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
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