Reservoir Simulation Gridding and Well

Outline
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
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
Δt (timestep)
Δx, Δy, Δz
φ, k, c, S, p
PVT

Gridding in Reservoir Simulation
Grids
Dimension
Structure
1D
2D
3D
Cartesian
Radial
Distorted
Hybrid, LGR

Gridding in Reservoir Simulation
Grids
Dimension
Structure
1D
2D
3D
Cartesian
Radial
Distorted
Hybrid, LGR

Gridding in Reservoir Simulation
1D Cartesian (Linear) Grids
Horizontal
Buckley-Leverett type water displacement
Vertical
gravity drainage
gravity stable gas displacement of oil
Qw
Qw
Qo
Qo
Qg
Qg

Gridding in Reservoir Simulation
2D 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
2D 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
3D 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
1D
2D
3D
Cartesian
Radial
Distorted
Hybrid, LGR

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

Gridding in Reservoir Simulation
Radial Grids
pressure change near well is steep
logarithmically-spaced grid
grid is divided such that

Gridding in Reservoir Simulation
Grids
Dimension
Structure
1D
2D
3D
Cartesian
Radial
Distorted
Hybrid, LGR

Gridding 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 features (faults)
horizontal well modelling

Gridding in Reservoir Simulation
Grids
Dimension
Structure
1D
2D
3D
Cartesian
Radial
Distorted
Hybrid, LGR

Gridding 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
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

Gridding in Reservoir Simulation
Local Grid Refinement (LGR)
Hybrid Grid Local Grid Refinement

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

Gridding in Reservoir Simulation
Numerical problems
Numerical dispersion
Grid orientation

Gridding in Reservoir Simulation
Numerical problems
Numerical dispersion
Grid orientation

Gridding in Reservoir Simulation
Numerical Dispersion
numerical error due to grid block approximation for solving the flow equations
Qw
Qo
Sw=Swc
t=
Δt
0
NX=5
Sw>Swc
krw>0

Gridding in Reservoir Simulation
Numerical Dispersion
numerical error due to grid block approximation for solving the flow equations
Qw
Qo
t=
2Δt
NX=5
Sw>Swc
krw>0

Gridding in Reservoir Simulation
Numerical Dispersion
numerical error due to grid block approximation for solving the flow equations
Qw
Qo
t=
3Δt
NX=5
Sw>Swc
krw>0

Gridding in Reservoir Simulation
Numerical Dispersion
numerical error due to grid block approximation for solving the flow equations
Qw
Qo
t=
4Δt
NX=5
Sw>Swc
krw>0

Gridding in Reservoir Simulation
Numerical Dispersion
numerical error due to grid block approximation for solving the flow equations
Qw
Qo
t=
5Δt
NX=5
Sw>Swc
krw>0
Qw
Water breakthrough after 5Δt
If NX=10 ?
Water breakthrough after 10Δt’

Gridding in Reservoir Simulation
Numerical Dispersion
saturation front spreading
Analytical solution
(Buckley-Leverett)
Numerical solution

Gridding 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)
Swcr

Gridding in Reservoir Simulation
Numerical problems
Numerical dispersion
Grid orientation

Gridding in Reservoir Simulation
Grid Orientation effect
arises when the fluid flow both oriented with the principal and diagonal grid direction

Gridding 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)

Gridding in Reservoir Simulation
Streamline Simulation

Gridding in Reservoir Simulation
Issues in Choosing Reservoir Simulation Grid
Grid Dimension
1D, 2D, 3D ?
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

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
Waterflooding

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 Simulators
Wells in Reservoir Simulation

Calculation of Block to Block Flows in Reservoir Simulators
Darcy Law for Inter Block Flow
x
Pi
Pi-1
?

