Fully implicit higher-order schemes applied to polymer flooding. K.-A. Lie(SINTEF), T.S. Mykkeltvedt(IRIS), X. Raynaud (SINTEF)

Transcription

1 Fully implicit higher-order schemes applied to polymer flooding K.-A. Lie(SINTEF), T.S. Mykkeltvedt(IRIS), X. Raynaud (SINTEF)

2 Motivation Challenging and complex to simulate water-based EOR methods, unresolved simulation can lead to misleading predictions Polymer flooding enhances the water s ability to push the oil through the reservoir, and can reduce channeling through high flow zones Crucial to capture polymer fronts sharply to resolve the nonlinear displacement mechanisms correctly Polymer fronts will in the worst case be linear waves and generally have significantly less self-sharpening effects than water fronts challenge when using standard low-order methods

3 General flow equations for two-phase flow Gathering the equations, we have (φρ α S α ) t + (ρ α u α ) = qα, α = {w, n} u α = Kk rα µ α ( pα ρ α g z ) p c = p n p w, S w + S n = 1

4 Model for polymer flooding Simple model: introduce extra immiscible component and mixture law Polymer transported in water: u w = k rw(s) µ w,eff (c) R k (c) K( p w ρ ) wg z u p = krw(s) µ p,eff (c)r k (c) K( p w ρ ) wg z Conservation of polymer component: [ t φ(1 Sipv)cb ws + ρ r C a (1 φ) ] + (cb ) w u p = qp

5 Model for polymer flooding Simple model: introduce extra immiscible component and mixture law Polymer transported in water: u w = viscosity enhancement k rw(s) µ w,eff (c) R k (c) K( p w ρ ) wg z Todd Longstaff mixing: 1 µ = 1 c/cm w,eff µm(c) ω µ 1 ω w + c/cm µm(c) ω µ 1 ω p u p = krw(s) µ p,eff (c)r k (c) K( p w ρ ) wg z Conservation of polymer component: t [ φ(1 Sipv)cb ws + ρ r C a (1 φ) ] + (cb w u p ) = qp

6 Model for polymer flooding Simple model: introduce extra immiscible component and mixture law Polymer transported in water: u w = viscosity enhancement k rw(s) µ w,eff (c) R k (c) K( p w ρ ) wg z permeability reduction Todd Longstaff mixing: 1 µ = 1 c/cm w,eff µm(c) ω µ 1 ω w + c/cm µm(c) ω µ 1 ω p u p = krw(s) µ p,eff (c)r k (c) K( p w ρ ) wg z Conservation of polymer component: t [ φ(1 Sipv)cb ws + ρ r C a (1 φ) ] + (cb w u p ) = qp

7 Model for polymer flooding Simple model: introduce extra immiscible component and mixture law Polymer transported in water: u w = viscosity enhancement k rw(s) µ w,eff (c) R k (c) K( p w ρ ) wg z permeability reduction Todd Longstaff mixing: 1 µ = 1 c/cm w,eff µm(c) ω µ 1 ω w + c/cm µm(c) ω µ 1 ω p u p = krw(s) µ p,eff (c)r k (c) K( p w ρ ) wg z Conservation of polymer component: t [ φ(1 Sipv)cb ws + ρ r C a (1 φ) ] + (cb w u p ) = qp inaccessible pore space

8 Model for polymer flooding Simple model: introduce extra immiscible component and mixture law Polymer transported in water: u w = viscosity enhancement k rw(s) µ w,eff (c) R k (c) K( p w ρ ) wg z permeability reduction Todd Longstaff mixing: 1 µ = 1 c/cm w,eff µm(c) ω µ 1 ω w + c/cm µm(c) ω µ 1 ω p u p = krw(s) µ p,eff (c)r k (c) K( p w ρ ) wg z Conservation of polymer component: t [ φ(1 Sipv)cb ws + ρ r C a (1 φ) ] + (cb w u p ) = qp inaccessible pore space adsorption

9 Numerical framework Dicrete cell-averages ( q = p, s, c ) q i (t) = 1 ZZ q(x, y, t)dx i i Discretized eq. for water: [ w (p i ) (p i )s i ] n+1 =[ w (p i ) (p i )s i ] n t X Z [ w (p) w (s, c)] m ij i (vm w n) ij ds i j=1 ij i j

10 Numerical framework Dicrete cell-averages ( q = p, s, c ) q i (t) = 1 ZZ q(x, y, t)dx i i Discretized eq. for water: [ w (p i ) (p i )s i ] n+1 =[ w (p i ) (p i )s i ] n t X Z [ w (p) w (s, c)] m ij i (vm w n) ij ds i j=1 ij i j How to compute the mass flux evaluated at the interface? Z ij (p) ij (s, c)(v n) ij ds 1 2 [ (p i)+ (p j )] v ij Z (s, c)ds ij ij

