Development of a special connection fracture model for reservoir simulation of fractured reservoirs

Journal of Petroleum Science and Engineering 183 (2019) 106390

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Development of a special connection fracture model for reservoir simulation of fractured reservoirs

T

Manuel Gomes Correia∗, João Carlos von Hohendorff Filho, Denis José Schiozer CEPETRO/FEM - University of Campinas (UNICAMP), Postal Code 6052, 13.083-970, Campinas, São Paulo, Brazil

ARTICLE INFO

ABSTRACT

Keywords: Fractured reservoir Special connections Dual porosity

The significant world oil and gas reserves related to naturally fractured carbonate reservoirs adds new frontiers to the development of upscaling and numerical simulation procedures for reducing simulation time. This work aims to accurately represent fractured reservoirs in reservoir simulators within a shorter simulation time when compared to dual porosity models, based on special connections between matrix and fracture mediums, both modeled in different grid domains of a single porosity flow model. For the definition of special connection fracture model (SCFM), four stages are necessary: (a) construction of a single porosity model with two symmetric structural grids, (b) geomodelling of fracture and matrix properties for the corresponding grid domain, (c) application of special connections through the conventional reservoir simulator to represent the fluid transfer between matrix and fracture medium, (d) calculation of the fracturematrix fluid-transfer. For a proper validation, we apply our methodology in a fractured reservoir type II (tight matrix with flow controlled by fractures) and consider a probabilistic framework regarding geological and dynamic uncertainties. The probabilistic approach of SCFM under several static uncertainties revealed a good dynamic matching with DP. Under three rock-wettability scenarios (water-wet, oil-wet and intermediate-wet) the dynamic matching with DP is preserved. Furthermore, SCFM did not present convergence issues, considering all probabilistic realizations. The results revealed that the new method can be applied to commercial flow simulators in fractured reservoirs and it presents itself as a solution to reduce simulation time without disregarding the upscaling and dynamic representation of dual porosity flow models.

1. Introduction Over the past years the interest in naturally fractured carbonate reservoirs, due to the significant quantities of oil and gas reserves, has led to an increasing concern in representing the multiscale behavior in reservoir simulation and in a reasonable simulation time. For such simulations, the dual-porosity flow model (DP), introduced by Barenblatt et al. (1960) and later by Warren and Root (1963), is the common approach for representing the matrix to fracture fluid transfer. However, the time-consuming simulation based on dual-porosity flow models is higher than single porosity flow models. An alternative for the implicit representation of fractures into single porosity mediums is the use of pseudo-relative permeability curves to represent both matrix and fracture system into a single representative relative permeability curve. Several studies (Al-Otaibi and Al-Majed, 1998; Hearn, 1971; Correia et al., 2015; Fayazi et al., 2016; Saalfeld

∗

et al., 2016) studied the approach of pseudo-relative permeability curves to represent heterogeneous reservoirs into single models. However, the use of single models to represent fractures through pseudo-relative permeability curves is not always practical. For example, if the simulation time required for the flow progress in the fracture system is shorter to the time necessary for the matrix-fracture fluid transfer to occur, then the system is not under a static equilibrium as different dynamic behaviors are occurring at the same time. This behavior endorses a dual-porosity model (Bourbiaux, 2010). Another alternative for implicit representation of fractures into single porosity mediums is the use of unstructured or hybrid grids, such as Voronoi, allowing the use of discrete fracture models (DFM). In addition to the fact that DFMs are numerically and computationally expensive (Mi et al., 2017; Moinfar, 2013; Moinfar et al., 2014), commercial black-oil flow simulators only allow structured grids based on finite difference methods.

Corresponding author. E-mail address: [email protected] (M.G. Correia).

https://doi.org/10.1016/j.petrol.2019.106390 Received 31 August 2018; Received in revised form 19 July 2019; Accepted 14 August 2019 Available online 21 August 2019 0920-4105/ © 2019 Elsevier B.V. All rights reserved.

Journal of Petroleum Science and Engineering 183 (2019) 106390

M.G. Correia, et al.

