A Numerical Validation of the Gnirk and Johnson Theoretical Solution for Petroleum Salt Wells

"The theoretical solution for a viscoelastic well was presented by Gnirk and Johnson in 1964. From this an exercise was executed to validate the numerical simulation of a cemented salt well. The exercise comprised the geomechanical behavior through time of a cemented salt well, this was achieved using a finite element commercial software. This simulation showed the accuracy of the theoretical solution for viscoelastic materials such as salt rock. The theoretical solution presented elastic and viscoelastic analytic equations for the tangential and radial stresses. The equations for the elastic material (casing and cement) and for the viscoelastic material (salt rock) are presented here: Where equations 1a & 1b are the radial and tangential stresses of the elastic material; equations 2a & 2b are the radial and tangential stresses of the viscoelastic material; and a, r and b are the geometrical parameters explained in Figure 1. These equations are function of Pb, a stress time dependent factor that comes from the Burgers model of viscoelasticity. This is well explained in Gnirk and Johnson (1964) and for the sake of simplicity is not disclosed here. Po is the stress at section A-A’ dependent of the depth h."