**4. Numerical algorithm**

laboratory setting is around 5 *ml* / *min*. The pressurization rate *β* before breakdown in the field will be taken as 60 *psi* /*s* due to higher pumping rates in the field whereas in the laboratory it is about 1.2 *psi* /*s*. The wellbore compressibility *U* results from the compressibility of the fluid in the wellbore and the injection lines, as well as the compressibility of the wellbore and injection lines themselves. It is expressed as the ratio between the constant flow rate and the pressurization rate *U* =*Qo* / *β*. The typical fluid used in the field is slick water [19] with a viscosity of 1 *cP*. We consider glycerin as the fluid used in the laboratory with viscosity

The compressibility length scale *L uk* and time scale *tuk* corresponding to these parameters are displayed in Table 1. It is shown in the previous section that the dimensionless parameters depend upon the two dimensionless numbers i.e. and *χ*. These dimensionless numbers are also given in Table 1. While *L uk* and *tuk* give the length and the timescale associated with the compressibility effects, the value of *χ* determine the infinite medium propagation regime after the dissipation of compressibility effects. Large value of *χ* i.e. (*χ* >1) means that the fracture will propagate in the viscosity regime whereas, smaller values of *χ* i.e. (*χ* <1) means that the fracture will propagate in the toughness regime after the dissipation of compressibility effects.

> Ratio of the timescales χ 14 0.0036 Dimensionless wellbore radius 0.0065 0.017

**Table 1.** Characteristic length scales and dimensionless parameters for the field and the laboratory conditions.

It is obvious from Table 1, that in the field, the fracture propagates in the viscosity dominated regime (*χ* >1) whereas in the laboratory for the parameters chosen here, the fracture propagates in the toughness dominated regime (*χ* <1) after the dissipation of the early-time compressi‐ bility effects. For the field conditions, the compressibility length scale is equal to 56 *ft*. Which means that the propagation in the field is dominated by the compressibility effect until the fracture reaches about 56 *ft* (i.e. 150 times the wellbore radius) in to the formation. Even though this is a large value as compared to the wellbore radius, the length scale is still small as compared to the final fracture length which may be of the order of 800 to 1000 *ft* in the field. For laboratory conditions, the length scale is 2.4 *ft* (i.e. about 60 times the wellbore radius) which is smaller as compared to the field conditions, still the length-scale is large enough as compared to the specimen size that entire fracture propagation is dominated by the injection

The previous example has emphasized the differences with field conditions for a particular set of experimental parameters. However, these laboratory parameters can be appropriately adjusted in order to study a given regime of propagation. There can be different goals for an

Length scale of transition from the compressibility effects to the infinite medium propagation

Timescale of transition from the compressibility effects to the infinite medium propagation

**Field Laboratory**

*L uk* (ft) 56 2.4

*tuk* (sec) 3 743

1000 *cP*.

410 Effective and Sustainable Hydraulic Fracturing

system compressibility.

The governing equations are solved in their dimensionless form described in Section 3. The elasticity equation is discretized by Displacement Discontinuity Method (DDM) using piecewise constant elements with the tip element correction [22] for better accuracy. The fluid flow is discretized by the finite volume method.

At time *tn*, the fracture length is given as *γn*, fracture opening as Ω*n* and net pressure Π*n*. The algorithm consists of two nested loops. In the outer loop (time-stepping loop), a fracture increment Δ*γ* =Δ*ρ* is specified and the time step Δτn, for which the tip asymptotic condition (15) is satisfied, is found iteratively. In the inner loop (i.e. the Reynolds solver), the coupled system ofelasticityandlubricationequationsaresolvedinordertofindthefractureopeningandpressure profilescorrespondingtoagivenfracturelengthand(trial)timestep.Detailsofthetimestepping loop are given in Appendix B and the Reynolds Solver is described in Appendix C.
