A Brief History of Frac Model Calibration – Cycle I: Fracture Growth Qualification

The Nolte Plot is a staple for hydraulic fracture treatment evaluation, but often overused in shale fracs

Four decades ago, Dr. Ken Nolte and Dr. Michael B. Smith led the way to incorporate observations into frac design decision making by qualifying fracture growth behavior based on observed pressure trends. It was a first effort to evaluate a measurement and observation – of pressure – to understand “what happens down there”.

Amoco started a program in 1978, led by Ken Nolte, to aid in the understanding of hydraulic fracture growth, with a goal of developing technology for data acquisition and improved frac design. This was in response to a scale increase in fracture treatment size from skin fracs to massive hydraulic fractures (MHFs). These MHFs initially disappointed, and the hunt was on to understand why. 

In a series of papers (Nolte and Smith, 1979; Nolte 1979, 1986, 1988), Nolte developed the iconic log-log plot and derived a version of the net pressure equation below: 

We can derive this equation if we seek a 1D-model for net pressure where the fracture dimension L and h remain static (only the fracture width changes), and where we only evaluate how net pressure changes through Darcy’s law for flow between parallel plates and Sneddon’s equation for fracture width.

This net pressure equation is very similar to that of a Perkins-Kern-Nordgren (PKN) model for fracture growth. This is no coincidence, as both assume a fixed fracture height. 

Now, let’s add to the above equation that in a PKN model the fracture length increases as a function of pump time to the power 4/5. This means that most of the injected volume helps to create fracture length, while the remainder is used to increase width (remember height is fixed).  When this length-to-time relationship is substituted for L in the equation above, net pressure becomes a function of pump time to the power 1/5. 

That relationship is also used in Nolte’s log-log plot. As parameters are plotted on a log-log scale, power law relationships become straight lines. Nolte’s defines Mode I fracture growth – shown in the list below with a 1/5 slope in a log net pressure vs log pump time plot – as growth of a perfectly confined fracture. 

As a design goal, Nolte used the observed (net) pressure trend to maintain this desirable state of growth. Once the observed net pressure reaches a “controlling pressure”, fractures may start growing out of zone. This behavior of fracture height growth, referenced as Mode II fracture growth, is associated with leveling of the net pressure slope with pump time, to a near-zero slope. 

A downside to this Mode II growth behavior is that fracture length growth becomes much less efficient – as a metric of target layer coverage – than in growth Mode I, as an increasing portion of the proppant in the job is placed outside the target zone. Nolte’s main remedy to maintain Mode I growth for longer is to lower net pressures. Referring to the net pressure equation above, he suggests this can primarily be achieved with viscosity, as the range in viscosity changes from crosslinked gels to slickwater spans several orders of magnitude. As a secondary parameter, with a much smaller application range, Nolte lists pump rate. A reduction in these parameters does provide more risk for successful proppant placement, and tradeoffs between viscosity and rate on one side, and successful proppant placement on the other side, therefore need to be carefully considered. Also, we have learned that rate and viscosity changes do not affect net pressure as much as the above equation indicates…but more on that next time.

Nolte’s Mode III fracture growth is associated with an arrest in fracture tip extension, or a tip screen-out (TSO). This growth behavior is the easiest to recognize in a Nolte plot, as it appears as a unit slope. Why is a TSO represented by a unit slope?

In the graph below we assume we have a pre-existing fracture in rock that is of a fixed length. If pressure in the frac is lower than closure stress, this pre-existing fracture does not inflate, and the fracture width remains zero. Once the pressure in the fracture exceeds the closure stress in the rock, fracture width is created. 

With the assumption of constant length (like a TSO), fracture width becomes directly proportional to the difference between frac pressure and closure stress, thus directly proportional to the net pressure.

As fracture width is the only growing dimension, it is directly proportional to fracture volume. Ignoring leakoff and assuming a constant pump rate, we can then extend this proportionality to pump time. That completes the loop and makes net pressure proportional to pump time – a unit slope relationship for the parameters of Nolte’s log-log axes. 

Identification of a unit slope in a Nolte log-log plot is therefore taken to represent a TSO, often a desired outcome for a frac job in high-permeability formations. In cases where a log-log slope steeper than a unit slope is observed, the cause is generally a premature screen-out closer to the wellbore, often associated with the connection between the wellbore and the fracture.

The plot below shows a Nolte plot in a highly permeable formation, where for about 15 final minutes this job remains in Mode III / TSO growth, while net pressure rises from 1,250 psi to about 1,850 psi.

Finally, Nolte’s Mode IV fracture growth is associated with a negative log-log slope of net pressure vs pump time. This behavior is associated with unrestricted fracture height growth.

Nolte was the first to admit that this implicit net pressure evaluation relies on idealized data. There are a few challenges associated with the log-log plots usability, especially in today’s shale fracs:

• Net pressure needs to be anchored by a closure stress measurement. In shales, we generally lack a closure stress measurement, apart from some wells where we collect the occasional DFIT. An improper guess at this anchor affects log-log slopes.
• The long laterals through which we pump in shale plays exhibit very high friction during pumping, which can completely overshadow net pressure. It is hard to see a net pressure of 500 – 1,500 psi when total friction during pumping is as high as 4,000 psi. Friction changes can easily distort a net pressure slope. 
• Other growth mechanisms may exhibit similar behavior as the designated growth modes. For example, the simultaneous growth of multiple fractures, may exhibit similar net pressure trends as confined growth.

For those reasons, I feel Nolte’s log-log plot is overused in today’s shale fracs. I walk into our datavans regularly and see company consultants have requested the Nolte plot up on a screen. What are they trying to see? With ~95% end-of-job slurry efficiencies in shale, are we expected to observe an unexpected TSO? Do they use this plot to identify a pre-mature screen-out?

What I like about the request to display this iconic plot is that Liberty Field Engineers need to dig into some frac theory and history. The concept of using observations to teach our frac models started with some of this early work, and it has pushed our industry forward.