PETE 603
Lecture Session #30
Thursday, 7/29/10
Accuracy of Solutions
30.1
•
•
•
•
•
•
•
Material Balance Error
Nonlinear Error
Instability Error
Truncation Error
Roundoff Error
Numerical Dispersion
Grid Orientation
30.2
Material Balance Error
• Over a single timestep
 V p So n +1  V p So n 
Local MBE =  
 -
  +  q o t
i, j 
 Bo   i, j
 Bo 
• Over entire simulation
 VpSo n 1  VpS o
Cum.MBE   
  
Bo
ij  Bo 


o
n 1

     q o t
  1 ij
30.3
Material Balance Error
Causes of material balance error:
- non-conservative equation formulation
- error in solution of nonlinear equations
(Newton-Raphson)
- error in matrix solutions
- roundoff errors (numerical precision)
- data errors (negative compressibilities)
- program bugs
Nonlinear Error
30.4
True PVT data
PVT
quantity
Data points in PVT table
Pressure
Interpolating PVT functions linearly is a source of error.
30.5
Nonlinear Error
• Nonlinear errors are especially a
problem near the bubblepoint due to
discontinuity in total compressibility.
• Chord slopes in IMPES
n1
p
• Solutions:
 
n1
i
V  V 1 c f p  p
n
p
– take smaller timesteps
– iterate on chord slope (IMPES)
n
i

Instability Error
30.6
Sw
Time
If a simulation run becomes unstable erroneous
saturations will occur.
Instability Error
30.7
• Caused by taking timesteps which are
too large in an IMPES method.
• Solution:
– Take smaller timesteps
– Use fully implicit method
30.8
Truncation Error

f(x + x) - f(x) x
x 2
f (x) =
f (x) f (x) + ....
x
2!
3!
f(x + x ) - f(x)
f (x) =
- Ox 
x
Truncating the Taylor series expansions is a source
of error.
The size of this error depends on the grid size and
the timestep.
Truncation Error
30.9
PRESSURE (X = 100 ll ), psia
1,000
DELT
800
10 days
600
5 days
1.25 days
400
0.15 days
200
0
20
40
60
80
TIME, days
Refining the grid and/or reducing the timestep
reduces the truncation error.
30.10
Numerical Dispersion
Numerical dispersion causes fluid fronts to arrive
sooner than they should. Fronts that should be sharp
become smeared.
30.11
Numerical Dispersion
• Solution:
– Use smaller gridblocks
– Upstream relative permeabilities also help
minimize numerical dispersion.
• Multipoint upstream sometimes used in tracer
studies
– Use pseudorelative permeabilities
(pseudofunctions)
– Choose timestep wisely in IMPES
(maximum stable timestep)
30.12
Numerical Dispersion
• Do we want to remove numerical
dispersion altogether?
• Buckley-Leverett flow does not apply for
many field cases.
– Capillary pressure spreads out fluid front,
especially in low permeability reservoirs.
– Permeability heterogeneity smears fluid
front (fingering).
• Some numerical dispersion may be
acceptable.
30.13
Grid Orientation
The orientation of the grid with respect to the well
locations can influence the simulation results.
Fluid moves preferentially along the grid lines. If the red
dot represented a water injector, the water would
break through fastest in the diagonal grid (right) - even if
the grid dimensions were the same.
Grid Orientation
30.14
To overcome the grid orientation effect a
different form of the finite-difference method can
be used.
•
• The usual method is called the “five-point”
approach.
• The “nine-point” approach to the flow equations
removes the grid orientation effect, however
more computational work is required.
• The “Control Volume Finite Element” approach
often uses triangular gridblocks (full permeability
tensor required).
30.15
Grid Orientation
30.16
Grid Orientation
30.17
Data Modification - 5 spot
Producer
Producer
Injector
Producer
Producer
30.18
Data Modification - 5 spot
Simulator: Quarter 5-spot
30.19
Data Modification - 5 spot
Data Modification: Quarter 5-spot
ky
2
kx
2

30.20
Data Modification - 5 spot
Final Result: Quarter 5-spot