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PEH:Casing Design

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Petroleum Engineering Handbook

Larry W. Lake, Editor-in-Chief

Volume II - Drilling Engineering

Robert F. Mitchell, Editor

Chapter 7 - Casing Design

By R.F. Mitchell, Landmark Graphics

Pgs. 287-342

ISBN 978-1-55563-114-7
Get permission for reuse

Casing and tubing strings are the main parts of the well construction. All wells drilled for the purpose of oil/gas production (or injecting materials into underground formations) must be cased with material with sufficient strength and functionality. Therefore, this chapter provides the basic knowledge for practical casing and tubing strength evaluation and design.


Casing is the major structural component of a well. Casing is needed to maintain borehole stability, prevent contamination of water sands, isolate water from producing formations, and control well pressures during drilling, production, and workover operations. Casing provides locations for the installation of blowout preventers, wellhead equipment, production packers, and production tubing. The cost of casing is a major part of the overall well cost, so selection of casing size, grade, connectors, and setting depth is a primary engineering and economic consideration.

Casing Strings

There are six basic types of casing strings. Each is discussed next.

Conductor Casing. Conductor casing is the first string set below the structural casing (i.e., drive pipe or marine conductor run to protect loose near-surface formations and to enable circulation of drilling fluid). The conductor isolates unconsolidated formations and water sands and protects against shallow gas. This is usually the string onto which the casing head is installed. A diverter or a blowout prevention (BOP) stack may be installed onto this string. When cemented, this string is typically cemented to the surface or to the mudline in offshore wells.

Surface Casing. Surface casing is set to provide blowout protection, isolate water sands, and prevent lost circulation. It also often provides adequate shoe strength to drill into high-pressure transition zones. In deviated wells, the surface casing may cover the build section to prevent keyseating of the formation during deeper drilling. This string is typically cemented to the surface or to the mudline in offshore wells

Intermediate Casing. Intermediate casing is set to isolate unstable hole sections, lost-circulation zones, low-pressure zones, and production zones. It is often set in the transition zone from normal to abnormal pressure. The casing cement top must isolate any hydrocarbon zones. Some wells require multiple intermediate strings. Some intermediate strings may also be production strings if a liner is run beneath them.

Production Casing. Production casing is used to isolate production zones and contain formation pressures in the event of a tubing leak. It may also be exposed to injection pressures from fracture jobs, downcasing, gas lift, or the injection of inhibitor oil. A good primary cement job is very critical for this string.

Liner. Liner is a casing string that does not extend back to the wellhead but instead is hung from another casing string. Liners are used instead of full casing strings to reduce cost, improve hydraulic performance when drilling deeper, allow the use of larger tubing above the liner top, and not represent a tension limitation for a rig. Liners can be either an intermediate or a production string. Liners are typically cemented over their entire length.

Tieback String. Tieback string is a casing string that provides additional pressure integrity from the liner top to the wellhead. An intermediate tieback is used to isolate a casing string that cannot withstand possible pressure loads if drilling is continued (usually because of excessive wear or higher than anticipated pressures). Similarly, a production tieback isolates an intermediate string from production loads. Tiebacks can be uncemented or partially cemented. An example of a typical casing program that illustrates each of the specified casing string types is shown in Fig. 7.1.


Tubing is the conduit through which oil and gas are brought from the producing formations to the field surface facilities for processing. Tubing must be adequately strong to resist loads and deformations associated with production and workovers. Further, tubing must be sized to support the expected rates of production of oil and gas. Clearly, tubing that is too small restricts production and subsequent economic performance of the well. Tubing that is too large, however, may have an economic impact beyond the cost of the tubing string itself because the tubing size will influence the overall casing design of the well.

Properties of Casing and Tubing

The American Petroleum Inst. (API) has formed standards for oil/gas casing that are accepted in most countries by oil and service companies. Casing is classified according to five properties: the manner of manufacture, steel grade, type of joints, length range, and the wall thickness (unit weight).

Almost without exception, casing is manufactured of mild (0.3 carbon) steel, normalized with small amounts of manganese. Strength can also be increased with quenching and tempering. API has adopted a casing "grade" designation to define the strength of casing steels. This designation consists of a grade letter followed by a number, which designates the minimum yield strength of the steel in ksi (103 psi). Table 7.1 summarizes the standard API grades.

The yield strength, for these purposes, is defined as the tensile stress required to produce a total elongation of 0.5% of the length. However, the case of P–110 casing is an exception where yield is defined as the tensile stress required to produce a total elongation of 0.6% of the length. There are also proprietary steel grades widely used in the industry, which do not conform to API specifications. These steel grades are often used in special applications requiring high strength or resistance to hydrogen sulfide cracking. Table 7.2 gives a list of commonly used non-API grades.

Pipe Strength

To design a reliable casing string, it is necessary to know the strength of pipe under different load conditions. Burst strength, collapse resistance, and tensile strength are the most important mechanical properties of casing and tubing.

Mechanical Properties

Each mechanical property of casing and tubing is discussed next.

Burst Strength. If casing is subjected to internal pressure higher than external, it is said that casing is exposed to burst pressure. Burst pressure conditions occur during well control operations, integrity tests, and squeeze cementing. The burst strength of the pipe body is determined by the internal yield pressure formula found in API Bull. 5C3, Formulas and Calculations for Casing, Tubing, Drillpipe, and Line Pipe Properties.[1]



PB = minimum burst pressure, psi,

Yp = minimum yield strength, psi,

t = nominal wall thickness, in.,


D = nominal outside pipe diameter, in.

This equation, commonly known as the Barlow equation, calculates the internal pressure at which the tangential (or hoop) stress at the inner wall of the pipe reaches the yield strength (YS) of the material. The expression can be derived from the Lamé equation for tangential stress by making the thin-wall assumption that D/t >> 1. Most casing used in the oilfield has a D/t ratio between 15 and 25. The factor of 0.875 appearing in the equation represents the allowable manufacturing tolerance of –12.5% on wall thickness specified in API Bull. 5C2, Performance Properties of Casing, Tubing, and Drillpipe.[2]

Because a burst failure will not occur until after the stress exceeds the ultimate tensile strength (UTS), using a yield strength criterion as a measure of burst strength is an inherently conservative assumption. This is particularly true for lower-grade materials such H-40, K-55, and N-80 whose UTS/YS ratio is significantly greater than that of higher-grade materials such as P-110 and Q-125. The effect of axial loading on the burst strength is discussed later.

Collapse Strength. If external pressure exceeds internal pressure, the casing is subjected to collapse. Such conditions may exist during cementing operations or well evacuation. Collapse strength is primarily a function of the material’s yield strength and its slenderness ratio, D/t. The collapse strength criteria, given in API Bull. 5C3, Formulas and Calculations for Casing, Tubing, Drillpipe, and Line Pipe Properties,[1] consist of four collapse regimes determined by yield strength and D/t . Each criterion is discussed next in order of increasing D/t.

Yield Strength Collapse. Yield strength collapse is based on yield at the inner wall using the Lamé thick wall elastic solution. This criterion does not represent a "collapse" pressure at all. For thick wall pipes (D/t < 15±), the tangential stress exceeds the yield strength of the material before a collapse instability failure occurs.


Nominal dimensions are used in the collapse equations. The applicable D/t ratios for yield strength collapse are shown in Table 7.3.

Plastic Collapse. Plastic collapse is based on empirical data from 2,488 tests of K-55, N-80, and P-110 seamless casing. No analytic expression has been derived that accurately models collapse behavior in this regime. Regression analysis results in a 95% confidence level that 99.5% of all pipes manufactured to API specifications will fail at a collapse pressure higher than the plastic collapse pressure. The minimum collapse pressure for the plastic range of collapse is calculated by Eq. 7.3.


The factors A, B, and C and applicable D/t range for the plastic collapse formula are shown in Table 7.4.

Transition Collapse. Transition collapse is obtained by a numerical curve fit between the plastic and elastic regimes. The minimum collapse pressure for the plastic-to-elastic transition zone, PT, is calculated with Eq. 7.4.


The factors F and G and applicable D/t range for the transition collapse pressure formula, are shown in Table 7.5.

Elastic Collapse. Elastic Collapse is based on theoretical elastic instability failure; this criterion is independent of yield strength and applicable to thin-wall pipe (D/t > 25±). The minimum collapse pressure for the elastic range of collapse is calculated with Eq. 7.5.


The applicable D/t range for elastic collapse is shown in Table 7.6.

Most oilfield tubulars experience collapse in the "plastic" and "transition" regimes. Many manufacturers market "high collapse" casing, which they claim has collapse performance properties that exceed the ratings calculated with the formulae in API Bull. 5C3, Formulas and Calculations for Casing, Tubing, Drillpipe, and Line Pipe Properties.[1] This improved performance is achieved principally by using better manufacturing practices and stricter quality assurance programs to reduce ovality, residual stress, and eccentricity. High collapse casing was initially developed for use in the deeper sections of high-pressure wells. The use of high collapse casing has gained wide acceptance in the industry, but its use remains controversial among some operators. Unfortunately, all manufacturers’ claims have not been substantiated with the appropriate level of qualification testing. If high collapse casing is deemed necessary in a design, appropriate expert advice should be obtained to evaluate the manufacturer’s qualification test data such as lengths to diameter ratio, testing conditions (end constraints), and the number of tests performed.

Equivalent Internal Pressure. If the pipe is subjected to both external and internal pressures, the equivalent external pressure is calculated as



pe = equivalent external pressure,

po = external pressure,

pi = internal pressure,


Δp = popi.

To provide a more intuitive understanding of the sense of this relationship, Eq. 7.6 can be rewritten as



D = nominal outside diameter,


d = nominal inside diameter.

In Eq. 7.7, we can see the internal pressure applied to the internal diameter and the external pressure applied to the external diameter. The "equivalent" pressure applied to the external diameter is the difference of these two terms.

Axial Strength. The axial strength of the pipe body is determined by the pipe body yield strength formula found in API Bull. 5C3, Formulas and Calculations for Casing, Tubing, Drillpipe, and Line Pipe Properties.[1]



Fy = pipe body axial strength (units of force),

Yp = minimum yield strength,

D = nominal outer diameter,


d = nominal inner diameter.

Axial strength is the product of the cross-sectional area (based on nominal dimensions) and the yield strength.

Combined Stress Effects. All the pipe-strength equations previously given are based on a uniaxial stress state (i.e., a state in which only one of the three principal stresses is nonzero). This idealized situation never occurs in oilfield applications because pipe in a wellbore is always subjected to combined loading conditions. The fundamental basis of casing design is that if stresses in the pipe wall exceed the yield strength of the material, a failure condition exists. Hence, the yield strength is a measure of the maximum allowable stress. To evaluate the pipe strength under combined loading conditions, the uniaxial yield strength is compared to the yielding condition. Perhaps the most widely accepted yielding criterion is based on the maximum distortion energy theory, which is known as the Huber-Hencky-Mises yield condition or simply the von Mises stress, triaxal stress, or equivalent stress.[3] Triaxial stress (equivalent stress) is not a true stress. It is a theoretical value that allows a generalized three-dimensional (3D) stress state to be compared with a uniaxial failure criterion (the yield strength). In other words, if the triaxial stress exceeds the yield strength, a yield failure is indicated. The triaxial safety factor is the ratio of the material’s yield strength to the triaxial stress.

The yielding criterion is stated as



Yp = minimum yield stress, psi,

σVME = triaxial stress, psi,

σz = axial stress, psi,

σϴ = tangential or hoop stress, psi,


σr = radial stress, psi.

The calculated axial stress, σz, at any point along the cross-sectional area should include the effects of self-weight, buoyancy, pressure loads, bending, shock loads, frictional drag, point loads, temperature loads, and buckling loads. Except for bending/buckling loads, axial loads are normally considered to be constant over the entire cross-sectional area.

The tangential and radial stresses are calculated with the Lamé equations for thick-wall cylinders.





pi = internal pressure,

po = external pressure,

ri = inner wall radius,

ro = outer wall radius,


r = radius at which the stress occurs.

The absolute value of σϴ is always greatest at the inner wall of the pipe and that for burst and collapse loads, where |pipo| >> 0, then |σϴ| >> |σr|. For any pi and po combination, the sum of the tangential and radial stresses is constant at all points in the casing wall. Substituting Eq. 7.10 and Eq. 7.11 into Eq. 7.9, after rearrangements, yields


in which





D = outside pipe diameter,


t = wall thickness.

Eq. 7.12 calculates the equivalent stress at any point of the pipe body for any given pipe geometry and loading conditions. To illustrate these concepts, let us consider a few particular cases.

