## Finding Linear Flow is more complex than you think!

What is linear flow and why is it important to know which portion of the historical data is in linear flow? When a multi-fractured horizontal well commences production, adjacent fractures are not communicating: flow into the fractures acts like the reservoir is infinite. This is termed linear flow. Ultimately the fractures will interfere with one another, linear flow ends, and the production rate decreases at a faster pace. On the graph in Figure 1, linear flow is represented by solid dots. The faster drop-in rate is easily observed. Without knowing when linear flow ends, it will be impossible to obtain an accurate estimate of future production.

Anderson and Mattar (Petroleum Society of CIM, Paper 2003-201) teach that for perfect linear flow, various log-log rate-time plots will have the same ½ slope regardless of the nature of the data: constant rate, constant pressure or material balance time (mbt). For me this has been confusing because when I plot log-log rate-time (constant pressure) and log-log rate-mbt on the same graph the ½ slope time range that identifies linear flow using rate-time differs from that for rate-mbt. Witness the Colorado Codell well API 05-123-37231-00 plotted in Figure 1 and hundreds of others like it that I have reviewed.

For me, this prompted two questions: why are these curves different and which is right? My thought process was influenced by hearing operators asking each other about b-values they were using and discussing that some of their wells have steeper declines than others. Wells in linear flow have the same b-value of 2.0 (½ slope). When the slope is fixed at ½, the nominal decline factor (slope) at time  will be the same, whether the linear flow period has a high or low rate. On the strength of these observations, I felt that rate-mbt was the proper method to identify linear flow. That would result in a steeper rate-time slope, supporting the notion of different b-values and decline factors between wells.

As I worked with more wells, I observed that the rate-time slopes tended to be steeper when the liquid content was higher. This led me to believe that the steeper slope was somehow related to multi-phase flow.

Before we get too far along this path, let me explain that I have recently come to believe that rate-time is, and has always been, the correct method to identify linear flow. Let me explain the journey I took to reach this conclusion and apologize to those who have remained steadfast in their belief in a rate-time diagnostic.

As I re-read Anderson and Mattar, my memory was prompted: in perfect linear flow, mbt should be twice the time obtained for a constant pressure solution. If the measured data is trending as a straight line without normalizing to pressure drawdown, then there is a reasonable expectation that pressure drawdown has stabilized, normalizing is not required and the 2:1 ratio should hold. I plot 2 times the measured time (adjusted to mid-month) with the material balance calculation (also at mid-month) as seen in Figure 2. The curves do not overlay.

What do Anderson and Mattar mean when they use the term perfect linear flow? I think it implies linear flow from the first minute of production. This would result in the calculation of material balance time using a greater value for cumulative production because perfect linear flow would not have lower rates associated with cleanup and stabilization.

To test this concept, I integrated the power law form of the linear flow equation, q = a t-n, to determine revised values of cumulative production up to and including the 1st mid-month of linear flow. Integration was to mid-month, and cumulative production for subsequent months was calculated by adding month volumes to mid-month. We see in Figure 3 that adjusting the cumulative production to reflect “perfect linear flow” results in linear flow identification that is the same whether using rate-time or rate-mbt.

There are some important lessons to be learned:

• Linear flow always has a ½ slope and hyperbolic b-value of two. If your forecast does not have this property, then it is likely that cleanup data or data from another flow regime is improperly included.
• The slope of the power law equation is fixed at ½. If you are in linear flow, at any specific time every well must have the same slope and therefore nominal decline factor. If this is not the case, then some of wells are not in linear flow, making your comparison of the apples to oranges variety.
• If using material balance time, the cumulative production used for that calculation must be re-determined each time there is a discontinuous change in rate related to revised operating conditions such as the installation of artificial lift.

If rate-time and rate-mbt result in the same linear flow indicator, you might be asking why material balance time needed. The answer, coming in a future white paper, is that it will allow independent evaluation of the end of linear flow and the boundary dominated b-value.