5. LA ESPERANZA COMO ACTITUD VITAL: CUATRO CUARTETOS
5.2. East Coker
There are three sources of tension (or compression) in a casing string.
• Gravitational forces (weight)
• Borehole friction
• Bending
The tension in a casing string at any point due to gravity or weight is a function of the buoyancy of the drilling fluid and by the inclination of the wellbore. The tension due to borehole friction is a function of gravity, buoyancy, wellbore inclination and curvature, and also the tension in the pipe. (In the case of a curved wellbore, the tension is a function of the friction, but the friction itself is also a function of the tension). We are not going to consider directional wells or borehole friction at this time.
Here are some of the considerations in designing casing to consider the axial load.
• Weight in air or buoyed weight?
• Over pull or design factor?
• Borehole friction?
When we work with casing in a wellbore we must consider its weight and the amount of tension in the string due to that weight. What measure we use for the weight? Do we use the weight of the casing in air or do we use the buoyed weight of the casing in the drilling fluid that is in the hole? As hard as it may be to believe, there is no universally accepted answer to that question in oilfield practice in the world today.
Many use the weight in air claiming that it gives an extra margin of safety. Others say the buoyed weight is more realistic.
Weight of Pipe
When it comes to the tensile design of casing there are two schools of thought. One is to use a design factor, e.g. 1.6, and the other is to use a specified amount of over pull, say 100,000 lb. It is quite common to use both and say that the design should incorporate whichever one leads to the strongest design. In cases where the design factor is the highest value that is usually the case only near the surface and the over pull will be greater near the bottom.
Design Factor
If we chose to use the weight of the casing in air, the design process is quite simple.
The problem is that it can lead to an over design of the string because the casing is almost never actually suspended in air. While this used to be a common practice it is seldom used by many today.
Axial Load
The buoyed weight of the casing in the drilling fluid is generally recognized as the standard approach for designing casing to withstand the axial loads in the pipe. There are two ways to go about this. One way is to use the true axial load and the other is to use the effective axial load.
• True axial load comes from the actual hydrostatic forces acting on the tube and is valid for all bodies
• Effective axial load comes from Archimedes principle which is only valid for rigid bodies
Let us examine a plot of the axial load of our surface casing string in air (un-buoyed), the true axial load in 9.2 ppg mud, and the effective axial load in 9.2 ppg mud.
Axial Load Curves
Notice the true axial load curve on the left. It is actually in compression at the bottom – because of the hydrostatic pressure on the cross-sectional are of the tube at the bottom. Notice also that at 2100 feet the curve shifts slightly. That is due to the difference in cross-sectional area of the 54.5 lb/ft and the 68 lb/ft casing at that point.
The tension increases, meaning that the net hydrostatic force is acting downward because the ID of the 68 lb/ft pipe below is smaller than the ID of the 54.5 lb/ft pipe above. Had the heavier pipe been on top, the curve would have shifted in the opposite direction.
Another thing to notice about these curves is that the un-buoyed load curve essentially parallels the true load curve. It is much easier to calculate manually since there are no differences in cross-sectional areas and hydrostatic pressures to calculate. That is why in the past many used this as the basis for their design (along with an appropriate design factor). In fact many still do use it, especially when doing manual calculations for calculating the axial load line.
We will now examine the method used for calculating the true axial load. Perhaps it is best to first look at a schematic of a casing string showing the change in cross-sectional areas and hydrostatic forces.
True Axial Load
We can write an equilibrium equation for the hydrostatic and gravitational forces in this diagram. Since the load is discontinuous at each change in cross-sectional are it requires two equations, one for the load just above each change of cross-sectional area and one just below each change. Above each change of area the equation is:
( )
1 1
1 1
for 0
k k
k o i i i i i
i i
P p A w L p A+ A k
= =
= − +
∑
+∑
− ≤ <n (5.1)Below each change in cross-sectional area and at the very top of the string the true axial load is given by:
( )
1
1 1
1 1
for 0
k k
k o i i i i i
i i
P p A w L − p A+ A k
= =
= − +
∑
+∑
− ≤ =n (5.2)where
node or section number
total number of sections in string k
n
=
=
For example if we desired to obtain the true axial load just above section 2 in the drawing above we would use Equation (5.1) as follows:
( ) ( )
2 o 1 1 1 2 2 1 2 1 2 3
P = −p A +w L +w L +p A −A + p A −A2
And the true axial load just below the top of section 2 is calculated using Equation (5.2) as follows:
( )
2 o 1 1 1 2 2 1 2
P = −p A +w L +w L +p A −A1
While that may seem a little complicated, it is very easy to do on a computer spread sheet.
