I would like to address about the shoe pressure concept while while circulating gas influx out of wellbore. This is the fist scenario – Top of the gas kick below the casing shoe
The hydrostatic pressure above the casing shoe remained constantly because the fluid column is the same. The over all hydrostatic pressure in the well will decrease because gas expansion when it is being circulated. In order to maintain the bottom hole pressure constant, casing pressure will increase to balance the loss of hydrostatic pressure in the wellbore due to expansion.
You can express in term of an equation below;
bottom hole pressure = casing pressure + hydrostatic pressure in the annulus
Let’s take a look at casing shoe and we can write the relationship as follows;
pressure at shoe = hydrostatic pressure above shoe + casing pressure
If the gas influx is still below the casing shoe, the hydrostatic pressure is the same and the casing pressure will increase; therefore, pressure at shoe will increase and the shoe pressure will reach the highest pressure when the gas influx reaches the casing shoe.
Conclusion:Shoe pressure will increase until the top of the bubble reaches the casing shoe.
Ps – assumptions : water base mud is used and kick is gas.
Reference book: Well Control Books