A reservoir with strong water drive has; a net pay $($h$)$ of $50ft,$ anisotropic permeability with $k_r,K_{H-\text{m}\text{i}\text{n}},$ and $k_{H-\text{m}\text{a}\text{x}}$ of $0\mathrm{.}1,1,$ and $10md,$ respectively, a wellbore radius $(r_m)$ of $3$ inch, and a drainage area of $40$ acres. An existing vertical well is producing above the bubble point with a total skin $(s_v)$ of $3.$

If a horizontal well of $1000ft$ in length is drilled in the same reservoir without partial penetration skin $(($:$s_R=0\}$ :

$($a$)$ What would be the horizontal well skin $(s_{tu})$ to deliver the same productivity index as the existing vertical well? The horizontal well has the same wellbore radius and equivalent drainage area of the vertical well. Also, the horizontal well is to drill in a direction that maximizes the productivity.

Note, $1$ Acre $=43\mathrm{.}560f{t}^2.$

$($b$)$ What is the ratio of horizontal well skin pressure drop $(p_{s-kz})$ to vertical well skin pressure drop $(p_{s-y})$ when both wells are producing at the same rate? Which completion would you choose? Note, you want to chose the well that has a higher skin pressure drop when both wells have the same productivity index.