Question

Three wells are located at the corners of an equilateral triangle inside an infinite acting reservoir. The reservoir is horizontal, homogeneous, isotropic, and of uniform thickness. The initial reservoir pressure is 3000 psi. If only one well had been producing under transient flow conditions, the pressure at the centre of the equilateral triangle would have dropped by 100 psi after 60 days. If all three wells start producing simultaneously under transient flow conditions at the same rate, then after 60 days, the pressure (in psi) at the centre of the triangle is ____________.

Hint:

Use superposition: since all three wells are equidistant from the centroid and produce at the same rate, the total drawdown is three times the single well drawdown.

Answer 1

Correct Answer: 2700

Step 1: Write the line source solution for a single well: 
For transient radial flow around one well in an infinite acting reservoir, the pressure drop at a distance $r$ from that well after it has produced at a constant rate $q$ for time $t$ is given by the exponential integral solution: $\Delta p(r,t) = -\dfrac{70.6 \, q \, \mu \, B}{k h} Ei\left(-\dfrac{948 \, \phi \, \mu \, c_t \, r^2}{k t}\right)$. This shows that the pressure drop at any point depends only on the rate, the fluid and rock properties, and the ratio $r^2/t$, since these are the quantities that can change from well to well or point to point. 

Step 2: Use the symmetry of the equilateral triangle: 
Let the side length of the triangle be $a$. The distance from each corner, where a well is located, to the centroid of an equilateral triangle is the same for all three corners, equal to $a/\sqrt{3}$. Since $q$, $\mu$, $B$, $k$, $h$, $\phi$, $c_t$ and $t$ are identical for all three wells, same rate, same reservoir, same 60 day production period, and $r$ is also identical for all three wells with respect to the centroid, the argument of the Ei function is exactly the same for each well. So each well produces an identical pressure drop $\Delta p_1$ at the centroid. 

Step 3: Assign the known single well pressure drop: 
It is given that a single well alone would cause a pressure drop of 100 psi at the centroid after 60 days, so $\Delta p_1 = 100$ psi. Since all three wells behave identically as far as the centroid is concerned, same rate, same distance, same time, each one independently contributes 100 psi of drawdown there. 

Step 4: Superpose the three equal contributions and solve for the final pressure: 
With simultaneous production, superposition means the individual drawdowns simply add: $\Delta p_{total} = \Delta p_1 + \Delta p_2 + \Delta p_3 = 100 + 100 + 100 = 300$ psi. Subtracting this total drawdown from the initial reservoir pressure of 3000 psi gives the pressure at the centre of the triangle: $p = p_i - \Delta p_{total} = 3000 - 300 = 2700$ psi. 

Final Answer: 
\[ \boxed{2700 \text{ psi}} \]