A Decomposed Fracture Network Model for Characterizing Well Performance of Fracture Networks on the Basis of an Approximated Flow Equation
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GeoScienceWorld Lithosphere Volume 2021, Article ID 5558746, 19 pages https://doi.org/10.2113/2021/5558746 Research Article A Decomposed Fracture Network Model for Characterizing Well Performance of Fracture Networks on the Basis of an Approximated Flow Equation Jiazheng Liu ,1 Xiaotong Liu ,2 Hongzhang Zhu,3 Xiaofei Ma ,3 Yuxue Zhang ,4 Jingying Zeng ,1 Wanjing Luo ,1 Bailu Teng ,1 Yang Li ,5 and Wan Nie 6 1 School of Energy Resources, China University of Geosciences (Beijing), Beijing 100083, China 2 Lushang Oilfield Operation Area, PetroChina Jidong Oilfield Company, Tangshan 063004, China 3 The Fifth Oil Production Plant, Changqing Oil Field, PetroChina, Xi’an 710000, China 4 Exploration & Production Research Institute, Southwest Oil & Gas Branch Company, Sinopec, Chengdu 610041, China 5 GWDC Drilling Engineering and Technology Research Institute, Panjin 124000, China 6 Liaohe Oilfield Safety and Environment Protection Technology Supervision Center, Panjin 124000, China Correspondence should be addressed to Bailu Teng; bailu@cugb.edu.cn Received 25 January 2021; Accepted 27 April 2021; Published 28 June 2021 Academic Editor: Zhongwei Wu Copyright © 2021 Jiazheng Liu et al. Exclusive Licensee GeoScienceWorld. Distributed under a Creative Commons Attribution License (CC BY 4.0). The gridless analytical and semianalytical methodologies can provide credible solutions for describing the well performance of the fracture networks in a homogeneous reservoir. Reservoir heterogeneity, however, is common in unconventional reservoirs, and the productivity can vary significantly along the horizontal wells drilled for producing such reservoirs. It is oversimplified to treat the entire reservoir matrix as homogeneous if there are regions with extremely nonuniform properties in the reservoir. However, the existing analytical and semianalytical methods can only model simple cases involving matrix heterogeneity, such as composite, layered, or compartmentalized reservoirs. A semianalytical methodology, which can model fracture networks in heterogeneous reservoirs, is still absent; in this study, we propose a decomposed fracture network model to fill this gap. We discretize a fractured reservoir into matrix blocks that are bounded by the fractures and/or the reservoir boundary and upscale the local properties to these blocks; therefore, a heterogeneous reservoir can be represented with these blocks that have nonuniform properties. To obtain a general flow equation to characterize the transient flow in the blocks that may exhibit different geometries, we approximate the contours of pressure with the contours of the depth of investigation (DOI) in each block. Additionally, the borders of each matrix block represent the fractures in the reservoir; thus, we can characterize the configurations of complex fracture networks by assembling all the borders of the matrix blocks. This proposed model is validated against a commercial software (Eclipse) on a multistage hydraulic fracture model and a fracture network model; both a homogeneous case and a heterogeneous case are examined in each of these two models. For the heterogeneous case, we assign different permeabilities to the matrix blocks in an attempt to characterize the reservoir heterogeneity. The calculation results demonstrate that our new model can accurately simulate the well performance even when there is a high degree of permeability heterogeneity in the reservoir. Besides, if there are high-permeability regions existing in the fractured reservoir, a BDF may be observed in the early production period, and formation linear flow may be indistinguishable in the early production period because of the influence of reservoir heterogeneity. 1. Introduction and shale oil/gas, for the last two decades over the world [1–5]. A complex fracture network often ensues after the Hydraulic fracturing technology has significantly stimulated fracturing treatment because of the stress anisotropy and pre- the production of unconventional resources, such as tight existing natural fractures in the unconventional formations Downloaded from http://pubs.geoscienceworld.org/gsa/lithosphere/article-pdf/doi/10.2113/2021/5558746/5351483/5558746.pdf by guest
2 Lithosphere [6, 7]. Evaluating the well performance of the fractured reser- voirs becomes a fundamental task for the petroleum indus- tries to optimize the production and maximize the profit, but the complex fracture networks accompanied with reser- voir heterogeneity pose a severe challenge for the engineers (a) to conduct the reservoir simulations for these reservoirs. Production of such reservoirs with fracture networks and k1 k2 k3 spatial heterogeneity is often evaluated with numerical simu- lation methods. The numerical methods discretize the k4 k5 k6 reservoir into grid cells to capture the spatial heterogeneities and the configurations of fracture networks. But the intrinsic (b) (c) deficiencies make these methods difficult to be applied in complex fracture network reservoirs. The use of the struc- Figure 1: A simple fracture network model. (a) The fractures divide tured grid system in conventional simulators is normally the reservoir into blocks. (b) The blocks have different properties. constrained to the modeling of planar fractures [8–10] and (c) The borders of the blocks represent the fracture network. orthogonal fracture networks [11–14]. Although the unstructured grid system provides a feasible way for simu- lating both the complex fracture networks and the reservoir heterogeneity [15–17], the heavy load of building such a the reservoir into blocks to model the pressure transient model and its low computation efficiency render this behavior in compartmentalized reservoirs. However, these method less attractive. methods can only model simple cases involving reservoir Analytical and semianalytical approaches, such as the heterogeneity. They cannot be applied to deal with a hetero- multiple-porosity model, Green function method, and stimu- geneous reservoir with a fracture network. lated reservoir volume (SRV) model, have been extensively Based on the aforementioned arguments, one can con- used to model the production of fracture networks. In the clude that an analytical method which can simultaneously multiple-porosity model, the fractures are upscaled into the model the fracture network and the reservoir heterogeneity multiple-porosity model to take hydraulic fractures and nat- is still lacking. In the real field cases, however, the fracture ural fractures into consideration [11, 18–20]. But this model network, accompanied with reservoir heterogeneity, can be cannot reflect the real configurations of the fracture net- frequently present in unconventional reservoirs. Hence, it is works. The Green function method simulates the complex imperative for us to develop analytical/semianalytical fracture network by discretizing the fractures into small methods, which are convenient and stable for computational panels and then coupling the contribution of these panels use, to model the fluid flow in heterogeneous fracture [7, 21]. This method can treat