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PETROLEUM EXPLORATION & PRODUCTION PETROVIETNAM JOURNAL Volume 10/2020, p. 20 - 40 ISSN 2615-9902 INFERRING INTERWELL CONNECTIVITY IN A RESERVOIR FROM BOTTOMHOLE PRESSURE FLUCTUATIONS OF HYDRAULICALLY FRACTURED VERTICAL WELLS, HORIZONTAL WELLS, AND MIXED WELLBORE CONDITIONS Dinh Viet Anh1, Djebbar Tiab2 1 PetroVietnam Exploration Production Corporation 2 University of Oklahoma Email: anhdv@pvep.com.vn; dtiab@ou.edu https://doi.org/10.47800/PVJ.2020.10-03 Summary A technique using interwell connectivity is proposed to characterise complex reservoir systems and provide highly detailed information about permeability trends, channels, and barriers in a reservoir. The technique, which uses constrained multivariate linear regression analysis and pseudosteady state solutions of pressure distribution in a closed system, requires a system of signal (or active) wells and response (or observation) wells. Signal wells and response wells can be either producers or injectors. The response well can also be either flowing or shut in. In this study, for consistency, waterflood systems are used where the signal wells are injectors, and the response wells are producers. Different borehole conditions, such as hydraulically fractured vertical wells, horizontal wells, and mixed borehole conditions, are considered in this paper. Multivariate linear regression analysis was used to determine interwell connectivity coefficients from bottomhole pressure data. Pseudosteady state solutions for a vertical well, a well with fully penetrating vertical fractures, and a horizontal well in a closed rectangular reservoir were used to calculate the relative interwell permeability. The results were then used to obtain information on reservoir anisotropy, high-permeability channels, and transmissibility barriers. The cases of hydraulically fractured wells with different fracture half-lengths, horizontal wells with different lateral section lengths, and different lateral directions are also considered. Different synthetic reservoir simulation models are analysed, including homogeneous reservoirs, anisotropic reservoirs, high-permeability-channel reservoirs, partially sealing barriers, and sealing barriers. The main conclusions drawn from this study include: (a) The interwell connectivity determination technique using bottomhole pressure fluctuations can be applied to waterflooded reservoirs that are being depleted by a combination of wells (e.g. hydraulically fractured vertical wells and horizontal wells); (b) Wellbore conditions at the observations wells do not affect interwell connectivity results; and (c) The complex pressure distribution caused by a horizontal well or a hydraulically fractured vertical well can be diagnosed using the pseudosteady state solution and, thus, its connectivity with other wells can be interpreted. Key words: Interwell connectivity, bottomhole pressure fluctuations, waterflooding, vertical wells, horizontal wells, hydraulically fractured wells. 1. Introduction Numerous studies on inferring interwell connectivity in a waterflood have been carried out. Some of these studies used statistical techniques that are very different Date of receipt: 12/10/2020. Date of review and editing: 12 - 14/10/2020. Date of approval: 15/10/2020. This article was presented at SPE Production and Operations Symposium and licensed by SPE (License ID: 1068761-1) to republish full paper in Petrovietnam Journal 20 PETROVIETNAM - JOURNAL VOL 10/2020 from the approach used in this study. Albertoni and Lake developed a technique that calculates the fraction of flow caused by each of the injectors in a producer [1, 2]. This method uses a constrained Multivariate Linear Regression (MLR) model similar to the model proposed by Refunjol [3]. The model introduced by Albertoni and Lake, however, considered only the effect of injectors on producers, not producers on producers. Albertoni and Lake also intro- PETROVIETNAM duced the concepts and uses of diffusivity filters to account for the time lag and attenuation that occur between the stimulus (injection) and the response (production). The procedures were proven effective for synthetic reservoir models, as well as real water flood fields. Yousef et al. introduced a capacitance model in which a nonlinear signal processing model was used [4, 5]. Compared to Albertoni and Lake’s model which was a steady-state (purely resistive) one, the capacitance model included both capacitance (compressibility) and resistivity (transmissibility) effects. The model used flow rate data and could include shut-in periods and bottom hole pressures (if available). Dinh and Tiab [6 - 9] used a similar approach as Albertoni and Lake [1, 2], however, bottom hole pressure data were used instead of flow rate data. Some constraints were applied to the flow rates such as constant production rate at every producer and constant total injection rate. Using bottom hole pressure data offers several advantages: (a) diffusivity filters are not needed, (b) minimal data is required and (c) flexible plan to collect data. All of the studies above only considered fully penetrating vertical wells. Dinh and Tiab only considered reservoirs with vertical wells without any hydraulic fractures or horizontal wells [6 - 9]. In this study, bottomhole pressure fluctuations were used to determine the interwell connectivity in a waterflood where horizontal wells, hydraulically fractured vertical wells or both are present. MLR model was used to determine the interwell connectivity coefficients from bottomhole pressure data. For the case of hydraulically fractured vertical wells, a late time solution for a well with a fully penetrating vertical fracture in a closed rectangular reservoir was used to calculate the influence functions and the relative interwell permeabilities. The case where the fractures are of different fracture half-lengths is also considered. Similarly, for the horizontal well cases, the late time solution for a horizontal well in a closed rectangular reservoir was used to calculate the influence functions and the relative interwell permeabilities. The cases in which the reservoir contains horizontal wells of different lengths and different directions were also considered. In order to quantify the effect of observation wells on the interwell connectivity coefficients, the case of different injector well lengths and unchanged producer well lengths was analysed. Results for different cases such as all wells are horizontal along the x-direction, along both x- and y-directions and different horizontal well lengths are provided. This study also provides the results for different cases where mixed wellbore conditions are present. 5 injector and 4 producer synthetic reservoirs containing hydraulic fractures and vertical wells, horizontal and vertical wells or all three types of wellbore conditions are used in the analysis. The results were then used to obtain information on reservoir anisotropy, high permeability channels and transmissibility barriers. Different synthetic reservoir models were analysed including homogeneous, anisotropic reservoirs, reservoirs with high permeability channel, partially sealing barrier and sealing barrier. 2. Analytical model and calculation approach Previous studies have developed a novel technique to determine interwell connectivity from bottom hole pressure fluctuation data. This study extends the application of the technique to hydraulically fractured, horizontal wells and mixed wellbore conditions. The technique was described in detail by Dinh and Tiab [6 - 9]. Key equations and definitions of dimensionless variables below are used throughout this study. 