# Real-Time Simulation of Hydraulic Fracturing Using a Combined Integrated Finite Difference and Discontinuous Displacement Method: Numerical Algorithm and Field Applications

^{*}

## Abstract

**:**

## 1. Introduction

_{2}as a fracturing fluid.

## 2. Mathematical Model

#### 2.1. Fracture Initiation and Propagation

_{ext}and n are rock internal properties.

_{b}is calculated as follows [30].

#### 2.2. Mass Transport Inside the Fracture

_{sl}is the gravity acceleration term. ${\mu}_{sl}$ represents slurry viscosity. Based on Equation (9), the fluid component velocity ${\overrightarrow{v}}_{fl}$ calculated as follows.

#### 2.3. Leak-Off Rate Calculation

_{N}, at the interface between the compressed formation fluids zone and the invaded zone can be calculated as follows [40].

_{cap}is the capillary pressure between the invading fluid and the reservoir fluid. ${P}_{S}$ is the pressure of the filter case zone. ${\alpha}_{v}$ is the overall leak-off coefficient of the invaded zone.

_{w}, which is the slope of a plot of filtrate volume with respect to the square root of time.

#### 2.4. Energy Transport Inside the Fracture

_{res}, ρ

_{res}and C

_{res}are the thermal conductivity, density and heat capacity of the reservoir formation, respectively. τ is the starting time of the heat loss, as used in Carslaw and Jaeger [41].

#### 2.5. Mass/Energy Transport Inside the Wellbore

_{i}, inside the wellbore during the simulation. The boundaries of the slurry segment are denoted by S

_{i}. The numbering of these slurry segment boundaries starts at one, the wellbore bottom, and increases as the wellbore is traversed toward the wellhead, with the last segment boundary being S

_{N}

_{+1}(where N is the number of well segments) that is located at the wellhead. The slurry is injected into the wellbore at the wellhead and leaves the wellbore through perforations.

_{ann}is the annular area, x denotes component volume fraction, T

_{ws}is transmissibility per unit length for slurry loss to the surroundings, P

_{w}and P

_{surr}are the wellbore pressure and the surrounding pressure, respectively. q

_{inj}is rate of the injected slurry. The transmissibility per unit length for slurry loss to the surroundings is only non-zero along completed intervals.

**g**is the gravitational term and

**S**denotes wellbore trajectory. P

_{w,fric}is the frictional pressure, which is obtained as follows.

_{h}is the hydraulic diameter. The friction factor f is calculated based on Reynolds number, as follows. For laminar flow, friction factor is 16/N

_{Re}and in correlations for turbulent flow the friction factor is given by the following equation [42].

_{loss}is the rate of heat loss per unit area from the slurry to the surroundings and D

_{o}is the outer wellbore diameter for fluid flow. Heat loss from the slurry to the surroundings is modeled as heat flow through a uniform medium. In this work we derive the overall heat transfer coefficient for the medium based on the analytical solution of transient heat flow with a point heat source [41].

_{th}and α

_{th}are the thermal conductivity and the thermal diffusivity of the media, respectively. α

_{th}is thermal diffusivity, Q

_{0}is the magnitude of the heat source (per unit length), t refers to time, τ is the time at which the heat source is activated, and E

_{i}is the exponential integral function. Using Equations (36) and (37), the heat flow between the wellbore and the surroundings can be calculated.

## 3. Numerical Approach

#### 3.1. Integrated Finite Difference Discretization

_{n}, we get

_{nm}is the interface area between V

_{n}and V

_{m}

_{.}The time derivative term is discretized by the forward difference quotient. There, Equation (39) becomes

#### 3.2. Combination with DDM

_{3}axis along the normal displacement direction.

#### 3.3. Real-Time Monitoring and Simulation

## 4. Validation and Results

#### 4.1. Comparison with Finite Element Method

^{3}/min in the center of the domain. The profiles of fracture width at the end of the injection simulated by our program and by the finite element code are compared in Figure 7. It can be seen that the two programs produce close results. Moreover, the radius of the fracture in this case can be analytically calculated. We compare our result with the analytical solution obtained by Geertsma and Klerk [3] in Figure 8. As shown by the figure, our result match the analytical solution well.