the fractures explicitly as well network reservoirs. as simulate the well performance efficiently. Besides, we can From the fracture network model provided in Nagel et al. also summarize the effect of fractures as an enhanced perme- [30] and microseismic information provided in Warpinski ability within the SRV, which, however, limits our insight et al. [14], one can see that the fractures and reservoir bound- into the transient flow between the fractures and the reservoir ary may divide the reservoir into a set of matrix blocks. Fol- matrix [22, 23]. With their efficiency and stability, these ana- lowing these real-life scenarios, we propose a new approach lytical and semianalytical methods enhance our understand- to simulate the flow behaviors of fracture networks in hetero- ing of the transient behavior of fractured wells and help the geneous reservoirs. Figure 1(a) shows a simple fracture net- industry optimize the well operations. However, these work. We first discretize the fractured reservoir into matrix approaches bear stringent restrictions because they require blocks as shown in Figure 1(b) and model the flow behavior the reservoir matrix to be homogeneous in most cases within each block, and then, we couple all of the flow behav- [24, 25]. The unconventional reservoirs, however, often ior in the matrix blocks to predict the pressure response exhibit a high degree of heterogeneity, and their productivity within the entire reservoir. As the flow behavior in each can vary significantly along the horizontal wells drilled for matrix block is individually characterized in our method, producing such reservoirs ([26]; Miller et al., 2011; Cipolla these blocks can have different properties, enabling us to take and Wallace, 2014). into account the reservoir heterogeneity. Additionally, the Many authors carried out different attempts to character- borders of the blocks can reflect the configuration of the ize the reservoir heterogeneity. Loucks and Guerrero, [27] fractures, as shown in Figure 1(c); thus, a complex fracture obtained an analytical solution to describe the pressure drop network can be represented by assembling all of the borders in a circular composite reservoir. Based on the reflection- of the matrix blocks. transmission concept of electromagnetics, Kuchuk and Habashy [28] presented a new general method for solving 2. Methodology the pressure diffusion equation in laterally composite reser- voirs. Liu and Wang [29] studied the transient behavior of When being applied for reservoir simulation purposes, the a two-dimensional flow in layered reservoirs. The reservoir semianalytical methods have several distinct advantages, heterogeneity was characterized by assigning different per- such as high computation efficiency and high stability. meabilities to different layers. Medeiros et al. [18] divided To obtain a semianalytical solution to predict the well Downloaded from http://pubs.geoscienceworld.org/gsa/lithosphere/article-pdf/doi/10.2113/2021/5558746/5351483/5558746.pdf by guest
Lithosphere 3 performance in heterogeneous reservoirs, we make the fol- It is worth mentioning that ωðrÞ is equal to 0, 1/r, and lowing fundamental assumptions: 2/r for linear flow, radial flow, and spherical flow, respec- tively, such that Equation (1) can be readily transformed (i) The reservoir model is assumed to be isotropic and into the well-known linear flow equation, radial flow equa- sealed by an upper and lower impermeable layer tion, and spherical flow equation. For a hydraulically frac- tured reservoir, where the reservoir can be discretized and (ii) The reservoir model is simplified into a two- represented with N m matrix blocks, we can have the fol- dimensional model, and the temperature variation lowing equation to characterize the fluid flow in different in the reservoir is neglected matrix blocks: (iii) All of the fractures penetrate the formation completely in the vertical direction, and the fluid 8 2 flows into the wellbore only through fractures > > ∂p ∂p 1 ∂p > > + ω ð r Þ = , > > ∂r 2 n ∂r α n ∂t (iv) Only single-phase oil flow is considered in this > > > > work, and the well is constrained with a constant > < p = pi,n , t = 0, production rate ð4Þ > ∂p qμ > > = − , r = 0, (v) Only the Newtonian fluid and laminar flow are > > ∂r 0:0853kC ðr Þh n > > considered in this work > > > : ∂p = 0, r = R : > (vi) The influence of gravity is neglected ∂r n (vii) The fluid flow in the fractures and matrix is described with Darcy’s law For a matrix block with a given geometry, the ω func- It is worth noting that, although only oil flow is consid- tion in Equation (4) can be readily obtained, and we can ered in this work, this proposed method can be readily analytically characterize the fluid flow in this matrix block extended to characterize gas flow or gas condensate flow by by solving Equation (4). Subsequently, the fluid flow in the use of the concepts of pseudopressure and pseudotime. The global reservoir matrix can be described by coupling the definitions of pseudopressure and pseudotime can be found flows from all the matrix blocks. In this work, we provide in Singh and Whitson [31]. the analytical pressure functions for characterizing the fluid flow in some typical geometries, and these analytical 2.1. Fluid Flow in the Matrix. In real-life scenarios, the matrix functions are tabulated in Tables 1, 2, and 3. Appendix B blocks that are divided by the fractures and/or reservoir shows the details of how these pressure functions are boundary may have different geometries, and it is difficult obtained. For convenience, the pressure within the nth to obtain a general flow equation to characterize the fluid matrix block is implicitly expressed as flow in these blocks. In this work, we propose a new method to overcome this difficulty. The core idea of our method is that we assume that the contours of pressure can be approx- pn = f n ðkn , μn , αn , ωn , Rn , h, qn , r, t Þ: ð5Þ imated with the contours of the depth of investigation (DOI) within each matrix block. It is noted that, although this assumption cannot capture the physical scenarios, based on 2.2. Pressure Equation of Fractures. In the unconventional this assumption, we derive a new flow equation that can be reservoirs, where the matrix permeability can be extremely applied in the matrix block with an arbitrary geometry. This low, the pressure drop along the fractures can be neglected idea is validated with case studies in the validation section, compared with that in the matrix; therefore, we characterize and the proposed flow equation can be expressed as the fracture pressure with an average pressure pf . This approximation is especially reliable when the matrix perme- ability is in a range of microdarcies or below. Based on our ∂2 p ∂p 1 ∂p + ωðr Þ = , ð1Þ assumption that the fluid flows from the matrix to the frac- ∂r2 ∂r α ∂t tures and then to the wellbore and there is no direct fluid transportation between the matrix and the wellbore, the where material balance equation for the fracture system can be expressed as 1 dC ðr Þ ωðr Þ = , ð2Þ C ðrÞ dr n=N m ðt ðt 〠 − qn dt + ctf V f ϕf ðpi − pf Þ = qsc Bdt: ð6Þ 0:0853k n=1 0 0 α= : ð3Þ μϕct The derivation details for Equation (1) can be referred to In the fractures, the convergence flow near the wellbore Appendix A. can be simplified with a radial flow (see Figure 2), and the Downloaded from http://pubs.geoscienceworld.org/gsa/lithosphere/article-pdf/doi/10.2113/2021/5558746/5351483/5558746.pdf by guest
