In this post, I address a topic that seems esoteric, but it has critical implications for understanding how DFITs respond to closure. In turn, this directly affects how we estimate stress and permeability. The question under investigation: do fractures store and conduct fluid after they close?

My answer: in most DFITs, yes. Two caveats: unless the formation is extremely soft or ductile (allowing residual aperture to be nearly zero) or the matrix permeability is sufficiently high (making the residual fracture conductivity so small relative to the matrix permeability that it is negligible). Note that as fluid pressure draws down after mechanical closure, conductivity decreases. So it is possible that fractures that are hydraulically conductive during a DFIT may effectively close when fluid pressure is drawn down during long-term production (this depends on the stiffness and strength of the asperities that allow the mechanically “closed” fracture to retain aperture).

Why does postclosure fracture conductivity matter to DFIT interpretation? Because it is the crux of the debate between the fracture compliance method and the conventional method of picking closure. I explain the linkage in the final section of this post.

First, I discuss observations that support the idea that fractures continue to store and conduct fluid after closure. Second, I discuss exceptions: two situations where fractures do not continue to conduct significant fluid after closure. Third, I discuss a notable paper in the literature on fracture closure in DFITs. The authors perform a fascinating experiment in which they perform DFITs at the lab scale. The experiments are fantastic, but their analysis has a serious flaw. They assume Carter leakoff, even though pressure in the fracture drops greatly during shut-in (decreasing from around 17 MPa to 6 MPa). Carter leakoff is valid only if pressure in the fracture is approximately constant. This assumption leads them to the erroneous conclusion that leakoff ceases to occur from the parts of the fracture where the walls have touched. Ironically, they observe that the fractures retain aperture after the walls touch, an observation that supports the opposite conclusion. I am addressing this paper because it has been used justify the conventional method of picking closure from DFITs, and I want to demonstrate why this position is not actually supported by the experimental data (for example, these results are discussed in a DFIT paper at this year’s upcoming ATCE). Finally, I discuss implications for DFIT interpretation.

## Observations that support the idea that fractures continue to conduct fluid after closure

Fractures walls are not perfectly smooth. They have roughness, and the opposing walls of the fracture do not perfect mate. As a result, even when the fracture walls are in contact, there are void spaces between the walls, which store and conduct fluid. There is a large scientific literature describing this phenomenon. A good introduction to the topic is provided in Chapter 12 of “Fundamentals of Rock Mechanics” by Jaeger, Cook, and Zimmerman. A canonical paper was written by Barton et al. (1985).

After walls come into contact, further increase in effective normal stress causes the fracture walls to squeeze together. Because the walls cannot interpenetrate, it becomes increasingly difficult to close the fracture as stress goes up. The aperture asymptotically approaches zero (or, Barton et al. (1985) experimentally observed fractures in which aperture asymptotically approached a small, nonzero value).

I have occasionally heard people claim that when a hydraulic fracture opens and closes, then it closes perfectly without offset. Therefore, they claim that when the hydraulic fracture closes, it has no residual aperture and has no ability to conduct fluid. Direct observations in both the lab and the field show otherwise. For example see Figure 7 from Warpinski et al. (1997), the section “Prediction of Roughness” from van Dam and de Pater (2001), or Figure 11 from Jung (1989).

In DFIT pressure transients, we routinely observe a “postclosure linear flow” period. It is identified by pressure scaling with time to the negative one half power. Postclosure linear flow can only occur if there is an “infinite conductivity” fracture in the formation, which requires that the closed fracture remains conductive.

## Exceptions where fractures may stop conducting fluid after closure

As noted above, the aperture of a closed fracture decreases nonlinearly as effective normal stress increases. The rate at which the aperture decreases is related to the stiffness of the rock and other factors (Barton et al., 1985). In rock with very low stiffness, it is conceivable that aperture could become practically zero. This is particularly true in rock that experiences significant ductile deformation.

