**Introduction**

The ‘system permeability’ in shale is probably the most important single parameter for optimizing frac design and well spacing in shale. An accurate permeability estimate can lead to better frac designs, cluster spacing, and well spacing, and substantial improvement in economic performance (Fowler et al., 2019).

Unfortunately, system permeability is tricky to estimate. In rate-transient analysis (RTA), production from shale is observed to be from a flow regime called ‘linear flow.’ Analysis of linear flow can provide an estimate for the product of ‘effective fracture surface area’ and ‘square root of permeability,’ but not both separately.

Within ResFrac, our preferred method for estimating system permeability is to use the DFIT analysis technique outlined by McClure et al. (2019). We’ve used this method as part of projects in nearly every major shale play in North America (plus Vaca Muerta) and consistently found that it gives a good initial guess for history matching a ResFrac simulation.

Other techniques exist for estimating permeability. Core analysis is often performed to estimate permeability. While this method can be effective, there are two potential issues: (a) different core analysis methods yield different results, and (b) shale matrix core may not sample fractures and other preferential fluid pathways.

In theory, DFIT permeability estimates should be the most accurate because they involve in-situ fluid flow through the reservoir. However, DFIT interpretations are indirect – they require mathematical analysis of pressure transient signals. If the analysis method is unreliable, then the DFIT permeability estimate will be unreliable.

‘Log-derived’ permeability values may come from a few different sources. In shale, if they come from a correlation that has not been calibrated to the particular formation, they cannot be considered reliable and should not be used. If they are calibrated to the formation, then you should always ask what data was used for calibration. Their accuracy can be no greater than the underlying calibration data.

To derive permeability from RTA, it is necessary to assume effective fracture length (surface area). Unfortunately, the effective fracture surface area is very difficult to know, and so RTA permeability values are uncertain.

In this post, I focus on a particular correlation that is sometimes used in the industry to estimate permeability from a shale DFIT: the ‘holistic permeability correlation’ (Barree et al., 2009; Barree, 2020). I show that this correlation can greatly overestimate permeability. When applied to optimize frac design, well spacing, and cluster spacing, overestimated permeability leads to significant loss of economic value (Fowler et al., 2019).

I discuss: (a) theoretical problems with the correlation, (b) theoretical problems with the method of estimating effective fracture length that is used to calibrate the correlation, (c) inconsistency of the correlation with field data, (d) economic implications, and (e) the connection between the holistic permeability correlation and the inaccuracy of the tangent method of estimating fracture closure.

**The Correlation**

There are at least two forms of the holistic permeability correlation. Barree et al. (2009) recommends estimating permeability from the equation:

Equation 1

Alternatively, Barree (2020) recommends estimating permeability from this equation:

Equation 2

Where *k* is permeability (md), *μ* is the viscosity of the *injection fluid *(not the reservoir fluid) (cp), *P* is the pressure in the fracture during propagation (approximately equal to the effective ISIP) (psi), Shmin is the minimum principal stress (psi), *ϕ* is the porosity (unitless), *c _{t}* is the total compressibility (psi^-1),

*G*is the G-time at closure (estimated using the tangent procedure) (unitless),

_{c}*E*is the Young’s modulus of the formation (MMpsi),

*r*is the ratio of effective leakoff area to total area,

_{p}*P*is the closure pressure (estimated using the tangent procedure) (psi), and

_{c}*P*is the pore pressure. Note that while there is disagreement on how to estimate stress from DFITs (McClure et al., 2019), Equations 1 and 2 were designed to be used with the ‘tangent’ method, and so only the tangent method should be used when plugging into these equations.

_{p}The holistic permeability correlation is an ‘empirical’ equation, which means that it is not derived from governing or constitutive equations. Empirical equations are created by plotting up measurements, and then testing different combinations of variables until the equation’s predictions adequately correlate with the calibration data.