#### 4.2. Comparison with Commercial Fracturing Software

## 5. Results and Discussion

#### 5.1. Numerical Study of Single Fracture in Multi-Layer Reservoirs

#### 5.2. Numerical Study of Multiple Fractures with Realistic Frictions

#### 5.3. Field Application Results

#### 5.4. Discussion

- The wellbore friction has a considerable impact on the hydraulic length of the fracture.
- The stress interference effect as well as the leak-off effect both affect the propped length of the fracture, via affecting the fracture width and the proppant-carrying capability of the slurry.
- The multi-process framework shows sound flexibility in dealing with real-time simulations.

## 6. Summary and Conclusions

- Stress offset along with permeability distribution not only affects the fracture propagation, but also the proppant distribution inside the fracture plane. Ignoring the impact of proppant concentration on the fluid flow may lead to overestimation of the propped height.
- The energy loss induced by friction and heat conduction within the wellbore and the formation plays an important role in the simulation of hydraulic fractures. The mass/energy transport must be comprehensively simulated to obtain realistic resutls of single as well as multiple fracture propagation.
- In general, the hydraulic fracturing operations involve complex thermal-hydraulic-mechanical processes, and thus multiphysical simulation techniques are crucial in real practice.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Nordgren, R.P. Propagation of a Vertical Hydraulic Fracture. Soc. Pet. Eng. J.
**1972**, 12, 306–314. [Google Scholar] [CrossRef] - Perkins, T.K.; Kern, L.R. Widths of Hydraulic Fractures. J. Pet. Technol.
**1961**, 13, 937–949. [Google Scholar] [CrossRef] - Geertsma, J.; De Klerk, F. A Rapid Method of Predicting Width and Extent of Hydraulically Induced Fractures. J. Pet. Technol.
**1969**, 21, 1571–1581. [Google Scholar] [CrossRef] - Advani, S.H.; Lee, T.S.; Lee, J.K. Three-Dimensional Modeling of Hydraulic Fractures in Layered Media: Part I—Finite Element Formulations. J. Energy Resour. Technol.
**1990**, 112, 1. [Google Scholar] [CrossRef] - Baca, R.G.; Arnett, R.C.; Langford, D.W. Modelling fluid flow in fractured-porous rock masses by finite-element techniques. Int. J. Numer. Methods Fluids
**1984**, 4, 337–348. [Google Scholar] [CrossRef] - Chen, Z. Finite element modelling of viscosity-dominated hydraulic fractures. J. Pet. Sci. Eng.
**2012**, 88–89, 136–144. [Google Scholar] [CrossRef] - Yang, T.H.; Tham, L.G.; Tang, C.A.; Liang, Z.Z.; Tsui, Y. Influence of Heterogeneity of Mechanical Properties on Hydraulic Fracturing in Permeable Rocks. Rock Mech. Rock Eng.
**2004**, 37, 251–275. [Google Scholar] [CrossRef] - Moës, N.; Belytschko, T. Extended finite element method for cohesive crack growth. Eng. Fract. Mech.
**2002**, 69, 813–833. [Google Scholar] [CrossRef][Green Version] - Moes, N.; Dolbow, J.; Belytschko, T. A finite element method for crack growth without remeshing. Int. J. Numer. Methods Eng.
**1999**, 46, 131–150. [Google Scholar] [CrossRef] - Sukumar, N.; Moes, N.; Moran, B.; Belytschko, T. Extended finite element method for three-dimensional crack modelling. Int. J. Numer. Methods Eng.
**2000**, 48, 1549–1570. [Google Scholar] [CrossRef] - Baghbanan, A.; Jing, L. Hydraulic properties of fractured rock masses with correlated fracture length and aperture. Int. J. Rock Mech. Min. Sci.
**2007**, 44, 704–719. [Google Scholar] [CrossRef] - Jing, L.; Stephansson, O. Fundamentals of Discrete Element Methods for Rock Engineering: Theory and Applications, 1st ed.; Elsevier: Amsterdamm, The Netherlands, 2007. [Google Scholar]
- Tan, Y.; Yang, D.; Sheng, Y. Discrete element method (DEM) modeling of fracture and damage in the machining process of polycrystalline SiC. J. Eur. Ceram. Soc.