Combined Collapse and Tension. Assuming that σz > 0 and σϴ >> σr and setting the triaxial stress equal to the yield strength results in the next equation of an ellipse.


This is the biaxial criterion used in API Bull. 5C3, Formulas and Calculations for Casing, Tubing, Drillpipe, and Line Pipe Properties,[1] to account for the effect of tension on collapse.



Sa = axial stress based on the buoyant weight of pipe,


Yp = yield point.

It is clearly seen that as the axial stress S a increases, the pipe collapse resistance decreases. Plotting this ellipse, Fig. 7.2 allows a direct comparison of the triaxial criterion with the API ratings. Loads that fall within the design envelope meet the design criteria. The curved lower right corner is caused by the combined stress effects, as described in Eq. 7.14.

Combined Burst and Compression Loading. Combined burst and compression loading corresponds to the upper left-hand quadrant of the design envelope. This is the region where triaxial analysis is most critical because reliance on the uniaxial criterion alone would not predict several possible failures. For high burst loads (i.e., high tangential stress and moderate compression), a burst failure can occur at a differential pressure less than the API burst pressure. For high compression and moderate burst loads, the failure mode is permanent corkscrewing (i.e., plastic deformation because of helical buckling). This combined loading typically occurs when a high internal pressure is experienced (because of a tubing leak or a buildup of annular pressure) after the casing temperature has been increased because of production. The temperature increase, in the uncemented portion of the casing, causes thermal growth, which can result in significant increases in compression and buckling. The increase in internal pressure also results in increased buckling.

Combined Burst and Tension Loading. Combined burst and tension loading corresponds to the upper right-hand quadrant of the design envelope. This is the region where reliance on the uniaxial criterion alone can result in a design that is more conservative than necessary. For high burst loads and moderate tension, a burst yield failure will not occur until after the API burst pressure has been exceeded. As the tension approaches the axial limit, a burst failure can occur at a differential pressure less than the API value. For high tension and moderate burst loads, pipe body yield will not occur until a tension greater than the uniaxial rating is reached.

Taking advantage of the increase in burst resistance in the presence of tension represents a good opportunity for the design engineer to save money while maintaining wellbore integrity. Similarly, the designer might wish to allow loads between the uniaxial and triaxial tension ratings. However, great care should be taken in the latter case because of the uncertainty of what burst pressure might be seen in conjunction with a high tensile load (an exception to this is the green cement pressure test load case). Also, connection ratings may limit your ability to design in this region.

Use of Triaxal Criterion for Collapse Loading. For many pipes used in the oil field, collapse is an inelastic stability failure or an elastic stability failure independent of yield strength. The triaxial criterion is based on elastic behavior and the yield strength of the material and, hence, should not be used with collapse loads. The one exception is for thick-wall pipes with a low D/t ratio, which have an API rating in the yield strength collapse region. This collapse criterion along with the effects of tension and internal pressure (which are triaxial effects) result in the API criterion being essentially identical to the triaxial method in the lower right-hand quadrant of the triaxial ellipse for thick-wall pipes.

For high compression and moderate collapse loads experienced in the lower left-hand quadrant of the design envelope, the failure mode may be permanent corkscrewing because of helical buckling. It is appropriate to use the triaxial criterion in this case. This load combination typically can occur only in wells that experience a large increase in temperature because of production. The combination of a collapse load that causes reverse ballooning and a temperature increase acts to increase compression in the uncemented portion of the string.

Most design engineers use a minimum wall for burst calculations and nominal dimensions for collapse and axial calculations. Arguments can be made for using either assumption in the case of triaxial design. Most importantly, more so than the choice of dimensional assumptions, is that the results of the triaxial analysis should be consistent with the uniaxial ratings with which they may be compared.

Triaxial analysis is perhaps most valuable when evaluating burst loads. Hence, it makes sense to calibrate the triaxial analysis to be compatible with the uniaxial burst analysis. This can be done by the appropriate selection of a design factor. Because the triaxial result nominally reduces to the uniaxial burst result when no axial load is applied, the results of both of these analyses should be equivalent. Because the burst rating is based on 87.5% of the nominal wall thickness, a triaxial analysis based on nominal dimensions should use a design factor that is equal to the burst design factor multiplied by 8/7. This reflects the philosophy that a less conservative assumption should be used with a higher design factor. Hence, for a burst design factor of 1.1, a triaxial design factor of 1.25 should be used.

Final Triaxal Stress Considerations. Fig. 7.3 graphically summarizes the triaxial, uniaxial, and biaxial limits that should be used in casing design along with a set of consistent design factors.

Because of the potential benefits (both cost savings and better mechanical integrity) that can be realized, a triaxial analysis is recommended for all well designs. Specific applications include saving money in burst design by taking advantage of the increased burst resistance in tension; accounting for large temperature effects on the axial load profile in high-pressure, high-temperature wells (this is particularly important in combined burst and compression loading); accurately determining stresses when using thick-wall pipe (D/t < 12) (conventional uniaxial and biaxial methods have imbedded thin-wall assumptions); and evaluating buckling severity (permanent corkscrewing occurs when the triaxial stress exceeds the yield strength of the material).

While it is acknowledged that the von Mises criterion is the most accurate method of representing elastic yield behavior, use of this criterion in tubular design should be accompanied by a few precautions.

First, for most pipe used in oilfield applications, collapse is frequently an instability failure that occurs before the computed maximum triaxial stress reaches the yield strength. Hence, triaxial stress should not be used as a collapse criterion. Only in thick-wall pipe does yielding occur before collapse.

Second, the accuracy of triaxial analysis is dependent upon the accurate representation of the conditions that exist both for the pipe as installed in the well and for the subsequent loads of interest. Often, it is the change in load conditions that is most important in stress analysis. Hence, an accurate knowledge of all temperatures and pressures that occur over the life of the well can be critical to accurate triaxial analysis.

API Connection Ratings

While a number of joint connections are available, the API recognizes three basic types: coupling with rounded thread (long or short); coupling with asymmetrical trapezoidal thread buttress; and extreme-line casing with trapezoidal thread without coupling.

Threads are used as mechanical means to hold the neighboring joints together during axial tension or compression. For all casing sizes, the threads are not intended to be leak resistant when made up. API Spec. 5C2, Performance Properties of Casing, Tubing, and Drillpipe,[2] provides information on casing and tubing threads dimensions.

Coupling Internal Yield Pressure

The internal yield pressure is the pressure that initiates yield at the root of the coupling thread.



PCIY = coupling internal yield pressure, psi,

Yc = minimum yield strength of coupling, psi,

W = nominal outside diameter of coupling, in.,


d1 = diameter at the root of the coupling thread in the power tight position, in.

This dimension is based on data given in API Spec. 5B, Threading, Gauging, and Thread Inspection of Casing, Tubing, and Line Pipe Threads,[4] and other thread geometry data. The coupling internal yield pressure is typically greater than the pipe body internal yield pressure. The internal pressure leak resistance is based on the interface pressure between the pipe and coupling threads because of makeup.



PILR = coupling internal pressure leak resistance, psi,

E = modulus of elasticity,

T = thread taper, in.,

N = a function of the number of thread turns from hand-tight to power-tight position, as given in API

Spec. 5B, Threading, Gauging, and Thread Inspection of Casing, Tubing, and Line Pipe Threads,[4]

pt = thread pitch, in.,


Es = pitch diameter at plane of seal, as given in API Spec. 5B, Threading, Gauging, and Thread Inspection of Casing, Tubing, and Line Pipe Threads.[4]

This equation accounts only for the contact pressure on the thread flanks as a sealing mechanism and ignores the long helical leak paths filled with thread compound that exist in all API connections.

In round threads, two small leak paths exist at the crest and root of each thread. In buttress threads, a much larger leak path exists along the stabbing flank and at the root of the coupling thread. API connections rely on thread compound to fill these gaps and provide leak resistance. The leak resistance provided by the thread compound is typically less than the API internal leak resistance value, particularly for buttress connections. The leak resistance can be improved by using API connections with smaller thread tolerances (and, hence, smaller gaps), but it typically will not exceed 5,000 psi with any long-term reliability. Applying tin or zinc plating to the coupling also results in smaller gaps and improves leak resistance.

Round-Thread Casing-Joint Strength

The round-thread casing-joint strength is given as the lesser of the fracture strength of the pin and the jump-out strength. The fracture strength is given by


The jump-out strength is given by



Fj = minimum joint strength, lbf,

Ajp = cross-sectional area of the pipe wall under the last perfect thread, in.2,

= π/4[(D – 0.1425)2d2],

D = nominal outside diameter of pipe, in.,

d = nominal inside diameter of pipe, in.,

L = engaged thread length, in., as given in API Spec. 5B, Threading, Gauging, and Thread Inspection of Casing, Tubing, and Line Pipe Threads,[4]

Yp = minimum yield strength of pipe, psi,


Up = minimum ultimate tensile strength of pipe, psi.

These equations are based on tension tests to failure on 162 round-thread test specimens. Both are theoretically derived and adjusted using statistical methods to match the test data. For standard coupling dimensions, round threads are pin weak (i.e., the coupling is noncritical in determining joint strength).

Buttress Casing Joint Strength

The buttress thread casing joint strength is given as the lesser of the fracture strength of the pipe body (the pin) and the coupling (the box). Pipe thread strength is given by


Coupling thread strength is given by



Uc = minimum ultimate tensile strength of coupling, psi,

Ap = cross-sectional area of plain-end pipe, in.2 ,


Ac = cross-sectional area of coupling, in.,

= π/4(W 2d12).

These equations are based on tension tests to failure on 151 buttress-thread test specimens. They are theoretically derived and adjusted using statistical methods to match test data.

Extreme-Line Casing-Joint Strength

Extreme-line casing-joint strength is calculated as



Fj = minimum joint strength, lbf,


Acr = critical section area of box, pin, or pipe, whichever is least, in.2.

When performing casing design, it is very important to note that the API joint-strength values are a function of the ultimate tensile strength. This is a different criterion from that used to define the axial strength of the pipe body, which is based on the yield strength. If care is not taken, this approach can lead to a design that inherently does not have the same level of safety for the connections as for the pipe body. This is not the most prudent practice, particularly in light of the fact that most casing failures occur at connections. This discrepancy can be countered by using a higher design factor when performing connection axial design with API connections.

The joint-strength equations for tubing given in API Bull. 5C3, Formulas and Calculations for Casing, Tubing, Drillpipe, and Line Pipe Properties,[1] are very similar to those given for round-thread casing except they are based on yield strength. Hence, the UTS/YS discrepancy does not exist in tubing design.

If API casing connection joint strengths calculated with the previous formulae are the basis of a design, the designer should use higher axial design factors for the connection analysis. The logical basis for a higher axial design factor (DF) is to multiply the pipe body axial design factor by the ratio of the minimum ultimate tensile strength, Up, to the minimum yield strength, Yp.


Tensile property requirements for standard grades are given in API Spec. 5C2, Performance Properties of Casing, Tubing, and Drillpipe,[2] and are shown in Table 7.7 for reference along with their ratio.

Proprietary Connections

Special connections are used to achieve gas-tight sealing reliability and 100% connection efficiency (joint efficiency is defined as a ratio of joint tensile strength to pipe body tensile strength) under more severe well conditions. Severe conditions include high pressure (typically > 5,000 psi); high temperature (typically > 250°F); a sour environment; gas production; high-pressure gas lift; a steam well; and a large dogleg (horizontal well). Also, efficiency in flush joint, integral joint or other special clearance applications improves connections; a large diameter (> 16 in.) pipe improves the stab-in and makeup characteristics; galling should be reduced (particularly in CRA applications and tubing strings that will be re-used); and connection failure under high torsional loads (e.g., while rotating pipe) should be prevented.

The improved performance of many proprietary connections results from one or more of these features not found in API connections: more complex thread forms; resilient seals; torque shoulders; and metal-to-metal seals. The "premium" performance of most proprietary connections comes at a "premium" cost. Increased performance should always be weighed against the increased cost for a particular application. As a general rule, it is recommended to use proprietary connections only when the application requires them. "Premium" performance may also be achieved using API connections if certain conditions are met. Those conditions are tighter dimensional tolerance; plating applied to coupling; use of appropriate thread compound; and performance verified with qualification testing.

The performance of a proprietary connection can be reliably verified by performing three steps: audit the manufacturer’s performance test data (sealability and tensile load capacity under combined loading); audit the manufacturer’s field history data; and require additional performance testing for the most critical applications. When requesting tensile performance data, make sure that the manufacturer indicates whether quoted tensile capacities are based on the ultimate tensile strength (i.e., the load at which the connection will fracture, commonly called the "parting load") or the yield strength (commonly called the "joint elastic limit"). If possible, it is recommended to use the joint elastic limit values in the design so that consistent design factors for both pipe-body and connection analysis are maintained. If only parting load capacities are available, a higher design factor should be used for connection axial design.