From the form of the correlation, it should be clear that it is empirical, not derived from theory. First, it isn’t dimensionally consistent. If you work out all the units on the right-hand side of the equations, they do not work out to the units of the left-hand side (md). Second, the equations use the viscosity of the *injected* fluid, not the *reservoir* fluid. This violates theory because theory predicts that the leakoff coefficient in shale is dominated by the *reservoir effect*, and not by the viscosity of the filtrate zone (which will be exceedingly thin). For this reason, theory would predict that, regardless of any calibration to data, the equations could not possibly provide reliable permeability estimates from closure time because they do not account for the effect of differences in the viscosity of the *reservoir fluid*. Third, Barree et al. (2009; 2020) recommends using 1 cp, the viscosity of the injected water at room temperature, rather than the viscosity at reservoir temperature (approx. 0.3 cp). Fourth, the definition of the Carter leakoff coefficient shows that a permeability estimate based on closure time should scale inversely with the square of the pressure difference between fracture fluid pressure and the formation fluid pressure (Section 6-4 from Economides and Nolte, 2000). But Equation 1 does not have a term accounting for formation fluid pressure, and Equation 2 (in a very unconventional approach) scales the permeability estimate with the inverse of the natural log of the pressure difference to the 9.6 power.

A thought experiment demonstrates why it is necessary to account for reservoir fluid viscosity and fluid pressure. Holding everything else constant, fluid will leak off more rapidly into a formation with lower fluid pressure and lower viscosity. If two DFITs have identical ‘tangent method’ time to closure, but one has lower fluid pressure and viscosity, then for the leakoff rate to be similar between the two tests, the formation with lower pressure and lower viscosity must have lower permeability (assuming, for this thought experiment, that other parameters like porosity, compressibility, and Young’s modulus are the same). If they had the same permeability, then leakoff would occur more rapidly in the test with lower pore pressure and viscosity, and the time to closure would be different between the tests. Equations 1 and 2 do not account for these effects, and so should not be able to reliably predict permeability, regardless of calibration to data.

To assess the reliability of an empirical correlation, we need to be able to assess: (a) the accuracy of the correlation at predicting the calibration data, and (b) the accuracy of the underlying calibration data. To my knowledge, the accuracy of the holistic permeability correlation at predicting the calibration data is not provided in any public document, and so we have no way of assessing its quality of fit. Also, to my knowledge, the source and reliability of the underlying calibration data is also not provided in any public document. However, during the Q&A session of the Barree (2020) webinar, it was stated that the equation was calibrated to RTA matches to production data. Based on this, we can assess whether the permeability estimates used to calibrate the holistic permeability correlation are reliable.

As mentioned above, in order to estimate permeability from an RTA history match, it is necessary to start with an estimate of effective fracture surface area (or, under simplifying assumptions, length). Barree (2016) explains how he and colleagues estimate effective fracture length in shale. The method is based on a large, high-quality set of laboratory experiments performed by the Stim-Lab consortium. These experiments are highly-regarded in the industry, and are comprehensive. However, it is critical to ask how Barree (2016) takes the leap from these laboratory experiments to something very different: predicting actual producing fracture length in shale. We can evaluate their ‘effective fracture length’ calculation from two perspectives: (a) theory, and (b) comparison of the results with field data. Barree (2016) does not provide a clear, step-by-step description of the calculation method. However, the information provided gives enough detail for us to evaluate key assumptions.

**Theoretical Considerations **

Page 19 from Barree (2016) says after estimating dimensionless fracture conductivity: “the flowing length and dimensionless conductivity are then entered into the modified Pratt’s relation.” The Pratt’s equation calculates an ‘effective infinite conductivity fracture length’ from an actual finite fracture conductivity fracture length. However, there is a key fallacy to the procedure from Barree (2016): the Pratt’s equation is only valid for ‘pseudoradial flow.’ For example, page 176 from Smith and Montgomery (2015) provides a worked example demonstrating that even in a highly idealized setup, the Pratt’s equation concept is inaccurate until radial flow is reached. The Pratt’s equation was developed to be used in conventional reservoirs, not shale. Radial flow is not reached during the economic lifetime of a multiple fracture horizontal well in shale.

