Tom Aage Jelmert
NTNU
Department of Petroleum Engineering
and Applied Geophysics
A NOTE ON UNIT SYSTEMS
Introductory remarks
So far, all equations have been expressed in a consistent unit system. The SI unit
system is consistent. A consistent unit system is one that respects the dimension of a
variable. For instance, dimensional analysis shows that the dimension of permeability
is L2 (like an area). The corresponding SI unit is m2. Such a unit system leads to
dimensionless constants in any equation. The advantage is transparency. It becomes
easy to check whether an equation is dimensional consistent or not.
No unit system can be flexible enough to suit all practical purposes. Hence key
variables may assume inconvenient numerical values. For example, pressure may
assume the value of 3000 psi or 2.1⋅108 Pa depending on the unit system. In the
same unit systems, the permeability could assume either the value of 1 mD or
1 ⋅10-16 m2. The magnitudes of the first values are considered more “practical” than
the latter. The reason is that the numerical values fall within a range that most people
are used to handle. This was an important aspect in the old days when the most
advanced engineering calculating machine was the slide rule. A practical unit system
is designed to give convenient numerical values. This objective may be achieved at
the expense of introducing dimensional constants into the equations.
The traditional way of finding permeability is by core analysis. It is not surprising
then that the unit for permeability, Darcy denoted by D, is based on the cgs
(centimeter, gram, second) system. This system is convenient for laboratory work. In
addition it has the advantage of accommodating dimensionless constants in many
problems involving flow in porous media. The resulting unit system has been called
Darcy units. The Darcy unit system is inconvenient for oilfield calculations. This is
because the basic units are too small. A practical unit system can be based on the
metric system or the traditional British system.
Variable
Formation Factor B
Compressibilty
c
h
Height
k
Permeability
p
Pressure
f
Porosity
q
Rate
r
Radius
t
Time
Viscocity
Darcy
dimless
1/atm
cm
D
atm
dimless
cc/s
cm
s
cp
American
dimless
1/psi
ft
mD
psi
dimless
STB/Day
ft
h
cp
SI
dimless
1/Pa
m
m^2
Pa
dimless
m^3/s
m
s
Pas
Table 1: Darcy, SI and American Field Units
1
Properties of the SI unit system
To illustrate the advantage of a consistent unit system let us check the dimensional
consistency of the drawdown equation for radial flow:
pi − pwf =

1.15q µ B 
kt
+ 0.351 + 0.87 S 
log
2
ϕµ ct rw
2π kh 

The units used for the SI unit system are listed table 1.
The units may be included in the equation. These are enclosed within square brackets
to distinguish them from the corresponding variable.
 m3 
1.15q   µ [ Pa ⋅ s] B 

k m2  t [ s]
s 


+
+
S
( pi − pwf ) [ Pa] =
log
0.351
0.87


2
2
−1
2π k m2  h [ m]
 ϕµ [ Pa ⋅ s] ct  Pa  rw m 

Inspection of the above equation will clarify the dimensional consistency of the
equation. By cancellation of common units, it becomes clear that the argument in the
logarithmic expression is dimensionless. By the same argument, the coefficient to the
parenthesis has the unit Pa which is the unit of pressure. The criterion for dimensional
consistency is that the left and right hand side of the equation is characterized by the
same unit. Thus the dimensional consistency of the equation is easily verified.
Dimensionless variables
The argument of the logarithmic term is a dimensionless function of time. The group
has been called dimensionless time.
tD =
kt
ϕµ ct rw2
A dimensionless pressure group may be defined in the same way. The ratio of the
pressure drawdown, ∆pwf, to the pressure group on the right hand side of the
equation is obviously dimensionless. Hence a dimensionless pressure function, pDfunction, may be defined as:
pD =
Tom Aage Jelmert
2π kh
∆pwf
qµ B
NOTES ON UNIT SYSTEMS 2
The drawdown equation becomes simplified when expressed in terms of
dimensionless variables.
p D = 1.15 {log t D + 0.351 + 0.87 S }
The use of dimensionless equations is important for the theory of type curve
matching.
Conversion from the Darcy unit system to American field units.
As an example we select the drawdown equation for radial flow in Darcy units. These
are listed in Table 1.:
pi − pwf =

