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# Shale Reservoir Drainage Visualized for a Wolfcamp Well (Midland Basin, West Texas, USA)

### Pressure Field

For a hydrostatic reservoir, the initial reservoir pressure P0 is determined by the hydrostatic pressure gradient [ΔPHydrostatic/dL] times reservoir depth (d):

The hydrostatic pressure gradient typically is [ΔPHydrostatic=0,45psi/ft]. The pressure change due to production or injection is obtained from:

μ is the viscosity (psi·day or Pa·s, or Poise), k is the permeability (ft2 or m2 or Darcy).

The reservoir pressure consequent to the well flow will be:

Each pressure field plot is generated by evaluating the pressure differential Equation (31) for a large number of complex-valued coordinates z, in conjunction with Matlab’s command ‘contourf’.

### Streamline Tracing

The streamline tracing method uses a first order Eulerian displacement scheme. The initial particle position is z0 (in complex coordinates) at time t0 (unit in (s); see Weijermars et al. [7] for a non-dimensional approach). The velocity field V(z,t) needs to be solved to obtain the velocity vector components (vx and vy) for all positions of tracer elements at times t1, denoted by z1(t1):

The term v(z0(t0)) in expression (33) is the velocity of the traced particle at time t0 and location z0, for which the velocity field V(z,t) is used. The size of the time step is Δt. The locations of tracer elements at time tj are:

In Matlab we evaluate expression (34) by using the corresponding velocity field V(z,t) as follows:

In order to maintain smooth streamlines and time-of-flight contours (TOFCs), the time step Δt needs to be smaller when the strength of point sources/sinks increases. The velocity field plots in this study map the spatial variation in the magnitude of the fluid phase velocity, which is obtained by taking the absolute value |V(z,t)| at a given time t. Each velocity magnitude contour plot uses a large number of complex-valued coordinates z, in conjunction with Matlab’s command ‘contourf’.

## Results

Our flow visualization method can show both the matrix drainage by the main fractures and any deeper reach into the matrix when micro-cracks occur as offshoots from the main fractures. First we show the drainage area when no micro-cracks develop. The locations of the main fractures for each stage of the multi-staged fractures in our prototype well were inferred from the micro-seismic image recorded by the operator during the frack job (Figure 9a). The images of Figure 9b–d show the reservoir response to drainage by a hydraulically fractured, horizontal well after 1 month of production. A summary of input data is given in Table 1. The table includes a description of hydraulic fracture geometry and dimensions, the well productivity based on Duong parameters, original oil in place (OOIP) parameters, and other reservoir parameters.

Figure 9. (a) Map view of imaged seismic events centered about a horizontal well during fracking of each stage (individual color bands). Adjacent seismic imaging well is highlighted (olive green) in left of image. (b) Velocity field (m/s) in after 1 month of well-life; field dimensions in m. (c) Pressure change (MPa) relative to the original reservoir pressure. (d) Detailed pressure gradient map of central region of the fracked well.

Table 1. Key parameters used in this study.

Table

The velocity plot (Figure 9b) reveals that the oil migrates fastest near the tips of the fractures, with an interior region between the fractures draining much slower. Classical analytical flow models [92] asserted that no fluid flow would occur from matrix into the fracture tips. Only flow of matrix normal to the fracture is assumed, which is clearly not what we see in our CAM simulations where flow near the fracture is fastest and not following sub-parallel or linear streamlines, but more complex particle paths. The simplification of linear flow would lead to overestimation of diffusion to the fracture, particularly when such fractures are short.

### Velocity Field Versus Pressure Plots

The velocity field near hydraulic fractures is rarely visualized in commercial reservoir simulators [93,94]. Pressure depletion plots (Figure 9c) are commonly used as a proxy for drainage of the reservoir. The reservoir pressure when production starts is replaced by an evolving pressure gradient. The steepest pressure gradients occur at the onset of production, with local variations of higher and lower gradients. Figure 9c has the steepest pressure gradients between the red and blue zones (with tighter contour spacing), which broadly coincide with the region of higher flow velocities peripheral to the frac tips (Figure 9b).

In commercial simulators, pressure plots resemble the analytical solution of Figure 9c but have limited resolution due to finite grid size and computational time limitations. Our model based on closed-form formulae provides infinite resolution and allows for the computation of a detailed pressure contour map for the central region between the fracs (Figure 9d).

Due to relatively shallow pressure gradients, flow in the central zone is slowest. Contoured for smaller changes, pressure gradients still exist in the central region (Figure 9d), but pressure differences are more subtle than elsewhere in the fracked region. The shallow pressure gradient is the reason why the flow is nearly stagnant in the central region between the frac clusters (Figure 9b).

