Hyperbolic hydraulic fracture with tortuosity M. R. R. Kgatle, D. P. - PowerPoint PPT Presentation

Modelling process Group invariant soln. Operating conditions Numerical soln. Averaged velocity Conclusion Hyperbolic hydraulic fracture with tortuosity M. R. R. Kgatle, D. P. Mason March 22, 2016 School of Computer Science and Applied Mathematics, University of the Witwatersrand. M. R. R. Kgatle, D. P. Mason Hyperbolic hydraulic fracture with tortuosity

Modelling process Group invariant soln. Operating conditions Numerical soln. Averaged velocity Conclusion Model formulation Problem description (a) Tortuous fracture (b) Two-dimensional symmetric model v x = v x ( x , z , t ) , v y = 0 , v z = v z ( x , z , t ) , p = p ( x , z , t ) , M. R. R. Kgatle, D. P. Mason Hyperbolic hydraulic fracture with tortuosity

Modelling process Group invariant soln. Operating conditions Numerical soln. Averaged velocity Conclusion • Reynolds flow law • General flow law ( Fitt et al ): Substitute h 3 by a n h n Q ( t , x ) = − 2 3 µ a n h n ∂ p ∗ Fluid flux: ∂ x ( t , x ) , v x ( t , x ) = − a n 3 µ h n − 1 ∂ p ∗ Width averaged fluid velocity: ∂ x ( t , x ) , ∂ h ∂ h n ∂ p ∂ t = a n � � ∗ Governing PDE: . 3 µ ∂ x ∂ x • Crack laws (a) Partially open fracture (b) Open fracture M. R. R. Kgatle, D. P. Mason Hyperbolic hydraulic fracture with tortuosity

Modelling process Group invariant soln. Operating conditions Numerical soln. Averaged velocity Conclusion E σ zz ( t , x ) = σ ( ∞ ) • PKN approximation: − Λ h ( t , x ) , Λ = zz (1 − ν 2 ) B • Linear crack law ( Pine et al [3], Fitt et al [1], Kgatle & Mason [2] ) • Hyperbolic crack law ( Goodman [4] ) � h max − h ( t , x ) � p ( t , x ) + p 2 ( t , x ) = − σ zz ( t , x ) , p 2 ( t , x ) = − k h ( t , x ) − h min where k < 0. ∗ h min << h max , ∴ assume h min = 0 ( Fitt et al [1], King and Please [9] ) ∂ p Λ − kh max � ∂ h � ∗ Pressure gradient: ∂ x ( t , x ) = ∂ x . h 2 x ∗ = x h t ∗ = Ut h ∗ = ∗ Transformation variables: , , , L o h max L o 1 − σ ( ∞ ) U = Λ h 3 L ∗ = L x = v x Q Q ∗ = � � zz max , v ∗ U , h max U , L o µ L o Λ h max M. R. R. Kgatle, D. P. Mason Hyperbolic hydraulic fracture with tortuosity

Modelling process Group invariant soln. Operating conditions Numerical soln. Averaged velocity Conclusion Governing equations ∂ h ∗ ∂ h ∗ n ∂ h ∗ ∂ x ∗ + φ h ∗ n − 2 ∂ h ∗ � � ∗ Governing PDE: ∂ t ∗ = K n ∂ x ∗ ∂ x ∗ ∗ BCs: h ∗ ( t ∗ , L ( t )) = 0 , � � h ∗ n ( t , 0) ∂ h ∗ ∂ x ∗ ( t ∗ , 0) + φ h ∗ n − 2 ( t ∗ , 0) ∂ h ∗ = dV ∗ − 2 K n ∂ x ∗ ( t ∗ , 0) dt ∗ , � � h ∗ n ∂ h ∗ ∂ x ∗ + φ h ∗ n − 2 ∂ h ∗ ∗ Fluid flux: Q ∗ ( t ∗ , x ∗ ) = − 2 K n , ∂ x ∗ � � h ∗ n − 1 ∂ h ∗ ∂ x ∗ + φ h ∗ n − 3 ∂ h ∗ ∗ Width averaged velocity: v ∗ x ( t ∗ , x ∗ ) = − K n , ∂ x ∗ � � K n = a n h n − 3 1 k max , φ = − , k < 0 . 1 − σ ( ∞ ) 3 Λ h max zz Λ h max M. R. R. Kgatle, D. P. Mason Hyperbolic hydraulic fracture with tortuosity