At high level, in shale wells, the flow geometry can be considered ‘linear flow’ from the matrix into the fracture (flow inward to the fractures perpendicular to their strike). The fractures are finite conductivity, and so fluid pressure in the fracture is increases with distance from the well. Consequently, the ‘deltaP’ driving production weakens with distance from the well. The pressure distribution in the fracture changes over time as fractures experience production in interference with their neighbors. This is fundamentally a different physical process than pseudoradial flow, which is why it is invalid to apply the Pratt’s equation.

There is a second key fallacy to the method described by Barree (2016). Pages 11-21 argue that during production through a fracture in shale, “Capillary and gravity forces dominate the fluid movement to such an extent that viscous gradients are negligible.” This statement is then used to argue that a high percentage of both the unpropped and *propped* hydraulic fracture surface area are entirely unproductive. This statement, and the rest of the discussion and analysis provided, appear to be based on a single phase, uniform-flux fracture flow model (page 11). This analysis is internally inconsistent: to calculate viscous pressure gradients, it assumes the entire fracture surface area is producing at *uniform flux* everywhere along the fracture; then, these uniform flux pressure gradients are used to conclude that most of the fracture must not be producing or is producing very little. The conclusion of the analysis is contradictory to its premise.

A thought experiment shows the fallacy of the logic. Using a uniform-flux model, Barree (2016) claims that there is effectively zero production from most of the fracture because capillary effects should overcome the viscous pressure gradient. However, if much of the fracture truly experienced zero production, then it would drain like a fixed-volume tank, drop down to the bottomhole pressure many 1000s of psi below the formation fluid pressure, and this huge deltaP would overwhelm any capillary effects. In reality, multiphase production from the fracture is determined by a complex equilibrium between physical processes. Real-life is something ‘in-between’ the two extreme cases considered by Barree (2016): the fracture is neither uniform flux, nor is it entirely unproductive across the great majority of its propped length. Analysis of this problem requires a full multiphase, transient flow calculation, such as would be performed in a multiphase reservoir simulator.

**Predictions of Effective Fracture Length**

Aside from theoretical considerations, we can look at the predictions of permeability and effective fracture length and ask if they are consistent with field observations.

Figure 33 from Barree (2016) shows an example output from their calculation of effective fracture length. It yields a prediction between 20 and 60 ft. This ‘effective fracture length’ correlation from Barree (2016) is embodied in the software module PredictK. Based on my conversations with users of PredictK, I have been told that it nearly always arrives at an estimate for effective fracture length that is on the order of a few 10s of feet. This is also consistent with the work presented by Barree et al. (2015), who on page 304 conclude that the effective frac length of a gas shale well is 32 ft.

Production interference tests provide constraints on effective fracture length. If we notice outside wells in a pad outproducing inner wells, we can infer that the effective frac length is at least half the well spacing. The same holds true for a well-to-well interference test: if a well is shut-in and soon after, there is an increase in production at a neighboring well, then the effective frac length must be at least half the well spacing. For example, refer to the offset-well pressure gauge measurements from Raterman et al. (2019), production interference tests showing communication between adjacent wells (Chu et al., 2018), or the rapid depletion seen in a vertical observation well 1000 ft from a Bakken producer in Figure 7 from Fowler et al. (2020).

As another example, let’s consider the Utica shale dataset described by Cipolla et al. (2018) and Fowler et al. (2019). Because of nonuniqueness, it is possible to history match the production data using either a low or high permeability estimate. The higher permeability value implies that the effective fracture length is extremely short, 10s of feet. The lower permeability value implies that the effective fracture length is much longer (hundreds of feet), so that fractures from adjacent wells (825 ft spacing) are overlapping and experience production interference due to fracture overlap.