**2009**, 29, 1029–1037. [Google Scholar] [CrossRef] - Li, L.C.; Tang, C.A.; Li, G.; Wang, S.Y.; Liang, Z.Z.; Zhang, Y.B. Numerical Simulation of 3D Hydraulic Fracturing Based on an Improved Flow-Stress-Damage Model and a Parallel FEM Technique. Rock Mech. Rock Eng.
**2012**, 45, 801–818. [Google Scholar] [CrossRef] - Tang, C.; Tham, L.; Lee, P.K.; Yang, T.; Li, L. Coupled analysis of flow, stress and damage (FSD) in rock failure. Int. J. Rock Mech. Min. Sci.
**2002**, 39, 477–489. [Google Scholar] [CrossRef] - Wang, S.; Lukyanov, A.; Wu, Y.-S. Application of Algebraic Smoothing Aggregation Two Level Preconditioner to Multiphysical Fluid Flow Simulations in Porous Media. In Proceedings of the SPE Reservoir Simulation Conference, Galveston, TX, USA, 10–11 April 2019; Society of Petroleum Engineers: Richardson, TX, USA, 2019. [Google Scholar] [CrossRef]
- Scholtès, L.; Donzé, F.-V. Modelling progressive failure in fractured rock masses using a 3D discrete element method. Int. J. Rock Mech. Min. Sci.
**2012**, 52, 18–30. [Google Scholar] [CrossRef] - Azevedo, N.M.; Lemos, J.V. Hybrid discrete element/finite element method for fracture analysis. Comput. Methods Appl. Mech. Eng.
**2006**, 195, 4579–4593. [Google Scholar] [CrossRef] - Mahabadi, O.K.; Lisjak, A.; Munjiza, A.; Grasselli, G. Y-Geo: New Combined Finite-Discrete Element Numerical Code for Geomechanical Applications. Int. J. Geomech.
**2012**, 12, 676–688. [Google Scholar] [CrossRef] - Munjiza, A.; Owen, D.R.J.; Bicanic, N. A combined finite-discrete element method in transient dynamics of fracturing solids. Eng. Comput.
**1995**, 12, 145–174. [Google Scholar] [CrossRef] - Wu, K.; Olson, J.E. Simultaneous Multifracture Treatments: Fully Coupled Fluid Flow and Fracture Mechanics for Horizontal Wells. SPE J.
**2015**, 20, 337–346. [Google Scholar] [CrossRef] - Wu, K.; Olson, J.E. Mechanics Analysis of Interaction Between Hydraulic and Natural Fractures in Shale Reservoirs. In Proceedings of the Unconventional Resources Technology Conference (URTEC), Denver, CO, USA, 25–27 August 2014. [Google Scholar] [CrossRef]
- Wu, K.; Olson, J.E. Investigation of the Impact of Fracture Spacing and Fluid Properties for Interfering Simultaneously or Sequentially Generated Hydraulic Fractures. SPE Prod. Oper.
**2013**, 28, 427–436. [Google Scholar] [CrossRef] - Guo, X.; Wu, K.; An, C.; Tang, J.; Killough, J. Numerical Investigation of Effects of Subsequent Parent-Well Injection on Interwell Fracturing Interference Using Reservoir-Geomechanics-Fracturing Modeling. SPE J.
**2019**, 24, 1884–1902. [Google Scholar] [CrossRef] - Kresse, O.; Cohen, C.; Weng, X.; Wu, R.; Gu, H. Numerical Modeling of Hydraulic Fracturing In Naturally Fractured Formations. 2011. Available online: https://onepetro.org/ARMAUSRMS/proceedings-abstract/ARMA11/All-ARMA11/ARMA-11-363/120401 (accessed on 7 January 2023).
- Weng, X. Modeling. of complex hydraulic fractures in naturally fractured formation. J. Unconv. Oil Gas Resour.
**2015**, 9, 114–135. [Google Scholar] [CrossRef] - Weng, X.; Kresse, O.; Cohen, C.E.; Wu, R.; Gu, H. Modeling of Hydraulic Fracture Network Propagation in a Naturally Fractured Formation. In SPE Hydraulic Fracturing Technology Conference; Society of Petroleum Engineers: Richardson, TX, USA, 2011. [Google Scholar] [CrossRef]
- Li, Q.; Wang, F.; Wang, Y.; Bai, B.; Zhang, J.; Lili, C.; Sun, Q.; Wang, Y.; Forson, K. Adsorption behavior and mechanism analysis of siloxane thickener for CO
_{2}fracturing fluid on shallow shale soil. J. Mol. Liq.**2023**, 376, 121394. [Google Scholar] [CrossRef] - Li, Q.; Wu, J. Factors affecting the lower limit of the safe mud weight window for drilling operation in hydrate-bearing sediments in the Northern South China Sea. Geomech. Geophys. Geo. Energy Geo. Resour.
**2022**, 8, 82. [Google Scholar] [CrossRef] - Yew, C.H.; Weng, X. Mechanics of Hydraulic Fracturing, 2nd ed.; Gulf Professional Publishing: Waltham, MA, USA, 2015. [Google Scholar]
- Olson, J. Fracture Mechanics Analysis of Joints and Veins; Stanford University: Stanford, CA, USA, 1991. [Google Scholar]
- Sheibani, F. Solving Three-Dimensional Problems in Natural and Hydraulic Fracture Development: Insight from Displacement Discontinuity Modeling; UT Austin: Austin, TX, USA, 2013. [Google Scholar]
- Mastrojannis, E.N.; Keer, L.M.; Mura, T. Growth of planar cracks induced by hydraulic fracturing. Int. J. Numer. Methods Eng.
