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Ξ(ΔΩ) ⋅ΔΩ=Γ(ΔΩ) (44)

Ξ=*δij* - *Bij* (45)

, *i* =2, *N* (48)

, *i* =1, *N* - 1 (49)

. (52)

Ξ(ΔΩk) ⋅ΔΩk+1 =Γ(ΔΩk) (50)

ΔΩk+1 =(1 - *η <sup>γ</sup>*)ΔΩ<sup>k</sup> + *η <sup>γ</sup>*ΔΩk+1 (51)

(46)

(47)

*A*(*i*, *<sup>j</sup>*) + *Ke*(*i*)*A*(*i*+1, *<sup>j</sup>*) - *Ke*(*i*)*A*(*i*, *<sup>j</sup>*), *i* =1 *j* =1, *N*

<sup>0</sup> , *i* =2, *N* - 1, *j* =1, *N*

<sup>0</sup> , *i* = *N* , *j* =1, *N*

<sup>0</sup> , *i* =1, *j* =1, *N*

*Kw*(*i*-1)*A*(*i*-1, *<sup>j</sup>*) - *Kw*(*i*-1)*A*(*i*, *<sup>j</sup>*), *i* = *N* , *j* =1, *N*

*Kw*(*i*-1)*A*(*i*-1, *<sup>j</sup>*) - (*Kw*(*i*-1) + *Ke*(*i*))*A*(*i*, *<sup>j</sup>*) + *Ke*(*i*)*A*(*i*+1, *<sup>j</sup>*), *i* =2, *N* - 1, *j* =1, *N*

<sup>2</sup>*πρ<sup>i</sup> A*(*i*, *<sup>j</sup>*) + *Ke*(*i*)(*A*(*i*+1, *<sup>j</sup>*) - *A*(*i*, *<sup>j</sup>*)) ⋅Ω *<sup>j</sup>*

*<sup>K</sup>*e<sup>=</sup> *<sup>ρ</sup>i*-1/2

*<sup>K</sup>*e<sup>=</sup> *<sup>ρ</sup>i*+1/2

The nonlinear system (45) is solved by fixed point iteration

where *η* is the relaxation parameter. The convergence criteria is

ΔΩ*<sup>k</sup>* +1 - ΔΩ*<sup>k</sup>* <sup>2</sup> ΔΩ*<sup>k</sup>* +1 <sup>2</sup>

<ϵ

, with

ϵ=10-5

*<sup>ρ</sup><sup>i</sup> Ki*-1/2

*<sup>ρ</sup><sup>i</sup> Ki*+1/2

where Ξ is expressed as

420 Effective and Sustainable Hydraulic Fracturing

is the Kronecker delta and

where *δij*

*Bij* <sup>=</sup>*α*{ *<sup>U</sup>* <sup>Δ</sup>*<sup>ρ</sup>* 2*π*Δ*τρ<sup>i</sup>*

where α<sup>=</sup> *r*Δ*<sup>τ</sup>*

<sup>Γ</sup>=*α*{ *q*Ψ*o*Δ*<sup>ρ</sup>*

<sup>Δ</sup>*<sup>ρ</sup>* 2 and

with under-relaxation

*Bij* ⋅Ω *<sup>j</sup>*

Ψ*mΔρ* + *BNj* ⋅Ω *<sup>j</sup>*

Safdar Abbas\* and Brice Lecampion

\*Address all correspondence to: sabbas10@slb.com

Schlumberger Doll Research, Cambridge, USA