11 Which quadrature rule to use for the integral? How to reconstruct the necessary one-sided point values from the cell-averages? How to approximate the mobility given point values that generally are different on each side of? ij

12 Which quadrature rule to use for the integral? ØMidpoint, Simpsons rule, 4 th order Gauss How to reconstruct the necessary one-sided point values from the cell-averages? How to approximate the mobility given point values that generally are different on each side of ij? ( (sij,c ij ), if v ij 0, ij(s, c) = (s + ij,c+ ij ), otherwise.

13 First-order scheme: constant reconstruction q(x) =q i Second-order slope-limiter scheme: linear reconstruction q(x) =q i + i x (x x i )+ y i (y y i) with slopes estimated from cell-averages x i x = q i q i (1,0),q i+(1,0) q i

14 First-order scheme: constant reconstruction q(x) =q i Second-order slope-limiter scheme: linear reconstruction q(x) =q i + i x (x x i )+ y i (y y i) with slopes estimated from cell-averages x i x = q i q i (1,0),q i+(1,0) q i 3 vanleer superbee minmod 5 (1,b) = b

15 WENO schemes WENO builds on ENO, which picks the least oscillatory stencil, uses a convex combination of local stencils Here: simplified version, with four linear reconstructions Polynomial: q NE (x) =q i + E i (x x i )+ N i (y y i ) Smoothness indicator: NE i = 1 h i x E 2 i + y N 2 l 4 i + Weights: w NE = NE / Linear combination: q i (x) = X v=ne,nw,se,sw NE + NW + SE + SW w v q v (x) E i = q i+(1,0) q i x N i = q i+(0,1) q i y

16 Solving the discretized systems System on residual form: F(y) =0 y =[p, s, c] Use Newton-Raphson method: y = y 0 + y 0 F(y 0 )+J y, J = df dy Structure of the Jacobian matrix J = R R R R R R R R R

17 Structure of the Jacobian matrix for the implicit WENO w Second order: WENO J = R R R R R R R R R 3 7

18 Structure of the Jacobian matrix for the implicit WENO w Second order: WENO When using lagged evaluation of the WENO weights, the structure of the Jacobian

19 A motivating example Consider φq t + q x = 0

20 A motivating example Consider φq t + q x = 0 τ=xφ transformed q t + q τ = 0 φ M

21 A motivating example Consider φq t + q x = 0 τ=xφ transformed q t + q τ = 0 φ M Modified equation (implicit/explicit first order) q t + q τ = 1 2 ( τ ± t)q ττ Smearing of discontinuity O ( t( τ ± t))

22 A motivating example Consider φq t + q x = 0 τ=xφ transformed q t + q τ = 0 φ M Modified equation (implicit/explicit first order) q t + q τ = 1 2 ( τ ± t)q ττ Smearing of discontinuity O ( t( τ ± t)) Hence: 9φ( ) xφ ± t + φ } 10 {{} 10M high-porosity region ( xφ M ± t ) } {{ } low-porosity region

23 A motivating example Consider φq t + q x = 0 τ=xφ transformed q t + q τ = 0 φ M Modified equation (implicit/explicit first order) q t + q τ = 1 2 ( τ ± t)q ττ Smearing of discontinuity O ( t( τ ± t)) Hence: 9φ( ) xφ ± t + φ } 10 {{} 10M high-porosity region ( xφ M ± t ) } {{ } low-porosity region = 9 xφ ( ν ) φ 2 x ( ) 1± + 1±ν 10 M 10M 2

24 1D: pure waterflooding with regions of different porosity = 0.01, if x 2 [0.1L, 0.2L], 0.2, otherwise. t = 8 < : 0.01, - 0.1, -.- 1, (exceeds the CFL condition in the low-porosity region) Explicit first-order scheme Implicit first-order scheme t = T t =8T t =8T t =3T t =3T t = T 0.1L 0.2L 0.1L 0.2L Explicit: oscillations (disappear as time evolves) Implicit: stable solutions for both time steps

25 1D: polymer flooding with regions of different porosity 0.8 = 0.01, if x 2 [0.1L, 0.2L], 0.2, otherwise. Saturation Explicit 1 st - order Implicit 1 st - order Implicit 2 nd - order / / Concentration t =0.01 t =0.1/ 1 1 t =5t t =5t t = T t = T 0 0.1L 0.2L 0.1L 0.2L the high-order implicit scheme resolves the displacement fronts as good as the explicit scheme the low-order implicitscheme fails to sharply resolve the structure of the oil bank that arises 0