### Implications for Production Efficiency

A prototype well that had 50 month historic production data was history matched with a decline curve using a Duong model [81,91,95]. The well type curve for Wolfcamp shale in the Midland Basin, West Texas, is given in Figure 10a. The production forecast was used for production allocation to individual fractures proportionally to each frac stage (see Section 3.6). The width of the drained matrix region around the fractures is visualized using the allocated well flux (Figure 10b).

Even after 40 years of production, large portions of the matrix between the fractures remain un-drained. The drainage area and pressure plots were generated using a flow reversal principle: the produced fluid was injected back into the reservoir at the same rate as produced, which allows the construction of time-of-flight contours for the drained rock volume (Figure 10b). All pressure plots show positive pressure, but are equivalent to pressure depletion plots by taking the negative of the pressure scale.

Figure 10. (a) History matching monthly production (bbls/month, red curve) for 50 months with Duong decline curve (blue curve). (b) Drained area next to the main hydraulic fractures after 480 months (40 years). (c) Cumulative production (in bbls; left axis) and recovery factor (in %; right axis) over time (in months, horizontal axis).

### Recovery Factors

Figure 10b shows the total drainage area for 40 years of production and the narrow region drained corresponds to the cumulative production given by the type curve (Figure 10a). OOIP of the prototype well was estimated for a well spacing of 320 acre/well resulting in a recovery factor of only 6% after 40 years, with 4% recovery already achieved after 5 years.

Figure 10c plots the cumulative production and the corresponding recovery factor appreciation over a 40 year field life. These results are based on the history matched and drained rock volume (Figure 10b) obtained by our simulation method. Large undrained (dead) zones occur between the principal fracture zones. However, when micro-cracks are present to drain the deeper matrix, the dead zones may be narrower, which was further investigated.

### Micro-Cracks

The CAM model can be adapted to account for drainage by micro-cracks, were these to occur, that reach deeper into the matrix. Such fractures are currently below the resolution of fracture diagnostics. Unknown matrix micro-cracks acting as hydraulic conduits may drain the matrix further away from the main fractures. Assuming micro-cracks may develop in hydraulically fractured shale reservoirs, the degree of micro-crack contribution to shale drainage will vary with the density and hydraulic conductivity of such cracks.

The flow support that hydraulic fractures receive from the microcracks was simulated using line dipoles (see Section 3.5). The strength of dipoles was systematically varied relative to the hydraulic fractures (modeled by interval sources) to show the effect on the drained region in our model. Examples of micro-crack assisted flow in the central region of the multistage fractured well of Figure 9 are shown in Figure 11.

What the dipole fracs drain around the micro-crack tips as a consequence of their conductivity being higher than that of the matrix is transported toward the main hydraulic fractures. The depth of the drained area and length of the dipoles representing the micro-cracks is determined by the assigned strength of the dipole relative to that of the interval source used to model the hydraulic fracs. The combined dipole strength of micro-cracks that are connected to a particular hydraulic fracture is variable: equal to the hydraulic fracture strength (Figure 11a), five times the hydraulic fracture strength (Figure 11b), and 10 times the hydraulic fracture strength (Figure 11c).

Figure 11. (a–c) Row 1: Pressure contour maps (in MPa) after 1 month drainage for the same central region of Figure 9d, now including flow through micro-cracks normal to the main fractures. Dipole strength of (a) is scaled to match hydraulic conductivity of the main fractures. Cases (b,c) are respectively 5 and 10 times stronger. (a–c) Row 2: Velocity field (log scale in m·s−1) for cases a-c in Row 1. Local velocities are higher near the micro-crack tips when conductivity is high. Overall area drained stays constant.

The pressure plots (Figure 11a–c, row 1) show pressure contours begin to bungle near the micro-cracks when the hydraulic conductivity is larger. Such relatively steep pressure gradients are absent in Figure 9d, which is free of micro-cracks. The flow velocities are clearly enhanced around the micro-cracks (Figure 11a–c, row 2), due to the steep pressure gradients induced near the cracks (Figure 11a–c, row 1).

As a result, the micro-cracks would extend the drainage region deeper into the matrix (Figure 11) as compared to situations where micro-cracks are absent (Figure 10b). The velocity peaks are confined to the immediate vicinity of the micro-cracks (Figure 11b,c, row 2), which concur with the locations were pressure gradient are steepest and pressure contours bungle (Figure 11b,c, row 1).

