectively optimize the design parameters of the hydraulically fractured horizontal well, and the NPV can be improved greatly after optimization so that the design parameters can approach to their optimal values. Keywords Shale gas · Ensemble optimization · Embedded discrete fracture model · Hough transformation 1 Introduction Shale gas development has received great attention in recent years because of the increasing demand for natural gas resources and its huge reserves worldwide (Dong et al. 2012, Edited by Yan-Hua Sun 2014). The matrix permeability of the shale gas reservoir is ultra-low due to the nanoscale pore size (Javadpour et al., * Liang Xue 2007; Liu et al. 2019a). With the advancement of horizontal xueliang@cup.edu.cn well drilling and hydraulic fracturing techniques, the natural gas resources in shale reservoir can be successfully unlocked State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum (Beijing), in a cost-effective manner. Beijing 102249, China Shale gas is considered as one of the unconventional Department of Oil-Gas Field Development Engineering, energy resources, and many special physical mechanisms College of Petroleum Engineering, China University are associated with the gas flow in shale reservoirs. The of Petroleum (Beijing), Beijing 102249, China natural gas in the shale gas reservoir can mainly exist in two State Key Laboratory of Shale Oil and Gas Enrichment forms, i.e., free gas and adsorbed gas (Curtis 2002). The Mechanisms and Effective Development, Beijing 100083, adsorbed gas will be desorbed into the matrix pores from China the organic matters when the reservoir pressure decreases Sinopec Petroleum Exploration and Production Research during the production process. In the shale matrix, the non- Institute, Beijing 100083, China Darcy flow can occur due to the nanoscale pore size, and School of Water Conservancy, North China University the Knudsen diffusion has been used to account for the of Water Resources and Electric Power, Zhengzhou 450046, effect of the molecular scale gas flow (Javadpour 2009; Xue Henan, China Vol.:(0123456789) 1 3 840 Petroleum Science (2021) 18:839–851 et al. 2019; Zhang et al. 2020). In the fracture system of the engineering. Brouwer and Jansen (2004) used a systematic shale gas reservoir, the natural fractures and the hydraulic dynamic optimization approach to control the valve setting fractures together form a complex fracture network, and the in the injector and producers for a water flooding project in shale gas will flow through these fractures with various sizes the heterogeneous reservoir. Sarma and Chen (2008) and into the horizontal well (Cipolla et al. 2010; Clarkson et al. Zhang et al. (2010) converted the discrete control variables 2011). The multistage hydraulic fracturing will create mul- to continuous ones and used the adjoints and gradient- tiple transverse fractures along the lateral direction of the based optimization method to determine the optimal the horizontal well in relatively intact reservoirs, or generate well locations. Volkov and Bellout (2018) approximated the complex fracture networks connected to the horizontal well well placement optimization gradient by using finite differ - in fractured reservoirs (Li et al. 2016, 2017; Li and Zhang ence approximations of augmented Lagrangian derivatives 2018; Liu et  al. 2019b). To improve the ultimate recov- with adjoint formulation to improve efficiency. Al Dossary ery of the shale gas in a cost-effective manner, the design and Nasrabadi (2016) proposed an imperialist competitive of the hydraulically fractured horizontal well needs to be algorithm (ICA) to optimize the well location. Hamida optimized. et  al. (2017) modified the traditional genetic algorithm Fractures in the hydraulically fractured shale gas reser- with a similarity operator to optimize the well placement. voir increase the contact surface between shale matrix and Jesmani et al. (2015) used the derivative-free particle swarm fractured horizontal well and form the primary flow path optimization algorithm to optimize the well location with for shale gas flow. Reservoir numerical simulation and data- well design constrains. Tukur et al. (2019) implemented the driven methods can forecast the shale gas production and aid genetic algorithm and simulated annealing methods to opti- the optimization of the fractured horizontal well (Yu et al. mize the well placement. In the optimization of a hydrauli- 2018; Xue et al. 2020, 2021). To take the fractured flow cally fractured horizontal well in a shale reservoir, Ma et al. into consideration, multi-continuum models, such as the dual (2013) optimized the hydraulic fracture placement with the porosity model (Warren and Root 1963; Kazemi 1969; Ros- gradient-based finite difference method (FD), discrete simul- tami et al. 2020) and multi-porosity model (Wu and Pruess taneous perturbation stochastic approximation (DSPSA), and 1988), have been used for shale gas reservoir simulation. genetic algorithm (GA). Yu and