Reservoir Simulation Gridding and Well Modelling Sergey Kurelenkov

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Reservoir Simulation Gridding and Well Modelling Sergey Kurelenkov 2011 (after Ken Sorbie) Reservoir Simulation Gridding and Well Modelling Sergey Kurelenkov 2011 (after Ken Sorbie)

Outline • Gridding in Reservoir Simulation • Calculation of Block to Block Flows in Reservoir SimulatorsOutline • Gridding in Reservoir Simulation • Calculation of Block to Block Flows in Reservoir Simulators • Wells in Reservoir Simulation

Gridding in Reservoir Simulation • Reservoir Simulation Model – grid block model of a petroleum reservoir…Gridding in Reservoir Simulation • Reservoir Simulation Model – grid block model of a petroleum reservoir… • Gridding – process of dividing up a reservoir into grid blocks ( spatial discretization ) • Discretization – process of dividing up a continuous quantity into discrete intervals

Gridding in Reservoir Simulation Discretization Spatial Temporal t. Q Δ t (timestep)Δ x,  Δ y,Gridding in Reservoir Simulation Discretization Spatial Temporal t. Q Δ t (timestep)Δ x, Δ y, Δ z φ , k, c, S, p PVT

Gridding in Reservoir Simulation Grids Dimension Structure 1 D 2 D 3 D Cartesian Radial Distorted.Gridding in Reservoir Simulation Grids Dimension Structure 1 D 2 D 3 D Cartesian Radial Distorted. H y b rid , L G R

Gridding in Reservoir Simulation Grids Dimension Structure 1 D 2 D 3 D Cartesian Radial Distorted.Gridding in Reservoir Simulation Grids Dimension Structure 1 D 2 D 3 D Cartesian Radial Distorted. H y b rid , L G R

Gridding in Reservoir Simulation • 1 D Cartesian (Linear) Grids – Horizontal • Buckley-Leverett type waterGridding in Reservoir Simulation • 1 D Cartesian (Linear) Grids – Horizontal • Buckley-Leverett type water displacement – Vertical • gravity drainage • gravity stable gas displacement of oil x. S w Q w. Q o Q g

Gridding in Reservoir Simulation • 2 D Cartesian Grids – Cross-sectional (x/z) • vertical sweep efficiencyGridding in Reservoir Simulation • 2 D Cartesian Grids – Cross-sectional (x/z) • vertical sweep efficiency • water/oil displacements in geostatistically generated cross-section • generation of pseudo-relative permeabilities (2 phase upscaling) • mechanism of gas displacement (importance of gravity)

Gridding in Reservoir Simulation • 2 D Cartesian Grids – Areal (x/y) • areal sweep efficiencyGridding in Reservoir Simulation • 2 D Cartesian Grids – Areal (x/y) • areal sweep efficiency (water/gas flooding) • benefits of infill drilling in an areal pattern flood • near-miscible gas injection in a homogeneous layer

Gridding in Reservoir Simulation • 3 D Cartesian Grids – Default grid type • easy toGridding in Reservoir Simulation • 3 D Cartesian Grids – Default grid type • easy to set up • less time consuming simulation (equations are derived for this grid) • wide range of field wide reservoir production processes (full field simulation of water/gas flooding and etc. )

Gridding in Reservoir Simulation Grids Dimension Structure 1 D 2 D 3 D Cartesian Radial Distorted.Gridding in Reservoir Simulation Grids Dimension Structure 1 D 2 D 3 D Cartesian Radial Distorted. H y b rid , L G R

Gridding in Reservoir Simulation • Radial Grids – 1 D • pressure front propagation – 2Gridding in Reservoir Simulation • Radial Grids – 1 D • pressure front propagation – 2 D (r/z), 3 D • near wellbore processes (water/gas coning)

Gridding in Reservoir Simulation • Radial Grids – pressure change near well is steep – logarithmically-spacedGridding in Reservoir Simulation • Radial Grids – pressure change near well is steep – logarithmically-spaced grid • grid is divided such that wr r r. Pln~)( const)()(1 iir. P 1 iiirrr const 1 i i r r 1 1 ln~)()( i i ii r r r. P