Local grid refinement is suitable for commercial flow simulators and is a proper solution to represent dynamic trends from a local heterogeneous dynamic behavior. Near-well refinements are widely used to calibrate well productivity from fractured systems (Ding, 2004). However, the extension of this technique to detail dynamic trends for nonwell regions is time-consuming, considering reservoir simulation, mainly regarding probabilistic frameworks (Correia et al., 2018). A method called Embedded Discrete Fracture Model (EDFM), developed by Li and Lee (2008), and extended by Moinfar (2013), is a robust approach and one of the modern techniques to represent fractured reservoir in flow simulation. It consists of representing large-scale fractures explicitly between matrix blocks through non-neighboring connections (NNC), whereas small-scale fractures are modeled implicitly in matrix using a dual continuum approach (Moinfar, 2013). Therefore, the use of EDFM focuses in geological scenarios regarding sporadic large-scale fractures, such as hydraulic fractures, within numerous small-scale fractures (Li and Lee, 2008; Moinfar, 2013). NNC is available in commercial finite difference reservoir simulators and therefore adds an important applicability of this method in existing approaches. Junior et al. (2016) applied this EDFM approach and integrated it with geomechanical models for three field studies, all of these are fractured reservoirs from the Santos Basin. One of the main achievements from this methodology is the shorter CPU time consumption compared with a dual permeability flow model. Ding et al. (2017), combines EDFM with MINC (Multiple interacting continua) approach for a shale reservoir to improve the interactions between fracture and matrix medium. Dachanuwattana et al. (2018), developed a workflow regarding EDFM approach for a history matching procedure. One of the main advantages of EDFM refers to the dualporosity model limitation in representing large-scale fractures such as conventional upscaling methods, as the Oda method (1965), underestimates the output permeability (Li and Lee, 2008).

reservoirs in flow simulation in shorter simulation time when compared to dual porosity (DP), based on special connections between matrix and fracture mediums, both modeled in different grid domains of a single porosity flow model. This model is called special connection fracture model, abbreviated as SCFM. For a detailed analysis of the benefits of applying special connections to a single porosity model to represent fractured reservoir in reservoir simulation we compare the results with the conventional procedure based on a DP. The comparison is made based on dynamic response from flow simulation and time consumption for the probabilistic framework, considering geostatistical realizations and reservoir simulation. This work relies on two assumptions: it assumes a fractured reservoir type II (tight matrix with flow controlled by fractures) and a well-connected fractured system. To the best of our knowledge, SCFM has not been used in the industry for field-scale reservoir simulation. 3. Methodology The proposed methodology follows five main steps: (1) Define a conventional DP, used as reference, (2) Define the SCFM, (3) Apply both procedures for a probabilistic framework considering static and dynamic uncertainties, (4) Compare the dynamic response from flow simulation for both conventional DP and SCFM, (5) Compare the time consumption for the probabilistic framework considering geostatistical realizations and reservoir simulation. The following sections focus on each step in more detail. This methodology assumes a fractured reservoir type II (tight matrix with flow controlled by fractures) and a well-connected fractured system.

1.1. Motivation

3.1. Defining the reference model (DP)

Disregarding this structural scenario, the dual-porosity flow model remains as the conventional approach for fractured reservoirs. Furthermore, EDFM still faces challenges in modeling small fractures in comparable size with matrix grids (Mi et al., 2017), and there is a lack of research regarding a probabilistic approach of EDFM in conventional reservoir simulators. The Oda method is the upscaling method to convert discrete fracture network (DFN) into an equivalent block property. However, it is only valid for highly connected and uniform fractured systems. For low density of discrete fractures, the Oda method overestimates the effective block permeability (Oda, 1985). Furthermore, the block size also influences the output permeability estimates from the Oda method. A small block size or a very large block size could underestimate the permeability (Ahmed Elfeel and Geiger, 2012). Delorme et al. (2008) suggests a routine based in a fracture connectivity index threshold for choosing the numerical (flow-based) or analytical (Oda) method. The block size for dual porosity flow models should represent the fracture connectivity of the system and therefore, as opposed to what occurs in matrix mediums, a small block volume may not guarantee a representative elementary volume (REV). Therefore, the use of a higher grid resolution for the matrix medium and a larger block for fracture system could add important remarks in reservoir simulation, especially to reduce simulation time. However, unlike a single porosity model, different grid resolutions for matrix and fracture system are not possible for DP. EDFM is a close solution for this issue but is used for specific geological scenarios and is not well validated for uncertainty approaches.