Modelling process Group invariant soln. Operating conditions Numerical soln. Averaged velocity Conclusion Group invariant solution ∂ h ∂ h n ∂ h ∂ x + φ h n − 2 ∂ h � � ∂ t = K n . ∂ x ∂ x • Other methods of solution ( Huppert [7], Spence and Sharp [10] ) • Lie point symmetry generator: X = ( c 1 + c 2 t ) ∂ ∂ t + ( c 3 + c 2 2 x ) ∂ ∂ x where c 1 , c 2 , and c 3 are constants. ∗ X ( φ − h ) | φ = h = 0 → a linear PDE → Group invariant solution: � 1 � 1 � 1 �� c 2 � c 2 x n 2 u , h = F ( ξ ) = f ( u ) , ξ = u = L ( t ) . c 1 2 K n c 1 h ∗ (0 , 0) = h (0 , 0) ∗ Important half-width condition: = β h max ∗ Partially open fracture: 0 ≤ β < 1 M. R. R. Kgatle, D. P. Mason Hyperbolic hydraulic fracture with tortuosity

Modelling process Group invariant soln. Operating conditions Numerical soln. Averaged velocity Conclusion The problem is to solve ∗ BVP: � � du + φ f 2 (0) d f n df f n − 2 df + d du ( uf ) − f = 0 , β 2 du du f (1) = 0 , � 1 1 f n (0) df du (0) = − f ( u ) du . � � 1 + φ 0 β 2 � 1 � β � 2 � n ∗ Length: L ( t ) = 1 + 2 , K n t f (0) � 1 f ( u ) ∗ Volume: V ( t ) = 2 β L ( t ) f (0) du , 0 h ( t , x ) = β f ( u ) ∗ Half-width: f (0) . M. R. R. Kgatle, D. P. Mason Hyperbolic hydraulic fracture with tortuosity

Modelling process Group invariant soln. Operating conditions Numerical soln. Averaged velocity Conclusion ∗ Fluid flux: � β � � f n + φ f 2 (0) � n +1 ∂ f Q ( t , x ) = − 2 K n f n − 2 ∂ u , β 2 L ( t ) f (0) ∗ Width averaged velocity: � β n �� � f n − 1 + φ f 2 (0) ∂ f v x ( t , x ) = − K n f n − 3 ∂ u , β 2 L ( t ) f n (0) where � � K n = a n h n − 3 1 k max , φ = − 3 1 − σ ( ∞ ) Λ h max zz Λ h max and 0 ≤ β < 1 . M. R. R. Kgatle, D. P. Mason Hyperbolic hydraulic fracture with tortuosity

Modelling process Group invariant soln. Operating conditions Numerical soln. Averaged velocity Conclusion Operating conditions Conservation laws • Double reduction theorem ( Sj¨ oberg [8] ) • Conservation law for a PDE D t T 1 + D x T 2 � PDE = 0 � � where D t and D x are total derivatives D t = ∂ ∂ t + h t ∂ ∂ h + h tt ∂ ∂ ∂ h t + h xt ∂ h x + ... ∂ x + h x ∂ ∂ ∂ h + h tx ∂ ∂ D x = ∂ h t + h xx ∂ h x + ... respectively and T = ( T 1 , T 2 ) is a conserved vector. • New conserved vector ( Kara and Mahomed [12]) : T ∗ = X ( T i ) + T i D k ( ξ k ) − T k D k ( ξ i ) , i = 1 , 2 . T ∗ = 0 . • Association ( Kara and Mahomed [13]) : M. R. R. Kgatle, D. P. Mason Hyperbolic hydraulic fracture with tortuosity

Modelling process Group invariant soln. Operating conditions Numerical soln. Averaged velocity Conclusion • Hyperbolic hydraulic fracture with tortuosity ∂ h ∂ h n ∂ h ∂ x + φ h n − 2 ∂ h � � ∂ t = K n ∂ x ∂ x ∗ From the elementary conservation law, the conserved vector is T (1) = ( h , − K n ( h n + φ h n − 2 ) h x ) , (1) = c 2 New conserved vector: T ∗ 2 T (1) . ∗ From the second conservation law, the conserved vector is � � �� ( n + 1) + φ h n − 1 h n +1 ( n − 1) − x ( h n + φ h n − 2 ) h x T (2) = xh , K n , New conserved vector: T ∗ (2) = c 3 T (1) + c 2 T (2) . ∗ Association for non-trivial solutions is not satisfied M. R. R. Kgatle, D. P. Mason Hyperbolic hydraulic fracture with tortuosity