**1980**, 15, 41–54. [Google Scholar] [CrossRef] - Tang, H.; Wang, S.; Zhang, R.; Li, S.; Zhang, L.; Wu, Y.-S. Analysis of stress interference among multiple hydraulic fractures using a fully three-dimensional displacement discontinuity method. J. Pet. Sci. Eng.
**2019**, 179, 378–393. [Google Scholar] [CrossRef] - Wu, Y.S. Hydraulic Fracture Modeling; Gulf Professional Publishing: Oxford, UK, 2017. [Google Scholar]
- Friehauf, K.E. Simulation and Design of Energized Hydraulic Fractures; The University of Texas at Austin: Austin, TX, USA, 2009. [Google Scholar]
- Wang, Y.; Ju, B.; Wang, S.; Yang, Z.; Liu, Q. A tight sandstone multi-physical hydraulic fractures simulator study and its field application. Petroleum
**2019**, 6, 198–205. [Google Scholar] [CrossRef] - Nicodemo, L.; Nicolais, L.; Landel, R.F. Shear rate dependent viscosity of suspensions in newtonian and non-newtonian liquids. Chem. Eng. Sci.
**1974**, 29, 729–735. [Google Scholar] [CrossRef] - Schechter, R.S. Oil Well Stimulation, 1st ed.; Prentice Hall Inc.: Old Tappan, NJ, USA, 1992. [Google Scholar]
- Economides, M.J.; Nolte, K.G. Reservoir Stimulation, 3rd ed.; Prentice Hall Inc.: Old Tappan, NJ, USA, 2000. [Google Scholar]
- Carslaw, H.S.; Jaeger, J.C. Conduction of Heat in Solids, 2nd ed.; Clarendon Press: Oxford, UK, 1959. [Google Scholar]
- Valko, P.; Economides, M.J. Hydraulic Fracture Mechanics; JOHN WILEY & SONS: New York, NY, USA, 1995. [Google Scholar]
- Narasimhan, T.N.; Witherspoon, P.A. An integrated finite difference method for analyzing fluid flow in porous media. Water Resour. Res.
**1976**, 12, 57–64. [Google Scholar] [CrossRef][Green Version] - Bonduà, S.; Battistelli, A.; Berry, P.; Bortolotti, V.; Consonni, A.; Cormio, C.; Geloni, C.; Vasini, E.M. 3D Voronoi grid dedicated software for modeling gas migration in deep layered sedimentary formations with TOUGH2-TMGAS. Comput. Geosci.
**2017**, 108, 50–55. [Google Scholar] [CrossRef] - Hu, L.; Zhang, K.; Cao, X.; Li, Y.; Guo, C. IGMESH: A convenient irregular-grid-based pre- and post-processing tool for TOUGH2 simulator. Comput. Geosci.
**2016**, 95, 11–17. [Google Scholar] [CrossRef] - Wang, S.; Huang, Z.; Wu, Y.-S.; Winterfeld, P.H.; Zerpa, L.E. A semi-analytical correlation of thermal-hydraulic-mechanical behavior of fractures and its application to modeling reservoir scale cold water injection problems in enhanced geothermal reservoirs. Geothermics
**2016**, 64, 81–95. [Google Scholar] [CrossRef] - Wang, S.; Xiong, Y.; Winterfeld, P.; Zhang, K.; Wu, Y.-S. Parallel Simulation of Thermal-Hydrological-Mechanic (THM) Processes in Geothermal Reservoirs. In Proceedings of the 39th Workshop on Geothermal Reservoir Engineering, Stanford, CA, USA, 24–26 February 2014; Stanford University: Stanford, CA, USA, 2014. [Google Scholar]
- Crouch, S.L. Solution of plane elasticity problems by the displacement discontinuity method. I. Infinite body solution. Int. J. Numer. Methods Eng.
**1976**, 10, 301–343. [Google Scholar] [CrossRef] - Tang, H.; Winterfeld, P.H.; Wu, Y.-S.; Huang, Z.; Di, Y.; Pan, Z.; Zhang, J. Integrated simulation of multi-stage hydraulic fracturing in unconventional reservoirs. J. Nat. Gas Sci. Eng.
**2016**, 36, 875–892. [Google Scholar] [CrossRef] - Ribeiro, L.; Sharma, M.M. A New Three-Dimensional, Compositional, Model for Hydraulic Fracturing with Energized Fluids. In Proceedings of the SPE Annual Technical Conference and Exhibition, San Antonio, TX, USA, 8–10 October 2012; Society of Petroleum Engineers: Richardson, TX, USA, 2012. [Google Scholar] [CrossRef]
- CARBO. FRACPRO Fracture Design &. Analysis Software; Pinach. Inc.: Denver, CO, USA, 2019. [Google Scholar]
- Barboza, B.R.; Chen, B.; Li, C. A review on proppant transport modelling. J. Pet. Sci. Eng.
**2021**, 204, 108753. [Google Scholar] [CrossRef] - Wang, L.; Tian, Y.; Yu, X.; Wang, C.; Yao, B.; Wang, S.; Winterfeld, P.H.; Wang, X.; Yang, Z.; Wang, Y.; et al. Advances in improved/enhanced oil recovery technologies for tight and shale reservoirs. Fuel
**2017**, 210, 425–445. [Google Scholar] [CrossRef] - Wang, L.; Wang, S.; Zhang, R.; Wang, C.; Xiong, Y.; Zheng, X.; Li, S.; Jin, K.; Rui, Z. Review of multi-scale and multi-physical simulation technologies for shale and tight gas reservoirs. J. Nat. Gas Sci. Eng.
**2017**, 37, 560–578. [Google Scholar] [CrossRef]

