Comment on “Shape factors for constant‐head double‐packer permeameters” by S. A. Mathias and A. P. Butler

[1] In their paper, Mathias and Butler [2007] revisited the old problem of constant-head tests using double packers, presented a new semianalytical solution and compared it to previous solutions. The new solution for the shape factor assumes that the boundary condition above the upper packer and below the lower packer is a constant-head condition similar to that existing everywhere as the initial condition and always acting on distant (infinite) boundaries. About 30 references on packer tests are quoted. The oldest is Hvorslev [1951], which was written to select the best methods for monitoring either hydraulic head or pore water pressure, and which is best known for its long list of shape factors gathered from older publications. The paper did not use many publications on packer tests and shape factors, especially those in mathematics, petroleum, civil, geological and mining engineering, which have studied both theoretical and practical problems with single-, double- and three-cell packer tests. Hence the paper may highlight a lack of communication between different domains interested in shape factors. This comment deals with the theoretical origin of the shape factors, previously published results and practical aspects that are not covered in the paper. [2] The Laplace equation describes steady state groundwater seepage and heat conduction among other phenomena. Solutions to Laplace equation are called harmonic functions. Hvorslev's solution for steady state is incomplete, providing only the flux value Q at the bounded boundary where the hydraulic head, h, is held constant. This constant head differs from the constant value that exists initially throughout the domain and is also maintained at all time on other distant boundaries that are not impervious. The solution for Q stems directly from the mathematical properties of harmonic functions [e.g., Chapuis, 1998], stating that the ratio Q/K Δh is a constant (the shape factor) that depends on the domain geometry and boundary conditions. In the ratio, Δh is the applied constant difference in hydraulic head and K is the hydraulic conductivity, either the single eigenvalue for an isotropic medium or the horizontal conductivity KH for a cross-anisotropic medium. Therefore, the shape factor finds its origin in properties of harmonic functions. Values of various shape factors can be found in many mathematics textbooks as well as in literature dealing with phenomena that obey Laplace equation, e.g., heat conduction [Lamé, 1861] and electric fields [Maxwell, 1873]. [3] In the work by Hvorslev [1951] as in many publications, the cylindrical injection zone has a diameter D and a length L, whereas the authors used its radius r0 and half-length z0. As a result, the aspect ratio is L/D = z0/r0. Hvorslev [1951] used m = (KH/KV)1/2, KH and KV being the horizontal and vertical hydraulic conductivities respectively. The authors used = = m−1. Equation (3) with L and D is for the case 8 of Figure 12 in the work by Hvorslev [1951], and corresponds to the third equation of Hvorslev [1951, p. 35], who used L, D and m. [5] Figure 1b of Mathias and Butler [2007] represents two well-known tests [Lefranc, 1937; Lugeon, 1933], the procedures of which have been standardized at least since the 1980s. A Lefranc test is performed as a constant-head test, a variable-head (slug) test, or a pulse test. It is done using a driven flush joint casing [e.g., Chapuis et al., 1981; Chapuis, 2001], a monitoring well [e.g., Chapuis, 1988; Chapuis and Sabourin, 1989; Chapuis and Cazaux, 2002], a self-boring permeameter [e.g., Baguelin et al., 1974, 1978; Chapuis et al., 1992], or a driven penetrometer equipped with either one lateral constant-head zone [e.g., Chapuis and Chenaf, 2003] or two constant-head zones to operate as a dipole test [e.g., De Leeuw and Silence, 1983]. A Lugeon test is performed in an open borehole using single-, double- or three-cell packers. It includes nine steps at constant head differences, namely, 20–40%–60–80%–100–80%–60–40% and 20% of the maximum head difference. A Lugeon test is used to assess not only the rock permeability, but also the in situ stress conditions, the need to grout the rock, and the hydraulic conditions that could bring the rock to failure. [6] The authors assumed that the horizontal plane passing through the center of the cylindrical injection zone is a plane of symmetry [Mathias and Butler, 2007] in an infinite medium. This is acceptable for a packer test performed at the midpoint of a long open borehole, which is a special case. This assumption is incorrect for other cases and in particular, for Lefranc tests, for which, below the injection zone, there is either soil or rock, or a limited section of the penetrometer. [7] The authors also assumed that the hydraulic head in the annular space between the rod and the borehole wall, above the upper packer and below the lower packer, is a constant-head condition similar to that existing everywhere as initial conditions and always acting on distant (infinite) boundaries. This is a simplification of real conditions in rock masses, where the almost constant hydraulic head in the open borehole corresponds to a complex fluid exchange between several aquifer zones that have been hydraulically connected by the open borehole [e.g., Chapuis, 2006]. In addition, the presence of either less or more pervious zones close to the injection zone has major effects on the shape factor value [e.g., Schneebeli, 1966; Chapuis, 1989]. [8] Major findings appear in Figures 3 and 4 of Mathias and Butler [2007], for different normalized packer lengths. According to Figure 4, the difference between the new solution and the old equation (4) of the ellipsoid approximation is very small, except when the normalized well screen aspect ratio, NWSAR, falls below 0.5, the relative difference reaching about 50% for NWSAR = 0.1. First, this is normal since the ellipsoid approximation is known to be a poor approximation for an aspect ratio less than 4. Second, the findings must be assessed in the context of real tests and known information. [9] Frequent users of packer tests identified many years ago several sources of problems with these tests. They include wall damage due to drilling, seepage around the packers, inadequate test design/implementation and equipment errors, abnormal opening of fissures, irregular data due to deformation of the rock mass, either erosion or clogging within fissures, etc. Short packers were shown to be prone to errors, not only with respect to the shape factor, but mainly for seepage losses around the packers. In order to reduce the latter, double packers with long sleeves and three-cell packers have been developed [e.g., Maini et al., 1972]. With parameters of Figure 1 of Mathias and Butler [2007], the normalized packer length is (zp − z0)/z0. For common long packers, z0 is about 0.5 m, 0.75 m or 1.5 m, whereas (zp − z0) is usually 0.5 to 1.0 m. Thus, the normalized packer length is between 0.33 and 2, a range for which Figure 3 of Mathias and Butler [2007] provides fairly close values (about 10%) for the normalized hydraulic conductivity. Taking some average would yield an accuracy of about ±5% for the hydraulic conductivity. In addition, the usual range of values for the normalized well screen aspect ratio, NWSAR = z0/(r0) can be evaluated. Most often, the borehole radius r0 is between 3 and 8 cm, which means that NWSAR = 7 to 30 for an isotropic rock. For a cross-anisotropic rock, since KV is usually smaller than KH, the ratio is less than unity, and therefore the NWSAR has a greater range than 7–30. [10] Consequently, considering the common ranges of values for the NWSAR and the normalized packer length, the difference between the proposed more complete solution and the old ellipsoid approximation is only a few percent. This is reassuring, knowing that the other sources of problems and repeating the tests in the same borehole yield much larger errors. [11] Finally, another practical question is worth asking. How long does it take to reach steady state conditions after a constant hydraulic head has been applied between the packers? The duration is known to depend on the hydraulic conductivity, the stress-strain relationship and geometrical factors. For tests in clay exhibiting delayed deformation during the test, a few days are needed to reach steady state [e.g., Gibson, 1970; Mieussens and Ducasse, 1977; Novakowski, 1993; Neville and Markle, 2000]. This may be a time constraint that was not discussed in the paper.

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