There has been a lot of controversy over the past five years over how to estimate stress/closure pressure from shut-in pressure transients from Diagnostic Fracture Injection Tests (DFITs). Today, I am reporting on an important new publication relevant to the discussion: “Hydromechanical insight of fracture opening and closure during in-situ hydraulic fracturing in crystalline rock” by Dutler, Valley, Gischig, Jalali, Brixel, Krietsch, Roques, and Amann, published in International Journal of Rock Mechanics and Mining Sciences. The paper can be downloaded free-access at:

https://www.sciencedirect.com/science/article/pii/S1365160920308194

The investigators performed a series of fracture injection/shut-in tests in wellbores drilled off a cavern at the Grimsel Test Site (www.grimsel.com) in Switzerland, which is operated by the Swiss National Cooperative for the Disposal of Radioactive Waste (NAGRA). They installed a series of displacement/strain gauges around the fractures as they opened, closed, and propagated. To quote the paper, there were “60 Fibre Bragg Grating (FBG) sensors in the three FBS boreholes (Fig. 1). 20 FBG sensors were installed along each FBS borehole to characterize the strain field in both intact and fractured rock.”

This is a big deal because in most conversations about ‘how to pick closure from a DFIT,’ we don’t have a ‘ground truth’ to compare against reality. But here, we have a direct physical measurement of the fractures closing and opening. This gives us a ‘ground truth’ to compare against the interpretations from the G-function plots.

The results are very consistent – the tangent method is too low. For example, the strain measurements from their interval HF3 show that stress is between 5-6 MPa, with their ‘best estimate’ tabulated as being 5 MPa (Figure 8e from Dutler et al., 2020). As labeled in their figure below, the tangent method arrives at a value of 2.2 MPa. This is a substantial underestimate. Results like this are repeated over and over throughout the paper.

Please note that the paper is published ‘open access,’ and so I am permitted to reproduce their figures in this blog post.

**Background**

The original concept of ‘picking fracture closure’ for stress estimation was to plot pressure versus sqrt of time (or G-time) and find the deviation from the straight line (Castillo, 1987; Nolte, 1988; Zoback, 2007). This method is based on stress estimation methods used successfully since the 1950s (Hubbert and Willis, 1957; Haimson and Fairhurst, 1967; Hickman and Zoback, 1983; Schmitt and Haimson, 2017).

Nolte (1991) hypothesized certain ‘non-ideal’ preclosure processes might also be able to cause deviation from a straight line. The ideas in his paper are reasonable. But they were subsequently misapplied.

For years, DFIT analysis has been done ‘preclosure’ and ‘postclosure,’ skipping over a mathematical description of closure itself (Craig and Blasingame, 2006; Barree et al., 2009). Classical frac simulations (Barree and Mukherjee, 1996) and analytical solutions (Nolte, 1997) assume that when a patch of walls contact, they have zero aperture and leakoff. But then, post-closure solutions assume closed fractures are infinite conductivity! It is inconsistent to assume that the fractures essentially disappear during closure, but then later assume that they are infinite conductivity. This theoretical problem was overlooked for years. Barree and Mukherjee (1996) and Barree et al. (2009) used a numerical simulator that *could only describe preclosure processes*, and so they tried to explain everything seen in a DFIT pressure transient as being *preclosure*. To do so, they adapted the ideas from Nolte (1991) to explain virtually everything apparently on a G-function plot as *non-ideal* *preclosure processes*. Because they assumed all the important features on the plot were *preclosure*, this led them to a critical assumption – the fracture ‘closes’ when G*dP/dG reaches near the peak. This method is called the “tangent method” and is widely used throughout the petroleum industry. The Barree et al. (2009) tangent method yields results inconsistent with all other mainstream stress estimation techniques (Hubbert and Willis, 1957; Haimson and Fairhurst, 1967; Hickman and Zoback, 1983; Castillo, 1987; Nolte, 1988; Zoback, 2007; Schmitt and Haimson, 2017). Proponents of the tangent method have never justified why they advocate a method that is inconsistent with what is done outside the petroleum industry and inconsistent with what was done within the petroleum industry until the mid to late 2000s.