Gridding in Reservoir Simulation Grids Dimension Structure 1 D 2 D 3 D Cartesian Radial Distorted.Gridding in Reservoir Simulation Grids Dimension Structure 1 D 2 D 3 D Cartesian Radial Distorted. H y b rid , L G R

Gridding in Reservoir Simulation • Distorted grids – Corner Point • account structure features (faults) 8Gridding in Reservoir Simulation • Distorted grids – Corner Point • account structure features (faults) 8 corners

Gridding in Reservoir Simulation • Distorted grids – Perpendicular Bisector (PEBI), Voronoi • account structure featuresGridding in Reservoir Simulation • Distorted grids – Perpendicular Bisector (PEBI), Voronoi • account structure features (faults) • horizontal well modelling

Gridding in Reservoir Simulation Grids Dimension Structure 1 D 2 D 3 D Cartesian Radial Distorted.Gridding in Reservoir Simulation Grids Dimension Structure 1 D 2 D 3 D Cartesian Radial Distorted. H y b rid , L G R

Gridding in Reservoir Simulation • Local Grid Refinement (LGR) – Simple grid refinement Refinement in areaGridding in Reservoir Simulation • Local Grid Refinement (LGR) – Simple grid refinement Refinement in area of interest Refinement where not required (e. g. aquifer)

Gridding in Reservoir Simulation • Local Grid Refinement (LGR) – Local Grid Refinement in area ofGridding in Reservoir Simulation • Local Grid Refinement (LGR) – Local Grid Refinement in area of interest Keep coarse grid where not required (e. g. aquifer)

Gridding in Reservoir Simulation • Local Grid Refinement (LGR) – Hybrid Grid Local Grid Refinement 20Gridding in Reservoir Simulation • Local Grid Refinement (LGR) – Hybrid Grid Local Grid Refinement

Gridding in Reservoir Simulation • Local Grid Refinement (LGR) – Hybrid Grid Local Grid Refinement 21Gridding in Reservoir Simulation • Local Grid Refinement (LGR) – Hybrid Grid Local Grid Refinement

Gridding in Reservoir Simulation • Distorted grids – Faults modelling • Non-neighbor connections (NNC) 22 Gridding in Reservoir Simulation • Distorted grids – Faults modelling • Non-neighbor connections (NNC)

Gridding in Reservoir Simulation Numerical problems Numerical dispersion Grid orientation 23 Gridding in Reservoir Simulation Numerical problems Numerical dispersion Grid orientation

Gridding in Reservoir Simulation Numerical problems Numerical dispersion Grid orientation 24 Gridding in Reservoir Simulation Numerical problems Numerical dispersion Grid orientation

Gridding in Reservoir Simulation • Numerical Dispersion – numerical error due to grid block approximation forGridding in Reservoir Simulation • Numerical Dispersion – numerical error due to grid block approximation for solving the flow equations Q w Q o i=1 i=2 i=3 i=4 i=5Δ x S w =S wct= Δ t 0 NX=5 S w >S wc k rw >

Gridding in Reservoir Simulation • Numerical Dispersion – numerical error due to grid block approximation forGridding in Reservoir Simulation • Numerical Dispersion – numerical error due to grid block approximation for solving the flow equations Q w Q o i=1 i=2 i=3 i=4 i=5 t= 2 Δ t NX=5 S w >S wc k rw >

Gridding in Reservoir Simulation • Numerical Dispersion – numerical error due to grid block approximation forGridding in Reservoir Simulation • Numerical Dispersion – numerical error due to grid block approximation for solving the flow equations Q w Q o i=1 i=2 i=3 i=4 i=5 t= 3 Δ t NX=5 S w >S wc k rw >

Gridding in Reservoir Simulation • Numerical Dispersion – numerical error due to grid block approximation forGridding in Reservoir Simulation • Numerical Dispersion – numerical error due to grid block approximation for solving the flow equations Q w Q o i=1 i=2 i=3 i=4 i=5 t= 4 Δ t NX=5 S w >S wc k rw >