The consequent drained region for each case is visualized in Figure 12. The higher resolution images show locally enhanced pressure gradients (Figure 11, row 1) and flow velocity peaks (Figure 11, row 2) occur near the micro-cracks, which facilitate the flow from the deeper matrix regions toward the main fractures.

Figure 12. (a–d) Drained region after 1 month of production (red contours). (a) micro-cracks absent, (b,c) micro-cracks with increasing hydraulic conductivity as in Figure 11a–c.

## Discussion

Our study suggests that matrix drainage by hydraulic fractures can be accelerated by micro-crack systems. Micro-crack tips drain deeper into the matrix as a consequence of their conductivity being higher than that of the matrix and thus may act as hydraulic fairways. The results show that micro-cracks will improve the drainage depth, but the existence of such micro-cracks is beyond the resolution of current fracture diagnostics.

Although our method can visualize flow around the main fractures and any micro-cracks at high resolution, uncertainty remains about the precise location and geometric complexity of the micro-cracks. The relative strengths of dipoles, in our model, may be systematically varied relative to the interval source to show the effect of the micro-crack drainage with different hydraulic conductivity. Darcy’s law diffusion based flow descriptions was used to characterize the drainage of the SRV.

This assumption implies nano-scale effects such as osmosis and capillary effects do not invalidate the macroscopic flow description. Future work should account for such nano-scale effects. In any case, micro-cracks would enhance and deepen the matrix area drained by the main hydraulic fractures.

Micro-cracks incorporated in the simulation method are synthetic and hypothetical. Such micro-cracks may contribute to the productivity of the well. However, any net gain in productivity is not accounted for in our model, because the net flux is history matched against the actual production, which fixes the 40 year production curve.

What the model shows is that the drained rock volume, in the absence of micro-cracks, remains confined to the immediate vicinity of the hydraulic main fractures. Consequently, we are not underestimating well productivity or recovery factors. The main issue is that the region that will be drained may shift to the interior of the matrix (away from the main fractures) due the presence of micro-cracks.

Several simplifying assumptions are acknowledged. Multiphase flow is not included in our current model, for Wolfcamp B wells, reservoir pressures do not stay above the bubble-point for the lifetime of a well. However, we think the effect on well productivity in the oil window is immaterial due to the low gas content of the wells and late life well productivity contributes little to the cumulative production of the well. Although multiphase flow is not included in our current model, bubble-point effects can certainly be incorporated (subject of new work).

Our conclusion that the highest gradients will occur around the fracture tips, may be nuanced when proppant density varies and premature screen out occurs at fracture tips during the fracture treatment. When the flux around the tip is limited by the connectivity from the tip of the fracture to the wellbore, the flux will shift inward, which is accounted for in a forthcoming paper from our research group [96].

## Conclusions

We conclude that even when micro-cracks develop minor splays into the matrix, the region between the larger master fractures will remain largely un-drained. The flow velocity and pressure gradients for both the main hydraulic fractures and micro-cracks will be largest near the fracture tips. The smallest velocities and flow stagnation occur in the zones between competing fractures, where the pressure gradients are shallower and become zero in pressure culmination points.

The consequent oil entrapment between fractures would not occur in conventional, high-permeability reservoirs, because even slowly moving reservoir fluids would still reach the well on the time scale of the field life (5–50 years). However, unconventional reservoir rocks such as shale with permeabilities in the nano-Darcy range have ultra-low flow rates, even near the fracture tips, in the order of 10−7 m·s−1, the rate at which finger nails grow.

The cluster spacing in the Wolfcamp well studied here is already tight (60 ft). Closer spacing of the fractures would accelerate early production but still creates flow stagnation points between the fractures. However, the un-drained oil occurring between the closely spaced fractures (in low permeability reservoirs with ultra-low flow rates) may be recovered by the refracking of shale wells with perforations placed midway between the prior cluster spacing and fracture initiation points. The second generation of hydraulic fractures will tap into the earlier, stagnant flow regions. The refracs will improve the recovery factors and this may boost well economics accordingly.

Author Contributions

Conceptualization, R.W.; Methodology, R.W.; Software, A.v.H. and R.W.; Validation, R.W. and A.v.H.; Formal Analysis, R.W. and A.v.H.; Investigation, R.W. and A.v.H.; Resources, R.W.; Data Curation, R.W.; Writing-Original Draft Preparation, R.W.; Writing-Review & Editing, R.W.; Visualization, R.W. and A.v.H.; Supervision, R.W.; Project Administration, R.W.; Funding Acquisition, R.W.

