A trigonometric interpolation approach to mixed‐type boundary problems associated with permeameter shape factors

Abstract

Hydraulic conductivity is a fundamental hydrogeological parameter, whose in situ measurement at a local scale is principally performed through injection tests from screened probes or using impermeable packers in screened wells. The shape factor F [L] is a proportionality constant required to estimate conductivity from observed flow rate to injection head ratios, and it depends on the geometric properties of the flow field. Existing approaches for determination of F are either based on geometric or mathematical simplifications and are limited to particular assumptions about the flow domain's external boundaries. The present work presents a general semianalytical solution to steady state axisymmetric flow problems, where external boundaries may be nearby and of arbitrary combinations of impermeable and constant head type. The inner boundary along the probe or well may consist of an arbitrary number of impermeable and constant head intervals resulting in a mixed‐type boundary value problem, for which a novel and direct solution method based on trigonometric interpolation is presented. The approach is applied to generate practical nondimensional charts of F for different field and laboratory situations. Results show that F is affected by less than 5% if a minimum distance of 10 probe or well diameters is kept between the injection screen and a nearby boundary. Similarly, minimum packer lengths of two well diameters are required to avoid increasing F by more than 10%. Furthermore, F is determined for laboratory barrel experiments giving guidelines for achieving equal shape factors as in field situations without nearby boundaries. F for the theoretical case of infinitely short packers is shown to be infinitely large.

A semi-analytical generalized Hvorslev formula for estimating riverbed hydraulic conductivity with an open-ended standpipe permeameter

“…The analytical solution produced by Hvorslev (1951) highly depends on the shape factor of the installed piezometer (F), which is considered a function of the geometric constants, i.e., the length-to-diameter ratio, of the piezometer (Silvestri et al, 2012). As indicated by Klammler et al (2011), most existing approaches used to determine F are based only on geometric or mathematical simplifications that neglect the effects of the boundaries of the flow domain. Therefore, the objectives of this study are to: (1) develop a semi-analytical expression for hydraulic resistance of an open-ended standpipe permeameter in the vicinity of a constant head boundary; (2) validate the obtained expression using numerical simulations of the falling head tests in the standpipe permeameter; (3) examine the influence of the natural vertical flow gradient in bottom sediments and medium elastic storage on the falling head test results; and (4) analyse the possibility of determining the hydraulic conductivity profiles of layered bottom sediments using falling head tests in a standpipe permeameter.…”

Sediment Bed Borehole Advection Method

“…The flow convergence factor F 0 depends on the geometry of the SBBAM casing and the hydraulic conductivities K h and K v of the sand in the horizontal and vertical directions, respectively. The factor is quantified based on existing work for the SBPFM [4,20], knowing that the hydraulic conductivity K c of the open casing (i.e., no sorbent medium) between both SBBAM screens is very much larger than both K h and K v . First, K v = 31.8 m/d was determined from two falling head conductivity tests through an impermeable pipe casing inserted into the sand to the depths of each screen [4] (p. 4).…”

Insights From Unsteady Flow Analysis of Underdamped Slug Tests in Fractured Rock

“…In this study, a zero storage model (Hvorslev, ) is used to determine T from slug tests using the following well‐known relationships: where flow rate Q (L 3 /T) is determined from the water level changes w ′ = dw / dt and riser pipe radius R r ( L ), h D is the deviation of formation head from static at the test interval (L), T is the transmissivity to be estimated ( L 2 / T ), and F is a dimensionless shape factor (e.g., Hvorslev, ; Klammler et al, ). A radial flow shape factor, 2π/ln ( r o / r w ), is used in this study for comparison to independent CH step tests using a radius of influence of r o = 30 m (e.g., Quinn et al, ), and the radius of the well, r w = 0.0635 m for a 5‐in.…”