My Favorite Model

“It is better to be roughly right than precisely wrong.”, John Maynard Keynes

Gary Wallace and Wyatt Donnelly built their favorite model back in 1985, as documented in the movie Weird Science. In the years following, a few of us also nerded out to build our favorite model. 

This model was simple. It was elegant. Was it a model or was it a supermodel?

In my opinion it is still a frac'ing supermodel. Chris, Liberty’s CEO, baptized it the “Back of the Envelope Model”, where it still easily fits: 

For much of the 1980s and 1990s fracture modelers were polarized along the lines of commercially available fracture models, of which there were several. Fracture modelers’ divisions by “faiths” was predominantly driven by their belief in the underlying assumptions in a frac model. The source of these divisions was the complicated coupling of physical processes in models, the lack of knowledge about “what happens down there”, and the stubbornness of a few big egos. Then, in the early 2000s, modelers were put to shame when it became possible to directly measure fracture geometries in real-time through commercialized downhole tiltmeter and micro-seismic fracture mapping (SPE papers 46194, 40014). 

Completion engineers run fracture growth models for optimization purposes, to determine how changes in fracture treatment design impact production response. For these considerations, we should initially only care about the “big picture”, and therefore only require an engineering approximation for an optimized fracture geometry - not a detailed scientific evaluation. Keynes captures this differing sentiment between science and engineering perfectly in his quote “It is better to be roughly right than precisely wrong.” In our opinion, a modeling approach with this philosophy can help to quickly optimize production. It is better to get the main physical mechanisms and calibration right and obtain an approximate answer, than to worry about the details of growth in a small area of the fracture system.

My favorite model captures the simplicity of this approach perfectly. Use the basic laws of physics for fracture propagation in a width equation and the conservation of mass. Then, conduct a few simple lab and field measurements. Et voila, you have a very rough approximation of fracture height, length and (combined multi-frac) width. 

The two main field measurements for this model can be obtained from (1) the difference in bottomhole ISIP and bottomhole closure stress σclosure, and (2) from the ratio of closure time and the sum of closure time and injection time:

Equation (2) is a significant simplification of the actual implied slurry efficiency, as it assumes no pressure dependence of leakoff behavior. In general, this equation delivers slurry efficiencies within about 10% of the actual implied efficiency value, and it delivers this with a much simpler equation. For those nerds who want to build a more complicated model to estimate of the implied slurry efficiency, use Nolte’s dimensionless closure time tcD in the following equation:

Let us now consider two equations that describe radial fracture growth. First, we consider the mass conservation equation, which basically states that the total volume of fluid remaining in the fracture after leak-off is equal to the product of the fracture dimensions:

On the left side of this equation, implied slurry efficiency multiplied with volume pumped (V) represents the total fluid volume of the fracture system. Typically, we want to know this volume at the end of a frac job, when overall frac dimensions are at their greatest. On the right side, it is assumed that the average fracture width over the fracture surface area is equal to 2/3 of the fracture width at the injection point at the well.

The second equation to consider is the fracture width equation rewritten for a radial fracture, using the fracture opening modulus , which value equals about one quarter of Young’s Modulus of the rock, and which can be measured on core in a lab or in the field using a sonic log:

As we can determine net pressure and slurry efficiency independently through a DFIT, we can now solve this system of two equations with two unknowns for the fracture radius and the fracture width at the wellbore:

That’s the Back-of-the-Envelop model for a radial fracture assuming elastic deformation. 

While this model is simple and elegant, two issues frac modelers have encountered in the early 2000s require us to make this model slightly more complicated, but more universally applicable.

The first issue is that fractures are not often radial. This issue has a complicated history, and I will give you the three-sentence synopsis. Early (2D) frac models through the 1970s generally assumed hydraulic fractures were perfectly confined in height, mostly due to high stress contrasts. Direct measurement of high net pressure (Shlyapobersky et al in SPE paper 18194) in the 1980s changed this assessment, now assuming that fracture growth was mostly radial as net pressures were significantly higher than stress contrasts in layers. Direct fracture geometry measurements in the early 2000s, however, showed much more height confinement, attributed mostly to layer debonding at layer interfaces. 

Bottom line: Microseismic mapping generally shows fractures are 2 to 10 times longer than they are tall (SPE paper 96080). Therefore, we need to account for elliptical frac shapes with a length-to-height ratio alpha obtained from mapping. 

Secondly, through the layer-debonding issue I referenced above, the assumption of continuous elastic width displacement along the fracture height is not accurate anymore, as this width profile likely comprises discontinuities at layer interfaces. 

Bottom line: we need a fudge factor for the relationship between fracture width and the pressure distribution along various rock layers across its height. Enter gamma in equation (5):

This changes equations (6) and (7) in the following way:

Luckily, this new model still fits on the back of an envelope as my updated favorite frac model. It incorporates two big lessons we have learned from the second fracture model calibration wave in the 2000s: fractures are more confined that we would assume based on vertical layer stresses alone, as vertical frac growth causes layer debonding and a general composite layering effect.

What we have now created is an elegant system of equations to determine approximate fracture dimensions from four independent measurements. One of the goals of net pressure history matching is to determine approximate fracture dimensions. This is the simplest model with real-world feedback to get you there. 

In the spreadsheet shown here we have incorporated these equations and tested them for a 2018 URTeC paper with micro-seismic data in the Wolfcamp formation in the Permian Basin. In URTeC 2902311, two of my previous Pinnacle colleagues measured fracture half-lengths of order 1,100 ft and fracture heights of order 950 ft. For the typical fracture volumes in the jobs and the general net pressures, slurry efficiencies and rock moduli that prevail in this area, we can easily match dimensions using a width coupling coefficient of 0.07. This is significantly lower that the default value of 0.636 (2/pi) applicable to pure elastic deformation. 

Note that the model cannot say anything about the extent of the fracture network. The cumulative width we obtain is likely distributed over many multiple fractures and other fracture complexities.

With this simple model calibration, we now have a scoping tool for approximate fracture dimensions for other volumes pumped.

Weird science is nothing unusual among our industry’s frac modelers. Having them all agree that this is an elegant and simple model that approximates what’s happening downhole…well, that would be even weirder.

More information

Liberty customers can request access to both Back-of-the-Envelope Models here.