Pressure Normalized Rate (PNR) DCA

1. Basic Principles

1.1. PNR for Oil

The Pressure-Normalized Rate (PNR) method relates production rate to pressure drawdown. We can calculate the PNR for oil using equation below:

$\begin{equation} PNR_{\text{oil}} = \frac{q_{\text{o}}}{p_i - p_{\text{wf}}} \end{equation}$

where, $q_o$ is the oil production rate, $p_i$ is the initial reservoir pressure, and $p_{wf}$ is the flowing-bottomhole pressure (BHP).

1.2. PNR for Gas

The PNR method for gas follows the same fundamental principles as for oil but incorporates pseudo-pressure terms to account for the nonlinear changes in gas properties with pressure. This adjustment is essential because gas compressibility and viscosity vary significantly with pressure, making conventional PNR-based calculations less reliable for gas reservoirs. The equation of PNR for gas is as below:

$\begin{equation} PNR_{\text{gas}} = \frac{q_{\text{g}}}{m(p_i) - m(p_{\text{wf}})} \end{equation}$

and, $m(p)$ represents the pseudo-pressure function, and can be defined as:

$\begin{equation} m(p) = \int_{p_i}^{p} \frac{2p}{\mu z} \, dp \end{equation}$

where, $\mu$ is the gas viscosity and $z$ is the gas compressibility factor.

1.3. PNR Flowrate

Xie (2023) identified that performing DCA in unconventional wells may lead to an overestimated EUR due to early-time production rates exhibiting a prolonged flat trend. A better approach is to use the PNR flowrate, which is obtained by multiplying the PNR by a constant pressure drawdown to ensure a consistent rate unit:

$\begin{equation} q_{PNR} = \frac{q}{p_i - p_{\text{wf}}} \times (p_i - p_{\text{wf,min}}) \end{equation}$

where, $q_{PNR}$ represents the PNR flowrate at a constant pressure drawdown and $p_{wf,min}$ is the minimum or abandonment BHP.

2. Practical Aspects

The PNR flowrate decline trend converges to the actual rate decline trend when BHP is considerably stabilize. In the figure below, the late-time trends of PNR flowrate and actual oil rate are converge. This result demonstrates that performing DCA on PNR flowrate is more reliable than using actual rates, which are highly uncertain, especially in early-time data.

Another key advantage of this method is that the PNR flowrate decline trend is formed at early-time, and when we extend the decline trend to the end of historical period, it aligns well with the actual production trend.

$\label{practical_aspects}$
Source: Xie, Xueying (2025)

2.1. PNR DCA Methodology

Performing DCA on PNR flowrate provides a more reliable EUR estimation as it integrates both rate and pressure decline,overcoming early-time challenges. The methodology involves:

Plotting the PNR flowrate vs cumulative production.
Performing a history match curve fit for the identified decline segment(s).
Forecasting production using the fitted decline parameters until the end of the well's life.

$\label{methodology}$
Source: Xie, Xueying (2025)

2.2. Practial Tips

Here are some practical guidelines for using PNR DCA

For a starting point, setting $b$ = 1 has shown to provide a good fit.
Be cautious when using data within the first 30 days, as early-time data can be highly influenced by the initial reservoir pressure ( $p_{i}$ ).
The minimum or limiting decline rate ( $D_{\text{min}}$ or $D_{\text{lim}}$ ) can be assumed in a range to obtain EUR uncertainty.

3. PNR DCA in whitson+

3.1. Autofit

The autofit algorithm identifies the last 50% data that is non-zero and fits the PNR DCA parameters to this portion. A sequence of zero rates in the data typically indicates a shut-in, requiring re-initialization of the PNR DCA fit.

3.1.1. Residual Function

The fitting routine uses the following residual function in the least squares routine:

$\begin{equation} r_i = w_i(q_i(1+ba_i(t-t_0))^{-1/b} - q) \end{equation}$

where $r_i$ is the residual at time $i$ , $w_i$ is the weight factor of time $i$, $t_0$ is the time index and $q$ is the production rate.