4 Lithosphere Table 1: General pressure functions for slab blocks. ω function: ωðrÞ = 0, 0 ≤ r ≤ R pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi General solution: Δpðr, sÞ = − qμ/ 0:0853 s3 /αkC 0 h s3 /α pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi exp s/αr / 1 − exp 2 s/αR + exp − s/αr / exp −2 s/αR − 1 Block geometry C0 R (1) a a b b (2) a a 2a b 2b Table 2: General pressure functions for block geometries that have an inscribed circle. ω function: ωðrÞ = 1/ðr − RÞ, 0 ≤ r ≤ R pffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffi General solution: Δpðr, sÞ = −ð1/0:0853Þðqμ/skC0 hÞ I 0 s/αðR − r Þ / s3 /αI 1 s/αR Block geometry C0 R c pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (1) b a+b+c ðC 0 /2 − aÞðC 0 /2 − bÞðC 0 /2 − cÞ/C 0 /2 a (2) a 4a a/2 a c 2 (3) b d a+b+c+d ab sin θ1 + cd sin θ2 /C 0 1 a (a + c = b + d) a (4) 2πa A (5) a/2 2a a/2 a (6) a/2 a a/2 a/2 (7) Regular n-side polygon (side length is a) na ða/2Þ cot ðπ/nÞ Downloaded from http://pubs.geoscienceworld.org/gsa/lithosphere/article-pdf/doi/10.2113/2021/5558746/5351483/5558746.pdf by guest
Lithosphere 5 Table 3: General pressure functions for rectangle blocks. ω function: ωðrÞ = 1/ðr − ðC 0 /8ÞÞ, 0 ≤ r ≤ R pffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi General solution: Δpðr, sÞ = − qμ/ 0:0853 s3 /αkC 0 h I 0 s/αðlu − r Þ K 1 s/αld + I 1 s/αld K 0 s/αðlu − r Þ / pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi I 1 s/αlu K 1 s/αld − I 1 s/αld K 1 s/αlu Þ, ld = ðC 0 /8Þ − R, lu = C 0 /8 Block geometry C0 R (1) b 2a + 2b b/2 a (a > b) (2) b/2 a+b b/2 a (a > b) (3) b/2 a+b a/2 a (a < b) (4) b/2 ða + bÞ/2 b/2 a/2 (a > b) p = pf p = pw Z (a) (b) Figure 2: The convergence flow near the horizontal wellbore in the fracture. (a) Real flow pattern. (b) Simplified flow patterns. Z direction refers to the vertical direction. relationship between the bottom-hole pressure (pw ) and the 2.3. Solution Methodology. We discretize the entire produc- fracture pressure (pf ) can be expressed as tion period into N t timesteps. Note that the solution of Equation (4) is based on a constant production rate con- dition, but the contribution of each matrix block to the 1 qsc Bμ h total production rate is time-dependent in nature. So, pw ≈ pf − ln : ð7Þ we apply the material balance time to make the constant 0:0853 2ns πkf w 2rw production rate solution applicable to a transient produc- tion rate condition. For the matrix blocks, r = 0 indicates In summary, the semianalytical model for describing the location of the fracture system; thus, we have the fol- the fluid flow in a fracture network reservoir includes lowing at r = 0: the following elements: j pf ≈ pnj ðr = 0Þ (i) The analytical solutions of the matrix block flow ð8Þ j which can be obtained by solving Equation (4) = f n kn , μn , αn , ωn , Rn , h, qn = qnj , r = 0, t = t MBn , (ii) The material balance equation of the fracture system given by Equation (6) k=j j ∑k=1 qkn Δt k (iii) The convergence flow equation near the wellbore t MBn = j : ð9Þ given by Equation (7) qn Downloaded from http://pubs.geoscienceworld.org/gsa/lithosphere/article-pdf/doi/10.2113/2021/5558746/5351483/5558746.pdf by guest
6 Lithosphere Equations (6) and (7) at the jth timestep can be 1 qksc Bk μ h rewritten, respectively, as j pwj ≈ pf − ln : ð11Þ 0:0853 2ns πkf w 2r w n=N m k=j k=j 〠 〠 − qkn Δt k + ctf V f ϕf ðpi − pf Þ = 〠 qksc Bk Δt k , ð10Þ The governing equation at the jth timestep can be n=1 k=1 k=1 expressed as 8 > > j j j j pf ≈ p1 ðr = 0Þ = f 1 k1 , μ1 , α1 , ω1 , R1 , h, q1 = q1 , r = 0, t = t MB1 , > > > > > > > > ⋮ > > > > > > j j j j p ≈ pN m ðr = 0Þ = f N m kN m , μN m , αN m , ωN m , RN m , h, qN m = qN m , r = 0, t = t MBN , > > f > > m > > > > j k=j ∑ qk Δt k > > t MB1 = k=1 j1 , > > > > q1 > < ⋮ ð12Þ > > > > k=j ∑k=1 qkN m Δt k > > j > > t MB = , > > Nm j qN m > > > > > > n=N m k=j > > k=j > > 〠 〠 − qkn Δt k + ctf V f ϕf ðpi − pf Þ = 〠 qksc Bk Δt k , > > > > n=1 k=1 > > k=1 > > > > 1 qk k B μ h > pwj ≈ pfj − : sc ln : 0:0853 2ns πkf w 2rw The unknowns at the jth timestep include the following: pseudopressure concept can be made to enable the proposed methodology to be capable of modeling single-phase gas (i) N m fluxes defined at matrix blocks: q j n, n = 1, flow. 2, ⋯, N m (ii) N m material balance times defined at matrix blocks: 3. Validation of the New Semianalytical Model t j MBn, n = 1, 2, ⋯, N m We first apply the new semianalytical method to a multi- (iii) Average fracture pressure: p j f stage hydraulic fracture model and a fracture network model, respectively, and then validate the calculated results (iv) Bottom-hole pressure: p j w against Eclipse. Each model considers two scenarios: (1) the reservoir matrix is homogeneous, and (2) the reservoir The governing equations at the jth timestep include the matrix is heterogeneous. Figure 3 shows the configurations following: of the multistage hydraulic fracture model, while Figure 4 shows the configurations of the fracture network model. (i) N m matrix flow equations at matrix blocks: Equation To take heterogeneity into consideration, the shadowed (8), which can be explicitly and analytically obtained matrix blocks shown in Figures 3(b) and 4(b) are set with by solving Equation (4) for each matrix block a higher permeability than the transparent blocks. As seen (ii) N m material balance time equations at matrix in Figure 3, the distance between two neighboring fractures blocks: Equation (9) is 200 m, the fracture’s half-length is 200 m, and the reser- voir thickness is 20 m. As seen in Figure 4, the dimension (iii) Material balance equation for the fracture system: of the fracture network model is 900 × 700 × 20 m, and Equation (10) the side length of each square matrix block is 100 m. As (iv) Convergence flow equation near the wellbore: Equa- for the four models mentioned above, the wells keep pro- tion (11) ducing for 2000 days at a constant rate of 15 m3/d. All these models have been also built with Eclipse; the grid dimen- This group of nonlinear equations can be readily solved sion used is 10 × 10 × 5 m, and the local grid refinement with the Newton method at each timestep. Furthermore, technique is applied to characterize the fractures. Table 4 although the proposed methodology is dedicated to modeling shows the other reservoir/fracture parameters used in single-phase oil flow, a minor modification on the basis of the these four models. Downloaded from http://pubs.geoscienceworld.org/gsa/lithosphere/article-pdf/doi/10.2113/2021/5558746/5351483/5558746.pdf by guest