26 1D: polymer flooding with regions of different porosity Comparison of the computational costs:

27 1D: polymer flooding with regions of different porosity Comparison of the computational costs: oscillations in the low porosity region the second-order implicit scheme is at least as effective as the explicit schemes

28 Implicit (50 control steps) Approximated solution with different reconstructions, on a uniform Cartesian grid with 50x50 cells Explicit (1600 control steps) constant: 93.2 sec minmod: 98.1 sec vanleer: sec superbee: sec weno: 111 sec constant: 9.1 sec minmod: 23.5 sec vanleer: 31.5 sec superbee: 71.2 sec weno: 42.2 sec constant: 9.1 sec minmod: 13.5 sec vanleer: 13.5 sec superbee: 13.4 sec weno: 16.6 sec

29 Implicit (50 control steps) Approximated solution with different reconstructions, on a uniform Cartesian grid with 50x50 cells Explicit (1600 control steps) constant: 93.2 sec minmod: 98.1 sec vanleer: sec superbee: sec weno: 111 sec constant: 9.1 sec minmod: 23.5 sec vanleer: 31.5 sec superbee: 71.2 sec weno: 42.2 sec constant: 9.1 sec minmod: 13.5 sec vanleer: 13.5 sec superbee: 13.4 sec weno: 16.6 sec

30 Approximated solution with different reconstructions, on a uniform Cartesian grid with 50x50 cells Explicit (1600 control steps) constant: 93.2 sec minmod: 98.1 sec vanleer: sec superbee: sec weno: 111 sec constant: 9.1 sec minmod: 23.5 sec vanleer: 31.5 sec superbee: 71.2 sec weno: 42.2 sec Implicit (50 control steps) steeper slopes constant: 9.1 sec minmod: 13.5 sec vanleer: 13.5 sec superbee: 13.4 sec weno: 16.6 sec The overall system more nonlinear and coupled

31 Implicit (50 control steps) Approximated solution with different reconstructions, on a uniform Cartesian grid with 50x50 cells Explicit (1600 control steps) constant: 93.2 sec minmod: 98.1 sec vanleer: sec superbee: sec weno: 111 sec constant: 9.1 sec minmod: 23.5 sec vanleer: 31.5 sec superbee: 71.2 sec weno: 42.2 sec constant: 9.1 sec minmod: 13.5 sec vanleer: 13.5 sec superbee: 13.4 sec weno: 16.6 sec

32 Implicit (50 control steps) Approximated solution with different reconstructions, on a uniform Cartesian grid with 50x50 cells Explicit (1600 control steps) constant: 93.2 sec minmod: 98.1 sec vanleer: sec superbee: sec weno: 111 sec constant: 9.1 sec minmod: 23.5 sec vanleer: 31.5 sec superbee: 71.2 sec weno: 42.2 sec constant: 9.1 sec minmod: 13.5 sec vanleer: 13.5 sec superbee: 13.4 sec weno: 16.6 sec Using a second-order reconstruction and improved spatial quadrature gives more accurate solution profiles both for the explicit and the implicit schemes (as expected)

33 Implicit (50 control steps) Implicit* (50 control steps) Approximated solution with different reconstructions, on a uniform Cartesian grid with 50x50 cells constant: 9.1 sec minmod: 23.5 sec vanleer: 31.5 sec superbee: 71.2 sec weno: 42.2 sec constant: 9.1 sec minmod: 13.5 sec vanleer: 13.5 sec superbee: 13.4 sec weno: 16.6 sec

34 Implicit (50 control steps) Implicit* (50 control steps) Approximated solution with different reconstructions, on a uniform Cartesian grid with 50x50 cells constant: 9.1 sec minmod: 23.5 sec vanleer: 31.5 sec superbee: 71.2 sec weno: 42.2 sec constant: 9.1 sec minmod: 13.5 sec vanleer: 13.5 sec superbee: 13.4 sec weno: 16.6 sec Implicit *: lagged evaluation of slope limiters and WENO weights reduces the number of nonlinear iterations no adverse effect on the stability and only reduces accuracy slightly

35 Number of iterations required by the explicit schemes constant minmod vanleer superbee weno

36 Number of iterations required by the explicit schemes constant minmod 0 steeper slopes vanleer The overall system more nonlinear and coupled 1 superbee weno

