Production Forecast with Decline Uncertainty

The business question

Well K-12 has 36 months of production history. The usual forecast fits a decline line through the history and extends it: one line, one number for the remaining volume. But the decline rate is estimated from scattered data, so slightly flatter or steeper lines fit almost as well, and over 12 years the difference adds up.

The history also cannot show what has not happened yet: water from the nearest injector may break through and steepen the decline. The model fits an exponential decline to the history by regression, turns the standard errors of the fit into a joint uncertainty for the starting rate and the decline rate, and adds the breakthrough risk.

The result is a forecast band rather than a line, a P90 / P50 / P10 range for the remaining volume, and the chance that the single best-fit forecast overstates what the well will produce.

Who it is for

Reservoir and production engineers, reserves auditors and planners who forecast producing wells.

Main outputs

Annual production with a P90 / P50 / P10 band · remaining recoverable volume and estimated ultimate recovery (EUR) · remaining producing life · the chance that the single best-fit forecast is too high · what drives the range.

How the model works

Four sheets after this one, read from left to right. Each one feeds the next.

1. 1

   Inputs sheet

   Assumptions the history cannot provide: the economic limit, the first forecast year and the water-breakthrough risk (its chance, when it happens and how much it steepens the decline).

2. 2

   History sheet

   Monthly rates for 36 months. An exponential decline is a straight line when the logarithm of the rate is plotted against time, so Excel's LINEST gives the fitted intercept and slope with their standard errors. With their covariance they define a bivariate Normal distribution; VoseMultiNormal draws one plausible line from it in every iteration.

3. 3

   Forecast sheet

   Each year starts at the rate where the previous year ended and declines exponentially. From the year of a breakthrough, the decline is multiplied by the breakthrough factor. Production stops when the rate reaches the economic limit. Annual volumes are the exact integral of the exponential decline.

4. 4

   Results sheet

   Percentile bands of annual production, the remaining volume and EUR, how the single best-fit forecast compares with the simulated range, and the inputs that drive it.

ModelRisk techniques

• Exponential decline by regression
• Fit uncertainty VoseMultiNormal
• Risk event, uncertain timing
• Economic limit
• Percentile fan chart

Find similar models

Product

• ModelRisk

Type

• Simulation

Application

• Forecasting

Uses

• PERT
• Bernoulli
• Multivariate Normal
• Sensitivity analysis (tornado)

Inputs and assumptions

Every assumption in the model, with its uncertainty and where it comes from. The figures are illustrative.

Units and conventions

Rates and volumes

bbl/d = barrels per day, as a monthly or annual average; kbbl = thousand barrels.

Decline

Exponential decline: rate(t) = rate(0) × exp(−D × t), with D per month. A decline of 1.6% per month is about 17% per year.

Percentiles

Petroleum convention: P90 is the low estimate (90% chance of being exceeded), P10 the high estimate.

Key assumptions and limitations

Decline type

Exponential (Arps b = 0). Many wells decline hyperbolically early in life; the same method works with a non-linear fit of the hyperbolic curve.

Fit

The regression's standard errors describe the uncertainty of the line. Month-to-month scatter in the future is ignored because it averages out over a year.

Breakthrough

The steeper decline applies to the whole year in which breakthrough happens, a slightly cautious simplification.

Operations

No workovers, stimulation or downtime beyond what is already in the history.

Data

The production history is synthetic, generated for this example.

How to use it

1. Replace the rates on the History sheet with your own well's monthly rates (36 months); the fit and everything after it update.
2. Press F9 and watch the Forecast sheet: each recalculation is one plausible decline line and breakthrough history.
3. Run a simulation from the ModelRisk ribbon (10,000 samples with a fixed seed are set).
4. Set 'Include the uncertainty of the fit' to 0 on the Inputs sheet and simulate again: what remains is the breakthrough risk alone.

The tornado chart fills in after a simulation with ModelRisk’s classic engine.

ModelRisk functions used

VoseMultiNormal

Draws the intercept and slope of the decline line together, with the correlation the regression implies.

VoseBernoulli, VosePERT

The water-breakthrough risk: whether it happens, when, and how much it steepens the decline.

VoseInput, VoseOutput

Name the sampled parameters and the forecast outputs; each year's production is an output for the fan chart.

VoseSimMean, VoseSimPercentile, VoseSimProbability

Statistics of the simulation for the tables and charts on the Results sheet.

VoseSimSensitivityMean

The tornado chart: every VoseInput ranked by how far the mean remaining volume moves across its range.

LINEST (Excel)

Regression of log-rate on time, with the standard errors of the intercept and the slope.

Further reading

• Sensitivity analysis in Excel: a tornado chart step by step (Vose guide)
• Arps, J. J. (1945). Analysis of decline curves. Transactions of the AIME 160, 228–247.
• Vose, D. (2008). Risk Analysis: A Quantitative Guide, 3rd edition. Wiley — fitting models to data and parameter uncertainty.
• SPE, WPC, AAPG, SPEE, SEG, SPWLA, EAGE (2018). Petroleum Resources Management System (PRMS).