The reference model is the conventional DP applied to fractured reservoirs. The upscaled properties from geological matrix and fracture networks are set into the matrix and fracture system, respectively. The fluid transfer between matrix and fracture system is intrinsically calculated from reservoir simulator through Warren and Root (1963) formulation, which is dependent on the following static parameters: (1) fracture connectivity, associated to fracture spacing, (2) block volume and (3) matrix permeability. 3.2. Defining the SCFM For the definition of SCFM, four stages are necessary: (a) construct a single porosity model with two symmetric structural grids, (b) geomodelling of fracture and matrix properties for the corresponding grid domain, (c) apply special connections through the conventional reservoir simulator to represent the fluid transfer between matrix and fracture medium, (d) calculate the correspondent fracture-matrix fluidtransfer, based on Warren and Root (1963) formulation. Fig. 1 shows an excerpt to define the SCFM throughout all stages, using as example a SCFM with a block volume four times larger for fracture domain. One special connection is exemplified for one fracture block with four matrix blocks. Flows in the individual fracture grid domain, matrix grid domain and between both domains are governed by Darcy's equation, implicitly solved by numerical simulator. The main difference is the input of special connections between domains where a transmissibility value (based on geometrical parameters) is required. The details over stages are described below:

2. Objective

3.2.1. Definition of two symmetric structural grids This step consists of defining two similar structural grids in the same

The purpose of this work is to accurately represent fractured 2

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Fig. 1. Stages to define SCFM.

Fig. 2. Production strategy applied to SCFM for one geostatistical realization.

Fig. 3. Production strategy applied to DP for one geostatistical realization.

model (single porosity model). It is important to observe the similarity between both grids for the further approach of special connections. Through conventional flow simulators, DP does not allow different blocks sizes for the fracture and matrix system. However, in a single porosity model, this is possible. Therefore, the block size for the fracture medium can be larger than matrix medium if the upscaling for DFN based on Oda method preserves the static and dynamic behavior for a larger block.

The main concern is the representativeness of fracture connectivity and consequently permeability for the chosen block. The upscaling match of discrete fracture networks to a larger block is validated by histograms and flow simulation by comparing these with DP. 3.2.2. Geomodelling of matrix and discrete fracture networks for each grid domain The geological modeling for matrix and DFN is applied separately 3

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Fig. 4. a. Relative Permeability curve for water-wet cases. b. Relative Permeability curve for intermediate-wet cases. c. Relative Permeability curve for oil-wet cases.

for each grid. Geomodelling is not the aim of this work. However, details over approaches in geomodelling of fractured reservoirs can be seen by Correia et al. (2017). After upscaling, the output properties are fracture spacing in all grid axis (I, J, K), fracture permeability and porosity, and matrix permeability and matrix porosity. These properties are used as input for flow simulation.

mediums into special connections. As is the case for DP, in SCFM the transmissibility between fracture and matrix block should be associated with the matrix-fracture fluid transfer formulation, described in equation (1).

tmf = V

3.2.3. Applying special connections to simulation model For applying the special connections, it is necessary to define which blocks are connected and the respective transmissibility between blocks. The special connections are applied to represent the matrixfracture fluid transfer. Therefore, the flow between fractured or matrix blocks is explicitly calculated from flow simulator. The calculation of transmissibility between fracture and matrix blocks is described below.

where: is the shape factor, representing the characteristic of the fractured rock; is the fluid density; pm is the matrix pressure and pf the fracture pressure; k r is the relative permeability; µ is the viscosity; V is the matrix block volume. However, in SCFM, several parameters (kr , pm , pf , µ , ) must be implicitly set in flow simulator. Therefore, the transmissibility value set in special connections must follow equation (2),

kr (p µ m

pf )

transmissibility = V ,

3.2.4. Transmissibility between fracture and matrix blocks For DP, the transmissibility between matrix and fracture blocks is explicitly calculated by flow simulator through a matrix-fracture fluid transfer term. For SCFM however, as this model is a single porosity flow model, it is necessary to assign the transmissibility between both

(1)

(2)

and, according to Kazemi et al. (1976),

=4 4

km (

1 1 1 + + ) Lx 2 Ly 2 Lz 2

(3)

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Fig. 5. Fracture permeability for SCFM and DP after applying the Oda method, for one geostatistical realization.

Fig. 6. Shape factor for SCFM and DP, for one geostatistical realization.

Fig. 7. Shape factor versus Fracture permeability, for DP.

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Fig. 8. Shape factor versus Fracture permeability, for SCFM.

Fig. 9. Nomenclature for the simulation models according to the probabilistic framework.

Fig. 10. Oil recovery factor for water-wet scenarios.

Fig. 11. Water-cut for water-wet scenarios.

where: km is the matrix permeability (mD) and L is the fracture spacing (m) defined in each direction. The transmissibility must be multiplied by the volume ratio between fracture and matrix block due to the differences in volume between blocks.

3.3. Probabilistic framework The accounting of uncertainty and probabilistic realizations adds an important response to validate our procedure, as different geological scenarios can induce different dynamic scenarios. Furthermore, most of 6

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Fig. 15. Reservoir pressure for intermediate-wet scenarios.