Lithosphere 7 Y Z X (a) (b) Figure 3: The multistage hydraulic fracture model. (a) Case #1: homogeneous matrix. (b) Case #2: heterogeneous matrix. The shadowed blocks have different permeabilities from the transparent blocks. 100 m 100 m Y Z X Y X (a) 100 m 100 m Y Z X Y X (b) Figure 4: The fracture network model. (a) Case #1: 3D view and top view of the homogeneous matrix. (b) Case #2: 3D view and top view of the heterogeneous matrix. In the 3D schematic, the shadowed blocks have different permeabilities from the transparent blocks. Table 4: Reservoir and fluid properties used for validating the new 100 100 semianalytical model. Property Value Reservoir pressure 30.0 MPa 10 10 d( p)/d(lnt) p (MPa) Reservoir thickness 20 m Matrix porosity 0.15 Matrix total compressibility 1:15 × 10−3 MPa−1 1 1 Matrix permeability (transparent block) 0.002 mD Matrix permeability (shadowed block) 0.200 mD Fracture width 0.001 m 0.1 0.1 Fracture conductivity 50 D∙cm 0.1 1 10 100 1000 10000 −3 −1 Fracture’s total compressibility 1:15 × 10 MPa Time (day) Formation volume factor (dead oil) 0.985 Oil viscosity 0.1 mPa∙s Pressure drop from eclipse Pressure derivative from eclipse Radius of the wellbore 0.05 m Pressure drop from proposed method Pressure derivative from proposed method Figure 5: Comparison between the pressure drops and pressure 3.1. Multistage Hydraulic Fracture Model. The fluid flow in derivatives calculated from the proposed method and those the multistage hydraulic fracture model can be characterized calculated by Eclipse for the homogeneous multistage hydraulic with the equations provided in Table 1. Adopting the solu- fracture model. tion methodology introduced in Section 2, we predict the well performance for both the homogeneous case and the hetero- geneous case. hydraulic fracture model. The pressure drop plot calculated Figure 5 compares the pressure drops and pressure by our method is in excellent agreement with that obtained derivatives calculated from the proposed method and those by Eclipse. Similar observation can be also found for the calculated by Eclipse for the homogeneous multistage pressure derivative plot. In the pressure derivative plot, the Downloaded from http://pubs.geoscienceworld.org/gsa/lithosphere/article-pdf/doi/10.2113/2021/5558746/5351483/5558746.pdf by guest
8 Lithosphere 100 100 100 100 10 10 10 10 d( p)/d(lnt) d( p)/d(lnt) p (MPa) p (MPa) 1 1 1 1 0.1 0.1 0.01 0.01 0.1 0.1 0.1 1 10 100 1000 10000 0.1 1 10 100 1000 10000 Time (day) Time (day) Pressure drop from eclipse Pressure derivative from eclipse Pressure drop from ECL Pressure drop from proposed method Pressure derivative from ECL Pressure derivative from proposed method Pressure drop from proposed method Pressure derivative from proposed method Figure 6: Comparison between the pressure drops and pressure Figure 7: Comparison between the pressure drops and pressure derivatives calculated from the proposed method and those derivatives calculated from the proposed method and those calculated by Eclipse for the heterogeneous multistage hydraulic calculated by Eclipse for the homogeneous fracture network model. fracture model. 100 100 half-unit slope indicates the linear flow from the matrix to the fractures, while the unit slope indicates the boundary- dominated flow (BDF). Figure 6 compares the pressure drops and pressure 10 10 d( p)/d(lnt) derivatives calculated from the proposed method and p (MPa) those calculated by Eclipse for the heterogeneous multi- stage hydraulic fracture model. Different from the pressure derivative plot for the homogeneous model (see Figure 5), 1 1 the pressure derivative plot for this heterogeneous model exhibits two unit slope periods. The high-permeability blocks exhibit the earlier BDF, corresponding to the first unit slope in the pressure derivative plot. The flow behavior in the high- 0.1 0.1 permeability blocks leads to the appearance of the first unit 0.1 1 10 100 1000 10000 slope period because the pressure response in the high- Time (day) permeability blocks tends to be more rapid than that in the Pressure drop from eclipse low-permeability blocks. The transition period between these Pressure derivative from eclipse two unit slopes represents the superposition of two flow Pressure drop from proposed method Pressure derivative from proposed method regimes, i.e., the BDF regime in the high-permeability blocks and the linear flow regime in the low-permeability blocks. Figure 8: Comparison between the pressure drops and pressure When the flow regimes in all the blocks enter the BDF derivatives calculated from the proposed method and those period, the second unit slope appears in the pressure deriv- calculated by Eclipse for the heterogeneous fracture network model. ative plot. 3.2. Fracture Network Model. In this fracture network model, those calculated by Eclipse for the homogeneous fracture net- the fractures divide the reservoir into two kinds of matrix work model, while Figure 8 provides the same comparison blocks, namely, the square matrix blocks sealed by fractures for the heterogeneous fracture network model. Again, we and the outer SRV matrix block sealed by the reservoir can achieve an excellent match between the pressure drops boundary and fractures (see Figure 4). Table 2 provides the and pressure derivatives calculated by our method and those pressure function that characterizes the fluid flow in the by Eclipse. It can be seen from Figure 7 that, in the homoge- square matrix blocks, while Appendix C shows the pressure neous case, there is no distinguishable half-unit slope or unit function that characterizes the fluid flow in the outer SRV slope appearing in the pressure derivative plot due to the matrix block. Appendix C also gives the details for deriving compound transient flows of the square blocks and the outer the pressure function. SRV block. In comparison, it can be observed from Figure 8 Figure 7 compares the pressure drops and pressure deriv- that a unit slope in the early production period appears in atives calculated from the new semianalytical method and the pressure derivative plot, which indicates that BDF Downloaded from http://pubs.geoscienceworld.org/gsa/lithosphere/article-pdf/doi/10.2113/2021/5558746/5351483/5558746.pdf by guest
Lithosphere 9 appears within the high-permeability blocks. Afterwards, we Discretize the reservoir into Nm cannot identify distinguishable flow regimes, again, due to matrix blocks and discretize the the compound transient flows of the square blocks and the production time into Nt timesteps outer SRV block. 4. Application of the New Semianalytical Model Upscale the local properties to the blocks Figure 9 shows a flowchart detailing the procedures used for applying the new semianalytical approach to heterogeneous fracture network reservoirs. The essential steps are as Predict the contours of DOI and obtain the follows: function for each block (i) Discretize the fractured reservoir into matrix blocks and upscale the local properties to the blocks Substitute functions into equation 4 and obtain the (ii) Predict the contours of DOI within each block and pressure function for each block obtain the ω function for each block (iii) Substitute the ω functions into Equation (4) and Construct the obtain the pressure function for each block equation system for the jth timestep (iv) Divide the entire production duration into discrete Set j = j+1 timesteps. At each timestep, construct and solve the governing equation (i.e., Equation (12)) that Solve the equation system Substitute the fluxes into describes the fluid flow in the heterogeneous fracture and obtain the flux of equation 9 to obtain the network reservoir each block and the material balance time bottom-hole pressure equation for the (j+1)th In this section, we apply this new method to a het- timestep erogeneous reservoir that