Connection Failures

Most casing failures occur at connections. These failures can be attributed to improper design or exposure to loads exceeding the rated capacity; failure to comply with makeup requirements; failure to meet manufacturing tolerances; damage during storage and handling; and damage during production operations (corrosion, wear, etc.).

Connection failure can be classified broadly as leakage; structural failure; galling during makeup; yielding because of internal pressure; jump-out under tensile load; fracture under tensile load; and failure because of excessive torque during makeup or subsequent operations. Avoiding connection failure is not only dependent upon selection of the correct connection but is strongly influenced by other factors, which include manufacturing tolerances; storage (storage thread compound and thread protector); transportation (thread protector and handling procedures); and running procedures (selection of thread compound, application of thread compound, and adherence to correct makeup specifications and procedures).

The overall mechanical integrity of a correctly designed casing string is dependent upon a quality assurance program that ensures damaged connections are not used and that operations personnel adhere to the appropriate running procedures.

Connection Design Limits

The design limits of a connection are not only dependent upon its geometry and material properties but are influenced by surface treatment; phosphating; metal plating (copper, tin, or zinc); bead blasting; thread compound; makeup torque; use of a resilient seal ring (many companies do not recommend this practice); fluid to which connection is exposed (mud, clear brine, or gas); temperature and pressure cycling; and large doglegs (e.g., medium- or short-radius horizontal wells).

Casing and Tubing Buckling


As installed, casing usually hangs straight down in vertical wells or lays on the low side of the hole in deviated wells. Thermal or pressure loads might produce compressive loads, and if these loads are sufficiently high, the initial configuration will become unstable. However, because the tubing is confined within open hole or casing, the tubing can deform into another stable configuration, usually a helical or coil shape in a vertical wellbore or a lateral S-shaped configuration in a deviated hole. These new equilibrium configurations are what we mean when we talk about buckling in casing design. In contrast, conventional mechanical engineering design considers buckling in terms of stability (i.e., the prediction of the critical load at which the original configuration becomes unstable).

Accurate analysis of buckling is important for several reasons. First, buckling generates bending stresses not present in the original configuration. If the stresses in the original configuration were near yield, this additional stress could produce failure, including permanent plastic deformation called "corkscrewing." Second, buckling causes tubing movement. Coiled tubing is shorter than straight tubing, and this is an important consideration if the tubing is not fixed. Third, tubing buckling causes the relief of compressive axial loads when the casing is fixed. This effect is not as recognized as the first two buckling effects but is equally important. The axial compliance of buckled tubing is much less than the compliance of straight tubing. Casing movement, because of thermal expansion or ballooning, can be accommodated with a lower increase in axial load for a buckled casing.

The accuracy and comprehensiveness of the buckling model is important for designing tubing. The most commonly used buckling solution is the model developed by Lubinski in the 1950s. This model is accurate for vertical wells but needs modification for deviated wells. Tubing bending stress, because of buckling, will be overestimated for deviated wells using Lubinski’s formula. However, Lubinski’s solution, applied to deviated wells, will also overpredict tubing movement. This solution overestimates tubing compliance, which might greatly underestimate the axial loads, resulting in a nonconservative design.

Casing Buckling in Oilfield Operations

Buckling should be avoided in drilling operations to minimize casing wear. Buckling can be reduced or eliminated by applying a pickup force before landing the casing; holding pressure, while weighing on cement (WOC), to pretension the string (subsea wells); raising the top of cement; using centralizers; and increasing pipe stiffness.

In production operations, casing buckling is not normally a critical design issue. However, a large amount of buckling can occur because of increased production temperatures in some wells. A check should be made to ensure that plastic deformation or corkscrewing will not occur. This check is possible using triaxial analysis and including the bending stress because of buckling. Corkscrewing occurs only if the triaxial stress exceeds the yield strength of the material.

Tubing Buckling in Oilfield Operations

Buckling is typically a more critical design issue for production tubing than for casing. Tubing is typically exposed to the hottest temperatures during production. Pressure/area effects in floating seal assemblies can significantly increase buckling. Tubing is less stiff than casing, and annular clearances can be quite large. Buckling can prevent wireline tools from passing through the tubing. Buckling can be controlled by tubing-to-packer configuration (latched or free, seal bore diameter, allowable movement in seals, etc.); slackoff or pickup force at surface; cross-sectional area changes in tubing; packer fluid density; pipe stiffness; centralizers; and hydraulic set pressure. As in casing design, a triaxial check should be made to ensure that plastic deformation or corkscrewing will not occur.

Buckling Models and Correlations

Buckling occurs if the buckling force, Fb, is greater than a threshold force, Fp , known as the Paslay buckling force. The buckling force, Fb, is defined as



Fb = buckling force, lbf,

Fa = axial force (tension positive), lbf,

pi = internal pressure, psi,

Ai = ri2 , where r i is the inside radius of the tubing, in.2,

po = external pressure, psi,


Ao = ro2 , where ro is the outside radius of the tubing, in.2

The Paslay buckling force, Fp , is defined as



Fp = Paslay buckling force, lbf,

wc = casing contact load, lbf/in.,

we = distributed buoyed weight of casing, lbf/in.,

Φ = wellbore angle of inclination, radians,

Θ = wellbore azimuth angle, radians,

EI = pipe bending stiffness, lbf-in.2 ,


r = radial annular clearance, in.

Table 7.8 gives the relationship between the buckling force Fb, the Paslay buckling force Fp, and the type of buckling expected for the tubing.

An increase in internal pressure acts on the buckling force in two ways. It increases F a because of ballooning, which tends to decrease buckling, and increases the piAi term, which tends to increase buckling. The second effect is much greater; hence, an increase in internal pressure will result in an increase in buckling.

A temperature increase results in a reduction in the axial tension (or increase in the compression). This reduction in tension results in an increase in buckling. The onset and type of buckling is a function of hole angle. Because of the stabilizing effect of the lateral distributed force of a casing lying on the low side of the hole in an inclined wellbore, a greater force is required to induce buckling. In a vertical well, Fp = 0, and helical buckling occurs at any Fb > 0. For production tubing that is free to move in a seal assembly, the upward force, because of pressure/area effects in the seal assembly, will decrease Fa, which, in turn, increases buckling.

In order to give the correlations for tubing stresses and movement, definitions are made. The lateral displacements of the tubing, shown in Fig. 7.4, are given by




where ϴ is the helix angle.

The quantity ϴ´, where ´ denotes d/dz , is important and appears often in the next analysis. It can be related to the more familiar quantity, pitch through Eq. 7.27.


where Phel = pitch of helically buckled pipe, in.

Other important quantities, such as pipe curvature, bending moment, bending stress, and tubing length change are proportional to the square of ϴ´. Nonzero ϴ´ indicates that the pipe is curving, while zero ϴ´ indicates that the pipe is straight.

Correlations for Maximum Buckling Dogleg

The correlation for the maximum value of ϴ´ for lateral buckling, with 2.8 Fp > Fb > Fp, can be expressed by


For Fb > 2.8 Fp, the corresponding helical buckling correlation is


The region 2.8 Fp > Fb > 1.4 Fp may be either helical or lateral; however, 2.8 Fp is believed to be the lateral buckling limit on loading, while 1.4 Fp is believed to be the helical buckling limit on unloading from a helical buckled state. An important distinction between Eq. 7.28 and Eq. 7.29 is that Eq. 7.28 is the maximum value of ϴ´, while Eq. 7.29 is the actual value of ϴ´.

The equation for a dogleg curvature for a helix is


assuming ϴʺ is negligible. The dogleg unit for Eq. 7.30 is radians per inch. To convert to the conventional unit of degrees per 100 ft, multiply the result by 68,755.

Correlations for Bending Moment and Bending Stress

Given the tubing curvature, the bending moment is determined by


The corresponding maximum bending stress is


where do is the outside diameter of the pipe.

The following correlations can be derived with Eqs. 7.28 and 7.29. M = 0, for Fb < Fp;



σb = 0, for Fb < Fp;



Correlations for Buckling Strain and Length Change

The buckling "strain," in the sense of Lubinski, is the buckling length change per unit length. The buckling strain is given by


For the case of lateral buckling, the actual shape of the ϴ´ curve was integrated numerically to determine the relationship,


for 2.8 Fp > Fb > Fp, which compares to the helical buckling strain given by


for Fb > 2.8 Fp. The lateral buckling strain is roughly half the conventional helical buckling strain. To determine the buckling length change, ΔLb, we must integrate Eqs. 7.38 and 7.39 over the appropriate length interval, which is written as


where z1 and z2 are defined by the distribution of the buckling force, F. For the general case of arbitrary variation of Fb over the interval ΔL = Z2Z1 , Eq. 7.40 must be numerically integrated. However, there are two special cases that are commonly used. For the case of constant force, Fb, such as in a horizontal well, Eq. 7.40 is easily integrated.


where eb is defined by either Eq. 7.28 or Eq. 7.29 . The second special case is for a linear variation of Fb over the interval.


The length change is given for this case by Eqs. 7.43 and 7.44.


for 2.8 Fp > F2 > Fp.


for F > 2.8 Fp.

Correlations for Contact Force

From equilibrium considerations only, the average contact force for lateral buckling is


The average contact force for the helically buckled section is


When the buckling mode changes from lateral to helical, the contact force increases substantially.

Sample Buckling Calculations

The basis of the sample calculations is the buckling of tubing (2 7/8 in., 6.5 lbm/ft) inside of casing (7 in., 32 lbm/ft). The tubing is submerged in 10-lbm/gal packer fluid with no other pressures applied. The effect of the packer fluid is to reduce the tubing weight per unit length through buoyancy. we = w + AiγiAoγo , where we is the effective weight per unit length of the tubing, Ai is the inside area of the tubing, γi is the density of the fluid inside the tubing, Ao is the outside area of the tubing, and γo is the density of the fluid outside the tubing. The calculation gives


Other information useful for the buckling calculations are radial clearance = r = 1.61 in.; moment of inertia = I = 1.611 in.4, and Young’s modulus = 30 × 106 psi.

Sample Buckling Length Calculations. From Eq. 7.24, we can calculate the Paslay force for a variety of inclinations. First, we calculate the value for a horizontal well, which is written as
This means that the axial buckling force must exceed 7,500 lbf before the tubing will buckle. We can evaluate other angles by multiplying the horizontal Fp by the square root of the sine of the inclination angle. Table 7.9 was developed with this procedure. Of particular notice in Table 7.9 is how large these buckling forces are for relatively small deviations from vertical. For a well 10° from the vertical, the buckling forces are nearly half of the horizontal well buckling forces.

With Table 7.9, the total buckled length of the tubing can be calculated, as well as maximum and minimum lateral buckling or helical buckling. Assume an applied buckling force of 30,000 lbf is applied at the end of the tubing in a well with a 60° deviation from vertical. The tubing will buckle for any force between 6,939 lbf and 30,000 lbf. The axial force will vary as we cosΦ (i.e., wa = we cos(60) = 5.56 lbf/ft (0.50) = 2.78 lbf/ft). Therefore, the total buckled length, Lbkl, is Lbkl = (30,000 – 6939)lbf/(2.78 lbf/ft) = 8,295 ft. The maximum helically buckled length, Lhelmax, is Lhelmax = (30,000 – 9,813)lbf/(2.78 lbf/ft) = 7,262 ft. The minimum helically buckled length, Lhelmin, is Lhelmin = (30,000 – 19,626)lbf/(2.78 lbf/ft) = 3,732 ft.

Sample Buckling Bending Stress Calculations. The maximum bending stress, because of buckling, can be evaluated with Eq. 7.38. σb = .25(2.875 in.)(1.61 in.)(30,000 lbf)/(1.611 in.4) = 21,550 psi. This stress is fairly large compared to tubing yield strengths of about 80,000 psi, so buckling bending stresses can be important for casing and tubing design. At the buckling load of 19,626 lbf, both helical and lateral buckling can occur. The lateral bending stress is given by Eq. 7.35. σb = .3151 (2.875 in.)(1.61 in.)/(1.611 in.4) (6,939 lbf).08(19,626 – 6,939 lbf)0.92 = 10,945 psi. The equivalent calculation for helical buckling gives σb = .25(2.875 in.)(1.61 in.)(19,626 lbf)/(1.611 in.4) = 14,097 psi, so helical buckling produces approximately 29% higher stresses than lateral buckling. This indicates that determination of buckling type can be important in casing design where casing strength is marginal.

