A double-continuum transport model for segregated porous media: Derivation and sensitivity analysis-driven calibration
Introduction
The development of accurate mathematical models to describe solute mass transport in porous media is particularly challenging when the medium is characterized by the presence of cavities, dead-end pores, stagnant zones, and a highly heterogeneous velocity field. The structure and the extent of low-velocity regions directly impact solute transport, potentially leading to long mass retention times. Accurately modeling such trapping effects is crucial, for example, in the context of remediation and risk assessment (e.g. de Barros et al., 2013). A sound understanding of the conditions and pore-scale processes that physically control the rate of exchange between stagnant and fast-flowing regions is needed to better understand solute spreading and mixing and subsequently the evolution of conservative and reactive transport processes (e.g. Alhashmi et al., 2015, Lichtner and Kang, 2007, Kitanidis and Dykaar, 1997, Wirner et al., 2014, Cortis and Berkowitz, 2004, Briggs et al., 2018, Baveye et al., 2018).
Solute transport has been widely studied by performing direct numerical simulations at the pore scale (see e.g. Scheibe et al., 2015, Bijeljic et al., 2013b, Hochstetler et al., 2013, Porta et al., 2013). Such techniques present the remarkable advantage of providing detailed information on solute concentration evolution at each point of the porous domain. However, the applicability of such methods, which are computationally demanding, is limited to small domains, typically much smaller than field scales of common interest (Dentz et al., 2011). Upscaled continuum models are consequently more suited to simulating larger-scale systems.
To upscale the effects of low-velocity regions on emerging transport features, different approaches have been proposed such as Multi-Rate Mass Transfer Model (MRMT, Tecklenburg et al., 2016, Haggerty and Gorelick, 1995, Carrera et al., 1998), various Continuous Time Random Walk approaches (CTRW, Berkowitz and Scher, 2009, Berkowitz and Scher, 1997, Le Borgne et al., 2008, Dentz and Castro, 2009), time and space-fractional models (fADE, Kelly et al., 2017, Berkowitz et al., 2002), Time Domain Random Walk approaches (TDRW, Banton et al., 1997, Delay and Bodin, 2001, Russian et al., 2016) and other nonlocal formulations (Neuman and Tartakovsky, 2009). In particular, double or multiple continua approaches are appealing due to their ability to explicitly distinguish stagnant zones from fast flowing channels.
In the classical double-continuum approach (Haggerty and Gorelick, 1995, Carrera et al., 1998, Bear and Cheng, 2010), a mobile and immobile continuum exchange mass as a first-order process with an effective mass transfer coefficient. Typically, effective parameters of double or multi-continuum models, need to be estimated via fitting against solute concentration data, e.g., measured breakthrough curves or solute concentration profiles.
Alternatively, upscaled dual continua models have been formally derived (see e.g., Souadnia et al., 2002) by means of volume-averaging techniques. While appealing due to their sound theoretical basis, as discussed by Davit et al. (2012), these formally derived double-continua formulations present practical limitations in terms of their applicability to real problems. Often such models can be nearly as difficult to solve as their pore-scale direct simulation counterparts, as they include complex non-local terms, which imply significant numerical implementation challenges (Porta et al., 2016).
As a consequence, a number of studies have proposed more parsimonious effective up-scaled formulations that, however, still exploit key pore scale information. These studies encompass Eulerian (Porta et al., 2015) and Lagrangian formulations (Sund et al., 2017a, Sund et al., 2017b, Dentz et al., 2018). These methods are designed to embed pore-scale characteristics into effective parameters which can be applied at a larger scale. For example, Sund et al., 2017a, Sund et al., 2017b) employ trajectories and travel times distributions measured numerically at pore-scale to infer the effective evolution of mixing and reaction rates in an effective Lagrangian spatial Markov model. The work of Porta et al. (2015) focuses on the use of pore-scale information to characterize a double-continuum transport model, from an Eulerian perspective. Porta et al. (2015) model relies on the cumulative distribution function of velocity measured from a pore-scale simulation of single-phase flow and assumes that the exchange time between high and low velocity regions is dictated by the characteristic diffusion time scale. The model reproduces observed transport behaviors in relatively well connected three-dimensional porous systems, i.e. a beadpack and a sandstone sample. However, as shown by Bénichou and Voituriez (2008), realistic cavities may be characterized by complex geometry such that it can take an extremely long time to exchange mass from slow regions to faster flowing channels. For this reason, the double-continuum approach proposed by Porta et al. (2015) might suffer limitations when the geometry of the porous medium is highly tortuous and presents significant stagnant cavities. Such features may arise in both three-dimensional (e.g. carbonate rocks, see Bijeljic et al., 2013a) and two dimensional porous media which are often employed in simulations and experiments (Acharya et al., 2007, Wirner et al., 2014, de Anna et al., 2014).