Fig. 12. Reservoir pressure for water-wet scenarios.

Fig. 16. Oil recovery factor for oil-wet scenarios.

Fig. 13. Oil recovery factor for intermediate-wet scenarios.

Fig. 17. Water-cut for oil-wet scenarios.

Fig. 14. Water-cut for intermediate-wet scenarios.

geostatistics and three types of rock wettability (water-wet, intermediate and oil-wet) to evaluate the proposed methodology against a different dynamic response, such as rock-wettability, which has a direct

the field development stages regarding reservoir simulation (history matching, production strategy optimization, etc.) follow probabilistic workflows. This work considers static uncertainties regarding 7

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4. Application The application section focuses on the first three steps of the methodology. We first describe the case study and the structural model for DP and SCFM, which will be used for the further flow simulation step, presented in the results section. 4.1. Description of case study We base our study on Field B, a fractured reservoir type II, from the Campos Basin, Brazil. However, we mask some data (fracture properties and structural data) to be present in this article. As the matrix permeability is generally small, we assume it as a constant of 0.1 mD. Furthermore, the matrix porosity is also assumed as a constant of 15%. DFN has an average length of 300 m and an aperture of 6.0E-04 m. The average intensity (P32) is 0.08. These values are for the base case, before the introduction of uncertainties. Fig. 18. Reservoir pressure for oil-wet scenarios.

4.1.1. Production strategy and operational constraints Figs. 2 and 3 shows the production strategy for the SCFM and DP, respectively, using permeability as example for one geostatistical realization. To prevent an early water breakthrough, eight producers are completed on the reservoir top and five injectors on the base. The bottom hole pressure for injector is limited by a maximum value of 400 kg/cm2 and the water injection rate is set constant at 2500 m³/day for all cases during the entire simulation. The initial pressure for reservoir is 327 kgf/cm2. The recovery mechanism is water injection. The only constraint in operational conditions for producers is the bottomhole pressure control of 220 kgf/cm2. This value is above the bubble point pressure in order to prevent gas production at reservoir conditions.

impact on matrix-fracture interaction (Correia et al., 2016). 3.4. Comparing the dynamic response from flow simulation In this step, we compare all cases for the DP and SCFM, based on the dynamic response from flow simulation (oil recovery, water cut and average reservoir pressure). Each SCFM scenario is compared to the correspondent DP. Furthermore, we add a table with the relative differences between DP and SCFM for the water cumulative production and oil cumulative production. 3.5. Evaluating the time consumption

4.1.2. Dynamic data The relative permeability curves used in this work (Fig. 4) are presented by Correia et al. (2016). The intermediate and oil-wet curves are from a laboratorial experiment by Faerstein (2010) and Ligero and Schiozer (2015), and the water-wet curve is from a real field, from the Campos Basin, Brazil. The relative permeability curve for fracture

The simulation time is compared for DP and SCFM. For a detailed evaluation, variance values (σ2) for simulation time were calculated, and linear regression curves were calculated for simulation time according to matrix-fracture transmissibility, fracture permeability and fracture porosity.

Table 1 Relative difference (%) for water and oil cumulative production between DP and SCFM.

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Fig. 19. a. Water-front for the fracture system of the DP after 9 years. b. Water-front for the matrix system of the DP after 9 years. Table 2 Relative difference (%) for simulation time between DP and SCFM.

Fig. 20. Water-front for the SCFM after 9 years.

4.2. Definition of DP and SCFM For the DP, the grid block measures 100 × 100 × 2 m. For the SCFM, the grid block measures 100 × 100 × 2 m for the matrix domain and 200 × 200 × 4 m for fracture domain. Given the small and continuous permeability values in the matrix, the geological grid resolution is the same as the simulation grid model. For the upscaling of DFN, we applied the Oda method. Fig. 5 shows the histogram with equivalent permeability after the DFN upscaling for one geostatistical realization (example). Despite considering different block sizes for the fracture blocks of DP and SCFM, it is possible to observe a good match between both models. Fig. 6 shows the shape factor for both methods, for the same geostatistical realization. Shape factor for DP is implicitly calculated by flow simulation. The shape factor shows a good match, as it was observed for equivalent fracture permeability. It is important to note that, the analysis through histograms is a static evaluation. In the results section, we present the dynamic comparison. Figs. 7 and 8 show the comparison between shape factor versus fracture permeability for DP and SCFM, respectively. Each point is

Fig. 21. Simulation time (ratio DP/SC) for all simulation models according to rock-wettability.

