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

vx = vx(x, z, t), vy = 0, vz = vz(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 h3 by anhn

∗ Fluid flux: Q(t, x) = − 2 3µanhn ∂p ∂x (t, x), ∗ Width averaged fluid velocity: vx(t, x) = − an 3µhn−1 ∂p ∂x (t, x), ∗ Governing PDE: ∂h ∂t = an 3µ ∂ ∂x

• hn ∂p

∂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

• PKN approximation:

σzz(t, x) = σ(∞)

zz

− Λh(t, x), Λ = E (1 − ν2)B

• Linear crack law (Pine et al [3], Fitt et al [1], Kgatle & Mason [2])
• Hyperbolic crack law (Goodman [4])

p(t, x) + p2(t, x) = −σzz(t, x), p2(t, x) = −k hmax − h(t, x) h(t, x) − hmin

• where k < 0.

∗ hmin << hmax, ∴ assume hmin = 0 (Fitt et al [1], King and Please [9]) ∗ Pressure gradient: ∂p ∂x (t, x) =

• Λ − khmax

h2 ∂h ∂x . ∗ Transformation variables: x∗ = x Lo , h∗ = h hmax , t∗ = Ut Lo , L∗ = L Lo , v∗

x = vx

U , Q∗ = Q hmaxU , U = Λh3

max

µLo

• 1 − σ(∞)

zz

Λhmax

• 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 ∗ Governing PDE: ∂h∗ ∂t∗ = Kn ∂ ∂x∗

• h∗n ∂h∗

∂x∗ + φh∗n−2 ∂h∗ ∂x∗

• ∗ BCs:

h∗(t∗, L(t)) = 0, − 2Kn

• h∗n(t, 0)∂h∗

∂x∗ (t∗, 0) + φh∗n−2(t∗, 0)∂h∗ ∂x∗ (t∗, 0)

• = dV ∗

dt∗ , ∗ Fluid flux: Q∗(t∗, x∗) = −2Kn

• h∗n ∂h∗

∂x∗ + φh∗n−2 ∂h∗ ∂x∗

• ,

∗ Width averaged velocity: v∗

x(t∗, x∗) = −Kn

• h∗n−1 ∂h∗

∂x∗ + φh∗n−3 ∂h∗ ∂x∗

• ,

Kn = anhn−3

max

3

• 1

1 − σ(∞)

zz

Λhmax

• ,

φ = − k Λhmax , k < 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

Group invariant solution

∂h ∂t = Kn ∂ ∂x

• hn ∂h

∂x + φhn−2 ∂h ∂x

• .
• Other methods of solution (Huppert [7], Spence and Sharp [10])
• Lie point symmetry generator:

X = (c1 + c2t) ∂ ∂t + (c3 + c2 2 x) ∂ ∂x where c1, c2, and c3 are constants. ∗ X(φ − h)|φ=h = 0 → a linear PDE → Group invariant solution: h = F(ξ) = c2 c1 1 2Kn 1

n

f (u), ξ = c2 c1 1

2 u,

u = x L(t). ∗ Important half-width condition: h∗(0, 0) = h(0, 0) hmax = β ∗ 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: d du

• f n df

du + φf 2(0) β2 f n−2 df du

• + d

du (uf ) − f = 0, f (1) = 0, f n(0)df du (0) = − 1

• 1 + φ

β2

• 1

f (u)du. ∗ Length: L(t) =

• 1 + 2

β f (0) n Knt 1

2

, ∗ Volume: V (t) = 2βL(t) 1 f (u) f (0)du, ∗ Half-width: h(t, x) = β f (u) 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: Q(t, x) = −2 Kn L(t) β f (0) n+1

• f n + φf 2(0)

β2 f n−2

• ∂f

∂u , ∗ Width averaged velocity: vx(t, x) = − Kn L(t) βn f n(0)

• f n−1 + φf 2(0)

β2 f n−3

• ∂f

∂u , where Kn = anhn−3

max

3

• 1

1 − σ(∞)

zz

Λhmax

• ,

φ = − k Λhmax 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¨
• berg [8])
• Conservation law for a PDE

DtT 1 + DxT 2

• PDE = 0

where Dt and Dx are total derivatives Dt = ∂

∂t + ht ∂ ∂h + htt ∂ ∂ht + hxt ∂ ∂hx + ...

Dx =

∂ ∂x + hx ∂ ∂h + htx ∂ ∂ht + hxx ∂ ∂hx + ...

respectively and T = (T 1, T 2) is a conserved vector.

• New conserved vector (Kara and Mahomed [12]):

T∗ = X(T i) + T iDk(ξk) − T kDk(ξi), i = 1, 2.