is, respectively, equipped with an orthogonal fracture network, a nonorthogonal fracture network, and a compound fracture network. Figure 10(a) Compare j j < Nt shows the permeability distribution of the heterogeneous with Nt reservoir, while Figures 10(b)–10(d) illustrate the three different fracture networks, respectively. Table 5 lists the j = Nt fluid/rock properties that are incorporated into the het- erogeneous model considered in this section. To illustrate End the influence of heterogeneity on the pressure response, we compare the pressure drops and pressure derivatives Figure 9: Flowchart showing the procedure used to implement the calculated for the heterogeneous reservoir against those proposed approach in a heterogeneous fracture network reservoir. calculated for a counterpart homogeneous reservoir with a uniform permeability of 0.002 mD. Using our newly developed semianalytical model, we perform calculations much faster in the high-permeability blocks than in the over a production period of 2000 days. Figures 11–16 low-permeability blocks. Figure 12 presents the pressure show the calculation results obtained for the three afore- drop and pressure derivative plots for the heterogeneous mentioned models. reservoir and the homogeneous reservoir. On the pressure derivative plots, a half-unit slope can be readily recognized 4.1. Fracture Network #1: Orthogonal Fracture Network in the homogeneous case, which indicates a linear flow Model. This orthogonal fracture network model divides the from matrix blocks to the fracture system, whereas in reservoir into a set of square matrix blocks characterized with the heterogeneous case, there is no distinguishable flow different permeabilities. Block geometries (2) and (6) pro- regime in the early production period; a reasonable expla- vided in Table 2 and block geometry (3) provided in nation is that the coupling of the BDF appearing in the Table 3 can be used to characterize this fracture network high-permeability blocks and the linear flow appearing in model. By using these block geometries to represent the the low-permeability blocks render the flow regime indis- entire fracture network reservoir, we can obtain the sim- tinguishable in the early production period. In the late plified reservoir model equipped with fracture network production period, the flow regime in all of the blocks #1 as shown in Figure 11(a). Blocks with different perme- enters BDF, and hence, the pressure derivative plot shows abilities can be observed in this fracture network model. a unit slope in both the homogeneous case and the hetero- Figures 11(b)–11(d) show the pressure drop distributions geneous case. on the 10th, 100th, and 1000th production days, respec- tively, which are calculated by our semianalytical model. 4.2. Fracture Network #2: Nonorthogonal Fracture Network From Figure 11, one can observe that the pressure drops Model. In the real field case, the fractures may not be Downloaded from http://pubs.geoscienceworld.org/gsa/lithosphere/article-pdf/doi/10.2113/2021/5558746/5351483/5558746.pdf by guest
10 Lithosphere 0.25 0.2 0.15 0.1 0.05 (a) (b) (c) (d) Figure 10: A heterogeneous reservoir and three fracture networks. (a) Permeability map of the matrix, mD. (b) The configuration of fracture network #1: orthogonal fracture network. (c) The configuration of fracture network #2: nonorthogonal fracture network. (d) The configuration of fracture network #3: compound fracture network. Table 5: Reservoir and fluid properties used for applying the new unsealed square blocks. Figures 13(b)–13(d) show the pres- semianalytical methodology to different fracture network models. sure drop distributions on the 10th, 100th, and 1000th produc- tion days that are calculated with our semianalytical model. A Property Value more rapid pressure drop can also be observed in the high- Reservoir pressure 30.0 MPa permeability blocks than in the low-permeability blocks. Reservoir dimension 800 m × 500 m × 20 m Figure 14 compares the pressure drops and pressure deriva- Matrix porosity 0.05 tives obtained for the heterogeneous reservoir with those obtained for the homogeneous reservoir; both reservoirs are Matrix total compressibility 1:2 × 10−3 MPa−1 equipped with fracture network #2. The pressure derivative Fracture width 0.001 m plots shown in Figure 14 are similar to those shown in Fracture conductivity 50 D∙cm Figure 12 which is dedicated to fracture network #1. In the Fracture’s total compressibility 5:0 × 10−3 MPa−1 early production period, a linear flow regime can be recog- Formation volume factor (dead oil) 0.985 nized on the pressure derivative plot in the homogeneous case, while the flow regime is indistinguishable in the hetero- Oil viscosity 0.1 mPa∙s geneous case because of the influence of heterogeneity. A unit Production rate 15 m3/d slope, which indicates BDF, can be observed in both the Radius of the wellbore 0.05 m homogeneous case and the heterogeneous case in the late Matrix permeability k1 0.002 mD production period. Matrix permeability k2 0.020 mD 4.3. Fracture Network #3: Compound Fracture Network Matrix permeability k3 0.200 mD Model. Fracture network #3 simulates such a case that the fracture network divides the reservoir into matrix blocks with different geometries as well as different permeabilities. orthogonal, dividing the reservoir into irregular matrix Figure 15(a) shows the simplified reservoir model after blocks. Fracture network #2 is used to simulate such a case. applying the block geometries provided in this work to char- We can use block geometries (1) and (6) provided in acterize fracture network #3. As seen from Figure 15(a), the Table 2 and block geometry (3) provided in Table 3 to char- reservoir is divided into triangle, rectangle, and square blocks acterize fracture network #2. Figure 13(a) presents the sim- with different dimensions. We characterize this fracture net- plified fracture network model that is built with these work with block geometries (1), (2), (5), and (6) listed in simple block geometries. As one can see in Figure 13(a), the Table 2 and block geometries (1) and (3) listed in Table 3. fractures divide the reservoir into triangle blocks and Figures 15(b)–15(d) show the pressure drop distributions Downloaded from http://pubs.geoscienceworld.org/gsa/lithosphere/article-pdf/doi/10.2113/2021/5558746/5351483/5558746.pdf by guest