medium is assumed, as two straight-lines function with endpoints at zero and 100% saturation. By adding a rock type map in simulation model, where fracture and matrix domain are different rock types, we set different relative permeability curves for each domain. 9

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Table 3 Average values for transmissibility, fracture permeability and fracture porosity for each simulation model.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Transmissibility ( V )

Frac Permeability (mD)

Frac Porosity (%)

DP

SCFM

DP

SCFM

DP

SCFM

342 411 101 326 137 360 321 208 251 301 96 201 158 330 195 253 96 171 183 264

335 399 100 315 134 357 317 196 244 295 93 193 152 322 189 246 89 164 175 259

687 491 526 741 129 630 277 1441 333 141 414 1462 331 82 126 300 241 820 200 96

685 490 524 739 129 628 276 1437 332 141 412 1458 330 82 126 300 240 818 200 96

0.005707 0.005662 0.003681 0.00479 0.002191 0.006172 0.004461 0.00531 0.004461 0.003201 0.003337 0.005083 0.003826 0.002439 0.002665 0.003944 0.002773 0.00417 0.003223 0.002451

0.00569 0.005647 0.003668 0.004777 0.002185 0.006154 0.004446 0.005297 0.004446 0.003191 0.003325 0.005069 0.003814 0.002431 0.002655 0.003932 0.002763 0.004159 0.003214 0.002444

Fig. 23. Simulation time according to transmissibility for DP.

Fig. 24. Simulation time according to fracture permeability for SCFM.

Fig. 22. Simulation time according to transmissibility for SCFM.

related to one grid block. So, the number of points is smaller for the SCFM as the number of blocks is smaller. Note that shape factor has a positive correlation with fracture permeability. The shape factor is inversely proportional with fracture spacing. The same as for fracture permeability against fracture spacing. Both graphics show that the enlargement of grid block size for SCFM do not mask the expected positive correlation between fracture permeability and shape factor after apply the Oda method. 4.3. Probabilistic framework We generated 20 realizations of petrophysical properties considering uncertainty in fracture variables (fracture aperture, fracture density and fracture length). These 20 realizations were combined with three types of rock wettability (oil-wet, water-wet and intermediate-wet). Therefore, it resulted in 60 simulation models for SCFM to be compared against 60 simulation models for DP (Fig. 9). The number is the corresponding geostatistical realization.

Fig. 25. Simulation time according to fracture permeability for DP.

For example, SCFM-IW1 and SCFM-WW1 are associated with a different rock-wettability, but for the same geostatistical realization. All geostatistical realizations result in high values of equivalent fracture 10

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(14 and 20) the results extrapolate an acceptable match, considering the three rock-wettability scenarios. This means that geostatistical properties have a direct impact on this discrepancy. Combining these results with Table 1, it is possible to observe that geostatistical realization 14 and 20 have the lowest average fracture permeability (< 100mD), which is associated to the small values of fracture aperture in both realizations, close to 2 mm. Therefore, for lower fracture permeabilities (< 100mD) SCFM did not reproduce the expected results. This divergence can be easily calibrated through pseudo-relative permeability curves. However, this approach is not the aim of this work. Nevertheless, for a type II fracture reservoir, it is not common for the average values to be below 100mD for fracture permeability data. Generally, for fractured reservoirs the fracture permeability is much greater than the matrix permeability (Van Golf-Racht, 1982). Furthermore, realization 14 has presented convergence issues for DP and for the three rock wettability scenarios as explained in the next section. Therefore, a small ratio between fracture and matrix permeability or convergence issues could mask the consistency of results for DP, used as reference. According to Correia et al., (2016), for oil-wet rocks the water saturation is more expressive in the fracture system but for water-wet rocks the matrix-fracture fluid transfer is more relevant for oil recovery due to the importance of imbibition forces. Therefore, the approach of special connections is more significant for water-wet scenarios. Nevertheless, SCFM presents the same response in reservoir simulation as the DP procedure, despite considering different kinetics in matrixfracture fluid transfer. For the calculation of matrix-fracture fluid transfer in SCFM the dynamic parameters (kr , pm , pf , µ , ) are implicitly set in flow simulation. However, considering a long-term production strategy we did not observe concerns on this approach. Figs. 19 and 20 show the water saturation for DP-IW20 and SCFMIW20 case, as example. Despite considering different grid resolution for fractured blocks, the water saturation is similar between both cases. As the assumptions from Oda method (1965) - valid for highly connected and uniform fractured systems – matches with the geological scenario addressed in this work, the results presented in this section were expected. The increase in grid block size did not reproduce discrepancies in the output permeability and shape factor and, therefore, similar outcomes for static properties were predictable. Furthermore, despite considering different kinetics in matrix-fracture fluid transfer by changing the rock wettability, a similar dynamic behavior is also obtained. The next section show the time consumption based on flow simulation.

