**5. Results and discussion**

The calculation of embedment drill string depth in the mud cake is a difficult process and mostly depends on the interaction between the mud cake and downhole tools. It also depends on the mechanical properties of the mud cakes.

It's more realistic and reliable to predict and estimate the minimum and maximum stresses. The contour plot in **Figure 7** indicates the values and direction of the minimum and the maximum vertical stress.

**Figure 7.** *Contour plot of maximum induced stresses.*

*Modeling and Analysis Techniques for Solving Mechanical Pipe Sticking Problems… DOI: http://dx.doi.org/10.5772/intechopen.107307*

In this contour, it's clear that the maximum stresses will be concentrated at the contact surfaces between the drill pipe and the wellbore house. This is due to the high contact pressure and high friction force between the pipe and mud.

The various failure criteria are resulting during overbalance pressure conditions, which caused mechanical pipe sticking. The contour plot of total deformation shown in **Figure 8** illustrates the severe deformation that happens during the downward movement of the drill pipe. In this case, the deformable part (mud cake) will deforms, while the non-deformable part (drill pipe and well bore sleeve) doesn't.

The equivalent strain in this process is illustrated in the contour plot as shown in **Figure 9**. The relative displacement of both drill pipe and mud cake is represented in their movement downward for drill pipe and upward for mud cake

**Figure 8.** *Contour plot of total deformation.*

**Figure 9.** *Contour plot of equivalent strain.*

The overall deformation occurrence according to the relative interaction between the parts shows that high deformation values will take place around the drill pipe and will cause high mechanical sticking with time as shown in **Figure 10**.

Regarding the high contact forces between the parts, the probability of pipe stick will become serious, and the probability of freeing the pipe against soaking with time will be very difficult. **Figure 11** shows the difficulty of freeing the drill pipe with soaking time. Consequently, calculating the required time before backing off and circulating out is very essential.

**Figure 11.** *The probability of freeing the pipe against soaking time.*

*Modeling and Analysis Techniques for Solving Mechanical Pipe Sticking Problems… DOI: http://dx.doi.org/10.5772/intechopen.107307*

Based on the foregoing, and due to high sticking forces between the contact surfaces along the drill pipe, so the drag force will be inversely proportional to pipe drill depth. **Figures 12** and **13** are an illustration of the relationship between pipe depth and both drag force and the percentage probability of getting stuck.

**Figure 12.** *The relation between pipe depth and drag force.*

**Figure 13.** *The relation between pipe depth and the percentage probability of getting stuck.*
