Applied Reservoir Simulation
Course
Workshop Problems
And NExT
Denver and Houston
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Reservoir Simulation Application Training
Course
and (Eclipse) Workshop
GeoQuest Training and Development,
Outline of Workshop Problems
•
Problem 1:
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IMPES and Implicit Comparison
Time Truncation Tests
Numerical Dispersion
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A.
B.
C.
Outline of Workshop Problems
•
2. Problem 2:
Water Coning – Critical Coning Rate
Water Influx – History Matching Kh
Water Coning
I.
II.
D.
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History Matching Kv
Creation of Pseudo Krw in Coarse Grid to
Match Coning
Vertical Equilibrium Comparison
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A.
B.
C.
Outline of Workshop Problems
•
3. Problem 3: History Matching
Primary Production
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A. Sensitivity Simulations
B. History Match
C. Predictions
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Problem 1
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Problem 1 Part A: IMPES Fully
Implicit Comparison Coarse Grid –
IMPES Sub-directory
Reservoir Details
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• We have a linear reservoir 5000 x 100
x 30 feet. Grid is 50 x 1 x 1
• Water injector at one end and
producer at other end
• Two phase (oil-water) system
• Oil viscosity = 2 cp and water
viscosity = 0.5 cp
• Simulate 2000 days.
FloViz View
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Choice in Solution of the
Equations
• IMPES – Implicit Pressure Explicit
Saturation
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– Linear problem smaller
– Through put limitations: 5 – 10% PV
– Stability problems
– Timestep size - small
Choice in Solution of the
Equations
• Fully Implicit
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– Unconditionally Stable – large timesteps
– Timesteps controlled by time truncation
error
– Few long timesteps
Comparison IMPES and Fully
Implicit Solution
– Number of linear iterations
– Number of non-linear (Newton)
iterations
– CPU time per time step
– Total CPU time
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• Amount of Numerical Dispersion
(Error) in the results
• Work to obtain solution
Iteration Process in Reservoir
Simulation
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Example of linear
and non-linear
iteration process:
4 non-linear
iterations
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Usually a non-linear iteration
requires 10 to 30 linears to
converge pressure and saturations
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Data Set Names
– Problem 1/IMPES
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• IMPES.DATA – uses IMPES solution
•
• FULLIMP.DATA – uses fully implicit
solution
• Location – sub-directory:
View Results: Comparison
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• Oil Production Rate
• Production Water Cut
• Map of Water Saturations
– At 500 Days
– At 1000 Days
• Number of Linear Iterations in Each
Timestep
• Sum of Linear Iterations
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View Results: Comparison
– Note: Fully Implicit takes 31 timesteps
– IMPES takes 207 timesteps
• CPU Time Per Timestep
• Total CPU Time
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• Newton (non-linear) Iterations in
each timestep
• Sum of Newton Iteration
• Timestep Length
Your Task
• View the .data sets
• Run the data sets with ECLIPSE 100
• Plot the results
• Study the results and answer the
following questions
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– Difference – IMPES (in Schedule
Section) Fully Implicit (E100 Default)
Questions
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• Which technique takes more linear
iterations per timestep?
• Which technique takes more Newton
iterations per timestep?
• Which technique requires more work
to solve the simulation?
• Why?
• We will discuss the results in class.
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Problem 1 Part A: IMPES Fully
Implicit Comparison Fine Grid –
IMPES 2 Sub-directory
Reservoir Details
• The length of the grid blocks are now
2.5 feet instead of the previous
length of 100 feet.
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• We have the same reservoir, fluids,
etc. except that now we will subdivide the grid into 2000 blocks in the
x-direction instead of the previous
50.
Run and Compare Results
• With these smaller grid blocks the
problem is numerically more difficult,
but the numerical dispersion is less.
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• Run the IMPES.DATA and
FULLIMP.DATA in the IMPES2 subdirectory and run Graf in the IMPES2
sub-directory.
Questions
• Magnitude of the numerical dispersion in
the water cut, etc.
• Instabilities in the IMPES run.
• Number of linear and non-linear iterations
• Sum of the linear and non-linear
• Total CPU time for these cases.
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• Answer the same questions from the
previous part.
• Note the differences in the IMPES
and Fully Implicit runs.
Conclusion
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• For a blackoil simulation with
ECLIPSE we will prefer to use the
default solution – Fully Implicit since
the results are much more stable and
the CPU time is less.
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Problem 1 Part B: Time Truncation
Test: Quarter 5-Spot Water Flood
Time Truncation Error
• When it is discretized we get terms
as follows
 φ   S on+1 - S on
( ∆ x ∆ y ∆ z ) i 
 