Sample Buckling Length Change Calculations – Tubing Movement. Tubing length change calculations involve two calculations for this case, tubing movement because of lateral buckling and tubing movement because of helical buckling. Eqs. 7.43 and 7.44 are used to calculate tubing movement, and these equations assume the minimum amount of helical buckling. A third calculation is made to show the movement because of pure helical buckling. The lateral buckling tubing movement is given by


The helical buckling tubing movement is given by


The total tubing movement is 0.297 ft plus 1.242 ft, which equals 1.539 ft. Pure helical buckling produces the length change,


Tubing movement is a design consideration for packer selection. Seal length is an important criterion for tubing well completion design. The use of pure helical buckling produces a 41% error in the calculation of tubing movement. When designing seal length in a deviated well, use of pure helical buckling can produce significant error.

Loads on Casing and Tubing Strings

In order to evaluate a given casing design, a set of loads is necessary. Casing loads result from running the casing, cementing the casing, subsequent drilling operations, production and well workover operations. Casing loads are principally pressure loads, mechanical loads, and thermal loads. Pressure loads are produced by fluids within the casing, cement and fluids outside the casing, pressures imposed at the surface by drilling and workover operations, and pressures imposed by the formation during drilling and production. Mechanical loads are associated with casing hanging weight, shock loads during running, packer loads during production and workovers, and hanger loads. Temperature changes and resulting thermal expansion loads are induced in casing by drilling, production, and workovers, and these loads might cause buckling (bending stress) loads in uncemented intervals.

Next, we discuss casing loads that are typically used in preliminary casing design. However, each operating company usually has its own special set of design loads for casing, based on their experience. If you are designing a casing string for a particular company, this load information must be obtained from them. Because there are so many possible loads that must be evaluated, most casing design today is done with computer programs that generate the appropriate load sets (often custom tailored for a particular operator), evaluate the results, and sometimes even determine a minimum-cost design automatically.

External Pressure Loads

Pressure Distributions

Pressure distributions are typically used to model the external pressures in cemented intervals.

Mud/Cement Mix-Water. Fluid pressure is given by the mud gradient above the top-of-cement (TOC) and by the cement gradient below TOC.

Permeable Zones: Good Cement. Again, fluid pressure is given by the mud gradient above TOC and by the cement gradient below TOC. The exception is that formation pore pressure is imposed over the permeable zone interval. This pressure profile is discontinuous.

Permeable Zones: Poor Cement, High Pressure. In this case, the formation pore pressure is felt at the surface through the poor cement. This pressure profile is continuous with depth.

Permeable Zones: Poor Cement, Low Pressure. In this case, the mud surface drops so that the mud pressure equals the formation pressure. This pressure profile is continuous with depth.

Openhole Pore Pressure: TOC Inside Previous Shoe. In this case, fluid pressure is given by mud gradient above TOC, cement gradient to the shoe, and the minimum equivalent mud weight gradient of the openhole below the shoe. This pressure profile is not continuous with depth; it is discontinuous at the previous shoe.

Openhole Pore Pressure: TOC Below Previous Shoe, Without Mud Drop. In this case, fluid pressure is given by the mud gradient above TOC and by the minimum equivalent mud weight gradient of the openhole below the shoe. This pressure profile is not continuous with depth but is discontinuous at TOC.

TOC Below Previous Show, With Mud Drop. In this case, the mud surface drops so that the mud pressure equals the minimum equivalent mud weight gradient of the openhole at the TOC. This pressure profile is continuous with depth.

Above/Below TOC External Pressure Profile. In this case, fluid pressure is given by mud gradient above TOC, cement gradient to the shoe, and a specified pressure profile below a specified depth. This external pressure distribution may be discontinuous at the specified depth. If a pressure gradient is specified, the pressure profile may also be continuous at the specified depth.

Internal Pressure Loads

Pressure Distributions

Pressure distributions are typically used to model the internal pressures. These pressure distributions are discussed next.

Burst: Gas Kick. This load case uses an internal pressure profile, which is the envelope of the maximum pressures experienced by the casing while circulating out a gas kick using the driller’s method. It should represent the worst-case kick to which the current casing can be exposed while drilling a deeper interval. Typically, this means taking a kick at the total depth (TD) of the next openhole section. If the kick intensity or volume cause the fracture pressure at the casing shoe to exceed, the kick volume is often reduced to the maximum volume that can be circulated out of the hole without exceeding the fracture pressure at the shoe. The maximum pressure experienced at any casing depth occurs when the top of the gas bubble reaches that depth.

Burst: Displacement to Gas. This load case uses an internal pressure profile consisting of a gas gradient extending upward from a formation pressure in a deeper hole interval or from the fracture pressure at the casing shoe. This pressure physically represents a well control situation, in which gas from a kick has completely displaced the mud out of the drilling annulus from the surface to the casing shoe. This is the worst-case drilling burst load that a casing string could experience, and if the fracture pressure at the shoe is used to determine the pressure profile, it ensures that the weak point in the system is at the casing shoe and not the surface. This, in turn, precludes a burst failure of the casing near the surface during a severe well-control situation.

Burst: Maximum Load Concept. This load case is a variation of the displacement-to-gas load case that has wide usage in the industry and is taught in several popular casing design schools. It has been used historically because it results in an adequate design (though typically quite conservative, particularly for wells deeper than 15,000 ft), and it is simple to calculate. The load case consists of a gas gradient extending upward from the fracture pressure at the shoe up to a mud/gas interface and then a mud gradient to the surface. The mud/gas interface is calculated in a number of ways—the most common being the "fixed endpoint" method. The interface is calculated on the basis of surface pressure typically equal to the BOP rating and the fracture pressure at the shoe and assuming a continuous pressure profile.

Burst: Lost Returns With Water. This load case models an internal pressure profile, which reflects pumping water down the annulus to reduce surface pressure during a well-control situation in which lost returns are occurring. The pressure profile represents a freshwater gradient applied upward from the fracture pressure at the shoe depth. A water gradient is used, assuming that the rig’s barite supply has been depleted during the well-control incident. This load case typically dominates the burst design when compared to the gas-kick load case. This is particularly the case for intermediate casing.

Burst: Surface Protection. This load case is less severe than the displacement-to-gas criteria and represents a moderated approach to preventing a surface blowout during a well-control incident. It is not applicable to liners. The same surface pressure calculated in the "lost returns with water" load case is used, but in this load case, a gas gradient from this surface pressure is used to generate the rest of the pressure profile. This load case represents no actual physical scenario; however, when used with the gas-kick criterion, it ensures that the casing weak point is not at the surface. Typically, the gas-kick load case will control the design deep, and the surface-protection load case will control the design shallow, leaving the weak point somewhere in the middle.

Burst: Pressure Test. This load case models an internal pressure profile, which reflects a surface pressure applied to a mud gradient. The test pressure typically is based on the maximum anticipated surface pressure determined from the other selected burst load cases plus a suitable safety margin. For production casing, the test pressure is typically based on the anticipated shut-in tubing pressure. This load case may or may not dominate the burst design depending on the mud weight in the hole at the time the test occurs. The pressure test is normally performed prior to drilling out the float equipment.

Collapse: Cementing. This load case models an internal and external pressure profile, which reflects the collapse load imparted on the casing after the plug has been bumped during the cement job and the pump pressure bled off. The external pressure considers the mud hydrostatic column and different densities of the lead and tail cement slurries. The internal pressure is based on the gradient of the displacement fluid. If a light displacement fluid is used, the cementing collapse load can be significant.

Collapse: Lost Returns With Mud Drop. This load case models an internal pressure profile, which reflects a partial evacuation or a drop in the mud level because of the mud hydrostatic column equilibrating with the pore pressure in a lost-circulation zone. The heaviest mud weight used to drill the next openhole section should be used along with a pore pressure and depth that result in the largest mud drop. Many operators make the conservative assumption that the lost-circulation zone is at the TD of the next openhole section and is normally pressured. A partial evacuation of more than 5,000 ft, because of lost circulation during drilling, is normally not seen. Many operators use a partial evacuation criterion in which the mud level is assumed to be a percentage of the openhole TD.

Collapse: Other Load Cases. Full Evacuation. This load case should be considered when drilling with air or foam. It may also be considered for conductor or surface casing where shallow gas is encountered. This load case would represent all of the mud being displaced out of the wellbore (through the diverter) before the formation bridged off.

Water Gradient. For wells with a sufficient water supply, an internal pressure profile consisting of a freshwater or seawater gradient is sometimes used as a collapse criterion. This assumes a lost-circulation zone that can only withstand a water gradient.

Burst: Gas Migration (Subsea Wells). This load case models bottomhole pressure applied at the wellhead (subject to fracture pressure at the shoe) from a gas bubble migrating upward behind the production casing with no pressure bleedoff at the surface. The pressure is the minimum of the fracture pressure at the shoe and the reservoir pressure plus the mud gradient. The load case has application only to the intermediate casing in subsea wells where the operator has no means of accessing the annulus behind the production casing.

Burst: Tubing Leak. This load case applies to both production and injection operations and represents a high surface pressure on top of the completion fluid because of a tubing leak near the hanger. A worst-case surface pressure is usually based on a gas gradient extending upward from reservoir pressure at the perforations. If the proposed packer location has been determined when the casing is designed, the casing below the packer can be assumed to experience pressure, based on the produced fluid gradient and reservoir pressure only.

Burst: Injection Down Casing. This load case applies to wells that experience high-pressure annular injection operations such as a casing fracture stimulation job. The load case models a surface pressure applied to a static fluid column. This is analogous to a screenout during a frac job.

Collapse Above Packer: Full Evacuation. This severe load case has the most application in gas lift wells. It is representative of a gas filled annulus that loses injection pressure. Many operators use the full evacuation criterion for all production casing strings regardless of the completion type or reservoir characteristics.

Collapse Above Packer: Partial Evacuation. This load case is based on a hydrostatic column of completion fluid equilibrating with depleted reservoir pressure during a workover operation. Some operators do not consider a fluid drop but only a fluid gradient in the annulus above the packer. This is applicable if the final depleted pressure of the formation is greater than the hydrostatic column of a lightweight packer fluid.

Collapse Below Packer: Common Load Case. Full Evacuation. This load case applies to severely depleted reservoirs, plugged perforations, or a large drawdown of a low-permeability reservoir. It is the most commonly used collapse criterion.

Fluid Gradient. This load case assumes zero surface pressure applied to a fluid gradient. A common application is the underbalanced fluid gradient in the tubing before perforating (or after if the perforations are plugged). It is a less conservative criterion for formations that will never be drawn down to zero.

Collapse: Gas Migration (Subsea Wells). This load case models bottomhole pressure applied at the wellhead (subject to fracture pressure at the prior shoe) from a gas bubble migrating upward behind the production casing with no pressure bleedoff at the surface. The pressure distribution is the minimum of the following two pressure distributions. The load case has application only in subsea wells where the operator has no means of accessing the annulus behind the production casing. An internal pressure profile consisting of a completion fluid gradient is typically used.

Collapse: Salt Loads. If a formation that exhibits plastic behavior, such as a salt zone, is to be isolated by the current string, then an equivalent external collapse load (typically taken to be the overburden pressure) should be superimposed on all of the collapse load cases from the top to the base of the salt zone.

Annulur Pressure Buildup. In offshore wells with sealed annuli, increases in fluid temperatures caused by production will cause fluid expansion, resulting in increased fluid pressures. For instance, for water at 100°F, a 1°F increase in temperature will produce a pressure increase of 38 ksi in a rigid container. Fortunately, the casing and formation are sufficiently elastic to greatly reduce this pressure. The equilibrium pressure produced by thermal expansion must be calculated to balance fluid volume change with annular volume change. Nevertheless, the annular pressure change produced by thermal expansion has proved to be a serious design consideration, especially in the North Sea and in deep water.

Mechanical Loads

Changes in Axial Load

In tubing and over the free length of the casing above TOC, changes in temperatures and pressures will have the largest effect on the ballooning and temperature load components. The incremental forces, because of these effects, are given here.



ΔFbal = incremental force because of ballooning, lbf,

υ = Poisson’s ratio (0.30 for steel),

gc = gravity constant, = 1 lbf/lbm,

Δpi = change in surface internal pressure, psi,

Δpo = change in surface external pressure, psi,

Ai = cross-sectional area associated with casing inside diameter (ID), in.,

Ao = cross-sectional area associated with casing outside diameter (OD), in.,

L = free length of casing, in.,

Δρi = change in internal fluid density, lbm/in.3,


Δρo = change in external fluid density, lbm/in.3.