McClure et al. (2014; 2016) performed simulations that unify preclosure, closure, and afterclosure into a single simulation. The simulations seamlessly handle contacting of fracture walls and nonlinearly evolving fracture compliance. The simulations show that *in low permeability rock* contacting of the fracture walls causes increase in pressure derivative. The implication is that the tangent method picks closure are too late, and underestimate stress. Ie – what Barree et al. (2009) term a non-ideal preclosure process can now be explained as being the natural consequence of closure itself.

There are some nuances and subtleties, but the basic concept is fairly simple to explain. Under ideal conditions, dP/dG should be constant prior to closure. This is because (McClure et al., 2016):

is the leakoff rate with respect to G-time. It is constant (prior to closure) as long as Carter leakoff is valid. Specifically: (Nolte, 1979). is the system storage coefficient, equal to (McClure et al., 2016). Until P << Shmin, fracture compliance () is the dominant term. At late time, wellbore storage is the dominant term. When the walls begin to contact (which occurs at a pressure close to Shmin), the fracture stiffness increases, and so storage decreases, and the pressure derivative increases. Note that the actually does change prior to closure, but it is negligibly small, and so the storage coefficient can be well-approximated as being constant prior to closure.

The tangent method arrives at a different interpretation. If you make a plot of G*dP/dG, the tangent to the curve usually happens near the peak. This often occurs at a pressure much lower than Shmin, and lower than the ‘compliance’ closure pick when dP/dG starts to increase. McClure (2017) showed how the peaking of G*dP/dG is caused by the transition from Carter leakoff to impulse flow. This occurs in all shut-in tests, even tests in tests where pressure never exceeds Shmin, and a fracture never forms!

In McClure et al. (2019), coauthors and I took the insights from McClure et al. (2016) and formed them into a complete step-by-step DFIT interpretation procedure that resolves issues caused by the problematic assumptions underlying the tangent method.

**Direct observational evidence**

When discussing different techniques for estimating stress, I’ve found it frustrating that we usually don’t have a ‘ground truth’ reality to compare against. In publications, I and coauthors have shown that mathematical theory demonstrates the fallacy of the tangent method. But this hasn’t been enough to persuade everyone. An alternative perspective was articulated by Barree and Miskimins (2016), in an SPE paper written to criticize our work: “instead of dwelling on derivative calculation methods and theoretical elastic fracture model derivations, the section presents some real physical mechanisms that might (and likely do) account for this observed pressure decline behavior.” These authors reject the use of mathematics to analyze the physics of the problem, evidently because the mathematics leads to the conclusion that contradicts their preexisting beliefs. Fair enough – I recognize that the math won’t be enough to convince everybody. We need direct observational evidence.

There is already a good amount of observational evidence out there. In McClure et al. (2016), we compared downhole tilt-meter results from Branagan et al. (1996) with the shut-in pressure transient and found that the stress was correctly estimated by a ‘compliance’ approach, and not the ‘tangent’ approach. Our interpreted reopening pressure was the same as a previously published interpretation from Gulrajani and Nolte (2000).

If you look at MDT microfrac tests in oil and gas wells, you’ll nearly always see that the observed reopening pressure is higher than the tangent method ‘closure’ estimate. For example, this is demonstrated by Figures 11 and 12 from Malik et al. (2014).

In a horizontal-well DFIT, near-wellbore tortuosity can be 1000-3000 psi. As a result, reopening pressures are problematic to interpret for stress. But in a vertical openhole MDT test, near-wellbore tortuosity should be near-zero and reopening pressure is a great way to cross-validate the stress measurement.

For completeness, I should also note that Craig et al. (2017) attempted to use observational evidence to support the tangent method. This paper contained serious technical and methodological errors, as summarized in a brief response I posted on arXiv (McClure, 2019).

**Dutler et al. (2020)**

The new paper from Dutler et al. (2020) is a powerful new contribution. They performed a series of injection/shut-in/flowback tests performed in six different wellbore intervals. The wells are drilled off a cavern, which makes it feasible to install arrays of strain gauges in closely-nearby offset wells. These strain gauges allow them to watch the fractures open and close.

They provide G-function plots for the injection/shut-in tests, and also provide plots of measured displacement versus pressure. Their Figure 7, below, shows measurements from interval HF8 during a reopening injection. The two plots on the right show displacement versus pressure. This type of plot allows visual identification of the reopening pressure. It is similar to Figure 9A-4 from Gulrajani and Nolte (2000). The changing slope of the line shows the changing stiffness as the fracture reopens. The stress estimate is between 4-5 MPa, with roughly 4.6 MPa as the best estimate.