2.1. Dimensionless variable Considering a multi-well system with producers or injectors and initial pressure pi, the solution for pressure distribution due to a fully penetrated vertical well in a close rectangular reservoir is as follows [10, 11]: nwell pD ( xD , yD , tDA ) = ∑qD ,i ai (xD , yD , xwD ,i , ywD ,i , xeD , yeD , [tDA − tsDA ])(1) i =1 Where the dimensionless variables are defined in field units as follows: xx (2) xx DD = = A A yy yy DD = = A A kh kh p − p(x, y, t)) pDD = = 141. 2q Bµ (( p pini ini − p(x, y, t )) 141. 2qref Bµ (3) (4) ref kt = 0. 0002637 kt ttDA DA = 0. 0002637 φ c φ ctt µ µA A (5) ai is the influence function equivalent to the dimensionless pressure for the case of a single well in a bounded reservoir produced at a constant rate. Assuming tsDA = 0, the influence function is given as: PETROVIETNAM - JOURNAL VOL 10/2020 21 PETROLEUM EXPLORATION & PRODUCTION ai (xD , yD , xwD ,i , ywD ,i , xeD , yeD , tDA ) 1 ∞ ∞ (x + x + 2nxeD ) + (yD + ywD ,i + 2myeD) = ∑ ∑E1  D wD ,i 2 m=−∞ n=−∞  4tDA 2 2     (x − xwD ,i + 2nxeD )2 + (yD + ywD ,i + 2myeD )2  + E1  D  4tDA   (6)  (x + xwD ,i + 2nxeD )2 + (yD − ywD ,i + 2myeD )2  + E1  D  4tDA   .2Bµ nwell ∑an[xD , yD , xwDn , ywDn, xeD , yeD ,tAD ]qn(7) kh i =1 Equation 7 is the pressure response at point (xD, yD) due to a well n at (xwDn, ywDn) in a homogeneous closed rectangular reservoir. The influence function (an) can be different for different wellbore conditions as well as flow regimes (horizontal well, partial penetrating vertical well, fractured vertical well, etc.). 2  1 yD y D2 + y wD  − + 2 3 y 2 y eD eD      (10)  1 1   xwD  x D  cos kπ G(xeD , yeD , ywD , yD , k) + sinkπ  cos kπ 2 π ∑  xeD  x eD   xeD k =1 k 2 xeD ∞   y − ( y D + ywD )   + cosh kπ  eD    xeD     yeD   sinh kπ xeD   y − y −y cosh kπ  eD D wD xeD  (11) For the case of infinite conductivity fractures, the dimensionless pressure can be obtained by evaluating the above equation at xD = 0.732 [14]. 2.3.2. Horizontal wells The pressure distribution equation for a horizontal well in a closed rectangular reservoir is [13]: pDh = ah = pDf + F1 (12) Where 2.2. Shape factor calculation Shape factors are used to calculate pressure at wells at different locations in a reservoir of a certain shape. Letting CA denote the shape factor, we have the well known shape factor equation: 4A e γ C A L2 (8) with L = rw, Lxf and Lh/2 for vertical well, vertically fractured well and horizontal well respectively and γ is Euler’s constant (γ = 0.5772…) Thus, the shape factor can be calculated using Equation 9 [12]: 4A   CA = Exp (4π tDA − 2 pwD ) + Log 2 γ  Le   (9) Where the L term in the definitions of dimensionless quantities is L = Lxf which is the fracture half-length. 2.3. Influence function 2.3.1. Hydraulically fractured well For a hydraulically fractured well, for simplicity, the late time solution for a uniform flux fracture in 22 y eD x eD G(xeD , yeD , ywD , yD , k ) = Equation 6 is valid for pseudosteady state flow and can be rewritten as below: pwD = 2π tDA + 0. 