In an earlier publication, Cipolla et al. (2018) discussed why Hess had considered, and discarded, using the higher permeability. They found that outer wells outproduced inner wells within weeks of going on production. If the effective fracture lengths were a few 10s of feet, production interference (across the 825 ft spacing) would have to occur through flow through the formation, rather than through the hydraulic fractures. Flow through the formation is much slower than flow through conductive hydraulic fractures, and so production interference would be delayed by *years *(Fowler et al., 2019). Conversely, if the fractures were much longer, then fractures from adjacent wells would overlap, and production inference would occur within weeks. Because field data showed that production interference happened within weeks, Cipolla et al. (2018) concluded that the short fracture/high permeability hypothesis was not correct.

**The Holistic Permeability Correlation and Effective Fracture Length**

Returning to Equations 1 and 2, these correlations were calibrated to RTA permeability estimates derived from the method of Barree (2016), and as discussed above, the method systematically underestimates the effective fracture length. The result is that the RTA permeability estimates will systematically overestimate the system permeability, and so Equations 1 and 2 will overestimate the system permeability.

I went back to the Utica dataset from Fowler et al. (2019) and used the holistic permeability correlation to estimate permeability. Equation 1 yielded 540 nd, and Equation 2 yielded 2300 nd. Conversely, the permeability that was found to be consistent with the production interference results was 20 nd (Cipolla et al., 2018; Fowler et al., 2019). 20 nd was also consistent with the permeability estimate from the method of McClure et al (2019).

For an oil shale comparison, I applied the holistic permeability correlation to the Lower Eagle Ford (vertical well) dataset in Figure 10 from McClure et al. (2019). As noted in the text of the paper, the postclosure DFIT analysis yielded an estimate of 162 nd. The operator reported that this result was consistent with their internal estimates. Equation 2 yields a permeability estimate of 31,000 nd.

This comparison with field data is consistent with the expectation (from theoretical considerations) that the holistic permeability correlation should yield large overestimates of permeability. Keep in mind that we cannot rely on the equations to *always* overestimate permeability. Probably, they overestimate permeability much more in some cases than in others.

**Economic Implications**

In Fowler et al. (2019), after doing the history match with both values of permeability, we did sensitivity analysis to optimize well spacing and cluster spacing. Using the overestimated permeability, the analysis suggests that the optimal well spacing is much tighter than it should be, and the optimal cluster spacing is much further than it should be. The consequence is a large loss of economic return due to using the suboptimal well and cluster spacing.

**Connection with Fracture Closure and Breaking the Nonuniqueness**

As reviewed in a recent post and by McClure et al. (2019), the method advised by Barree et al. (2009) for estimating stress from a DFIT systematically underestimates the magnitude of the minimum principal stress and overestimates the net pressure. If the net pressure is overestimated, the length of the fracture created during the DFIT will be underestimated, and so a ‘before closure’ analysis or a frac simulator history match will overestimate permeability.

Because of this, it is plausible that the procedures recommended by Barree et al. (2009; 2020) may give results that are approximately consistent with each other. Underestimated fracture length from the DFIT and during production are both consistent with overestimated permeability. This may allow a history match to both DFIT and production data, giving the impression that the parameter estimates have been validated.

However, these parameter estimates are inconsistent with other field-scale measurements. Direct in-situ observations of fracture closure through strain measurements show that the holistic/tangent closure pick is inaccurate (discussed in this blog post), and as discussed above, production interference tests and downhole pressure measurements during production demonstrate that the permeability and effective fracture length estimates are inaccurate.

As discussed by McClure et al. (2019) and Fowler et al. (2019), numerical simulations of the governing and constitutive equations predict that a ‘false radial’ signature will be observed in shale gas reservoirs. This prediction is confirmed by field data, where false radial is, in fact, observed in the significant majority of gas shale DFITs. False radial in gas shale DFITs lead to greatly overestimated permeability. To my knowledge, false radial signatures were not used in the calibration of Equations 1 and 2. Nevertheless, we should keep in mind that because they yield overestimated permeability values, false radial estimates have the potential to reinforce an apparent, but false, ‘validation’ of the permeability estimates from Equations 1 and 2.

