SIAM PDE San-Diego, November 14-18,2011

In Collaboration with :

E.Aulisa, M.Toda, A.Cakmak, A.Solyinin, L. Bloshanskaya,L.Hoang (Texas Tech)

This well, with upgrades made by Dupuit in the

early 1850s, was one of the major municipal

supplies of water for the SW side of Paris, along

with the Perrier pumping station at Chaillot,

which supplied water from the Seine

History of the underground fluid flow

starts by Darcy with clean water

supply

Comprehensive principles ….established by Darcy and Dupuit, by R.W. Ritzi, and P. Bobeck,

Water resources research,,v44,2008.

Stratigraphy and structural geology of the Paris basin

[from Darcy, 1856]

q P Darcy approximation

1 2 , k P P P

J P

2q q P Forchheimer Two terms approximation

n nq c q P

Forchheimer Power approximation , 1 2n nq c q P n

Forchheimer Three terms (cubic) approximation 2 3Aq Bq Cq P

Muskat, M., The Flow of Homogeneous Fluids through Porous Media, 1937, latest edition, Springer 1982

Bear, J., Dynamics of Fluids in Porous Media ,Dover Publications, Inc., New York, 1972

1'

Two term Forchheimer Law

Power Forchheimer Law

Three term cubic Forchheimer Law

' ,v v v v p (1)

1

' ,n

nv c v v v p

(2)

, ' ,v B K v v v C v v v p (3)

Dimensionless

permeability tensor K

1

,...,nx x

p p p

1

0 1( ) ... n

n

vg v p

g s a a s a s

Algebraic polynomial

with positive coefficients

E. Aulisa, L. Bloshanskaya, L. Hoang, A. I. , Analyses of Gen. Forchheimer Flows of Compressible fluid …, J. Math. Phys. 50,

(2009).

Proposition. For any algebraic polynomial g(s) exists G positive function

such that , generalized nonlinear Darcy vector field (5) , solves generalized

Forchheimer equation.

v K p Darcy vector field

( ( ))v K p p Generalized Nonlinear Darcy Vector Field

( , )v x t velocity ( , )p x t pressure

Bilinear form ( ) ,p p p

E. Aulisa, L. Bloshanskaya, L. Hoang, A. I. , Analysis of Gen. Forchheimer Flows of Compressible Fluid in the porous media…, J.

Math. Phys. 50, (2009).

( ( ))v K p p (5)

tp v v p (6)

1 (7)

Momentum equation –Generalized Nonlinear Darcy Vector Field

Continuity equation

Equation of state for slightly compressible fluids

tp v v p

Then (6) is reduced to

w

1

( )= ( , ) S

S

Sp t p x t dx

If ( ) and ( )Q t Q const PDD t const

Definition : Pseudo-Steady State (PSS) regime

( ) ( , ),w

xQ t v x t n ds

( ) ( ) ( )w

UPDD t p t p t

( )( , , ) ( )

( )K

Q tJ p v t J t

PDD t

Total Flux on the well

Pressure drawdown

Average Pressure

( , , )G

J p v t constConsequently

Productivity Index/ Diffusive Capacity

A. I., Khalmanova, D., Valk´o, P. P., and Walton, J. R., “On a Mathematical Model of the Productivity Index

of a Well from Reservoir Engineering,” SIAM J. Appl. Math. 65, 1952 (2005).

( ( )) in the domain

( ) ( , ),

( , ), 0 on

( ,0) ( )

w

x

e

pK p K p U

t

Q t v x t n ds

v x t n

p x f x

K p

2 2

2

2 1 0 2 1 2 1( , ) 1

a aL L

a

Up p dx C p p p p

Weighted monotonicity: :

1 2 1 1 2 2 1 2

( , ) ,u u K u u K u u u u

2 1 ,

11 1

N

a a

N

C CK a

1/2 2

0

( )H K s ds K

E. Aulisa, L. Bloshanskaya, L. Hoang, A. I. , Analysis of Gen. Forchheimer Flows of Compressible Fluid in the porous media…, J.

