Stochastic reservoir: the scalar Buckley-Leverett model. 17/12/ Pau momas group Peppino Terpolilli Total-Pau

Transcription

1 Stochastic reservoir: the scalar Buckley-Leverett model 7/2/ Pau momas group Peppino Terpolilli Total-Pau

2 OUTLINE Why stochasticity? Mathematical issues Some models: Dead-oil oil,buckley-leverett New approach for upscaling Program Conclusions 2

3 Stochastic model Hard Data: wells : core : geology,, scanning logs petrophysic geological scheme scale problems 3

4 Stochastic model Soft Data: extension: geophysic, geology scale problems and uncertainty (geostatistic) 4

5 profil radial de résistivité Zone non contaminée (formation vierge) R t R w R xo Zone lavée (envahie) Mud cake S w R mf R mc R m R t S xo Eponte R s R xo Résistivité R mc Rayon Boue de forage R m Axe du trou Exa-Plans

6 0 GAPI GAPI GAPI 200 WIRE.CALI_ 6 IN DEGF DEGF DEGF DEGF OHMM 0.0 OHMM 0.0 OHMM 0.0 OHMM V/V 0 G/C G/C G/C G/C3 0 G/C3 0 G/C3 0 G/C3 2.5 G/C G/C G/C G/C G/C3 3 V/V 0 V/V 0 V/V 0 V/V V/V V/V V/V 0 V/V 0 0 V/V V/V 0 V/V 0 0 V/V V/V 0 0. MD 0000 PARAMETERS.RHOHCMAX_ PARAMETERS.RHOB_HC_ PARAMETERS.RHOHCMIN_ PETROLAN.VCL_ PETROLAN.RHO_HC_ PETROLAN.PHIT_ PETROLAN.VCL_NDCOR_ PARAMETERS.ROMAMAX_ PETROLAN.M_DCL_ PARAMETERS.GR_CL_ PARAMETERS.GR_MA_ PETROLAN.GR_ DEPTH METRES PETROLAN.NP_CL_ PARAMETERS.RT_CL_ PRECALC.FTEMP PARAMETERS.RW_DEF_ PARAMETERS.RMF_DEF_ PETROLAN.VCL_ND_ PARAMETERS.RHOB_MA_ PETROLAN.NP_ PETROLAN.RT_ PARAMETERS.TEMP_MF_ PARAMETERS.RW_ PARAMETERS.RMF_ PETROLAN.VCL_NP_ PETROLAN.IDM_ PARAMETERS.ROMAMIN_ PETROLAN.RHOB_CL_ RS PARAMETERS.TEMP_W_ PETROLAN.RW_A_ PETROLAN.RMF_A_ PETROLAN.VCL_RT_ PETROLAN.NUM_METH_ PETROLAN.RHO_MA_ DEPTH METRES PETROLAN.BADHOLE_FLAG PETROLAN.SXOMIN_ PETROLAN.VOL_UWATER_ PETROLAN.SXOMAX_ PETROLAN.VOL_UWATER_ PETROLAN.VOL_XWATER_ PETROLAN.SXOE_ PETROLAN.VOL_XWATER_ PETROLAN.VCL_ PETROLAN.RHOB_ PETROLAN.RXO_ PETROLAN.TEMP_ PETROLAN.RW_L_ PETROLAN.RMF_L_ PETROLAN.VCL_GR_ PETROLAN.ERRF_METH_ PETROLAN.RHO_DSOL_ PETROLAN.SWE_ PETROLAN.PHIE_ PETROLAN.PHIE_ PETROLAN.PERM_

7 2D/4C Post-stack time migration km km 0.5 t PP PP 3.0 PS 7

8 8

9 CASE B Model size (42209 active cells) Major parameters: Faults transmissivities - Xytrans 9

10 CASE H - new HM with RFT observations 0

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12 Darcy law Navier-Stokes equations: v + v v = p + ν v + f t ρ Darcy law: q = K µ p K is the matrix of permeability: porous media characteristic 2

13 Darcy law Extended Darcy law: Kkrp qp = ( pp + ρ pg D) µ p k rp relative permeability of phase p D the depth 3

14 Darcy law Continuum mechanics: at a REV located at : x S owg,, kr ( S) p ( ) c S saturation: fraction of pore volume relative permeability capillary pressure REV 4

15 Kr-pc 5

16 Kr-pc 6

17 Math issues For single-phase flows Darcy law leads to linear equation: p Φ µ C div( K( x). p) = f t For multi-phase flow we recover nonlinear equtions: hyperbolic, degenerate parabolic etc.. 7

18 Math issues The mathematical model is a system of PDE with appropriate initial and boundary conditions the coefficients of the equations are poorly known stochastic approach geology + stochastic = geostatistic K( x, ω) 8

19 Uncertainty SPDE: p div ( K ( x, ω ). p) = f t These problems are difficult: experimental design approach Grand projet incertitude Industrial tools 9

