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Evolution of Microstructure: An Example
Abstract
We consider a continuum model for the flow of two phases of different mobility and density in a Hele-Shaw cell or a porous medium. As a consequence of the Saffman-Taylor instability, the phase distribution is thought to develop a microstructure, so that its evolution is effectively unpredictable. We identify the constraints on the macroscopic quantities, like the averaged volume fraction of the phases, and show that these constraints allow to derive some predictions on how the macroscopic quantities change over time. Furthermore, we investigate a class of closure hypothesis, which complement these constraints and thereby determine an evolution of the macroscopic quantities themselves, by analyzing the stability of this evolution.
Our analysis uses a combination of tools from nonlinear scalar conservation laws and ideas from the theory of effective moduli.
... These subsolutions are intended to be a kind of coarse-grained solutions to IPM, withm representing the relaxation of the momentumρv. The subsolutions are very related to the relaxed solutions appearing in the Lagrangian relaxation approach of Otto [65,66] (see also [51]). ...
We construct mixing solutions to the incompressible porous media equation starting from Muskat type data in the partially unstable regime. In particular, we consider bubble and turned type interfaces with Sobolev regularity. As a by-product, we prove the continuation of the evolution of IPM after the Rayleigh–Taylor and smoothness breakdown exhibited in (Castro et al. in Arch Ration Mech Anal 208(3):805–909, 2013, Castro et al. in Ann Math. (2) 175(2):909–948, 2012). At each time slice the space is split into three evolving domains: two non-mixing zones and a mixing zone which is localized in a neighborhood of the unstable region. In this way, we show the compatibility between the classical Muskat problem and the convex integration method.
We show the existence of infinitely many admissible weak solutions for the incompressible porous media equations for all Muskat-type initial data with -regularity of the interface in the unstable regime and for all non-horizontal data with -regularity in the stable regime. Our approach involves constructing admissible subsolutions with piecewise constant densities. This allows us to give a rather short proof where it suffices to calculate the velocity and acceleration at time zero - thus emphasizing the instantaneous nature of non-uniqueness due to discontinuities in the initial data.
We prove that for an open domain with , for every (measurable) uniformly elliptic tensor field a and for almost every point , there exists a unique Green's function centred in y associated to the vectorial operator in D. In particular, when this result also implies the existence of the fundamental solution for elliptic systems, i.e. the Green function for in . Moreover, introducing an ensemble over the set of uniformly elliptic tensor fields, under the assumption of stationarity we infer for the fundamental solution G some pointwise bounds for , and . These estimates scale optimally in space and provide a generalization to systems of the bounds obtained by Delmotte and Deuschel for the scalar case.
An ill-posed sharp interface equation, describing the evolution of two well separated fluids in a porous medium, is replaced by a linearly well-posed Cauchy problem for the evolution of a mixing strip described in terms of level curves of volume proportion of one fluid.
We consider dynamic scaling in gravity driven miscible viscous fingering. We prove rigorous one-sided bounds on bulk transport and coarsening in regimes of physical interest. The analysis relies on comparison with solutions to one-dimensional conservation laws, and new scale-invariant estimates. Our bounds on the size of the mixing layer are of two kinds: a naive bound that is sharp in the absence of diffusion, and a more careful bound that accounts for diffusion as a selection criterion in the limit of vanishingly small diffusion. The naive bound is simple and robust, but does not yield the experimental speed of transport. In a reduced model derived by Wooding [20], we prove a sharp upper bound on the size of the mixing layer in accordance with his experiments. Woodings model also provides an example of a scalar conservation law where the entropy condition is not the physically appropriate selection criterion.
In this paper we show that there exist analytic initial data in the stable
regime for the Muskat problem such that the solution turns to the unstable
regime and later breaks down i.e. no longer belongs to .
In this note we survey some recent results for the Euler equations in
compressible and incompressible fluid dynamics. The main point of all these
theorems is the surprising fact that a suitable variant of Gromov's
h-principle holds in several cases.
Published in Petroleum Transactions, AIME, Volume 216, 1959, pages 188–194.
Introduction
The purposes of this paper are to present theoretical and experimental evidence for occurrence of macroscopic instabilities in displacement of one viscous fluid by another immiscible with it through a uniform porous medium and to compare available experimental data with some predictions of a theory of instability developed by the first author.
The instabilities are referred to as macroscopic in the sense that spatially quasi-sinusoidal, growing fingers of the displacing liquid are formed, the width and peak-to-peak separation (wavelength) of which is large relative to a characteristic length of the particular permeable medium such as grain size.
Visual models of two kinds have been used to obtain observations: displacement of oil by water-glycerine solutions through the flow channel formed by closely spaced parallel plates and displacement of oil by water with and without initial interstitial water through unconsolidated glass powder packs, employing the technique of matching indices of refraction.
In all cases we have observed macroscopic instabilities or fingers under conditions predicted by the theory to be favorable for their occurrence.