**Final Reflections**

This post demonstrates the value of conventional hydraulic fracturing theory, as embodied in textbooks such as Smith and Montgomery (2015) and Economides and Nolte (2000). Governing and constitutive equations are based on systematic comparison with observation, by many scientists and engineers over many years. They are subject to simplifying assumptions; they are not infallible, and their predictions should be consistently reexamined against empirical data and scrutinized. Having said that, theoretical predictions are very often correct. Even in situations with uncertain physics, they can be very helpful in identifying where to look in empirical data to find opportunities for improvement.

Application of theory finds fallacies to the analysis of Barree et al. (2009) for closure estimation and for permeability estimation, and the analysis for Barree (2016) for estimating effective fracture length. Empirical observations confirm these theoretical predictions – that the interpretation methods from Barree et al. (2009; 2016; 2020) are inaccurate.

Data-driven and physics-based approaches are most effective when used together synergistically (McClure et al., 2020). In this case, the payoff is more accurate permeability estimation, and consequently, better designs and significantly improved economic performance.

**References**

Barree, R. D., V. L. Barree, and D. P. Craig. 2009. Holistic fracture diagnostics: consistent interpretation of prefrac injection tests using multiple analysis methods. SPE Production & Operations 24 (3): 396-406.

Barree, R. D. 2016. Fracture conductivity and cleanup in GOHFER software.

Barree, R. D. 2020. Practical fundamentals of DFIT execution and interpretation. Webinar delivered December 8, 2020.

Barree, R. D., S. A. Cox, J. L. Miskimins, J. V. Gilbert, and M. W. Conway. 2015. Economic optimization of horizontal-well completions in unconventional reservoirs. SPE Production & Operations 3 (4): 293-311.

Chu, Weichun, Kyle Scott, Ray Flumerfelt, and Chih-Cheng Chen. 2018. A new technique for quantifying pressure interference in fractured horizontal shale wells. Paper SPE-191407-MS presented at the Annual Technical Conference and Exhibition, Dallas, TX.

Cipolla, Craig, Constance Gilbert, Aviral Sharma, and John LeBas. 2018. Case History of Completion Optimization in the Utica. 2018. Paper SPE-189838-MS presented at the SPE Hydraulic Fracturing Technology Conference & Exhibition in The Woodlands, TX.

Economides, Michael, and Kenneth Nolte, eds. 2000. Reservoir Stimulation. Wiley.

Fowler, Garrett, Mark McClure, and Craig Cipolla. 2019. A Utica case study: the impact of permeability estimates on history matching, fracture length, and well spacing. SPE-195980-MS. Paper presented at the SPE Annual Technical Conference and Exhibition, Calgary, Alberta.

Fowler, Garrett, Mark McClure, and Craig Cipolla. 2020. Making Sense Out of a Complicated Parent/Child Well Dataset: A Bakken Case Study. Paper SPE-201566 presented at the Annual Technology Conference and Exhibition.

McClure, Mark, Vidya Bammidi, Craig Cipolla, Dave Cramer, Lucas Martin, Alexei A. Savitski, Dave Sobernheim, and Kate Voller. 2019. A collaborative study on DFIT interpretation: integrating modeling, field data, and analytical techniques. URTeC 2019-123. Paper presented at the Unconventional Resources Technology Conference, Denver, CO.

McClure, Mark. Matteo Picone, Garrett Fowler, Dave Ratcliff, Charles Kang, Soma Medam, and Joe Frantz. 2020. Nuances and frequently asked questions in field-scale hydraulic fracture modeling. SPE-199726-MS. Paper presented at the SPE Hydraulic Fracturing Technology Conference and Exhibition, The Woodlands, TX.

Raterman, Kevin T., Yongshe Liu, and Logan Warren. 2019. Analysis of a drained rock volume: An Eagle Ford example. Paper URTeC-2019-263 presented at the Unconventional Resources Technology Conference, Denver, CO.

Smith, Michael Berry, and Carl T. Montgomery. 2015. Hydraulic Fracturing. CRC Press.