Peaceman and Thiem well models or how to remove a logarithmic singularity fr. your numerical solution

Transcription

1 Peaceman and Thiem well models or how to remove a logarithmic singularity from your numerical solution Department of Mathematics Oregon State University AMC, 3/2/2007 Current work supported by NSF and DOE Acknowledgement: Cristiano Garibotti

2 Outline Problem statement: what are wells 1 Problem statement: what are wells Steady state, rate specified wells Idealized and real well models 2 Not enough regularity for classical FE... Why subtracting singularities is not sufficient 3 Effective well radius idea: Peaceman correction Beyond Peaceman correction 4 Closeness of solutions: elliptic and parabolic estimates

3 What are wells Problem statement: what are wells Steady state, rate specified wells Idealized and real well models

4 Steady state, rate specified wells Idealized and real well models Well in a confined aquifer/hydrocarbon reservoir

5 Steady state, rate specified wells Idealized and real well models Pressure profiles around the well: side view

6 Steady state, rate specified wells Idealized and real well models Pressure profiles around the well: aerial view

7 Mathematical model: geometry Steady state, rate specified wells Idealized and real well models

8 Steady state, rate specified wells Idealized and real well models Problem statement for steady state, rate specified, R 2 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 is known. Put it together K p = q(x), x Ω p Ω = p GIVEN = 0

9 Steady state, rate specified wells Idealized and real well models Mathematical model for wells: idealized and real Geometry Ω w B(x 0, r w ) with boundary Γ w 1 (R)Real model of injection/production wells K p = 0, x Ω \ Ω w K p ν = q, x Ω w 2πr w 2 (ID) Idealization of a true real-life situation (K p δ ) = q(x) = qδ x0 (x), x Ω where by δ x0 (x) we mean the Dirac-δ distribution δ Rx0 (x) = 0, x x 0 D δx 0 (x) = 1, and where q (total rate) is given as data.

10 Steady state, rate specified wells Idealized and real well models 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 ( p δ ) = qδ 0, x Ω Change variable to polar coordinates p δ = p δ (r, θ), assume radial solution p δ = p δ (r): 1 r pδ (Kr r r ) = qδ 0, r > 0 Solution has a logarithmic singularity p δ (r) = C 1 log(r) + C 2, r > 0 Fix the constants (use total input q and boundary conditions on Ω C 1 = q 2πK, C 2 = p GIVEN (easiest case C 2 = 0) to get p δ (r) = q 2πK log(r)

11 Steady state, rate specified wells Idealized and real well models Analytical solution to (ID) is not very useful... when (bottom hole) pressure in the well is needed Problem!!!! p δ (x 0 ) = when multiple wells are present: K p = i q iδ xi, x Ω

12 Steady state, rate specified wells Idealized and real well models Solution to the real single well problem (R) Known: K, r w, q. Solve K p = 0, x Ω \ Ω w K p ν = q, x Ω w 2πr w Similarly as in (ID) model we can derive p(r) p(r) = q 2πK log( r R ) 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

13 Steady state, rate specified wells Idealized and real well models Try numerical solution for (R) real well model... Very fine grid around (multiple) wells would be necessary. Still no good way to get p(r w )

14 Not enough regularity for classical FE... Why subtracting singularities is not sufficient Numerical handling of idealized model using FE Functional spaces... notation W m,p (Ω) := {u : m u L p (Ω)} W 0,2 (Ω) L 2 (Ω) W m,2 (Ω) H m (Ω) Assume smooth q for the moment... derive weak form of (K p) = q K p w = qw, w H0 1 (Ω) Ω Ω (Classical FE Galerkin method:) Find p h V h which approximates p and solves: K p h w h = qw h, w h V h H0 1 (Ω) Ω Ω Error estimate p p h H1 (Ω) Ch p H2 Ω But q is not smooth... and p is not smooth...

15 Not enough regularity for classical FE... Why subtracting singularities is not sufficient How nonsmooth are solutions to (ID)...? Elliptic theory[giltru,dautray-lions] tells us that weak solution to (K p) = q has the following regularity q L 2 (Ω) = W 0,2 (Ω) = p H 2 (Ω) q H 1 (Ω) = W 1,2 (Ω) = p H 1 (Ω) q L 1 (Ω) = W 0,1 (Ω) W 1,p (Ω) = p W 1,p (Ω) (p < 2). But δ x0 L p (Ω)!! Note: Ω δ x 0 w = w(x 0 ) requires w C 0 (Ω) or that δ 0 (C 0 (Ω)) = M(Ω) (dual space to C 0 (Ω)). However, we have W 1,p (Ω) C 0 ( Ω) so q (C 0 ( 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