• Association (Kara and Mahomed [13]):

T∗ = 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

• Hyperbolic hydraulic fracture with tortuosity

∂h ∂t = Kn ∂ ∂x

• hn ∂h

∂x + φhn−2 ∂h ∂x

• ∗ From the elementary conservation law, the conserved vector is

T(1) = (h, −Kn(hn + φhn−2)hx), New conserved vector: T∗

(1) = c2

2 T(1). ∗ From the second conservation law, the conserved vector is T(2) =

• xh, Kn
• hn+1

(n + 1) + φ hn−1 (n − 1) − x(hn + φhn−2)hx

• ,

New conserved vector: T∗

(2) = c3T(1) + c2T(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 = (c1 + c2t) ∂ ∂t + (c3 + c2 2 x) ∂ ∂x

• Linear hydraulic fracture:

X = (c1 + c2t) ∂ ∂t + (c3 + c4x) ∂ ∂x + 1 n(2c4 − c2)h ∂ ∂h X = c1 c2 + t ∂ ∂t + c3 c2 + αx ∂ ∂x + 1 n(2α − 1)h ∂ ∂h η = 1 n(2α − 1)h = 0, provided α = 1 2 X = (c1 + c2t) ∂ ∂t + (c3 + c2 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

∗ Transformation variables: u = γu, f = γ− 2

n f .

• Asymptotic solution, (as

u → 1) f (u) ∼

• (n − 2)

β2 φf 2(0)

• 1

n−2

(1 − u)

1 n−2 ,

for 2 < n < 5, h(t, x) ∼ β

• (n − 2)

β2 φf 2(0)

• 1

n−2

1 − x L(t)

• 1

n−2 ,

∂h ∂x (t, L(t)) ∼      −∞, n > 3 −

1 φL(t)

• β

f (0)

3 , n = 3 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)

0.5 1 1.5 2 2.5 3 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

x h Kn

L t=0

Kn

L t=4

Kn

L t=20

Kn

L t=10

(b)

1 2 3 4 5 6 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

x h Kn

L t=20

Kn

L t=9

Kn

L t=3

Kn

L t=0

(c)

1 2 3 4 5 6 7 8 9 10 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

x h Kn

L t=0

Kn

L t=2

Kn

L t=8

Kn

L t=20

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 Knt 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)

1 2 3 4 5 6 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

x h φ=0 φ=0.1 φ=0.5 φ=1

(b)

10 20 30 40 50 60 70 80 90 100 1 2 3 4 5 6 7 8 9 10 11 12

Knt L(t) φ=0 φ=0.1 φ=0.5 φ=1

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 Knt = 20 (b) The length of the fracture plotted against Knt.

• M. R. R. Kgatle, D. P. Mason

Hyperbolic hydraulic fracture with tortuosity

Modelling process Group invariant soln. Operating conditions Numerical soln. Averaged velocity Conclusion

Width averaged fluid velocity

Velocity ratio : vx dL/dt = −f n−1

• 1 + φ

f (0) βf 2

• df

du

(a)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

u Vx/(dL/dt) φ=0.1 φ=0 φ=1

(b)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

u Vx/(dL/dt) φ=0 φ=0.1 φ=1

(c)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

u Vx/(dL/dt) φ=0 φ=0.1 φ=1

Velocity ratio curves

v x dL/dt plotted againt u for a partially open fracture

(β = 0.5) propagating with fluid injected at the fracture entry at a constant pressure 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

Width averaged fluid velocity

Velocity ratio : vx dL/dt = −f n−1

• 1 + φ

f (0) βf 2

• df

du = (1 − A)u + A

(a)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

u Vx/(dL/dt) φ=0.1 φ=0 φ=1

(b)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

u Vx/(dL/dt) φ=0 φ=0.1 φ=1

(c)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

u Vx/(dL/dt) φ=0 φ=0.1 φ=1

Velocity ratio curves

v x dL/dt plotted againt u for a partially open fracture

(β = 0.5) propagating with fluid injected at the fracture entry at a constant pressure 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

Approximate analytical solution

• Problem for 2 < n < 5:

f n + pf n−2 + q = 0, where p = nφf 2(0) β2(n − 2), q = n (1 − A) 2 u2 + Au − (A + 1) 2

• and

f (0) = (A + 1) 2 1

n

• n(n − 2)β2

β2(n − 2) + φn 1

n

.

• Special case of n=3:

f 3 + pf + q = 0, f (u) = ( √ 3

• 4p3 + 27q2 − 9q)1/3

21/332/3 −

• 2

3

1/3 p ( √ 3

• 4p3 + 27q2 − 9q)1/3 .
• Special case of n=4:

f 4 + pf 2 + q = 0, f (u) =

• −p +
• p2 − 4q

√ 2

• M. R. R. Kgatle, D. P. Mason

Hyperbolic hydraulic fracture with tortuosity

Modelling process Group invariant soln. Operating conditions Numerical soln. Averaged velocity Conclusion

• General case 2 < n < 5:

f (u) =

• [(A + 1) − 2Au − (1 − A)u2]β2n(n − 2)f 2(u)

