In a disordered media, certain regions allow the movement of interfaces more easily than others. In fact, due to capillary forces the interfaces can only move if a certain pressure is applied. Similarly to the yield stress fluid case, the physical reason behind this observation lies in the presence of the heterogeneity. 2021) have shown the existence of a nonlinear flowing regime at low flow rate. ![]() In the last decade, however, a series of experiments and simulations (Tallakstad et al. It is then surprising that, during many decades, the models predicting the mean flow rate as function of the mean applied pressure have assumed linear relations (Bear 1988 Dullien 1991). One of the main difficulties lies in the presence of numerous interfaces exerting capillary forces on the fluids present, which makes the dynamic very nonlinear. Multi-phase flow in porous media is a very old and rich subject, and is still the topic of many ongoing researches. If the flow of yield stress fluids in porous media is already a challenging problem, in many situations the complexity is increased by the presence of different immiscible fluids. The flow curve above the threshold depends then on the details of the opening distribution (Nash and Rees 2016). This is, however, not the case in 1D, if one describes the porous media by a series of uniform bundle of capillaries. ( 2019) for 2D porous media is independent from the type of disorder. The origin of this flowing regime is an effect of the disorder, but remarkably the value of the exponent \(\beta =2\) found in a recent work by Liu et al. Where \(\Delta P_c\) is a pressure drop threshold and \(\beta\) a characteristic exponent to be determined. As a consequence, the flow rate Q increases with the applied pressure drop \(\Delta P\) according to a power-law: 2019), a progressive increase of flowing paths occurs. Above this pressure threshold, as demonstrated by several studies (Roux and Herrmann 1987 Talon and Bauer 2013 Chevalier and Talon 2015 Waisbord et al. There is then a strong coupling between the rheology of the fluid and the disorder of the porous structure, implying that some regions are easier to yield than others. ![]() Because of the presence of a yield stress, the fluid is able to flow only if a certain amount of pressure is imposed (Roux and Herrmann 1987 Chen et al. 2005 Sochi and Blunt 2008 Talon and Bauer 2013 RodrÃguez de Castro and Radilla 2017 Liu et al. 1973 Al-Fariss and Pinder 1987 Chen et al. Yield stress fluid in porous media is a challenging and interesting problem which has been the subject of many studies in the last decades (Entov 1967 Park et al. 2013) or ground reinforcement by cement injection. 2016), stabilization of bone fractures in biomedical engineering (Widmer Soyka et al. These fluids are involved in many practical applications, such as drilling for oil extraction, where proppant fluids are injected in the soil for the fracking process (Barbati et al. Here, we are interested in yield stress fluids, which require a minimal applied stress to flow. 1989 Coussot 2005) or some biological fluids like blood (Popel and Johnson 2005 Bessonov et al. Indeed many complex fluids present a nonlinear rheology as, for example, slurries, heavy oils, suspensions (Barnes et al. In many industrial, geophysical or biological applications related to porous media, non-Newtonian fluids are frequently encountered. We perform numerical simulations confirming our analytical results. We consider two geometries: tubes of sinusoidal shape, for which we derive explicit expressions, and triangular-shaped tubes, for which we find that essential singularities are developed. For a capillary fiber bundle of identical parallel tubes, we calculate the probability distribution of the threshold pressure and the mean flow rate. Furthermore, in the presence of many blobs the threshold value depends on the number of blobs and their relative distances which are randomly distributed. We find that the capillary effects emerge from the non-uniformity of the tube radius and contribute to the threshold pressure for flow to occur. We compute here the yield pressure drop and the mean flow rate in two cases: (i) When a single blob is injected, (ii) When many blobs are randomly injected into the tube. We consider here a generalization of previously obtained results to blobs of non-Newtonian fluids. Fluid blobs in an immiscible Newtonian fluid flowing in a capillary tube with varying radius show highly nonlinear behavior.
0 Comments
Leave a Reply. |