Question

Consider the diffusivity equation for radial flow of a fluid in an infinite and homogeneous reservoir: \[ \frac{1}{r}\frac{\partial}{\partial r}\left( r \frac{\partial P}{\partial r} \right) = \frac{1}{\eta} \frac{\partial P}{\partial t} \] where \(P\) denotes pressure, \(r\) is the radial distance from the centre of the wellbore, \(t\) denotes time, and \(\eta\) is the diffusivity constant. The initial pressure of the reservoir is \(P_i\). The condition(s) used in the derivation of analytical solution of the above equation for pressure transient analysis in an infinite acting reservoir is/are:

• A. At time \(t=0\), \(P = P_i\) for all \(r\).
• B. Wellbore is treated as a line source.
• C. As \(r \to \infty\), \(P \to P_i\) for all \(t\).
• D. At any radius \(r\) and time \(t\), the pressure gradient \(\frac{\partial P}{\partial r}\) is constant.

Hint:

In pressure transient analysis, always apply: - Initial condition: \(P=P_i\) at \(t=0\). - Boundary condition: \(P \to P_i\) as \(r \to \infty\). - Line-source well approximation at \(r=0\).

Answer 1

The Correct Option is A, B, C

Step 1: Initial condition.
At \(t=0\), the reservoir pressure everywhere is equal to the initial pressure \(P_i\). This is the initial condition required for solving the PDE. Hence (A) is correct. Step 2: Wellbore assumption.
For analytical solution, the wellbore radius is assumed negligible compared to reservoir size. Hence, the well is modeled as a line source at \(r=0\). This is the standard line-source approximation. So (B) is correct. Step 3: Infinite reservoir boundary condition.
For an infinite-acting reservoir, at very large distances, the pressure remains undisturbed at initial pressure \(P_i\), irrespective of time: \[ \lim_{r\to\infty} P(r,t) = P_i \] Therefore, (C) is correct. Step 4: Pressure gradient condition.
It is not assumed that \(\frac{\partial P}{\partial r}\) is constant for all \(r\). The gradient depends on both \(r\) and \(t\) and decays with radial distance. Hence (D) is false. Final Answer: \[ \boxed{\text{(A), (B), and (C)}} \]