2[β2(n − 2)f 2(u) + φnf 2(0)] 1

n

fi+1 =

• [(A + 1) − 2Au − (1 − A)u2]β2n(n − 2)f 2

i

2[β2(n − 2)f 2

i + φnf 2(0)]

1

n

, i = 0, 1, 2, ...s, Ai+1 =

• 2

1 n β2n(n − 2)[β2(n − 2) + φn] 2 n

1

n 1

(1 − u)

1 n ×

• f 2

i

2

2 n β2(n − 2)[β2(n − 2) + φn] 2 n f 2

i + φn(Ai + 1)

2 n [n(n − 2)β2] 2 n

1

n

du, s + 1 = No. of iterations for convergence to be achieved

• M. R. R. Kgatle, D. P. Mason

Hyperbolic hydraulic fracture with tortuosity

Modelling process Group invariant soln. Operating conditions Numerical soln. Averaged velocity Conclusion

(a)

0.5 1 1.5 2 2.5 3 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

x h Kn t=20 Knt=10 Knt=4 Knt=0

(b)

1 2 3 4 5 6 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

x h Kn t=20 Knt=9 Knt=3 Knt=0

(c)

1 2 3 4 5 6 7 8 9 10 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

x h Knt=20 Kn t=8 Kn t=2 Kn t=0

Comparison of the numerical (—) and the approximate analytical (− − −) half-width solutions plotted against x for increasing time scales Knt 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

Conclusions

• Earlier paper: the linear crack model was found to give both

fluid injection and extraction solutions.

• The hyperbolic crack law model was found to admit only one

solution of fluid injected at constant pressure at the fracture entry.

• All solutions for the linear crack law model were found converge

to the constant pressure solution.

• An analytical solution could not be derived. A numerical solution

was therefore investigated and obtained.

• The width averaged fluid velocity was obtained to increase approx

linearly along the fracture length.

• The approximate analytical solution may be required in practice.
• M. R. R. Kgatle, D. P. Mason

Hyperbolic hydraulic fracture with tortuosity

Modelling process Group invariant soln. Operating conditions Numerical soln. Averaged velocity Conclusion

References

1 A. D.Fitt, A. D. Kelly and C. P. Please, Crack propagation models for rock fracture in a geothermal energy reservoir, SIAM J Appl Math. 55 (1995) 1592-1608. 2 M. R. R. Kgatle and D. P. Mason, Effects of tortuosity on the propagation of a linear two-dimensional hydraulic fracture,

• Inter. J Non-Linear Mech. 61 (2014) 39-53.

3 R. J. Pine, P. A. Cundall, Applications of the fluid-rock interaction programme (FRIP) to the modelling of hot dry geothermal energy systems, Proc. Int. S Fundamentals, September 1985, pp 293-302. 4 R. E. Goodman, The mechanical properties of joints, Proc. 3rd Congr ISRM, Denver, 1A. (1947) 127-140.

• M. R. R. Kgatle, D. P. Mason

Hyperbolic hydraulic fracture with tortuosity

Modelling process Group invariant soln. Operating conditions Numerical soln. Averaged velocity Conclusion

5 T. Perkins and L. Kern, Widths of hydraulic fracture, Journal

• f Petroleum Technology. 222 (1961) 937-949.

6 R. Nordgren, Propagation of vertical hydraulic fractures, Journal of Petroleum Technology. 253 (1972) 306-314. 7 H. E. Huppert, The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface,J. Fluid Mech. 121 (1982) 43-58. 8 A. Sj¨

• berg, Double reduction of PDEs from the association of

symmetries with conservation laws with applications,Appl.

• Math. Comput. 184 (2007) 608-616.

9 R. J. King and C. P. Please, Diffusion of dopant in crystalline silicon: An asymptotic analysis, IMA J Appl Math. 37 (1986) 185-197.

• M. R. R. Kgatle, D. P. Mason

Hyperbolic hydraulic fracture with tortuosity

Modelling process Group invariant soln. Operating conditions Numerical soln. Averaged velocity Conclusion

10 D. A. Spence and P. Sharp, Self-similar solutions for elasto-hydrodynamic cavity flow, Proc.R.Soc.Lond.A. 400 (1985) 289-313. 11 A. G. Fareo and D. P. Mason, Group invariant solutions for a pre-existing fluid-driven fracture in permeable rock, Nonlinear Analysis: Real World Applications. 12 (2011) 767-779. 12 A. H. Kara and F. M. Mahomed, A basis of conservation laws for partial differential equations, J Nonlinear Math Phys. 9 (2002) 60-72. 13 A. H. Kara and F. M. Mahomed, Relationship between symmetries and conservation laws, International J of Theoretical Physics. 39 (2000) 23-40.

• M. R. R. Kgatle, D. P. Mason

Hyperbolic hydraulic fracture with tortuosity

Modelling process Group invariant soln. Operating conditions Numerical soln. Averaged velocity Conclusion

Thank You

• M. R. R. Kgatle, D. P. Mason

Hyperbolic hydraulic fracture with tortuosity