Sepehrnoori (2013) used the Zhang et al. (2009) conducted the sensitivity analysis of the response surface method to conduct the optimization of the hydraulic fracture parameters in the shale gas reservoir by multiple hydraulically fractured horizontal wells in shale upscaling the discrete natural fractures to dual porosity sys- gas reservoirs. Wilson and Durlofsky (2013) introduced tem. Cipolla et al. (2010) simulated the shale gas production a direct search algorithm to optimize the shale gas field by using the dual porosity model with the consideration of development by combining with reduced physics model to gas adsorption effect. Rubin (2010) used local grid refine- improve computational efficiency. Rammay and Awotunde ment (LGR) method with dual porosity model to improve (2016) used differential evolution algorithm to optimize the the characterization of the hydraulic fractures. Essentially, hydraulic fracturing parameters and horizontal well length. the multi-continuum model represents the fractures by using Zhang and Sheng (2020) optimized the fractured horizontal an averaged property in the grid block. The discrete frac- well in the shale gas reservoir by considering the effect the ture modeling method can characterize the fracture geometry stimulated reservoir volume. The ensemble optimization explicitly (Karimi-Fard et al. 2004; Gong et al. 2008), but method (EnOpt) is powerful optimization method proposed the unstructured grid has to be used and the computational by Chen et al. (2009). It is a stochastic gradient optimiza- burden can be too heavy to be implemented in the field- tion method, which approximates the gradient through the scale application. The embedded discrete fracture modeling ensemble computation, and thus it is capable of integrating method (EDFM) was proposed by Li and Lee (2008) and with any simulator. In addition, compared to other gradient- improved by Moinfar et al. (2014), which can represent the based optimization method (e.g., finite difference method geometry of each fracture explicitly but with structure grid and adjoint method) and gradient-free optimization method system, and this method has been applied to simulate the (e.g., genetic algorithm and particle swarm method), it is shale gas production (Dai et al. 2017; Yu et al. 2018). Due more efficient when the dimension of the control variables to the uncertainty associated with the geological condition is high. Leeuwenburgh et al. (2010) applied this method to including the conceptual model uncertainty (Xue and Zhang optimize the settings of inflow control valve in a water flood- 2014; Xue et al. 2015), the production data can be used to ing project and compared its performance with the adjoint reduce the predictive uncertainty and obtain more accurate method. However, this method has not been fully investi- prediction results (Dachanuwattana et al. 2018; Kang et al. gated in the optimization problem of hydraulically fractured 2020; Xue et al. 2020). horizontal well in shale gas reservoir. Many optimization methods have been investigated In this research, we propose to use ensemble optimi- to deal with the well placement problem in reservoir zation method to maximize the net present value (NPV) 1 3 Petroleum Science (2021) 18:839–851 841 in the shale gas development project. The EDFM is used f f f f k k g rg to simulate the shale gas production. The optimization f well mf (3) ∇ ⋅ ∇p − q + q = g g g parameters are the location of horizontal well drilled in a shale reservoir with considering the effect of the natu - f f where k is the absolute permeability of fractures, k is the ral fractures, the optimal hydraulic fracturing parameters rg (i.e., number of stages, length of the hydraulic fractures, relative permeability of shale gas in fractures, p is the gas well and conductivity of hydraulic fractures) and the opera- pressure in fractures, q is the mass flow rate of shale gas mf tional parameter (i.e., bottom hole pressure). Usually, to well, q is the mass flow rate of shale gas from matrix to these parameters are optimized independently step by f f fractures,  is the fracture porosity, S is the gas saturation step. However, in our research, all these parameters are in fractures, and  is a coefficient to account for non-Darcy optimized simultaneously so that the most optimal shale flow. gas development plan can be designed. When optimizing The water flow equation in the shale matrix is: the horizontal well location, the coordinates of the well are m m m m discrete variables. Here, we further propose to use Hough k k   S w rw w m mf ∇ ⋅ ∇p − q = (4) transformation-based parameterization to transform the w w Cartesian space into Hough space. This transformation can improve the Gaussianity of the parameter so that it where  is the water density, k is the apparent permeability can be better integrated with EnOpt method. of water in the shale matrix, k is the relative permeability rw of water in the shale matrix,  is the water viscosity, p is w w mf the water pressure in the shale matrix, q is the mass flow rate from matrix to fracture, and S is the water saturation. 