Alexander M. LinkovRzeszow University of Technology, Eurotech (Poland)

Institute for Problems of Mechanical Engineering(Russian Academy of Sciences)

Gennady S. Mishuris Aberystwyth University, UK

Modified Formulation, e-Regularization and the Efficient Solution of Hydraulic Fracture Problems

The supports of EU Marie Curie IAPP program (Grant # 251475)and Russian Fund for Fundamental Researches (Grant # 12-05-00140)

are gratefully acknowledged

2

Objective

WellWell Rock

Rock

The importance of hydraulic fracturing has grown last years because huge resources

of gas are found in low permeable shales

The key element of technology, used in shales, is hydraulic fracturing

Analytical models and numerical simulations are important means to increase understanding and enhance efficiency of hydraulic fracturing

(e.g. Mack & Warpinski, 2000; Adachi, Siebrits et al., 2007)

2

The authors emphasized the need “to dramatically speed up … simulators”

The objective: to develop improved strategyfor solving hydraulic fracture problems

(i) Revisiting fundamentals

(ii) Disclosing significant features of HF problem

(iii) Suggesting modified formulation

(iv) Discussion of its applications

Scope

4

For any volume V(t) of moving particles, the RTT involves the normal component vn of particle velocity at its boundary S(t).

The shear component does not contribute into the boundary change.

ρ(x,t) is an arbitrary functionvΔt nV(t+Δt)

V(t)S(t)

∫ =)(

),(tV

dVtdtd xρ

⎭⎬⎫

⎩⎨⎧

−Δ+Δ ∫∫

Δ+→Δ

VVVt

dVtdVttt

),(),(1lim0

xx ρρ +∂∂

= ∫)(tV

dVtρ

0lim→Δt ∫Δ )(tV

dVρ

vΔt n

dS

vnΔtdV ∫)(tS

ndSvρdV = vndS

Hence, 0lim→Δt

=∫Δ )(tV

dVρ Finally,

+∂∂

∫)(tV

dVtρ

∫ =)(

),(tV

dVtdtd xρ ∫

)(tSndSvρ (RTT)

Revisiting Fundamentals: Reynolds Transport Theorem (RTT)

Temporal derivative of an integral over a moving volume of material particles

By the derivation, the normal component of the particle velocity equals to the speed of the surface propagation

5

Still the generality and significance of the SE have not been comprehended and used systematically for further 20 years after 1989.

On page 289 of his paper of 1989, L.F. Kemp wrote: “Nowhere is (5) mentioned.

(5)… is always present in moving boundary problem”

Speed Equation (SE)V* = vn

When applied to the entire volume of moving fluid, the RTT implies that the speed V* of its surface propagation equals to the normal component vn of the particle velocity:

vn = V*

S

The SE is Eqn (5) of the Kemp’s paper of 1989(L.F. Kemp, SPE 18959, 1989, 287-298)

It is reasonable to systematically employ the SE for HF problems,in particular, because it is the key eqn for tracing front propagation

by Level Set and Fast Marching Methods

Speed Equation

6

At SL: dSL = wLdL SLL

wLn

n

n

n-S-

S+n+

SL

SL

Smz

∫=)( tSn

e dSvdt

dV RTT = MC = VC for incompressible fluid

S(t) = S+ + S- + SLAt S+: tuv zn ∂∂= + / At S- : tuv zn ∂−∂= − /

Now both the speed of propagation V* and the particle velocity vn are averaged over the opening at L

where is the channel width (opening)−+ −= zz uuw

Hence for flow in narrow channel ∫ ∫∫ +∂∂

=)( )()( tS tL

nLtS

nm

dSvwdStwdSv

The RTT becomes: RTT = MC = VC∫ ∫+∂∂

=)( )(tS tL

nLe

m

dLvwdStw

dtdV

V* = vn SE

Reynolds Transport Theoremfor flow of incompressible fluid in narrow channel

7

RTT = MC = VC∫ ∫+∂∂

=)( )(tS tL

ne dLwvdS

tw

dtdV

S(t) L(t)

V* = vn

The Gauss theorem implies: ∫ +∂∂

=)(

)]([tS

e dSwvdivtw

dtdV

This yields the integral form of the VC for flow in a narrow channelFor distributed sinks (sources), the fluid out- (in-)flow is ∫−=

)(tSe

e dSqdt

dV

0])([)(

=++∂∂

∫tS

e dSqwvdivtw RTT = MC = VC

Being true for any part of moving fluid, it yields the differential form:

