Statistical scaling of pore-scale Lagrangian velocities in natural porous media

Abstract

We investigate the scaling behavior of sample statistics of pore-scale Lagrangian velocities in two different rock samples, Bentheimer sandstone and Estaillades limestone. The samples are imaged using x-ray computer tomography with micron-scale resolution. The scaling analysis relies on the study of the way qth-order sample structure functions (statistical moments of order q of absolute increments) of Lagrangian velocities depend on separation distances, or lags, traveled along the mean flow direction. In the sandstone block, sample structure functions of all orders exhibit a power-law scaling within a clearly identifiable intermediate range of lags. Sample structure functions associated with the limestone block display two diverse power-law regimes, which we infer to be related to two overlapping spatially correlated structures. In both rocks and for all orders q, we observe linear relationships between logarithmic structure functions of successive orders at all lags (a phenomenon that is typically known as extended power scaling, or extended self-similarity). The scaling behavior of Lagrangian velocities is compared with the one exhibited by porosity and specific surface area, which constitute two key pore-scale geometric observables. The statistical scaling of the local velocity field reflects the behavior of these geometric observables, with the occurrence of power-law-scaling regimes within the same range of lags for sample structure functions of Lagrangian velocity, porosity, and specific surface area.

Scalable statistics of correlated random variables and extremes applied to deep borehole porosities

“…Dashtian et al (2011) concluded that these formations are layered based on complete suites of well logs at each of the three sites. We note further that a similar dual-scaling phenomenon has recently been reported by Siena et al (2014) vis-á-vis porosities and specific surface areas imaged using X-ray computer microtomography throughout a millimeter-scale block of Estaillades limestone, at a spatial resolution of 3.3 µm, as well as Lagrangian velocities computed by solving the Stokes equation in the sample pore space.…”

“…Statistical scaling of hydrogeological data such as permeability or hydraulic conductivity has been studied amongst others by Painter (2001), Meerschaert et al (2004), Kozubowski et al (2006), Siena et al (2012Siena et al ( , 2014, Riva et al (2013bRiva et al ( , 2013c, and Guadagnini et al (2012Guadagnini et al ( , 2013Guadagnini et al ( , 2014. Whereas research in the subsurface hydrology literature has not addressed specifically the distribution and statistical scaling of extreme incremental values, spatial correlations between values significantly in excess of the mean have been studied vis-à-vis variables such as transmissivity and their relevance to transport processes has been highlighted.…”

“…Work by our group has demonstrated theoretically (Neuman, 2010(Neuman, , 2011Guadagnini and Neuman, 2011;Siena et al, 2012;Neuman et al, 2013), computationally Neuman et al, 2013) and on the basis of varied pedological, hydrological and hydrogeological data (Siena et al, 2012(Siena et al, , 2014Riva et al, 2013b, c;Guadagnini et al, 2012Guadagnini et al, , 2013Guadagnini et al, , 2014) that statistical scaling behaviors of the kind traditionally attributed to multifractals can be interpreted more simply and consistently by viewing the data as samples from stationary sub-Gaussian random fields subordinated to truncated fBm (tfBm) or fGn (tfGn). Such sub-Gaussian fields are scale mixtures of stationary Gaussian fields with random variances (Andrews and Mallows, 1974;West, 1987) that we model as being lognormal or Lévy-stable (Samorodnitsky and Taqqu, 1994).…”

Anti-correlated Porosity–Permeability Changes During the Dissolution of Carbonate Rocks: Experimental Evidences and Modeling

“…Experimental studies are notably challenging in the case of carbonate systems, due to the complex topological setting of the pore space (e.g., Siena et al 2014 and references therein), which is often associated with low porosity values (Hebert et al 2014 and references therein). A solid body of experimental evidences illustrates that the porosity increase resulting from carbonate dissolution is usually correlated with the increase of permeability.…”

Adaptive POD model reduction for solute transport in heterogeneous porous media

“…Practical difficulties associated with the way one can rigorously include pore-scale modeling in the analysis of typical laboratory and field-scale settings lead to depicting of solute transport in such media through effective models [14,27,38,39,45]. Several alternative modeling options are available in this context, a common choice being a continuum-based representation grounded on the standard advection-dispersion equation (ADE) on which we focus here.…”