**Figure 1.**Conceptual model of a propagating fracture plane. The fracture is allowed to propagate along x-direction and z-direction and to deviate along y-direction which is perpendicular to the fracture plane.

**Figure 2.**A sketch of the three-zone leak-off model, following [39].

**Figure 3.**Schematic of a wellbore with two wellbore segments and three slurry segments. S refers to cross-section area and the numbers refer to the index of fluid segments.

**Figure 4.**Conceptual model of fracture on the grid in FracCSM. Green grid blocks refer to newly activated (cracked) grid blocks on the edge of the fracture at the current time step. Blue grid blocks refer to grid blocks that were cracked in previous time steps, while grid blocks refer to intact rocks.

**Figure 7.**Simulated fracture width after 30 min injection. The right sub-figure is modified from [50].

**Figure 8.**Comparison between simulation and analytical solution [3] for fracture radius versus time.

**Figure 9.**Comparison of the fracture geometry and proppant concentration distribution predicted by FracCSM and Fracpro PT. Upper: fracture plane simulated by FracCSM. Lower: fracture plane simulated by Fracpro PT.

**Figure 12.**The stress profile, fracture width, proppant concentration and fluid pressure across the fracture of Case 1.

**Figure 13.**The stress profile, fracture width, proppant concentration and fluid pressure across the fracture of Case 2.

**Figure 14.**The stress profile, fracture width, proppant concentration and fluid pressure across the fracture of Case 3.

**Figure 15.**The stress profile, fracture width, proppant concentration and fluid pressure across the fracture of Case 4.

