Predicting Productivity Index of Horizontal Wells

The popular Joshi model slightly overestimated the flow resistance of a horizontal well. As a result of this, the Joshi model underpredicts the productivity index (PI) of a horizontal well by a few percent. In the extreme case in which vertical permeability goes to zero, the Joshi model predicts a 0.0 stb∕day-psi PI, which is wrong. In this paper, the flow for a horizontal well is divided into three flows: the flow in the reservoir above the horizontal wellbore, the flow in the reservoir with a thickness of 2$rw$ containing the horizontal wellbore, and a flow in the reservoir below the horizontal well bore. The second flow is assumed to be pure horizontal flow. The first and third flows can be further divided into a horizontal flow and a vertical flow. In this paper, the equation for each flow is provided, and then combining these flows we give the equation to calculate the effective PI of horizontal wells. In addition, when the horizontal wellbore is not located at the h∕2 midpoint of a reservoir, the Joshi model predicts an increasing PI, which is intuitively and mathematically an incorrect trend. This paper derives a new equation to compute the PI of horizontal wells when the wellbore is eccentric relative to the reservoir midpoint. The new equation generates the correct trend.