**4. Comparing simulation results with analytical solution and experiment**

The complete validation of a hydraulic fracturing simulator is enormously challenging, because of the difficulty in obtaining the general analytical solutions or constructing realistic laboratory experiments. A carefully conducted laboratory hydraulic fracturing experiment has been published [32]. Figure 4 shows our simulation of the experiment. The comparison is excellent not only in the created fracture geometry, but also the pumping parameters.

**Figure 4.** Comparing with experimental result of [32]

stress state of the leading element at the fracture tip. These assumptions exclude the application for certain geological formations. In particular, geological stress regimes as a result of reverse faulting may present difficulties in modelling as the hydraulic fracture would most likely grow in a horizontal plane. However, the assumptions made are sufficient and adequate for typical deep unconventional resource formations where the vertical stress is generally the maximum principal stress, or at least greater than the minimum horizontal principal stress – resulting in

Fractures advance when the maximum tensile stress ahead of the crack tip exceeds the intrinsic tensile strength of the rock. In linear elastic fracture mechanics, the stress at the crack tip is singular, therefore the stress intensity factor, which correlates to the strength of such a singular stress field, is generally used to determine fracture tip propagation [29]. In practice, hydraulic fracturing processes take place under relatively large compressive in-situ earth stress that is normally several orders of magnitude higher than the rock strength. Therefore, the process is dominated by the balance of the compressive stress and fluid pressure at the fracture tip region. A group of 'virtual' elements is placed along the crack front and at each time step a check is made on the stress status of these elements. The virtual elements are allowed to be active as part of the new fracture surface if the potential fluid pressure can overcome the compressive

Importantly, the interaction of multiple fractures may result in significant shearing displace‐ ments along the fracture surface, causing the fracture growth to follow a curved path, and the computational technique leads these virtual elements to be oriented according to the stress field so that the local minimum principal stress is perpendicular to the new fracture surface as

An outline of the mathematical description of the fracture stiffness matrix, the displacement discontinuity method is given in [19], and will not be repeated here. The numerical approach is robust and efficient but less accurate than the variational boundary integral method which employs second order elements [30, 31]. For the current application, computational efficiency is deemed to outweigh the importance of a moderate improvement in accuracy. [19] also describes the coupling of derived non-linear fracture growth to flow equation and mass

The fractures are assumed to be initiated at the specified locations of a cluster of perforations. Near wellbore formation stress concentration is not considered. The downhole pressure is defined as the fluid pressure just upstream of all injection points. It is obvious that the distribution of the corresponding injection at each fracture follows wellbore fluid mechanics, depending on the injection area connected with the fracture and the fluid pressure within the fracture. An approximation is made by using Bernoulli equation to capture the fluid distribu‐ tion. This wellbore hydraulic model is able to account directly for the empirically derived or calibrated frictional pressure drop at the perforation and along the wellbore. Therefore, it is possible to account for limited entry perforation designs commonly employed in multiple

vertical fracture growth.

668 Effective and Sustainable Hydraulic Fracturing

stress plus the strength of the rock.

the fracture front advances.

**3.2. Wellbore hydraulic model**

hydraulic fracture design.

conservation.

#### **5. Frac fluid viscosity and fractures height growth**

A single stage of four hydraulic fractures is considered. Parameters chosen broadly resemble a typical shale gas development field case. The horizontal wellbore is placed at the relative depth of 0m as shown in Figure 5. Fracture injection points are spaced at 25m (82ft) apart and each fracture has 12 perforations. Fluid is injected into the wellbore at constant injection rate of 0.2 m3 /s (~75 bbl/min). Formation heterogeneities are limited to only in-situ stress variation. In this instance, the minimum horizontal stress gradient is set to 14kPa/m (0.62psi/ft) with no upper stress barrier to restrict height growth. The difference between minimum and maximum horizontal stress is chosen to be more than 20%, so that we do not cause the fracture direction to re-orient drastically. The key parameters are shown in Figure 5. The fluid leak-off factor is set to 1 with no spurt loss.

We consider two cases of distinctly different frac fluids, one with low viscosity (slick water) and the other a very high viscosity (gel frac). This corresponds to setting *n* = 0.8 and *K* = 0.05 for the low viscosity fluid case, and *n* = 0.8 and *K* = 0.5 for the high viscosity fluid in the powerlaw fluid model of Equation 7.

The injection parameters and the calculated total fracture volumes are shown in Figure 6. The simulated results are shown in Figure 7. As expected, the low viscosity fluid creates larger fractures with more fracture surface area at the expense of fracture aperture/width. Signifi‐ cantly, the low viscosity fluid creates a much larger fracture height. It is now interesting to see how the multiple fractures evolve in length and height as they compete in the presence of stress perturbation. Figure 8 shows fracture development at the end of the pumping of high viscosity fluid. The outer two fractures grew slightly outward and are longer than the inner fractures. This is due to stress shadowing. Interestingly, the inner two fractures developed more height, and this behaviour will become even more pronounced when we look at low viscosity fluid.

