IMPERIAL COLLEGE LONDON

Understanding solute transport is essential to assess carbon capture storage safety or miscible gas injection efficiency during enhanced oil recovery. Transport has been shown to be non-Fickian, or anomalous, at different scales of observation. However the classical description based on the advection-diffusion equation (ADE) is still used to quantify the transport in porous media. The last few years have seen emerging a number of approaches to model the anomalous behaviour of solute transport. The most elegant approaches are based on random walks. These methods allow simulating non-Fickian transport at the porescale. Concentration distributions as a function of displacement of the tracer, also called propagators, are derived from these simulations. The dispersion of the solute results from those propagators. Breakthrough curves (BTC), on the other hand, give the transit time the tracer needs to travel across a given distance. Both propagators and BTC are drastically different in an anomalous transport than in a Fickian transport at the pore-scale. It is now fundamental to understand the impact of this behaviour at a larger scale. This study seeks out to overcome this challenge by modelling anomalous transport at meter scale by using propagators derived by pore-scale simulations. Solute transport at meter scale is simulated in a cubic orthogonal network of volume-less nodes and one-dimensional links. The pore-scale propagators are obtained by direct simulation at the pore-scale, without assuming a priori the governing microscopic transport equations. Flow and transport are computed through the pore-scale voxels obtained by three-dimensional imaging (X-rays). Resulting pore-scale propagators contain all the advective and diffusive components of the displacement of a solute particle. Each small-scale propagator depends on Péclet number, thus on the velocity. Hence modelling the macroscopic displacement, using the correct pore-scale propagator in each link of the network, permits to capture the grid-block-scale behaviour of the transport. At the meter-scale, solute transport appears to be controlled by macroscopic heterogeneities. Microscopic heterogeneities also have an effect on the concentration distribution as they retard the occurrence of an asymptotic regime at the grid-blockscale for an increasing complexity of the structure. The microscopic heterogeneity increases the transit time of tracer particles in the network, giving a later breakthrough of the solute than in the homogeneous case. The results are depending on the available data at the pore-scale so that more small-scale simulations are needed to quantify the transport. Nevertheless, this method offers advantages by its simplicity and is very promising by its efficiency to model a complex behaviour in highly complex structures. Introduction Solute transport plays an important role in a number of scientific applications such as hydrology, enhanced oil recovery or carbon storage (Sahimi, 1995). Until recently, transport has been modelled and quantified using the so-called advectiondispersion equation (ADE). However, this model assumes that dispersion of the tracer follows Fick’s law (Adams and Gelhar, 1992). This makes the ADE valid only in the case of a perfectly homogeneous structure and gives a concentration normally distributed with a mean proportional to t and a standard deviation proportional to √ . But real porous rocks are never perfectly homogeneous and there are heterogeneities at each scale. The question of the suitability of the ADE framework thus, naturally arises. From the 1950s, experimental measurements displayed systematic errors from the analytical solution of the ADE predictions. Scheidegger (1959) wrote: “The deviations are systematic which appears to point toward an additional, hitherto unknown, effect. ” Such inadequacies between the measurements and the theory obliged the scientists to reconsider their understanding of solute transport in porous formations. Montroll and Scher (1973) first introduced the concept of anomalous transport in semiconductors. Also called non-Fickian transport, anomalous transport refers to transport behaviours that differ from the one given by the ADE description. Heterogeneous media, by their structure, contain preferential paths that influence fluid flow and transport. Hence, the velocity is not uniformly distributed in the medium and there are stagnant zones where solute can be stuck. This gives rise to concentration distributions showing a peak of concentration at low displacement and a tail of fast moving particles. Moreover, another characteristic shape derived from an anomalous transport is the long tailing breakthrough curve (BTC). This displays “anomalous early times (i.e., later than Fickian) and late time tails” (Levy and Berkowitz, 2003). The transport is sensitive to heterogeneity at each scale so that, even geologies that can be thought as homogeneous, exhibit non-Fickian