Question

Pressure (p) as function of radius (r) and time (t) can be found from the analytical solution of radial diffusivity equation for homogenous and isotropic reservoir following Darcy law. For this solution, if dp/dt is treated as constant, which of the following correctly describe(s) the state(s) of pressure change?

• A. Transient
• B. Pseudo-steady
• C. Steady
• D. Unsteady

Hint:

Ask where dp/dt does not change with time, only pseudo-steady (constant nonzero decline) and steady (dp/dt = 0) qualify.

Answer 1

The Correct Option is B, C

Step 1: Recall the radial diffusivity equation and its flow regimes:
The radial diffusivity equation, \( \frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial p}{\partial r}\right) = \frac{\phi \mu c_t}{k} \frac{\partial p}{\partial t} \), describes how pressure p varies with radius r and time t in a reservoir. Its solution is generally classified into three flow regimes depending on the outer boundary condition and elapsed time, transient (infinite acting), pseudo-steady state, and steady state.
Step 2: Analyze option (A) Transient:
In the transient (infinite acting) regime, the pressure disturbance from the well has not yet reached the outer boundary, and the rate of pressure decline dp/dt keeps changing with both radius and time, it is not constant anywhere in the reservoir. So option (A) does not match the condition of dp/dt being constant.
Step 3: Analyze option (B) Pseudo-steady:
Once the pressure transient reaches a closed (no flow) outer boundary, the reservoir enters the pseudo-steady state (also called semi-steady state or boundary dominated flow). In this regime, every point in the reservoir depletes at the same constant rate, so dp/dt becomes constant with respect to time and is the same at every radius, even though pressure itself still varies with r. This matches the given condition, so option (B) is correct.
Step 4: Analyze option (C) Steady:
In the steady state regime, the outer boundary pressure is held constant, typically by strong aquifer support or a constant pressure boundary, so pressure at every point in the reservoir stops changing with time once equilibrium is reached, meaning dp/dt equals zero everywhere. A value of zero is itself a constant, so steady state also satisfies dp/dt being constant. So option (C) is correct.
Step 5: Analyze option (D) Unsteady:
Unsteady state is essentially the same as the transient regime, where dp/dt varies with time and position and is not constant. So option (D) does not satisfy the given condition.
Final Answer:
\[ \boxed{\text{(B) Pseudo-steady and (C) Steady}} \]