Kinematic structure of minipermeameter flow

Abstract

Abstract. Minipermeameters are rapidly becoming a popular tool for collecting localized measurements of permeability in both laboratory and field studies. While one of the main advantages of minipermeameters is their ability to collect data on various support volumes, there have been only limited attempts to analyze their size and geometry. We define the support volume of minipermeameter measurements as a region containing 90% of the total gas flow, i.e., a region bounded by the 10% streamline. Using our new semianalytical solutions for the Stokes' stream function, we demonstrate that the support volume has a shape of the semitoroid adjacent to the sample surface. Hence there is a blind spot directly below the minipermeameter, which is not probed by the measurement. We demonstrate that the support volume of the minipermeameter measurements decreases with the tip-seal's ratio (a ratio of the inner tip-seal radius to the outer tip-seal radius), while the size of the corresponding blind spot increases. IntroductionDelineation of the spatial distribution of permeability in water-and oil-bearing formations is one of the major challenges in hydrogeology and petroleum engineering. Specifically, this is an ill-posed inverse problem, and hence it is inherently difficult to solve. Mathematical models that provide a means to extract permeability data indirectly from experimental measurements of dependent quantities (e.g., pressure head and flow rates) do so by defining a related well-posed problem through some form of regularization. The necessary presence of this regularization, which may not be stated explicitly, is likely a critical factor in the recent debate over the scale dependence of permeability measurements. Consequently, there is a growing interest in experimental procedures that possess well-defined regions of investigation or support volumes.Minipermeameters seem well suited for this purpose because they induce a localized flow by injecting gas into a sample through a small tip seal. Although these devices were first described by Dykstra and Parsons [1950], it was not until recently that Goggin et al. [1988] proposed a mathematical model for the application of the minipermeameter to localized permeability measurements. In particular, for the case of steady state gas flow, Goggin et al. [1988] introduced a coefficient of proportionality into an integral form of Darcy's law. Dubbed the geometric factor, this coefficient allowed the permeability to be inferred from the injection rate and the corresponding gas pressure. The experimental aspect of this work focused on measuring the permeability of core samples; thus the support volume was defined by the sensitivity of the geometric factor to the sample size. Specifically, the support volume was deter-

Extended power-law scaling of air permeabilities measured on a block of tuff

“…The latter is about 12 times the inner radius of the MSP. Albeit one should consider all the approximations involved in this estimate, we note that it is consistent with a definition of MPS support volume by Tartakovsky et al (2000) as a region containing 90 % of total gas flow (see their Fig. 6).…”

“…Measurements were conducted at intervals of = 0.85 cm on a grid of 36 × 36 points along each face using four tip-seal sizes having inner radii r i = 0.15, 0.31, 0.63, 1.27 cm and outer radii 2 r i . As the precise nature and size of the support volume associated with each measurement is the subject of debate (Goggin et al, 1988;Molz et al, 2003;Tartakovsky et al, 2000;Neuman and Di Federico, 2003), we consider the inner radius of the tip-seal to represent a nominal measurement scale (data support) as proposed by Tidwell and Wilson (1999). We conclude from their analysis that measurements on face 6 of the block are less reliable than the rest and therefore limit our analysis below to those on faces 1-5.…”

Surface permeability tests: experiments and modelling for estimating effective permeability

“…Although the integral equation (3.21) can be expressed in terms of elliptic integrals, the steady-state flow as defined by equation (3.16) is divergent since the function has a non-integrable singularity .23)). The dots indicate results due to Goggin et al (1988) and Tartakovsky et al (2000).…”

“…Approximate solutions to the triple system were presented by Goggin et al (1988) and, in a subsequent analysis, Tartakovsky et al (2000) provided an informative discussion of the errors and accuracy involved in the development of numerical solutions. The objective here is to develop an analytical solution, based on the series approximation techniques, to the triple integral equations along the lines similar to those proposed by Selvadurai & Singh (1984a,b) and Selvadurai (1996) and summarize the results for completeness.…”

“…The use of localized steady flows for permeability testing has been reported by Dykstra & Parsons (1950) and these experimental techniques were rigorously modelled by Goggin et al (1988). Innovative research along these lines was also conducted by Tidwell & Wilson (1997) in connection with the measurement of air permeability of a large block of Berea sandstone and the errors involved in the numerical evaluation of the intake shape factors were examined by Tartakovsky et al (2000). This paper briefly discusses the experimental procedure, provides a mathematical treatment of special cases of the experimental configuration and uses computational models to interpret the data for general situations.…”