Flow in Binary Media with Heterogeneous Air-Entry Pressure
In this chapter, the influence of heterogeneity in the air-entry pressure on the field-scale flow is examined. This issue is particularly important for media containing disconnected coarse-textured inclusions with low entry pressure embedded in a continuo
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Flow in Binary Media with Heterogeneous Air-Entry Pressure
The upscaled models presented in Chap. 6 are based on the assumption that the Richards equation is a valid model of water flow at the local (Darcy) scale. This, however, is not necessarily the case, especially during transition between unsaturated and water-saturated conditions in porous media showing distinct and locally variable values of the entry pressure. As the continuity of the air phase and its connection to the atmosphere may be lost, the assumptions underlying the Richards model no longer hold, and the description should be based on the full two-phase flow model. This chapter presents the development of an upscaled model which accounts for heterogeneity in the air entry pressure and which is applicable to capillary-dominated flow in media showing moderate permeability contrast. It is shown that, after appropriate modification, the upscaled Richards equation shows a better agreement with the reference two-phase model than the non-upscaled Richards equation solved for explicitly represented heterogeneous structure. The following presentation is based on papers [8, 9].
7.1 Upscaled Model of Two-phase Capillary Flow 7.1.1 Basic Assumptions The porous medium is characterized by the same binary structure as considered in Chap. 6, see Fig. 6.1, i.e. it is composed of a continuous background material, denoted by superscript I and disconnected inclusions, denoted by superscript II. Furthermore, it is assumed that the condition of separation of scales, given by Eq. (6.1), is satisfied. At the local scale, the flow of water and air in each region is described by Eqs. (2.50)– (2.51), with the storage terms written according to Eq. (2.52):
A. Szymkiewicz, Modelling Water Flow in Unsaturated Porous Media, GeoPlanet: Earth and Planetary Sciences, DOI: 10.1007/978-3-642-23559-7_7, © Springer-Verlag Berlin Heidelberg 2013
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7 Flow in Binary Media with Heterogeneous Air-Entry Pressure
I I kw I I − ∇ ρw ∇ pw − ρw g = ∂t ∂t μw I I ∂ paI I I ∂θa I ka I I θa ca + ρa − ∇ ρa ∇ pa − ρa g = ∂t ∂t μa ∂ p II ∂θ II k II θwII cw w + ρwII w − ∇ ρwII w ∇ pwII − ρwII g = ∂t ∂t μw II II ∂p ∂θ k II θaII ca a + ρaII a − ∇ ρaII a ∇ paII − ρaII g = ∂t ∂t μa ∂pI θwI cw w
∂θ I + ρwI w
0 in I ,
(7.1)
0 in I ,
(7.2)
0 in II , (7.3) 0 in II , (7.4)
where cw and ca are the compressibility coefficients of water and air. The conditions at the interface vary, depending on the saturation of the two materials, as discussed in the following sections. Similarly to the previous chapter, the characteristic time of the process is chosen as the time of flow at the macroscopic scale in the background material. However, in contrast to the previous chapter, the analysis is limited to the case when the permeabilities of the two materials are of the same order of magnitude. As far as the local balance of driving forces is considered, we assume that at the scale of a single periodic cell the capillary forces dominate over the viscous and gravitation
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