3.1.2. Default Bounds

Parameter	Lower Bound	Higher Bound	Unit
$q_i$	40% of max observed q	20% higher than max observed q	volume unit / time
$a_i$	0	10	/ yr
$b$	0.5	1.2	-

3.1.3. Adjusting Default Autofit Bounds on PNR DCA Parameters

You can adjust the autofit settings by modifying the decline parameter ranges for the default modified Arps decline.

$\label{autofittingPNRDCAsegments}$

In the modified Arps decline, the function automatically adds a final exponential-decline segment once the minimum or limiting effective decline rate is reached.
You can also lock specific DCA Segment parameters, forcing that the autofit function calculates using the provided input values.

3.1.4. Weighting factors

By default, all days have a weighting factor of 1, except when $q$ = 0, where the weighting factor is set to 0.

3.2. General tips

3.2.1. Manually Adjusting PNR DCA Parameter Graphically

In whitson⁺, users can manually adjust specific points on the PNR DCA plot, each influencing different parameters that define the production decline trend.

$\label{manually_adjust_pnr_dca}$

Adjustment points:

First point: modifies both the $q_i$ and decline rate. This adjustment impacts the early production and the overall scale of the decline curve.
Second point: adjusts the $b$ -factor, which controls the curvature of the decline trend. A higher $b$ -factor leads to a slower decline in production, while a lower $b$ -factor results in a steeper drop.
Third point: alters only the decline rate, influencing the rate at which production decreases over time.

Why manual adjustments matter?

Manually refining these parameters allows users to create more accurate production forecasts, particularly in unconventional reservoirs where decline trends may not fit standard analytical models perfectly. This flexibility helps in optimizing well performance analysis and forecasting long-term production more effectively.

3.2.2. Rate-Cum Decline vs Rate-Time Decline

You can view the decline fit in:

Rate vs. cumulative production plot
Rate vs. time plot

$\label{rate-time}$

3.2.3. Saving the PNR DCA fit

You can save the current PNR DCA fit as a saved case by using the SAVE CASE button. This allows you to adjust the fit as needed while keeping a checkpoint to revert to if necessary.

$\label{addingDCAsegments}$

To overlay saved cases in the plot, check the box under the "Show" column for each saved case.
To reload PNR DCA parameters (and the corresponding fit), use the "Load Case" option available for each saved case by clicking the three vertical dots.
If a saved case name already exists, you will be prompted to confirm whether to overwrite the existing case to prevent duplicates.

3.3. Hotkeys

$\label{hotkeys}$

The following hotkeys help streamline workflow and enhance navigation within the software:

3.3.1. Decline Curve Analysis (DCA)

Perform Autofit: Press Alt + Enter to automatically fit the decline curve to the available data.
Save Case: Use Ctrl + S to save the current case, ensuring no changes are lost.

Shift Entire DCA Segment: Hold Alt and drag the mouse to shift the entire decline curve analysis segment on the graph.
Zoom: Click and drag the mouse over a region to zoom in for a more detailed analysis.
Reset Zoom: Double-click anywhere on the plot to reset the zoom to its default view.

Switch Well: Use PgUp/PgDn to navigate between wells in the dataset.
Cycle to Next Phase: Press Shift + S to switch between different production phases.
Change Phase: Use Shift + O/G/W to toggle between oil, gas, and water phases.

References

[1] Xie, Xueying and Shunhua Liu. 2023. Consistent EUR Forecast in Permian Multiphase Unconventional Reservoirs with Pressure Normalized Rate Method. SPE-215173-MS.

[2] Xie, Xueying, Liu, Shunhua, and Courtney Leiker. 2023. Early EUR Indicator for Permian Basin Unconventional Resources Based on Hourly Flowback Data. URTEC-3854193-MS.

[3] Xie, Xueying, Amadi, Samuel U., Leiker, Courtney S. W., Liu, Shunhua, Kinzler, Erik A., Han, Mei, Castillo, Maria G. Melendez, and Santiago P. Rivera. 2021. Flowback Strategy Optimization for Permian Unconventional Bone Spring Sands and Wolfcamp Wells. URTEC-2021-5088-MS.

[4] Xie, Xueying, Fairbanks, Michael D., Fox, Kevin S., and Rena L. Koinis. 2012. A New Method for Earlier and More Accurate EUR Prediction of Haynesville Shale Gas Wells. SPE-159273-MS.