Funding

Funds for this study were provided by the Texas A&M Engineering Station (TEES) and the Crisman-Berg Hughes Consortium. Proprietary well data were made available by Pioneer with support of both the Texas Oil and Gas Institute and University Lands.

Conflicts of Interest

The authors declare no conflict of interest.

References

Weijermars, R. US shale gas production outlook based on well roll-out rate scenarios. Appl. Energy 2014, 124, 283–297.

Weijermars, R.; Paradis, K.; Belostrino, E.; Feng, F.; Lal, T.; Xie, A.; Villareal, C. Re-appraisal of the Bakken Shale Play: Accounting for Historic and Future Oil Prices and applying Fiscal Rates in North Dakota, Montana and Saskatchewan. Energy Strategy Rev. 2017, 16, 68–95.

Weijermars, R.; Sorek, N.; Seng, D.; Ayers, W. Eagle Ford Shale Play Economics: U.S. versus Mexico. J. Nat. Gas Sci. Eng. (JNGSE) 2017, 38, 345–372.

Weijermars, R.; Sun, Z. Regression Analysis of Historic Oil Prices: A Basis for Future Mean Reversion Price Scenarios. Glob. Finance J. 2018, 35, 177–201.

Task Force. Unconventional Reserves Taskforce—Report to Participating Societies. Final Report—1st March 2016. Available online: https://www.spwla.org/Documents/SPWLA/TEMP/Unconventional%20Taskforce%20Final%20Report.pdf (accessed on 25 June 2018).

Nelson, R.; Zuo, L.; Weijermars, R.; Crowdy, D. Outer boundary effects in a petroleum reservoir (Quitman field, east Texas): Applying improved analytical methods for modelling and visualization of flood displacement fronts. J. Pet. Sci. Eng. 2017, 166, 1018–1041.

Weijermars, R.; van Harmelen, A.; Zuo, L. Controlling flood displacement fronts using a parallel analytical streamline simulator. J. Pet. Sci. Eng. 2016, 139, 23–42.

Weijermars, R.; van Harmelen, A.; Zuo, L.; Nascentes, I.A.; Yu, W. High-Resolution Visualization of Flow Interference between Frac Clusters (Part 1): Model Verification and Basic Cases. In Proceedings of the Unconventional Resources Technology Conference, Austin, TX, USA, 24–26 July 2017.

Van Harmelen, A.; Weijermars, R. Complex analytical solutions for flow in hydraulically fractured hydrocarbon reservoirs with and without natural fractures. Appl. Math. Modell. 2018, 56, 137–157.

Yu, W.; Wu, K.; Zuo, L.; Tan, X.; Weijermars, R. Physical Models for Inter-Well Interference in Shale Reservoirs: Relative Impacts of Fracture Hits and Matrix Permeability. In Proceedings of the SPE Unconventional Resources Technology Conference, San Antonio, TX, USA, 1–3 August 2016.

Cipolla, C.L.; Wallace, J. Stimulated Reservoir Volume: A Misapplied Concept? Presented at the SPE Hydraulic Fracturing Technology Conference, The Woodlands, TX, USA, 4–6 February 2014.

Wu, R.; Kresse, O.; Weng, X.; Cohen, C.; Gu, H. Modeling of Interaction of Hydraulic Fractures in Complex Fracture Networks. Presented at the SPE Hydraulic Fracture Technology Conference, The Woodlands, TX, USA, 6–8 February 2012.

Grechka, V.; Li, Z.; Howell, B.; Garcia, H.; Wooltorton, T. High-resolution microseismic imaging. Leading Edge 2017, 36, 822–828.

Karimi-Fard, M.; Durlofsky, L.J.; Aziz, K. An efficient discrete fracture model applicable for general purpose reservoir simulators. In Proceedings of the SPE Reservoir Simulation Symposium, Houston, TX, USA, 3–5 February 2003.

Berkowitz, B. Characterizing flow and transport in fractured geological media: A review. Adv. Water Resour. 2002, 25, 861–884.

Neuman, S.P. Trends, prospects and challenges in quantifying flow and transport through fractured rocks. Hydrogeol. J. 2005, 13, 124–147.

Geiger, S.; Dentz, M.; Neuweiler, I. A Novel Multi-rate Dual-porosity Model for Improved Simulation of Fractured and Multi-porosity Reservoirs. In Proceedings of the SPE Reservoir Characterisation and Simulation Conference and Exhibition, Abu Dhabi, UAE, 9–11 October 2011.

Flemisch, B.; Berre, W.; Boon, A.; Fumagalli, N.; Schwenck, A.; Scotti, I.; Stefanson, A. Tatomir. Benchmarks for single-phase flow in fractured porous media. arXiv, 2017.