Gridding in Reservoir Simulation • Numerical Dispersion – numerical error due to grid block approximation forGridding in Reservoir Simulation • Numerical Dispersion – numerical error due to grid block approximation for solving the flow equations Q w Q o i=1 i=2 i=3 i=4 i=5 t= 5 Δ t NX=5 S w >S wc k rw >0 Q w Water breakthrough after 5 Δ t If NX=10 ? Water breakthrough after 10 Δ t’

Gridding in Reservoir Simulation • Numerical Dispersion – saturation front spreading Analytical solution (Buckley-Levere tt) NumericalGridding in Reservoir Simulation • Numerical Dispersion – saturation front spreading Analytical solution (Buckley-Levere tt) Numerical solution

Gridding in Reservoir Simulation • Numerical Dispersion – Methods of reducing • Grid refinement (increase inGridding in Reservoir Simulation • Numerical Dispersion – Methods of reducing • Grid refinement (increase in number of grid blocks) • Use of Pseudo-relative permeabilities (2 phase upscaling) • Use of alternative simulation scheme (Streamline simulation) S wcr

Gridding in Reservoir Simulation Numerical problems Numerical dispersion Grid orientation 32 Gridding in Reservoir Simulation Numerical problems Numerical dispersion Grid orientation

Gridding in Reservoir Simulation • Grid Orientation effect – arises when the fluid flow both orientedGridding in Reservoir Simulation • Grid Orientation effect – arises when the fluid flow both oriented with the principal and diagonal grid direction L L P 1 P

Gridding in Reservoir Simulation • Grid Orientation effect – Methods of reducing • Grid refinement (increaseGridding in Reservoir Simulation • Grid Orientation effect – Methods of reducing • Grid refinement (increase in number of grid blocks) • Use of distorted grids (PEBI) • Use of improved numerical schemes (9 -point) • Use of alternative simulation scheme (Streamline simulation) 5 -point 9 -point

Gridding in Reservoir Simulation • Streamline Simulation Pressure Streamlines Saturation 35 Gridding in Reservoir Simulation • Streamline Simulation Pressure Streamlines Saturation

Gridding in Reservoir Simulation • Issues in Choosing Reservoir Simulation Grid – Grid Dimension • 1Gridding in Reservoir Simulation • Issues in Choosing Reservoir Simulation Grid – Grid Dimension • 1 D, 2 D, 3 D ? – Grid Geometry/Structure • Cartesian, Distorted, LGR ? – Grid Fineness/Coarseness • Grid block size (number of grid blocks) ? should be related to the simplicity/complexity of the problem

Gridding in Reservoir Simulation • Issues in Choosing Reservoir Simulation Grid – Example 1 37 Gridding in Reservoir Simulation • Issues in Choosing Reservoir Simulation Grid – Example

 • Issues in Choosing Reservoir Simulation Grid – Example 1 Gridding in Reservoir Simulation 38 • Issues in Choosing Reservoir Simulation Grid – Example 1 Gridding in Reservoir Simulation

Gridding in Reservoir Simulation • Issues in Choosing Reservoir Simulation Grid – Example 1 MWAG WaterfloodingGridding in Reservoir Simulation • Issues in Choosing Reservoir Simulation Grid – Example 1 MWAG Waterflooding

 • Issues in Choosing Reservoir Simulation Grid – Example 2 Gridding in Reservoir Simulation 40 • Issues in Choosing Reservoir Simulation Grid – Example 2 Gridding in Reservoir Simulation

Outline • Gridding in Reservoir Simulation • Calculation of Block to Block Flows in Reservoir SimulatorsOutline • Gridding in Reservoir Simulation • Calculation of Block to Block Flows in Reservoir Simulators • Wells in Reservoir Simulation

Calculation of Block to Block Flows in Reservoir Simulators • Darcy Law for Inter Block FlowCalculation of Block to Block Flows in Reservoir Simulators • Darcy Law for Inter Block Flow x P B k AQ p p p x P i-1 ? 2 1 1 ii ii p p xx PP B k