**Figure 7.** Calculated fracture parameters

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**Figure 8.** Fracture geometry with high viscosity frac fluid at 150 mins

**Figure 5.** Formation input parameters for the case studies

**Figure 6.** Injection volume and calculated fracture volumes

**Figure 7.** Calculated fracture parameters

We consider two cases of distinctly different frac fluids, one with low viscosity (slick water) and the other a very high viscosity (gel frac). This corresponds to setting *n* = 0.8 and *K* = 0.05 for the low viscosity fluid case, and *n* = 0.8 and *K* = 0.5 for the high viscosity fluid in the power-

The injection parameters and the calculated total fracture volumes are shown in Figure 6. The simulated results are shown in Figure 7. As expected, the low viscosity fluid creates larger fractures with more fracture surface area at the expense of fracture aperture/width. Signifi‐ cantly, the low viscosity fluid creates a much larger fracture height. It is now interesting to see how the multiple fractures evolve in length and height as they compete in the presence of stress perturbation. Figure 8 shows fracture development at the end of the pumping of high viscosity fluid. The outer two fractures grew slightly outward and are longer than the inner fractures. This is due to stress shadowing. Interestingly, the inner two fractures developed more height, and this behaviour will become even more pronounced when we look at low viscosity fluid.

law fluid model of Equation 7.

670 Effective and Sustainable Hydraulic Fracturing

**Figure 5.** Formation input parameters for the case studies

**Figure 6.** Injection volume and calculated fracture volumes

**Figure 8.** Fracture geometry with high viscosity frac fluid at 150 mins

**Figure 9.** Fracture geometry of low viscosity frac fluid at 150 mins

Figure 9 shows the final fracture geometry of pumping low viscosity fluid (note that the scale in Figure 9 is different from that in Figure 8). Perhaps against intuition, the inner fractures grew appreciably in height. To understand this, we look at the evolution of fracture length and height growth over time as depicted in Figures 10 and 11. It becomes clear that as the inner fractures are constrained from growing in length because of competition with the outer fractures, they found relative freedom to grow upward instead.

**Figure 10.** Fracture geometry of low viscosity frac fluid after 20 mins

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**Figure 10.** Fracture geometry of low viscosity frac fluid after 20 mins

**Figure 9.** Fracture geometry of low viscosity frac fluid at 150 mins

672 Effective and Sustainable Hydraulic Fracturing

fractures, they found relative freedom to grow upward instead.

Figure 9 shows the final fracture geometry of pumping low viscosity fluid (note that the scale in Figure 9 is different from that in Figure 8). Perhaps against intuition, the inner fractures grew appreciably in height. To understand this, we look at the evolution of fracture length and height growth over time as depicted in Figures 10 and 11. It becomes clear that as the inner fractures are constrained from growing in length because of competition with the outer

**Acknowledgements**

**Author details**

Sau-Wai Wong1

**References**

21-24.

The authors recognize the valuable input from Alexei Savitski, Anastasia Dobroskok, Mauricio Farinas, Ernesto Fonseca, Kaiming Xia, and Zongyu Zhai. Permission by Shell International

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and Guanshui Xu2

[1] United States Department of Energy(2009). Modern shale gas development in the

[2] Mutalik, P. N, & Gibson, B. (2008). Case History of Sequential and Simultaneous Fracturing of the Barnett Shale in Parker County. Paper SPE 116124 presented at the SPE Annual Technical Conference and Exhibition, Denver, September. DoiMS.,

[3] Waters, G, Dean, B, Downie, R, Kerrihard, K, Austbo, L, & Mcpherson, B. (2009). Si‐ multaneous Fracturing of Adjacent Horizontal Wells in the Woodford Shale. Paper SPE119635 presented at the SPE Hydraulic Fracturing Technical Conference, Wood‐

[4] Sierra, J, Kaura, J, Gualtieri, D, Glasbergen, G, Sarkar, D, & Johnson, D. (2008). DTS Monitoring Data of Hydraulic Fracturing: Experiences and Lessons Learned. Paper SPE 116182 presented at the SPE Annual Technical Conference and Exhibition in

[5] Molenaar, M. M, Hill, D. J, Webster, P, Fidan, E, & Birch, B. (2012). First Downhole Application of Distributed Acoustic Sensing for Hydraulic-Fracturing Monitoring

[6] Molenaar, M. M, Fidan, E, & Hill, D. J. (2012). Real-Time Downhole Monitoring of Hydraulic Fracturing Treatments Using Fibre Optic Distributed Temperature and Acoustic Sensing. Paper SPE 152981 presented at the SPE/EAGE European Uncon‐

ventional Resources Conference and Exhibition in Vienna, Austria, March.

and Diagnostics. SPE Drilling & Completion, March, , 32-38.

2 FrackOptima Inc., and The University of California Riverside, California, USA

Exploration and Production Inc. to publish is gratefully acknowledged.

, Mikhail Geilikman1

lands, TX, USA, January. Doi:MS., 19-21.

Denver, Colorado, USA. September.

United States: A Primer.

1 Shell Exploration and Production Inc., Houston, Texas, USA

**Figure 11.** Fracture geometry of low viscosity frac fluid after 40 mins

#### **6. Discussion**

At present, multiple fracture stimulations in horizontal wells often assume a uniform growth of all fractures, with the total volume of frac fluid more or less evenly distributed amongst the fractures. We have pointed out some of the potential impacts of stress shadows, which can affect treatment cost and production. Crucially, it has consequences in the understanding of subsurface development. For instance, if, instead of stimulating four fractures uniformly, only two or three of the fractures grow disproportionally, then some of the fractures may be unknowingly over-stimulated, resulting in excessive length or height growth. Normally this does not pose any major issue unless there are near-by wells or geological faults.

The non-planar 3D simulator presented here is capable of capturing the influence of key parameters such as injection rate, fracture spacing, formation properties, etc. on fracture design. Attempts are being made to extend the present numerical framework to include the interaction of hydraulic fractures with natural fractures.