Lithosphere 11 500 0.4 0.36 400 0.32 k3 0.28 300 0.24 0.2 200 0.16 0.12 k1 100 0.08 0.04 k2 0 0 0 100 200 300 400 500 600 700 800 (a) (b) 500 500 2.05 15.8 1.9 400 400 15.6 1.75 15.4 1.6 15.2 300 1.45 300 15 1.3 14.8 200 1.15 200 14.6 1 14.4 100 0.85 100 14.2 0.7 14 0 0.55 0 13.8 0 100 200 300 400 500 600 700 800 0 100 200 300 400 500 600 700 800 (c) (d) Figure 11: Reservoir model and pressure drop maps at different times. (a) The simplified heterogeneous reservoir model equipped with fracture network #1: k1 = 0:002 mD, k2 = 0:020 mD, and k3 = 0:200 mD. (b) Pressure drop map on the 10th production day. (c) Pressure drop map on the 100th production day. (d) Pressure drop map on the 1000th production day. 100 100 drops and pressure derivatives for both the heterogeneous reservoir and the homogeneous reservoir that are equipped 10 10 with fracture network #3. The pressure drop plots and pres- sure derivative plots shown in Figure 16 exhibit similar d ( p)/d (lnt) trends as observed in Figures 12 and 14 that are dedicated p (MPa) 1 1 to fracture network #1 and fracture network #2, respectively. A linear flow regime can be identified in the homogeneous case, while it cannot be observed in the heterogeneous case. 0.1 0.1 The BDF leads to a unit slope in the late production period in both cases. 0.01 0.01 0.01 0.1 1 10 100 1000 10000 5. Conclusions Time (day) In this work, we propose a novel semianalytical approach to Pressure drop in the heterogeneous case simulate the well performance in heterogeneous fracture net- Pressure drop in the homogeneous case work reservoirs. This approach is dedicated to the cases that Pressure derivative in the heterogeneous case Pressure derivative in the homogeneous case the hydraulically fractured reservoir can be discretized into matrix blocks which are sealed with the fractures and/or res- Figure 12: Comparison between the pressure drops and pressure ervoir boundary. By coupling the fluid flow in these matrix derivatives for the heterogeneous reservoir and those for the blocks, we develop a new semianalytical methodology that homogeneous reservoir; both reservoirs are equipped with fracture can be used to model a fracture network in a heterogeneous network #1. reservoir. The key features of this new method can be sum- marized as follows: on different production days. Again, the pressure drops much faster in the high-permeability blocks than in the (i) By approximating the contours of pressure with the low-permeability blocks. Figure 16 compares the pressure contours of DOI, we derive a novel general flow Downloaded from http://pubs.geoscienceworld.org/gsa/lithosphere/article-pdf/doi/10.2113/2021/5558746/5351483/5558746.pdf by guest
12 Lithosphere 500 0.34 k3 400 0.3 0.26 300 0.22 0.18 k1 0.14 200 0.1 100 0.06 0.02 k2 0 –0.02 0 100 200 300 400 500 600 700 800 (a) (b) 500 500 2 15.9 1.9 1.8 15.7 400 1.7 400 15.5 1.6 15.3 1.5 300 1.4 300 15.1 1.3 14.9 1.2 14.7 200 1.1 200 1 14.5 0.9 14.3 100 0.8 100 14.1 0.7 0.6 13.9 0 0.5 0 13.7 0 100 200 300 400 500 600 700 800 0 100 200 300 400 500 600 700 800 (c) (d) Figure 13: Reservoir model and pressure drop maps at different times. (a) The simplified heterogeneous reservoir model equipped with fracture network #2: k1 = 0:002 mD, k2 = 0:020 mD, and k3 = 0:200 mD. (b) Pressure drop map on the 10th production day. (c) Pressure drop map on the 100th production day. (d) Pressure drop map on the 1000th production day. 100 100 voir can be simulated by discretizing the reservoir into a series of blocks 10 10 (ii) Compared with the semianalytical methodologies that discretize the fracture network into fracture d( p)/d(lnt) p (MPa) panels to capture the fracture network configuration, 1 1 our method discretizes the fractured reservoir into matrix blocks 0.1 0.1 (iii) As the fluid flow in each matrix block is character- ized individually, the properties of different blocks do not have to be uniform, such that we can take res- 0.01 0.01 ervoir heterogeneity into consideration by upscaling 0.01 0.1 1 10 100 1000 10000 the local properties to the matrix blocks Time (day) We apply our method to several synthetic reservoirs and Pressure drop in the heterogeneous case Pressure drop in the homogeneous case validate our method against the commercial simulator Pressure derivative in the heterogeneous case Eclipse. The simulated results provide us with the flow Pressure derivative in the homogeneous case regime information for these complex fracture networks. Figure 14: Comparison between the pressure drops and pressure The findings include the following: derivatives for the heterogeneous reservoir and those for the homogeneous reservoir; both reservoirs are equipped with fracture (i) If there are high-permeability blocks existing in the network #2. fractured reservoir, a BDF may be observed in the early production period equation that can be applied to characterize the fluid (ii) Linear flow may be indistinguishable in the early pro- flow in different block geometries. As such, the well duction period because of the influence of reservoir performance in a complex fracture network reser- heterogeneity Downloaded from http://pubs.geoscienceworld.org/gsa/lithosphere/article-pdf/doi/10.2113/2021/5558746/5351483/5558746.pdf by guest
Lithosphere 13 500 0.44 400 0.38 k3 0.32 300 0.26 k2 0.2 200 0.14 k1 100 0.08 0.02 0 –0.04 0 100 200 300 400 500 600 700 800 (a) (b) 500 500 16.4 2.4 2.2 16.1 400 2 400 15.8 1.8 15.5 300 1.6 300 15.2 1.4 14.9 1.2 200 200 14.6 1 0.8 14.3 100 0.6 100 14 0.4 13.7 0 0.2 0 13.4 0 100 200 300 400 500 600 700 800 0 100 200 300 400 500 600 700 800 (c) (d) Figure 15: Reservoir model and pressure drop maps at different times. (a) The simplified heterogeneous reservoir model equipped with fracture network #3: k1 = 0:002 mD, k2 = 0:020 mD, and k3 = 0:200 mD. (b) Pressure drop map on the 10th production day. (c) Pressure drop map on the 100th production day. (d) Pressure drop map on the 1000th production day. 100 100 Appendix A. Derivation of the New Flow Equation 10 10 In the unconventional reservoirs, the pressure “front” can d ( p)/d (lnt) p (MPa) reach the furthest tip of the fractures in a few seconds. 1 1 Compared with the time of a few months or even years it may take for the pressure response to cover the entire matrix block, the pressure response within the fractures 0.1 0.1 can be regarded as instant. Therefore, we divide the prop- agation of DOI into two periods: first, the DOI expands in 0.01 0.01 the fracture system; second, the DOI travels from the bor- 0.01 0.1 1 10 100 1000 10000 ders of the matrix blocks to the interior of the blocks, Time (day) which is illustrated in Figure 17. In Figure 17, r is the DOI, t is the time, and R is the minimum DOI that can Pressure drop in the heterogeneous case cover the entire matrix block. Figure 17(a) shows a matrix Pressure drop in the homogeneous case Pressure derivative in the heterogeneous case block in a fracture network. In Figures 17(b)–17(d), the Pressure derivative in the homogeneous case fracture in the two-dimensional plane is considered to be composed of a set of point sources, and the DOIs simulta- Figure 16: Comparison between the pressure drops and pressure neously expand outwards from the center of these sources. derivatives for the heterogeneous reservoir and those for the homogeneous reservoir; both reservoirs are equipped with fracture Figure 17(b) shows that the DOIs just start to expand and network #3. the matrix block has not been investigated at all. Figure 17(c) shows some point sources along the fractures; at this time, the matrix blocks have been partially investi- gated. Figure 17(d) illustrates that the matrix block has been totally investigated when the DOIs reach a certain Downloaded from http://pubs.geoscienceworld.org/gsa/lithosphere/article-pdf/doi/10.2113/2021/5558746/5351483/5558746.pdf by guest
14 Lithosphere
Fracture
Non-investigated
region
Matrix unit
(a) (b)
N
inve on-
sti
reg gated
ion
2r 2R
(c) (d)
Figure 17: The expansion of DOI in a matrix block. (a) A matrix block in a fracture network. (b) At time t = 0, the DOI equals 0. (c) At time
t = t 1 , the investigated region expands. (d) At a certain later time, the investigated region covers the entire matrix block.
