Equations of state (EoS) are essential in the modeling of a wide range of industrial and natural processes. Desired qualities of EoS are accuracy, consistency, computational speed, robustness, and predictive ability outside of the domain where they have been fitted. In this work, we review present challenges associated with established models, and give suggestions on how to overcome them in the future. The most accurate EoS available, multiparameter EoS, have a second artificial Maxwell loop in the two-phase region that gives problems in phase-equilibrium calculations and excludes them from important applications such as treatment of interfacial phenomena with mass-based density functional theory. Suggestions are provided on how this can be improved. Cubic EoS are among the most computationally efficient EoS, but they often lack sufficient accuracy. We show that extended corresponding state EoS are capable of providing significantly more accurate single-phase predictions than cubic EoS with only a doubling of the computational time. In comparison, the computational time of multiparameter EoS can be orders of magnitude larger. For mixtures in the two-phase region, however, the accuracy of extended corresponding state EoS has a large potential for improvement. The molecular-based SAFT family of EoS is preferred when predictive ability is important, for example, for systems with strongly associating fluids or polymers where few experimental data are available. We discuss some of their benefits and present challenges. A discussion is presented on why predictive thermodynamic models for reactive mixtures such as CO2–NH3 and CO2–H2O–H2S must be developed in close combination with phase- and reaction equilibrium theory, regardless of the choice of EoS. After overcoming present challenges, a next-generation thermodynamic modeling framework holds the potential to improve the accuracy and predictive ability in a wide range of applications such as process optimization, computational fluid dynamics, treatment of interfacial phenomena, and processes with reactive mixtures.
Based on thermodynamic considerations we derive a set of equations relating the seepage velocities of the fluid components in immiscible and incompressible two-phase flow in porous media. They necessitate the introduction of a new velocity function, the co-moving velocity. This velocity function is a characteristic of the porous medium. Together with a constitutive relation between the velocities and the driving forces, such as the pressure gradient, these equations form a closed set. We solve four versions of the capillary tube model analytically using this theory. We test the theory numerically on a network model.
We study three different time integration methods for a dynamic pore network model for immiscible two-phase flow in porous media. Considered are two explicit methods, the forward Euler and midpoint methods, and a new semi-implicit method developed herein. The explicit methods are known to suffer from numerical instabilities at low capillary numbers. A new time-step criterion is suggested in order to stabilize them. Numerical experiments, including a Haines jump case, are performed and these demonstrate that stabilization is achieved. Further, the results from the Haines jump case are consistent with experimental observations. A performance analysis reveals that the semi-implicit method is able to perform stable simulations with much less computational effort than the explicit methods at low capillary numbers. The relative benefit of using the semi-implicit method increases with decreasing capillary number Ca, and at Ca ∼ 10 −8 the computational time needed is reduced by three orders of magnitude. This increased efficiency enables simulations in the low-capillary number regime that are unfeasible with explicit methods and the range of capillary numbers for which the pore network model is a tractable modeling alternative is thus greatly extended by the semi-implicit method. method [2, 3] to keep track of the fluid interface locations, have been used. The lattice-Boltzmann method is another popular choice, see e.g. [4]. These methods can provide detailed information on the flow in each pore. They are, however, computationally intensive and this restricts their use to relatively small systems.Pore network models have proven to be useful in order to reduce the computational cost [5], or enable the study of larger systems, while still retaining some pore-level detail. In these models, the pore space is partitioned into volume elements that are typically the size of a single pore or throat. The average flow properties in these elements are then considered, without taking into account the variation in flow properties within each element.Pore network models are typically classified as either quasi-static or dynamic. The quasi-static models are intended for situations where flow rates are low, and viscous pressure drops are neglected on the grounds that capillary forces are assumed to dominate at all times. In the quasi-static models by Lenormand et al. [6], Wilkinson and Willemsen [7] and Blunt [8], the displacement of one fluid by the other proceeds by the filling of one pore at the time, and the sequence of pore filling is determined by the capillary entry pressure alone.The dynamic models, on the other hand, account for the viscous pressure drops and thus capture the interaction between viscous and capillary forces. As three examples of such models, we mention those by Hammond and Unsal [5], Joekar-Niasar et al.[9] and Aker et al. [10]. A thorough review of dynamic pore network models was performed by Joekar-Niasar and Hassanizadeh [11].The pore network model we consider here is of the dynamic type that was first pre...
We present in detail a set of algorithms for a dynamic pore-network model of immiscible two-phase flow in porous media to carry out fluid displacements in pores. The algorithms are universal for regular and irregular pore networks in two or three dimensions and can be applied to simulate both drainage displacements and steady-state flow. They execute the mixing of incoming fluids at the network nodes, then distribute them to the outgoing links and perform the coalescence of bubbles. Implementing these algorithms in a dynamic pore-network model, we reproduce some of the fundamental results of transient and steady-state two-phase flow in porous media. For drainage displacements, we show that the model can reproduce the flow patterns corresponding to viscous fingering, capillary fingering and stable displacement by varying the capillary number and viscosity ratio. For steady-state flow, we verify non-linear rheological properties and transition to linear Darcy behavior while increasing the flow rate. Finally we verify the relations between seepage velocities of two-phase flow in porous media considering both disordered regular networks and irregular networks reconstructed from real samples.
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