Fig. 26. Simulation time according to fracture porosity for SCFM.

Fig. 27. Simulation time according to fracture porosity for DP.

5.2. Evaluating the time consumption

permeability but with a tight matrix, the petrophysical characteristic from a fractured reservoir type II.

Fig. 21 shows the relative simulation time (ratio DP/SCFM) for all cases. Ratio below 1 means that SCFM has a higher simulation time than DP. Generally, the DP cases have a higher simulation time. Furthermore, for the geostatistical realization 14 the DP presents convergence issues for the three rock wettability scenarios. The average relative simulation time (ratio DP/SCFM), disregarding the geostatistical realization 14, is 1.03 for water-wet, 1.28 for intermediate-wet and 1.16 for oil-wet. Therefore, for water-wet scenarios the SCFM presents a close number of simulation models with higher or lower simulation time when compared to DP. This means that under imbibition forces, which are more expressive for water-wet rocks, SCFM did not present advantages regarding simulation time, with an exception of geostatistical realization 14, which presents numerical convergence issues for DP. For oil-wet and intermediate-wet scenarios, most DP cases require a higher simulation time for numerical convergence. While conventional reservoirs are often water-wet, fractured reservoirs are mostly intermediate to oil-wet (Chilingar and Yen, 1983). Therefore, SCFM presents a better computational response for the essential wettability scenarios in fractured reservoirs. Table 2 shows the relative difference (%) for simulation time between DP and SCFM. The green values are the relative differences below 10%; yellow, between 10 and 50%; red, higher than 50%. The

5. Results and discussion The results and discussion section explores the two final steps of the methodology and presents the dynamic response and time consumption, based on flow simulation, for all images. 5.1. Comparison of dynamic response from flow simulation for DP and SCFM Figs. 10–18 show the flow simulation results (oil recovery factor, water cut and reservoir pressure), comparing the SCFM and DP cases for each rock wettability scenario. To simplify, we exemplify the results for five cases. Despite considering different rock wettability scenarios all five cases present good matches regarding these objective functions. Table 1 shows the relative differences for water and oil cumulative production between DP and SCFM. The green values are the relative differences below 5%; yellow, between 5 and 10%; red, higher than 10%. The smaller relative difference means that DP and SCFM are quite similar for both objective functions. Note that only for two geostatistical realizations 11

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Fig. 28. Linear regression curves for simulation time according to fracture permeability of the DP and SCFM, for each rock-wettability scenario.

Fig. 29. Linear regression curves for simulation time according to fracture porosity of the DP and SCFM, for each rock-wettability scenario.

50% are positive, which means that for all these cases the simulation time for DP is significantly higher than SCFM. Table 3 shows the average values for transmissibility, fracture permeability and fracture porosity for each geostatistical realization. Close values are expected for DP and SCFM for each geostatistical realization, as for each geostatistical realization the DFN is the same. The small differences are associated to the different block size for fractured blocks. Based on Table 3, it is possible to make a further analysis over the

Table 4 Comparison of DP and SCFM according to variance of simulation time. Variance (σ2)

DP-WW 142 SCFM-WW 2

DP-IW 639 SCFM-IW 11

DP-OW 284 SCFM-OW 7

smaller relative difference means that DP and SCFM are quite similar in simulation time. Positive difference means that DP has a higher simulation time than SCFM. Note that all values for a difference higher than 12

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variability of simulation time according to geostatistical parameters. Figs. 22–27 show the simulation time according to the different uncertainty attributes (transmissibility, fracture permeability, fracture porosity) for DP and SCFM, separately. The values from attributes in horizontal axis are ranked in increasing order. Figs. 22 and 23 show that transmissibility between fracture and matrix blocks do not have a linear correlation with simulation time. However, the simulation time has a linear correlation with average fracture permeability and fracture porosity (Figs. 24–27). Figs. 28 and 29 present the linear regression curves of simulation time according to fracture permeability and fracture porosity, respectively. The geostatistical realization 14 was removed from linear regression analysis as it is a spurious sample, due to numerical convergence issues. It is relevant to observe that, for DP cases, the simulation time has a close fit (higher values of R-squared) with the increasing values of fracture permeability and fracture porosity. The simulation time for SCFM does not present a high correlation with permeability and porosity. These results show that SCFM is less numerically susceptible to changes in average permeability and porosity in comparison to DP. The variance values (σ2) for simulation time were calculated (Table 4). The variance is much higher for DP than for SCFM, showing that simulation time for DP is more sparsely distributed from their average value when compared to SCFM. This proves that time consumption is more susceptible to the probabilistic uncertainties for DP cases. Also, the geostatistical realization with numerical converge issues, related to a DP case, was removed from this analysis.