value R. If the properties of a given matrix block are 2(
r+
homogeneous, the expansion velocities of the DOIs along dr
the borders can be regarded as identical, enabling us to )
predict the contours of DOI within each block. r + dr, p (r + dr), q (r + dr), (r + dr)
The Darcy equation in a matrix block can be expressed as
r, p (r), q (r), (r)
0:0853k ∂pðr, t Þ 2r
qðr, t Þ = − A : ðA:1Þ
μ ∂r
Figure 18: A matrix element (grey region) which is sealed by the
contours of DOI.
It should be noted that, based on the assumption that
the contours of pressure can be approximated with the con-
tours of DOI, the parameter r in Equation (A.1) indicates For a slightly compressible fluid, we have
the DOI. The inner boundary (r = 0) represents the block
borders (fractures), while the outer boundary (r = R) repre-
sents the minimum DOI that can cover the entire matrix ρ = ρi exp ½cl ðp − pi Þ = ρi ½1 + cl ðp − pi Þ: ðA:4Þ
block.
Figure 18 shows a matrix element (grey region) that
is sealed by the contours of DOI. The single-phase mass The porosity of the matrix will change during the
balance equation for this matrix element can be written production as per
as
∂ðρϕÞ ϕ = ϕi ½1 + cm ðp − pi Þ: ðA:5Þ
½ρðr ÞqðrÞ − ρðr + drÞqðr + dr Þdt = Aðr Þdrdt: ðA:2Þ
∂t
Inserting Equation (A.1) into Equation (A.2), we can Inserting Equations (A.4) and (A.5) into Equation
have (A.3), we can obtain
0:0853 ∂ðρðk/μÞAð∂p/∂r ÞÞ ∂ðρϕÞ 0:0853 dAðr Þ k ∂p 0:0853∂ k ∂p ∂p
= : ðA:3Þ + = ϕct , ðA:6Þ
A ðr Þ ∂r ∂t Aðr Þ dr μ ∂r ∂r μ ∂r ∂t
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by guestLithosphere 15 where ct is the total compressibility which can be written Inserting Equation (B.2) into Equation (4) gives as 8 2 > > ∂p 1 ∂p 1 ∂p ct = cm + cl : ðA:7Þ > 2 + > = , > > ∂r r − R ∂r α ∂t > > in > > In a 2D model, the flow area AðrÞ can be expressed > < p = pi , t = 0, as ∂p 1 qμ ðB:3Þ > > =− > > ∂r , r = 0, > > 0:0853 kC 0h Aðr Þ = hC ðr Þ: ðA:8Þ > > > > : ∂p = 0, r = Rin : > If the permeability within a matrix block is homoge- ∂r neous and isotropic and the viscosity can be treated as a constant, Equation (A.6) can be rewritten as We now set ∂2 p ∂p 1 ∂p Δp = pi − p, ðB:4Þ + ωðr Þ = , ðA:9Þ ∂r 2 ∂r α ∂t where l = Rin − r, 0 ≤ l ≤ Rin : ðB:5Þ 1 dCðr Þ Inserting Equations (B.4) and (B.5) into Equation (B.3) ωðr Þ = , ðA:10Þ C ðr Þ dr and applying Laplace transformation to Equation (B.3), we can obtain α= 0:0853k : ðA:11Þ 8 2 μϕct > > ∂ Δp 1 ∂Δp s > > + = Δp, > > ∂l2 l ∂l α > > < B. Derivations of the Analytical Solutions ∂Δp 1 qμ =− , l = Rin , ðB:6Þ > > ∂l 0:0853 skC 0h On the basis of our assumption that the contours of pressure > > > > > can be approximated with the contours of DOI within each : ∂Δp = 0, l = 0, > matrix block, we can characterize the fluid flow within each ∂l matrix block if we can predict the contours of DOI in that block. In a fractured reservoir, the fractures divide the reser- where s is the Laplace operator. The solution of Equation voir into matrix blocks with different geometries. We provide (B.6) is the pressure functions for some typical block geometries, and the derivations are detailed below. pffiffiffiffiffiffi 1 qμ I 0 s/αl Δpðl, sÞ = − pffiffiffiffiffiffiffiffi pffiffiffiffiffiffi , ðB:7Þ B.1. Block Geometries That Have an Inscribed Circle. In a 0:0853 kC0 h s3 /αI 1 s/αRin homogeneous matrix block, the propagation velocity of DOI along the borders is approximated to be identical in this work, enabling us to predict the contours of DOI in where I 0 and I 1 are the modified Bessel functions of the first some specific block geometries. Figure 19 shows the kind. Inserting Equation (B.5) into Equation (B.7), we have approximated contours of DOI in a triangle block, a square block, and a quadrilateral block, respectively. In pffiffiffiffiffiffi 1 qμ I 0 s/αðRin − rÞ Figure 19, the dark dash lines are the approximated con- Δpðr, sÞ = − pffiffiffiffiffiffiffiffi pffiffiffiffiffiffi , ðB:8Þ tours of DOI, the solid circles are the inscribed circles 0:0853 kC0 h s3 /αI 1 s/αRin for these blocks, and Rin are the radii of the inscribed cir- cles. For these block geometries, the investigated region where Equation (B.8) is the general pressure function for the can cover the entire block if the DOI equals Rin . Accord- block geometry that has an inscribed circle. ing to the principle of similarity, B.2. Rectangle Block. Figure 20 shows the approximated con- tours of DOI in a rectangle matrix block, while a and b (a > b) Rin − r r C ðr Þ = C0 = 1 − C , 0 ≤ r ≤ Rin , ðB:1Þ are the side lengths of the block. When the DOI equals b/2, Rin Rin 0 the entire block can be covered by the investigated region. The circumference of the contours can be expressed as where C 0 is the circumference of the matrix block. b C ðr Þ = 2ða − 2rÞ + 2ðb − 2r Þ = 2a + 2b − 8r = C0 − 8r, 0≤r≤ , 1 dCðr Þ 1 2 ωðr Þ = = , 0 ≤ r ≤ Rin : ðB:2Þ C ðr Þ dr r − Rin ðB:9Þ Downloaded from http://pubs.geoscienceworld.org/gsa/lithosphere/article-pdf/doi/10.2113/2021/5558746/5351483/5558746.pdf by guest
