## The Role for Material Balance Time

Material balance time (mbt) is the time it would take to produce a specified volume at constant rate and is defined as cumulative production divided by rate. Traditionally it is used to identify flow regimes for unconventional wells and has the effect of transforming variable rate production data in boundary dominated flow (bdf) to constant rate for comparison with analytical solutions. It has the unique property that the transformed rate-time profile for bdf has a hyperbolic b‑exponent of one.

Until recently, I have been a strong supporter of using mbt to detect flow regimes and assist in the forecasting of unconventional wells. I am now of the opinion that mbt calculations require complex adjustments, and there is no well-defined method for determining the timing for and magnitude of those adjustments. To explain my journey in reaching this opinion, I will use the Colorado Codell well API 05-123-37231-00.

A friend says that “

material balance time has a memory”.

By this he means that sharp changes in rate and extended trends where the rates are above or below average will take a long time to be averaged out of the calculation. This is especially true for linear flow where the cumulative production to date is small and has limited capacity to moderate early time rates that are low due to cleanup and stabilization. The result is that the rate-mbt profile is shifted to the left and the slope is reduced.

The problem is illustrated in Figure 1, where we plot rate-time and rate-mbt side by side. We see that the ½ slope diagnostic for linear flow occurs over a different rate and time interval. When the data is fit to modified hyperbolic equations, the switch point differs by more than 2-fold. Only one of these equations can be correct. We will initially assume the correct equation is rate-time.

__Using a reference point for mbt calculation__

When a value is added or subtracted from a straight-line logarithmic plot, the previous straight line will become curved. Could the difference in transition time be related to a lateral translation of mbt caused by lower early time rates? If yes, how might that translation be reversed?

Think of the mbt calculation as being relative to a reference point and not the traditional calculation that begins at time zero. If the reference rate and time is the first point of linear flow, then there will be no translation of linear flow data. The reference mbt is calculated using the reference cumulative volume that would have occurred had all prior data been in linear flow. All other material balance times will be calculated by adding or subtracting monthly volumes from the reference cumulative production. The integration to calculate the reference cumulative volume follows.

*Q = a t* ^{(1+n)}* / *(*1 + n*)*, from q = a t*^{-n}

Does the concept of a reference point work? For a well operating at constant pressure, Anderson and Mattar (Petroleum Society of CIM, paper 2003-201) tell that material balance time in “perfect linear flow” will be double the variable rate producing time. To explore the effect of calculating mbt from a reference point, we plot rate against mbt and twice time as shown in Figure 2.

The new reference point truly causes the mbt to align with double the producing time, resulting in the selection of the same date range for the linear flow period. We observe that the lateral shift is slightly reducing over time through the linear flow period, but this does not seem to be cause for concern.

In Figure 3, we show the best fit modified hyperbolic equations for rate-time and rate-mbt (using reference point). Both fits look good, but the transition times used to obtain the fits still differ by almost 2-fold. Although the use of a reference time helped align the linear flow period it has not resolved the issue of different transition times.

__Reversing the reference point adjustment__

Now we will examine the suitability of a mbt reference point from a different perspective. Historically material balance has been thought to be accurate over longer time frames, but it *“*…. *has a memory*” and takes time to adjust to changes in rate. If that is true, then the adjustment in mbt cumulative recovery used to align linear flow must be reversed. This was done with two changes:

- The mbt recovery in linear flow was calculated by assuming every linear flow data point had its own reference calculated in the same manner as the first linear flow point.
- The remaining difference in cumulative recovery was reduced in equal monthly amounts over the next 10 months. This time frame was arbitrary.

Having made the adjustments, the rate-time profiles for the reference adjusted mbt and reversed reference adjusted mbt were determined assuming the transition time calculated from the rate-time fit. The results are shown in Figures 4a and 4b.

In Figure 4a, the material balance times are above the fit required to align with the rate-time transition. This implies that the post transition mbt is too high. In Figure 4b, the mbt has been reduced and data fit is pretty good. I believe this confirms the increase in recovery used to match linear flow must be reduced later.

The need to adjust cumulative recovery to offset discrete differences between measured rates and rates expected for the flow regime make the use of material balance time for forecasting untenable. My recommendation is to use only rate-time to determine modified hyperbolic parameters when forecasting unconventional wells. Any forecasting method, like Duong, that uses mbt as an integral part of the forecasts should be reviewed to decide whether the mbt deficiencies described in this white paper will influence Duong results. I believe they do because the Duong “m” value is never one, which is what is should be for linear flow.

__Outlier detection__

Does that mean material balance time is no longer useful? No, mbt is useful in the identification of outliers. As we have seen, a large discrete change in rate (an outlier) will change mbt. Thus, a plot of mbt versus rate is a good visual and numeric diagnostic for either bad data or change in operating condition. We have found that a log-log display is most useful because data lies on a straight line.

We recommend mbt be used in conjunction with a log-log plot of the linear flow coefficient, *a*. Here we calculate the value for *a* for all data points where *a = q t ** ^{0.5}* and is the intercept of the linear flow line at a time of one. Linear flow will plot as a horizontal line. Outliers and pre-linear flow data will not fall on the line.

In Figure 5, we illustrate some of the features discussed.

- The line through linear flow does not identify the first two data points as being unsuitable for analysis, but the plot of the
*a*coefficient clearly shows which data should be analyzed for linear flow. - There is a change in slope when data moves out of linear flow.
- In this example, outliers represent months when the rate fell below trend. The mbt increases because of the lower rate and decreases because of the lower cumulative production. Rate is dominant resulting in a net increase in mbt. The outliers are clearly identified.
- When rates return to normal, the mbt remains higher than the previous trend because the loss in cumulative production has a more lasting effect. Over time data will return to the original trend.

We conclude that mbt does not provide additional information in respect of determining modified hyperbolic parameters for forecasting unconventional wells. However, it is an excellent tool in detecting outliers that should be omitted from the evaluation.

## About the Author

Randy Freeborn is an SPE Distinguished Lecturer on type wells and a subject matter expert in empirical forecasting. Currently, he is a Fellow at Aucerna where he is a thought leader, responsible for identifying and inventing engineering technology and assisting clients. He has been a professional engineer for 45 years and is a member of SPEE and SPE. Freeborn has prepared numerous technical papers for presentation at conferences, workshops and industry meetings and has been a guest lecturer at the University of Houston and Texas A&M.