ABSTRACT
In the present contribution we discuss the basic problem of a solid-liquid phase-transformation process which is known in the geomechanical literature as erosion, elutriation or abrasion. These terms describe a large variety of phenomena like the sand production problem in well bores in the North-Sea sandstone reservoir (Vardoulakis et al. 1996), outbursts in coalmining (Paterson 1986) or the selective removal of fine dust from unconsolidated media, e. g. in granular beds or columns, c. f. Carman (1956). What is the common mechanism in the mentioned, quite different, applications? To answer this question, we have a closer look in the microstructure of an (artificial) geomaterial where one can observe four stages in a Representative Element Volume (REV) , with a domain R and a boundary ∂R. Here, the so-called branch vector x(i)M is defined from the center of mass of R to its boundary ∂R. The involved stages are characterized as follows: Figure 1(a) shows the initial configuration of a fluid-saturated porous medium, e. g. a granular bed. The pore-fluid is shown as the (light gray) phase in the background and is abbreviated by ϕf while the polydisperse granular material is sketched as gray circles and denoted by ϕs. As our aim is a macroscopic, i. e. homogenized, description of the governing process we apply the mixture theory (Truesdell & Toupin 1960; Bowen 1980; Ehlers 2002) as an appropriate tool to formulate the governing equations of the process. Thus, we introduce volume fractions nα(x, t) as additional field quantities on the macroscale to describe the local composition of the mixture at the level of the REVs. The volume
fractions for the phases or constituents ϕα are defined as nα = dvα/dv, where dvα is the volume occupied by the constituent ϕα while dv is the total volume of the mixture. It is obvious that
∑ α n
α ≡ 1. The latter relation is the so-called saturation-condition. Coming back to 1(a), we observe that no solid-liquid phase transition occurs. Thus, we investigate only an intact solid
skeleton ϕs and the pore-fluid ϕf. This is the classical case in non-erosive poro-mechanical applications. In Figure 1(b) all three phases are active, characterized by volume fractions 0 < nα < 1 with α={s, f, a} larger than zero. Now we observe eroded fines, i. e. particles which are in non-permanent contact to neighboring particles, sketched as dark gray circles. Anyway, the phase-transition process cannot be observed on the macroscale, as the fines are just hovering in small cavities. Thus, no diffusion or convection of fines is possible at this stage. Transportation of fines, sometimes denoted as mass-flux, is possible in 1(c), if the erosion process has built-up channel-like formations with diameters which are larger than the characteristic diameter of the fines. Having a closer look at 1(d), the contradiction of 1(a) is observed. Thus, the solid skeleton is completely destroyed, leading to ns = 0, whereas all particles are fluidized. It is obvious that the Cauchy extra stress tensor of the solid constituents is zero in that case, denoted as TsE = 0, as no contactforces can be transmitted from one solid particle to another. The latter stage (d) is studied intensively in the fluid mechanical literature, (Jackson 2000; Davidson & Harrison 1971; Vardoulakis 2004), and will not be considered within the present model.