0)( =++∂∂

eqwvdivtw the common continuity equation (CE)

of hydraulic fracture problems

Note that the starting forms (both integral and differential) of the MC for a flow of an incompressible fluid in a narrow channel

contain the particle velocity rather than the flux

Integral and Differential Forms of RTT for flow of incompressible fluid in narrow channel

8

S(t) L(t)

V* = vn 0)( =++∂∂

eqwvdivtw continuity equation (CE)

Emphasize that, similar to Poiseuille-type eqns, the primary physical quantity is the particle velocity v,

while the flux q appears only by definition: q = wv

Physically, the product wv expresses the flux: q = wv

This beneficial property has been clearly recognized and employed in the review by M.G. Mack & N.R. Warpinski (2000). On page 6-21 of their paper, they wrote:

The particle velocity enters the SE and has the advantage over the flux:commonly, it is a smooth function, which changes slowly in the flow region

“Detailed numerical calculations have shown that the velocity varies much more slowly than flow rate”

The modified formulation employs the particle velocity in(i) SE, (ii) CE, and (iii) Poiseuille-type eqn.

Significance of Particle Velocity

9

Significance of Modified Opening

Hydraulic fractureFront

x1s

n

x2

r x*

O

O’ε instead of the pressure p, which is commonly singular

at the front when neglecting the lag

(i) The particle velocity ppwDw

v grad),(1=

We have noticed that the SE and the starting form of the CE suggest using:

Note now that the SE, taken together with an elasticity eqn, implies that commonly the opening has power asymptotics w = C(t)rα, with 0 < α < 1. Hence its derivative is singular near the front: dw/dr → ∞, when r → 0.

(ii) Modified opening y = w1/α instead of the opening wThus, it is reasonable to use:

In addition to the particle velocity, the modified formulation employs also the modified opening

Emphasize that the new variables y and v have ‘good’ properties:Modified opening is linear near the frontParticle velocity is non-zero finite smooth function up to the front

10

xξξ = V*t – x

In the system, moving with the fluid front, the CE reads:

0)( =++∂∂

eqwvdivtw continuity equation (CE)

0)( * =+∂∂

−∂∂

−+∂∂

eqvwwvVtw

ξξCE near the fluid front

In particular, this makes dubious extending the Carter leak-off approximation to the fluid front

This implies that the leak-off qe should be either finite at the crack tip or its singularity less than that of . ξ∂∂ /w

Otherwise, the assumption that the flow occurs predominantly in the channel plane is drastically violated both in the CE and Poiseuille-type eqn.

ξ∂∂+=

/*

wq

Vv ehence,

Neglecting the lag, the opening tends to zero when approaching the front (crack tip). In the limit w → 0, the CE becomes:

,0)]([ *=+

∂−∂

eqvVwξ

Continuity Equation in System Moving with Fluid Front. Its Implications

11

We see that at any point of a fluid, where the opening is zero, the CE degenerates to the equation, expressing the SE: CE → SE

the boundary condition of zero opening automatically leads to the

physically consistentadditional boundary condition, expressing the SE

Conclusion: under common assumptions of HF modeling,

ξ∂∂+=

/*

wq

Vv eWe could see that for zero lag, near the fluid front where w(t) = 0, in the limit w → 0, the CE becomes:

where for predominantly in-plane flow → 0 . In the limit w → 0:ξ∂∂ /wqe

*Vv = CE when w(t) = 0

Specific Feature of HF Problem: Identically Satisfied Speed Equation

Consider the common case of zero lag and predominantly in-plane flow

0)( * =+∂∂

−∂∂

−+∂∂

eqvwwvVtw

ξξCE in moving systemxξ

ξ = V*t – x

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Thus for the elliptic (in spatial coordinates) operatorwe have two, rather than one, physically consistent

boundary conditionsThis indicates that there might be difficulties

BC : at fluid pumping part at propagating fluid front (zero lag)But ! As shown, the BC (5)2 automatically yields additional BC at LC

∗∗∗

∗ =∂

−= Vdn

ppwDw

vn ),(1BC=SE ! (6)

qn = q0

S(t)Lq

w = 0

Initial condition (zero opening): 0)0,( =xw

Poiseuille equationppwD grad),(−=q

Continuity equation (local form)0/ =−∂∂+ eqtwdivq

0/]grad),([ =+∂∂− eqtwppwDdivReynolds equation (using (2) in (1) )

(1)

(2)

(3)(4)

0)( =xw (5)2

)()( 0 xx qq n = (5)1

For a fixed front, the boundary value problem appears ill-posed

Conventional Formulation. Its Specific Feature for Zero Lag

13

Hadamard Definition and Tychonoff Regularization

The SE being physically consistent, we need a proper method of regularization for the problem of hydraulic fracturing