Math. Phys. 50, (2009).

(1)

(2)

(3)

(1s-2s)

(3s)

(4s)

Theorem . Psudo Steady State solution exist and unique ff

( , ),w

xv x t n ds Q const

U

w

E. Aulisa, A. I. P. P. Valko, J. R. Walton, “Math. Frame-Work For PI of The Well for Fast Forchheimer (non-Darcy) Flow in

porous media”, Mathematical Models & Methods in Applied Science, v.. 19, (2009),

E. Aulisa, L. Bloshanskaya, A. I. Long-term dynamics for well productivity index for nonlinear flows , J. Math. Phys. 52 (2011)

(4)

/A Q U K w w

( , ) 0U

v n

0( ) ( ) on

ww x x

( ) 0 on e

w x

( ,0) ( )s

p x w x

0)

wsp At x

( ( ))s s st

p K p K p

Definition :

Is called pseudo-steady state solution

, here is integral average.

Theorem . PSS regime exists and is unique

( , )sp x t

Proof.

Indeed, pressure function solves transient problem (1-3)

And corresponding pressure drawdown is equal to

E. Aulisa, A. I. P. P. Valko, J. R. Walton, “Math. Frame-Work For PI of The Well for Fast Forchheimer (non-Darcy) Flow in

porous media”, Mathematical Models & Methods in Applied Science, v.. 19, (2009),

E. Aulisa, L. Bloshanskaya, A. I. Long-term dynamics for well productivity index for nonlinear flows , J. Math. Phys. 52 (2011),

( , ) ( )sp x t w x At

0( )PDD t w const

f0

( , , ) K s s

QJ p v t const

w

1( ) ( ) )

2F F K

Proposition 4 Equation /Q U div K w w

Serves as Euler-Lagrange equation for functional

21( ) ( )

2K

I u F u u Au dx

a) in linear Darcy case b) in case of Forchheimer

Power Law, Two terms Law

Theorem 5. Variational interpretation of diffusive capacitance

1

0 1 1( ) , 1g s a a s 0 1( )g s a a s

_ _1/

Two Terms Two TermsI J( ) 1/Darcy DarcyI W J 1/

Power PowerI J

E. Aulisa, A. Cakmak, A. I. A. Solynin, “Variational principle and steady state invariant for non-linear hydrodynamic interactions

in porous media Advances in Dynamical . Systems 14 (S2) (2007).

Gap Between Variational Interpretation in

Linear and Non-linear Cases

1; = 1/

0 , 0 w U

Darcy D

D

D D

J w Uw

w w n

( ) ; ( ) = /

0 , 0 w U

g F F

F

F F

QJ Q K w w Q U

w

w w n

Comparison Theorem

,( )

1 maxg PSS

Darcy

Darcy

J Qw QR

J , here

1

0 1

max( )

ik

jj

R ag

1 0/R a a2 0

1 0

2 0 1 0

//

2 / /

a aR a a

a a a a

Two terms (quadratic)Forchheimer Three terms (cubic) Forchheimer

E. Aulisa, L. Bloshanskaya, A. I. Long-term dynamics for well productivity index for nonlinear flows , J. Math. Phys. 52 (2011)

Impact of the non-linearity on the deviation , from g PSS DarcyJ J

Application of the mathematical framework for general Forchheimer polynomial

Radial flow

er

wr

02 /, for

ln 3 / 4Darcy e w

e

w

aJ r r

r

r

02 /, for

ln 3 / 4Forch e w

e

w

aJ r r

rskin

r

Routine empirical approach, skin factor obtained by matching

0, 1 1

1

0

22 2

1

1 12( 2)1 0

2 /1, for

ln 3 / 4 2 /

2

jj e

j jj

g PSS e weDarcy F

w

rkej

j e

aJ r r

rJ Sa S

r

r ra QS dr

rr

E. Aulisa, A. I., P. P. Valko, J. R. Walton, “A new method for evaluation the productivity index…”, SPE, v.14, n., 2009 .

