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An Ising-Based Simulator for Capillary Action in Porous Media

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Abstract

Multiphase flows in porous media are encountered in several contexts—e.g., hydrocarbon recovery operations, battery electrodes, microfluidic devices, etc. Capillary-dominated flows are interesting due to the complex interplay of interfacial properties and pore geometries. Conventional hydrodynamic flow solvers are computationally inefficient in the capillary-dominated regime, particularly in complex pore structures. The algorithm developed here specifically targets this regime to reduce simulation times. We minimise the fluid–fluid and fluid–solid interaction energies through an approach inspired by the ferromagnetic Ising model. We validate the algorithm on (1) model pore geometries with analytical solutions for capillary action, and (2) rocks with available mercury porosimetry data. We validate its predictions for model geometries and sandstones using (1) curvatures calculated from theories developed by Mayer–Stowe–Princen, Ma and Morrow, and Mason and Morrow; (2) predictions from GeoDict, a commercial software package, which also includes a state-of-the-art drainage simulator; (3) mercury porosimetry data. Drainage capillary pressure curves predicted for Bentheimer and Fontainebleau rocks reasonably match porosimetry data.

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The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

We thank Steffen Berg, Nishank Saxena and Cor van Kruijsdijk from Shell for going through the manuscript and providing valuable suggestions. We also thank Steffen Berg and Nishank Saxena for providing us the sandstone images and associated MICP data. We are also grateful to Shell for funding this work and providing permission to publish this paper.

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Correspondence to Nitish Nair.

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Appendix

Appendix

1.1 A1 Link with Work of Invasion

We will now connect our invasion scheme to the thermodynamics of drainage as laid out by Mayer and Stowe (1965) and Morrow (1970). Figure 13 shows the oil meniscus (red) advancing through a water-filled pore with solid walls. A differential displacement of dn leads to changes in the areas of interfaces between oil–water (dAOW), oil-solid (dAOS) and water–solid (dAWS). The dotted red line denotes the new location of the oil–water interface.

Fig. 13
figure 13

Differential displacements and changes in surface area during drainage

If PO is the pressure in the oil phase and PW is the pressure in the water phase, the work expended in advancing the oil–water interface through a volume dV under quasi-static conditions is

$$ \left( {P_{O} - P_{W} } \right){\text{d}}V = \gamma_{\text{OW}} {\text{d}}A_{\text{OW}} + \gamma_{\text{OS}} {\text{d}}A_{\text{OS}} + \gamma_{WS} {\text{d}}A_{\text{WS}} $$
(A1a)

where the right-hand side is the energy required to modify the various interfaces in the system. The different interfacial energies are products of the respective interfacial tensions, γ, and the change in interfacial area, dA. As the oil phase moves forward, the increase in the oil-solid contact area equals the decrease in the water–solid contact area, i.e., dAOS = − dAWS. The balance of interfacial forces at the three-phase contact point gives us γOS − γWS = γOW cosθ, where θ is the contact angle measured in the water phase.

The difference in pressure across the oil–water interface is the capillary pressure, i.e., Pc = POPW. Applying these 3 simplifications to Eq. A1a gives

$$ P_{c} {\text{d}}V = \gamma_{\text{OW}} \left( {{\text{d}}A_{\text{OW}} + {\text{d}}A_{\text{OS}} \cos \theta } \right) $$
(A1b)