The phenomenon discussed here is not the production of streamers due to gross inhomogeneities such as permeability stratification of the porous medium.It is our object to show, on the contrary, that a quantitative prediction of finger spacing is possible in a porous medium known to be macroscopically homogeneous and isotropic throughout.
The importance of the phenomenon in its influence on the configuration of oil and water with respect to oil production behavior was noted earlier by Engelberts and Klinkenberg who coined the term "viscous fingering".
Theory
Necessary and Sufficient Conditions for Instability and Initial Kinematics
There are several levels of increasing complexity in the theoretical description of instability of fluid displacements in permeable media. Of these, the simplest description, adapted to low permeability systems, is selected for presentation. More inclusive descriptions are reserved for separate publication.
The homogenization method gives the possibility of finding equations satisfied by macroscopic quantities from equations satisfied by the physical quantities and from information on the microscopic composition (or on the microscopic structure).
When a viscous fluid filling the voids in a porous medium is driven forwards by the pressure of another driving fluid, the interface between them is liable to be unstable if the driving fluid is the less viscous of the two. This condition occurs in oil fields. To describe the normal modes of small disturbances from a plane interface and their rate of growth, it is necessary to know, or to assume one knows, the conditions which must be satisfied at the interface. The simplest assumption, that the fluids remain completely separated along a definite interface, leads to formulae which are analogous to known expressions developed by scientists working in the oil industry, and also analogous to expressions representing the instability of accelerated interfaces between fluids of different densities. In the latter case the instability develops into round-ended fingers of less dense fluid penetrating into the more dense one. Experiments in which a viscous fluid confined between closely spaced parallel sheets of glass, a Hele-Shaw cell, is driven out by a less viscous one reveal a similar state. The motion in a Hele-Shaw cell is mathematically analogous to two-dimensional flow in a porous medium. Analysis which assumes continuity of pressure through the interface shows that a flow is possible in which equally spaced fingers advance steadily. The ratio lambda = (width of finger)/ (spacing of fingers) appears as the parameter in a singly infinite set of such motions, all of which appear equally possible. Experiments in which various fluids were forced into a narrow Hele-Shaw cell showed that single fingers can be produced, and that unless the flow is very slow lambda = (width of finger)/(width of channel) is close to 1/2, so that behind the tips of the advancing fingers the widths of the two columns of fluid are equal. When lambda =1/2 the calculated form of the fingers is very close to that which is registered photographically in the Hele-Shaw cell, but at very slow speeds where the measured value of lambda increased from 1/2 to the limit 1\cdot 0 as the speed decreased to zero, there were considerable differences. Assuming that these might be due to surface tension, experiments were made in which a fluid of small viscosity, air or water, displaced a much more viscous oil. It is to be expected in that case that lambda would be a function of mu U/T only, where mu is the viscosity, U the speed of advance and T the interfacial tension. This was verified using air as the less viscous fluid penetrating two oils of viscosities 0\cdot 30 and 4\cdot 5 poises.
In this paper we construct a theory of generalized solutions in the large of Cauchy's problem for the equations in the class of bounded measurable functions. We define the generalized solution and prove existence, uniqueness and stability theorems for this solution. To prove the existence theorem we apply the “vanishing viscosity method”; in this connection, we first study Cauchy's problem for the corresponding parabolic equation, and we derive a priori estimates of the modulus of continuity in of the solution of this problem which do not depend on small viscosity.Bibliography: 22 items.
The fingering instability of an interface between two immiscible fluids in a Hele Shaw cell is simulated numerically. The algorithm used is based on a transcription of the equations of motion for the interface in which it formally becomes a generalized vortex sheet. The evolution of this sheet is computed using a variant of the vortex-in-cell method. The resulting scheme and code make it possible to follow the collective behaviour of many competing and interacting fingers well into the nonlinear, large-amplitude regime. It is shown that in this regime the evolution is controlled essentially by just one dimensionless parameter, the ratio of fluid viscosities. The effects of varying this parameter are studied and the results compared with experimental investigations. Scaling properties of the average density profile across the evolving mixed layer between the two homogeneous fluid phases are investigated. Many phenomena are observed that must be characterized as collective interactions and thus cannot be understood in terms of flows with just a single finger.
. We consider the flow of two immiscible fluids of different mobility in a porous medium. If the more mobile fluid displaces the other, a macroscopically sharp interface is unstable. By growing a network of fingers on a mesoscopic scale, the two phases mix on a macroscopic scale. We are interested in the evolution of this mixing zone. We show that the effect of a large but finite mobility ratio is strong enough to limit the growth rate of the mixing zone. This is done by rigorously deriving an a priori estimate for the Saffman--Taylor model. In this geometry of an infinite channel, the estimate essentially states that the mobility ratio itself (in the nondimensionalized stetting with unit velocity imposed at infinity) is the optimal bound on the velocity by which the penetrating phase progresses in direction of the channel. Since the introduction of DLA, various stochastic algorithms simulating this two-- phase flow have been developed. The generated clusters, which correspond to th...
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