# Modeling of Simultaneous Propagation of Multiple Blade-Like Hydraulic Fractures from a Horizontal Well

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## Abstract

We explore different aspects of the multi-stage fracturing process such as stress interaction between growing hydraulic fractures, perforation friction, as well as the wellbore flow dynamics using a specifically developed numerical solver. In particular, great care is taken to appropriately solve for the fluid partition between the different growing fractures at any given time. We restrict the hydraulic fractures to be fully contained in the reservoir (fractures of constant height) thus reducing the problem to two dimensions. After discussions of the numerical algorithm, a number of verification tests are presented. We then define via scaling arguments the key dimensionless parameters controlling the growth of multiple hydraulic fractures during a single pumping stage. We perform a series of numerical simulations spanning the practical range of parameters to quantify which conditions promote uniform versus non-uniform growth. Our results notably show that, although large perforations friction helps to equalize the fluid partitioning between fractures, the pressure drop in the well along the length of the stage has a pronounced adverse effect on fluid partitioning as a result on the uniformity of growth of the different hydraulic fractures.

## Keywords

Multi-stage fracturing Well stimulation Fluid partitioning## List of Symbols

- \(\sigma ^{in},\, \tau ^{in}\)
Normal and shear tractions at \(\mathbf {x}\) located on the fracture surface

- \(\delta ^{n},\, \delta ^{s}\)
Normal and shear displacement discontinuities

- \(\varSigma\)
Union of all fracture surfaces

- \(N_{\text {frac}}\)
Number of fractures in the stage

- \(K^{nn},\, K^{ns},\,K^{ss},\, K^{ss}\)
Elastic fundamental influence function for normal and shear components of tractions due to unit displacement discontinuities

- \(\sigma ^o,\,\tau ^o\)
Normal and shear stresses at \(\mathbf {x}\) located on the fracture surface

- \(E, E^\prime ,\,\nu\)
Elastic Young’s modulus, plane-strain elastic modulus, and Poisson’s ratio

*H*Fractures height

*p*Fluid pressure

*w*Fracture opening \(w=\delta ^n\)

*s*Curvilinear coordinate (along the fracture or along the wellbore)

*t*Time

*q*Fluid flux

- \(c_\text {f}\)
Fluid compressibility

- \(\rho\)
Fluid density

- \(\mu\)
Fluid viscosity

*V*Cross-sectional average fluid velocity

*A*Cross-sectional area of the wellbore tubing

*a*Wellbore tubing radius

- \(\epsilon\)
Wellbore tubing roughness

*g*Gravitational earth acceleration

*Re*Reynolds number in the well

- \(\theta\)
Wellbore local deviation

- \(Q_o\)
Surface pump fluid flow injection rate

- \(Q_{I}\)
Flow rate entering fracture \(\#\,I\)

- \({\tilde{Q}}_{I}=Q_I/H\)
Flow rate entering fracture \(\#\,I\) divided by fracture height

- \(s_{I}\)
Curvilinear coordinate of the entrance to fracture \(\#\,I\) (along the fracture or along the wellbore)

- \(p_{w,I}\)
Fluid pressure in the wellbore in front of the entrance to fracture \(\#\,I\)

- \(p_{{\text {in}},I}\)
Fluid pressure inside the fracture at the entrance to fracture \(\#\,I\)

- \(\beta\)
Near-wellbore tortuosity exponent

- \(f_t\)
Near-wellbore tortuosity coefficient

- \(f_{\text {p}}\)
Perforation pressure drop coefficient

- \(n_{\text {p}}\)
Number of perforations for a given fracture entry

- \(D_{\text {p}}\)
Diameter of perforations

- \(K_{\text {Ic}}\)
Rock fracture toughness

- \(K_{I}\)
Mode I stress intensity factor

- \(v_{\text {tip}}\)
Local fracture velocity

- \(\ell\)
fracture half-length

- \(p_i,\,\sigma _i^o,\,\tau _i^o\)
Fluid pressure, in-situ normal, and shear stress in element