2 Methodology The water flow equation in the fracture is: f f f f k k   S 2.1 Shale gas flow simulation w rw w f well mf ∇ ⋅ ∇p − q + q = (5) w w w The EDFM is used to solve the simulation of shale gas f well production. The detailed equations to characterize the mul- where p is the pressure of water phase in fractures and q w w tiscale shale gas flow have been presented in Xue et al. is the mass flow rate of water to well. (2020), and the design of the simulation software was In the EDFM, the mass flow rate from matrix to fractures, mf introduced in Li et al. (2015). The brief governing equa- q , and two fracture segments connections in the same tions of the gas–water two phase flow model of shale gas matrix block or in two adjacent matrix blocks can be com- production through EDFM method are introduced here. puted through the non-neighboring connection (NNC) The gas flow equation in shale matrix is: (Moinfar et al. 2014): m m m m S  m f nnc m f k k N N g nnc nnc k p − p g a rg  k k p − p  rg g g rg m s mf j g g (1) mf nnc nnc nnc ∇ ⋅ ∇p + q − q = q = q = A = T g c g g g,c j nnc j g g g j=1 j j=1 (6) where  is the density of the shale gas, k is the apparent g a where p is the gas pressure in fractures, N is the number nnc permeability of shale matrix, k is the relative permeability rg of the non-neighboring connection grids, T is the NNC nnc of gas in the shale matrix, is the gas viscosity,  is the nnc transmissibility factor, A is the contact area of fracture and nnc matrix porosity, S is the gas saturation in the shale matrix, g matrix, k is the harmonic mean of matrix and fracture nnc q is the desorption/adsorption mass flow rate, which can be permeability, and d is the characteristics distance between characterized by Langmuir isothermal adsorption model, matrix block and fracture plane. mf and q is the mass flow rate from matrix to fractures. Together with the NNC between matrix and facture in To account for the Knudsen diffusion, the apparent per - EDFM mentioned above, two more NNCs need to be con- meability k can be evaluated by (Tang et al. 2005): sidered, i.e., the two fracture segments connections in the same matrix block and two adjacent matrix blocks. The m m 2 k = k 1 + 8C Kn + 16C Kn (2) 1 2 a ∞ NNC transmissibility, T , can be written as: nnc nnc nnc where k is the absolute permeability, C and C are con- T T 1 2 k A ∞ 1 2 T = = (7) nnc stant coefficients, and Kn is the Knudsen number. nnc d T + T 1 2 The shale gas flow equation in fractures is: 1 3 842 Petroleum Science (2021) 18:839–851 with where  is the updated control variable,  is the control k+1 k variable before updating,  is the coefficient determining the k w L f1 f1 int updating step size,  is the prior covariance matrix of the T = , xx f1 control variables, and  is the sensitivity of NPV function k w L f2 f2 int g(x) to the control variables. T = The prior covariance matrix of the control variables f2 xx is: where L is the length of the fracture intersection, k is the int f fracture permeability, w is the fracture aperture, and d is the f f (11) xx N − 1 normal distance between the center of the fracture and the fractures intersection. The developed EDFM model has been where N is the ensemble size. validated against the commercial software CMG to show its The control variable matrix  can be expressed as: accuracy (Dai et al. 2017). x − x … x − x ⎛ ⎞ 1,1 1 N ,1 N x x 2.2 Ensemble optimization method ⎜ ⎟ = ⋮⋱ ⋮ (12) ⎜ ⎟ x − x … x − x ⎝ ⎠ 1,N 1 N N N e x e x The ensemble optimization method was proposed by Chen 1 N et al. (2009), and it has been applied in several production where x = x is the ensemble mean of the control i ij j=1 optimization problems (Chen and Oliver 2010; Fonseca et al. variables. 2014; Tueros et al. 2018). A vector of the control variable is The product term   can be approximated by xx defined in the EnOpt method, which contains all the variables that need to be optimized. The control variable vector can be (13) xx xg defined as: where  is the covariance between X  and G. It can be xg = x , x , x , … , x (8) 1 2 3 N expressed as: where N is the number of control variables. 1 (14) xg The NPV is used as the objective function during the opti- N − 1 mization process here, and the formula is written as: And the NPV vector G can be expressed as: Q(i)P g(x)= − w P − 2hf hf P (9)  = g x − g, g x − g, … , g x )− g (15) L hw xf stage hf 1 2 N (1 + r) i=1 � � where g = g x is the ensemble mean of the NPV where i is the time step index, N is the total production time, t N l=1 Q(i) is the cumulative shale gas production within the given values. time step which can be computed from the above EDFM- In the EnOpt method, we can substitute Eq. (13) in Eq. based shale gas production model, P is the price of shale (10) and use  as the filtering matrix. Then the optimiza- xx gas which is set as 3 CNY/m , r is the discount rate which is tion parameters are updated by set