Let us see how it works for our surface casing so far selected:
Type Casing Interval Section Length
13 3/8” 54.5# K55 ST&C 0 – 2100 ft 2100 ft 13 3/8” 68# K55 ST&C 2100 – 3000 ft 900 ft
We will make a table of the variables that go into the equations first to organize things a bit.
Section Weight Length Area Depth at
bottom Pressure at bottom
Lb/ft ft in2 ft psi
2 54.5 2100 15.513 2100 1005
1 68 900 19.445 3000 1435
Calculate the true axial load at the bottom of each section and at the top of the well using Equation (5.2)
( )
33, 296 54.5 2100 1005 19.445 15.513 151,698 lb
Next we use Equation (5.1) to calculate the true axial load just above the places where the cross-sectional area changes, and in this case there is only one at the top of the first section.
( )
( ) (
1 1 1 1 1 2 1
27904 68 900 1005 19.445 15.513 37, 248 lb
2 (btm) 2,100 1,005 15.513 37,248 1 (top) 2,100 1,005 19.445 33,296 1 (btm) 3,000 1,435 19.445 -27,904
This direct method appears in a number of sources, and is the generally accepted method for calculating the true axial load. We will see later that it has a flaw as it is presented here and in other sources.
Now the question will likely arise, what is the purpose if any of the effective load curve? If it is not the actual load in the casing, why do we even consider it? This load curve requires some discussion even though the load it portrays is strictly fictitious; in other words, except at the top of the string it does not even exist. First, you should be aware of what it is because it makes an appearance in a lot of oilfield applications. As we said before it comes from Archimedes principle, and usually involves the calculation of a buoyancy factor based on the difference between the density of the
Effective Axial Load
steel and the liquid it is in. It is very easy to calculate and we are going to use it for exactly that reason. Second, you should be aware that it is frequently misused in some oilfield applications. Archimedes principle was intended for use with rigid bodies and not with extensible bodies (things that deform). You can sometimes use it with extensible bodies, but you can also get into a lot of trouble if you are not careful.
Third, you should be aware that it does have legitimate uses that are quite important.
One legitimate use for the effective load curve is for determining lateral buckling of tubes in a wellbore due to axial loads. Contrary to what you might initially think the hydrostatic force on the end of a tube has no effect whatsoever on the lateral buckling of pipe as long as the density of the pipe is greater than the density of the fluid it is in.
Believe it or not, there was a time that some engineers were so confused by this that they thought it would be impossible to run a wire line into a deep well because the hydrostatic force on the bottom of the wire would cause it to buckle and not go below a certain depth. Most of us now know intuitively that such cannot possibly be the case, but there are still many who harbor some notion that it can cause drill pipe, drill collars, or even casing to buckle laterally. A typical misapplication along these lines is in the selection of the number of drill collars required to prevent buckling of the drill pipe.
For many years drillers calculated a buoyancy factor for the drill collars and calculated how many would be required to maintain a certain weight on the bit plus a few extras for safety. That was a method based on the effective axial load. Along in the 1970’s someone looked at the true axial load and decided that drillers had been doing it wrong all those years. They showed us all how we should calculate the true axial load as opposed to the effective axial load, and of course the net result was that we were not running enough drill collars. A lot of us affected by that revelation realized we had really been dumb in the past so we all changed our calculation method. Of course the new method always required more drill collars than the old method. Then some began to ask the question, if the old method was so obviously wrong why did it work?
Well, it worked because it was correct. A static liquid cannot sustain a shear force or a moment on a buckled tube in a wellbore as long as the density of the tube is greater than the density of the liquid. In other words, the hydrostatic forces acting axially along the tube can neither cause nor sustain a laterally buckled configuration. In fact use of the true axial load leads to the absurd result that you still need drill collars even when there is no weight on the bit. Unfortunately, some are still using the true axial load to determine the number of drill collars required. The rental tool companies love it.
Now there is another use of the effective axial load and that is to simplify the manual calculation of the true axial load. We can use it to calculate the true axial load with just a few steps. Some find it easier to calculate the effective load first then the true load from those results, and it has another advantage which we will mention later. Let us see how this procedure works.