5 ln p Df = 2π t DA + 2π Where the G-function is:  (x − xwD ,i + 2nxeD )2 + (yD − ywD ,i + 2myeD )2  + E1  D  4tDA   pini − p(x, y)= a closed rectangular reservoir provided by Ozkan was used [13]. The influence function for hydraulically fractured well becomes: PETROVIETNAM - JOURNAL VOL 10/2020 F1 = 2 x eD L D ∞ 1 ∑n cos(nπ z D )cos(nπ zwD ) n=1     + cosh nπ  y eD − ( y D + y wD )     x eD    ×  y  sinh  nπ eD  (13) x eD    1  x   x  sin kπ  cos kπ wD cos kπ D ∞ ∞ 1  xeD  x eD  xeD + 4 cos(nπ zD)cos(nπ zwD) k b n=1 k =1  y eD − y D − y wD cosh nπ   x eD  ∑ ∑ cosh b (y eD − y D − y wD ) + cosh b ( y eD − ( y D + y wD )) ( sinh b y eD ) 2 Where b = n2π 2 L2D + k 2π 2/xeD and the L term in the dimensionless definition is the horizontal well half-length L = Lh/2, and zD = z/h and LD = 1/hD = L/2h. xwD and ywD are at the midpoint of the well length for the uniform flux horizontal well case. For the infinite conductivity horizontal well case, Ozkan showed that the point xD = 0.732 used to calculate pressure distribution for an infinite conductivity fracture can also be used for an infinite conductivity horizontal well [13]. The term F1 can be rewritten as follows: PETROVIETNAM F1 = 2 x eD LD ∞ ∞ 1 ∑n cos(nπ z )cos(nπ z D wD n=1 ∑ +4 )G (xeD , yeD , ywD , yD , nπ ) ∞ 1   xwD  coskπ    xeD  eD    1 ∑k sinkπ x cos(nπ zD )cos(nπ zwD ) n=1 k =1  x cos  kπ D  x eD  Where: (  GH y eD , y wD , y D , b   2 2 b = n2π 2L2D + k π ( ) ) G (1, y eD b( X − a ) b( X + a)  b (X − α )2 dα = 1  K (u )du − K (u )du K 0 0 0 ∫ ∫ ∫    b 0  0 −a  (21) If X = a then +a xeD2 GH yeD , ywD , yD , b = (14) , ywD , yD , b ) +a K 0 b ∫  −a b To calculate F1 as suggested by Ozkan [13]: F1 = F + F b1 + F b 2 + F b3 (15) Where +1 ∞ 2 2 F =∑cos (nπ zD )cos(nπ zwD )∫K0 nπLD (xD- xwD - α) + ( yD - ywD )  dα (16)   n =1 −1 Fb1 = [ 2 xeD LD 1 ∑n cos(nπ z )cos(nπ z ) ∞ D wD n =1 ]  e− nπ LD ( yD + y wD ) + e− nπLD (2 yeD − y D − y wD ) + e− nπ LD (2 yeD − ( y D + y wD ))  (17)   ∞ ∞   − nπ LD y D − y wD − 2 mnπLD y eD  − 2 mnπ LD yeD e ∑  1 + ∑e +e  m =1    m =1 1 x  x  sinkπ cos kπ D  cos k wD 1  xeD  xeD  xeD Fb 2 = 4∑cos(nπ zD)cos(nπ zwD )∑ 2 k π2 n =1 k =1 k n2π 2 LD + 2 xeD (18) 2 2 2 2 2 2  − n 2π 2 L2D + k π ( yD + ywD ) − n 2π 2 L2D + k π (2 yeD − ( yD + ywD )) − n 2π 2L2D + k π (2 yeD − yD − ywD ) 2 2 2  xeD xeD xeD  +e +e × e    k 2π 2 k 2π 2 k 2π 2  − n 2π 2 L2D + 2 y D − y wD ∞ − 2 m n 2π 2 L2D + 2 y eD  ∞ − 2 m n 2π 2 L2D + 2 y eD  xeD xeD x eD   × 1 + ∑e +e  ∑e   m=1 m=1  ∞ Fb3 = ∑ cos nπ zD cos nπ zwD n=1 + 1 2  ∫ K nπ L x + x − α 2 + y − y  dα + D wD  −1 0  D D wD  ∞ ∞ ) ( ( ( ) ) ( For the case of yD = ywD, if X ≤ a then X ≤a b(a + X ) b(a −X) +a  +a (a −X) b (X − α )2 d1α =b(a1+X) K (u)bdu  + K 2 0 0 0 (u)du ∫ ∫  dα =  b K (u)du + K (∫uK b (X   ) ) K du − α ∫ 0−a ∫0 0 0    0 0  b  ∫ −a 0  If X ≥ a (20) If X ≥ a then )    K nπ L (x − x − 2kx − α )2 + (y − y ) 2    eD D wD     0  D D wD     2 2     ∞ +1+ K0 nπ LD (xD + xwD − 2kxeD − α ) + (yD − ywD )   (19)    + ∑ ∫  dα  k =1 −1 + K nπL (x − x + 2kx −α)2 + (y − y )2    eD D wD 0       D D wD     nπ L (x + x + 2kx −α ) 2 + (y − y )2     + K  0 D D wD eD D wD       (X 1 2 − α )  dα =  b 2 ab ∫K (u)du 0 (22) 0 Where a = 1, b = nπLD Table 1 presents the dimensionless coordinates for all the vertically fractured wells in the 5 × 4 synthetic field (5 injectors: I1, I2, I3, I4 and I5 and 4 producers: P1, P2, P3 and P4 as shown on Figure 1). All wells have the same fracture half-length of 145 ft. Other data include xeD = yeD = 21.38 and rwD = 0.0049. Table 2 shows the shape factors for all the wells in the 5 × 4 synthetic field calculated using PwD results (influence functions) from the different calculation techniques and Equation 9. As shown in Table 2, the shape factors are in good agreement. These shape factors can be used to calculate the influence functions using Equation 8. Table 3 presents the dimensionless coordinates for all the wells in the 5 × 4 homogeneous synthetic field. Other data include xeD = yeD = 20.67 and rwD = 0.004733. Table 4 shows the shape factors for the horizontal wells in the 5 × 4 synthetic field calculated using PwD results (influence functions) from Equations 9 and 12. Table 1. Dimensionless coordinates of the fractured wells in the 5 × 4 synthetic field Wells I01 I02 I03 I04 I05 P01 P02 P03 P04 xwDf 3.7931 17.5862 10.6897 3.7931 17.5862 10.6897 3.7931 17.5862 10.6897 ywDf 17.5862 17.5862 10.6897 3.7931 3.7931 17.5862 10.6897 10.6897 3.7931 PETROVIETNAM - JOURNAL VOL 10/2020 23 PETROLEUM EXPLORATION & PRODUCTION Table 2. Shape factors for the fractured wells in the 5 × 4 synthetic field calculated for different fracture types Wells I01 I02 I03 I04 I05 P01 P02 P03 P04 CAf Infinite Conductivity 0.2665 0.1606 7.5580 0.2665 0.1606 1.6560 1.9678 1.3396 1.6560 Uniform Flux 0.1144 0.1140 4.1698 0.1144 0.1140 0.9083 0.9026 0.9003 0.9083 Table 3. Dimensionless coordinates of the horizontal wells in the 5 × 4 synthetic field Wells I01 I02 I03 I04 I05 P01 P02 P03 P04 xwDh 3.6667 17.0000 10.3333 3.6667 17.0000 10.3333 3.6667 17.0000 10.3333 ywDh 17.0000 17.0000 10.3333 3.6667 3.6667 17.0000 10.3333 10.3333 3.6667 Figure 1. Top view of the simulation model showing the LGRs at the fractured wells in the 5 × 4 homogeneous synthetic field. Table 4. Shape factors for uniform flux and infinite conductivity horizontal wells in 5 × 4 synthetic reservoir Wells CAh I01 I02 Uniform Flux 0.0404 0.0403 Infinite Conductivity 0.0950 0.0563 I03 I04 1.4741 0.0404 2.6713 0.0950 I05 P01 0.0403 0.3212 0.0563 0.5857 P02 0.3190 0.6997 P03 0.3182 0.4699 P04 0.3212 0.5857 3. Simulation results for hydraulically fractured wells 3.1. Model descriptions for hydraulically fractured wells The grids in the small areas containing the wells were refined using the Local Grid Refinement (LGR) options. Thus, there are nine LGRs in this model [15]. Figure 1 shows the top view of the permeability distribution for this case. The LGRs can be seen at each well. Figure 2 is a permeability distribution plot showing the cross-sectional view through three wells. The hydraulic fractures are represented in red indicating high permeability. The LGR areas are 300 ft × 20 ft each with a global grid configuration of 13 × 1 which is refined to a grid configuration of 65 × 25. 24 PETROVIETNAM - JOURNAL VOL 10/2020 Figure 2. Cross sectional view showing three wells and the hydraulic fractures in the 5 × 4 homogeneous synthetic reservoir. No refinement in the vertical direction was applied. Thus, the number of layers in the LGRs stayed at five layers. Figure 3 presents a zoom-in top view of a LGR containing a high permeability strip representing a hydraulic fracture. Notice that the permeability of the cell at the tips of the fracture was set to zero following the assumption that there was no flow through the tips of the fracture. The permeability of the fractures was set to 8,000 Darcys. The width of the fractures was 0.8 ft, and the fracture half-lengths were the same at 145 ft. Thus, the dimensionless fracture conductivity for every fracture, which is the product of fracture permeability and fracture width divided by the product of formation permeability and fracture half-length, is equal to 441. Thus, according to previous studies [16, 17], the fractures can be considered as infinite conductivity fractures (dimensionless fracture conductivity is larger than 300). The porosity of the fracture was input as 0.6 which is higher than the porosity of the formation of 0.3. I01 I02 P01 PETROVIETNAM I01 P02 I02 P01 P03 I03 P02 I04 P04 P03 I03 I05 Figure 3. A zoom-in view of a LGR showing a high permeability strip representing a hydraulic fracture - 5 × 4 homogeneous system. I01 I02 P01 I04 P04 I05 Figure 4. Representation of the interwell connectivity coefficients for the 5 × 4 homogeneous system with hydraulically fractured wells. P02 I01 P01 I02 I01 P01 I02 P03 I03 P03 P02 I03 P02 I04 P04 P03 I03 I05 Figure 5. Representation of the relative interwell permeability for the 5 × 4 homogeneous reservoir with hydraulically fractured wells. 