behaviour (Levy and Berkowitz, 2003). This will last until reaching an asymptotical regime where the transport becomes Fickian. In theory, for a time long enough, depending on the level of heterogeneity of the structure, the transport tends to become Fickian where ADE can be applied and dispersion of the solute is finite (Berkowitz et al., 2000). However, this may never occur in highly heterogeneous structures. Imperial College London Upscaling of non-Fickian transport using pore-scale propagators 2 Once the anomalous concept was introduced, researchers tried to tackle the challenge of modelling this behaviour. To date, there are a number of approaches to model it. Namely, multi-rate mass transfer (MRMT) (Haggerty an Gorelick, 1995), averaged version of ADE (Bear, 1972) or continuous time random walk (CTRW) (Berkowitz and Scher, 1995) attempted to describe non-Fickian transport more or less successfully. Berkowitz et al. (2006) has done a thorough review development of the CTRW framework and its approach to solute transport in geological media. The same paper also compares the previously mentioned methods to model anomalous transport. CTRW has proved to give a satisfactory approach to quantify anomalous transport. Berkowitz et al. (2006), describes CTRW as “a probabilistic approach for calculating the plume motion and/or BTC across the heterogeneity scale, based on a probability density function (pdf) of transition times generated by the range of heterogeneities. The nature of the transport, non-Fickian or Fickian, is determined by the functional shape of the pdf. ” With a better understanding of the non-Fickian transport, characterisation of concentration distribution shapes became essential to apprehend anomalous transport in a wide variety of porous structures and to implement this to industrial applications. Scheven et al. (2005) first established distribution shapes for bead pack, sandstone and carbonate rocks through nuclear magnetic resonance (NMR) measurements. This provides an experimental support to validate pore-scale simulations of solute transport in heterogeneous media. Bijeljic et al. (2011) then obtained generic pore-scale propagators (concentration as a function of travelled distance) by direct simulation at pore-scale, without assuming apriori the governing transport equations. With all the implications of assessing non-Fickian transport effect at field-scale, for industry like carbon dioxide sequestration or improved oil recovery, the next challenge to be tackled is the upscaling of anomalous transport in complex heterogeneous structures. Indeed, carbonate rocks represent more than half of the reserves on the planet (Ahlbrandt et al., 2005) and it is essential to understand the impact of anomalous transport at the field-scale. Rhodes et al. (2008) published a hierarchical multi-scale approach using CTRW framework and demonstrated that “the macroscopic behaviour is affected by the small-scale transport even when a very heterogeneous field-scale reservoir description is used. ” In spite of this previous study, the use of pore-scale propagators for upscaling has not been attempted and this is what the following paper proposes. A model has been built to analyse the non-Fickian transport at the grid-block-scale. The methodology used for upscaling the solute transport is described first. In a further section, the model is applied to the “perfectly homogeneous” case to verify its consistency in the simplest case. A comparison of the different types of simulations depending on the pore-scale available data is then displayed. The injection methods and the elapsed time pore-scale propagators are compared in this section to define which one is the most suitable for simulating more complex structures. Finally, the impact of both microscopic and macroscopic heterogeneities is assessed by comparing different rock types (Mt Gambier limestone and Estaillades carbonate) of different microscopic heterogeneities at the two scales. The paper ends with a discussion on the results and a summary of the results found from the study, before giving advices for later work on this subject. The terms “non-Fickian” and “anomalous” are interchangeably used in this report, as well as “propagators” and “solute concentration distribution”. Propagator plots are displayed as a normalized probability density function P(ζ) against dimensionless displacement as in the CTRW framework. Upscaling methodology This aim of this study is to understand the behaviour of solute transport in heterogeneous media at a meter-scale, and to assess the impact of microscopic heterogeneities on solute displacement at the larger scale of the grid-block. This is done through the upscaling of pore-scale propagators, and by analysing the transit time of the particles through the medium with BTC. The project is adjacent to another study simulating pore-scale propagators using random walks, which results are about to be published (Bijeljic et al., 2012). These microsco