ΔFtemp = incremental force because of temperature change, lbf,

α = thermal expansion coefficient (6.9 × 10–6 °F–1 for steel), °F–1,

E = Young’s modulus (3.0 × 10 7 psi for steel), psi,

As = cross-sectional area of pipe, in.2,


ΔT = average change in temperature over free length, °F.

Axial: Running in Hole

This installation load case represents the maximum axial load that any portion of the casing string experiences when running the casing in the hole. It can include effects such as: self-weight; buoyancy forces at the end of the pipe and at each cross-sectional area change; wellbore deviation; bending loads superimposed in dogleg regions; shock loads based on an instantaneous deceleration from a maximum velocity [this velocity is often assumed to be 50% greater than the average running speed (typically 2 to 3 ft/sec)]; and frictional drag (typically, the maximum axial load experienced by any joint in the casing string is the load when the joint is picked up out of the slips after being made up).

Axial: Overpull While Running

This installation load case models an incremental axial load applied at the surface while running the pipe in the hole. Casing designed using this load case should be able to withstand an overpull force applied with the shoe at any depth if the casing becomes stuck while running in the hole. Certain effects must be considered, such as self-weight; buoyancy forces at the end of the pipe and at each cross-sectional area change; wellbore deviation; bending loads superimposed in dogleg regions; frictional drag; and the applied overpull force.

Axial: Green Cement Pressure Test

This installation load case models applying surface pressure after bumping the plug during the primary cement job. Because the cement is still in its fluid state, the applied pressure will result in a large piston force at the float collar and often results in the worst-case surface axial load. The effects that should be considered are self-weight; buoyancy forces at the end of the pipe and at each cross-sectional area change; wellbore deviation; bending loads superimposed in dogleg regions; frictional drag; and piston force because of differential pressure across float collar.

Axial: Other Load Cases

Air Weight of Casing Only. This axial load criterion has been used historically because it is an easy calculation to perform, and it normally results in adequate designs. It still enjoys significant usage in the industry. Because a large number of factors are not considered, it is typically used with a high axial design factor (e.g., 1.6+).

Buoyed Weight Plus Overpull Only. Like the air weight criterion, this load case has wide usage because it is an easy calculation to perform. Because a large number of factors are not considered, it is typically used with a high axial design factor (e.g., 1.6+).

Axial: Shock Loads

Shock loads can occur if the pipe hits an obstruction or the slips close while the pipe is moving. The maximum additional axial force, because of a sudden deceleration to zero velocity, is given by the equation,



Fshock = shock loading axial force, lbf,

νrun = running speed, ft/sec,

As = pipe cross-sectional area, in.2,

E = Young’s modulus for pipe, lbf/in.2,

ρs = density of pipe, lbm/ft3,


gc = gravity constant, ft/sec2.

The shock load equation is often expressed as



wa = pipe weight per unit length in air, lbm/ft,


vsonic = speed of sound in pipe, ft/sec,

= RTENOTITLE (For steel, vsonic is 16,800 ft/sec.)

For practical purposes, some operators specify an average velocity in this equation and multiply the result by a factor that represents the ratio between the peak and average velocities (typically 1.5).

Axial: Service Loads

For most wells, installation loads will control axial design. However, in wells with uncemented sections of casing and where large pressure or temperature changes will occur after the casing is cemented in place, changes in the axial load distribution can be important because of effects such as self-weight; buoyancy forces; wellbore deviation; bending loads; changes in internal or external pressure (ballooning); temperature changes; and buckling.

Axial: Bending Loads

Stress at the pipe’s OD because of bending can be expressed as



σb = stress at the pipe’s outer surface, psi,

E = modulus of elasticity, psi,

D = nominal outside diameter, in.,


R = radius of curvature, in.

This bending stress can be expressed as an equivalent axial force as



Fbnd = axial force because of bending, lbf,

α/L = dogleg severity (°/unit length),


As = cross-sectional area, in2.

The bending load is superimposed on the axial load distribution as a local effect.

Thermal Loads and Temperature Effects

In shallow normal-pressured wells, temperature will typically have a secondary effect on tubular design. In other situations, loads induced by temperature can be the governing criteria in the design. Next, we discuss how temperature can affect tubular design.

Temperature Effects on Tubular Design

Annular Fluid Expansion Pressure. Increases in temperature after the casing is landed can cause thermal expansion of fluids in sealed annuli and result in significant pressure loads. Most of the time, these loads need not be included in the design because the pressures can be bled off. However, in subsea wells, the outer annuli cannot be accessed after the hanger is landed. The pressure increases will also influence the axial load profiles of the casing strings exposed to the pressures because of ballooning effects.

Tubing Thermal Expansion. Changes in temperature will increase or decrease tension in the casing string because of thermal contraction and expansion, respectively. The increased axial load, because of pumping cool fluid into the wellbore during a stimulation job, can be the critical axial design criterion. In contrast, the reduction in tension during production, because of thermal expansion, can increase buckling and possibly result in compression at the wellhead.

Temperature Dependant Yield. Changes in temperature not only affect loads but also influence the load resistance. Because the material’s yield strength is a function of temperature, higher wellbore temperatures will reduce the burst, collapse, axial, and triaxial ratings of the casing.

Sour Gas Well Design. In sour environments, operating temperatures can determine what materials can be used at different depths in the wellbore.

Tubing Internal Pressure. Produced temperatures in gas wells will influence the gas gradient inside the tubing because gas density is a function of temperature and pressure.

Casing Design

To design a casing string, one must know the purpose of the well, the geological cross section, available casing and bit sizes, cementing and drilling practices, rig performance, as well as safety and environmental regulations. To arrive at the optimal solution, the design engineer must consider casing as a part of a whole drilling system. A brief description of the elements involved in the design process is presented next.

Design Objective

The engineer responsible for developing the well plan and casing design is faced with a number of tasks that can be briefly characterized.

  • Ensure the well’s mechanical integrity by providing a design basis that accounts for all the anticipated loads that can be encountered during the life of the well.
  • Design strings to minimize well costs over the life of the well.
  • Provide clear documentation of the design basis to operational personnel at the well site. This will help prevent exceeding the design envelope by application of loads not considered in the original design.

While the intention is to provide reliable well construction at a minimum cost, at times failures occur. Most documented failures occur because the pipe was exposed to loads for which it was not designed. These failures are called "off-design" failures. "On-design" failures are rather rare. This implies that casing-design practices are mostly conservative. Many failures occur at connections. This implies that either field makeup practices are not adequate or the connection design basis is not consistent with the pipe-body design basis.

Design Method

Phases of Design Process

The design process can be divided into two distinct phases.

Preliminary Design. Typically the largest opportunities for saving money are present while performing this task. This design phase includes data gathering and interpretation; determination of casing shoe depths and number of strings; selection of hole and casing sizes; mud-weight design; and directional design. The quality of the gathered data will have a large impact on the appropriate choice of casing sizes and shoe depths and whether the casing design objective is successfully met.

Detailed Design. The detailed design phase includes: Selection of pipe weights and grades for each casing string. Connection selection. The selection process consists of comparing pipe ratings with design loads and applying minimum acceptable safety standards (i.e., design factors). A cost-effective design meets all the design criteria with the least expensive available pipe.

Required Information

The items listed next are a checklist, which is provided to aid the well planners/casing designers in both the preliminary and detailed design.

  • Formation properties: pore pressure; formation fracture pressure; formation strength (borehole failure); temperature profile; location of squeezing salt and shale zones; location of permeable zones; chemical stability/sensitive shales (mud type and exposure time); lost-circulation zones, shallow gas; location of freshwater sands; and presence of H2S and/or CO2.
  • Directional data: surface location; geologic target(s); and well interference data.
  • Minimum diameter requirements: minimum hole size required to meet drilling and production objectives; logging tool OD; tubing size(s); packer and related equipment requirements; subsurface safety valve OD (offshore well); and completion requirements.
  • Production data: packer-fluid density; produced-fluid composition; and worst-case loads that might occur during completion, production, and workover operations.
  • Other: available inventory; regulatory requirements; and rig equipment limitations.

Preliminary Design

The purpose of preliminary design is to establish casing and corresponding drill-bit sizes, casing setting depths and, consequently, the number of casing strings. Casing program (well plan) is obtained as a result of preliminary design. Casing program design is accomplished in three major steps. First, mud program is prepared; second, the casing sizes and corresponding drill-bit sizes are determined; and next, the setting depths of individual casing strings are found.

Mud Program

The most important mud program parameter used in casing design is the "mud weight." The complete mud program is determined from: pore pressure; formation strength (fracture and borehole stability); lithology; hole cleaning and cuttings transport capability; potential formation damage, stability problems, and drilling rate; formation evaluation requirement; and environmental and regulatory requirements.

Hole and Pipe Diameters

Hole and casing diameters are based on the requirements discussed next.

Production. The production equipment requirements include tubing; subsurface safety valve; submersible pump and gas lift mandrel size; completion requirements (e.g., gravel packing); and weighing the benefits of increased tubing performance of larger tubing against the higher cost of larger casing over the life of the well.

Evaluation. Evaluation requirements include logging interpretation and tool diameters.

Drilling. Drilling requirements include a minimum bit diameter for adequate directional control and drilling performance; available downhole equipment; rig specifications; and available BOP equipment.

These requirements normally impact the final hole or casing diameter. Because of this, casing sizes should be determined from the inside outward starting from the bottom of the hole. Usually the design sequence is as described next.

Based upon reservoir inflow and tubing intake performance, proper tubing size is selected. Then, the required production casing size is determined considering completion requirements. Next, the diameter of the drill bit is selected for drilling the production section of the hole considering drilling and cementing stipulations. Next, one must determine the smallest casing through which the drill bit will pass, and the process is repeated. Large cost savings are possible by becoming more aggressive (using smaller clearances) during this portion of the preliminary design phase. This has been one of the principal motivations in the increased popularity of slimhole drilling. Typical casing and rock bit sizes are given in Table 7.10.

Casing Shoe Depths and the Number of Strings. Following the selection of drillbit and casing sizes, the setting depth of individual casing strings must be determined. In conventional rotary drilling operations, the setting depths are determined principally by the mud weight and the fracture gradient, as schematically depicted in Fig. 7.5, which is sometimes called a well plan. Equivalent mud weight (EMW) is pressure divided by true vertical depth and converted to units of lbm/gal. EMW equals actual mud weight when the fluid column is uniform and static. First, pore and fracture gradient lines must be drawn on a well-depth vs. EMW chart. These are the solid lines in Fig. 7.5. Next, safety margins are introduced, and broken lines are drawn, which establish the design ranges. The offset from the predicted pore pressure and fracture gradient nominally accounts for kick tolerance and the increased equivalent circulating density (ECD) during drilling. There are two possible ways to estimate setting depths from this figure.

Bottom-Up Design. This is the standard method for casing seat selection. From Point A in Fig. 7.5 (the highest mud weight required at the total depth), draw a vertical line upward to Point B. A protective 7 5/8-in. casing string must be set at 12,000 ft, corresponding to Point B, to enable safe drilling on the section AB. To determine the setting depth of the next casing, draw a horizontal line BC and then a vertical line CD. In such a manner, Point D is determined for setting the 9 5/8-in. casing at 9,500 ft. The procedure is repeated for other casing strings, usually until a specified surface casing depth is reached.

Top-Down Design. From the setting depth of the 16-in. surface casing (here assumed to be at 2,000 ft), draw a vertical line from the fracture gradient dotted line, Point A, to the pore pressure dashed line, Point B. This establishes the setting point of the 11¾-in. casing at about 9,800 ft. Draw a horizontal line from Point B to the intersection with the dotted frac gradient line at Point C; then, draw a vertical line to Point D at the pore pressure curve intersection. This establishes the 9 5/8 -in. casing setting depth. This process is repeated until bottom hole is reached.

There are several things to observe about these two methods. First, they do not necessarily give the same setting depths. Second, they do not necessarily give the same number of strings. In the top-down design, the bottomhole pressure is missed by a slight amount that requires a short 7-in. liner section. This slight error can be fixed by resetting the surface casing depth. The top-down method is more like actually drilling a well, in which the casing is set when necessary to protect the previous casing shoe. This analysis can help anticipate the need for additional strings, given that the pore pressure and fracture gradient curves have some uncertainty associated with them.

In practice, a number of regulatory requirements can affect shoe depth design. These factors are discussed next.

Hole Stability. This can be a function of mud weight, deviation and stress at the wellbore wall, or can be chemical in nature. Often, hole stability problems exhibit time-dependent behavior (making shoe selection a function of penetration rate). The plastic flowing behavior of salt zones must also be considered.