Second, we should note that the relevant question is not whether fractures conduct fluid after closure, but rather, whether or not fractures conduct a nonnegligible amount of fluid after closure. This can be quantified with the “dimensionless fracture conductivity” (CFD), which is the fracture conductivity divided by fracture half-length and matrix permeability. If CFD is high (greater than 100-300), then pressure drop along the fracture is negligible relative to pressure drop in the matrix. In this case, pressure will be nearly constant along the fracture. If CFD is very low (lower than 0.1), the fracture has such low conductivity, it doesn’t make any contribution to flow because it is easier for fluid to flow through the matrix. At intermediate values of CFD, flow occurs through the fracture, but pressure gradient along the fracture is not negligible.

The permeability for flow through porous media varies over approximately 12 orders of magnitude, from unconsolidated sand that could reach 1000 darcies to ultratight shale with permeability approaching 1 nanodarcy. Holding fracture half-length and conductivity constant, we can see that the difference in fracture conductivity for CFD of 0.1 and 100 is three orders of magnitude. Clearly, matrix permeability (ie, rock type) has a very strong effect on whether a closed fracture will conduct a negligibly small amount of fluid or is infinite conductivity. If permeability is relatively high (as in groundwater aquifers and conventional oil and gas reservoirs), then a closed fracture may indeed have very low CFD and host negligible flow. But DFITs are performed in very low permeability rock. On page 1323 of SPE 179725, we show a calculation that for a fracture with half-length of 50 m and matrix permeability of 100 nd, the fracture will have CFD > 300 (infinite conductivity) as long as aperture is greater than 26 microns, narrower than a strand of hair.

Thus, it is true that in many types of rock, fracture conductivity becomes negligible when the fracture closes. But DFITs are typically performed in low permeability rock. DFITs are performed in formations that experience some ductile deformation, and I am open-minded to the possibility that sometimes dimensionless fracture conductivity stops being “infinite conductivity” at some point after closure, once pressure has decreased sufficiently below the minimum principal stress. But if this occurred, it would take place significantly after closure has occurred (once net compressive stress is significantly greater than zero) and so should not impact the interpretation of closure, except in the very softest formations.

I will digress to mention a recent SPE paper that addressed this topic of fracture conductivity after closure. The paper is “Comparison among Fracture Calibration Test Models” by Ehlig-Economides and Liu (2017). The main purpose of the paper was to state disagreement with the fracture compliance method of picking closure. Using the concept of dimensionless fracture conductivity, they calculate the aperture that a fracture would need to have in order to be infinite conductivity in rock with permeability of 1000 nd, similar to the calculation that I mentioned above from SPE 179725 (they use slightly different assumptions). However, they make the unit conversion error that 0.001 md = 10^-15 m^2. In fact, 0.001 md = 10^-18 m^2. Because of this error, they calculate that a fracture must have aperture of 260 microns in order to be infinite conductivity. In fact, without the error, the result is 26 microns. When the paper came out, I emailed the authors to notify them of the error and reminded them that the SPE allows corrections of typos and other errors in its conference proceedings papers. But eight months later, I see the online version has not been corrected. The error is important because their calculation seems to suggest that fractures quickly stop being infinite conductivity after closure. However, if the calculation is performed correctly, it leads to to the opposite conclusion.

Also, Ehlig-Economides and Liu (2017) suggest that the residual aperture of a fracture in shale should be 1 micron or less because that is the “typical pore radius” in shale. This is a head scratcher. Aperture is controlled by roughness and asperity mismatch, which can be much larger than pore radius, as discussed by Barton et al. (1985), Jaeger et al. (2007), and Jung (1989), among countless others. Van Dam and de Pater (2001) explain some processes that create this roughness and mismatch in a hydraulic fracture.

Ehlig-Economides also advocates a very problematic (and popular) method of plotting DFITs, as discussed in my recent publication SPE 186098.

## Discussion of “Analysis of Hydraulic Fracture Closure in Laboratory Experiments” by van Dam et al. (2000)

The paper “Analysis of Hydraulic Fracture Closure in Laboratory Experiments” by van Dam et al. (2000) (SPE 65066) outlines a very impressive set of experiments in which they essentially performed DFITs at the laboratory scale. Fractures with radius of approximately 0.1 m were created in blocks of rock, cement, or plaster, and then injection was stopped to allow fluid to leak off into the rock. Closure was observed with direct strain measurements and ultrasonic imaging. The laboratory experiments are outstanding and a great contribution to the literature. However, the paper uses a flawed analysis of the results, which leads to a problematic conclusion. They conclude that as the fractures close, leakoff ceases to occur from the “closed” part of the fractures. This particular conclusion regarding leakoff is not a direct experimental observation. Rather, it is based on their analysis of the data.