First we calculate a buoyancy factor.
1 65.43
This formula can be used in any system of units as long as the density of steel is in the same units as the density of the mud. Other units are given in Chapter 12.
To illustrate the process we will use our 13 3/8 in. surface casing string we have thus far designed for collapse and burst. So for our casing string the buoyancy factor is:
1 9.2 0.8594
65.43
fb = − =
To calculate the buoyed weight of a section of pipe we multiply the weight in air by the buoyancy factor.
In our casing string we have:
Weight Section
54.5 2100 114,450 175,650 0.8594 150,954
68 900 61,200 61,200 0.8594 52,595
Next we calculate the true axial load using the following equation.
m t
P P= ′−p A (5.4)
where
(
2 2)
true axial load, lb effective axial load, lb
0.052 pressure of the mud, psi density of the mud, ppg
true vertical depth, ft
cross sectional area of tube, in 4
inside diameter of tube, in.
di =
2
We will use a table to show the calculations.
Section Effective
Whether it is easier to do the calculations this way or directly using Equations (5.1) and (5.2) is a matter of personal preference. However, when designing a casing string for directional wells we find that the axial load has to be calculated by some type of torque and drag software using the pipe weights, mud density, directional surveys, and friction factors. All of the commercial software models give the resulting axial load in the form of the effective axial load. (If your torque and drag software shows a zero axial load at the bottom of the string it is definitely the effective axial load, not the true axial load.) So for casing design in a highly deviated well, it becomes necessary to transform that into a true axial load (though many are not aware of that), and this method is the easiest way to accomplish that manually.
You may also notice this last method of calculating the true axial load gives slightly different results from Equations (5.1) and (5.2). This is not the fault of the method, but rather the way that the API specifies its casing weights, and an error on the part of
whoever came up with the direct method. When we calculated the true axial load directly we used the dimensions and weights given by API assuming that the hydrostatic forces only acted on changes in the cross-sectional area of the pipe. As it turns out there is a problem with the direct method as given in this manual and other sources. Although it is the recognized industry standard method it does not account for the buoyancy of the couplings. For instance 13 3/8 in. 68 lb/ft casing does not really weigh 68 lb/ft, it weighs 66.1 lb/ft. The extra weight in the API specification is to account for the weight of the couplings. The API calculates the nominal weight by adding the weight of a coupling less the material removed in cutting the threads to the weight of a 20 ft length of pipe. It is then divided by 20 to give the nominal weight per foot. That is obviously not very accurate because a short coupling does not weigh the same as a long coupling and integral joints like API Extreme-line have an upset and an integral connection. Also, almost no one uses casing joints of 20 feet in length;
typically casing is run in joints that are range 2 or range 3, which average about 30 ft and 40 ft respectively. The direct method we employed accounts for the weight of the couplings in air because they are approximately included in the 68 lb/ft API nominal weight, but it does not account for any buoyancy of the couplings. In our example the actual buoyant force on the couplings would be about 1042 lb total, assuming 20 ft joints in the string. So the truth should be clear; the method of calculating the true axial load from the effective axial load is actually more accurate than the direct method which appears in many sources, e.g. Bourgoyne, et al. (1991) because the direct method (as published) does not take into account the buoyant effects on the couplings. Of course, the issue is rather trivial on the whole, but you should understand the difference. One of the problems we have had for years in the petroleum industry has been a lack of understanding of simple hydrostatics.
Now that we have shown the various methods of calculating the axial load curve let us proceed with the design of the surface casing string. We have already stated that we are going to use a design factor of 1.6 and 100,000 lb over pull, whichever is greater.
Surface Casing Axial Load
In this case the design factor of 1.6 is less than the 100,000 lb over pull at all points so we use the over pull line as the design line. When we plot the casing we have already selected to meet the collapse and burst requirements we find that it easily exceeds the tension requirements also. This is fairly typical of many surface strings, but the tensile design should always be checked to be certain.
Casing Design Summary
13 3/8" Surface Casing
Collapse Burst Joint Strength
2 13.375 12.615 54.5 K-55 ST&C 2100 2100 114450 175650 1.125 1.128 3.6
1 13.375 12.415 68 K-55 ST&C 3000 900 61200 61200 1.359 1.916 26.135
0 0 0
Minimum Safety Factors Mud Weight: 9.2
Collapse: 1.125