Peaceman and Thiem well models or how to remove a …sites.science.oregonstate.edu/~mpesz/amc/amc_wells.pdf · 2007-03-02 · Małgorzata Peszy´nska Peaceman and Thiem well models

Problem statement: what are wellsNumerical solution of idealized model

Peaceman correction and beyondIdealized (with point sources) versus real model

Peaceman and Thiem well modelsor

how to remove a logarithmic singularity fromyour numerical solution

Małgorzata Peszynska

Department of MathematicsOregon State University

AMC, 3/2/2007

Current work supported by NSF-0511190 and DOE-98089Acknowledgement: Cristiano Garibotti

Małgorzata Peszynska Peaceman and Thiem well models or how to remove a logarithmic singularity from your numerical solution



Outline

1 Problem statement: what are wellsSteady state, rate specified wellsIdealized and real well models

2 Numerical solution of idealized modelNot enough regularity for classical FE ...Why subtracting singularities is not sufficient

3 Peaceman correction and beyondEffective well radius idea: Peaceman correctionBeyond Peaceman correction

4 Idealized (with point sources) versus real modelCloseness of solutions: elliptic and parabolic estimates




Steady state, rate specified wellsIdealized and real well models

What are wells





Well in a confined aquifer/hydrocarbon reservoir





Pressure profiles around the well: side view





Pressure profiles around the well: aerial view





Mathematical model: geometry





Problem statement for steady state, rate specified, R2

Conservation equations:

u = −K∇p, x ∈ Ω Darcy′slaw

∇ · u = q, conservation of mass

q(x) represents source/sink terms,and is known,K is the conductivity tensor and isknown.

Put it together

−∇ · K∇p = q(x), x ∈ Ω

p|∂Ω = pGIVEN = 0





Mathematical model for wells: idealized and real

Geometry Ωw ≈ B(x0, rw ) with boundary Γw

1 (R)Real model of injection/production wells

−∇ · K∇p = 0, x ∈ Ω \ Ωw

K∇p · ν =q

2πrw, x ∈ ∂Ωw

2 (ID) Idealization of a “true” real-life situation

−∇(K∇pδ) = q(x) = qδx0(x), x ∈ Ω

where by δx0(x) we mean the Dirac-δdistribution

δx0(x) = 0, x 6= x0RD δx0(x) = 1,

and where q (total rate) is given as data.





Solution to the mathematically idealized problem (ID)

Analytical solution in 2D, steady state, with one well at 0,single-phase flow with homogeneous isotropic K = K I

−K∇ · (∇pδ) = qδ0, x ∈ Ω

Change variable to polar coordinates pδ = pδ(r , θ), assume radialsolution pδ = pδ(r):

−1r

∂

∂r(Kr

∂pδ

∂r) = qδ0, r > 0

Solution has a logarithmic singularity

pδ(r) = C1log(r) + C2, r > 0

Fix the constants (use total input q and boundary conditions on ∂ΩC1 = q

2πK , C2 = pGIVEN (easiest case C2 = 0) to get

pδ(r) =q

2πKlog(r)





Analytical solution to (ID) is not very useful ...

when (bottom hole) pressure in the well is needed

Problem !!!! pδ(x0) = ∞

when multiple wells are present: −∇ · K∇p =∑

i qiδxi , x ∈ Ω





Solution to the “real” single well problem (R)

Known: K , rw , q. Solve

−∇ · K∇p = 0, x ∈ Ω \ Ωw

K∇p · ν =q

2πrw, x ∈ ∂Ωw

Similarly as in (ID) model we can derive

p(r)− p(R) =q

2πKlog(

rR

)

where p(R) is given at some R > 0.For example, p(R) can be given at some distance away from wells(Thiem solution) [Guenther,Lee 8.1/#12,Marsilly’86]

Analytical solution still not useful when multiple wells are present





Try numerical solution for (R) real well model ...

Very fine grid around (multiple) wells would be necessary.

Still no good way to get p(rw )




Not enough regularity for classical FE ...Why subtracting singularities is not sufficient

Numerical handling of idealized model using FEFunctional spaces ... notation

W m,p(Ω) := u : ∂mu ∈ Lp(Ω)W 0,2(Ω) ≡ L2(Ω)

W m,2(Ω) ≡ Hm(Ω)

Assume smooth q for the moment ... derive weak form of−∇(K∇p) = q ∫

Ω

K∇p∇w =

∫Ω

qw , ∀w ∈ H10 (Ω)

(Classical FE Galerkin method:) Find ph ∈ Vh which approximates pand solves:∫

Ω

K∇ph∇wh =

∫Ω

qwh, ∀wh ∈ Vh ( H10 (Ω)

Error estimate ‖ p − ph ‖H1(Ω)≤ Ch ‖ p ‖H2Ω

But q is not smooth ... and p is not smooth ...





How nonsmooth are solutions to (ID) ...?

Elliptic theory[GilTru,Dautray-Lions] tells us that weak solution to

−∇(K∇p) = q

has the following regularityq ∈ L2(Ω) = W 0,2(Ω) =⇒ p ∈ H2(Ω)q ∈ H−1(Ω) = W−1,2(Ω) =⇒ p ∈ H1(Ω)q ∈ L1(Ω) = W 0,1(Ω) ⊂ W−1,p(Ω) =⇒ p ∈ W 1,p(Ω) (p < 2).