Starting from the model developed in Porta et al. (2015), we develop a double-continuum model, which explicitly accounts for a characteristic time for the exchange process between high and low velocity regions which may be larger than the diffusion time scale. This model lumps the effect of exchange process at pore-scale into a single effective parameter, which is defined as the ratio of the time required by the solute to escape/explore the stagnant regions of the porous medium to the characteristic diffusive time scale. Our main objective is to derive a closed form double continuum model and to test it against numerical pore-scale simulations of solute transport performed in a disordered synthetic two-dimensional porous medium, considering different initial conditions.
We explore the flexibility of the model by means of a sensitivity analysis. We assess the effectiveness of the model by means of i) a qualitative inspection of concentration profiles predicted considering different initial conditions and ii) quantification of the Sobol’ indices of appropriately defined target metrics. We also investigate the role of a Global Sensitivity Analysis (GSA) in defining a tailor-made objective function to increase the efficacy of model calibration.
The paper is structured as follows. In Section 2 we present the problem setup that will be used as the test bed for the proposed double continuum model, along with details on the pore-scale model. In Section 3, we derive the proposed closed form double continuum model. In Section 4, the flexibility of the model is explored via a GSA. Calibration and validation of the model are discussed in Section 5 and conclusions are presented in Section 6.
Section snippets
Pore scale domain
In this work, we consider a two-dimensional porous medium made up of repeating periodic unit cells, Ω′. The cell configuration is the same as that of Porta et al. (2016). The geometry of the unit cell is generated by the disordered superposition of circular grains of uniform diameter w = 8 × 10−5 m and then discretized into pixels of 2 × 10−5 m. The resulting pixelated image, characterized by porosity ϕ = 0.5948, constitutes the reference cell configuration for the pore-scale numerical
Dual continuum model formulation
The development of the double-porosity model proposed here is built starting from the procedure originally developed by Porta et al. (2015) and it is schematically outlined in Fig. 2. We start with the 2D-porous medium introduced in Section 2 (Fig. 1). We define the average Péclet number where Dm[m2s−1] is the molecular diffusion coefficient and L [m] a characteristic length scale of the system that is considered unknown a priori and should be properly determined.
In the
Characterization of the mass transfer at the continuum scale
In this Section, we elucidate how the proposed model accounts for the exchange process between fast and slow regions and the impact of the exchange process on solute evolution depending on the initial conditions.
Model calibration and validation
In Section 4, we showed that the proposed model is flexible and able to reproduce symmetric, highly skewed profiles or entrapped solute for an extremely long time by opportunely setting two effective parameters L and RD. In this Section, we discuss calibration and validation of the model against pore-scale simulations performed in the two-dimensional porous medium from Section 2.
Conclusions
The present work is devoted to the formulation, calibration, and validation of a double-continuum model for solute transport in porous media, which aims to embed characteristics of the pore-scale geometry and velocity field.
As opposed to available non-Fickian transport models (Neuman and Tartakovsky, 2009) or those rigorously derived through Volume Averaging (Davit et al., 2012), our model does not include any nonlocal term, which makes the model implementation straightforward. In spite of this
Acknowledgments
G. Ceriotti and G.M. Porta would like to thank the EU and MIUR for funding, in the frame of the collaborative international Consortium (WE-NEED) financed under the ERA-NET WaterWorks2014 Co- funded Call. This ERA-NET is an integral part of the 2015 Joint Activities developed by the Water Challenges for a Changing World Joint Programme Initiative (Water JPI).
D. Bolster greatly acknowledges financial support from the U.S. National Science Foundation via grants EAR 1351625 and CBET 1705770.
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