Barenblatt, G.I.; Zheltov, I.P.; Kochina, I.N. Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks. J. Appl. Math. 1960, 24, 1286–1303.

Warren, J.E.; Root, P.J. The behavior of naturally fractured reservoirs. SPE J. 1963, 3, 245–255.

Lu, H.; Di Donato, G.; Blunt, M.J. General Transfer Functions for Multiphase Flow in Fractured Reservoirs. SPE J. 2008, 13.

Al-Kobaisi, M.; Ozkan, E.; Kazemi, H. A hybrid numerical-analytical model of finite-conductivity vertical fractures intercepted by a horizontal well. SPE Res. Eval. Eng. 2006, 9, 345–355.

Mason, G.; Fischer, H.; Morrow, N.R. Correlation for the effect of fluid viscosities on counter-current spontaneous imbibition. J. Pet. Sci. Eng. 2010, 72, 195–205.

Kazemi, H.; Merrill, L.S.; Porterfield, K.L.; Zeman, P.R. Numerical Simulation of Water-Oil Flow in Naturally Fractured Reservoirs. Soc. Pet. Eng. J. 1976, 16.

Lim, K.T.; Aziz, K. Matrix-fracture transfer shape factors for dual-porosity simulators. J. Pet. Sci. Eng. 1995, 13, 169–178.

Rangel-German, E.R.; Kovscek, R.A. Time-dependent matrix-fracture shape factors for partially and completely immersed fractures. JPSE 2006, 54, 149–163.

Sarma, P.; Aziz, K. New transfer functions for simulation of naturally fractured reservoirs with dual-porosity models. SPEJ 2006, 11, 328–340.

Saboorian-Jooybari, H.; Ashoori, S.; Mowazi, G. Development of an Analytical Time-Dependent Matrix/Fracture Shape Factor for Countercurrent Imbibition in Simulation of Fractured Reservoirs. Transp. Porous Med. 2012, 92, 687–708.

Olorode, O.; Freeman, C.M.; Moridis, G.; Blasingame, T.A. High-Resolution Numerical Modeling of Complex and Irregular Fracture Patterns in Shale-Gas Reservoirs and Tight Gas Reservoirs. SPE Reserv. Eval. Eng. 2013, 16.

Gale, J.; Laubach, S.; Olson, J.; Eichhubl, P.; Fall, A. Natural fractures in shale: A review and new observations. AAPG Bull. 2014, 98, 2165–2216.

Li, B. Natural Fractures in Unconventional Shale Reservoirs in US and their Roles in Well Completion Design and Improving Hydraulic Fracturing Stimulation Efficiency and Production. In Proceedings of the SPE Annual Technical Conference and Exhibition, Amsterdam, The Netherlands, 27–29 October 2014.

Ouenes, A.; Bachir, A.; Khodabakhshnejad, A.; Aimene, Y. Geomechanical modeling using poro-elasticity to prevent frac hits and well interferences. First Break 2017, 35, 71–76.

Ouenes, A.; Hartley, L.J. Integrated Fractured Reservoir Modeling Using Both Discrete and Continuum Approaches. In Proceedings of the SPE Annual Technical Conference and Exhibition, Dallas, TX, USA, 1–4 October 2000.

Alfred, D.; Ramirez, B.; Rodriguez, J.; Hlava, K.; Williams, D. An integrated approach to reservoir characterization and geo-cellular modeling in an unconventional reservoir: The Woodford play. In Proceedings of the 2013 Unconventional Resources Technology Conference, Denver, CO, USA, 26–31 October 2013.

Elfeel, A.M.; Jamal, S.; Enemanna, C.; Arnold, D.; Geiger, S. Effect of DFN Upscaling on History Matching and Prediction of Naturally Fractured Reservoirs. In Proceedings of the EAGE Annual Conference & Exhibition incorporating SPE Europec, London, UK, 10–13 June 2013.

Ouenes, A.; Richardson, S.; Weiss, W.W. Fractured Reservoir Characterization and Performance Forecasting Using Geomechanics and Artificial Intelligence. In Proceedings of the SPE Annual Technical Conference and Exhibition, Dallas, TX, USA, 22–25 October 1995.

Ouenes, A. Practical Application of Fuzzy Logic and Neural Networks to Fractured Reservoir Characterization. Comput. Geosci. 2000, 26, 953–962.

Jenkins, C.; Ouenes, A.; Zellou, A.; Wingard, J. Quantifying and predicting naturally fractured reservoir behavior with continuous fracture models. AAPG Bull. 2009, 93, 1597–1608.