Differential Sticking. The probability of becoming differentially stuck increases with increasing differential pressure between the wellbore and formation, increasing permeability of the formation, and increasing fluid loss of the drilling fluid (i.e., thicker mudcake).

Zonal Isolation. Shallow freshwater sands must be isolated to prevent contamination. Lost-circulation zones must be isolated before a higher-pressure formation is penetrated.

Directional Drilling Concerns. A casing string is often run after an angle building section has been drilled. This avoids keyseating problems in the curved portion of the wellbore because of the increased normal force between the wall and the drillpipe.

Uncertainty in Predicted Formation Properties. Exploration wells often require additional strings to compensate for the uncertainty in the pore pressure and fracture gradient predictions.

Another approach that could be used for determining casing setting depths relies on plotting formation and fracturing pressures vs. hole depth, rather than gradients, as shown in Fig. 7.6 and Fig. 7.5. This procedure, however, typically yields many strings and is considered to be very conservative. See the chapter on geoscience principles in this volume of the handbook.

The problem of choosing the casing setting depths is more complicated in exploratory wells because of shortage of information on geology, pore pressures, and fracture pressures. In such a situation, a number of assumptions must be made. Commonly, the formation pressure gradient is taken as 0.54 psi/ft for hole depths less then 8,000 ft and taken as 0.65 psi/ft for depths greater than 8,000 ft. Overburden gradients are generally taken as 0.8 psi/ft at shallow depth and as 1.0 psi/ft for greater depths.

TOC Depths. TOC depths for each casing string should be selected in the preliminary design phase because this selection will influence axial load distributions and external pressure profiles used during the detailed design phase. TOC depths are typically based on zonal isolation; regulatory requirements; prior shoe depths; formation strength; buckling; and annular pressure buildup in subsea wells. Buckling calculations are not performed until the detailed design phase. Hence, the TOC depth may be adjusted, as a result of the buckling analysis, to help reduce buckling in some cases.

Directional Plan. For casing design purposes, establishing a directional plan consists of determining the wellpath from the surface to the geological targets. The directional plan influences all aspects of casing design including mud weight and mud chemistry selection for hole stability, shoe seat selection, casing axial load profiles, casing wear, bending stresses, and buckling. It is based on factors that include geological targets; surface location; interference from other wellbores; torque and drag considerations; casing wear considerations; bottomhole assembly [(BHA) an assembly of drill collars, stabilizers, and bits]; and drill-bit performance in the local geological setting.

To account for the variance from the planned build, drop, and turn rates, which occur because of the BHAs used and operational practices employed, higher doglegs are often superimposed over the wellbore. This increases the calculated bending stress in the detailed design phase.

Detailed Design

Load Cases

In order to select appropriate weights, grades, and connections during the detailed design phase using sound engineering judgment, design criteria must be established. These criteria normally consist of load cases and their corresponding design factors that are compared to pipe ratings. Load cases are typically placed into categories that include burst loads; drilling loads; production loads; collapse loads; axial loads; running and cementing loads; and service loads.

Design Factors

In order to make a direct graphical comparison between the load case and the pipe’s rating, the DF must be considered.



DF = design factor (the minimum acceptable safety factor), and

SF = safety factor.

It follows that


Hence, by multiplying the load by the DF, a direct comparison can be made with the pipe rating. As long as the rating is greater than or equal to the modified load (which we will call the design load), the design criteria have been satisfied.

Other Considerations

After performing a design based on burst, collapse and axial considerations, an initial design is achieved. Before a final design is reached, design issues (connection selection, wear, and corrosion) must be addressed. In addition, other considerations can also be included in the design. These considerations are triaxial stresses because of combined loading (e.g., ballooning and thermal effects)—this is often called "service life analysis"; other temperature effects; and buckling.

Sample Design Calculations

In the examples that are discussed next, burst, collapse, and uniaxial tension failure criteria are examined. Triaxial stresses are calculated for a variety of load situations to demonstrate how the casing strength formulas and the load formulas are actually used.

Example 7.1: Sample Burst Calculation With Triaxal Comparison

Assume that we have a 13 3/8-in., 72-lbm/ft N-80 intermediate casing set at 9,000 ft and cemented to surface. The burst differential pressure for this casing is given by Eq. 7.1.


The load case we will test against is the burst displacement-to-gas case, with formation pressure of 6,000 psi, formation depth at 12,000 ft, and gas gradient equal to 0.1 psi/ft.


According to this calculation, the casing is strong enough to resist this burst pressure. As an additional test, let us calculate the von Mises stress associated with this case. Surface axial stress is the casing weight divided by the cross-sectional area (20.77 in.2) less pressure loads when cemented (assume 15 lbm/gal cement).


The radial stresses for the internal and external radii are the internal and external pressures.


The hoop stresses are calculated by the Lamé formula ( Eq. 7.10 ).


The von Mises equivalent stress or triaxial stress is given as Eq. 7.9. Evaluating Eq. 7.9 at the inside radius and at the outside radius, we have




The maximum von Mises stress is at the inside of the 13 3/8-in. casing with a value that is 66% of the yield stress. In the burst calculation, the applied pressure was 89% of the calculated burst pressure. Thus, the burst calculation is conservative compared to the von Mises calculation for this case.

Example 7.2 Sample Collapse Calculation For the sample collapse calculation, we will test the collapse resistance of a 7-in., 23-lbm/ft P-110 liner cemented from 8,000 to 12,000 ft. Comparing the 7-in. liner properties against the various collapse regimes, it was found that transition collapse was predicted for this liner. The collapse pressure for this liner is calculated from Eq. 7.4 with the following values for F and G, as taken from Table 7.5.


The collapse pressure is then given by


To evaluate the collapse of this liner, we need internal and external pressures. Internal pressure is determined with the full evacuation above packer.


The external pressure is based on a fully cemented section behind the 7-in. liner. The external pressure profile is given by the mud/cement mix-water external pressure profile where the liner is assumed to be cemented in 10-lbm/gal mud with an internal mix-water pressure gradient of 0.45.


An equivalent pressure is calculated from pi and po for comparison with the collapse pressure, pc , through use of Eq. 7.6.


Because pe exceeds pc (4,440 psi), the liner is predicted to collapse. It is not appropriate to calculate a von Mises stress for collapse in this case because collapse in the transitional region is not strictly a plastic yield condition.

Example 7.3: Sample Uniaxal Tension Calculation For this example, consider a 9 5/8-in. 43.5-lbm/ft N-80 production casing in an 11,000-ft vertical well, with TOC at 8,000 ft. The casing is run in 11-lbm/gal water-based mud. The hanging weight in air for the casing is


The casing stress at the surface is Fair divided by the cross-sectional area of the casing, less the hydrostatic pressure at the bottom of the casing when cemented. If we assume 15-lbm/gal cement and 11-lbm/gal displaced mud, this bottomhole pressure is


Therefore, the surface hanging stress is


For N-80 casing, a stress of 31,181 psi leaves a large margin of safety. Next, consider the effects of a stimulation treatment on this surface stress. Assume that the average temperature change in the 0–8,000-ft interval is –50°F. The change in axial stress, because of this temperature increase, is given by Eq. 7.48.


where α is the coefficient of thermal expansion (6.9 × 10 6 /°F for steel) and E is Young’s modulus (30 × 106 psi for steel). The net surface stress in the casing is


Arctic Well Completions

The surface formations in the Arctic, called permafrost, may be frozen to depths in excess of 2,000 ft. In addition to addressing concerns about the freezing of water-based fluids and cement, the engineer must also design surface casing for the unique loads generated by the thawing and refreezing of the permafrost. There are also road and foundation design problems, associated with ice-rich surface permafrost, that are not addressed here.

The following is a qualitative description of the loading mechanism in permafrost. If we consider a block of permafrost before thaw, the overburden and lateral earth pressures surrounding this block are balanced by the intergranular stresses between the soil panicles and the pore pressure in the ice. Upon thaw, the ice changes to water; the volume of the pore fluid decreases by about 9%; and the pore pressure decreases. To maintain equilibrium, the soil compacts, increasing intergranular forces until a new stress state is reached that balances the surrounding earth pressures.

The loading of the permafrost is the pore-pressure change caused by the phase change of the pore ice, illustrated in Fig. 7.7. The pore pressure is discontinuous at the thaw boundary and equal to Δp. Associated with the thaw is a body force or "gravity like" loading caused by the gradient of the pore-pressure change. This loading is equivalent to the loss of the buoyant pressure of the ice on the soil particles.

The mechanical response of the permafrost to the pore-pressure loads determines the casing loads. Experiments on simulated deep-frozen permafrost show that it can be characterized as a linear, isotropic elastic material with coefficients corresponding to the compressibility, C, and shear modulus, G. These moduli are functions of the mean normal effective stress, the soil type, and the degree of consolidation of the soil.

Determining the pore-pressure loading requires knowledge of the pore pressure before and after thaw. Thaw subsidence and freeze-back field tests at Prudhoe Bay suggest that the initial pore pressure is hydrostatic. The following mechanisms influence the final pore pressure. First, water may flow into the thawed zone from the surface, the base of the permafrost, or horizontally through the permafrost. Second, water may flow from one part of the thaw zone to another. Third, dissolved or trapped gases within the frozen ice may evolve and maintain some pressure upon thaw. Finally, the soil may compact so that the pore spaces are no longer undersaturated. If the compaction is sufficient to remove voidage and recompress the pore water, then the pressure within the pore space will rise. This limiting compaction is particularly important near the base of the permafrost, where the permafrost contains initially unfrozen water. Unfrozen water leads to a smaller amount of voidage upon thaw; hence, compaction and repressurization occur at lower soil strains. Unfrozen water may occur as a result of the effects of salinity and of adsorption in fine-grained materials. These effects not only depress the initial freezing point but also cause freezing to occur over a range of several degrees.

Internal Freeze-Back

Most of the discovery wells in the Prudhoe Bay field were lost because of the freezing of annular fluids. This failure mode is called internal freeze-back, to distinguish it from the refreezing of the permafrost, called external freeze-back. The solution to internal freeze-back is to replace freezeable fluids in the annuli with nonfreezeable fluids, such as oil-based fluids or alcohol-based fluids, such as glycol. Complete displacement of water-based fluids is essential for successful mitigation of internal freeze-back.

Permafrost Cementing

Experience has shown that a cement system used for permafrost cementing must meet a minimum set of requirements:

  • Provide an ample thickening time.
  • The ability to set at bottomhole temperatures without requiring external heat.
  • The ability to set with a low heat of hydration.
  • Provide an acceptable WOC time.
  • The ability to set without freezing.
  • The ability to attain adequate compressive strength for the well conditions.
  • Provide stability to freeze/thaw cycling.
  • Other desirable qualities of a permafrost cement system include:
  • The ability to be bulk blended and easily handled by field equipment and personnel.
  • Provide controlled rheology.
  • Provide the ability to be easily mixed in a continuous process at Arctic temperatures.
  • Have no free water.

As with any cementing system, once the slurry is in place, the major consideration of system design becomes long-term performance of the cement. In permafrost cementing, considerations are compressive strength development and stability to freeze/thaw cycling.

Experience with permafrost cementing has shown the value of using high-alumina cements for this application. A high-alumina cement marketed under the name of Ciment Fondu has been used extensively in Arctic/North Slope operations.

Through the use of chemical extenders and freeze depressants, a high-alumina cement can be used to make a permafrost cement system. The system exhibits heat of hydration high enough to enhance the setting process. However, the large quantity of water in an extended system absorbs heat generated during hydration, eliminating the need for fly ash.

A high-alumina cement cannot be blended with Portland cement because blending the two causes extreme acceleration of the high-alumina cement, resulting in severe gelation or "flash" setting. Operators must use extreme caution to prevent contamination of a high-alumina cement system with Portland cement. The chance of contamination can be minimized with astringent cleaning of field bins, bulk trucks, and storage facilities before and after each job using a high-alumina cement system. However, under normal operations, it becomes almost impossible to eliminate the chance of alumina cement and Portland cement contacting each other.

A permafrost cementing system using Portland cement and appropriate cement additives eliminates the chance of this problem occurring. An extended Class G permafrost cement may offer the same performance as the high-alumina cement except that it is compatible with conventional permafrost tail-in cement systems, whereas the high-alumina cement is not. Another feature of extended Class G permafrost cement is superior compressive strength after freeze/thaw cycling. The extended Class G system eliminates the storage and handling problems previously associated with a high-alumina cement system. These attributes make an extended system using Class G Portland cement more cost effective than a high-alumina cement system.