as 6%, w is the length of horizontal well, P is the drill- L hw ing cost per unit well length which is set as 30,000 CNY/m, = x + k+1 k xx xg (16) hf is the half-length of the hydraulic fractures, hf is the k xf stage stage number of the hydraulic fractures, and P is the well hf complete cost per unit fracture length, which is set as 20,000 CNY/m. All these parameter setting values are inferred from 2.3 Hough transform parameterization a laboratory report to describe the shale gas development in for horizontal well Zhaotong shale gas field in China. During the optimization process, the NPV is maximized by optimizing the control Hough transformation was proposed by Hough (1962), which variables. was used to detect lines. Duda and Hart (1972) extended In each optimization step, the control variable is updated Hough transformation to detect any arbitrary objects in the through image analysis. Hough transform converts the parameters from the Cartesian coordinate space to the Hough space. Hough =  + space is basically an accumulator space, and it uses the vot- k+1 k xx (10) ing method in the accumulation space to find the local maxi- mum value to conduct feature detection. The accumulator to 1 3 Petroleum Science (2021) 18:839–851 843 transform x–y Cartesian coordinate system space into ρ–θ 0 if t < a 𝛿 (u − a)du = (18) polar coordinate system of Hough space can be expressed as: 1 if t > a −∞ +∞ +∞ As shown in Fig.  1, the straight line y =−x + 5 in the A(, )= I(x, y)( − x cos  − y sin )dxdy −∞ −∞ Cartesian coordinate system can be transformed into a point � � � (17) in the Hough space π∕4, 5 2 2 . where A(, ) is recording the how many sinusoidal curves The accumulator in the Hough space provides the infor- actually pass through the (, ) point in the Hough space, mation on the total number of sinusoidal curves that pass I(x, y) is the point in the Cartesian coordinate space. (⋅) is through the point (, ) , and all the grid blocks along the the Dirac delta function, and it can be defined as: line segment, characterized by (, ) in Hough space, will contribute to the accumulator. The Hough transform has the capability to transform any line to the parameter set that y ρ can take a better continuity than the discrete point in the Cartesian coordinate (Lu and Zhang 2015; Yao et al. 2018); therefore, it can be used to represent a line segment in the 4 Cartesian coordinate space. By using this method, the hori- y = -x + 5 zontal well is not characterized by its endpoint coordinates x , y , z and x , y , z , but it is represented by a parameter 1 1 1 2 2 2 set (as shown in Fig. 2): (, , D, L, ,  , ) (19) x 01 (a) Cartesian coordinate space (b) Hough space where  is the vertical line distance between the origin to line segment in the x–y projection plane;  is the angle between Fig. 1 Transformation of a straight line in Cartesian space to Hough x-axis and the vertical line in the x–y projection plane; D is space A″′ B″′ A″ B″ Fig. 2 Parameterization of a horizontal well in the Hough space 1 3 B′ A′ L 844 Petroleum Science (2021) 18:839–851 the distance between the vertical line and the center point of e the line segment in the x–y projection plane; L is the length North e of the line segment in the x–y projection plane;  is the angle between x-axis and the line segment in the x–z projection plane; and  is the angle between z-axis and the line segment in the y–z projection plane;  is the angle between y-axis and the line segment in the x–y projection plane. The horizontal well drilled in the shale gas reservoir, which can be regarded as the line segment. When defining West East the horizontal well with the Hough transformation-based α parameterization method, the horizontal well is within the x–y plane in the coordinate system, and  = 0 and  = 0 in this case. In addition, it can be seen from the triangu- lar relationship that  =  . Therefore, the horizontal well South can be represented by the parameter set in the optimization problem: (, , D, L) (20) Fig. 3 Characterization of the fracture plane (Song et al. 2019) With consideration of the shale matrix permeability, the 2.4 Equivalent permeability conversion of natural final permeability tensor can be computed by: fractures K + K K K ⎛ ⎞ eN11 m eN12 eN13 In the shale gas reservoir, there may exist a large set of natu- ⎜ ⎟ K K + K K (23) eN21 eN22 m eN23 ral fractures, which results in the shale gas reservoir with ⎜ ⎟ ⎝ K K K + K ⎠ severe heterogeneous permeability distribution (Khanal eN31 eN32 eN33 m and Weijermars 2019). The hydraulic fractures can be rep- resented explicitly with its full geometrical properties in the EDFM method. However, the number of the natural fractures is very large, which is infeasible to obtain the information on 3 Results and discussion each natural fracture. Even we can know the exact distribu- tion of the natural fractures, the simulation model can be too In this research, the EnOpt method based on Hough trans- time-consuming to be used in practice. Here, we establish form is used to