Aimene, Y.E.; Nairn, J.A. Modeling Multiple Hydraulic Fractures Interacting with Natural Fractures Using the Material Point Method. In Proceedings of the SPE/EAGE European Unconventional Resources Conference and Exhibition, Vienna, Austria, 25–27 February 2014.

Aimene, Y.; Ouenes, A. Geomechanical modeling of hydraulic fractures interacting with natural fractures—Validation with microseismic and tracer data from the Marcellus and Eagle Ford. Interpretation 2015, 3, SU71–SU88.

Belytschko, T.; Krongauz, Y.; Organ, D.; Fleming, M.; Krysl, P. Meshless Methods: An Overview and Recent Developments. Comput. Appl. Mech. Eng. 1996.

Ouenes, A.; Umholtz, N.; Aimene, Y. Using geomechanical modeling to quantify the impact of natural fractures on well performance and microseismicity: Application to the Wolfcamp, Permian Basin, Reagan County, Texas. Interpretation 2016, 4.

Paryani, M.; Poludasu, S.; Sia, D.; Bachir, A.; Ouenes, A. Estimation of Optimal Frac Design Parameters for Asymmetric Hydraulic Fractures as a Result of Interacting Hydraulic and Natural Fractures—Application to the Eagle Ford. In Proceedings of the SPE Eastern Regional Meeting, Canton, OH, USA, 13–15 September 2016.

Du, S.; Yoshida, N.; Liang, B.; Chen, J. Dynamic Modeling of Hydraulic Fractures Using Multisegment Wells. SPE J. 2016, 21.

Sun, J.; Schechter, D.S. Optimization-Based Unstructured Meshing Algorithms for Simulation of Hydraulically and Naturally Fractured Reservoirs with Variable Distribution of Fracture Aperture, Spacing, Length and Strike. In Proceedings of the SPE Annual Technical Conference and Exhibition, Amsterdam, The Netherlands, 27–29 October 2014.

Sun, J.; Schechter, D. Investigating the Effect of Improved Fracture Conductivity on Production Performance of Hydraulically Fractured Wells: Field-Case Studies and Numerical Simulations. J. Can. Pet. Technol. 2015, 54.

Singh, G.; Amanbek, Y.; Wheeler, M.F. Adaptive Homogenization for Upscaling Heterogeneous Porous Medium. In Proceedings of the SPE Annual Technical Conference and Exhibition, San Antonio, TX, USA, 9–11 October 2017.

Houze, O.P.; Tauzin, E.; Allain, O.F. New Methods to Deconvolve Well-Test Data Under Changing Well Conditions. In Proceedings of the SPE Annual Technical Conference and Exhibition, Florence, Italy, 19–22 September 2010.

Moridis, G.J.; Blasingame, T.A.; Freeman, C.M. Analysis of Mechanisms of Flow in Fractured Tight-Gas and Shale-Gas Reservoirs. In Proceedings of the SPE Latin American and Caribbean Petroleum Engineering Conference, Lima, Peru, 1–3 December 2010.

Wang, S.; Chen, S. A Novel Bayesian Optimization Framework for Computationally Expensive Optimization Problem in Tight Oil Reservoirs. In Proceedings of the SPE Annual Technical Conference and Exhibition, San Antonio, TX, USA, 9–11 October 2017.

Yu, W.; Luo, Z.; Javadpour, F.; Varavei, A.; Sepehrnoori, K. Sensitivity analysis of hydraulic fracture geometry in shale gas reservoirs. J. Pet. Sci. Eng. 2014, 113, 1–7.

Yu, W.; Xu, Y.; Weijermars, R.; Wu, K.; Sepehrnoori, K. Impact of Well Interference on Shale Oil Production Performance: A Numerical Model for Analyzing Pressure Response of Fracture Hits with Complex Geometries. In Proceedings of the SPE Hydraulic Fracturing Technology Conference and Exhibition, The Woodlands, TX, USA, 24–26 January 2017.

Chen, S.; Li, H.; Yang, D. Optimization of Production Performance in a CO2 Flooding Reservoir under Uncertainty. J. Can. Pet. Technol. 2010, 49.

Ma, M.; Chen, S.; Abedi, J. Equation of State Coupled Predictive Viscosity Model for Bitumen Solvent-Thermal Recovery. In Proceedings of the EUROPEC 2015, Madrid, Spain, 1–4 June 2015.

Onwunalu, J.E.; Durlofsky, L. A New Well-Pattern-Optimization Procedure for Large-Scale Field Development. SPE J. 2011, 16.