Recasting Eq. A1b along the lines of Eq. 2a, 2b, 2c in the main text, we get

$$ \frac{{P_{\text{c}} }}{{\gamma_{\text{OW}} }} = C = \frac{{{\text{d}}A_{\text{OW}} }}{{{\text{d}}V}} + \frac{{{\text{d}}A_{\text{OS}} }}{{{\text{d}}V}}\cos \theta $$
(A1c)

where C is the curvature of the meniscus and Pc/γOW is the scaled capillary pressure. Let us focus on the average interfacial energy, \( \bar{E} \). We stated earlier that Pc/γOW scales as \( \bar{E} \). Starting with this notion, we use Eqs. 45 from the main text to get

$$ \frac{{P_{c} }}{{\gamma_{\text{OW}} }}\sim \frac{{\sum\limits_{i = 1}^{{N_{\text{flip}} }} {\Delta E_{i} } }}{{N_{\text{flip}} }} = \frac{1}{{N_{\text{flip}} }}\sum\limits_{i = 1}^{{N_{\text{flip}} }} {\sum\limits_{{j \in {\text{nbrs}}\left( i \right)}} {s_{j} } } $$
(A2a)

The innermost sum over the neighbourhood of cell i is decomposed into three more sums: one over water-like neighbours, the second over oil-like neighbours, and the third over rock-like neighbours. We denote these constituent neighbourhoods as W(i), O(i) and R(i), respectively.

$$ \frac{{P_{\text{c}} }}{{\gamma_{\text{OW}} }}\sim \frac{1}{{N_{\text{flip}} }}\sum\limits_{i = 1}^{{N_{\text{flip}} }} {\left[ {\sum\limits_{j \in W\left( i \right)} {s_{j} } + \sum\limits_{j \in O\left( i \right)} {s_{j} } + \sum\limits_{j \in R\left( i \right)} {s_{j} } } \right]} $$
(A2b)

The spins of water, oil and rock cells—1, − 1 and cosθ, respectively—are inserted in Eq. A2b to facilitate the easy calculation of the three inner sums.

$$ \frac{{P_{\text{c}} }}{{\gamma_{\text{OW}} }}\sim \frac{1}{{N_{\text{flip}} }}\sum\limits_{i = 1}^{{N_{\text{flip}} }} {\left[ {N_{W} \left( i \right) - N_{O} \left( i \right) + N_{R} \left( i \right)\cos \theta } \right]} $$
(A2c)

where NX(i) is the number of X-like (X = W, O, R) neighbours near cell i. Using the relation ΔNWO′(i) = NW(i) − NO(i) and teasing apart the fluid and rock terms, we get

$$ \frac{{P_{\text{c}} }}{{\gamma_{\text{OW}} }}\sim \frac{{\sum\nolimits_{i = 1}^{{N_{\text{flip}} }} {\Delta N_{\text{WO}} \left( i \right)} }}{{N_{\text{flip}} }} + \frac{{\sum\nolimits_{i = 1}^{{N_{\text{flip}} }} {N_{R} \left( i \right)} }}{{N_{\text{flip}} }}\cos \theta $$
(A2d)

A comparison of Eqs. A1c and A2d suggests that

$$ \frac{{{\text{d}}A_{\text{OW}} }}{{{\text{d}}V}}\sim \frac{{\sum\nolimits_{i = 1}^{{N_{\text{flip}} }} {\Delta N_{\text{WO}} \left( i \right)} }}{{N_{\text{flip}} }} $$
(A3a)
$$ \frac{{{\text{d}}A_{\text{OS}} }}{{{\text{d}}V}}\sim \frac{{\sum\nolimits_{i = 1}^{{N_{\text{flip}} }} {N_{R} \left( i \right)} }}{{N_{\text{flip}} }} $$
(A3b)

Recall that Nflip is the number of interface cells that are flipped from water to oil during an invasion step. It represents the incremental progress of the interface and hence the differential volume, dV, associated with this progress. On the LHS in Eqs. A3a, A3b, dAOW and dAOS are differential areas that denote changes in the fluid and rock interactions during invasion. Similarly, the RHS in Eqs. A3a, A3b represent the net contributions from the fluid and rock neighbours of the interfacial cells in an invasion step. This completes the equivalence between the thermodynamics of invasion and our computational formalism.

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Nair, N., Koelman, J.V. An Ising-Based Simulator for Capillary Action in Porous Media. Transp Porous Med 124, 413–437 (2018). https://doi.org/10.1007/s11242-018-1075-5

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