Calculation of Block to Block Flows in Reservoir Simulators • Averaging k-A Product, k A Calculation of Block to Block Flows in Reservoir Simulators • Averaging k-A Product, k A 2 1 1 ii ii p p p xx PP B k AQ

Calculation of Block to Block Flows in Reservoir Simulators • Averaging k-A Product,   Calculation of Block to Block Flows in Reservoir Simulators • Averaging k-A Product, 2 1 111 i ifii x PPAk Q 1 1 2 i i if Ak x QPP block i-1 44 k

Calculation of Block to Block Flows in Reservoir Simulators • Averaging k-A Product,   Calculation of Block to Block Flows in Reservoir Simulators • Averaging k-A Product, 2 i fiii x PPAk Q i i fi Ak x QPP 1 2 block i 45 k

    i i ii Ak x. Q PP 1 1 1 2 Calculation i i ii Ak x. Q PP 1 1 1 2 Calculation of Block to Block Flows in Reservoir Simulators • Averaging k-A Product, 1 1 2 i i if Ak x QPP i i fi Ak x QPP 1 2 + 46 k

    i i ii Ak x. Q PP 1 1 1 2 Calculation i i ii Ak x. Q PP 1 1 1 2 Calculation of Block to Block Flows in Reservoir Simulators • Averaging k-A Product, 1 1 1 21 ii i i PP Ak x Q 47 k

   2 1 1 ii ii xx PPk A Q Calculation of Block to 2 1 1 ii ii xx PPk A Q Calculation of Block to Block Flows in Reservoir Simulators • Averaging k-A Product, 1 1 1 21 ii i i PP Ak x Q i iii Ak xxx k A 1 11 2 2 48 k

Calculation of Block to Block Flows in Reservoir Simulators • Averaging k-A Product,   Calculation of Block to Block Flows in Reservoir Simulators • Averaging k-A Product, i iii Ak xxx k A 1 11 2 2 i i ii Ak x xx k A 1 1 1 Harmonic average 49 k

Calculation of Block to Block Flows in Reservoir Simulators • Averaging k-A Product,   Calculation of Block to Block Flows in Reservoir Simulators • Averaging k-A Product, 2 1 1 ii ii p p p xx PP B k AQ Harmonic average 50 k

Calculation of Block to Block Flows in Reservoir Simulators • Averaging Two-Phase Mobility, p  Calculation of Block to Block Flows in Reservoir Simulators • Averaging Two-Phase Mobility, p 2 1 1 ii ii p p p xx PP B k AQ p rp p k 2 1 1 ii ii pp rp xx PP B k k A 2 1 ii p BB B Arithmetic average

Calculation of Block to Block Flows in Reservoir Simulators • Averaging Two-Phase Mobility,   Calculation of Block to Block Flows in Reservoir Simulators • Averaging Two-Phase Mobility, 2 1 1 ii ii pp rp p xx PP B k k AQ Harmonic average Arithmetic average 52 p

Calculation of Block to Block Flows in Reservoir Simulators • Averaging Two-Phase Mobility, – Physically QCalculation of Block to Block Flows in Reservoir Simulators • Averaging Two-Phase Mobility, – Physically Q w >0 Q o =0 53 p

 • Averaging Two-Phase Mobility, – Harmonic average (kh ) ? Calculation of Block to Block • Averaging Two-Phase Mobility, – Harmonic average (kh ) ? Calculation of Block to Block Flows in Reservoir Simulators k hrw =0 k hro =0 54 p

 • Averaging Two-Phase Mobility, – Arithmetic average (ka ) ? Calculation of Block to Block • Averaging Two-Phase Mobility, – Arithmetic average (ka ) ? Calculation of Block to Block Flows in Reservoir Simulators k arw >0 k aro >0 55 p

 • Averaging Two-Phase Mobility, – Upstream value ? Calculation of Block to Block Flows in • Averaging Two-Phase Mobility, – Upstream value ? Calculation of Block to Block Flows in Reservoir Simulators k urw >0 k uro =0 56 p