External Freeze-Back

Drilling and production in the Arctic thaws the permafrost. If thawed permafrost is allowed to freeze back, significant collapse loads near the bottom of the permafrost will be generated and must be considered in casing design. The loading mechanism is associated with the phase-change expansion of pore water in the thawed permafrost. The magnitude of the pressure buildup depends on the mechanical response of the frozen permafrost.

The following analytic model was effective in predicting freeze-back pressures. The permafrost is initially thawed to radius rb and then allowed to freeze back to radius ra. These two radii serve to determine the amount of phase-change expansion at each instant in time. At the beginning of freeze-back, the thawed permafrost is nearly saturated because of vertical drainage, water influx from drilling fluids, and compaction of the soil structure. The freeze-back process occurs in three stages: relief of effective stress, elastic behavior, and elastic-yield behavior (see Fig. 7.8). In the first stage, as the pore water freezes, the ice expands into the fluid-filled pores, increasing the porosity and, at the same time, compressing the pore water. The grain size and permeability of the solids and the pressure conditions on the solids and fluids determine which of the two situations will occur. In either case, however, the pressure in the thawed zone will increase until the effective stress between grains is relieved and the material is fluidized (can no longer support shear). The freeze-back radius at which this occurs is denoted by re. Further freezing generates a zone of excess ice between re and ra together with higher pressures within this zone. The second stage of the freeze-back process then occurs, as the frozen permafrost outside re is loaded and responds elastically. Elastic behavior continues until the third stage, when the stress in the permafrost reaches the yield point. A yielded region between rp and re is created, as shown in Fig. 7.8 and grows as freeze-back proceeds. In the model, each of the three stages of freeze-back is treated as a separate boundary-value problem. The model predicts pressures along the entire length of casing through the permafrost at any instant in time during the freeze-back process.

This analytical freeze-back model and its correlation with freeze-back field-test data from Prudhoe Bay yielded the conclusions that are listed next.

  • The 13-in., 72-lbm/ft N-80 casing used in the field test and commonly used at Prudhoe Bay can safely withstand the maximum freeze-back pressures.
  • For freeze-back from large thaw radii (50 ft of production thaw), the maximum pressure is not significantly greater than that for freeze-back from small radii (3 ft of drilling thaw).
  • The maximum freeze-back pressure depends on the elastic and yield properties of permafrost but is most sensitive to the Young’s modulus of frozen permafrost.
  • Based on laboratory studies and supported by field-test data, the creep or viscoelastic behavior of permafrost subject to freeze-back is negligible compared with the purely elastic and yield behavior.
  • To limit the freeze-back pressure, the model is useful in the design of methods to limit the amount of initial thaw or to limit the extent of freeze-back.

Thaw Subsidence

Thaw subsidence is the soil compaction resulting from the thawing of permafrost by a producing oil well. Thaw subsidence should be considered in well design because of the strains induced on well casing by this compaction. Thaw-subsidence effects are influenced considerably by the geometry of the thawed zone. A typical thaw zone is roughly cylindrical and, even after 20 years of production, the radius of this cylinder is less than 2% of the length. The consequences of this geometry are that one-dimensional, vertical compaction is not applicable and that the full 3D geometry must be considered in the analysis. Further, the permafrost loading illustrated in Fig. 7.7 shows radial inward loading applied to the surface of the thawed zone. Thus, any resulting compaction of the permafrost should be predominantly in the radial direction with the gravity like loads carried by the arching support of the surrounding permafrost.

The lateral loading produced some very interesting effects in the thaw-subsidence field test. From 400 to 1,300 ft, the measured strains along the casing alternated between compression and tension. In Fig. 7.9, the alternating strain behavior is explained in terms of layering in the permafrost. A sand layer is bounded above and below by a fine silt layer. As the pore pressure decreases in the thawed zone, the thawed/frozen interface moves inward and the sand layer, which is relatively incompressible compared with the silt layers, elongates along the casing, at the expense of the compressible silts, which contract. The casing experiences tension adjacent to the elongating sands and compression opposite the contracting silts.

Another interesting effect occurs at the base of the permafrost. Below the base, the casing experiences tension, while above the base, the casing experiences compression; this indicates uplifting of the permafrost base. The decrease in pore pressure (as shown in Fig. 7.7) not only causes the thawed/frozen interface to move inward but also causes the permafrost base to move upward.

Thaw-subsidence strain is the most difficult arctic well design quantity to evaluate. The problem is complex and very dependent on lithology and permafrost mechanical properties. On the basis of numerous sensitivity studies, Prudhoe Bay operators developed "bounding curves" for tensile and compressive thaw-subsidence strains. At Prudhoe Bay, for single wells assuming no thaw interference from adjacent wells, calculations give upper-bound tensile strains of 0.5% and upper-bound compressive strains of 0.7 to 0.9%, depending on production variables.

Calculated maximum strains are much higher than those measured in the ARCO/Exxon field test. Maximum field-test strains are 0.08% tension and 0.13% compression. The principal reason for this difference is that the field test did not have a worst-case lithology near the permafrost base where loading mechanisms are greatest. Recall that sand/silt layering is required for maximum strain generation, which was not present at depth in the field test. These values are considerably less than 13 3/8-in. L-80 buttress-casing strain limits. Safety factors are 2.3 in compression and 8.8 in tension.

Risk-Based Casing Design


Oilfield tubulars have been traditionally designed using a deterministic working stress design (WSD) approach, which is based on multipliers called safety factors (SFs). The primary role of a safety factor is to account for uncertainties in the design variables and parameters, primarily the load effect and the strength or resistance of the structure. While based on experience, these factors give no indication of the probability of failure of a given structure, as they do not explicitly consider the randomness of the design variables and parameters. Moreover, the safety factors tend to be rather conservative, and most limits of design are established using failure criteria based on elastic theory. In contrast, reliability-based approaches are probabilistic in nature and explicitly identify all the design variables and parameters that determine the load effect and strength of the structure. Moreover, they use a limit-states approach to the design of tubulars, rather than elasticity-based initial yield criteria to predict structural failure. Such probabilistic design methodologies allow either the computation of the probability of failure (Pf) of a given structure or the design of a structure that meets a target probability of failure.

Reliability-based techniques have been formally applied to the design of load-bearing structures in several disciplines. However, their application to the design of oilfield tubulars is relatively new. Two different reliability-based approaches have been considered: the more fundamental quantitative risk assessment (QRA) approach and the more easily applied load and resistance factor design (LRFD) format. Comparison of SF to the estimated design reliability offers a reliability-based interpretation of WSD and gives insight into the design reliabilities implicit in WSD.


In all design procedures, a primary goal is to ensure that the total load effect of the applied loads is lower than the strength of the tubular to withstand that particular load effect, given the uncertainty in the estimate of the load effect, resistance, and their relationship.

The load effect is related to the resistance of the tubular by means of a relationship, often known as the "failure criterion," which is thought to represent the limit of the tubular under that particular load effect. Thus, the failure criterion is specific to the response of the tubular to that load effect. Three conventional design procedures are considered: WSD, QRA, and LRFD.

Clearly, the relationship between the load effect and resistance and the means of ensuring safety or reliability are different in each of these procedures. In what follows, zi are the variables and parameters (such as tension, pressure, diameter, yield stress, etc.) that determine the load effect and resistance; Q is the total load effect; and R is the total resistance in response to the load effect, Q.

Working Stress Design

WSD is the conventional casing design procedure, as discussed earlier in this chapter, that is, the familiar deterministic approach to the design of oilfield tubulars. In WSD, the load effect is separated from the resistance by means of an arbitrary multiplier, the SF. The estimated load effect is often the worst-case load, Qw, based on deterministic design values for the parameters, zi, that determine the load effect. The estimated resistance is often the minimum resistance, Rmin, based on deterministic design values for the parameters that determine the resistance. The design values chosen in formulating the resistance are such that the resulting resistance is a minimum. In most cases, the limits of design are established using failure criteria based on elastic theory. In some cases, such as collapse, WSD employs empirical failure criteria. In general, the design procedure can be represented by the relationship


The ratio Rmin/SF is called the safe working stress of the structure, hence, the name of the procedure.

The role of the SF is to account for uncertainties in the design variables and parameters, primarily the load effect and the strength or resistance of the structure. The magnitude of the SF is usually based on experience, though little documentation exists on their origin or impact. Different companies use different acceptable SFs for their tubular design. SFs give little indication of the probability of failure of a given structure, as they do not explicitly consider the randomness of the design variables and parameters. Some other limitations of this approach are listed in brief next.

  • WSD designs to worst-case load, with no regard to the likelihood of occurrence of the load.
  • WSD mostly uses conservative elasticity-based theories and minimum strength in design (though this is not a requirement of WSD).
  • WSD gives the engineer no insight into the degree of risk or safety (though the engineer assumes that it is acceptably low), thus making it impossible to accurately assess the risk-cost balance.
  • SFs are based on experience and not directly computed from the uncertainties inherent in the load estimate (though these uncertainties are implicit in the experience).
  • WSD sometimes makes the design engineer change loading or accept smaller SFs to fit an acceptable WSD, without giving him the means to evaluate the increased risk.

Reliability-Based Design Approaches

Both QRA and LRFD are reliability-based approaches. The general principles of reliability-based design are given in ISO 2394, International Standard for General Principles on Reliability of Structures,[5] and a detailed discussion of the underlying theory is given by Kapur and Lamberson.[6] In reliability-based approaches, the uncertainty and variability in each of the design variables and parameters is explicitly considered. In addition, a limit-states approach is used rather than elasticity-based criteria. Thus, the "failure criterion" of WSD is replaced by a limit state that represents the true limit of the tubular for a given load effect. Such probabilistic design approaches allow the estimation of a probability of failure of the structure, thus giving better risk-consistent designs.

Quantative Risk Assessment. In QRA, the limit state is considered directly. The limit state is the relationship between the load effect and resistance that represents the true limit of the tubular. Conceptually, the limit state G(Zi) is written as


where Zi are the random variables and parameters that determine the load effect and resistance for the given limit state. G(Zi) is known as the limit-state function (LSF). In Eq. 7.56, the upper case Z is used to represent the parameters to remind us that the parameters are treated as random variables in QRA. The LSF usually represents the ultimate limit of load-bearing capacity or serviceability of the structure, and the functional relationship depends upon the failure mode being considered. G(Zi) < 0 implies that the limit state has been exceeded (i.e., failure). The probability of failure can be estimated if the magnitude and uncertainty of each of the basic variables, Zi, is known and the mechanical models defining G(Zi) are known through the use of an appropriate theory. The uncertainty in Q(Zi) and R(Zi) is calculated from the uncertainty in each of the basic variables and parameters, Zi , through an appropriate uncertainty propagation model, such as Monte Carlo simulation. Fig. 7.10 illustrates the concept, with the load effect and resistance being shown as random variables. The shaded region shows the interference area, which is indicative of Pf, the probability of failure. It is the area where the loads exceed the strength, hence, this is the area of failure. The interference area can be estimated using reliability theory.

Thus, the probability that any given design may fail can be estimated, given an appropriate limit state and estimated magnitude and uncertainty of each of the basic variables and a reliability analysis tool. The approach previously mentioned, although simple in concept, is usually difficult to implement in practice. First, the LSF is not always a manageable function and is often cumbersome to use. Second, the uncertainty in the load and resistance parameters must be estimated each time a design is attempted. Third, the probability of failure must be estimated with an appropriate reliability analysis tool. It is tempting to treat each of the parameters, Zi, as normal variates and use a first-order approach to the propagation of uncertainty. However, such an analysis would be in error because the variables are usually not normal, and first-order propagation gives reliable information only on the central tendencies of the resultant distributions and is erroneous in estimating the tail probabilities.

From Fig. 7.10, it is clear that it is the tail probabilities that are of interest in our work. Therefore, it is important to do a full Monte Carlo simulation to estimate the probability of failure of any real design with real variables. However, this too, is not easy because to obtain probability of failure information of the order 10–n, the simulation has to go through l0n+2 iterations. Clearly, this is a computer-intensive effort. See the chapter on risk assessment in Emerging and Peripheral Technologies, Vol. VI of this Handbook, for more discussion on the Monte Carlo method.

Load and Resistance Factor Design. The load and resistance factor design approach is a reliability-based approach that captures the reliability information characteristic of quantitative risk assessment and presents it in a design format far more amenable to routine use, just like WSD. The limit state is the same as the one considered by QRA. However, the design approach is simplified by the use of a design check equation (DCE).

