Question

The general equation for the production rate decline can be expressed as \[ \frac{1}{q} \frac{dq}{dt} = -bq^d \] where, b and d are empirical constants, and q is the production rate.
Match the value of \(d\) (Group 1) with the appropriate decline curves (Group 2).

• A. I – P; II – Q; III – R
• B. I – P; II – R; III – Q
• C. I – Q; II – R; III – P
• D. I – Q; II – P; III – R

Hint:

For production rate decline models, \(d = 1\) results in an exponential decline, \(d = 0\) represents a harmonic decline, and \(0<d<1\) represents hyperbolic decline.

Answer 1

The Correct Option is D

The general equation for the production rate decline is a mathematical model used to describe the decline in the production rate of a reservoir over time. The variable \(d\) in the equation affects the type of decline curve produced. Step 1: Understanding the different values of \(d\).
- When \(d = 0\), the equation becomes \( \frac{1}{q} \frac{dq}{dt} = -bq^0 \), which simplifies to \( \frac{dq}{dt} = -b \). This is a constant decline rate and corresponds to a harmonic decline curve. Therefore, the correct matching for \(d = 0\) is P. Harmonic decline.
- When \(d = 1\), the equation becomes \( \frac{1}{q} \frac{dq}{dt} = -bq \), which describes an exponential decline curve, as the rate of decline remains constant over time. Thus, the matching for \(d = 1\) is Q. Exponential decline.
- When \(0<d<1\), the equation results in a hyperbolic decline curve, where the decline rate decreases over time as the production rate decreases, but it does not approach zero as rapidly as the exponential decline. Therefore, the matching for \(0<d<1\) is R. Hyperbolic decline.
Step 2: Final matching of values.
- For \(d = 0\), the correct match is P. Harmonic decline.
- For \(d = 1\), the correct match is Q. Exponential decline.
- For \(0<d<1\), the correct match is R. Hyperbolic decline.
Thus, the correct combination is I – Q; II – P; III – R.