 • Averaging Two-Phase Mobility, – Upstream value ? Calculation of Block to Block Flows in • Averaging Two-Phase Mobility, – Upstream value ? Calculation of Block to Block Flows in Reservoir Simulators k urw =0 k uro >0 57 p

Calculation of Block to Block Flows in Reservoir Simulators • Summary    x PCalculation of Block to Block Flows in Reservoir Simulators • Summary x P B k k AQ pp rp p Harmonic average Arithmetic average Upstream value P i-

Outline • Gridding in Reservoir Simulation • Calculation of Block to Block Flows in Reservoir SimulatorsOutline • Gridding in Reservoir Simulation • Calculation of Block to Block Flows in Reservoir Simulators • Wells in Reservoir Simulation

 • Basic Idea of Well Model Wells in Reservoir Simulation Δ P f-w ( Δ • Basic Idea of Well Model Wells in Reservoir Simulation Δ P f->w ( Δ P skin )Δ P well Δ P f->sΔ P s P wh P wf

Wells in Reservoir Simulation • Well Models for Single and Two Phase Flow)(wfe. PPPIQ Givens ControlsWells in Reservoir Simulation • Well Models for Single and Two Phase Flow)(wfe. PPPIQ Givens Controls P wf Q P wf. Q or Limits

Wells in Reservoir Simulation • Well Models for Single and Two Phase Flow)(wfe. PPPIQ  Wells in Reservoir Simulation • Well Models for Single and Two Phase Flow)(wfe. PPPIQ wr r k h Q r. Pln 2 )( ln 2 wfe w e PP r r k h Q P=P e at r e : w e wfe r r k h Q PPln

Wells in Reservoir Simulation • Well Models for Single and Two Phase Flow)( ln 2 wfeWells in Reservoir Simulation • Well Models for Single and Two Phase Flow)( ln 2 wfe w e PP r r k h Q Analytical model Grid block model k h μ r er w P wf P e k h μ r w P wf Δ x Δ y P? ?

Wells in Reservoir Simulation • Well Models for Single and Two Phase Flow)( ln 2 wfeWells in Reservoir Simulation • Well Models for Single and Two Phase Flow)( ln 2 wfe w e PP r r k h Q Analytical model Grid block model r e P e Δ x Δ y P = ? P r e

Wells in Reservoir Simulation • Well Models for Single and Two Phase Flow – Peaceman EquivalentWells in Reservoir Simulation • Well Models for Single and Two Phase Flow – Peaceman Equivalent radius, re 2 D Areal grid r er w r ij i, j P 0 P ij Δ xΔ x w wf r r k h Q Pr. Pln 2 )( e ij oij r r k h Q PPln 2 e ijoij r r k h Q PP ln

Wells in Reservoir Simulation • Well Models for Single and Two Phase Flow – Peaceman EquivalentWells in Reservoir Simulation • Well Models for Single and Two Phase Flow – Peaceman Equivalent radius, re e ijoij r r k h Q PP ln 2 1 x r r x x r r reij e ij lnln x r k h Q PPeijoi j lnln

Wells in Reservoir Simulation • Well Models for Single and Two Phase Flow – Peaceman EquivalentWells in Reservoir Simulation • Well Models for Single and Two Phase Flow – Peaceman Equivalent radius, re x r k h Q PPeijoi j lnln 2 1 y = k · ( x – a ) r ij / Δ x 0. 2 x reΔxre 0.

Wells in Reservoir Simulation • Well Models for Single and Two Phase Flow – Peaceman EquivalentWells in Reservoir Simulation • Well Models for Single and Two Phase Flow – Peaceman Equivalent radius, re Δxre 0. 2 ( Δ x= Δ y) 2 yre 2 x 0. 14 ( Δ x≠ Δ y)

Wells in Reservoir Simulation • Well Models for Single and Two Phase Flow – Peaceman EquivalentWells in Reservoir Simulation • Well Models for Single and Two Phase Flow – Peaceman Equivalent radius, re Δxre 0. 196 Δxre 0. 2 Δxre 0. 433 Δxre 0. 193 Δxre 0.