E. Aulisa, L. Bloshanskaya, A. I. “Long-term dynamics for well productivity index for nonlinear flows “, J. Math. Phys. 52 (2011)

Theorem. Let

Then function ( )u exists, such that

2

(w u u u

2

1

1

Qu

Uu

Constant Mean Curvature Equation

2

2 Qw

Uw

G w

Steady state Two terms

Forchheimer Equation

M. Toda, M, A.I., A, Aulisa, E, “Geometric Frame-Work for Modeling Non-Linear Flows …”, J. of Non-linear Anal. v.. 11, 3, 2010.

E, Aulisa, A.I. , Toda, M. “Geom. Meth. in the Anal. of Non-linear…” Proc. of the AMS, Spectral Theory and Geometric Analysis,2011

w

( , )wA

p x t Is specified

Bottom hole Pressure

(w

q q xK p p n ds

Is specified

Total well production

Total

Flux Dirichlet

Satisfies simultaneously two conditions on the boundary

total flux and Dirichlet (given pressure)

Goal

( , , ) ( , , )K q q K s s

J p v t J p v t const ( , , ) ( , , )K A A K s s

J p v t J p v t const

( ) , and initial Data ( ,0) ( )s s s st

p K p p p x w x

0( , ) ( ) on

s wp x t At x ( ( )

w

s s xK p p n ds Q t

0

( , )

( , ) ( )

wA

H

p At x t

x t x

( ( )

w

q q xK p p n ds Q t Q

Asymptotic and structural stability of the Diffusive capacity

with respect to boundary Data

Then

Theorem . Dirichlet BC Assume degree condition

Let

1,2 1,2( ) ( ): || '( ) || , || ( ) || 0 as , for some

w wW WH t t t t t k

L.Hoang, A.I.” Structural stab. of gen. Forchheimer equations for compressible fluids in porous media”, Nonlinearity, v.24, 1 , 2011

E. Aulisa, L. Bloshanskaya, A. I. “Long-term dynamics for well productivity index for nonlinear flows “, J. Math. Phys. 52 (2011)

Convergence follows from the inequality

and asymptotical regularity with respect to time derivative:

2

( ) t L

U

Q t Q p ds Q C q

Theorem on “asymptotical regularity”. If H condition is satisfied then

2

0s t

U

p p dx

4deg( )

2g

n

0( , ) ( )

Hx t x

( , , ) ( , , )K A A K s s

J p v t J p v t const

In case of total Flux condition solution is not unique

Assume ( , ) is stabilising at time infinity to some

and total flux ( ) is stabilising to constant .

Then pair , generates a problem, with

time independent diffusive capacity

x t x

Q t Q

x Q PSS

Theorem ( , , ) ( , , )

K q q K s sJ p v t J p v t const

Class of the traces on the boundary is introduced in terms of the deviation

from the averages on the boundary

0

0

0

( , ) ( , )

( ) ( , )

( , ) ( , ) ( )

Then the trace

( , ) ( , ) ( )

p x t x t

t x t

x t x t t

p x t x t t

The following Theorem is true under degree constraint and some

assumptions on deviation from the average:

No explicite conditions on ( ) !t

0For example if Dirchlet Data are split as ( , ) ( ), and ( , ) ( ), and total flux ( )f x t r t f x t f x Q t Q

Average of the trace on the boundary

( )Q t

0( , )x t

• Generalized non-linear potential flows can be used as an effective framework to study non-linear flows in porous media.

• In case of Two Terms Law if the product between the Forchheimer coefficient and the production rate is constant then the well productivity index for PSS regime is invariant with respect to any variation of the production rate and the Forchheimer coefficient.

• PSS regime serves as a global attractor for a class of IBVP with arbitral initial pressure distribution, and diffusive capacitance characterizes the deviation between regimes of production