optimize the economic benefits of shale gas a method to convert the discrete natural fractures to their reservoir produced by fractured horizontal wells. The loca- equivalent permeability. Let us define the azimuth angle of tion parameters of fractured horizontal wells are transformed the fracture is  , the dip angle of the fracture is  , and per- into Hough space, and the EnOpt algorithm is used for the meability parallel to the fracture is K (as shown in Fig. 3). integrated optimization of the design parameters. The Hough The permeability tensor of each fractures can be com- transformation-based parameterization can be suitable to puted by (Song et al. 2019): improve the continuity of the discrete design parameters, 2 2 2 2 ⎛ cos  ⋅ cos  + sin  sin  ⋅ cos  ⋅ sin  cos  ⋅ sin  ⋅ cos  ⎞ i i i i i i i i i 2 2 2 2 ⎜ ⎟ (21) =  sin  ⋅ cos  ⋅ sin  cos  ⋅ sin  + cos  − cos  ⋅ sin  ⋅ sin ei i i i i i i i i i i ⎜ ⎟ cos  ⋅ sin  ⋅ cos − cos  ⋅ sin  ⋅ sin  sin ⎝ ⎠ i i i i i i i When a number of N fractures exist in a single grid block, such as the location or central point coordinate of the hori- the permeability tensor can be expressed by: zontal well, and can be helpful to the linearization of the 2 2 2 2 N N cos  ⋅ cos  + sin  sin  ⋅ cos  ⋅ sin  cos  ⋅ sin  ⋅ cos ⎛ ⎞ i i i i i i i i i � � 2 2 2 2 ⎜ ⎟ (22) =  =  sin  ⋅ cos  ⋅ sin  cos  ⋅ sin  + cos  − cos  ⋅ sin  ⋅ sin eN ei i i i i i i i i i i ⎜ ⎟ i=1 i=1 cos  ⋅ sin  ⋅ cos  − cos  ⋅ sin  ⋅ sin  sin ⎝ ⎠ i i i i i i i 1 3 Fracture Petroleum Science (2021) 18:839–851 845 nonlinear parameter in the set optimization algorithm. The distribution. For example, the natural fracture apertures transformed parameters can improve the Gaussian assump- follow a Gaussian distribution (characterized by the mean tion of the EnOpt process and thus provide a more reason- and variance in the generation function), the lengths of the able optimization performance. natural fractures follow a uniform distribution (character- ized by the minimal and maximal values in the generation 3.1 Synthetic model construction of the shale gas function), and the azimuth follows a Gaussian distribution. reservoir The random discrete natural fracture model is generated by using the parameter setting listed in Table  2. The central −8 To demonstrate the workflow and performance of the pro- point density of the natural fractures are set as 4.0 × 10 / posed EDFM-EnOpt method, a synthetic model of shale gas m . According to the relationship of the generated fracture reservoir simulation is constructed here. In the synthetic endpoints coordinate data and the computational grid, the model, a hydraulically fractured horizontal well is located fracture endpoint coordinate information is converted into in a 3D shale gas reservoir. The shale gas simulation model computational grid property, i.e., the equivalent grid block is constructed using EDFM method. During the simulation permeability. Then, the shale gas reservoir geological model process, it considers the special gas flow mechanisms in the with natural fractures can be constructed as shown in Fig. 4. shale gas reservoir. The simulation model is constructed For the hydraulic fractures, the number of hydraulic frac- based on gas–water two-phase flow model. In the shale tures is small and the fracture conductivity is large. They matrix flow, it considers the adsorption/desorption of the directly connect with the horizontal wellbore and the shale shale gas on the organic content in the shale reservoir and matrix, which form the primary flow path for shale gas and the Knudsen diffusion effect of the shale gas flow caused by has a great impact on the flow field. Therefore, the hydraulic the nanoscale pore structure. In the fracture flow, it consid- fractures should be accurately described by considering their ers the gas flow in natural fractures by upscaling the natural explicit fracture geometry. In this paper, the hydraulic frac- fractures using equivalent permeability method and the gas tures are established by EDFM, and the hydraulic fractures flow in hydraulic fractures by explicitly taking the fracture are transverse fractures under the reservoir condition, that is, geometry into account. the hydraulic fractures are rectangular plates perpendicular The parameter values used in the synthetic model to char- to the horizontal stratum. The parameter values are listed in acterize the shale gas reservoir properties are summarized Table 3, which are used as the initial design of the hydraulic in Table 1. These data are collected from a laboratory report fracturing project before optimization. to study the Zhaotong shale gas field in China and represent Once combining the properties of shale matrix, natural the state of the shale gas reservoir before the well drilling fractures and hydraulic fractures, the reservoir model for and gas production. the shale gas production simulation can be generated, as A large number of natural fractures can be developed shown in Fig. 5. in the shale gas reservoir. It is hard