Applied Reservoir Simulation

Course
Workshop Problems
Reservoir Simulation Application Training

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Course
and (Eclipse) Workshop
GeoQuest Training and Development,
And NExT
Denver and Houston

March 06 Applied Reseervoir Simulation Day 1 1

 Outline of Workshop Problems

Problem 1:
A. IMPES and Implicit Comparison
B. Time Truncation Tests

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C. Numerical Dispersion

March 06 Applied Reseervoir Simulation Day 1 2

 Outline of Workshop Problems

2. Problem 2:
A. Water Coning Critical Coning Rate
B. Water Influx History Matching Kh

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C. Water Coning
I. History Matching Kv
II. Creation of Pseudo Krw in Coarse Grid to
Match Coning
D. Vertical Equilibrium Comparison

March 06 Applied Reseervoir Simulation Day 1 3

 Outline of Workshop Problems

3. Problem 3: History Matching

Primary Production
A. Sensitivity Simulations

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B. History Match
C. Predictions

March 06 Applied Reseervoir Simulation Day 1 4

 Schlumberger Public
Problem 1

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 Problem 1 Part A: IMPES Fully

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Implicit Comparison Coarse Grid
IMPES Sub-directory

March 06 Applied Reseervoir Simulation Day 1 6

 Reservoir Details

We have a linear reservoir 5000 x 100

x 30 feet. Grid is 50 x 1 x 1
Water injector at one end and

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producer at other end
Two phase (oil-water) system
Oil viscosity = 2 cp and water
viscosity = 0.5 cp
Simulate 2000 days.

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 FloViz View

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 Choice in Solution of the
Equations
IMPES Implicit Pressure Explicit
Saturation
Linear problem smaller

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Through put limitations: 5 10% PV
Stability problems
Timestep size - small

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 Choice in Solution of the
Equations
Fully Implicit
Unconditionally Stable large timesteps
Timesteps controlled by time truncation

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error
Few long timesteps

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 Comparison IMPES and Fully
Implicit Solution
Amount of Numerical Dispersion
(Error) in the results
Work to obtain solution

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Number of linear iterations
Number of non-linear (Newton)
iterations
CPU time per time step
Total CPU time

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 Iteration Process in Reservoir
Simulation

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Example of linear
and non-linear
iteration process:
4 non-linear
iterations Usually a non-linear iteration
requires 10 to 30 linears to
converge pressure and saturations

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 Data Set Names

IMPES.DATA uses IMPES solution

FULLIMP.DATA uses fully implicit

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solution
Location sub-directory:
Problem 1/IMPES

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 View Results: Comparison

Oil Production Rate

Production Water Cut
Map of Water Saturations

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At 500 Days
At 1000 Days
Number of Linear Iterations in Each
Timestep
Sum of Linear Iterations

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 View Results: Comparison

Newton (non-linear) Iterations in

each timestep
Sum of Newton Iteration

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Timestep Length
Note: Fully Implicit takes 31 timesteps
IMPES takes 207 timesteps
CPU Time Per Timestep
Total CPU Time

March 06 Applied Reseervoir Simulation Day 1 15

 Your Task

View the .data sets

Difference IMPES (in Schedule
Section) Fully Implicit (E100 Default)

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Run the data sets with ECLIPSE 100
Plot the results
Study the results and answer the
following questions

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 Questions

Which technique takes more linear

iterations per timestep?
Which technique takes more Newton

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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

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Implicit Comparison Fine Grid
IMPES 2 Sub-directory

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 Reservoir Details

We have the same reservoir, fluids,

etc. except that now we will sub-
divide the grid into 2000 blocks in the

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x-direction instead of the previous
50.

The length of the grid blocks are now

2.5 feet instead of the previous
length of 100 feet.

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 Run and Compare Results

Run the IMPES.DATA and

FULLIMP.DATA in the IMPES2 sub-
directory and run Graf in the IMPES2

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sub-directory.

With these smaller grid blocks the

problem is numerically more difficult,
but the numerical dispersion is less.

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 Questions

Answer the same questions from the

previous part.
Note the differences in the IMPES

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and Fully Implicit runs.
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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 Conclusion

For a blackoil simulation with

ECLIPSE we will prefer to use the
default solution Fully Implicit since

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the results are much more stable and
the CPU time is less.

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Problem 1 Part B: Time Truncation

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Test: Quarter 5-Spot Water Flood

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 Time Truncation Error

We have the accumulation term in

our equations, for example
So

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t B o
When it is discretized we get terms
as follows S on+1 - S on
( x y z ) i


B w i t i

(1 - S oi ) P o in+1 - P o in
- ( x y z )i (1 )C f
B wi t

d (1/ B w )i P on+1 - P on P cow n+1 - P cow n

+ i (1 - S oi )
-
d p t t i

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 Time Truncation Error

This discretization process yields

and error that is

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O(t)
in the solution of the equations.
This error is called Time Truncation
Error

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 Simulation Situation

11x11x1 quarter 5-spot involving

water injection
Diagonal Grid

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1452 x 1452 x 50 meter flow field
K = 830 mD, = 0.17
Water Injector on water rate control
Producer on liquid rate control
Simulate 3660 days

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 Grid

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March 06 Applied Reseervoir Simulation Day 1 27
 Time stepping
Three data sets are provided with
different time stepping regimes
maxts.data initial timestep = 10 days,
grows to max timestep = 183 days, 53

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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

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 Questions to Consider While
Viewing the Results
What is the effect of solving the flow
field with a few large timesteps?
What is the most accurate solution?

