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The *Influx Management Envelope* (IME) Culen et al. (2016), is a decision making tool for how to deal with a influx during
managed pressure drilling (MPD) operations that offers a substantial improvement over the traditional MPD Well Control
Matrix (WCM). However, in the case study by Gabaldon et al. (2017), it has been found that the simplified analytical solution
introduced by Culen et al. (2016) makes the IME inaccurate for many real-world applications. This paper extends the original
IME that considers the gas migration in the annulus as a single bubble, without making the simplifying assumptions that are
required to make the equations explicit and analytically solvable. The underlying equations that are required to develop the
IME is derived from first principles and it is shown that using realistic equations of state (EOS) for gas, and a single-bubble
type model, the equations for the IME can be numerically solved yielding less conservative limits. The proposed approach is
significantly faster than constructing the IME through high-fidelity simulations. The proposed method allows the calculation
of peak circulating pressure at the surface and the maximum weak point pressure for a given kick size and initial shut-in
pressure, as well as a kick envelope with respect to formation limits. This makes the contributions of this paper relevant for
traditional well control decision making as well. The effect of considering various simplifying assumptions on the resulting
IME is studied and different scenarios that compare the results of the proposed approach with that of the original single
bubble equation for the IME by Culen et al. (2016) is presented