Numerical Modelling in Geosciences

Lecture 2 Equation of mass conservation

Continuum mechanics Geological media are considered as continuous mass rather than as discrete particles. In reality the matter is made of atoms, and so it is not continuous. However, on length scales much greater than inter-atomic distances, models assuming an object as continuous are highly accurate. In continuum mechanics, the medium completely fills the space it occupies: no voids! For example, cracks and pores are considered to be filled by fluids/air. Displacement in a continuous medium implies conservation of mass described by the continuity equation.

Continuity equation

∂ρ∂t+∇⋅ ρ

v( ) = 0DρDt

+ ρ∇⋅v = 0

Eulerian formulation Lagrangian formulation

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L

Flow

L

Eulerian continuity equation An Eulerian point is immobile, not connected with any material point. Time variation of density due to outward/inward of mass flux. Volume is constant. The rate of density change is equal to minus the divergence of the mass flux. ∂ρ∂t+∇⋅

qmass( ) = ∂ρ∂t+∇⋅ ρ

v( ) = 0

Mass  flux   qmass = ρv   kg /m2 / s( )

Eulerian time derivative ⇒ ∂f∂t

∂ρ∂t

= −∇⋅ ρv( )

∂ρ∂t

=∂∂tmV=1V∂m∂t

≈1VΔmΔt

=1

ΔxΔyΔzmt+Δt −mt

Δt...

Let’s derive it…

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L

Flow

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Lagrangian continuity equation A Lagrangian point is connected with a given material point and is moving with this point. Variation of density due to variation of volume (expansion/contraction). Mass is constant. The rate of logarithmic density change is equal to minus the divergence of the velocity vector

∂ρ∂t+∇⋅ ρ

v( ) = ∂ρ∂t+v ⋅∇ρ + ρ∇⋅ v = 0

Lagrangian time derivative⇒ DfDt

=∂ f∂t

+v ⋅∇f

DρDt

+ ρ∇⋅v = 0

D lnρDt

= −∇⋅v

Vx Vx

Vy

Vy

Vx Vx

Vy

Vy

€

DlnρDt

> 0→∇⋅ v < 0

€

DlnρDt

< 0→∇⋅ v > 0

Contraction Expansion

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L

Flow

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Derive Lagrangian continuity equation

1)

2)

3)

4)

for Δt à 0, this term goes to 0

for Δt à 0, this term goes to 1

DρDt

=DDt

mV=m D

Dt1V≈ m

Δ 1VΔt

=mΔt

1Δx1Δy1Δz1

−1

Δx0Δy0Δz0

$

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'

()

Incompressibility We can derive the continuity equation for incompressible media from the Lagrangian formulation. The material point density should not change otherwise we would expect compaction/dilation (i.e., volume variation of the material point). The incompressible continuity equation is valid for both Eulerian and Lagrangian formulations

DρDt

+ ρ∇⋅v = 0

∇⋅v = − 1

ρDρDt

= −D ln ρ( )Dt

∇⋅v = 0

E

L

Flow

L

Advective term: rate of change of density for an immobile, Eulerian point due to displacement of a medium with density gradients. If material is incompressible:

EULERIAN LAGRANGIAN A Lagrangian point is connected with a given material point and is moving with this point. Variation of density due to variation of volume (expansion/contraction). Mass is constant

A Eulerian point is immobile, not connected with any material point. Variation of density due to variation of mass flux (outward/inward). Volume is constant

∂ρ∂t

= −∇⋅ ρv( ) = −ρ∇⋅

v − v ⋅∇ρ

∂x∂t

(Eulerian time derivative)

Difference between Eulerian and Lagrangian formulations

DρDt

= −ρ∇⋅v

DxDt

=∂x∂t+v ⋅∇x (Lagrangian time derivative)

DρDt

=∂ρ∂t+v ⋅∇ρ

DρDt

= 0  ⇒Advectivetransportequation

 ∂ρ∂t

= −v ⋅∇ρ

Exercise

z

Numerical Modelling in Geosciences

Practice 2 Equation of mass conservation

Exercise

Homework

Read the chapter 1 of textbook: Gerya, T. Introduction to numerical geodynamic modelling. Cambridge University Press, 345 pp. (2010) Finish Exercise 1.2