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.
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.Design methodology
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.
Background
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)
WSD is the conventional casing design procedure, 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
....................(1)
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.
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,[1] and a detailed discussion of the underlying theory is given by Kapur and Lamberson.[2] 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 (QRA)
In quantative risk assessment, 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
....................(2)
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. 2, 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. 1 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.
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. 1, 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.
Load and resistance factor design (LFRD)
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
....................(3)
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. 3 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. 3 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. 3. 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. 3 that the partial factors are, in a sense, similar to the SF used in WSD. Comparing Eq. 3 to Eq. 1, we notice that both equations are based on deterministic values, and the SF in Eq. 1 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
....................(4)
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 a design within this target probability. In brief, the procedure may be summarized as follows:
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 has the same problem that the WSD user has—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 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.
To produce wells that are as safe as those designed using WSD, a risk-based design system needs to include:
Nomenclature
| G(Zi) | = the limit state function |
| Qchar | = characteristic value for the load effect, lbf |
| Zi | = the random variables and parameters that determine the load effect and resistance for the given limit state |
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