3.2. Homogeneous reservoir with hydraulic fractures Table 5 and Figure 4 show the results for the interwell connectivity coefficients. Similar to previous cases, the results are as good as the results obtained in the case of homogeneous reservoir with vertical wells only with asymmetry coefficient of 0.0048. Table 6 and Figure 5 present the corresponding relative interwell permeabilities with the equivalent time of 5.66 days, and the reference permeability of 100 mD. The difference between the high and low interwell connectivity coefficients is more significant than in the case of vertical wells suggesting an observation well is less affected by a far away active fractured well than by a vertical unfractured well of the same distance away. This is reasonable because with the same flow rate, the pressure drop in a fractured well is less than its unfractured counterpart. I04 P04 I05 Figure 6. Representation of the connectivity coefficients for the case of 5 × 4 anisotropic I05 I04 reservoir - hydraulically fractured wells. P04 3.3. Anisotropic reservoir with hydraulic fractures Similar to the anisotropic case in the previous chapter, the effective permeability in the x direction is tenfold the fracture permeability in the y direction. Table 7 and Figure 6 show the results for the interwell connectivity coefficients. As expected, the results are good indications of the anisotropy with large coefficients for well pairs in the direction of high permeability. Table 8 and Figure 7 present the corresponding relative interwell permeabilities with the equivalent time of 5.66 days, and the reference permeability of 316 mD. PETROVIETNAM - JOURNAL VOL 10/2020 25 PETROLEUM EXPLORATION & PRODUCTION Table 5. Interwell connectivity coefficient results from simulation data for the 5 × 4 homogeneous synthetic field with hydraulic fractured wells (As = 0.0048) β0j (psia) I1 I2 I3 I4 I5 Sum P1 -223.6 0.32 0.32 0.24 0.06 0.06 1.00 P2 -226.1 0.31 0.06 0.25 0.31 0.06 1.00 P3 -225.7 0.06 0.31 0.26 0.06 0.31 1.00 P4 -223.6 0.06 0.06 0.25 0.32 0.32 1.00 Sum -899 0.75 0.75 1.01 0.75 0.75 Table 7. Interwell connectivity coefficient results from simulation data for the 5 × 4 anisotropic synthetic field - hydraulically fractured wells P1 P2 P3 P4 Sum β0j (psia) I1 -69.6 0.43 -96.5 0.13 -96.5 0.10 -69.6 0.02 -332 0.67 I2 I3 0.43 0.11 0.10 0.55 0.13 0.55 0.02 0.11 0.67 1.32 I4 0.02 0.13 0.10 0.42 0.67 I5 0.02 0.10 0.13 0.43 0.67 Sum 1.00 1.00 1.00 1.00 I01 Table 6. Relative interwell permeability results for the 5 × 4 homogeneous synthetic field with hydraulic fractured wells (kref = 100 mD, Δteq = 5.66 days) I1 I2 I3 I4 I5 Average P1 114 114 92 92 91 101 P2 112 91 96 111 92 101 P3 90 111 99 91 113 101 P4 91 91 93 114 117 101 Average 102 102 95 102 103 Table 8. Relative interwell permeability results for the 5 × 4 anisotropic synthetic field hydraulically fractured wells (kref = 316 mD, Δteq = 5.66 days) I1 I2 I3 I4 I5 Average P1 353 351 90 80 77 190 P2 75 152 444 75 153 180 P3 152 76 444 151 77 180 P4 78 80 90 350 357 191 Average 164 164 267 164 166 I02 P01 P02 P03 I03 I04 P04 I05 Figure 7. Representation of relative interwell permeability for the case of 5 × 4 synthetic reservoir - hydraulically fractured wells. Figure 8. Top view of the simulation model showing the permeability in x direction for the high permeability channel case of the 5 × 4 synthetic field with fractured wells. 