to obtain the fracture geometry for each natural fracture, but the statistical dis- 3.2 Optimization results of the hydraulically tribution of the natural fractures can be obtained by geo- fractured horizontal well physical method, such as imaging well logging. Usually, the properties of the natural fractures follow a certain random When the shale gas production simulation model is estab- lished, the EnOpt proposed in this research can be used to optimize the hydraulic fracturing parameters. The objective Table 1 Values of the reservoir properties in the shale reservoir is to maximize the NPV shown in Eq. (9) by optimizing the parameters of well drilling, well completion and reservoir Reservoir properties Value operator simultaneously. Traditionally, the optimization pro- Reservoir dimension, m × m × m 1000 × 580 × 30 cess is conducted in a sequential way where these parameters Buried depth, m 2000 Temperature, °C 85 Rock density, kg/m 2.579 Table 2 Values to generate random discrete natural fracture param- Initial pressure, MPa 40 eter Initial gas saturation, % 85 Natural fracture Generation parameter Distribution Matrix permeability, mD 0.0005 property Matrix porosity, % 6.0 3 −3 −6 Langmuir volume, m /kg 0.0035 Aperture, m N (4.0 × 10, 10 ) Gaussian Langmuir pressure, MPa 3 Azimuth, degree N (45, 400) Gaussian Planned production time, year 10 Length, m U (40, 150) Uniform 1 3 846 Petroleum Science (2021) 18:839–851 Permeability, mD 0.0001 0.0117 0.0184 0.0251 0.0318 0.0385 (a) Discrete natural fractures in shale gas reservoir (b) Converted permeability distribution Fig. 4 Conversion of discrete fracture model to equivalent permeability model realizations can be sampled from any prescribed distribution Table 3 Values of hydraulic fracture properties to avoid Gaussian assumption, but the updating process only Hydraulic fracture property Value uses the first moment (i.e., mean) and second moment (i.e., covariance) in the EnOpt method. If the distribution of the Stage number 11 horizontal well parameter is far from the Gaussian distribu- Half-length of fractures, m 160 tion, the updating or the optimizing performance will be Fracture conductivity, mD m 400 deteriorated step by step. In the EDFM simulation method, the horizontal well is characterized by each discrete grid block in the background matrix grid system. The Hough transformation-based parameterization method here is used to transform the discrete horizontal well grid block to the continuous Hough parameter space. The location of the hori- zontal well can be defined by the angle and radius (i.e.,  and ) in the Hough space. The parameters after Hough trans- formation distribution are more continuous and smoother, and thus the Hough transformation-based parameterization is beneficial to improve the performance of the EnOpt when they are combined together. During the optimization process, the parameters to char- Permeability, mD 0.00010.01170.01840.02510.0318 0.0385 acterize the horizontal well with Hough transformation- based parameterization are updated through the optimiza- tion method. Once the horizontal well parameter updating is Fig. 5 Shale gas reservoir model with fractured horizontal well done and a shale gas simulation process is required to evalu- ate the NPV, the horizontal well parameters in the Hough space are transformed back to its original space by: are optimized one by one independently. This method can- not guarantee to find the optimal design scheme. Due to x =  cos  + D − sin the ensemble feature of our proposed method and the joint � (24) optimization paradigm, the obtained solution is expected to be optimal globally. Before starting the optimization process, we need to pre- y =  sin  − D − cos (25) pare the parameters that will be optimized. The parameters to characterize the horizontal well placement are the param- eters (, , D, L) shown in Eq. (20). The well location is a x � =  cos  + D + sin (26) crucial parameter that controls the well performance of the shale gas development. The EnOpt method requires a Gauss- ian updating during the optimization process. The initial 1 3 Petroleum Science (2021) 18:839–851 847 L iterations. It can be seen that the design of the fractured y =  sin  − D + cos � (27) 2 horizontal well has been changed dramatically. Figure 6a visualizes the initial design of fractured horizontal well with With these two endpoints, the horizontal well can be con- the design parameters shown in the Table 4. Under the ini- structed into the matrix grid system by setting the grid tial condition, the associated cumulative shale gas produc- blocks that intersect with the straight line between these two 7 3 tion is 4.57 × 10 m and the NPV value is 34.84 million endpoints as horizontal well. CNY. Figure 6b shows the optimization results after 4 itera- The parameters to characterize the hydraulic fracturing tions. The parameters (, ,D) characterizing the horizontal design are the number of fracturing stage hf , the half- stage well location have been changed to 