**Figure 16.**The stress profile, fracture width, proppant concentration and fluid pressure across the fracture of Case 5.

**Figure 19.**Deviation of the three fractures. Positive values refer to deviation pointing to the left direction in Figure 18. Left: deviation profile across the fracture plane. Right: top view of the fracture deviation.

**Figure 21.**The real-time simulation of a slug of proppant. The photo was taken at Sulige gas field. (

**a**) Data measured on the ground. The black curve is the real-time proppant concentration measured at the surface. Upper-left in (

**b**) real-time downhole pressure calculated by FracCSM. Upper-right in (

**b**): real-time proppant distribution inside the fracture predicted by FracCSM. Lower-right in (

**b**): real-time width distribution inside the fracture predicted by FracCSM.

Properties | Values | Units |
---|---|---|

Young’s modulus of sandstone | 41 | GPa |

Poisson’s ratio of sandstone | 0.22 | dimensionless |

Biot’s coefficient of sandstone | 1.0 | dimensionless |

Fracture toughness of sandstone | 2 | MPa·m^{1/2} |

Permeability of sandstone | 2 | mD |

Porosity of sandstone | 0.1 | dimensionless |

Thermal conductivity sandstone | 1.75 | W/m·hr |

Specific heat of sandstone | 0.88 | kJ/kg °C |

Young’s modulus of caprock | 48 | GPa |

Poisson’s ratio of caprock | 0.22 | dimensionless |

Biot’s coefficient of caprock | 1.0 | dimensionless |

Fracture toughness of caprock | 3 | MPa·m^{1/2} |

Permeability of caprock | 0.01 | mD |

Porosity of caprock | 0.01 | dimensionless |

Thermal conductivity of caprock | 1.57 | W/m·hr |

Specific heat of caprock | 0.84 | kJ/kg°C |

Grid block length (x-direction) of FracCSM | 1 | m |

Grid block length (z-direction) of FracCSM | 1 | m |

Gel (initial) viscosity | 170 | cp |

Gel density | 1100 | kg/m^{3} |

Formation pore pressure | 27 | MPa |

Formation temperature | 90 | °C |

Maximum principal stress | 79 | MPa |

Minimum horizontal principal stress of sandstone | 55 | MPa |

Minimum horizontal principal stress of caprock | 65 | MPa |

Vertical stress | 90 | MPa |

Injection temperature | 20 | °C |

Proppant density | 2600 | kg/m^{3} |

Proppant diameter | 20/40 | mesh |

Proponent flow exponent of FracCSM (n in Equation (17)) | 4.1 | dimensionless |

Maximum proppant fraction of FracCSM ( ${c}_{\mathrm{max}}$ in Equation (17)) | 0.8 | dimensionless |

**Table 2.**Pumping schedule used for the validation with Fracpro PT software. The stage index refers to the sequence of injected segments.

Stage Index | Injection Volume (m^{3}) | Injection Rate (m^{3}/min) | Proppant Concentration (kg/m^{3}) |
---|---|---|---|

1 | 40.0 | 2.4 | 0 |

2 | 10.0 | 2.4 | 50 |

3 | 16.0 | 2.4 | 100 |

4 | 16.0 | 2.4 | 200 |

5 | 20.4 | 2.4 | 300 |

6 | 25.3 | 2.4 | 400 |

7 | 35.3 | 2.4 | 500 |

Results of FracCSM (m) | Results of Fracpro PT (m) | |
---|---|---|

Total (hydraulic) length | 224 | 212 |

Total (hydraulic) height | 34 | 28 |

Propped length | 210 | 205 |

Propped height | 28 | 28 |

**Table 4.**Formation properties of the five cases. (‘Stress’ refers to the minimum principal stress.) The stage index refers to the sequence of injected segments.

