In one of my previous projects we had to perform some pressure tests of pipelines at 27.0
bar for 12 hrs using water. Being located on the south of the North Sea the temperature was
fluctuating during the day therefore the pressure tests have been influenced dramatically
although no leak was noticed. The test results were in jeopardy of not being accepted by the
client. In order to avoid a redo of the tests (with eventually the same results) and to
convince the client that the pipe lines have no leaks we did the mathematical model and we
compared the theoretical values with the recorder readings, as follows:
A.
Theoretical Considerations
Input data:
P0– initial pressure (N/M2)
P1 – final pressure (N/M2)
T0– initial temperature (oC)
T1 – final temperature (oC)
ρ0 – Water initial density
ρ1– Water final density
β - volumetric temperature expansion coefficient (m3/m3 oC)
E - bulk modulus fluid elasticity (N/m2)
The equation which expresses the relation between temperature and density of a fluid can
be expressed as:
ρ1 = ρ0 / (1 + β (T1 - T0))
(1)
The equation which expresses the relation between pressure and density of a fluid can be
expressed as
ρ1 = ρ0 / (1 - (P1 - P0) / E)
(2)
Some can write
(1)=(2) =>
ρ0 / (1 + β (T1 - T0)) = ρ0 / (1 - (P1 - P0) / E) (3) or
1 + β (T1 - T0)= 1 - (P1 - P0) / E (4) or
β (T1 - T0)= - (P1 - P0) / E (5) or
β (T1 - T0) + (P1 - P0) / E = 0 (6) or
P1= P0 - β E (T1 - T0) (7)
B.
Practical considerations of the pressure test.
1.
The pressure parameters (recorder readings):
•
P0– initial pressure (N/sqm) – 2740000
•
P1 – final pressure (N/sqm) - 2414000
•
T0 – initial temperature (oC) – 9.7
•
T1 – final temperature (oC) - 8
2. The recorders used in pressure test were class 0.6 which means accuracy of +/- 0.6%.
We can consider the worst case scenario meaning:
T0= 9.7C +(9.7C x 0.006)=9.7582C
T1 = 8C -(8C x 0.006)=7.952C
P0= 2740000 + (2740000 x 0.006)= 2756440 N/sqm
3. Conducting the mathematical calculations in equation (7) the result is P1= 2418360
N/sqm (theoretical)
4. Due to visual accuracy in reading the diagram the input readings may be erroneous
leading to less accuracy in input data
5. The small bore diameter of the hose (1/4 inch) and its length, between the manifold
(where the recorder was connected) and the tested pipeline, the pressure recording could
have been affected.
6. Water volumetric temperature expansion coefficient (β) is influenced by the
temperature as it is shown below. Note that at 20C it is more than 2 times bigger than at
10C.
water at 0C: -0.000050 (1/C)
·
water at 4oC: 0 (1/C)
·
water at 10C: 0.000088 (1/C)
·
water at 20C: 0.000207 (1/C)
·
water at 30C: 0.000303 (1/C)
·
water at 40C: 0.000385 (1/C)
·
water at 50C: 0.000457 (1/C)
·
water at 60C: 0.000522 (1/C)
·
water at 70C: 0.000582 (1/C)
·
water at 80C: 0.000640 (1/C)
·
water at 90C: 0.000695 (1/C)
7. Bulk Modulus fluid elasticity for water is influenced by the temperature as it is shown
below:
C. Conclusions
Although the pressure test starts at 27.4 bar after 12 hrs the pressure drops to 24.14 bar
due to temperature drop. The mathematical model shows a theoretical pressure drop to
24.18 bar which is consistent with the pressure reading on the recorder. The difference
between the mathematical model and the real conditions was only 0.04 bar. Client accepted
the results and the test.
The model works only with water as test fluid. For other fluids (liquids) adequate volumetric
temperature expansion coefficient and bulk modulus fluid elasticity shall be used. The model
does not work for gases.
Lower start/stop temperatures, lower the pressure drop. That is understandable
considering that volumetric temperature expansion coefficient at 10C is 3.4 times smaller
than at 30C. We recommend that the tests to be performed at lower temperatures (during
the night in hot areas like ME) to avoid big pressure drops and possible suspicions about the
accuracy and the result of the tests.
The total volume of the water does not have any influence in pressure fluctuation.