3.4. Reservoir with a high permeability channel Table 9 and Figure 9 show the results for the interwell connectivity coefficients. Similar to previous cases of high permeability channels, the results reflect well the presence of the channel. Different from the previous cases, well I03 has much higher connectivity with producers P02 and P04. The reason for this is that in the previous cases, well I03 was not connected to the high permeability channel while in this case, due to the extension provided by the hydraulic fracture, it is directly connected to the channel and has better connectivity with the producers. Figure 8 shows the top view of the permeability distribution for this case. The cells in yellow color have high permeability in both x and y direction. Similar to the high permeability channel cases in the previous chapters, the permeability of the channel was ten-fold (1,000 mD) of that in the other areas of the reservoir (100 mD). There are nine vertically fractured wells with the same fracture half-length of 145 ft. 26 PETROVIETNAM - JOURNAL VOL 10/2020 PETROVIETNAM Table 9. Interwell connectivity coefficient results from simulation data for the 5 × 4 synthetic reservoir with a high permeability channel - hydraulically fractured wells β0j (psia) I1 I2 I3 I4 I5 Sum P1 -153.5 0.46 0.23 0.26 0.02 0.03 1.00 P2 -54.1 0.42 0.02 0.45 0.07 0.04 1.00 P3 -194.2 0.10 0.28 0.33 0.03 0.25 1.00 P4 -65.4 0.16 0.02 0.53 0.13 0.16 1.00 Sum -467 1.14 0.55 1.57 0.25 0.48 Table 10. Relative interwell permeability results for the 5 × 4 synthetic reservoir with high permeability channel - hydraulically fractured wells. (kref = 300 mD, Δteq = 5.66 days) P1 P2 P3 P4 Average I1 I2 369 162 337 77 153 210 200 84 265 133 I3 I4 202 79 347 24 256 92 412 69 304 66 118 I5 90 94 184 104 Average 180 176 179 174 I01 P01 I02 P02 P03 I03 I04 P04 I01 I05 P01 I02 P03 P02 I03 I04 P04 I05 Figure 9. Representation of the connectivity coefficients for the case of 5 × 4 synthetic reservoir with a high permeability channel - hydraulically fractured wells. Figure 10. Representation of relative interwell permeability for the 5 × 4 synthetic reservoir with a high permeability channel - hydraulically fractured wells. Table 10 and Figure 10 present the corresponding relative interwell permeabilities with the equivalent time of 5.66 days, and the reference permeability of 300 mD. Table 11 and Figure 12 show the results for the interwell connectivity coefficients. The presence of the partially sealing barrier is well established by the results. Table 12 and Figure 13 present the corresponding relative interwell permeabilities with the equivalent time of 5.66 days, and the reference permeability of 100 mD. The relative interwell permeability for well pair I01-P01 was negative because the influence function for the pair was calculated using the late time solution. When the interwell connectivity coefficients are small, they are translated to early 3.5. Reservoir with a partially sealing barrier Figure 11 shows the top view of the x-direction permeability distribution for this case. The permeability for the cells in grey color were set to zero and thus, those cells served as a partially sealing barrier. The formation permeability was 100 mD. PETROVIETNAM - JOURNAL VOL 10/2020 27 PETROLEUM EXPLORATION & PRODUCTION I01 I02 P01 P02 P03 I03 I04 Figure 11. Top view of the simulation model showing the permeability distribution in x direction for the case of 5 × 4 synthetic field with a partially sealing barrier - hydraulically fractured wells. P04 I05 Figure 12. Representation of the connectivity coefficients for the case of 5 × 4 dualporosity reservoir with a partially sealing barrier - hydraulically fractured wells. Table 11. Interwell connectivity coefficient results from simulation data for the 5 × 4 synthetic field with partially sealing barrier - hydraulically fractured wells β0j (psia) I1 I2 I3 I4 I5 Sum P1 -440.1 0.01 0.79 0.06 0.04 0.11 1.00 P2 -204.0 0.34 0.02 0.25 0.32 0.07 1.00 P3 -306.9 0.01 0.49 0.08 0.05 0.37 1.00 P4 -226.1 0.06 