Isebor, O.J.; Echeverría Ciaurri, D.; Durlofsky, L.J. Generalized Field-Development Optimization with Derivative-Free Procedures. SPE J. 2014, 19.

Witherspoon, P.A.; Wang, J.S.Y.; Iwai, K.; Gale, J.E. Validity of Cubic Law for fluid flow in a deformable rock fracture. Water Resour. Res. 1980, 16, 1016–1024.

Zimmerman, R.W.; Bodvarsson, G.S. Hydraulic conductivity of rock fractures. Transp. Porous Media 1996, 23, 1–30.

Zimmerman, R.W.; Kumar, S.; Bodvarsson, G.S. Lubrication theory analysis of the permeability of roughwalled fractures. Int. J. Rock. Mech. Min. Sci. Geomech. Abstr. 1991, 28, 325–331.

Zimmerman, R.W.; Chen, D.W.; Cook, N.G.W. The effect of contact area on the permeability of fractures. J. Hydrol. 1992, 139, 79–96.

Pyrak-Nolte, L.J.; Morris, J.P. Single fractures under normal stress: The relation between fracture specific stiffness and fluid flow. Int. J. Rock Mech. Min. Sci. 2000, 37, 245–262.

Sisavath, S.; Al-Yaarubi, A.; Pain, C.C.; Zimmermann, R.W. A simple model for deviations from the cubic law for a fracture undergoing dilation or closure. Pure Appl. Geophys. 2003, 160, 1009–1022.

Zimmerman, R.W. A simple model for coupling between the normal stiffness and the hydraulic transmissivity of a fracture. In Proceedings of the 42nd US Rock Mechanics, 2nd US–Canada Rock Mechanics Symposium, San Francisco, CA, USA, 29 June–2 July 2008.

Zimmerman, R.W.; Al-Yaarubi, A.H.; Pain, C.C.; Grattoni, C.A. Nonlinear regimes of fluid flow in rock fractures. Int. J. Rock Mech. Min Sci. 2004, 41, 163–169.

Vilarrasa, V.; Koyama, T.; Neretnieks, I.; Jing, L. Shear-induced flow channels in a single rock fracture and their effect on solute transport. Transp. Porous Media 2011, 87, 503–523.

Chen, Z.; Qian, J.; Qin, H. Experimental study of the non-Darcy flow and solute transport in a channeled single fracture. Hydrodynamics 2011, 23, 745–751.

Yasuhara, H.; Polak, A.; Mitani, Y.; Grader, A.S.; Halleck, P.M.; Elsworth, D. Evolution of fracture permeability through fluid–rock reaction under hydrothermal conditions. Earth Planetary Sci. Lett. 2006, 244, 186–200.

Chaudhuri, A.; Rajaram, H.; Viswanathan, H. Fracture alteration by precipitation resulting from thermal gradients: Upscaled mean aperture-effective transmissivity relationship. Water Resour. Res. 2012, 48, W01601.

Fujita, Y.; Data-Gupta, A.; King, M.J. A comprehensive reservoir simulator for Unconventional reservoirs based on the fast-marching method and diffusive time of flight. In Proceedings of the SPE Reservoir Simulation Symposium, Houston, TX, USA, 23–25 February 2015.

King, M.J.; Wang, Z.; Datta-Gupta, A. Asymptotic Solutions of the Diffusivity Equation and Their Applications. In Proceedings of the 78th EAGE Conference and Exhibition, Vienna, Austria, 30 May–2 June 2016.

Kuchuk, F.J. Radius of Investigation for Reserve Estimation from Pressure Transient Well Tests. In Proceedings of the SPE Middle East Oil and Gas Show and Conference, Manama, Bahrain, 15–18 March 2009.

Xie, J.; Yang, C.; Gupta, N.; King, M.; Datta-Gupta, A. Depth of Investigation and Depletion Behavior in Unconventional Reservoirs Using Fast Marching Methods. In Proceedings of the SPE Europec/EAGE Annual Conference, Copenhagen, Denmark, 4–7 June 2012.

Yang, C.; Sharma, V.; Datta-Gupta, A.; King, M. A Novel Approach for Production Transient Analysis of Shale Gas/Oil Reservoirs. In Proceedings of the SPE Unconventional Resources Technology Conference, San Antonio, TX, USA, 20–22 July 2015.

Weijermars, R.; Nascentes Alves, I. High-Resolution Visualization of Flow Velocities Near Frac-Tips and Flow Interference of Multi-Fracked Eagle Ford Wells, Brazos County, Texas. J. Pet. Sci. Eng. 2018, 165, 946–961.