37 Number of iterations required by the implicit schemes 4 3 constant 2 1 minmod van Leer superbee WENO Control step Ok iterations Failed iterations

38 Number of iterations required by the implicit schemes constant minmod vanleer superbee weno Time [days]

39 Number of iterations required by the implicit* schemes constant minmod vanleer superbee weno Time [days]

40 Test of large-time-step capability constant constant t 8 7 minmod Implicit vanleer superbee minmod vanleer superbee d r e weno weno o constant constant CFL number t CFL number 8 7 minmod Implicit* vanleer superbee minmod vanleer superbee d r e weno weno o CFL number CFL number Simulating a single time-step, starting from a well-established displacement profile, 1 day

41 Test of large-time-step capability constant constant t 8 7 minmod Implicit vanleer superbee minmod vanleer superbee d r e weno weno o constant constant CFL number t CFL number 8 7 minmod Implicit* vanleer superbee minmod vanleer superbee d r e weno weno o CFL number CFL number Simulating a single time-step, starting from a well-established displacement profile, 1 day (left) and 5 days (right)

42 Spatial convergence 20 x x x 80 Implicit first-order constant: 1.6 sec constant: 5.9 sec constant: 26.7 sec Implicit second-order minmod: 7.9 sec minmod: 15.8 sec minmod: 74.5 sec

43 Spatial convergence 20 x x x 80 Implicit first-order constant: 1.6 sec constant: 5.9 sec constant: 26.7 sec Implicit second-order minmod: 7.9 sec minmod: 15.8 sec minmod: 74.5 sec

44 Spatial convergence s Explicit CFL=1 CFL=2 CFL=4 CFL=8 CFL=16 CFL=32 CFL=64 CFL= x

45 Temporal convergence s Explicit CFL=1 CFL=2 CFL=4 CFL=8 CFL=16 CFL=32 CFL=64 CFL= x

46 Temporal convergence s Explicit CFL=1 CFL=2 CFL=4 CFL=8 CFL=16 CFL=32 CFL=64 CFL= x

47 Temporal convergence

48 Temporal convergence Observe: can safely increase the CFL number for the implicit scheme to one order of magnitude beyond the stability limit before the numerical dissipation causes significant smearing

49 Test of grid-orientation errors Producer Injector Quarter five-spot Rotated five-spot

50 Test of grid-orientation errors Producer Injector Quarter five-spot Rotated five-spot Original setup: overestimates the frontal movement into stagnant regions, and underestimates the movement in the high-flow zones along the diagonal by smearing the tip of the finger Rotated setup: overestimates the movement of the front in the high-flow zone, and underestimates the movement towards stagnant zones

51 Test of grid-orientation errors 1 st order constant Colors: rotated setup setup Solid lines: original

52 Test of grid-orientation errors 1 st order constant 2 nd order minmod 2 nd order van Leer 2 nd order WENO Colors: rotated setup setup Solid lines: original

53 Test of grid-orientation errors 1 st order constant 2 nd order minmod 2 nd order van Leer 2 nd order WENO Colors: rotated setup setup Solid lines: original

54 Test of grid-orientation errors st order: rotated minmod: rotated van Leer: rotated st order: original minmod: original 0.2 van Leer: original Time [days] Water saturation in cell containing the producer

55 Channelized reservoir (K, )= ( (200 md, 0.25), for x 2 [100, 200] and x 2 [300, 400], (100 md, 0.20), otherwise. x

56 Channelized reservoir (K, )= ( (200 md, 0.25), for x 2 [100, 200] and x 2 [300, 400], (100 md, 0.20), otherwise. x

57 Channelized reservoir Explicit: constant Explicit: vanleer Implicit: constant Implicit: vanleer Explicit: constant Explicit: vanleer Implicit: constant Implicit: vanleer ) x = 150 2) x = 250

58 Channelized reservoir ) y = 700 4)y = 300

59 Including buoyancy Does including buoyancy make the nonlinear convergence more challenging? t =0days t = 110 days t = 1000 days 0

60 Including buoyancy Cummulative numberof iterations required by the nonlinear solver: implicit (solid) and implicit* (dashed) constant minmod vanleer superbee weno Time [days]

61 Summary Studied explicit and fully implicit schemes with second-order formal spatial accuracy applied to polymer flooding The use of high-resolution spatial stencils improves the accuracy and reduces grid-orientation effects Demonstrated that implicit time discretization is more suitable than explicit time integration through numerous numerical experiments To make high-resolution methods amenable for implicit discretizations, preference should be given to spatial stencils and nonlinear limiter functions that are as smooth as possible to avoid exacerbating the nonlinearity of the implicit flow equations