The most commonly used equation to assess the remaining strength of a pipeline with metal loss is the Modified ASME B31G. This is dependent on several variables such as diameter, wall thickness and length and depth of the anomaly. For reference the equation is as follows.

\[\sigma_f=(\sigma_y+10ksi)\frac{1-0.85\frac{d}{t}}{1-0.85\frac{d}{t}\frac{1}{M}}\] and

\[M=\sqrt{1+0.6275\left(\frac{L}{Dt}\right)^2-0.003375\left(\frac{L}{Dt}\right)^4}\] where: d=depth L=length of anomaly D = Diameter t= wall thickness M=folias factor

In calculus, the derivative of the function represents the rate of change of the equation relative to another. Once the derivative is known it is possible to quantify how fast an equation changes when the other variable changes. In this paper we won’t discuss how a derivative is calculated but start from the premises that the derivatives are already given and show how much influence the change in several of the variables have on the failure pressure.

Derivative of B31G w.r.t. depth (d)

The first one is the derivative of B31G with respect to (w.r.t.) the depth. While these derviatives look complicated, in reality several of the terms are used multiple times making it relatively easy to program into something like a spreadsheet or some sort of math software.

\[\frac{\partial B31G}{\partial d}= \frac{0.85 \sigma_S \left(- \frac{0.85 d}{t} + 1\right)}{t \left(- \frac{0.85 d}{t \left(- \frac{0.003375 L^{4}}{\left(D t\right)^{2.0}} + \frac{0.6275 L^{2}}{\left(D t\right)^{1.0}} + 1\right)^{0.5}} + 1\right)^{2} \left(- \frac{0.003375 L^{4}}{\left(D t\right)^{2.0}} + \frac{0.6275 L^{2}}{\left(D t\right)^{1.0}} + 1\right)^{0.5}} - \frac{0.85 \sigma_S}{t \left(- \frac{0.85 d}{t \left(- \frac{0.003375 L^{4}}{\left(D t\right)^{2.0}} + \frac{0.6275 L^{2}}{\left(D t\right)^{1.0}} + 1\right)^{0.5}} + 1\right)}\]
Figure 1

Figure 1

From Figure 1 it can be seen that the rate of change in the failure stress is approximately linear in relation to change in the anomaly depth.

Derivative of B31G w.r.t. wall thickness (t)

\[\frac{\partial B31G}{\partial t}= \frac{0.85 \sigma_S d}{t^{2} \left(- \frac{0.85 d}{t \left(- \frac{0.003375 L^{4}}{\left(D t\right)^{2.0}} + \frac{0.6275 L^{2}}{\left(D t\right)^{1.0}} + 1\right)^{0.5}} + 1\right)} + \frac{\sigma_S \left(- \frac{0.85 d}{t} + 1\right) \left(\frac{0.85 d \left(- \frac{0.003375 L^{4}}{t \left(D t\right)^{2.0}} + \frac{0.31375 L^{2}}{t \left(D t\right)^{1.0}}\right)}{t \left(- \frac{0.003375 L^{4}}{\left(D t\right)^{2.0}} + \frac{0.6275 L^{2}}{\left(D t\right)^{1.0}} + 1\right)^{1.5}} - \frac{0.85 d}{t^{2} \left(- \frac{0.003375 L^{4}}{\left(D t\right)^{2.0}} + \frac{0.6275 L^{2}}{\left(D t\right)^{1.0}} + 1\right)^{0.5}}\right)}{\left(- \frac{0.85 d}{t \left(- \frac{0.003375 L^{4}}{\left(D t\right)^{2.0}} + \frac{0.6275 L^{2}}{\left(D t\right)^{1.0}} + 1\right)^{0.5}} + 1\right)^{2}}\]

Figure 2

Figure 2

Derivative of B31G w.r.t. Length (L)

\[\frac{\partial B31G}{\partial L}= \frac{0.85 \sigma_S d \left(\frac{0.00675 L^{3}}{\left(D t\right)^{2.0}} - \frac{0.6275 L}{\left(D t\right)^{1.0}}\right) \left(- \frac{0.85 d}{t} + 1\right)}{t \left(- \frac{0.85 d}{t \left(- \frac{0.003375 L^{4}}{\left(D t\right)^{2.0}} + \frac{0.6275 L^{2}}{\left(D t\right)^{1.0}} + 1\right)^{0.5}} + 1\right)^{2} \left(- \frac{0.003375 L^{4}}{\left(D t\right)^{2.0}} + \frac{0.6275 L^{2}}{\left(D t\right)^{1.0}} + 1\right)^{1.5}}\]

The following plot shows the rate of change in the failure stress relative to the length of anomaly. This shows that the biggest changes in predicted failure stress occur when the anomaly is initially shorter with the maximum change occurring around 2".

Figure 2

Figure 2