This work shows that SCFM has a considerable performance regarding a dynamic matching response with DP but within smaller simulation time. This work aims to contribute with a new method that can be applied in commercial flow simulators concerning fractured reservoirs and it presents itself as a solution to reduce simulation time without disregarding the upscaling and dynamic representation of dual porosity flow models. Acknowledgements The authors are grateful to the Center of Petroleum Studies (Cepetro-Unicamp/Brazil), PETROBRAS S/A, Grant No. 0050.0100204.16.9, UNISIM, ANP, Petroleum Engineering Department (DEP-FEM-Unicamp/Brazil) and Energi Simulation Foundation (formerly FCMG) for their support of this work. The authors are also grateful to Schlumberger Information Solution for the use of Petrel® and the Computer Modeling Group for the use of Black Oil reservoir simulator IMEX™. Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.petrol.2019.106390. References Ahmed Elfeel, M., Geiger, 2012. Static and dynamic assessment of DFN permeability upscaling. In: SPE Paper 154369 Presented at the SPE Europec/EAGE Annual Conference, 4-7 June. Al-Otaibi, S.S., Al-Majed, A., 1998. Factors affecting pseudo relative permeability curves. J. Pet. Sci. Eng. 21 (3), 249–261. https://doi.org/10.1016/S0920-4105(98)00070-9. Barenblatt, G.E., Zheltov, I.P., Kochina, I.N., 1960. Basic concepts in the theory of homogeneous liquids in fissured rocks. J. Appl. Math. Mech. (USSR) (24), 1286–1303. Bourbiaux, B., 2010. Fractured reservoir simulation: a challenging and rewarding issue. Oil Gas Sci. Technol. Rev. IFP 65 (2), 227–238. Chilingar, G.V., Yen, T.F., 1983. Some notes on wettability and relative permeabilities of carbonate rocks, II. Energy Sources 7 (1), 67–75. Correia, M.G., Maschio, C., Schiozer, D.J., 2015. Integration of multiscale carbonate reservoir heterogeneities in reservoir simulation. J. Pet. Sci. Eng. 131, 34–50. https:// doi.org/10.1016/j.petrol.2015.04.018. Correia, M.G., Maschio, C., Hohendorff Filho, J.C.V., Schiozer, D.J., 2016. The impact of time-dependent matrix-fracture fluid transfer in upscaling match procedures. J. Pet. Sci. Eng. 146, 752–763. https://doi.org/10.1016/j.petrol.2016.07.039. Correia, M.G., Maschio, C., Schiozer, D.J., 2017. Development of complex layered and fractured reservoir models for reservoir simulation. J. Braz. Soc. Mech. Sci. Eng. 39, 219–233. Correia, M.G., Maschio, C., Schiozer, D.J., 2018. Flow simulation using local grid refinements to model laminated reservoirs. Oil Gas Sci. Technol. Rev. IFP 73, 5. https:// doi.org/10.2516/ogst/2017043. Dachanuwattana, S., Xia, Z., Yu, W., Qu, L., Wang, P., Liu, W., Miao, J., Sepehrnoori, K., 2018. Application of proxy-based MCMC and EDFM to history match a shale gas condensate well. J. Pet. Sci. Eng. 167, 486–497. https://doi.org/10.1016/j.petrol. 2018.04.025. Delorme, M., Atfeh, B., Allken, V., Bourbiaux, B., 2008. Upscaling Improvement for Heterogeneous Fractured Reservoir Using a Geostatistical Connectivity Index. Geostats conference, Santiago, Chile. Ding, D.Y., 2004. Near-well upscaling for reservoir simulations. Oil Gas Sci. Technol. 59 (2), 157–165. Ding, D.Y., Farah, N., Bourbiaux, B., Wu, Y.-S., Mesteri, I., 2017. Simulation of MatrixFracture Interaction in Low-Permeability Fractured Unconventional Reservoirs. SPE Reservoir Simulation Conference, Texas, USA. Faerstein, M., 2010. Impact of Rock Wettability in Production and Oil Recovery. Dissertation, UFRJ, Brazil. Fayazi, A., Bagherzadeh, H., Shahrabadi, A., 2016. Estimation of pseudo relative permeability curves for a heterogeneous reservoir with a new automatic historymatching algorithm. J. Pet. Sci. Eng. 140, 154–163. https://doi.org/10.1016/j.petrol. 2016.01.013. Hearn, C.L., 1971. Simulation of stratified waterflooding by pseudo relative permeability curves. J. Pet. Technol. 23 (7), 805–813. https://doi.org/10.2118/2929-PA. Junior, L.C.S., Santos, L.O.S., Rios, V.S., Araújo, C., Celes, W., Sepehrnoori, K., 2016. Methodology for geomechanically controlled transmissibility through active natural fractures in reservoir simulation. J. Pet. Sci. Eng. 147, 7–14. https://doi.org/10. 1016/j.petrol.2016.04.040. Kazemi, H., Merril, J.R., Porterfield, K.L., Zeman, P.R., 1976. Numerical simulation of water-oil flow in naturally fractured reservoirs. SPE J. 16 (6), 317–326 December. Li, L., Lee, S.H., 2008. Efficient field-scale simulation of black oil in naturally fractured reservoir through discrete fracture networks and homogenized media. SPE Reserv.

