Practice and answer pack
Decline curve teaching dataset and worked answer pack
Use this synthetic decline dataset to practice rate evaluation, cumulative integration and time-basis checks. The values are generated from stated parameters and are not production observations.
By PetroCalcHub Team · Updated October 24, 2026 · Free to read and use
Record the generating equation
The baseline exponential case uses 1,000 STB/day and 0.001/day nominal decline. Every sample follows that same equation; no noise or downtime has been added.
This makes it suitable for checking a calculation or teaching integration, while deliberately excluding the preprocessing and fitting challenges of a real production history.
Work the analytic and numerical paths
Evaluate rate at the requested dates and calculate cumulative analytically. Then compare a sample-based approximation using a stated rectangle or trapezoid rule.
Convert the same curve to an annual nominal basis and verify that its rate is unchanged when elapsed time is transformed consistently.
Explain the forecast interpretation
The finite-horizon integral is a model cumulative, not a reserves estimate. The exercises ask for the effect of a terminal rule or stopping condition without assuming commerciality.
Print the answer pack or download the table. Each answer includes units and the relevant assumption so it can be used as a transparent class exercise.
Explore the example
Original synthetic teaching calculation. Equations and fixed controls are stated here; these are not field measurements.
Select one of 5 recorded scenarios. Intermediate states are not interpolated.
Elapsed time (day) = 365. Analytical cumulative (STB): 305803.349; Endpoint rectangle (STB): 253381.778; Endpoint rate (STB/day): 694.196651.
- Analytical cumulative (STB)
- Endpoint rectangle (STB)
Worked example
Rate, integral and endpoint rectangle
Exponential qi = 1,000 STB/day, nominal D = 0.001/day. The finite elapsed duration varies.
Relation and unit basis
q = 1,000 exp(−0.001t); Np = 1,000[1 − exp(−0.001t)]/0.001
t in days gives rate in STB/day and cumulative in STB. Endpoint rectangle q(t)t is shown only as a deliberately approximate comparison.
1. Define the baseline
Elapsed time (day) = 365
Exponential qi = 1,000 STB/day, nominal D = 0.001/day. The finite elapsed duration varies.
2. Select the relation
q = 1,000 exp(−0.001t); Np = 1,000[1 − exp(−0.001t)]/0.001
t in days gives rate in STB/day and cumulative in STB. Endpoint rectangle q(t)t is shown only as a deliberately approximate comparison.
3. Substitute the values
Np(365) = 1,000[1 − exp(−0.365)]/0.001; endpoint rectangle = [1,000 exp(−0.365)] × 365
Retain intermediate precision. The displayed values are rounded only after evaluation.
4. Read the result
Analytical cumulative (STB) = 305803.349; Endpoint rectangle (STB) = 253381.778; Endpoint rate (STB/day) = 694.196651
The endpoint rectangle misses earlier higher rates. The analytical cumulative remains conditional on the finite horizon.
Result and interpretation
Analytical cumulative (STB) = 305803.349; Endpoint rectangle (STB) = 253381.778; Endpoint rate (STB/day) = 694.196651
The endpoint rectangle misses earlier higher rates. The analytical cumulative remains conditional on the finite horizon.
The dataset is deliberately exact. Use it to isolate arithmetic and integration choices before working with noisy histories.
Exact scenario data
Original synthetic teaching calculation. Equations and fixed controls are stated here; these are not field measurements.
| Elapsed time (day) | Analytical cumulative (STB) | Endpoint rectangle (STB) | Endpoint rate (STB/day) |
|---|---|---|---|
| 30 | 29,554.47 | 29,113.37 | 970.4455 |
| 90 | 86,068.81 | 82,253.81 | 913.9312 |
| 365 | 305,803.3 | 253,381.8 | 694.1967 |
| 730 | 518,091 | 351,793.6 | 481.909 |
| 1,825 | 838,782.4 | 294,222.2 | 161.2176 |
Scroll the table to see all columns. The highlighted row is the worked baseline. Downloads retain full numerical precision.
Practice problems and answer key
Work through the question before revealing its answer. Use the stated assumptions and include units in every numerical step.
1. Why is this exact dataset unsuitable as evidence of a model's field prediction accuracy?
Show the worked answer
It was generated by the same stated curve. Recovering it verifies arithmetic, not predictive performance on independent observations.
2. Reproduce the baseline Analytical cumulative (STB) without rounding intermediate values.
Show the worked answer
Np(365) = 1,000[1 − exp(−0.365)]/0.001; endpoint rectangle = [1,000 exp(−0.365)] × 365. Analytical cumulative (STB) = 305803.349. The endpoint rectangle misses earlier higher rates. The analytical cumulative remains conditional on the finite horizon.
3. Compare the first and last recorded scenarios. Which input changes?
Show the worked answer
Elapsed time (day) changes from 30 to 1825. Other controls remain as stated in the premise. Analytical cumulative (STB) changes from 29554.4665 to 838782.356.
4. What would be needed to apply this example to a different physical system?
Show the worked answer
Replace the synthetic controls with compatible documented inputs and check the relation's assumptions. t in days gives rate in STB/day and cumulative in STB. Endpoint rectangle q(t)t is shown only as a deliberately approximate comparison. The endpoint rectangle misses earlier higher rates. The analytical cumulative remains conditional on the finite horizon.
5. Convert 20% effective annual exponential decline to nominal annual decline.
Show the worked answer
D = −ln(0.8) = 0.223143551/year. On a 365-day year, D = 0.000611352/day.
6. For monotonically decreasing rate, which endpoint rectangle sum overstates cumulative?
Show the worked answer
The left-endpoint sum overstates it; the right-endpoint sum understates it. State interval widths and sampling convention.
Before using the result
- Retain generating parameters.
- Use matching rate and time units.
- State the finite horizon and model scope.
The endpoint rectangle misses earlier higher rates. The analytical cumulative remains conditional on the finite horizon.
Continue learning
Start with the broader workflow guide
Sources and method
The explanation and synthetic examples are original PetroCalcHub teaching material. The references below support the definitions and model context. They do not supply field validation for the synthetic scenarios.