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,
block i-1

Calculation of Block to Block Flows in Reservoir Simulators
Averaging k-A Product,
block i

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,

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,
Harmonic average

Calculation of Block to Block Flows in Reservoir Simulators
Averaging k-A Product,
Harmonic
average

Calculation of Block to Block Flows in Reservoir Simulators
Averaging Two-Phase Mobility,
Arithmetic average

Calculation of Block to Block Flows in Reservoir Simulators
Averaging Two-Phase Mobility,
Harmonic
average
Arithmetic
average

Calculation of Block to Block Flows in Reservoir Simulators
Averaging Two-Phase Mobility,
Physically
Qw>0
Qo=0

Averaging Two-Phase Mobility,
Harmonic average (kh) ?
Calculation of Block to Block Flows in Reservoir Simulators
khrw=0
khro=0

Averaging Two-Phase Mobility,
Arithmetic average (ka) ?
Calculation of Block to Block Flows in Reservoir Simulators
karw>0
karo>0

Averaging Two-Phase Mobility,
Upstream value ?
Calculation of Block to Block Flows in Reservoir Simulators
kurw>0
kuro=0

Averaging Two-Phase Mobility,
Upstream value ?
Calculation of Block to Block Flows in Reservoir Simulators
kurw=0
kuro>0

Calculation of Block to Block Flows in Reservoir Simulators
Summary
Harmonic
average
Arithmetic
average
Upstream
value
Pi
Pi-1

Outline
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
ΔPf->w
(ΔPskin)
ΔPwell
ΔPf->s
ΔPs
Pwh
Pwf

Wells in Reservoir Simulation
Well Models for Single and Two Phase Flow
Givens
Controls
Pwf
Q
Pwf
Q
or
Limits

Wells in Reservoir Simulation
Well Models for Single and Two Phase Flow
P=Pe at re:

Wells in Reservoir Simulation
Well Models for Single and Two Phase Flow
Analytical model
Grid block model
k
h
μ
re
rw
Pwf
Pe
k
h
μ
rw
Pwf
?
?

Wells in Reservoir Simulation
Well Models for Single and Two Phase Flow
Analytical model
Grid block model
re
Pe
=
?
re
re

Wells in Reservoir Simulation
Well Models for Single and Two Phase Flow
Peaceman Equivalent radius, re
2D Areal grid
re
rw
rij
i, j
P0
Pij
Δx
Δx

Wells in Reservoir Simulation
Well Models for Single and Two Phase Flow
Peaceman Equivalent radius, re

Wells in Reservoir Simulation
Well Models for Single and Two Phase Flow
Peaceman Equivalent radius, re
y
=
k
· (
x
–
a
)
rij/Δx

Wells in Reservoir Simulation
Well Models for Single and Two Phase Flow
Peaceman Equivalent radius, re
(Δx=Δy)
(Δx≠Δy)

Wells in Reservoir Simulation
Well Models for Single and Two Phase Flow
Peaceman Equivalent radius, re

Wells in Reservoir Simulation
Well Models for Single and Two Phase Flow
Peaceman Equivalent radius, re
(kx≠ky)
(Δx≠Δy)
kx
ky

Wells in Reservoir Simulation
Well Models for Single and Two Phase Flow
Analytical model
Grid block model
re
=
?
re
re
Pe
=

Wells in Reservoir Simulation
Well Models for Single and Two Phase Flow
Two phase flow

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..4)
+

Wells in Reservoir Simulation
Well Modelling in a Multi-Layer System
For each layer (k=1..4)
Total
For simplicity

Wells in Reservoir Simulation
Modelling Horizontal Wells

Wells in Reservoir Simulation
Modelling Horizontal Wells

Wells in Reservoir Simulation
Hierarchies of Wells and Well Controls
Well Controls
Rate constrained injection
Pressure constrained production
Set Qwi
Set Pwf

Wells in Reservoir Simulation
Hierarchies of Wells and Well Controls
Well Controls
Rate or Pressure constrained production
Voidage replacement (injection)
variable Qwi
Qop + Qwp
Qwires=Qwi·Bw
Qpres=Qop·Bo+Qwp·Bw

Wells in Reservoir Simulation
Hierarchies of Wells and Well Controls
Well Controls
Rate or Pressure constrained production
Voidage replacement (injection)
variable Qwi
Qop + Qwp
Qwires=Qwi·Bw
Qpres=Qop·Bo+Qwp·Bw
Qwi·Bw=Qop·Bo+Qwp·Bw
Qwi=Qop·(Bo/Bw)+Qwp