315.16 m, 89.01°, and length of the hydraulic fractures hf , and the conductivity xf 562.69 m. Due to the existence of the natural fractures, the of hydraulic fractures hf . The operational parameter con- condc distribution of the natural distribution makes the shale reser- sidered in the shale gas development process is the bottom voir anisotropic and the adjustment of the well location is to hole flowing pressure p . Augmenting all these parameters wf improve the well productivity. The length of the horizontal together, the optimization parameter set is: well, L, has been increased to 676.45 m. The number of the hydraulic fracture stages has been slightly decreased to , , D, L, hf , hf , hf , p (28) stage xf condc wf 10, but the half-length of the fracture has been increased to 184 m and the conductivity of the hydraulic fractures has To initialize the ensembles of the optimization parameter set, 50 initial ensemble realizations are generated by sam- been increased to 427.42 mD m. These parameter changes show that the optimization algorithm tends to find the most pling the parameters in the synthetic shale gas production model with a uniformly distributed random perturbation. economical scenario to design the fractured horizontal well and improve the productivity. The bottom hole flow - The parameter values used in the synthetic model and the optimization process are listed in Table 4. ing pressure value has been decreased to 25.7 MPa. After four optimization iterations, the corresponding cumulative During the optimization process, the EDFM-based shale 7 3 gas production model is used to compute the 10-year cumu- shale gas production is 6.46 × 10  m and the NPV value is 61.52 million CNY. Figure  6c shows the optimization lative shale gas production. The computation is conducted for each ensemble member in the generated initial parameter results after 11 iterations. The parameters (, ,D) charac- terizing the horizontal well location have been changed to set, and the NPV is evaluated with considering the shale gas production revenue and the cost of the well drilling and 353.56 m, 85.79º, and 586.57 m. The number of the hydrau- lic fracture stages remains the same as 10. The length of completion. The covariance between the prior optimization parameters themselves,  , and the covariance between the the horizontal well, and the half-length and the conductivity xx of the hydraulic fractures have been increased to 702.93 m, prior optimization parameters and the NPV values,  , are xg computed by Eqs. (11) and (14), respectively. Then the opti- 214 m, and 522.4 mD m. The bottom hole flowing pressure has been decreased to 21.6 MPa. After 11 optimization itera- mization parameters in the initial ensembles can be updated by Eq. (16). The stopping criterion is that the increment rate tions, the corresponding cumulative shale gas production is 7 3 8.02 × 10  m and the NPV value is 89.69 million CNY. Fig- of the NPV in two consecutive iterations is < 1%. Figure 6 depicts the optimization process of the design ure 6d shows the optimization results after 18 iterations. The parameters (, ,D) characterizing the horizontal well loca- parameter for fractured horizontal well in the shale gas res- ervoir with natural fractures. Here, the iteration step is 33, tion have been changed to 343.86 m, 85.68º, and 602.21 m. The number of the hydraulic fracture stages remains the which means that the optimization process stops after 33 same as 10. The length of the horizontal well, and the half-length and the conductivity of the hydraulic fractures Table 4 Values of design parameters of fractured horizontal well have been increased to 743.02 m, 219 m, and 542.7 mD m. The bottom hole flowing pressure has been decreased to Optimization parameter Value in syn- Parameter perturbation 21.02  MPa. After 18 optimization iterations, the corre- thetic model range in initialization 7 3 sponding cumulative shale gas production is 8.32 × 10  m ρ, m 290 100–600 and the NPV value is 95.87 million CNY. Figure 6e shows θ, ° 90 0–180 the optimization results after 25 iterations. The parameters D, m 500 200–800 (, ,D) characterizing the horizontal well location have L, m 620 250–900 been changed to 338.93 m, 85.18°, and 594.07 m. The length hf 11 5–18 stage of the horizontal well, and the stage number, half-length hf , m 160 80–230 xf and the conductivity of the hydraulic fractures have been hf , mD m 400 100–800 condc increased to 782.7 m, 13, 221 m, and 586.7 mD m. The bot- p , MPa 30 19–35 wf tom hole flowing pressure has been decreased to 19.22 MPa. 1 3 848 Petroleum Science (2021) 18:839–851 (a) Initial optimization (b) 4th optimization (c) 11st optimization (d) 18th optimization (e) 25th optimization (f) 33rd optimization Fig. 6 Optimization process of the hydraulically fractured horizontal well After 25 optimization iterations, the corresponding cumula- half-length and the conductivity of the hydraulic fractures 7 3 tive shale gas production is 9.42 × 10  m and the NPV value have been