20 Uncertainty We need to compute too much flow simulation Grids are large: millions of cells Hindered ensemble-based based prediction 20

21 Uncertainty Work in progress : Zhang, Tchelepi Tom Russell, Jarnan Cho, Lindquist J Glimm et al. 2

22 Math issue Cho, Lindquist : ensemble-based based prediction using Buckley-Leverett equation 22

23 Dead-oil model Two immiscible phases: water and oil one component in each phase incompressibility p c p = pressure of the oil phase p capillary pressure water/oil cw ρw and ρo are constant 23

24 Dead-oil model Dead-oil equations: Q = Q + Q w o pressure equation krw kro krw krw kro div[ K ( + ) p ] = div ( K pc ) + div ( K ( + ) g D) + Q µ µ µ µ µo w o w w saturation equation S Φ + div q w = t Q w 24

25 Buckley-Leverett No capillary pressure, no gravity,, no source term in pressure equation: p = 0, g = 0, Q = 0, q + q = q = const c w o T Buckley-Leverett Leverett: dim= x krw kro p [ K ( + ) ] = 0 ; µ µ x w w o krw kro p K ( + ) = q µ µ x o T 25

26 Buckley-Leverett We obtain: with: S Φ + qt Fw( S) = 0 t x krw µ w Fw ( S) = = water fractional flow krw kro + µ µ w o 26

27 Buckley-Leverett A particular case of: with: f S + f ( S) = 0 t x S( x,0) = S ( x) (.) the flux function Method of characteristics o 27

28 Buckley-Leverett Quasilinear equation: S f S f + = 0; a( S) = t S x S S S + a( S) = 0 t x a(.) and S (.) are piecewise C o 28

29 Buckley-Leverett Characteristic curves : dx ( ) dt = as quasilinear equation means S constant along characteristic curves slope of characteristic curves constant: curves are straight lines the value of S in (x,t) equal the value at t=0 29

30 Buckley-Leverett When characteristic curves intersect, we obtain shoks weack solutions, existence but unicity? Entropy conditions to select the physical solution: Lax, Oleinik.. 30

31 Buckley-Leverett Some conclusions obtained by Cho,Lindquist Lindquist: for oil-cut cut, peak production uncertainty shortly after mean breackthrough accuracy of ensembles mean results behave according to central limit theorem history matching improve the relative error,, but not that much 3

32 upscaling We present now a new approch for upscaling first applied for single phase flow and a possible strategy to tackle Buckley Leverett model 32

33 Local Problems div + B. C. Linear B.C. Confined B.C. ( K( x) P ) Periodic B.C. Upscaling methods = P i 0 i 3 Γ = x sur on ω i Pi = 0 sur on ω Γ n Pi = xi sur on Γi Pi = wi + xi wi H p( ω ) Stochastic Homog. i on ω = Γ in ω (homogenization) periodic on ω 2 33 Γ = Γb Γ Γ2 = Γ a Γ i d c Γ d Γ a ω Γ c Γb * Kij = K( x) Pi. Pj dx ω ω i, j 3 50/2/2003 -

34 New Approach Constraints i) K ( x) satisfy ii) K(x) is symmetric 0 < α α ξ 2 β K( x) ξ. ξ β ξ 2 ξ d IR h = K ( x) dx ω ω Harmonic average a = K( x) dx ω ω Arithmetic average M ( h, a, ω ) = constant symmetric matrix H such that h ξ 2 H ξ. ξ a ξ 2 34

35 New Approach (energy function) ω div ( K( x) P ) = f in ω div( H U ) P i = g i Min f i, g i general i Eij D ( K) = K( x) Pi. Pjdx fipjdx 2 ω ω Eij D ( H) = i, j H Ui. Ujdx fiujdx 2 ω ω H i on ω M( h, a, ω ) i U m g i m D ij D ij D i, j= [ ] I ( H) = E ( H) E ( K) i = i = fi in on ω 2 ω m 35 70/2/2003 -

36 New Approach (velocity function) ω div P ( K( x) P ) = f in ω div( H U ) i = g i on ω Vi D ( K) = K( x) Pi dx ω ω i i g i general i m U f i, i m i = g i = fi in on ω Vi D ( H) = H Ui dx ω ω i ω H Min M( h, a, ω ) m D i D i D i= J ( H) = V ( H) V ( K) /2/2003 -

37 Special instances g = x, f = 0 i i i i d div P ( K( x) P ) i = x i i = f i on ω in ω i, j d div U ( H U ) i = x i i = f i on in ω ω Eij D ( K) = K( x) Pi. Pjdx E 2 ij D ( H) = H Ui. Ujdx = ω ω H 2 ω 2 ij H inf I ( ) D H = M( h, a, ω ) i, j= d ω H 2 ij E D ij ( K ) 2 Convex quadratic function 37 40/2/2003 -