Muskat, M. Physical Principles of Oil Production; McGraw-Hill: New York, NY, USA, 1949.

Muskat, M. The Theory of Potentiometric Models. Trans. AIME 1949, 179, 216–221.

Doyle, R.E.; Wurl, T.M. Stream Channel Concept Applied to Waterflood Performance Calculations for Multiwell, Multizone, Three-Component Cases. J. Pet. Tech. 1971, 23, 373–380.

Higgins, R.V.; Leighton, A.J. Matching Calculated with Actual Waterflood Performance by Estimating Some Reservoir Properties. J. Pet. Tech. 1974, 26, 501–506.

Weijermars, R.; van Harmelen, A. Breakdown of doublet re-circulation and direct line drives by far-field flow: Implications for geothermal and hydrocarbon well placement. Geophys. J. Int. (GJIRAS) 2016, 206, 19–47.

Weijermars, R.; Van Harmelen, A. Advancement of Sweep Zones in Waterflooding: Conceptual Insight and Flow Visualizations of Oil-withdrawal Contours and Waterflood Time-of-Flight Contours using Complex Potentials. J. Pet. Explor. Prod. Technol. 2017, 7, 785–812.

Weijermars, R.; van Harmelen, A.; Zuo, L. Flow Interference Between Frac Clusters (Part 2): Field Example From the Midland Basin (Wolfcamp Formation, Spraberry Trend Field) with Implications for Hydraulic Fracture Design. In Proceedings of the SPE/AAPG/SEG Unconventional Resources Technology Conference, Austin, TX, USA, 24–26 July 2017.

Olver, P.J. Complex Analysis and Conformal Mappings, Lecture Notes 26 January 2017. Available online: http://www-users.math.umn.edu/~olver/ln_/cml.pdf (accessed on 28 March 2017).

Pólya, G.; Latta, G. Complex variables, 1st ed.; Wiley: New York, NY, USA, 1974.

Brilleslyper, M.A.; Dorff, J.M.; McDougall, J.M.; Rolf, J.S.; Schaubroeck, L.E.; Stankewitz, R.L.; Stephenson, K. Explorations in Complex Analysis; American Mathematical Society: Providence, RI, USA, 2012.

Weijermars, R. Visualization of space competition and plume formation with complex potentials for multiple source flows: Some examples and novel application to Chao lava flow (Chile). J. Geophys. Res. Solid Earth 2014, 119, 2397–2414.

Weijermars, R.; van Harmelen, A. Quantifying Velocity, Strain Rate and Stress Distribution in Coalescing Salt Sheets for Safer Drilling. Geophys. J. Int. (GJIRAS) 2015, 200, 1483–1502.

Sato, K. Complex Analysis for Practical Engineering, 1st ed.; Springer International Publishing: New York, NY, USA, 2015.

Dugdale, D. Essentials of Electromagnetism, 1st ed.; American Institute of Physics: New York, NY, USA, 1993.

Moran, J. An Introduction to Theoretical and Computational Aerodynamics, 1st ed.; John Wiley & Sons: New York, NY, USA, 1984.

Graebel, W. Advanced Fluid Mechanics, 1st ed.; Academic Press (Elsevier Inc.): London, UK, 2007.

Duong, A.N. Rate-Decline Analysis for Fracture-Dominated Shale Reservoirs. SPE Reserv. Eval. Eng. 2011, 14, 377–387.

Guppy, K.H.; Cinco-Ley, H.; Ramey, H.J.; Samaniego-V, F. Non-Darcy Flow in Wells with Finite-Conductivity Vertical Fractures. Soc. Pet. Eng. J. 1982, 22.

Datta-Gupta, A.; King, M.J. Streamline Simulation: Theory and Practice; SPE Textbook Series; Society of Petroleum Engineers: Richardson, TX, USA, 2007; Volume 11.

Blunt, M. Multiphase Flow in Permeable Media. A Pore-Scale Perspective; Cambridge University Press: Cambridge, UK, 2017.

Duong, A.N. An Unconventional Rate Decline Approach for Tight and Fracture-Dominated Gas Wells. In Proceedings of the Canadian Unconventional Resources and International Petroleum Conference, Calgary, AB, Canada, 19–21 October 2010.

Parsegov, S.G.; Nandlal, K.; Schechter, D.S.; Weijermars, R. Physics-Driven Optimization of Drained Rock Volume for Multistage Fracturing: Field Example from the Wolfcamp Formation, Midland Basin. In Proceedings of the Unconventional Resources Technology Conference, Houston, TX, USA, 23–25 July 2018.

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