Wells in Reservoir Simulation • Well Models for Single and Two Phase Flow – Peaceman EquivalentWells in Reservoir Simulation • Well Models for Single and Two Phase Flow – Peaceman Equivalent radius, re (k x ≠ k y ) 4 1 2 1 22 y x x y e k k Δy k k Δx k k r 0. 28( Δ x≠ Δ y) k x k y

Wells in Reservoir Simulation • Well Models for Single and Two Phase Flow)( ln 2 wfeWells in Reservoir Simulation • Well Models for Single and Two Phase Flow)( ln 2 wfe w e PP r r k h Q Analytical model Grid block model r e Δ x Δ y = ? P r e P e P 22ΔyΔx 0. 14 =

Wells in Reservoir Simulation • Well Models for Single and Two Phase Flow – Two phaseWells in Reservoir Simulation • Well Models for Single and Two Phase Flow – Two phase flow. P )(wfoo. PPPIQ)(wfww. PPPIQ w e o oro o r r Skk h PI ln )(2 w e w wrw w r r Skk h PI ln )(

Wells in Reservoir Simulation • Well Modelling in a Multi-Layer System 73 Wells in Reservoir Simulation • Well Modelling in a Multi-Layer System

Wells in Reservoir Simulation • Well Modelling in a Multi-Layer System For each layer (k=1. .Wells in Reservoir Simulation • Well Modelling in a Multi-Layer System For each layer (k=1. . 4))(wfkkokok. PPPIQ w ek o koro ok r r Skk h PI ln ))((2 )(wf kkwkwk. PPPIQ w ek w kwrw wk r r Skk h PI ln ))((2 +)()(wfkkwkok. Tk. PPPIPIQ

Wells in Reservoir Simulation • Well Modelling in a Multi-Layer System For each layer (k=1. .Wells in Reservoir Simulation • Well Modelling in a Multi-Layer System For each layer (k=1. . 4))(wfkkokok. PPPIQ )(wf kkwkwk. PPPIQ )()(wfkkwkok. Tk. PPPIPIQ 4 1 )()( k wfkkwkok. TPPPIPIQ Total 4 1 )( k wfkkoko. PPPIQ 4 1 )( k wfkkwkw. PPPIQ For simplicity 4321 wfwf. PPPP

Wells in Reservoir Simulation • Modelling Horizontal Wells)(, , , , kjwfikjikjoi. PPPIQ i, j, kWells in Reservoir Simulation • Modelling Horizontal Wells)(, , , , kjwfikjikjoi. PPPIQ i, j, k

Wells in Reservoir Simulation • Modelling Horizontal Wells 77 Wells in Reservoir Simulation • Modelling Horizontal Wells

Wells in Reservoir Simulation • Hierarchies of Wells and Well Controls – Well Controls • RateWells in Reservoir Simulation • Hierarchies of Wells and Well Controls – Well Controls • Rate constrained injection • Pressure constrained production Set Q wi Set P wf

Wells in Reservoir Simulation • Hierarchies of Wells and Well Controls – Well Controls • RateWells in Reservoir Simulation • Hierarchies of Wells and Well Controls – Well Controls • Rate or Pressure constrained production • Voidage replacement (injection) variable Q wi Q op + Q wp Q wi res =Q wi ·B w Q p res =Q op ·B o +Q wp ·B w

Wells in Reservoir Simulation • Hierarchies of Wells and Well Controls – Well Controls • RateWells in Reservoir Simulation • Hierarchies of Wells and Well Controls – Well Controls • Rate or Pressure constrained production • Voidage replacement (injection) variable Q wi Q op + Q wp Q wi res =Q wi ·B w Q p res =Q op ·B o +Q wp ·B w. Q wi ·B w =Q op ·B o +Q wp ·B w. Q wi =Q op ·(B o /B w ) +Q wp

Thank you for attention! Reservoir Simulation Gridding and Well Modelling Sergey Kurelenkov 2011 Thank you for attention! Reservoir Simulation Gridding and Well Modelling Sergey Kurelenkov




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