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9. Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . flow within (drained) pore throats. In this chapter, we present a new formulation for pore-network modeling of two-phase flow, which explicitly accounts for the resistant to the flow within the drained pore bodies and calculates different fluxes within drained pore bodies. The results of Chapter 5 shows the significance of this effect under primary drainage conditions, by applying our formulation into a MDPN model. Resulting saturation-relative permeability relationships, with taking account the resistance due to the drained pore bodies, show a better agreement with experimentally measured values. The difference obtained using our method could be considerable especially at higher saturations when larger pore bodies are getting invaded by the non-wetting phase. We have also examined the concept of equivalent pore conductance and compared the results with our method. The comparison shows that making the correction terms for conductances of pore throats as a function of saturation can improve the results, however, using the concept of equivalent pore conductance, we do not solve for fluid flow within pore bodies. In this way, we do not get information on fluxes within pore bodies. This information is needed for an accurate simulation of solute transport, where because of the presence of non-wetting phase, there is limited mixing within the pore bodies. Chapter 6 intended to study dispersion coefficient under partially saturated conditions using a new formulation. It is known that in unsaturated porous media, the dispersion coefficient depends on the Darcy velocity as well as saturation. The dependency of dispersion on velocity is fairly studied, however, there is not much known about its dependence on saturation and the underlying process. The purpose of this chapter is to investigate how the longitudinal dispersivity varies with saturation. In our formulation, both pore bodies and pore throats have volumes and we assign separate concentrations to each of them. Further, since pore geometry and corner flows greatly influence transport properties, efforts are made to include different angular cross sectional shapes for the pore throats. This includes circular, rectangular, and all kind of irregular triangular cross sections, which are important especially under unsaturated/two-phase flow and reactive transport. We have introduced a new formulation of solute transport within pore network which helps to capture the effect of limited mixing under partially-saturated conditions. In this formulation we refine the discretization on the basis of saturation state of pores. We assign separate concentration for different corners of a given drained pore body and also we assign different concentrations for different edges of a drained angular pore throat. This formulation allows a very detailed description of 198
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pore-scale solute transport processes by accounting for limitations in mixing as a result of reduced water content. The numerically computed dispersivities could successfully explain the results obtained through experimental studies, and shows the underplaying pore scale processes contributing to dispersion under unsaturated conditions. Results of Chapter 6 show that, under partially saturated conditions, dispersivity coefficient is strongly depended on saturation. The relation could be non-monotonic, with the maximum dispersivity (α max ) corresponding to a specific saturation, which we refer to it as the critical saturation (S cr ). We could explain the α − S w relation by investigating pore-scale properties such as fraction of percolating saturated pores and coefficient of variation of pore velocities under different saturations. These pore scale observations could clearly explain the relation between solute dispersivity and saturation under partially saturated conditions. We have shown that saturation-relative permeability (k r − S w ) curve may be utilized to get insight into the behavior of α − S w curve and to approximate the critical saturation at which the maximum dispersivity (α max ) occurs. Chapter 7 is specified to transport of adsorptive solute under partially saturated conditions. Under partially saturated conditions, the system contains three phases: air, water, and solid. The principal interactions usually occur at the solid-water interfaces (SW) and air-water interfaces (AW), thus greatly influenced by water content. All of the modeling capabilities which were implemented through last chapters are also included in this chapter. In addition, we have formulated various types of adsorption such as i) two site (SW and AW interfaces) kinetic, ii) two site equilibrium, and iii) one site (SW or AW interfaces) kinetic and one site equilibrium. For the macro scale representation of equilibrium adsorptive transport, we have chosen two models: Advection-Dispersion Equation (ADE) with equilibrium adsorption term and Mobile-Immobile Model (MIM) which models the non-equilibrium transport. Comparing the results from analytical solutions of these two macro-scale models with the results obtained from pore network model show that the Mobile- Immobile Model (MIM) provides a better fit to the results of the pore network modeling. Applying equilibrium macro-scale model (ADE), we need to apply higher values of dispersivity which in addition to saturation will depend on the value of pore scale adsorption coefficient. However, using MIM we could use the dispersion coefficients which were independently determined through tracer simulations to simulate adsorptive transport. The results of this chapter show that, even if there is equilibrium adsorption at the solid-water (SW) and 199
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Reactive/Adsorptive Transport in (P
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text Reactive/Adsorptive Transport
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Promoter: Prof.dr.ir. S.M. Hassaniz
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My colleagues and other staff in th
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CONTENTS 2.2.3 Connection Matrix .
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CONTENTS 5.3.3 Regular hyperbolic p
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CONTENTS 8.3.1 Non-adsorptive solut
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LIST OF FIGURES 3.6 The relation be
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LIST OF FIGURES 6.5 Breakthrough cu
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LIST OF TABLES 2.1 Expressions for
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1. Introduction . . . . . . . . . .
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1. Introduction . . . . . . . . . .
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1. Introduction . . . . . . . . . .
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1. Introduction . . . . . . . . . .
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1. Introduction . . . . . . . . . .
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1. Introduction . . . . . . . . . .
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1. Introduction . . . . . . . . . .
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Part I Generation of Multi-Directio
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2. Generating MultiDirectional Pore
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3.1 Introduction . . . . . . . . .
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3.1 Introduction . . . . . . . . .
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3.2 Theoretical upscaling of adsorp
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3.3 Numerical upscaling of adsorbin
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3.4 Discussion of results . . . . .
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3.5 Conclusion . . . . . . . . . .
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Published as: Raoof, A. and Hassani
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4.1 Introduction . . . . . . . . .
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4.1 Introduction . . . . . . . . .
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4.2 Description of the Pore-Network
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4.3 Simulating flow and transport w
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4.3 Simulating flow and transport w
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4.4 Macro-scale adsorption coeffici
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4.5 Discussion . . . . . . . . . .
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4.5 Discussion . . . . . . . . . .
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4.6 Conclusions . . . . . . . . . .
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Raoof, A. and Hassanizadeh S. M.,
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5.1 Introduction . . . . . . . . .
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5.1 Introduction . . . . . . . . .
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5.2 Network Generation . . . . . .
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5.2 Network Generation . . . . . .
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5.3 Modeling flow in the network .
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5.4 Results . . . . . . . . . . . .
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5.4 Results . . . . . . . . . . . .
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5.4 Results . . . . . . . . . . . .
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5.5 Conclusion Our approach is also
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6. Dispersivity under Partially-Sat
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Page 256 and 257: REFERENCES D.F. Yule and W.R. Gardn
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