The flaw in the analysis is that they use the Carter leakoff model to calculate the leakoff rate from the fracture. The Carter leakoff model is derived assuming the pressure in the fracture is constant over time. If pressure changes only slightly, it is still approximately valid. But if pressure changes significantly, the Carter leakoff model is totally inapplicable. Decreasing fluid pressure in the fracture greatly decreases the leakoff rate from the fracture, relative to the prediction from Carter leakoff.

Figure 7 from van Dam et al. (2000) shows that pressure changes greatly during their experiments, varying from around 17 MPa to 6 MPa (prior to the tests, the blocks were at atmospheric pressure). On page 154, van Dam et al. (2000) apply the Carter leakoff model and note that the measured fluid leakoff is occurring more slowly than their calculations predict. To account for the discrepancy, they propose that the surface area of the fracture where leakoff occurs must be decreasing over time. They are forced to invoke “decreasing leakoff area” to account for the discrepancy that occurs because they do not account for the decrease in leakoff rate due to deviation from Carter leakoff. If they had accounted for deviation from Carter leakoff, they would have been able to explain the data without resorting to decreasing leakoff area.

On pages 3-5 of SPE 186098, I provide mathematical description of this process of deviation from Carter leakoff as pressure drops in a closing fracture. Below is a figure showing the calculation from the paper. It shows numerical and approximate analytical solutions to a problem setup where fluid is leaking off one-dimensionally from a constant storage coefficient fracture (ie, to keep things simple, the problem is set up so that there is no ‘closure’). Fluid flow is initially Carter leakoff (square root of time scaling). Once pressure has dropped about 10% of the way to the formation pressure, it starts to deviate from Carter leakoff. The peak in the log-derivative curve occurs about halfway between initial pressure and formation pressure. Then it subsequently settles into a -1/2 slope impulse linear solution. Typically, we call this a “postclosure” linear, but the calculation shows that this can occur even without closure. This calculation looks very similar to the data from Figure 8a from van Dam et al. (2000), which is not surprising because they have qualitatively similar setup.

The equivalent “G-function plot” (which is actually a square root of time plot for this particular simplified problem setup) is also shown below. You can see that the derivative curve peaks and curves downward. This calculation demonstrates that even in problems with no closure – constant fracture storage coefficient and constant leakoff surface area – the G*dP/dG curve deviates from linearity and peaks due to deviation from Carter leakoff. In this case, the problem behavior – the peaking and decline of G*dP/dG and t*dP/dt – is controlled entirely by the deviation from Carter leakoff.

A paper by Hanyi Wang and Mukul Sharma at this year’s upcoming SPE ATCE, “New Variable Compliance Method for Estimating In-Situ Stresses and Leak-off from DFIT data,” emphasizes the importance of accounting for deviation from Carter leakoff on G-function plots. They suggest this is a dominant effect in the interpretation of many DFITs. I agree with them that this is a critical point that has been widely overlooked!

Ironically, all the direct measurements from van Dam et al. (2000) support the notion that the fractures retained aperture and were infinite conductivity after closure (except the test performed in diatomite, which is an exceptionally soft rock). The aperture measurements in their Figs. 7 and 8 indicate that aperture did not go to zero after closure. This is noted by the authors on page 153 of their paper. They also note that residual aperture is common in field-scale measurements.

Figs 8b-d from van Dam et al. (2000) show the pressure derivative increasing when the aperture stiffens due to closure. This is consistent with the analysis presented in the appendix to SPE 179725 for the fracture compliance method. The increase in stiffness is relatively limited because (as noted by the authors), the equivalent wellbore storage of the system of about 35% of the total system storage. This is relatively large wellbore storage, compared with field-scale DFITs.

## Why postclosure fracture conductivity matters to DFIT interpretation

Why does postclosure fracture conductivity matter to DFIT interpretation? Because it is the crux of the debate between the “fracture compliance” method and the conventional method of picking closure. As explained in a previous post, the key equation for understanding the DFIT pressure transient is:

This equation is equivalent to saying that the pressure derivative is equal to the leakoff rate times the system stiffness. The premise of the fracture compliance method is that in very low permeability rock, the fracture remains infinite conductivity after closure (except maybe in oddball rocks like diatomite). But closure increases stiffness. So closure causes derivative to increase (G*dP/dG curves upward).