*i*- \(A_{il}^{kl}\)
Displacement discontinuity methods influence matrices (\(k,\,l=n,\,s\))

- \(h_i\)
Size of element

*i*-fracture mesh- \(q_{i-1/2},\,q_{i+1/2}\)
Left- and right-edge flux for element

*i*- \(C_{i-1/2},\,C_{i+1/2}\)
Left and right fluid conductivity for element

*i*- \(\mathbb {A}\)
Displacement discontinuity matrix

- \(\mathbf {T}_o\)
Initial traction vector on all elements

- \(\mathbb {L}\)
Finite-difference lubrication matrix

- \(\mathbb {I}_{\text {p}}\)
Matrix related to fluid pressure increment

- \(\mathbf {Q}\)
Entry fluxes vector

- \(\mathbb {I}_s\)
Matrix related to fracture volume increment

- \(\mathbb {I}_c\)
Matrix related to fracture volume

- \(\ell _{mk}\)
Near-tip viscosity-toughness transition lengthscale

- \({\tilde{h}}_{i\pm 1/2}\)
Size of element

*i*-wellbore mesh- \(A_{i-1/2},\,A_{i+1/2}\)
Left and right cross-sectional area of wellbore for element

*i*of the wellbore mesh- \(a_{i-1/2},\,a_{i+1/2}\)
Left and right wellbore radius value for element

*i*of the wellbore mesh- \(V_{i-1/2},\,V_{i+1/2}\)
Left and right fluid velocity for element

*i*of the wellbore mesh- \(p_{\text {H}}\)
Hydrostatic fluid pressure

- \({\tilde{p}}\)
Fluid pressure in excess of the hydrostatic pressure

- \(Re_{i-1/2},\,Re_{i+1/2}\)
Left and right Reynolds number for element

*i*of the wellbore mesh- \(C^w_{i-\nicefrac {1}{2}},\,C^w_{i+\nicefrac {1}{2}}\)
Left and right fluid conductivity for element

*i*of the wellbore mesh- \(\mathbb {L}_w\)
Finite-difference matrix for the wellbore mesh

- \(\mathbb {I}_{cw}\)
Matrix for wellbore volume

- \(\mathscr {M}\)
Dimensionless viscosity for plane-strain hydraulic fracture

- \(\mathscr {K}^{KGD}\)
Dimensionless viscosity for plane-strain hydraulic fracture

*S*Spacing between fractures along the wellbore

- \(\sigma _{\text {H}}\, \sigma _{\text {h}}\)
Maximum and minimum principal horizontal stresses magnitude

- \(Q_{n}=Q_o/N_{\text {frac}}\)
Evenly divided surface flow rate

- \(\varGamma\)
Ratio between the characteristic stress interaction and characteristic pressure drop through perforation

*L*Fracture characteristic lengthscale

- \(\varGamma _k^{(KGD)}\)
Expression of \(\varGamma\) in the toughness-dominated regime for plane-strain fractures

- \(\varGamma _m^{(KGD)}\)
Expression of \(\varGamma\) in the viscosity-dominated regime for plane-strain fractures

- \(\varGamma _m\)
Expression of \(\varGamma\) for a PKN fracture

- \(\varPi\)
Ratio between the characteristic pressure drop in the wellbore along the length of the stage and the characteristic pressure drop through perforation

- \(\varPi _\beta\)
Expression of \(\varPi\) under the assumption of fully turbulent flow in the wellbore

## Notes

### Acknowledgements

D.N. acknowledges partial funding from the EPFL Fellows European Union’s Horizon 2020 Research and Innovation programme under the Marie Sklodowska-Curie grant agreement 665667. We would like to thank Total E&P for additional funding and permission to publish this work. Numerous discussions with Atef Onaisi and Hamid Pourpak are greatly acknowledged.

### Compliance with Ethical Standards

### Conflict of interest

The authors have no conflict of interest.

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