Final Exam TPG4160 Reservoir Simulation, June 4, 2005 page 1 of ...

Final Exam
TPG4160 Reservoir Simulation, June 4, 2005
page 1 of 4
NORGES TEKNISK-NATURVITENSKAPELIGE UNIVERSITET
INSTITUTT FOR PETROLEUMSTEKNOLOGI OG
ANVENDT GEOFYSIKK
Contact person during exam:
Jon Kleppe 73 59 49 33
918 97 300 (mobil)
FINAL EXAM IN COURSE TPG4160 RESERVOIR SIMULATION
Friday, May 30, 2008
Time: 0900-1300
Grades will be available on: June 20, 2008
Allowed material D: -Type-approved calculator, with empty memory, according to
NTNU´s list of approved calculators
-No printed or hand-written material

Final Exam
TPG4160 Reservoir Simulation, June 4, 2005
page 2 of 4
Question 1 (27 points)
Explain briefly the following terms as applied to reservoir simulation (short sentence and/or a
formula for each):
a) Control volume
b) Mass balance
c) Taylor series
d) Numerical dispersion
e) Explicit
f) Implicit
g) Stability
h) Upstream weighting
i) Variable bubble point
j) Harmonic average
k) Transmissibility
l) Storage coefficient
m) Coefficient matrix
n) IMPES
o) Fully implicit
p) Cross section
q) Coning
r) PI
s) Stone´s relative permeability models
t) Discretization
u) History matching
v) Prediction
w) Black Oil
x) Compositional
y) Dual porosity
z) Dual permeability
Question 2 (22 points)
For a completely water-wet system, make sketches of saturation functions (including labels
for important points/areas)
a) Oil-water system: imbibition and drainage k rw
, k row
, P cow
vs. S w
b) Oil-gas system: imbibition and drainage k rg
, k rog
, P cog
vs. S g
c) Typical contours of three-phase k ro
in a ternary (triangular) diagram (axes S o
, S w
, S g
)
Then, answer the following questions:
d) Make sketches of Black Oil fluid properties B
o, Bg
, Bw,
Rso,
µ
o,
µ
g
, µ
w
. Label bubble
point pressure, and saturated and undersaturated regions.
e) Write expressions for reservoir densities for the three fluids (oil, gas, water) in terms of
B , B , B , R , ! , ! , ! .
o
g
w
so
oS
gS
wS
f) Write the expression for the density of the part of the reservoir oil that remains liquid at the
surface.
g) Write the expression for the density of the part of the reservoir oil that becomes gas at the
surface.
h) Derive an expression for the gas density based on the real gas equation.
i) Write the definition for fluid compressibility.

Final Exam
TPG4160 Reservoir Simulation, June 4, 2005
page 3 of 4
j) Write an expression for pore compressibility
k) Write Darcy´s equations for oil, gas and water in linear coordinates for an inclined system
(angle α)
Question 3 (14 points)
For the three-dimensional grid system:
1
i,j,k-1
k
i-1,j,k
i,j,k
i+1,j,k
i,j,k+1
N z
1
i
N x
1
j
N y
the set of linear equations to be solved for pressures has 7 non-zero diagonals on the left hand
side. If the grid block numbering sequence is first along the x-axis, then along the y-axis,
and finally along the z-axis, answer the following questions:
a) Write the set of linear equations in terms of pressures and coefficients a, b, c, d, e, f, g, h,
all with 3 subscripts.
b) Sketch the coefficient matrix using solid lines marked with coefficient name (the
coefficients used in a) above should be used in the sketch).
c) Give an expression for the bandwidth of the system.
Question 4 (14 points)
Answer the following questions related to the derivation of reservoir fluid flow equations:
a) Write the mass balance equation (one-dimensional, one-phase)
b) List 3 commonly used expressions for relating fluid density to pressure
c) Write the most common relationship between velocity and pressure, and write an
alternative relationship used for high fluid velocities.
d) Write the expression for the relationship between porosity and pressure.
e) Derive the following partial differential equation (show all steps):
!
!x
" k !P%
"
$ ' = ( c r
# µB !x & B + d(1/B) %
# dP &
!P
!t

Final Exam
TPG4160 Reservoir Simulation, June 4, 2005
page 4 of 4
Question 5 (13 points)
The discretized form of the oil equation may be written as
Txo i +1/ 2
(P oi +1
! P oi
) + Txo i!1/ 2
(P oi!1
! P oi
) ! q " oi
= C poi
(P oi
! P t t
oi
) + C soi
(S wi
! S wi
)
a) What is the physical significance of each of the 5 terms in the equation?
Using the following transmissibility as example,
Txo i !1 / 2
= 2k i !1 / 2" oi!1 / 2
#x i (#x i
+ #x i !1 )
b) What type of averaging method is normally applied to absolute permeability between grid
blocks? Why? Write the expression for average permeability between grid blocks (i-1) and
(i).
c) Write an expression for the selection of the conventional upstream mobility term for use in
the transmissibility term of the oil equation above for flow between the grid blocks (i-1) and
(i).
d) Make a sketch of a typical Buckley-Leverett saturation profile resulting from the
displacement of oil by water (ie. analytical solution). Then, show how the corresponding
profile, if calculated in a numerical simulation model, typically is influenced by the choice
of mobilities between the grid blocks (sketch curves for saturations computed with upstream
or average mobility terms, respectively).
Question 6 (10 points)
Thise questions are related to exercises 3 and 5, where gas is injected into an undersaturated
oil reservoir (ODEH case).
a) For exercise 3, make a qualitative sketch shoving pressure close to the injector vs. time for
the following two cases: 1) the injected gas is dissolved in the oil instantaneously until oil
pressure and bubble point pressure are equal, and 2) the injected gas is NOT dissolved in
the oil. Explain the differences.
b) In exercise 5 you compared results of simulations for the course grid (10x10x3) and a
refined grid (20x20x10), where in both cases injected gas is not dissolved in the oil. Make
a qualitative sketch over producert oil and gas vs. time for the course grid simulation (3
layers) and the fine grid simulation (10 layers). Explain the differences.