Petroleum Engineering 603
Final Exam
August 13, 2002
Name:_________________________________________
You may use three sheets of handwritten notes, attach to exam when turned in.
You may use a calculator.
Do not fold or unstaple the examination booklet.
Time allotted for the examination is 120 minutes.
When you have completed the examination, read and sign the statement below, then
turn in the examination booklet with handwritten note sheets attached.
STATEMENT OF ACADEMIC INTEGRITY
I pledge that I have neither given nor received aid in completing this examination. I
have followed the strictures of the Texas A&M University Aggie Code of Honor during
this examination.
Signature:
____________________________________
1
1) (20 points) We have previously considered the “Black Oil” fluid model. For this fluid model,
the liquid hydrocarbon phase (reservoir oil) is made up two mass species, stock tank oil and stock
tank gas (remember, stock tank conditions are standard temperature and pressure). For this fluid
model, the vapor hydrocarbon phase (reservoir gas) is made up of a single mass species, stock
tank gas. The amount of stock tank gas contained in the liquid hydrocarbon phase depends on
the fluid property, Rs (units of scf/STB).
For volatile oil reservoirs at temperatures near the critical temperature, this fluid model is not
adequate, due to the vaporization of stock tank oil in the vapor hydrocarbon phase. One
approach is to use fully compositional simulation. A simpler approach that can sometimes be
applied very successfully is the use of a “Volatile Oil” fluid model.
For the Volatile Oil fluid model, the liquid hydrocarbon phase is made up of two mass species,
stock tank oil, and stock tank gas. For the Volatile Oil fluid model, the vapor hydrocarbon phase
is also made up of the same two mass species, stock tank oil and stock tank gas. The amount of
stock tank gas contained in the liquid hydrocarbon phase depends on the fluid property, Rs (units
of scfgas/STBoil). The amount of stock tank oil contained in the vapor hydrocarbon phase depends
on the fluid property, Rso (units of STBoil/scfgas).
Your task is to derive the gridblock material balance equations that describe the flow of
stock tank oil and stock tank gas in the reservoir.
Assumptions:
- Reservoir is 1-D linear
- Two phase reservoir flow (liquid hydrocarbon phase and vapor hydrocarbon phase)
- Reservoir is isothermal (Rs and Rso depend on pressure only)
Use an approach that conserves mass. Your result should be two equations relating the pressures
and saturations in gridblocks: i-1, i, and i+1. You do not need to write this out in a matrix
format or discuss how the mass balance equations could be solved. Note that 1 STB = 5.615 scf.
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(Extra Page)
3
(Extra Page)
4
(Extra Page)
5
2a) (10 points) What is the fundamental difference between iterative and direct matrix solvers?
Discuss in 2 or 3 sentences.
2b) (10 points) What are the advantages and disadvantages of direct and iterative matrix solvers?
Consider accuracy, stability, computational cost and any other issues you see as relevant.
Discuss in sentences/bullet points. If you use equations make it clear that you understand the
implications of the equations.
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3) (20 points) Upscaling is a process by which we hope to match reservoir performance
calculated on a fine grid using a more coarsely gridded model. In this case we are going to
consider a two-phase (oil and water) model and attempt to match the well production rates
between the fine and coarse grids. Porosity, water saturation, absolute permeability and relative
permeability for the well connection will be upscaled.
Consider flow from a coarse grid block into a fully penetrating well centered in the gridblock.
The coarse grid block is comprised of three fine grid layers.
Fine Grid Block 1
Fine Grid Block 2
Fine Grid Block 3
Upscaling
Coarse Grid Block
Hint: Begin by expressing the flow rate of oil and water from each fine grid block into the well
as a function of the fine grid block properties. Next, express the total flow rate as a function of
the coarse grid block properties. Setting the rate in both models to be the same will allow you to
determine the coarse grid block properties.
Use overbars to represent upscaled properties, e.g. k is the permeability of the coarse grid block,
and k 1 is the permeability of fine grid block #1.
Assumptions:
 Upscale the horizontal permeability of the coarse grid block assuming parallel, horizontal,
radial flow toward the well
 Assume that the pressure of the coarse grid block is the pore volume weighted average
pressure of the fine grid blocks.
 Flowing wellbore pressure, pwf =1980. psia
 Each fine grid block has x = y = 100 ft, and thickness as shown below.
 All pressures corrected to a datum (no gravity correction is necessary in reservoir or in the
wellbore).
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

Wellbore radius = 0.25 ft, Water viscosity = 1 cp, oil viscosity = 10 cp, Bo=1.2 RB/STB,
Bw=1.0 RB/STB, no dimensionless skin (s = 0).
Recall for uniform grid (square grid blocks, isotropic permeability):
J model 
2 0.00633kh
ln ro rw   s
 cp  rcf
 d  psi


 ; ro  0.2 x

We have the following fine grid data (at a certain timestep). Note that only using data from one
timestep means we can only upscale one point on the relative permeability curve. Also note that
the upscaled relative permeability would be rate dependent.
Block
1
2
3
h, ft
100.
150.
50.
Porosity
0.15
0.17
0.20
k, md
100.
150.
200.
P, psia
2030.
2021.
2010.
You may find this blank table useful as workspace.
Bloc
k
1
2
3
Bloc
k
1
2
3
Bloc
k
1
2
3
Enter your final answers here (show units):
Upscaled Porosity, 
Upscaled Water Saturation, S w
Upscaled Horizontal Permeability, k
Upscaled Well Relative Permeabilities,
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Sw
0.70
0.75
0.80
kro
0.30
0.20
0.10
krw
0.45
0.55
0.70
k ro,well and k rw,well
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(Extra Page)
10
(Extra Page)
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4a) (10 points) What is the fundamental principle that allows streamline simulation to model 3-D
reservoir flow?
4b) (10 points) List the major reservoir/production parameters that are advantages and
disadvantages with regard to suitability of streamline modeling. Consider accuracy,
computational cost and any other issues you see as relevant. Discuss in sentences/bullet points.
If you use equations make it clear that you understand the implications of the equations.
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5) (20 points) Perform two complete Gauss Seidel Method matrix solution iterations (sweeps) for
the following system of linear equations. Reordering of the equations may be necessary. Use
initial values of (x1=x2=x3=1).
x1 + 4x2 + 2x3 = 4
x1 - x2 - 3x3 = -6
4x1 + 2x2 - x3 = 3
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