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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?
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 Results to be Viewed with Graf

Timestep length
Water cut effect of time truncation
error

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Oil production rate effect of time
truncation error
Water saturation maps
CPU time per timestep
Total CPU time

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 Your Task

View the .data sets

Difference in the TUNING Keyword
timestep control

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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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 My New Computer and ECLIPSE
2004a runs so fast has trouble

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storing timestep lengths

Following are results from old

Pentium III computer

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 CPU Time per Timestep in
Seconds

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March 06 Applied Reseervoir Simulation Day 1 33
 Total CPU Time

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March 06 Applied Reseervoir Simulation Day 1 34
 Problem 1 Part C: Numerical

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Dispersion: Radial Model with Gas
Coning

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 Numerical Dispersion

Definition: The error that occurs as

we discritize the partial differential
equations on a grid to create the

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finite difference equations.

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 Finite Difference Approximations
to First Derivatives
forward difference
f f (x + x) f (x)
=
x x

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Error term:
x f x f 2 2 3
.....
2! x 2
3! x 3
forward error

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 Objective

To see the effect of number of grid

block on a simulation model, a 2-D
radial (r, z) model was set up

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The reservoir has constant
properties; see data set RAD4.DATA
(rad4.data)

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 Description

Metric Units
Oil, Water, Gas, Dissolved Gas
The reservoir is 200 meters thick with

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the top 50 meters in the gas cap
Flat Top at 2950 meters
The width (radius) of the model is
500 meters

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 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

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 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

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 Data Sets Provided
Five data sets are provided for you:

RAD4.DATA 4x1x4 grid

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RAD8.DATA 8x1x8 grid
RAD16.DATA 16x1x16 grid
RAD32.DATA 32x1x32 grid
RAD64.DATA 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

Have the option of letting simulator

calculate the radial grid spacing

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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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 Equations for Calculating
Exponential Growth Grid Spacing
The outer cell radius for cells
logarithmically spaced can be
calculated by: ni

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ni re r nc
ri = rw exp ln ri = rw
e
n
c r
w OR
rw

ri = outer radius of cell i

rw = well radius
re = external radius
nc = number of cells
ni = number of cell i
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 Creation of the Radial Grid
Spacing
For our case:
INRAD = 0.22 meters
OUTRAD = 500 meters

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The r results for the 5 grids follow

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 r Values for NR = 4 Increasing
Exponentially

Next to the well

1.299 8.969 61.928 427.584

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Outer most grid block

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 r Values for NR = 8 Increasing
Exponentially
Next to the well

0.358 0.941 2.472 6.497 17.071

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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

0.137 0.221 0.359 0.582

0.943 1.529 2.479 4.018

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6.513 10.558 17.114 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

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0.415 0.528 0.673 0.856 1.090 1.388 1.768 2.250

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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 r Values for NR = 64 Increasing
Next to the well Exponentially
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

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0.1728 0.1950 0.2200 0.2482 0.2801 0.3161 0.3566 0.4024

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

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

Outer most grid block
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 z Values for the 4 Grids

nz = 4 z = 50 meters
nz = 8 z = 25 meters
nz = 16 z = 12.5 meters

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nz = 32 z = 6.25 meters
nz = 64 z = 3.125 meters

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 Grids for Radial Coning Problem

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March 06 Applied Reseervoir Simulation Day 1 52
 64 x 1 x 64 Grid

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March 06 Applied Reseervoir Simulation Day 1 53
 RAD4.DATA from FloViz

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March 06 Applied Reseervoir Simulation Day 1 54
 RAD8.DATA from FloViz

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March 06 Applied Reseervoir Simulation Day 1 55
 RAD16.DATA from FloViz

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March 06 Applied Reseervoir Simulation Day 1 56
 RAD32.DATA from FloViz

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March 06 Applied Reseervoir Simulation Day 1 57
 RAD64.DATA from FloViz

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March 06 Applied Reseervoir Simulation Day 1 58
 Numerical Dispersion

Numerical Dispersion can occur in

both the r and z direction
By decreasing r and z the

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numerical dispersion from both
discretization errors will decrease

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 Your Task

View the 5 data sets with an editor

Run the 5 data sets with ECLIPSE
Plot the results

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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?

Refinement in the radial direction?
OR refinement in the vertical direction?

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For this gas coning situation?

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 To Answer the Question

We assume that the 64 x 1 x 64

numerical is close to the solution.

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We will run two cases:
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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 Guess Which Refinement is More
Important
Before you run the 2 cases try to
estimate / guess which refinement
case will be closer to the 64 x 1 x 64

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very fine grid results.

Then run the cases, plot with fine.grf

Did you guess correctly?

March 06 Applied Reseervoir Simulation Day 1 64

 Schlumberger Public
End of Workshop Problem 1

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