Modelling and Upscaling of Transport in Carbonates During Dissolution: Validation and Calibration with NMR Experiments

We present an experimental and numerical study of transport in carbonates during dissolution and its upscaling from the pore ( ∼ µ m) to core ( ∼ cm) scale. For the experimental part, we use nuclear magnetic resonance (NMR) to probe molecular displacements (propagators) of an aqueous hydrochloric acid (HCl) solution through a Ketton limestone core. A series of propagator proﬁles are obtained at a large number of spatial points along the core at multiple time-steps during dissolution. For the numerical part, ﬁrst, the transport model—a particle-tracking method based on Continuous Time Random Walks (CTRW) by [1]—is validated at the pore scale by matching to the NMR-measured propagators in a beadpack, Bentheimer sandstone, and Portland carbonate [2]. It was found that the emerging distribution of particle transit times in these samples can be approximated satisfactorily using the power law function ψ ( t ) ∼ t − 1 − β , where 0 < β < 2. Next, the evolution of the propagators during reaction is modelled: at the pore scale, the experimental data is used to calibrate the CTRW parameters; then the

shape of the propagators is predicted at later observation times. Finally, a numerical upscaling technique is employed to obtain CTRW parameters for the core. From the NMR-measured propagators, an increasing frequency of displacements in stagnant regions was apparent as the reaction progressed. The present model predicts that non-Fickian behaviour exhibited at the pore scale persists on the centimetre scale. voirs by acidization [3], water and contaminant management [4], and geo-port laws that can be derived for homogeneous media, see [8,9,10,11,12,13], 23 such as the classical advection-diffusion-reaction equations. Because reactive 24 transport modeling is typically applied at large scales, it necessarily ignores 25 spatial heterogeneities at scales smaller than the size of model discretization 26 , see [14,15]. Several techniques have been introduced as a remedy, i.e to 27 compute effective parameters which capture subscale effects, see [16,17]. 28 Furthermore, while under limited circumstances the homogeneity assumption 29 is reasonable, the pore-scale heterogeneities can result in a significant scaling is extended to reactive transport in carbonates.

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To rid geological transport simulation of uncertainties due to upscaling 56 it is imperative that a numerical model undergoes rigorous laboratory vali-57 dations. In our study, the model and its validation are built upon pore-scale 58 information. The distribution of molecular displacement (or propagators) in 59 the preasymptotic dispersion regime can provide the basis for validation of 60 transport models that are based on X-ray microtomography images of the 61 pore space-see [24,25] PTM-CTRW is explained, in which transport is seen as a series of random 110 hops from one node in a 3D lattice to its neighbouring node. Particles move 111 between a series of discrete nodes or sites with a probability ψ(t : i, j) that 112 a particle that first arrives at site i will move to site j in a time t + dt.
where A is a normalization constant such that t 0 ψ(t )dt = 1, and β ≤ 2 is 127 a power-law coefficient. The pore-to-core simulation technique. Transport is modelled as a series of hops between nodes via links with a known transit time distribution ψ(t). At the smallest scales, advective and diffusive transport is simulated through a lattice representing the porous medium of interest. Transport from one pore to another is described by ψ P that is averaged over all possible statistical realizations of the structure. This ψ P (t) is then input into a simulation at the core-plug scale to compute ψ CP (t) for transitions of particles over the mm scale. Finally transport at a core scale can be represented as a single hop governed by the transit-time distribution function ψ C (t). This figure is adapted from [1].
acts as a measure of heterogeneity. In figure 2 the pore-to-core transport 140 simulation framework is described.

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To clarify the implementation of our method, in figure 3 we show the 142 behaviour of ψ given the variety of its parameters. We plot equation 1 where 143 ψ, is a function of the normalized time τ = t/t 1 for several Péclet numbers be calibrated a priori. Numerical upscaling will then be used again to obtain 156 ψ C (t) (see Section 6).

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In the pore-scale simulation, transport is simulated on a homogeneous 3D Newtonian flow. At each node the mass-flux (q) conservation k q k = 0 is 163 applied for each node connected to links k by which the velocity field at each 164 link can be known, see Appendix A for details. Assuming complete mixing at 165 each node, the probability p(i, j) that a particle landing at pore i will move 166 to one of its neighbours is calculated where q ij is the flux in a link connecting node i and j, and G is a normalization Then a number of particles are released either at the inlet face, or randomly 170 in the lattice.

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At an intersection, a random number a is generated. P (i, j) is then read 172 from memory, and defined as P (i, j) = m p(i, m); m≤ j. The process is When (4) is satisfied, the particle will move along the link i−j. A random 175 number z is generated and the time t required to move along the link i − j 176 is found by solving, using a root-finding method, and t 1 = l/v, l is the link length and v is the fluid velocity within that (1). This is illustrated in figure 4 where ψ s (t) is the transit time distribution 188 function at a scale larger than where transport is governed by ψ r (t). This 189 methodology is applied to obtain both ψ CP (t) from ψ P (t), and ψ C (t) from 190 ψ CP (t).

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In this paper, pulsed field gradient nuclear magnetic resonance (PFG-193 NMR) is used to obtain propagator measurements, i.e. probability distri-  Figure 9: Transport in each slice along the core is represented as a single hop governed by ψ CP (t). Each slice is represented as a 3D lattice consisting of 100 × 100 × 100 links.
In each link, transport is governed by ψ P (t) = ψ(t 1 , t 2 , β P ). t 1 , t 2 are computed using the knowledge of flow rate, Q = 8.3 × 10 −7 m 3 s −1 , and porosity φ. For each slice, β P has been calibrated by matching the NMR-measured propagators. Next, we use the upscaling methodology [1] to obtain ψ CP (t). ψ CP (t) for every slice is tabulated in Appendix B.
ψ CP (t) from ψ P (t) are obtained using the upscaling methodology pre-316 sented in [1], which is illustrated in figure 9. First, we run a particle tracking 317 simulation in each core-plug lattice described above. Then, the emergent ) 2 m 2 . Transport in each link is governed by ψ CP (t). Then the upscaling method [1] is used to obtain ψ C (t).
At the core scale, transport can be interpreted as a single hop with corre-338 sponding ψ C . For transport at this scale a cylindrical lattice is used, see figure   339 10, with length and diameter similar to the core plug used in the experiments.     each node i such that for each link k adjacent to node i, k q k = 0 applies, .

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The