increased, and the bottom hole flowing pressure is 115.87 million CNY. Figure 6f shows the optimization has been decreased. All the adjustments are conducive to results after 33 iterations. The parameters (, ,D) charac- the improvement of well productivity. In the meanwhile, the terizing the horizontal well location have been changed to associated location adjustment of fractured horizontal well 354.66 m, 78.36º, and 528.15 m. The length of the hori- tends to maximize the connections with the natural fractures zontal well, and the stage number, half-length and the con- so that they can form an effective fracture network. It creates ductivity of the hydraulic fractures have been increased to an advantageous condition for the shale gas to flow from the 817.2 m, 17, 229 m, and 621.4 mD m. The bottom hole matrix to the wellbore. flowing pressure has been decreased to 19.02 MPa. After The objective of the optimization is to maximize the NPV 33 optimization iterations, the corresponding cumulative value through the obtained 10-year shale gas cumulative 7 3 shale gas production is 10.01 × 10  m and the NPV value is production from the simulation model with a given frac- 143.42 million CNY. tured horizontal design. Figure 7 shows the changes of the During the entire optimization process, it can be seen cumulative production of shale gas and the NPV during the that the length of the horizontal well, and the stage number, entire optimization process. Both the shale gas production 1 3 Petroleum Science (2021) 18:839–851 849 10 150 8 120 6 90 4 60 2 30 0 0 05 10 15 20 25 30 05 10 15 20 25 30 Iteration Iteration (a) Cumulative production change (b) NPV value change Fig. 7 Optimization process of the hydraulically fractured horizontal well and the NPV values are increasing in the optimization pro- shale gas reservoir are optimized simultaneously to achieve cess. The cumulative production changes from its initial a global optimal performance. The horizontal well is param- 7 3 7 3 value 4.57 × 10  m to 10.01 × 10  m after optimization, eterized through Hough transform method, which improves and the NPV value changes from its initial value 34.84 mil- the continuity of the parameters and the Gaussianity required lion CNY to 143.42 million CNY after optimization. Since by the ensemble optimization algorithm. The optimization the objective function is NPV, it can be found that the NPV results show that the ensemble optimization algorithm is has been increased to 4.1 times more than its original value effective, and the NPV can be greatly increased to approach after optimization. to its maximal value after optimization. Acknowledgements This work is funded by the National Science and Technology Major Project of China (Grant Nos. 2016ZX05037003-003 4 Conclusions and 2017ZX05032004-002), PetroChina Innovation Foundation (Grant No. 2020D-5007-0203), the National Natural Science Foundation of An ensemble-based optimization method is proposed to cope China (Grant No. 51374222), the Sinopec fundamental perspective research project (Grant No. P18086-5) Joint Funds of the National Nat- with the optimization design problem of the hydraulically ural Science Foundation of China (U19B6003-02-05), and supported fractured horizontal well in the shale gas reservoir. When by Science Foundation of China University of Petroleum, Beijing (Nos. building the simulation model of shale gas reservoir, the 2462018QZDX13 and 2462020YXZZ028). effects of gas–water two phase flow, gas absorption/desorp- tion, and Knudsen diffusion in the shale matrix are consid- Open Access This article is licensed under a Creative Commons Attri- bution 4.0 International License, which permits use, sharing, adapta- ered. The equivalent permeability tensor method is used to tion, distribution and reproduction in any medium or format, as long consider the effect of the large number of developed natural as you give appropriate credit to the original author(s) and the source, fractures, which greatly improves the calculation efficiency. provide a link to the Creative Commons licence, and indicate if changes The embedded discrete fracture (EDFM) is used to explicitly were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated account for the geometry of the hydraulic fractures so that otherwise in a credit line to the material. If material is not included in the fracture flow can be better characterized. Based on the the article’s Creative Commons licence and your intended use is not steepest ascent method, the ensemble optimization method permitted by statutory regulation or exceeds the permitted use, you will is used to approximate the update gradient by covariance, 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/. which greatly reduces the difficulty of obtaining the update gradient. 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Journal

Petroleum ScienceSpringer Journals

Published: Feb 24, 2021

Keywords: Shale gas; Ensemble optimization; Embedded discrete fracture model; Hough transformation

References