∆t
 B w i 
 (1 - S oi )
(1 − φ )C f
- ( ∆ x ∆ y ∆ z )i 
 B wi


i
 P o in+1 - P o in 


∆t


 d (1/ B w )i   P on+1 - P on P cow n+1 - P cow n  
 
+ φ i (1 - S oi ) 
 

t
t
∆
∆
d

i 
p


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• We have the accumulation term in
our equations, for example
∂  So 
φ
∂ t  B o 
Time Truncation Error
• This discretization process yields
and error that is
• in the solution of the equations.
• This error is called Time Truncation
Error
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• O(∆t)
Simulation Situation
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• 11x11x1 quarter 5-spot involving
water injection
• Diagonal Grid
• 1452 x 1452 x 50 meter flow field
• K = 830 mD, ϕ = 0.17
• Water Injector on water rate control
• Producer on liquid rate control
• Simulate 3660 days
Grid
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Time stepping
• Three data sets are provided with
different time stepping regimes
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– maxts.data – initial timestep = 10 days,
grows to max timestep = 183 days, 53
total timesteps
– smallts.data – Initial timestep = 1 day,
max timestep = 10 days, 389 total
timesteps
– mints.data – Initial timestep = 1 day,
maximum timestep = 1 day, 3666 total
timesteps
Questions to Consider While
Viewing the Results
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• What is the effect of solving the flow
field with a few large timesteps?
• What is the most accurate solution?
• Which solution requires the most
CPU time?
• Given a balance between CPU time
and errors in the solution – which
time stepping system would you
prefer?
Results to be Viewed with Graf
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• Timestep length
• Water cut – effect of time truncation
error
• Oil production rate – effect of time
truncation error
• Water saturation maps
• CPU time per timestep
• Total CPU time
Your Task
• View the .data sets
• Run the data sets with ECLIPSE 100
• Plot the results
• Study the results and answer the
following questions
• We will discuss the results in class
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– Difference – in the TUNING Keyword –
timestep control
• Following are results from old
Pentium III computer
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• My New Computer and ECLIPSE
2004a runs so fast has trouble
storing timestep lengths
CPU Time per Timestep in
Seconds
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Total CPU Time
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Problem 1 Part C: Numerical
Dispersion: Radial Model with Gas
Coning
Numerical Dispersion
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• Definition: The error that occurs as
we discritize the partial differential
equations on a grid to create the
finite difference equations.
Finite Difference Approximations
to First Derivatives
• forward difference
• Error term:
• forward error
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f (x + ∆x) − f (x)
∂f
=
∂x
∆x
∆x ∂ f ∆x ∂ f
−
−
−.....
2
3
2! ∂x
3! ∂x
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3
Objective
• The reservoir has constant
properties; see data set RAD4.DATA
(rad4.data)
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• To see the effect of number of grid
block on a simulation model, a 2-D
radial (r, z) model was set up
Description
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• Metric Units
• Oil, Water, Gas, Dissolved Gas
• The reservoir is 200 meters thick with
the top 50 meters in the gas cap
• Flat Top at 2950 meters
• The width (radius) of the model is
500 meters
Description
• The well is completed in the bottom
50 meters of the model
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• The oil production rate is set at 6000
Sm3 / day
Description
• The flow field was sub-divided into grids
that were 4x1x4, 8x1x8, 16x1x16, 32x1x32
and 64x1x64 blocks
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• The well completions are located in the
proper layers so that the simulation
problems are identical except for the nr
and nz (number of grid blocks in the r and
z directions) values
Data Sets Provided
Five data sets are provided for you:
RAD4.DATA
RAD8.DATA
RAD16.DATA
RAD32.DATA
RAD64.DATA
4x1x4 grid
8x1x8 grid
16x1x16 grid
32x1x32 grid
64x1x64 grid
• 128 x 1 x 128 grid takes half a day to
run – so not provided
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•
•
•
•
•
•
•
Creation of the Radial Grid
Spacing
• In ECLIPSE – radial grid
• Required input: INRAD – internal (well)
radius and OUTRAD – outside radius and
NR – number of grid blocks in the radial
direction
• ECLIPSE will then calculate the radial grid
spacing with exponential growth
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• Have the option of letting simulator
calculate the radial grid spacing
Equations for Calculating
Exponential Growth Grid Spacing
ri
rw
re
nc
ni
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= outer radius of cell i
= well radius
= external radius
= number of cells
= number of cell i
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• The outer cell radius for cells
logarithmically spaced can be
calculated by:
ni
nc
 ni
 re  


