Non-linear rheology and direct numerical simulation of pressure propagation in gel
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Complex fluids are expected to demonstrate rheological behavior intermediate between solids (completely elastic) and fluids (completely viscous). However, despite abundant existence of complex fluids their rheological behavior is still a much debated topic in literature. Time dependent thixotropy rheology has long been considered for complex fluids which does not appropriately account for the compressibility of the fluids. Furthermore, prediction of pressure propagation in gel for the study of flow restart is challenging in absence of appropriate rheological representation of gel. Hence, we propose strain/deformation dependent non-linear shear thinning, viscous hardening and viscoelasticplastic thixotropy rheologies. Deformation in the gel is a function of applied stress, compressibility and time. The proposed rheologies are used in the numerical simulation for the prediction of pressure wave propagation in gel and flow restart. In particular, the effect of the propagation of sequential inertial, viscous and gel degradation waves on the shear melting of the gel are investigated. Moreover, the effect of thermal melting of the gel and heterogeneous initial gel conditions on the pressure wave propagation and flow restart are also analyzed. Three extreme compressibility regimes are identified namely; high compressibility (XΘ ~ 10-6pa-1) where is the isothermal compressibility), low compressibility (XΘ ~10 -6 pa-1) and intermediate compressibility (XΘ ~ 10 -6pa-1). Finite volume on the staggered grid as a numerical method is developed. A fully implicit solver is used for the solution of non-linear set of equations. For validation of numerical algorithm, the pressure wave speed for low viscosity Newtonian fluids is calculated, The pressure wave speed in Newtonian fluid is found to be same as acoustic speed. The rheological model is compared with experimental data. Furthermore, qualitative comparison of pressure wave propagation with literature data and with the data from the flow loop at IFE are obtained. Shear thinning thixotropy rheology, approximates initial deformation in the gel by high viscosity. Viscoelastic-plastic rheology assumes initial deformation in the gel as elastic deformation, followed by creep flow which results in the gel breakage. In viscous hardening rheology, initially viscosity increases with deformation then it decreases similar to shear thinning thixotropy gel. In addition to rheology, compressibility plays a significant role in the pressure wave propagation beyond critical length(critical length Lc = 4pD=Ty, where p is pressure in the gel, D is diameter of pipeline and Ty is the yield stress of the gel). In high compressibility gel, breakage is found to be sequential in nature. In this case both shear thinning and viscoelastic thixotropic rheology have similar prediction. In the cases of intermediate and high compressibility gel, the high initial viscosity in shear thinning rheology results in an attenuated pressure propagation. Subsequent creep flow degrades the gel after a very long time. The thermal melting of the gel and the cohesive failure in the heterogeneous gel makes pressure wave propagation and flow restart faster. The penetration of fresh fluid in the gel is found to be similar to finger (only central gel is displaced by incoming fresh fluid). For low compressibility fluids, the deformation by compression is very small. Hence,for viscoelastic–plastic fluid, the elastic force is much lower than the static yield value that allows pressure wave to propagate beyond critical length . However, the deformation slowly increases and when the absolute value of strain approaches the critical value of strain throughout the pipelines simultaneously, then flow does not restart despite of pressure propagation. In the case of intermediate compressible fluids, deformation in the gel by compression and its subsequent shearing effects (creep) is of the order of critical strain and pressure may not propagate beyond critical length. The pressure wave propagation beyond critical length depends on applied pressure, gel strength, and gel degradation constant in addition to compressibility. Furthermore, transition velocity profile is found as parabolic due to the initial compressional flow followed by the Bingham type intermediate flow profile and finally after gel degradation, velocity profile becomes similar to poiseuille flow. A water hammering kind of effect is observed at virtual boundary (at critical length). We have also proposed a new heuristic rule to determine flow restart and pressure wave propagation beyond critical length. If XΘΔ p2=8Ty >> Υs (where Υs is the strain at static yield stress(maximum stress point on stress–strain curve)) then pressure wave propagates and flow starts for pipeline with length larger than critical length, if XΘp2/8Ty << Υs pressure wave propagates beyond critical length without flow start and in the case of XΘΔp2/8y s neither pressure wave propagates nor flow restarts for the pipeline larger than critical length. Finally, three different approaches are recommended for faster flow restart of the gelled pipeline longer than critical length.