16 Not enough regularity for classical FE... Why subtracting singularities is not sufficient Classical FE solution may not converge. What now? We do not have p H 1 (Ω), hence, p h may fail to converge to p. Idea: subtract the singular part of the solution Consider a general problem K p = i q iδ xi, x Ω Construct p S (x) = i q i 2πK i log(r i (x)) and instead of p approximate p REGULAR := p p S, that is, solve K ph REGULAR w h = K p S νw h, w h V h Ω i Ω and so p REGULAR H 2 (Ω) (K homogeneous). If K i are different, then one can show p REGULAR H 2 ɛ (Ω). and almost optimal convergence rates for p REGULAR ph REGULAR are obtained. [Ewing 82,Wheeler/Russell 82]

17 Not enough regularity for classical FE... Why subtracting singularities is not sufficient Why subtracting singularities is not sufficient The above idea (solve for (p REGULAR ) h, add p S ) works well at a sufficiently large distance from wells. However, at wells it is STILL designed to give p(x i ) =. good for rate-specified wells (when q is given) when p(x i ) is not needed not applicable with pressure-specified wells (p(x i ) is given) Idea: consider the idea of an effective well radius... where we exploit the familiar equation from (R) real well model p(r) p(r w ) = q 2πK log( r ) r w

18 Knowing q, how to get p w? Effective well radius idea: Peaceman correction Beyond Peaceman correction Given q, we can hypothetically compute a numerical solution p h p = (p 0, p 1,... p N ) where p 0 p(x 0 ) we have a numerical method in which we solve for p h using q Ap = f but clearly p w p 0!!! but... we have an analytical formula p(r) p(r w ) = q 2πK log( r r w ) we hope this formula applies already to p 1,2,3,4 Idea: combine these two methods to get an expression for p w

19 Effective well radius idea: Peaceman correction Beyond Peaceman correction Peaceman correction [Peaceman 77] (homogeneous isotropic K, isotropic grid) Cell Centered Finite Difference Mixed Finite Element with RT0 1 1 h q h 2 ( K p 1 p 0 h + K p 2 p 0 h + K p 3 p 0 h + K p 4 p 0 ) h = 2 p 1,2,3,4 p 0 == q 2πK log( h r w ) Combining 1,2 we get p 0 = p w + q 2πK (log( h ) π r w 2 ) = p w + q 2πK (log( r 0 ) r w where r 0 = exp( π/2)h 0.208h is the effective well radius.

20 Effective well radius idea: Peaceman correction Beyond Peaceman correction But what if K is not isotropic homogeneous? K = K T (permeability, conductivity, mobility,...) is in general anisotropic and heterogeneous. And grids are not uniform and isotropic... Peaceman models allow for K = diag(k 11, K 22 ), h x h y homogenization for Darcy flow around wells [Zijl, Trykozko 01] multiscale FE method for Darcy flow near wells[z. Chen et al 03] extension to non-darcy flow, K const [Lazarov, Ewing et al 99] CG & MP... in progress

21 Effective well radius idea: Peaceman correction Beyond Peaceman correction Results: Darcy with rate-wells and BHP wells

22 Idealized model versus real model Closeness of solutions: elliptic and parabolic estimates But.. good in many situations when diam(ω) >>>>>>> diam(ω w ) How close is p δ to the true p? What if there are more wells and flow is not single phase but multiphase? use numerical solution: but how to guarantee convergence? We need to know p(r w ) (pressure in well)! in order to determine phase behavior What if K is not constant and not isotropic?

23 How close is p δ to p? Closeness of solutions: elliptic and parabolic estimates For non-stationary version of the model, as r w 0, one has (if coefficients are extended properly) [G. Chavent, J. Jaffré 86] p δ p L2 (Ω (0,T )) 0 For the original stationary problem: [Li Ta Tsien, A. Damlamian 80], [Da Qian Li, Shu Xing Chen (in Chinese) 78], Recent result [Zhiming Chen, Xinye Yue 03] max x Ω pδ p Cr w log(r w )

24 Back to recent estimate Closeness of solutions: elliptic and parabolic estimates [Zhiming Chen, Xinye Yue 03] Assume: K C 0,1 (Ω) with Lipschitz constant Λ Thus for any p > 0 the solution satisfies (p δ p) W 2,p (Ω) = (p δ p) C 1 (Ω) and we have the following estimate max x Ω pδ p CΛ(1 + log(λ)) 1/2 r w log(r w )

25 Closeness of solutions: elliptic and parabolic estimates