A.N. Tychonoff (1963) Solution of incorrectly formulated problems and the regularization method, Soviet Mathematics 4, 1035-1038. [Transl. from Russian: А. Н. Тихонов, ДАН СССР, 1963, 151, 501-504]

A.N. Tychonoff (1943) clearly recognized significance of ill-posed problems for applications. He was the first to suggest a means to solve them numerically by using regularization:

Hadamard considered that ill-posed problems had no physical sense

By Hadamard, a problem is well-posed whenA solution exists The solution is unique The solution depends continuously on the data, in a reasonable metric

Otherwise, a problem is ill-posedOtherwise, a problem is ill-posed

Jacques Hadamard (1902), Sur les problemes aux derivees partielles et leur signification physique, Princeton Univ. Bul. 49-52

Hadamard considered that ill-posed problems had no physical sense

1414

Clear Evidence that BVP is Ill-Posed: Nordgren Problem

Initial condition: w(x,t0)=w0(x)

BC at liquid front x = x*

BC at inlet x = 000

4q=

∂∂

−=x

xw

0),( =∗ tw x

Boundary conditions:

In dimensionless variables, the problem becomesNordgren’s PDE02

42=

∂∂

−∂

∂tw

x

w

Well

)(tx∗

x

y

Ow(x,t)

hReynolds equation (Newtonian fluid)

03 =∂∂

−⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

tw

xpw

xkl

Elasticity equation for plane-strain in vertical cross-sections wkp r=

Speed Equation:+ )(3

** *3

4txxx

wdt

dxV =∂

∂−==

There are three rather than two BC for the PDE of second orderin spatial variable x.

For any fixed x*, the problem is ill-posed

15

Even More Clear Evidence that BVP is Ill-Posed

Thus, there are two, rather than one, BC at the fluid front. By Picard’s theorem, the Cauchy conditions (3), (4) uniquely define y(ξ), dy/dξ and consequently influx at the inlet. Hence, a solution of BVP (1)-(3) does not exist for an arbitrary influx.

+ SPEED EQUATION, which is met identically by a solution of ODE satisfying BC (3): ∗

∗=−=

∂ξ

ξ ξξ6.0dy SE at fluid front ξ = ξ* (4)

)()( 3 ξψξ =yDenote The problem is reduced to ODEODE0

203),/,(

2

2=−

∂+

ξξξ

ξ

dyddyyad

yd

where a(y,dy/dξ,ξ)=(dy/dξ+0.6ξ)/(3y) is finite at fluid front ξ=ξ*

(1)

Well

)(tx∗

xO

w

The Nordgren problem is self-similar. Introduce self-similar variables

)()( , 5/45/15/4 −== xttxwtx ψξ 5/4tx ∗∗ = ξ

BC at fluid front ξ = ξ*

BC at inlet ξ= 030

0 )0(75.0

yqdy

−=∂ =ξξ

0)( =∗ξy

(2)(3)

Boundary conditions for the ODE of second order:

By Hadamard definition, the BV problem (1)-(3) is ill-posed

1616

Regularization Method for Hydraulic Fracturing

We suggest the regularization method employing the very cause of the difficulty

The regularization method consists in using the BC (5) at a smalldistance ε behind the front rather than the BC (2) and (3) on the front

We call this approach ε - regularizationIt appears really efficient for solving HF problems

PDFwith two BC at a point x* of the fluid front

( ) 0grad),(div =−−∂∂

etw qppwD (1)

(2)

(3))()( *0* xx pp =

∗=∂∂

−=

nvnppwD

w*xx*x

),()(

1

*

Prescribed for a problem

Speed Equation

We have:Hydraulic fracture

Front

x1s

n

x2

r x*

O

O’ε

(4)

(5)

Integration of (3) and accounting for (2) yield

By using (4) we impose the BC at a small distance ε behind the front:

εε

*0

),(1 vdppwDw

p

p=∫

rvdppwDw

p

p*

0

),(1≈∫

1717

Illustration: Solution of Nordgren Problemwithout regularization

Well

)( tx ∗

x

yO

w(x,t)h

We solved both the starting and self-similar BV Nordgren problem by finite differences

without regularizationBy no means could we have more than two correct digits

Using fine meshes (with the step less than 10-5x*) led to complete deterioration of the solution in the entire region

The results always deteriorated near the frontFurthermore,

This clearly shows that a proper regularization method is needed to have accurate and reliable numerical results

