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: Set up the problem using the principle of superposition: 
In an infinite acting reservoir under transient, unsteady state flow, the pressure drop caused by a single producing well at any point in the reservoir follows the line source solution, also called the exponential integral or Ei function solution. When several wells produce at the same time, the total pressure drop at any observation point is simply the sum of the pressure drops that each well would have caused on its own at that same point and after the same amount of time. This is the principle of superposition, and it applies here because transient radial flow in a homogeneous, isotropic reservoir of uniform thickness behaves as a linear system. 

Step 2: Use the geometry of the equilateral triangle: 
The three wells sit at the corners of an equilateral triangle, and the pressure is required at the centroid of that triangle. For an equilateral triangle, the centroid is equidistant from all three corners. Since the line source solution for pressure drop depends only on the distance from the producing well to the observation point, together with time, rate, and reservoir properties which are identical for all three wells here, each of the three wells produces exactly the same pressure drop at the centroid as a single well alone would produce.
 
Step 3: Apply superposition to find the combined pressure drop: 
It is given that if only one well had been producing, the pressure at the centroid would drop by 100 psi after 60 days. Because all three wells are equidistant from the centroid and each produces at the same rate under the same transient conditions, the pressure drop contributed by each individual well at the centroid is also 100 psi. Adding the contributions of the three wells by superposition gives the total pressure drop at the centroid: \[ \Delta p_{total} = \Delta p_1 + \Delta p_2 + \Delta p_3 = 100 + 100 + 100 = 300 \text{ psi} \] 

Step 4: Calculate the pressure at the centre of the triangle after 60 days: 
The initial reservoir pressure is 3000 psi, and the centroid pressure drops by a total of 300 psi once all three wells are producing together. So the pressure at the centre after 60 days is: \[ p = 3000 - 300 = 2700 \text{ psi} \] 

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