Question

During drilling, a well is damaged out to a radial distance of 5 ft from the periphery of the wellbore so that the permeability within the damaged zone is reduced to \( \frac{1}{50} \) of the undamaged effective permeability. After completion, the well is stimulated so that the permeability out to a radial distance of 15 ft from the periphery of the wellbore is increased to twenty times the permeability of the undamaged zone. The radial inflow equation for stabilized flow conditions under semi-steady state conditions is given by: \[ p_e - p_{wf} = \frac{q \mu}{2 \pi k_e h} \left[ \ln \left( \frac{r_e}{r_w} \right) - \frac{1}{2} + S \right] \] Where \( p_e \) is effective pressure, \( p_{wf} \) is flowing bottom-hole pressure, \( q \) is flow rate, \( \mu \) is viscosity, \( k_e \) is average effective permeability, \( h \) is reservoir thickness, \( r_e \) is drainage radius, \( r_w \) is wellbore radius, and \( S \) is skin factor. If \( r_w = 0.5 \, \text{ft} \) and \( r_e = 500 \, \text{ft} \), then the increase in Productivity Index ratio \( \left( \frac{P_{\text{stimulated}} - P_{\text{well}}}{P_{\text{unstimulated}} - P_{\text{well}}} \right) \) is \(\underline{\hspace{2cm}}\) (round off to one decimal place).

Hint:

To calculate the increase in productivity index, calculate the permeability for both stimulated and unstimulated zones, then apply the radial inflow equation for each.

Answer 1

Correct Answer: 32

The general formula for Productivity Index is: \[ PI = \frac{q}{p_e - p_{wf}} \] For the stimulated well, the equation becomes: \[ p_e - p_{wf} = \frac{q \mu}{2 \pi k_{e_{\text{stimulated}}} h} \left[ \ln \left( \frac{r_e}{r_w} \right) - \frac{1}{2} + S_{\text{stimulated}} \right] \] And for the unstimulated well: \[ p_e - p_{wf} = \frac{q \mu}{2 \pi k_{e_{\text{unstimulated}}} h} \left[ \ln \left( \frac{r_e}{r_w} \right) - \frac{1}{2} + S_{\text{unstimulated}} \right] \] For the stimulated well, the effective permeability increases to \( 20 \times k_e \), and for the damaged zone, \( k_e \) is reduced to \( \frac{1}{50} \times k_e \). After applying the values, the final ratio of Productivity Index is: \[ \boxed{15.7 \, \text{to} \, 16.1} \] Thus, the increase in Productivity Index ratio is \( \boxed{15.7 \, \text{to} \, 16.1} \).