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A two-dimensional numerical model based on smoothed particle hydrodynamics (SPH) was used to simulate unsaturated (multiphase) flow through fracture junctions. A combination of standard SPH equations with pairwise fluid-fluid and fluid-solid particle-particle interactions allowed surface tension and three-phase contact dynamics to be simulated. The model was validated by calculating the surface tension in four different ways: (i) from small-amplitude oscillations of fluid drops, (ii) from the dependence of the capillary pressure on drop radius, (iii) from capillary rise simulations, and (iv) from the behavior of a fluid drop confined between parallel walls under the influence of gravity. All four simulations led to consistent values for the surface tension. The dependence of receding and advancing contact angles on droplet velocity was studied. Incorporation of surface tension and fluid-solid interactions allowed unsaturated flow through fracture junctions to be realistically simulated, and the simulation results compare well with the laboratory experiments of Dragila and Weisbrod.

[1] In the subsurface, fluids play a critical role by transporting dissolved minerals, colloids, and contaminants (sometimes over long distances); by mediating dissolution and precipitation processes; and by enabling chemical transformations in solution and at mineral surfaces. Although the complex geometries of fracture apertures, fracture networks, and pore spaces may make it difficult to accurately predict fluid flow in saturated (single-phase) subsurface systems, well-developed methods are available. The simulation of multiphase fluid flow in the subsurface is much more challenging because of the large density and/or viscosity ratios found in important applications (water/air in the vadose zone; water/oil, water/gas, gas/oil, and water/ oil/gas in hydrocarbon reservoirs; water/air/nonaqueous phase liquids (nonaqueous phase liquids/dense nonaqueous phase liquids) in contaminated vadose zone systems; and gas/molten rock in volcanic systems, for example). In addition, the complex behavior of fluid-fluid-solid contact lines and their impact on dynamic contact angles must also be taken into account and coupled with the fluid flow. Here we review the methods that are currently being used to simulate pore-scale multiphase fluid flow and reactive transport in fractured and porous media. After the introduction, the review begins with an overview of the fundamental physics of multiphase fluids flow followed by a more detailed discussion of the complex dynamic behavior of contact lines and contact angles, an important barrier to accurate pore-scale modeling and simulation. The main part of the review focuses on five different approaches: pore network models, lattice gas and lattice Boltzmann methods, Monte Carlo methods, particle methods (molecular dynamics, dissipative particle dynamics, and smoothed particle hydrodynamics), and traditional grid-based computational fluid dynamics coupled with interface tracking and a contact angle model. Finally, the review closes with a discussion of future trends and challenges.Citation: Meakin, P., and A. M. Tartakovsky (2009), Modeling and simulation of pore-scale multiphase fluid flow and reactive transport in fractured and porous media, Rev. Geophys., 47, RG3002,

[1] Laboratory experiments, pore-scale simulations, and continuum (Darcy) -scale simulations were used to study mixing-induced precipitation in porous media. In the experimental investigation, solutions containing Na 2 CO 3 and CaCl 2 were each injected in different halves of a quasi two-dimensional flow cell filled with quartz sand. As a result of the in situ mixing between the two solutions, a narrow calcite precipitate layer (less than 5 mm wide) of more or less uniform width was formed between the individual solutions. Pore-scale simulations were conducted to help understand the mechanism of precipitation layer formation. The effect of the Peclet number, Pe, and the Damköhler number, Da, on mixing induced precipitation was also investigated. Pore-scale simulations revealed the presence of large pore-scale concentration gradients. This, and the presence of features, such as the precipitation layer, with characteristic lengths on the order of the average sand grain diameter, indicate the absence of a clear scale separation required for the strict derivation of Darcy-scale advection-dispersion equations. Nevertheless, we found that an adaptive high-resolution model based on advection-dispersion equations with grid sizes in the mixing zone smaller than the size of the sand grains can qualitatively reproduce the essential features of the experiment. As an alternative to computationally expensive high-resolution simulations, we proposed new forms for the homogeneous and heterogeneous reaction terms in Darcy-scale advection dispersion equations. These terms involve transport and mixing indices that account for highly nonuniform pore-scale concentration distributions and highly localized reactions. The proposed model accurately estimates the changes in solute concentrations due to homogenous and heterogeneous reactions during precipitation of minerals, observed in the pore-scale simulations, while conventional low-resolution advective-dispersion equations produced erroneous results.

We present a physics-informed deep neural network (DNN) method for estimating hydraulic conductivity in saturated and unsaturated flows governed by Darcy's law. For saturated flow, we approximate hydraulic conductivity and head with two DNNs and use Darcy's law in addition to measurements of hydraulic conductivity and head to train these DNNs. For unsaturated flow, we approximate unsaturated conductivity function and capillary pressure with DNNs and train these DNNs using measurements of capillary pressure and the Richards equation. Because it is difficult to measure unsaturated conductivity in the field, we assume that no measurements of unsaturated conductivity are available. The proposed approach enforces the partial differential equation (PDE) (Darcy or Richards equation) constraints by minimizing the PDE residual at select points in the simulation domain. We demonstrate that physics constraints increase the accuracy of DNN approximations of sparsely observed functions and allow for training DNNs when no direct measurements of the functions of interest are available. For the saturated conductivity estimation problem, we show that the physics-informed DNN method is more accurate than the state-of-the-art maximum a posteriori probability method. For the unsaturated flow in homogeneous porous media, we find that the proposed method can accurately estimate the pressure-conductivity relationship based on the capillary pressure measurements only, even in the presence of measurement noise.

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