Exponential Production Decline
Falling production rate of a well over time at a constant decline rate.
Formula
Variables
Example
q_0=1000 bpd, D=0.2/yr, t=3: q ≈ 549 bpd
How it works
This equation says that if a well declines by the same fraction of its current output in each unit of time, its rate follows an exponential curve rather than a straight line. Mathematically, it comes from the differential law dq/dt = -Dq, meaning the drop in production is proportional to the amount still being produced. Solving that relation gives q = q_0 e^{-Dt}, so the rate falls quickly at first and then more slowly as time goes on.
When to use it
Use this model when a well is behaving like a simple natural decline with an approximately constant decline constant, especially in early reservoir analysis or quick forecasting. It is useful for estimating future production, comparing wells, and back-calculating the initial rate or decline rate from measured data.
Worked example, step by step
Given: q_0 = 850 bpd, D = 0.15 /yr, t = 4 yr Step 1: Write the formula: q = q_0 e^{-Dt} Step 2: Substitute the values: q = 850 e^{-0.15(4)} Step 3: Simplify the exponent: q = 850 e^{-0.6} Step 4: Evaluate the exponential: e^{-0.6} \approx 0.5488 Step 5: Multiply: q \approx 850 \times 0.5488 = 466.5 Final answer: q \approx 467 bpd
Common mistakes to avoid
- Using a positive exponent, which would make production increase instead of decline
- Mixing up the units of D and t, such as using months with a per-year decline rate without converting
- Treating D as a percent number directly without converting to a decimal rate in the calculation
Did You Know?
Arps' decline curves, from 1945, still forecast the lifetime output of most oil wells.
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