SPE 106179 Multiscale Mixed Finite Element Modeling of Coupled - PowerPoint PPT Presentation

SPE 106179 Multiscale Mixed Finite Element Modeling of Coupled Wellbore / Near- Well Flow Stein Krogstad 1 , Louis J. Durlofsky 2 1 SINTEF ICT, 2 Stanford University RSS07 Feb 26-28, 2007 1

Motivation:  Near-well region extremely important  Cannot fully resolve all scales in typical simulation  Multiscale methods incorporate fine scales in coarse scale equations RSS07 Feb 26-28, 2007 2

Outline Motivation Multiscale Mixed Methods Drift-Flux Wellbore Flow Modeling Multiscale – Drift-Flux Coupling Numerical Experiments Conclusions / Further Work RSS07 Feb 26-28, 2007 3

Multiscale Methods for Reservoir Simulation  Multiscale Finite Element Method  T. Hou, X. H. Wu, 1997  Multiscale Mixed FEM  Z. Chen, T. Hou, 2003  T. Arbogast et al., 2000  J. Aarnes et al., 2004 (group at SINTEF)  Multiscale Finite Volume Method  P. Jenny, S. H. Lee, H.A. Tchelepi, 2003+ RSS07 Feb 26-28, 2007 4

Standard / Multiscale Discretization Coarse model Fine model Local flow: Local flow: Multiscale Standard Solve coarse equations / Solve coarse equations form fine scale flow RSS07 Feb 26-28, 2007 5

Mixed Finite Elements (MFEM) Model equations: Choose basis: = − λ ∇ u k p ≈ ∑ ψ u v ∇ ⋅ = i i u q ≈ ϕ ∑ p p j j Weak formulation: Mixed discretization: − ⋅ λ − ∇ ⋅ = 1 ∫ ∫ u ( k ) u p u 0 ˆ ˆ       A B v 0 ∇ ⋅ = =       ∫ ∫ p u p q ˆ ˆ − T       B 0 p q ∈ × ( u , ) . ˆ p ˆ U V for all RSS07 Feb 26-28, 2007 6

Mixed / Mimetic / Multiscale  Raviart-Thomas  Mimetic Finite Differences ∆ p = − ∆ p = 1 1  Multiscale Mixed FEM RSS07 Feb 26-28, 2007 7

Multiscale MFEM (MsMFEM) ψ Flux basis function , :  Initially compute basis = − ∇ ψ functions k p for n=1 to N ∈  w for x T i i  Solve coarse system ∇ ⋅ = ψ  − ∈ w for x T  based on current j j saturation ⋅ n = ∂ ψ 0 on ( T T ) i j  Form fine scale fluxes  Advance fine scale ∆ t saturation by end RSS07 Feb 26-28, 2007 8

Drift-Flux Wellbore Flow Model  Mixture velocity (oil/water): V w V o = α + α α = V V V A o / A m o o w w o α = A w / A V V w so sw  Oil velocity: = + θ V C V m ( ) V 0 o m d Shi et al. (2005): 0 = C 1 = − α V 1 . 53 V ( 1 ) d c o = m ( 0 ) 1 . 07 RSS07 Feb 26-28, 2007 9

Governing Equations for Wellbore Flow  Wellbore pressure: ρ 2 ∂ ρ 2 f V θ p 2 q V = ρ θ + + tp m m m m m g cos( ) ∂ m x D A hydrostatic friction acceleration  In-situ volume fraction: ∂ α ∂ V − = o so q ∂ ∂ o t x RSS07 Feb 26-28, 2007 10

Well – Reservoir Linkage Fine grid to the annulus, well segments included in the grid Well segments treated as coarse blocks - no well model is used RSS07 Feb 26-28, 2007 11

Well – Reservoir Coupling Saturation / Holdup: Pressure:  Implicit finite volume  Linearize wellbore  Optimal ordering of pressure equation  Couple to MsMFEM cells  Newton iteration in equations  Fixed-point iteration for sequence for each cell / small cluster initial pressure Prototype implementation: Enhancements required for full generality RSS07 Feb 26-28, 2007 12

Numerical Example 1: Validation  Homogeneous permeability  Compressibility ( psi -1 ), 3x10 -4  Wellbore radius (inch), 2.0  Pipe roughness ( inch ), 0.001  Initial saturation, 0.5  Quadratic relative perms, µ o / µ w = 1 600 ft 4200 ft  7056 fine cells �  284 coarse blocks t f 0 0 2 4  12 well segments RSS07 Feb 26-28, 2007 13

Numerical Example 1: Pressure profiles: Total rate: 1,600 STB/d Total rate: 20,000 STB/d 5400 Wellbore pressure (psia) 5700 Wellbore pressure (psia) 5200 5600 5000 5500 4800 GPRS 5400 GPRS 4600 Fine Fine 5300 4400 MsMFEM MsMFEM 5200 4200 1 3 5 7 9 11 1 3 5 7 9 11 Segment number Segment number RSS07 Feb 26-28, 2007 14

Numerical Example 1: In-situ oil fraction: Total rate: 1,600 STB/d Total rate: 20,000 STB/d 0.600 0.600 In-situ oil fraction In-situ oil fraction 0.500 0.500 0.400 0.400 0.300 0.300 GPRS GPRS 0.200 0.200 Fine Fine 0.100 0.100 MsMFEM MsMFEM 0.000 0.000 1 3 5 7 9 11 1 3 5 7 9 11 Segment number Segment number RSS07 Feb 26-28, 2007 15

Numerical Example 2: Long inclined producer in heterogeneous reservoir  Two-phase oil/water  85,000 fine cells  62 well segments incompressible flow  Prescribed total flowrate  θ between 70° and 80°  Quadratic relative perms, µ o / µ w = 10  Initially saturated with oil producer 600 ft 5 0 0 0 f t 3500 ft injector RSS07 Feb 26-28, 2007 16

Numerical Example 2: Pressure profiles at 0.12 PVI:  Coarse grid: 2856 blocks (factor of 30 coarsening) Flowrate: 4,000 STB/d Flowrate: 60,000 STB/d Wellbore pressure (psia) Wellbore pressure (psia) 5000 5000 4900 4800 4000 4700 3000 4600 Fine Fine 2000 4500 MsMFEM MsMFEM 4400 1000 1 11 21 31 41 51 61 1 11 21 31 41 51 61 Segment number Segment number RSS07 Feb 26-28, 2007 17

Numerical Example 2: Low flowrate: In-situ oil fraction: 1 Fine - low rate In-situ oil fraction MsMFEM - low rate 0.8 Fine - high rate MsMFEM - high rate 0.6 0.4 High flowrate: 0.2 0 1 11 21 31 41 51 61 Segment number RSS07 Feb 26-28, 2007 18

Numerical Example 2:  Coarsening factor varied from 10 to 850  Accuracy degrades with coarsening, but physically reasonable results in all cases Wellbore pressure (psia) 4950 1 Fine Fine 4900 In-situ oil fraction 17x24x21 0.8 17x24x21 4850 5x5x4 5x5x4 0.6 4800 4750 0.4 4700 0.2 4650 4600 0 1 11 21 31 41 51 61 1 11 21 31 41 51 61 Segment number Segment number RSS07 Feb 26-28, 2007 19

Conclusions / Further Work  Extended MsMFEM for oil-water systems to include drift-flux wellbore flow model  Demonstrated and validated through numerical experiments involving vertical and deviated wells  Achieved accurate results for significantly coarsened models  Extend to three-phase flow RSS07 Feb 26-28, 2007 20