38 Special instances I D d ( H ) = i, j= ω H 2 ij E D ij ( K ) 2 ID( H) H αβ ω Hαβ = 2 E D αβ ( K) ω α, β d I ( H ) D = 0 H = αβ K ( x) Pα P H ω. ω αβ Constraints are verified by H and I H D ( ) = 0 β dx We find the classical tensor with linear B.C /2/2003 -

39 New Approach ( ) Proposition: Cost functions I D, J D, FD are continuous. Proposition: Minimization problems have solutions. Proposition: The function I D is differentiable and we have the following expression for α, β d : 39 0/2/2003 -

40 Properties: G-convergence Definition (Spagnolo): where K G K ε 0 ( H 0 ω) If f H ( ω ) P ε P 0 div ( K ( x) P ) = f in ω div( K ( x) P ) ε P ε = 0 ε on ω 0 P 0 = 0 0 = f on ω in ω 40

41 Properties: G-stability Theorem: Let K (, β, ω) such that ε M α K G ε K 0 (A) ε > 0 K * ε arg min H M ( h, a, ω ) I D ( H ) = arg min H M ( α, β, ω ) I D ( H ) Then: * ε K G K * 0 Remark: For linear, confined or periodic boundary conditions Assumption (A) is satisfied. 4 90/2/2003 -

42 Properties: Upscaled permeabilities G-convergence Homogenization theory: ε = l L, aggregation rate We have: K G ε K hom, K ( = K hom eff ) constant tensor G-stability: * ε K G K * hom But * hom K = K hom Conclusion: G * ε K eff K /2/2003 -

43 upscaling a possible strategy to tackle Buckley Leverett model using the velocity field extend the approach by optimization and control 43

44 Conclusion Stochastic model Ensemble-based prediction Buckley-Leverett model New approach for upscaling 44

45 Darcy law Continuum mechanics: at a REV located at : x Φ( x) K( x) porosity: : ratio of void to bulk volume permeability: Darcy law REV 45

46 Darcy law Darcy law: empirical law (Darcy in 856) theoretical derivation: Scheidegger,, King Hubbert, Matheron (heuristic) Tartar (homogeneization theory) G Stokes Darcy law 46

47 Darcy law Different scale: pore level: : Stokes equations lab: measures numerical cell: upscaling field: heterogeneity G Darcy law Darcy law 47

48 Black-oil model Hypotesis: three phases: 2 hydrocarbon phases and water hydrocarbon system: 2 components a non-volatile oil a volatile gas soluble in the oil phase 48

49 Black-oil model PVT behaviour: : formation volume factor ( + ) ( V ) ( ) ( Vg ) ( ) ( V ) V V V V Bo= ; B = ; B = o dg RC g RC W RC g w o STC W STC STC where: V RC volume of a fixed mass at reservoir conditions V STC volume of a fixed mass at stock tank conditions 49

50 Black-oil model Mass transfer between oil and gas phases: R S V dg = V o STC V dg : gas component in the oil phase V o : oil component in the oil phase functions of the oil phase pressure 50

51 Black-oil model Thermo functions for oil: Rs(m3/m3) P Bo (Sm 3/m 3),4,35,3,25,2,5,,05 Bo muo,4,3,2, 0,9 0,8 0,7 0,6 0,5 0, P (bars) muo (cp) 5

52 Black-oil model Water: Sw Kkrw φ div Pw wgz t Bw + = Bwµ w ( ρ ) 0 oil: t φ So Bo div Kkro Bo µ o ( Po + ρ ogz ) = 0 gaz: Sg So Kkrg φ + φrs div ( Pg ρggz ) t Bg Bo + Bgµ g KkroRs div ( Po + ρogz ) = Boµ o 0 52

53 Black-oil model saturation: S o + S w + S g = capillary pressures: p w= po pcow p g = p o + p co g we obtain 3 equations with 3 unknowns: p, S, S if p = p o w g o b p, S, R if p > p o w s o b 53

54 Black-oil model:boundary conditions Boundaries closed: : no flux at the extreme cells aquifer: : source term in corresponding cells wells: Dirichlet condition: bottom pressure imposed Neumann condition: production rate imposed source terms for perforated cells (PI) 54

55 Black-oil model: initial conditions capillary and gravity equilibrium pressure imposed in oil zone at a given depth oil pressure in all cells and then Pc curves 55

56 New Approach ID( H) H αα ID( H) H αβ m = [ E H E K ] ij D ( ) ij D ( ) * i, j= m = [ E H E K ] ij D ( ) ij D ( ) * i, j= ω ω ω ω U U xα x dx U i j j Ψi β ω xα x dx + β U Ψ i j ω xα x dx β U U U U i j i j + dx + xα xβ xβ xα U j U i j Ψ Ψ + i dx xα xβ xβ xα U U i j i j Ψ Ψ + dx x x x x α β β α where is solution of the local problems ψ i U i and is solution of: div Ψ ( H Ψ ) Si α = β Si α β 56 20/2/ i i = f i in = 0 on ω ω i m