LRFD allows the designer to check a design using a simplified DCE. The DCE is usually chosen to be a simple and familiar equation (for instance, the von Mises criterion in tubular design). Appropriate characteristic values of the design parameters are used in the DCE, along with partial factors that account for the uncertainties in the load and resistance and the difference between the DCE and the actual limit state. Thus, if Qchar(zi) and Rchar(zi), respectively, represent the characteristic value of the load effect and of resistance, with zi being the characteristic values of each of the parameters and variables, the DCE can be represented by the inequality


where load factor (LF) and resistance factor (RF) are the partial factors required. In the literature, LF and RF are usually referred to as the load factor and resistance factor, respectively. The LF takes into account the uncertainty and variability in load effect estimation, while the RF takes into account the uncertainty and variability in the determination of resistance, as well as any difference between the LSF and DCE. Any design that satisfies Eq. 7.57 is a valid design. The design check equation can be functionally identical to the LSF, or the functional relationship can be a simple formula specified by the design code or familiar WSD formulas. Note that Eq. 7.57 is merely a conceptual representation. In practice, it might not be possible to separate the load effects and resistance in the way suggested by Eq. 7.57. Moreover, several load effects and resistance terms may be present in the DCE, with varying uncertainties, requiring the use of several partial factors.

Similarity to WSD. We observe, from Eq. 7.57 that the partial factors are, in a sense, similar to the SF used in WSD. Comparing Eq. 7.57 to Eq. 7.55, we notice that both equations are based on deterministic values, and the SF in Eq. 7.55 is replaced by two partial factors. Indeed, the ratio LF/RF is analogous to the SF used in WSD, if the DCE happens to be identical to the WSD failure criterion. Thus, in concept, it may be said that


Despite these similarities, however, there are three crucial differences. First, the loads and resistances are estimated using a set methodology. Second, the load effect and the resistance are treated separately, thus allowing the partial factors to separately account for the uncertainties in each. And third, the magnitude of loads and resistances is based on reliability, rather than being arbitrarily set.

Partial factors are chosen through a process of calibration, where the deterministic DCE with partial factors is calibrated against the probabilistic LSF. Partial-factor values are chosen such that their use in the DCE results in a design that has a preselected target reliability or target probability of failure, as determined from the LSF using reliability analysis. For the partial factors to do so, the calibration process should prescribe a scope of the application of LRFD, and the values of the partial factors should be optimized to ensure a uniform reliability across the scope. The objective is to obtain a set of factors that results in designs of this target probability. In brief, the procedure may be summarized as follows. First, choose a desired target probability of failure. Second, identify the characteristic values of each of the parameters, and the uncertainty and variability about these values. Third, for an assumed set of load and resistance factors, generate a set of "passed" designs from the DCE, across the scope of the structure, for all possible load magnitudes. In other words, all designs that pass the DCE are valid designs. The passing of a design is, of course, controlled by the assumed value of the load and resistance factors. Fourth, for each of the passed designs, estimate the probability of failure from the LSF, taking into account the uncertainty in each of the variables. Fifth, determine the statistical minimum reliability assured by the assumed set of load and resistance factors. This is the reliability (or equivalently, probability of failure) that results from the use of these partial factors. In other words, the probability of failure of any design that results from the use of these partial factors in the DCE will, statistically, be less than or equal to the probability of failure. Sixth, repeat until the set of partial factors results in the desired target probability of failure.

At the end of the process, we have a set of partial factors and their corresponding design reliability. If several target reliabilities are to be aimed for, the procedure is repeated, until a new set of partial factors is obtained.

It must be noted that this is a very brief summary of the approach. Calibration is usually the most time-consuming and rigorous step in devising an LRFD procedure. Several reliability-theory and statistical details such as uncertainty estimation, preprocessing of high-reliability designs, zonation, uniformity of reliability, multiple partial factor calibration, etc. have been omitted for brevity.

Critique of Risk-Based Design

WSD has been used successfully for many years to design casing. It is a simple system, understood by the average drilling engineer, of comparing a calculated worst-case load against the rating of the casing. The safety factors used may neither be based on strict logic nor be the same across industry, but the concept is simple and the numbers are similar. Generally, the system has served the industry well. Risk-based design advocates criticize WSD because the failure models do not always use the ultimate load limit as the failure criterion, but this is not inherent to WSD. In an ideal world, where casing is always within specification, using average safety factors and worst-case estimates of loads, the casing should always be overdesigned.

However, WSD makes no allowance for casing manufactured below minimum specification. The SF used may or may not compensate for the fact that a below-strength joint is in a critical location. The risks cannot be quantified, so there is no way of comparing the relative risks of different designs. It can also lead to a situation in which it is impossible to produce a practical design under extreme downhole conditions. There would be a temptation in this case either to try to justify a reduction in the SF, perhaps by relying on improved procedures, or to re-estimate the loads downward. Also, the system does not usually consider low levels of H2S, causing brittle failure in burst. Improvements such as better quality control, more accurate failure equations, and considering brittle burst could be utilized within a WSD system.

It is reasonable for the nonstatistician to accept that the strengths of joints of casing of the same weight and grade from the same mill will vary symmetrically around a mean value. The product is manufactured from nominally the same materials and by the same process, with the aim of producing identical properties. The predictability of the "resistance" side of the equation has been confirmed by large-scale examination and testing of the finished product.

The "load" side of the equation, such as formation pressures and kick volumes, may not be so predictable. There is also a much smaller data bank available for estimating probabilities. Further, human factors may influence the size of a kick by such things as speed of reaction in closing the well in and choosing the correct choke pressures when killing a kick.

The designer using risk-based casing design, thus has the same problem that the WSD user has—namely, which loads to consider in the design. The risk-based designer has an additional task, the assignment of probabilities to these loads. One could argue further that these loads should be weighted according to the severity of the resulting failure.

If risk-based designs are used to justify thinner/lower-grade casing and pipe manufactured to the same quality standards as used as with WSDs, the wells will not be safer. If risk-based design systems are used by people who do not understand the system, or only use partial factors rather than the full system, wells will not be safer. If the load data have been underestimated, the wells will not be safer, especially in high-temperature/high-pressure wells.

A risk-based design system with more accurate failure equations; account taken of brittle fracture in low levels of H2S; improved quality control of tubulars and connections; accurate load data; engineers who understand the system and the well; and a full training and competence assurance program may produce wells that are as safe as those designed using WSD.


A = constant in plastic collapse equation, dimensionless
Ac = cross-sectional area of coupling, in.2
Acr = critical section area of box, pin, or pipe, whichever is least, in.2
Ai = the inside area of the tubing, πri2, in.2
Ajp = cross-sectional area of the pipe wall under the last perfect thread, π/4[( D – 0.1425)2 – d2 ], in2
Ao = the outside area of the tubing, πro2, in.2
Ap = cross-sectional area of plain-end pipe, in.2
As = cross-sectional area of pipe, in.2
B = constant in plastic collapse equation, dimensionless
C = constant in plastic collapse equation, psi
d1 = diameter at the root of the coupling thread in the power tight position, in.
d = nominal inside diameter of pipe, in.
D = nominal outside pipe diameter, in.
D/t = slenderness ratio, dimensionless
eb = buckling strain, in./in.
ebavg = average buckling strain, in./in.
E = Young’s modulus (3.0 × 107 psi for steel)
Es = pitch diameter at plane of seal, in.
f1, f2, f3 = terms in combined stress effects for collapse, psi
F = constant in transition collapse equation, dimensionless
Fa = axial force (tension positive), lbf
Fb = buckling force (compression positive), lbf
Fbnd = bending stress equivalent force, lbf
Fj = minimum joint strength, lbf
Fp = Paslay buckling force, lbf
Fy = pipe-body axial strength, lbf
gc = gravity constant, 32.2 ft/sec2
G = constant in transition collapse equation, dimensionless
G = shear modulus, psi
G(Zi) = the limit state function
I = moment of inertia, in.4
L = engaged thread length, in.
Lbkl = buckled length of tubing, ft
Lhelmax = maximum helically buckled length, ft
Lhelmin = minimum helically buckled length, ft
M = bending moment, lbf-ft
N = API-defined thread-turns from Spec. 5B[4], dimensionless
pe = equivalent external pressure, psi
pi = internal pressure, psi
po = external pressure, psi
pt = thread pitch, in.
PB = minimum burst pressure, psi
PCIY = coupling internal yield pressure, psi
PE = elastic collapse pressure, psi
Pf = probability of failure, dimensionless
Phel = pitch of helically buckled pipe, ft
PILR = coupling internal leak resistance pressure, psi
PP = plastic collapse pressure, psi
PYp = yield strength collapse pressure, psi
PT = transition collapse pressure, psi
Qchar = characteristic value for the load effect, lbf
r = radial annular clearance, in.
ra = permafrost fluid-zone radius, ft
re = permafrost excess-ice-zone radius, ft
ri = inside radius of the pipe, in.
ro = outside radius of the pipe, in.
rp = permafrost plastic-zone radius, ft
R = radius of curvature
Rchar = characteristic value of the resistance, lbf
Sa = axial stress based on the buoyant weight of pipe, psi
t = nominal wall thickness, in.
T = thread taper, in./in.
u = tubing buckling displacement, in.
Uc = minimum ultimate tensile strength of coupling, psi
Up = minimum ultimate tensile strength of pipe, psi
w = distributed buoyed weight of casing, lbm/in.
wa = weight per unit length of pipe in air, lbm/ft
we = the effective (buoyant) weight per unit length of the tubing, lbm/ft
W = nominal outside diameter of coupling, in.
Wn = lateral contact force, lbf/in.
Yc = minimum yield strength of coupling, psi
Yp = minimum yield stress of pipe, psi
Zi = the random variables and parameters that determine the load effect and resistance for the given limit state
α = thermal-expansion coefficient (6.9 × 10–6 °F–1 for steel), °F–1
α/L = dogleg severity (°/unit length)
γi = the density of the fluid inside the tubing, lbm/ft3
γo = the density of the fluid outside the tubing, lbm/ft3
ΔFbal = incremental force caused by ballooning, lbf
ΔFtemp = incremental force caused by temperature change, lbf
ΔLb = buckling length change, ft
Δp = po – pi , psi
Δpi = change in surface internal pressure, psi
Δpo = change in surface external pressure, psi
ΔT = average change in temperature over free length, °F
Δρi = change in internal fluid density, lbm/ft3
Δρo = change in external fluid density, lbm/ft3
ϴ´ = rate of change of helix angle with respect to pipe length, radians/ft
ϴ = helix angle, radians
κ = curvature, radians/ft
σb = stress at the pipe’s outer surface, psi
σr = radial stress, psi
σVME = triaxial stress, psi
σz = axial stress, psi
σϴ = tangential or hoop stress, psi
υ = Poisson’s ratio (0.30 for steel), dimensionless
φ = wellbore angle with the vertical, radians


  1. 1.0 1.1 1.2 1.3 1.4 1.5 Bull. 5C3, Bulletin for Formulas and Calculations for Casing, Tubing, Drillpipe, and Line Pipe Properties, fourth edition. 1985. Dallas: API.
  2. 2.0 2.1 2.2 Bull. 5C2, Bulletin for Performance Properties of Casing, Tubing, and Drillpipe, eighteenth edition. 1982. Dallas: API.
  3. Crandall, S.H. and Dahl, N.C. 1959. An Introduction to the Mechanics of Solids. New York City: McGraw-Hill Book Company.
  4. 4.0 4.1 4.2 4.3 4.4 Spec. 5B, Specification for Threading, Gauging, and Thread Inspection of Casing, Tubing, and Line Pipe Threads, 14th edition. 1996. Dallas: API.
  5. ISO 2394, International Standard for General Principles on Reliability of Structures, second edition. 1986. Geneva: International Organization for Standardization.
  6. Kapur, K.C. and Lamberson, L.R. 1977 Reliability in Engineering Design. New York City: John Wiley & Sons.

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Wooley, G.R., Christman, S.A., and Crose, J.G. 1977. Strain Limit Design of 13 3/8-in., N-80 Buttress Casing. J. Pet Tech 29 (4): 355–359. SPE-6061-PA.

SI Metric Conversion Factors

ft × 3.048* E – 01 = m
ft/sec × 3.048* E – 01 = m/s
°F   (°F – 32)/1.8   = °C
gal × 3.785 412 E – 03 = m3
in. × 2.54* E + 00 = cm
in.2 × 6.451 6* E + 00 = cm2
ksi × 6.894 757 E + 03 = kPa
lbf × 4.448 222 E + 00 = N
lbm × 4.535 924 E – 01 = kg
psi × 6.894 757 E + 00 = kPa


Conversion factor is exact.