18

Illustration: Solution of Nordgren Problemwith ε – regularization

We have obtained that near the front:1 ζ0

1 ζ0 ζεε

)1)(()(75.0),( ςς −≈ ∗∗ tvtxtYHence, we may impose the BC

at the relative distance ε behind the frontεςε )()(75.0),( tvtxtY ∗∗=

We solved both the starting and self-similar BV Nordgren problem by finite differences with ε -regularization

This shows that ε - regularization is really beneficialThe modified formulation uses ε-regularization, when appropriate

For ODE of self-similar formulation, there are six correct digits, at least;For PDE, the error is less than 0.03% even for 100 000 steps

The results are accurate in a wide range of ε (10-2 > ε >10-5), size (10-2 >Δζ >10-5) and number (up to 100 000) of time steps

Even for 100 000 steps, the time expense does not exceed 15 s

There are no signs of instability in specially designed experiments

Conclusions obtained:

Small time expense on a conventional laptop

1919

Summary on Modified Formulation

The particle velocity, as a variable smooth near the front, instead of the pressure;

The modified opening, which is linear near the front, instead of the opening itself;

The SE at each point of the front, instead of the single equation of global mass balance;

ε-regularization to exclude solution deterioration caused by the fact that the boundary value problem is ill-posed for a fixed position of the front Moving spatial coordinates;Reformulation of the common system of equations and BC in terms of the suggested variables complimented, when appropriate, with ε-regularization.

Summarizing, we come to the modified formulation of HF problem. In contrast with the conventional formulation, it uses:

20

Analytical Advantages of the Modified Formulation

Analytical advantages are evident when revisiting the classical problems

For both problems, the modified formulation provides simple analytical solutionsSee: Linkov, IJES, 2012, 52, 77-88; Linkov, Journal of Mining Science, 2013, 1; and Linkov & Mishuris, Proceedings of this Conference, 2013

PKN model: plane-strain state in vertical cross sectionWell

)( tx∗

x

y

w(x,t)h

The conventional formulation requires involved calculationsSee: Nordgren, Soc. Pet. Eng., 1972, August, 306-314

KGD model: plane-strain state in horizontal cross-section

xp

σn

σn

See: Spence & Sharp, Proc. Roy. Soc. London, A, 1985, 400, 289-313;Adachi & Detournay, Int. J. Numer. Anal. Meth. Geomech., 2002, 26, 579-604

Again, the conventional formulation requires involved calculations

21

Analytical solution of Nordgren problem for a fluid with the power viscosity law n

nf x

pwkv/1

1 ⎟⎠⎞

⎜⎝⎛

∂∂

−= +

The solution shows that in the case of low-permeable formation, the particle velocity is almost constant for any thinning fluid

in the entire flow region

Illustration: Analytical Solution for Power-Type Viscosity Law

22

Computational Advantages of the Modified Formulation

Possibility to use well-established Level Set and Fast Marching methods of the theory of propagating interfaces;Avoiding deterioration of numerical solution;Avoiding singularities at the fluid front;Highly efficient simulators for the Nordgren and P3D problemsSee: Mishuris et al. IJES, 2012, 61, 10-23; and Proceedings of this Conference, 2013

The main computational advantages are:

23

After spatial discretization, we arrive at a dynamic system,efficiently solved by standard solvers

This option has appeared ONLY due to employing the SE

ε-regularization:

The only difference with the Nordgren problem is the presence of the function H(y). It is evaluated in advance, smooth and equals to the unit on the fracture front.

IC:

PDE:

BC:SE:

Illustration: Efficient Solution of P3D Problem

24

Standard MATLAB R-K solver ode15s

is used to solve the dynamic system

Results of numerical experiments (i) Guaranteed accuracy: 0.01%(ii) Run-time to cover the time range of 12 orders: 15 seven in MATLAB environment (iii) Extreme stability:

no signs of instability

ConclusionThe numerical scheme based on the modified formulation,

being accurate, robust and stable, is an efficient means for solving P3D problems

ε = 10-5

Illustration: Using Standard Solver for Nordgren Problem with Carter Leak-off

2525

Further Work

25

Development of enhanced simulators for solving truly and P3D problems in real timeCoupling simulation of hydraulic fracture propagation withmodeling of accompanying seismicity (in real time, as well)

Further work may employ new options provided by the modified formulation. They include:

Further results are to be presented next year at theInternational Conference

“Recent Advances in Numerical Simulation of Hydraulic Fracture“ (Rzeszow, Poland, 14 – 16 July, 2014)

The work is in progress

The colleagues are kindly invited

262626

Thank you!