16 Lithosphere where C0 b ld = − , ðB:13Þ 8 2 C0 Rin Rin r 2 lu = : ðB:14Þ r2 r1 r1 8 (a) (b) The solution of Equation (B.12) is qμ Δpðl, sÞ = − pffiffiffiffiffiffiffiffi 0:0853 s3 /αkC 0 h pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi I s/αl K s/αl + I s/αl K s/αl 0 pffiffiffiffiffiffi 1 pffiffiffiffiffiffi d 1 pffiffiffiffiffiffi d 0 pffiffiffiffiffiffi , I 1 s/αlu K 1 s/αld − I 1 s/αld K 1 s/αlu Rin ðB:15Þ r2 r1 where K 0 and K 1 are the modified Bessel functions of the (c) second kind. Inserting Equations (B.11) and (B.14) into Equation (B.15) yields Figure 19: The approximated contours of DOI in the geometries that have an inscribed circle. (a) Contours of DOI in a triangle qμ block. (b) Contours of DOI in a square block. (c) Contours of Δpðr, sÞ = − pffiffiffiffiffiffiffiffi DOI in a quadrilateral block. The dark dash lines are the contours 0:0853 s3 /αkC0 h pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi of DOI, while the solid circles are the inscribed circles. I 0 s/αðlu − r Þ K 1 s/αld + I 1 s/αld K 0 s/αðlu − rÞ pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi : I 1 s/αlu K 1 s/αld − I 1 s/αld K 1 s/αlu ðB:16Þ A unique scenario is described in Figure 21, showing b two unsealed matrix blocks divided by the fractures and r2 reservoir boundary. These two matrix blocks can be con- r1 verted into a rectangle block by applying the method of images, and the corresponding flow equations can be read- a ily obtained. Figure 20: The approximated contours of DOI in a rectangle block. B.3. Slab Block. In the multistage hydraulic fracture model, the fractures and reservoir boundary divide the reservoir into 1 dC ðr Þ 1 b slab blocks, and the contours of DOI can be approximated to ωðr Þ = = , 0≤r≤ : ðB:10Þ be parallel with the planar fracture, as shown in Figure 22. In C ðrÞ dr r − ðC0 /8Þ 2 this figure, a is the length of fracture and b is the interval between the fracture and the reservoir boundary. The length We set of the contours equals the fracture length: C0 C0 b C C ðr Þ = a, 0 ≤ r ≤ b, ðB:17Þ l= − r, − ≤l≤ 0: ðB:11Þ 8 8 2 8 1 dCðr Þ ω ðr Þ = = 0, 0 ≤ r ≤ b: ðB:18Þ Inserting Equations (B.4), (B.10), and (B.11) into C ðr Þ dr Equation (4) and applying Laplace transformation, we can obtain Inserting Equations (B.4) and (B.18) into Equation (4) and applying Laplace transformation, we can obtain 8 2 8 2 > > ∂ Δp 1 ∂Δp s > ∂ Δp s > > + = Δp, > > = Δp, > > ∂l2 l ∂l α > > > > > ∂r 2 α > < > < ∂Δp 1 qμ ∂Δp 1 qμ =− , l = lu , ðB:12Þ =− , r = 0, ðB:19Þ > > ∂l 0:0853 skC 0h > > ∂r > > > > 0:0853 skah > > > > ∂Δp > > : ∂Δp = 0, l = l , > > : = 0, r = b: ∂l d ∂r Downloaded from http://pubs.geoscienceworld.org/gsa/lithosphere/article-pdf/doi/10.2113/2021/5558746/5351483/5558746.pdf by guest
Lithosphere 17 Image Image Image block Reservoir block boundary block Real Image Real block Fracture block block Figure 21: The matrix blocks that can be converted into a rectangle block. r1 Ro r2 r2 a r1 r3 b Figure 23: The approximated contours of DOI in the outer SRV block. Figure 22: The approximated contours of DOI in a slab matrix block. The circumference of the contours in the outer SRV The solution of Equation (B.19) is matrix block can be expressed as qμ C o ðr Þ = 2ða2 + b2 + πr Þ, r ∈ ½0, Ro , ðC:3Þ Δpðr, sÞ = − pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 0:0853 s3 /αkah s3 /α " pffiffiffiffiffiffi pffiffiffiffiffiffi # exp s/αr exp − s/αr and the ω function for the outer SRV matrix block can be pffiffiffiffiffiffi + pffiffiffiffiffiffi : written as 1 − exp 2 s/αb exp −2 s/αb − 1 ðB:20Þ 1 dC o ðr Þ 1 ωo ðr Þ = = : ðC:4Þ C o ðr Þ dr r + ðða2 + b2 Þ/πÞ The flow equations as given by Equations (B.8), (B.16), and (B.20) are obtained in the Laplace domain. They can be We set readily inverted into the solutions in the physical time domain by using the algorithm proposed by Stehfest (1970). Δpo = pi − p, ðC:5Þ C. Pressure Function of the Outer SRV Matrix Block a2 + b2 R1 = , ðC:6Þ Figure 23 shows the approximated contours of DOI in the π outer SRV matrix block. We approximate the reservoir a 2 + b2 boundary with the outmost contour that has the same area R2 = Ro + : ðC:7Þ as that of the reservoir. The real reservoir area equals the area π within the outmost contour, such that The governing equation for the outer SRV matrix block can be expressed as a1 b1 = a2 b2 + 2a2 Ro + 2b2 Ro + πR2o , ðC:1Þ 8 2 > > ∂p 1 ∂p 1 ∂p > 2 + > = , where a1 is the length of the reservoir, b1 is the width of the > > ∂r r + ðð 2 a + b 2Þ Þ /π ∂r α o ∂t > > reservoir, a2 is the length of SRV, b2 is the width of SRV, > > and Ro is the DOI of the outmost contour. Ro can be obtained > < p = pi , t = 0, ðC:8Þ from Equation (C.1): > ∂p 1 qμ > > =− , r = 0, > > ∂r 0:0853 2ða2 + b2 Þhk o qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > 1 > Ro = ða2 + b2 Þ2 + πða1 b1 − a2 b2 Þ − ða2 + b2 Þ : ðC:2Þ : ∂p = 0, r = Ro : > π ∂r Downloaded from http://pubs.geoscienceworld.org/gsa/lithosphere/article-pdf/doi/10.2113/2021/5558746/5351483/5558746.pdf by guest
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