r
e
ri = rw ⋅ exp ⋅ ln   
r
=
r
⋅
i
w
 
OR
n
r
 w 
 c
 rw 
Creation of the Radial Grid
Spacing
• The ∆r results for the 5 grids follow
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• For our case:
• INRAD = 0.22 meters
• OUTRAD = 500 meters
∆r Values for NR = 4 – Increasing
Exponentially
Next to the well
8.969
61.928
427.584
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1.299
Outer most grid block
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∆r Values for NR = 8 – Increasing
Exponentially
Next to the well
0.941
2.472
6.497 17.071
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0.358
44.857 117.868 309.716
Outer most grid block
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∆r Values for NR = 16 – Increasing
Exponentially
Next to the well
0.221
0.359
0.943
1.529
2.479
6.513 10.558
17.114
0.582
4.018
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0.137
27.742
44.971 72.897 118.167 191.549
Outer most grid block
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∆r Values for NR = 32 – Increasing
Exponentially
Next to the well
0.060 0.077 0.097 0.124 0.158 0.201 0.256 0.326
2.865 3.648 4.645 5.913 7.529 9.586 12.204
15.538 19.783 25.187 32.068
40.829 51.983 66.184 84.265 107.285
Outer most grid block
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0.415 0.528 0.673 0.856 1.090 1.388 1.768 2.250
∆r Values for NR = 64 – Increasing
Exponentially
Next to the well
0.0282 0.0319 0.0360 0.0406 0.0458 0.0516 0.0583 0.0658
0.0742 0.0837 0.0945 0.1066 0.1203 0.1357 0.1531
0.4540 0.5123 0.5781 0.6523 0.7360 0.8305 0.9371
1.0574 1.1931 1.3462 1.5190 1.7140 1.9340 2.1822 2.4623
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0.1728 0.1950 0.2200 0.2482 0.2801 0.3161 0.3566 0.4024
2.7784 3.1350 3.5374 3.9914 4.5037 5.0818 5.7341
6.4701 7.3006 8.2376 9.2950 10.4880 11.8342 13.3532 15.067
17.0012 19.1834 21.6457 24.4240 27.5590 31.0963 35.0877
39.5914 44.6732 50.4073 56.8773
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Outer most grid block
50
∆z Values for the 4 Grids
nz = 4
∆z = 50 meters
nz = 8
∆z = 25 meters
nz = 16 ∆z = 12.5 meters
nz = 32 ∆z = 6.25 meters
nz = 64 ∆z = 3.125 meters
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•
•
•
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Grids for Radial Coning Problem
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64 x 1 x 64 Grid
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RAD4.DATA from FloViz
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RAD8.DATA from FloViz
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RAD16.DATA from FloViz
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RAD32.DATA from FloViz
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RAD64.DATA from FloViz
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Numerical Dispersion
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• Numerical Dispersion can occur in
both the r and z direction
• By decreasing ∆r and ∆z the
numerical dispersion from both
discretization errors will decrease
Your Task
View the 5 data sets with an editor
Run the 5 data sets with ECLIPSE
Plot the results
Analyze the effect of numerical
dispersion on the Gas-Oil Ratio
• We will discuss the results in class
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•
•
•
Results with 128x1x128 Grid
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Question?
• Which refinement is more important?
• For this gas coning situation?
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– Refinement in the radial direction?
– OR refinement in the vertical direction?
To Answer the Question
• We assume that the 64 x 1 x 64
numerical is “close to” the solution.
– radial-fine.data – refined in the radial
direction only – 64 x 1 x 8 grid
– vertical-fine.data – refined in the vertical
direction only – 8 x 1 x 64 grid
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• We will run two cases:
Guess Which Refinement is More
Important
• Then run the cases, plot with fine.grf
• Did you guess correctly?
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• Before you run the 2 cases – try to
estimate / guess which refinement
case will be closer to the 64 x 1 x 64
very fine grid results.
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End of Workshop Problem 1