Below is the G*dP/dG plot from one of the shut-in tests in that same interval, HF8. According to the tangent method, the fracture never closes, even as pressure goes below 3 MPa. Keep in mind – the displacement measurements show that the stress is around 4.6 MPa.

The Dutler et al. (2020) paper shows many of these examples over and over, and the tangent method is consistently shown to be an underestimate of the stress.

**Broader implications for stress measurement**

The shut-in transients from Dutler et al. (2020) do not show strong ‘compliance’ closure transients. This appears to be caused by very rapid closure, a topic discussed in Section 4.1 from McClure et al. (2019). Rapid closure appears to be typical of microfrac tests. Fractures in microfrac tests are relatively small, and smaller fractures have greater stiffness. They open with a smaller aperture and so close more rapidly. Rapid closure can also happen if there is leakoff into high natural fracture conductivity or relatively high matrix permeability. As long as these ‘rapid closure’ tests are performed under conditions that avoid significant near-wellbore tortuosity, it is possible to achieve good stress estimates. However, under ‘rapid closure’ conditions, we should not expect that the ‘compliance’ method will be usable. The fracture closes fast enough that a pressure gradient develops within the fracture, and the pressure transient within the fracture overprints on the behavior of the system. In this case, no dP/dG deflection may be is apparent.

Dutler et al. (2020) found that their most accurate interpretations came from plots of pressure versus dP/dt, such in their Figure 5, below, from interval HF3.

This dP/dt versus P technique was recommended by Lee and Haimson (1989). Prof. Haimson has been a legend of rock mechanics and stress measurement since his PhD work at the University of Minnesota in the 1960s (Haimson and Fairhurst, 1967), so it shouldn’t come as a surprise that his recommended approach turned out to be the best.

To me, the takeaway is that when interpreting microfrac tests or other tests where the fracture closes quickly, and as long as we can safely neglect near-wellbore tortuosity, then I am inclined to use the Lee and Haimson (1989) plot: pressure versus dP/dt. Alternatively, application of the Castillo (1987) method of G-function analysis would be likely to yield a similar result. Plots of dP/dG versus G may be challenging to interpret due to lack of clear compliance response.

It is unclear to me whether the Lee and Haimson (1989) method would be effective in typical oil and gas DFITs in shale. They are typically performed from horizontal wells, and the interpretation must be adapted to deal with the impact of near-wellbore tortuosity. The dP/dG compliance method from McClure et al. (2019) is designed to do that. So, for oil and gas DFIT tests from horizontal wells, I will continue to use the compliance method from McClure et al. (2019). In the substantial majority of these tests, a clear dP/dG closure response is apparent.

I see no circumstance when it would be advisable to use the Barree et al. (2009) tangent method. It systematically underestimates stress, often by a lot. It’s refuted by mathematical analysis, and, over and over, it is refuted by physical observations.

The tangent method does not always lead to severe inaccuracy. In general, the tangent method becomes more inaccurate as the difference between pore pressure and Shmin becomes larger. In a normally pressured reservoir, the Shmin error may be 500-1000 psi out of 5000-10,000 psi – roughly 5-10%. But when you consider net pressure – fracture pressure minus Shmin – error may be 100-300%+.

Overall, this paper from Dutler et al. (2020) is a really outstanding contribution. It is difficult and complex to make these kinds of measurements. We are very fortunate to have these new results. The paper (and their overall work at Grimsel) concerns much more than just this G-function closure discussion, and I encourage you, the reader, to check out more of their work. For example, I recommend the papers: Gischig et al. (2019), and Krietsch et al. (2020), and Villiger et al. (2020).

I know of at least one more dataset out there, not yet published, that has a pretty similar study design as this work from Dutler et al. (2020), and that data suggests similar findings with respect to the tangent method. I’ll post about that dataset once it is public.

**References**

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Barree, R. D. and J. L. Miskimins. 2016. Physical explanations of non-linear derivatives in diagnostic fracture injection test analysis. SPE 179134-MS. Paper presented at the SPE Hydraulic Fracturing Technology Conference, The Woodlands, TX.

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