Modelling process Group invariant soln. Operating conditions Numerical soln. Averaged velocity Conclusion Comparison of Lie point symmetries • Hyperbolic hydraulic fracture: X = ( c 1 + c 2 t ) ∂ ∂ t + ( c 3 + c 2 2 x ) ∂ ∂ x • Linear hydraulic fracture: X = ( c 1 + c 2 t ) ∂ ∂ t + ( c 3 + c 4 x ) ∂ ∂ x + 1 n (2 c 4 − c 2 ) h ∂ ∂ h � ∂ � ∂ � c 1 � c 3 ∂ x + 1 n (2 α − 1) h ∂ X = + t ∂ t + + α x c 2 c 2 ∂ h η = 1 α = 1 n (2 α − 1) h = 0 , provided 2 X = ( c 1 + c 2 t ) ∂ ∂ t + ( c 3 + c 2 2 x ) ∂ ∂ x • Constant pressure working condition M. R. R. Kgatle, D. P. Mason Hyperbolic hydraulic fracture with tortuosity

Modelling process Group invariant soln. Operating conditions Numerical soln. Averaged velocity Conclusion Numerical solution Method of solution • BVP → 2 IVPs f = γ − 2 n f . ∗ Transformation variables: u = γ u , • Asymptotic solution, (as u → 1) 1 β 2 � � n − 2 1 n − 2 , f ( u ) ∼ ( n − 2) (1 − u ) for 2 < n < 5 , φ f 2 (0) 1 β 2 1 � � x n − 2 � n − 2 , � h ( t , x ) ∼ β ( n − 2) 1 − φ f 2 (0) L ( t )  −∞ , n > 3 ∂ h  � 3  � β 1 ∂ x ( t , L ( t )) ∼ − , n = 3 φ L ( t ) f (0)  0 , 2 ≤ n < 3  M. R. R. Kgatle, D. P. Mason Hyperbolic hydraulic fracture with tortuosity

Modelling process Group invariant soln. Operating conditions Numerical soln. Averaged velocity Conclusion Numerical results (a) (b) (c) 0.5 0.5 0.5 0.45 0.45 0.45 0.4 0.4 0.4 0.35 0.35 0.35 0.3 0.3 0.3 0.25 0.25 0.25 h L t=20 h h K n 0.2 0.2 0.2 K n L t=10 L t=20 K n 0.15 0.15 L t=9 0.15 K n K n L t=20 L t=4 K n L t=8 0.1 0.1 K n L t=3 0.1 K n K n L t=0 K n L t=2 0.05 0.05 0.05 L t=0 K n L t=0 K n 0 0 0 0 0.5 1 1.5 2 2.5 3 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 9 10 x x x Partially open fracture ( β = 0 . 5) propagating with fluid injected at the fracture entry at a constant pressure. The numerical solution for the half-width h ( t , x ) plotted against x for increasing values of the scaled time K n t and for (a) n = 4, (b) n = 3, (c) n = 2 . 5. M. R. R. Kgatle, D. P. Mason Hyperbolic hydraulic fracture with tortuosity

Modelling process Group invariant soln. Operating conditions Numerical soln. Averaged velocity Conclusion Variation of φ (a) (b) 0.5 12 φ =1 11 0.45 10 0.4 9 0.35 φ =0.5 8 0.3 7 L(t) 0.25 6 h φ =0.1 5 0.2 4 0.15 3 φ =0 φ =1 0.1 φ =0.5 2 φ =0.1 0.05 φ =0 1 0 0 0 1 2 3 4 5 6 0 10 20 30 40 50 60 70 80 90 100 x K n t Partially open fracture ( β = 0 . 5) propagating with fluid injected at the fracture entry at a constant pressure for (i) φ = 0, (ii) φ = 0 . 1, (iii) φ = 0 . 5, (iv) φ = 1 and for n = 3. (a) The half-width of the fracture plotted against x for the time scale K n t = 20 (b) The length of the fracture plotted against K n t . M. R. R. Kgatle, D. P. Mason Hyperbolic hydraulic fracture with tortuosity