6. Conclusions This work proposed special connections to reduce simulation time in fractured reservoirs by modeling fractures and matrix properties into different grid domains through a single porosity model. The comparison between the conventional DP and SCFM showed that:

• Under a probabilistic approach, for an average fracture permeability • • • •

• •

higher than 100 mD, SCFM revealed to be a good dynamic matching with DP; Under three rock-wettability scenarios (water-wet, oil-wet and intermediate-wet) the dynamic matching with DP is preserved, showing that the kinetics of wettability alteration does not affect the SCFM viability; Simulation time for SCFM is less scattered from their average value when compared to DP, showing the minor influence of static and dynamic uncertainties in the numerical convergence; For the intermediate-wet and oil-wet scenarios, most SCFM cases present a shorter simulation time when compared to DP, showing that rock-wettability has a direct impact on SCFM performance regarding time consumption; Simulation time for DP has a higher linear correlation (R-squared) with fracture permeability and fracture porosity compared to SCFM, showing a geostatistical time-dependence effect on reservoir simulation. Once the fracture permeability increases, small pressure changes makes very large changes in saturation and therefore, by increasing the grid block size the oscillation in saturation value is smoothed. Therefore, SCFM reduces the geostatistical time-dependence effect; SCFM did not present convergence issues considering all probabilistic realizations as for larger grid blocks the maximum changes in pressure and saturation are smoothed and, timestep cuts are reduced for each Newtonian iteration; Considering geomodelling approaches, SCFM has the same flexibility as DP to model discrete fracture networks and matrix properties, as both mediums are separated into two domains of a single porosity model; 13

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M.G. Correia, et al. Eval. Eng. 11 (4), 750–758. https://doi.org/10.2118/103901-PA. Ligero, E.L., Schiozer, D.J., 2015. Efeito da Molhabilidade da Rocha Reservatório na Produção de Óleo Leve por Métodos de EOR com CO2. 3rd Brazilian Congress on CO2 in the Oil. Gas and Biofuels Industries, Rio de Janeiro. Mi, L., Yan, B., Jiang, H., An, C., Wang, Y., Killough, J., 2017. Enhanced Discrete Fracture Network model to simulate complex fracture distribution. J. Pet. Sci. Eng. 156, 484–496. https://doi.org/10.1016/j.petrol.2017.06.035. Moinfar, A., 2013. Development of an Efficient Embedded Discrete Fracture Model for 3D Compositional Reservoir Simulation in Fractured Reservoirs Dissertation. University of Texas. Moinfar, A., Varavei, A., Sepehrnoori, K., Johns, R.T., 2014. Development of an efficient embedded discrete fracture model for 3D compositional reservoir simulation in

fractured reservoirs. SPE J. 19, 2. https://doi.org/10.2118/154246-PA. Oda, M., 1985. Permeability tensor for discontinuous rock mass. Geotechnique 35 (4), 483–495. Saalfeld, R., Hohendorff Filho, J.C.V., Schiozer, D., 2016. Simulation of naturally fractured reservoirs using single-porosity equivalent models. In: CILAMCE. XXXVII Iberian Latin American Congress on Computational Methods in Engineering. Brasília. Brazil. Van Golf-Racht, T.D., 1982. Fundamentals of Fractured Reservoir Engineering, Developments in Petroleum Science, vol. 12 Elsevier Scientific Publishing Company, Netherlands. Warren, J.E., Root, P.J., 1963. The behavior of naturally fractured reservoirs. SPE J. 245–255 September.

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