Nonlinear waves in compacting media

Barcilon, Victor; Richter, Frank M.

doi:10.1017/s0022112086002628

Cited by 160 publications

(189 citation statements)

References 7 publications

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“…where 7], e are the solid phase shear and bulk viscosities and !1p = p. -P! is the buoyancy difference.…”

Section: Matrix Deformationmentioning

confidence: 99%

Solid-fluid interactions in porous media : processes that form rocks

Aharonov¹

1996

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“…where 7], e are the solid phase shear and bulk viscosities and !1p = p. -P! is the buoyancy difference.…”

Section: Matrix Deformationmentioning

confidence: 99%

Solid-fluid interactions in porous media : processes that form rocks

Aharonov¹

1996

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“…The fluid rheology is that of an incompressible inviscid fluid while the solid rheology is treated as viscously compressible fluid. We refer the reader to [2,3,4,5,6] for expositions on these relationships and their simplifications. When reduced to 1-D, as in [3,4,5,7,8,9], this transport can be described by a class of degenerate nonlinear dispersive partial differential equations of the form:…”

Section: Introduction and Overviewmentioning

confidence: 99%

“…Previous studies, [5,7,12,13,14,15,16,17], of the equations did not successfully address the issue of well-posedness and the related issue of a lower bound on φ, the (scaled) melt fraction, although [12] did point out several reasons why solutions for which φ went to zero would be non-physical.…”

Section: Introduction and Overviewmentioning

confidence: 99%

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Degenerate dispersive equations arising in the study of magma dynamics

2006

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Abstract. An outstanding problem in Earth science is understanding the method of transport of magma in the Earth's mantle. Two proposed methods for this transport are percolation through porous rock and flow up conduits. Under reasonable assumptions and simplifications, both means of transport can be described by a class of degenerate nonlinear dispersive partial differential equations of the form:, where φ(z, 0) > 0 and φ(z, t) → 1 as z → ±∞. Although we treat arbitrary n and m, the exponents are physically expected to be between 2 and 5 and 0 and 1, respectively.In the case of percolation, the magma moves via the buoyant ascent of a less dense phase, treated as a fluid, through a denser, porous phase, treated as a matrix. In contrast to classical porous media problems where the matrix is fixed and the fluid is compressible, here the matrix is deformable, with a viscous constitutive relation, and the fluid is incompressible. Moreover, the matrix is modeled as a second, immiscible, compressible fluid to mimic the process of dilation of the pores. Flow via a conduit is modeled as a viscously deformable pipe of magma, fed from below.Analog and numerical experiments suggest that these equations behave akin to KdV and BBM; initial conditions evolve into a collection of solitary waves and dispersive radiation. As φ → 0, the equations become degenerate. A general local well-posedness theory is given for a physical class of data (roughly H 1 ) via fixed point methods. The strategy requires positive lower bounds on φ(z, t). The key to global existence is the persistence of these bounds for all time. Furthermore, we construct a Lyapunov energy functional, which is locally convex about the uniform porosity state, φ ≡ 1, and prove (global in time) nonlinear dynamic stability of the uniform state for any m and n. For data which are large perturbations of the uniform state, we prove global in time well-posedness for restricted ranges of m and n. This includes, for example, the case n = 4, m = 0, where an appropriate uniform in time lower global bound on φ can be proved using the conservation laws. We compare the dynamics to that of other problems and discuss open questions concerning a larger range of exponents, for which we conjecture global existence.

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“…In order to simplify the calculations, most of the previous work assumed a constant viscosity. The 1D-depth dependent models of Richter and McKenzie [1984], Scott and Stevenson [1984], and Barcilon and Richter [1986] show that local melt pulses develop in response to the non-linear relation between permeability and melt concentration. Numerical modeling reveals that the 1D pulses are unstable in 2D to cylindrical solitary waves (Scott and Stevenson, [1986], Barcilon and Lovera, 1989) Recent experiments of Hirth and Kohlstedt [1995] suggest that the effective viscosity of the partially molten aggregates may drop abruptly for a melt concentration fc close to 5%, when full interconnectivity at the grain scale is established.…”

Section: Introductionmentioning

confidence: 99%

2D modeling of melt percolation in the mantle: The role of a melt dependent mush viscosity

Khodakovskii

Rabinowicz

Genthon

et al. 1998

Geophysical Research Letters

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Abstract. 2D-melt percolation models in the partially molten mantle are presented. Previous work has shown that melt solitary waves develop in response to the non-linear dependence of mush permeability to melt concentration. The present study focuses on the wave evolution when the mush viscosity abruptly drops, around a critical concentration fc-2D-waves develop with a diameter, that shortens as the melt concentration increment for viscosity drop narrows. They are initiated inside 1D-maturing horizontal waves at melt concentration of about 4 X fc. Both 1D-and 2D-waves collect all the melt whose concentration exceeds fc. These features may help to explain the distribution and organization of some melt impregnations observed in mantle outcrops.

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Nonlinear waves in compacting media

Cited by 160 publications

References 7 publications

Solid-fluid interactions in porous media : processes that form rocks

Solid-fluid interactions in porous media : processes that form rocks

Degenerate dispersive equations arising in the study of magma dynamics

2D modeling of melt percolation in the mantle: The role of a melt dependent mush viscosity

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