Case Index | Thickness of Upper Shale Layer (m) | Minimum Principal Stress of Upper Shale Layer (MPa) | Thickness of Lower Shale Layer (m) | Minimum Principal Stress of Lower Shale Layer (MPa) |
---|---|---|---|---|

1 | 15 | 58 | 15 | 58 |

2 | 20 | 58 | 20 | 58 |

3 | 25 | 58 | 25 | 58 |

4 | 10 | 62 | 12 | 62 |

5 | 10 | 72 | 12 | 62 |

Properties | Values | Units |
---|---|---|

Young’s modulus of sandstone | 41 | GPa |

Poisson’s ratio of sandstone | 0.22 | dimensionless |

Biot’s coefficient of sandstone | 1.0 | dimensionless |

Fracture toughness of sandstone | 2 | MPa·m^{1/2} |

Permeability of sandstone | 0.5 | mD |

Porosity of sandstone | 0.05 | dimensionless |

Thermal conductivity sandstone | 1.75 | W/m·hr |

Specific heat of sandstone | 0.88 | kJ/kg °C |

Young’s modulus of shale | 48 | GPa |

Poisson’s ratio of shale | 0.22 | dimensionless |

Biot’s coefficient of shale | 1.0 | dimensionless |

Fracture toughness of shale | 1.2 | MPa·m^{1/2} |

Permeability of shale | 0.01 | mD |

Porosity of shale | 0.05 | dimensionless |

Thermal conductivity of shale | 1.57 | W/m·hr |

Specific heat of shale | 0.84 | kJ/kg°C |

Grid block length (x-direction) | 2 | m |

Grid block length (z-direction) | 1 | m |

Gel viscosity | 170 | cp |

Gel density | 1100 | kg/m^{3} |

Formation pore pressure | 24 | MPa |

Formation temperature | 90 | °C |

Minimum principal stress of sandstone layer | 52 | MPa |

Vertical stress | 90 | MPa |

Injection temperature | 20 | °C |

Proppant density | 2600 | kg/m^{3} |

Proppant diameter | 20/40 | mesh |

Proponent flow exponent $($ in Equation (17)) | 4.1 | dimensionless |

Maximum proppant fraction $({c}_{\mathrm{max}}$ in Equation (17)) | 0.8 | dimensionless |

Perforation diameter | 10 | mm |

Tubing diameter | 62 | mm |

**Table 6.**Pumping schedules of the five cases. The stage index refers to the sequence of injected segments.

Stage Index | Injection Volume (m^{3}) | Injection Rate (m^{3}/min) | Proppant Concentration (kg/m^{3}) |
---|---|---|---|

1 | 68.0 | 2.4 | 0 |

2 | 12.5 | 2.4 | 120 |

3 | 16.1 | 2.4 | 240 |

4 | 20.0 | 2.4 | 360 |

5 | 24.0 | 2.4 | 460 |

6 | 23.4 | 2.4 | 540 |

7 | 21.3 | 2.4 | 580 |

**Table 7.**Pumping schedules of the multiple fracture cases. The stage index refers to the sequence of injected segments.

Stage Index | Injection Volume (m^{3}) | Injection Rate (m^{3}/min) | Proppant Concentration (kg/m^{3}) |
---|---|---|---|

1 | 155.0 | 6.4 | 0 |

2 | 22.5 | 6.4 | 120 |

3 | 33.1 | 6.4 | 240 |

4 | 45.0 | 6.4 | 360 |

5 | 57.0 | 6.4 | 460 |

6 | 65.4 | 6.4 | 540 |

7 | 28.3 | 6.4 | 580 |

8 | 20.0 | 6.4 | 600 |

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**MDPI and ACS Style**

Wang, S.; Yu, X.; Winterfeld, P.H.; Wu, Y.-S.
Real-Time Simulation of Hydraulic Fracturing Using a Combined Integrated Finite Difference and Discontinuous Displacement Method: Numerical Algorithm and Field Applications. *Water* **2023**, *15*, 938.
https://doi.org/10.3390/w15050938

**AMA Style**

Wang S, Yu X, Winterfeld PH, Wu Y-S.
Real-Time Simulation of Hydraulic Fracturing Using a Combined Integrated Finite Difference and Discontinuous Displacement Method: Numerical Algorithm and Field Applications. *Water*. 2023; 15(5):938.
https://doi.org/10.3390/w15050938

**Chicago/Turabian Style**

Wang, Shihao, Xiangyu Yu, Philip H. Winterfeld, and Yu-Shu Wu.
2023. "Real-Time Simulation of Hydraulic Fracturing Using a Combined Integrated Finite Difference and Discontinuous Displacement Method: Numerical Algorithm and Field Applications" *Water* 15, no. 5: 938.
https://doi.org/10.3390/w15050938