But δx0 6∈ Lp(Ω) !!

Note:∫Ω

δx0w = w(x0) requires w ∈ C0(Ω) or thatδ0 ∈ (C0(Ω))′ = M(Ω) (dual space to C0(Ω)).However, we have W 1,p′

(Ω) ⊂ C0(Ω) soq ∈ (C0(Ω))′ ≡M(Ω) ⊂ W−1,p(Ω) if p < 2 (Sobolev imbedding Thm)hence the solution p ∈ W 1,p(Ω) for any p < 2.

This is not enough regularity for FE solution to converge





Classical FE solution may not converge. What now ?

We do not have p ∈ H1(Ω), hence, ph may fail to converge to p.

Idea: subtract the singular part of the solution

Consider a general problem −∇ · K∇p =∑

i qiδxi , x ∈ ΩConstruct

pS(x) =∑

i

qi

2πKilog(ri(x))

and instead of p approximate pREGULAR := p − pS, that is, solve∫Ω

K∇pREGULARh ∇wh = −

∑i

∫∂Ω

K∇pS · νwh, ∀wh ∈ Vh

and so pREGULAR ∈ H2(Ω) (K homogeneous).

If Ki are different, then one can show pREGULAR ∈ H2−ε(Ω). andalmost optimal convergence rates for pREGULAR − pREGULAR

h areobtained. [Ewing’82,Wheeler/Russell’82]Małgorzata Peszynska Peaceman and Thiem well models or how to remove a logarithmic singularity from your numerical solution




Why subtracting singularities is not sufficient

The above idea (solve for (pREGULAR)h, add pS) works well at asufficiently large distance from wells.However, at wells it is STILL designed to give p(xi) = ∞.

good for rate-specified wells (when q is given) when p(xi) is notneeded

not applicable with pressure-specified wells (p(xi) is given)

Idea: consider the idea of an “effective well radius” ... where weexploit the familiar equation from (R) real well model

p(r)− p(rw ) =q

2πKlog(

rrw

)




Effective well radius idea: Peaceman correctionBeyond Peaceman correction

Knowing q, how to get pw ?

Given q, we can hypothetically compute a numerical solutionph ≡ p = (p0, p1, . . . pN) where p0 ≈ p(x0)

we have a numerical method in which we solve for ph using q

Ap = f

but clearly pw 6= p0 !!!

but ... we have an analytical formula p(r)− p(rw ) = q2πK log( r

rw)

we hope this formula applies already to p1,2,3,4

Idea: combine these two methods to get an expression for pw





Peaceman correction [Peaceman’77] (homogeneousisotropic K, isotropic grid)

Cell Centered Finite Difference ≈ Mixed Finite Element with RT0

1

− 1h

(K p1−p0

h + K p2−p0h + K p3−p0

h + K p4−p0h

)=

qh2

2 p1,2,3,4 − p0 == q2πK log( h

rw)

Combining 1,2 we get

p0 = pw +q

2πK(log(

hrw

)− π

2) = pw +

q2πK

(log(r0

rw)

where r0 = exp(−π/2)h ≈ 0.208h is the effective well radius.





But what if K is not isotropic homogeneous ?

K = KT (permeability, conductivity, mobility,...) is in generalanisotropic and heterogeneous. And grids are not uniform andisotropic ...

Peaceman models allow forK = diag(K11, K22), hx 6= hy

homogenization for Darcy flow around wells[Zijl, Trykozko’01]

multiscale FE method for Darcy flow nearwells[Z. Chen et al ’03]

extension to non-Darcy flow, K ≡ const[Lazarov, Ewing et al’99]

CG & MP ... in progress





Results: Darcy with rate-wells and BHP wells




Closeness of solutions: elliptic and parabolic estimates

Idealized model versus real model

good in many situations whendiam(Ω) >>>>>>> diam(Ωw )

But..

How close is pδ to the true p ?What if there are more wells and flow is notsingle phase but multiphase ?

use numerical solution: but how toguarantee convergence ?

We need to know p(rw ) (pressure in well)! inorder to determine phase behavior

What if K is not constant and not isotropic ?





How close is pδ to p ?

For non-stationary version of the model, as rw → 0, one has (ifcoefficients are extended properly) [G. Chavent, J. Jaffre ’86]

‖ pδ − p ‖L2(Ω×(0,T ))→ 0

For the original stationary problem:[Li Ta Tsien, A. Damlamian’80], [Da Qian Li, Shu Xing Chen (inChinese)’78],Recent result [Zhiming Chen, Xinye Yue’03]

maxx∈Ω

|pδ − p| ≤ Crw |log(rw )|





Back to recent estimate

[Zhiming Chen, Xinye Yue’03]Assume: K ∈ C0,1(Ω) with Lipschitz constant ΛThus for any p > 0 the solution satisfies

(pδ − p) ∈ W 2,p(Ω) =⇒ (pδ − p) ∈ C1(Ω)

and we have the following estimate

maxx∈Ω

|pδ − p| ≤ CΛ(1 + log(Λ))1/2rw |log(rw )|






Documents

Peaceman and Thiem well models or how to remove a …sites.science.oregonstate.edu/~mpesz/amc/amc_wells.pdf · 2007-03-02 · Małgorzata Peszy´nska Peaceman and Thiem well models