0.06 0.22 0.33 0.33 1.00 Sum -1177 0.42 1.36 0.61 0.73 0.87 Table 12. Relative interwell permeability results for the 5 × 4 synthetic field with partially sealing barrier - hydraulically fractured wells (kref = 100 mD, Δteq = 5.66 days) I1 I2 I3 I4 I5 Average P1 -40 347 23 80 115 105 P2 127 71 95 114 95 101 P3 68 199 29 88 141 105 P4 90 92 83 119 125 102 Average 62 177 58 100 119 time periods and thus the late time solution becomes inaccurate. Solutions that are good for both early time and late time should be used for better results. two compartments. Based on the change in average reservoir pressure calculated from each producer, this compartmentalisation can be inferred. 3.6. Reservoir with a sealing barrier Table 13 and Figure 15 show the results for the interwell connectivity coefficients. Similar to previous cases, the results clearly reflect the presence of the sealing barrier. Some connectivity coefficients are very small and even negative. They indicate poor connectivity or no connectivity at all. Small connectivities were still observed for some pairs of wells on different sides of the sealing barrier. Figure 14 shows the top view of the x-direction permeability distribution with a sealing barrier case. The permeability of the cells in grey color was set to zero and thus, those cells served as a sealing barrier. As seen in the figure, the barrier completely divides the reservoir into 28 PETROVIETNAM - JOURNAL VOL 10/2020 PETROVIETNAM I01 I02 P01 P03 P02 I03 I04 P04 I05 Figure 13. Representation of relative interwell permeability for the case of 5 × 4 dualporosity reservoir with a partially sealing barrier - hydraulically fractured wells. Figure 14. Top view of the simulation model showing the permeability in x direction for the case of 5 × 4 synthetic field with a sealing barrier - hydraulically fractured wells. Table 13. Interwell connectivity coefficient results from simulation data for the 5 × 4 synthetic field with a sealing barrier - hydraulically fractured wells β0j (psia) I1 I2 I3 I4 I5 Sum P1 -336.6 0.00 0.87 0.05 -0.02 0.07 0.97 P2 -266.0 0.35 -0.01 0.27 0.36 0.04 1.01 P3 -225.4 0.00 0.60 0.05 -0.02 0.35 0.97 P4 -365.7 0.10 -0.01 0.35 0.53 0.05 1.02 Sum -1194 0.45 1.44 0.73 0.84 0.51 Table 14. Relative interwell permeability results for the 5 × 4 synthetic field with a sealing barrier - hydraulically fractured wells (kref = 100 mD, Δteq = 5.66 days) I1 I2 I3 I4 I5 Average P1 0.00 385.6 0.00 0.00 98.2 97 P2 131.5 0.00 101.7 137.4 0.00 74 As explained before, these non-zero connectivity coefficients are due to the noises in the data as the injection rates were generated randomly. This problem can be resolved by increasing the number of data points. For this case, the interwell connectivity coefficients should be analysed with the average reservoir pressure change results. If the pressure changes indicate reservoir compartmentalisation, then the small interwell connectivity coefficients can be evaluated to decide whether the injectors and producers are on different side of the barrier. P3 0.00 253.1 0.00 0.00 132.6 77 P4 112.5 0.00 132.6 216.6 0.00 92 Average 61.01 159.7 58.6 88.5 57.7 Table 14 and Figure 16 present the corresponding relative interwell permeabilities with the equivalent time of 5.66 days, and the reference permeability of 100 mD. A cut-off coefficient of 0.06 was applied to eliminate the low connectivity coefficients. Thus, the relative interwell permeability corresponding to the coefficients lower than 0.06 were set to zeros. The resulting relative interwell permeabilities show a clear presence of the sealing barrier. Table 15 shows the results for the average reservoir pressure change for all producers in each case described PETROVIETNAM - JOURNAL VOL 10/2020 29