Nonparametric Method for Transmissivity Distributions Along Boreholes

The transmissivities of individual fractures along a borehole are difficult to obtain unless each fracture is tested. To estimate a fracture‐transmissivity distribution from section transmissivities, a method was developed based on fixed‐interval‐length transmissivities and the corresponding number of fractures for each interval. The method is nonparametric and iterative, and the fractures are viewed as two‐dimensional features, in which the total transmissivity of a borehole is equal to the sum of individual fracture transmissivities. Initially, a linear a priori assumption of the transmissivity distribution is made, and from this a so‐called mean transmissivity function is derived. Subsequently, the mean transmissivity of the N j fractures within a section, j, of the borehole is estimated, and the same value of the mean transmissivity function represents N j ‐ possible fracture transmissivities from the initial distribution. This is repeated for each borehole section, and, eventually, all fracture transmissivities are sorted to give the next iteration's transmissivity distribution and the corresponding mean transmissivity function. Finally, the distributions converge, yielding a possible fracture‐transmissivity distribution. The method was verified for a synthetic data sample and then tested on a sample from a borehole at the Aspo Hard Rock Laboratory, Sweden. For the synthetic data, the method gave a distribution that was fairly close to the original one; for the Aspo data, 15% of the fractures had a transmissivity larger than the measurement limit (1×10 −9 m 2 /sec), and these transmissivities follow a log‐normal distribution.