In contrast, the conventional method of picking closure is based on numerical simulations by Barree and Mukherjee (1996). Their simulator assumed that when the cracks close, they have zero aperture and conductivity. In that case, then the loss of fracture conductivity (and leakoff area) with closure causes the leakoff rate from the system to go down, decreasing the pressure derivative. This causes G*dP/dG to peak and then curve downward. Because these authors believe that closure always occurs at the peak in G*dP/dG, when they observe a curving upward G*dP/dG trend, they interpret it as a variety of “non-ideal” processes that could happen before closure.

Barree and Mukherjee (1996) are making the same mistake as van Dam et al. (2000). Both authors are trying to explain the peak in G*dP/dG (which occurs at the same time as the peak on the log-log derivative plot). As outlined on page 3-5 of SPE 186098, this peak occurs due to the deviation from Carter leakoff as pressure in the fracture drops. But they are using the Carter leakoff model, and are not considering the deviation from Carter leakoff as pressure declines. They are forced to reconcile the discrepancy by assuming that leakoff surface area is decreasing. This is a counter-intuitive result because it: (1) contradicts the field and laboratory observation that closed hydraulic fractures have residual aperture and (2) contradicts the field observation of postclosure linear flow in DFITs. Now we can see the origin of this counter-intuitive result – it arises from the invalid application of Carter leakoff. An analysis that accounts for decreasing fracture pressure will not need to resort to decreasing leakoff surface area with closure. In that case, compliance effect dominates the response to closure, as indicated by the fracture compliance method.

It occurs to me that much of the conventional theory of fracture pressure analysis was developed from the 1970s to the 1990s, when there was less interest in producing from very low permeability formations. In conventional (higher permeability) formations, closure really does cause a fracture to become negligible conductivity, as explained above in the discussion of dimensionless fracture conductivity. Perhaps the idea of zero closed fracture conductivity took hold in an era where this was a reasonable assumption (because most formations of interest were relatively high permeability), and then when people starting applying it to low permeability formations, they didn’t realize that the underlying assumptions are no longer valid. This sort of fallacy seems to be common. Another example is how van Dam et al. (2000) used the Carter leakoff model on a problem where pressure dropped by 2/3, even though Carter leakoff assumes constant fluid pressure in the fracture. A third example is how Ehlig-Economides uses superposition time derivative plots, which were developed for non-fracturing well tests, to analyze DFITs, even though the propagation of the fracture during the DFIT invalidates the assumption of rate-superposition that underpins the justification for the plot. All three are examples of taking an approach that is valid in one context and applying it in a different context without recognizing that its underlying assumptions are no longer valid.

This raises a final point. There are plenty of DFITs in low permeability rock where G*dP/dG does not curve upward. This seems to contradict the compliance method, which predicts that it should curve upward at closure. But keep in mind – the deviation from Carter leakoff is tending to decrease leakoff rate. If the difference between fracture pressure at shut-in and closure pressure is small relative to the difference between fracture pressure at shut-in and the formation fluid pressure, then Carter leakoff will be approximately valid during the entire preclosure to closure period of the test and G*dP/dG will curve upward at closure. But what if the closure pressure is relatively further from fracture pressure at shut-in and closer to formation pressure? For example, this could happen in formations where fluid pressure is very high, relatively close to minimum principal stress. In this case, closure may happen at about the same time as the peak caused by deviation from Carter leakoff. Because the two things are happening at the same time, their effects overprint on each other, and G*dP/dG may not have the upward curving shape. I have seen this in field data and have performed ResFrac simulations of exactly this process. I’ll save those for a future post.

The van Dam et al. (2000) experiments are an extreme case, where fluid pressure at shut-in is triple the difference between the minimum principal stress and the initial matrix pressure. In this unusual situation, the deviation from Carter leakoff (and the peak in the derivative curves) occurs before closure. This is seen in their plots that show the log-log derivative curving peaking and bending down prior to closure.