Decline Curve Statistics

Decline curve statistics quantify how well a forecast model matches observed production data. These objective functions are essential for:

Comparing multiple decline models to select the best fit
Validating curve-fitting results
Identifying periods where model assumptions break down

Sum of Squared Errors (SSE)

Theory

The Sum of Squared Errors measures the total squared deviation between observed and predicted values:

SSE = \sum_{i=1}^{n} (q_{predicted,i} - q_{observed,i})^2

Property	Value
Range	0 to ∞
Best value	0 (perfect fit)
Units	(rate units)²

Interpretation

SSE	Interpretation
0	Perfect fit (rare with real data)
Low	Good model match
High	Poor fit or wrong model type

!!! warning "Scale Sensitivity" SSE is sensitive to the scale of the data. A well producing 10,000 bbl/day will have much larger SSE than one producing 100 bbl/day, even for the same relative fit quality.

Function

ComputeSse — Sum of squared errors

Parameters:

Parameter	Description	Type
`observedValues`	Actual production rates	Range or array
`predictedValues`	Model-predicted rates	Range or array

Returns: SSE value (scalar)

Log-Space SSE

Theory

Log-space SSE applies the SSE calculation to the logarithms of the values:

SSE_{log} = \sum_{i=1}^{n} \left(\ln(q_{predicted,i} + \varepsilon) - \ln(q_{observed,i} + \varepsilon)\right)^2

Where $\varepsilon = 10^{-9}$ is a small constant to prevent $\ln(0)$ .

Why Log-Space?

Log-space error offers several advantages for decline curve analysis:

Benefit	Explanation
Scale invariance	Treats 10% error the same at 10,000 or 100 bbl/day
Handles wide dynamic range	Decline data spans orders of magnitude
Emphasizes late-time fit	Low-rate tails contribute more relatively
Matches visual assessment	Log-log plots weight errors more evenly

Function

ComputeLogSse — Log-space sum of squared errors

Parameters:

Parameter	Description	Type
`observedValues`	Actual production rates	Range or array
`predictedValues`	Model-predicted rates	Range or array

Returns: Log-space SSE value (scalar)

Weighted SSE

Theory

Weighted SSE allows assigning different importance to each data point:

SSE_{weighted} = \sum_{i=1}^{n} w_i \cdot (q_{predicted,i} - q_{observed,i})^2

Where $w_i \geq 0$ is the weight for sample $i$ .

Weight Applications

Weighting Strategy	Use Case
Time-based	$w_i = t_i$ emphasizes recent data
Inverse variance	$w_i = 1/\sigma_i^2$ for heteroscedastic data
Quality flags	$w_i = 0$ to exclude bad data points
Uniform	$w_i = 1$ (equivalent to standard SSE)

Function

ComputeWeightedSse — Weighted sum of squared errors

Parameters:

Parameter	Description	Type
`observedValues`	Actual production rates	Range or array
`predictedValues`	Model-predicted rates	Range or array
`weightsValues`	Per-sample weights (must be ≥ 0)	Range or array

Returns: Weighted SSE value (scalar)

Weighted Log-Space SSE

Theory

Combines the benefits of log-space error with sample weighting:

SSE_{weighted,log} = \sum_{i=1}^{n} w_i \cdot \left(\ln(q_{predicted,i} + \varepsilon) - \ln(q_{observed,i} + \varepsilon)\right)^2

This is the default objective function used by Petroleum Office's curve fitting algorithms.

Function

ComputeWeightedLogSse — Weighted log-space SSE

Parameters:

Parameter	Description	Type
`observedValues`	Actual production rates	Range or array
`predictedValues`	Model-predicted rates	Range or array
`weightsValues`	Per-sample weights (must be ≥ 0)	Range or array

Returns: Weighted log-space SSE value (scalar)

Model Selection Guidelines

When to Use Each Metric

Metric	Best For
SSE	Quick comparisons, same well/scale
Log SSE	Wide dynamic range, visual log-log match
Weighted SSE	Data quality issues, emphasis control
Weighted Log SSE	Production forecasting (default for fitting)

Converting SSE to Other Metrics

To derive commonly-used metrics from SSE:

Mean Squared Error (MSE):

MSE = \frac{SSE}{n}

Root Mean Squared Error (RMSE):

RMSE = \sqrt{\frac{SSE}{n}}

Normalized RMSE:

NRMSE = \frac{RMSE}{\bar{q}_{observed}}

Troubleshooting

Large SSE Values

Symptom	Likely Cause	Solution
SSE >> 10⁶	Wrong model type	Try different decline model
SSE increases late-time	Interference/restimulation	Fit subset of data
SSE high early	Transient flow period	Exclude early points

Array Length Mismatch

All input arrays must have the same length. If you get an error:

Check for blank cells in your data
Ensure observed and predicted ranges match exactly
Verify weight array (if used) has the same length

Decline Curve Statistics

Sum of Squared Errors (SSE)

Theory

Interpretation

Function

Log-Space SSE

Theory

Why Log-Space?

Function

Weighted SSE

Theory

Weight Applications

Function

Weighted Log-Space SSE

Theory

Function

Model Selection Guidelines

When to Use Each Metric

Converting SSE to Other Metrics

Troubleshooting

Large SSE Values

Array Length Mismatch

See Also

Related Excel Functions