1.15q µ B 
kt
+ 0.351 + 0.87 S 
 log
2
2π kh 
ϕµ ct rw

As noted previously, all the constants in the above equation are dimensionless.
We recognize the argument of the logarithmic term as dimensionless time. The first
task is to calculate dimensionless time in American Field Units.
From any table of conversion factors, the following relations may be obtained:
1 [ atm ] = 14.7 [ psi ]
1
1  cm3  =
[bbl ]
15900
1 [ D ] = 1000 [ mD ]
1 [ cm ] = 1/30.48
1 [s] =
1
[h]
3600
[ ft ]
Note that a permeability expressed in mD may be converted to Darcy by
multiplication with the factor 0.001. This factor has hidden units. These are:
 D 
0.001 
 md 
All conversion factors have associated units. This is an important aspect when
converting from one unit system to another.
For the sake of clarity we specify the unit of each variable in dimensionless time. The
units are listed in Table 1.
tD =
k [ D]t [ s]
ϕµ [ cp ] ct  atm −1  rw2 cm 2 
The inclusion of the units highlights the disadvantage of any practical unit system. By
inspection of the units alone, it is not obvious that the group on the right hand side of
the above equation is dimensionless.
Tom Aage Jelmert
NOTES ON UNIT SYSTEMS 3
Next we introduce the American field units in the dimensionless time. With each
variable we associate a conversion constant with the objective of converting the
practical unit back to the corresponding Darcy unit.
1  D 
s
t
h
3600
[
]
 h 
0.000264k [ mD ] t [ h ]
1000  mD 
=
tD =
2
psi  2
ϕµ [ cp ] ct  psi −1  rw2  ft 2 

−1
2
2  cm 
ϕµ [ cp ] ct  psi  14.7 
rw  ft  30.48  2 
 atm   
 ft 
k [ mD ]
It may seem like a paradox that dimensional time involves all the units listed above.
The reason that the group is dimensionless is that the constant, 0.000264, involves a
complicated expression of units that will make the group dimensionless.
The equation for dimensional time is correctly (since it is dimensionless) presented
without the confusing units. Then the equation becomes:
tD =
0.000264kt
ϕµ ct rw2
The next objective is to convert the coefficient to the logarithmic term by the same
procedure. Including the Darcy units in the equation:
 cm3 
1.15q 
 µ [ cp ] B
 s 
2π k [ D ] h [ cm ]
Substitution of the conversion constants yields:
 cm3 
 STB 
1
 Day 
µ [ cp ] B 11.061⋅ q  STB  µ [ cp ] B
0.183 ⋅ q 
159000





 Day 
 STB  24 ⋅ 3600  s 
 D 
=
k [ mD ] h [ ft ]
 cm 
1  D 
k [ mD ]
h [ ft ] 30.48  


1000  mD 
 ft 
Again the disadvantage of a practical unit system is conspicuous. Understanding the
overall unit of the group of variables by inspection is difficult.
The best way to find out is to look back to the original drawdown equation. The
terms within bracket are clearly dimensionless. This is a necessary conclusion since
the skin factor, S, is dimensionless. The pressure drawdown on the left hand side of
the equation has the unit [ atm ] . On the right hand side of the equation we have we
have sum of dimensionless terms multiplied by a group of constant with overall unit
[ atm] .
Tom Aage Jelmert
NOTES ON UNIT SYSTEMS 4
The last term to convert is the pressure drawdown on the left hand side of the
equation.
∆pwf [ psi ]
1  atm 
= 0.068∆pwf [ psi ]
14.7  psi 
Substitution of the above expressions into the original drawdown equation yields:
pi − pwf =

162.6qµ B   0.000264kt 
log 
 + 0.351 + 0.87 S 
2
kh
  ϕµ crw 

In the same way one may convert the dimensionless pressure group to find
pD =
0.0071kh
∆pwf
qµ B
Conclusion
The SI unit system is consistent. As such it has the advantage of dimensional
transparency and the use of dimensionless constants. Key variables may assume
inconvenient numerical values.
A practical system has the advantage of convenient numerical values. This advantage
is obtained at the expense of reduced dimensional transparency.
Tom Aage Jelmert
NOTES ON UNIT SYSTEMS 5