On the origin of Darcy’s law - Stanford Universitycsoulain/PRESENTATION/E221_DARCY_STO… · On...
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On the origin of Darcy’s law Stanford February 26, 2015 Contact : [email protected] http://web.stanford.edu/~csoulain/ Energy 221 – Fundamentals of Multiphase flow
Transcript of On the origin of Darcy’s law - Stanford Universitycsoulain/PRESENTATION/E221_DARCY_STO… · On...
On the origin of Darcy’s law
Stanford
February 26, 2015Contact: [email protected]
http://web.stanford.edu/~csoulain/
Energy 221 – Fundamentals of Multiphase flow
E221 – Multiphase flow in porous media – [email protected]
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Darcy and its sand column experiment
Empirically proposed by Henri Darcy in 1856*
Porous media = porosity, permeability, Darcy’s law
What are the links with Darcy’s law and the theory of fluid mechanics?
Permeability, constant intrinsic to each porous material
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Outline
From Stokes to Darcy
Fluid flow in the pore space
Exercises: analytical solutions of Stokes(demonstration of Hagen-Poiseuille law)
Two different representations of physics of fluid flow in porous media
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Zoom in a porous medium
2D cross section of a sandstone
We see clearly a sharp delineation at the solid/fluid interface. A lot of physico-
chemical phenomena, such as surface reaction may occur at this interface.
Highly complex pore network topology.
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Flow at the pore scale
For a given location, the instantaneous velocity is always the same.
Water seeded with micro-particles to enhance the flow visualization in the pore
space (Sophie Roman, SUPRI-A)
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Flow in fully saturated micromodel
Pressure, Velocity,
The local heterogeneities of the connected pore structure strongly influence
the flow pathways.
For a given pressure difference and a given geometry, the velocity profile in
the pore space will be always the same
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The Stokes equations
The velocity and pressure fields within
the void space of the porous medium
can be obtained by solving the equations
of fluid mechanics (for slow flow),
namely the Stokes equations.
For an incompressible single-phase
fluid, they read
Mass balance equation
Momentum balance equation
Non-slip condition at the solid boundary
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Stokes equations are not “pore scale” only!!!
Driven cavity flow
fixedWalls
fixedW
alls
fixe
dW
alls
MovingWall : Ux=1m/s
20 μm*
500 m
Same phenomenon modeled with the same
equations... even though there are several
orders of magnitude between them!!
* Roman, Soulaine, Tchelepi, Kovscek. Measurement of the pore-scale velocity distributions in a two-dimensional
porous medium (submitted)
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Two representations of the physics of fluid
for every point of the domain
fluid OR solid fluid AND solid
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The direct modeling approach
When working with the direct modeling you have
an explicit representation of the solid and of the
solid/fluid interface, which means that you need to specify boundary conditions at the solid walls.
and obey the Stokes equations,
Often referred to as,
- Pore scale
- Direct numerical simulation (DNS)
- microscale
This representation of the physics of flow in
porous media is restricted to small domains
The pore network geometry and the fluid
properties are the only input (no assumption)
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The continuum modeling approachWith such a representation of the physics
of flow in porous media, we deal with
averaged quantities (like and )
and averaged equations,
Information related to the pore network
topology is included in effective
parameters such as porosity and
permeability.
Often referred to as,
- Darcy scale
- Macro-scale
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Notion of averageLet’s define a function that has value in the pore space only. It can be a scalar
(like the pressure field ) or a vector (like the velocity field )
We now introduce the average over the control volume, V, known as superficial
average and defined as
and the intrinsic phase average as
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Definition of porosityNow that the average operators are defined, and assuming that the REV exists, we
can give a mathematical definition of porosity.
Let’s define the mask function, , in the entire domain V as,
The volume integral of this function gives the volume occupied by the void,
The superficial average of the phase indicator function gives the porosity
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nFrom Stokes to Darcy (1/2)
can be seen as an average operator that can be applied to the Stokes equationsin order to derive the governing equations for the average quantities,
This homogeneization process (passage from direct to continuum modeling approach)
is not straightforward. In particular, the boundary conditions at the pore scale need to
be included in the averaged equations at the Darcy scale.
This is achieved when computing the average of a gradient. Indeed, the result is the
gradient of the average PLUS the contribution of the value at the solid boundaries,
Transformation of the interfacial effects into a continuum representation
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From Stokes to Darcy (2/2)
Actually, when following the theoretical homogeneisation from Stokes momentum
equation*, the resulting equation is not Darcy, but a more comprehensive conservation
law known as Darcy-Brinkman equation**,
** Brinkman, HC. A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. 1947, A1, 27-34
Drag force due to the friction of the fluid with the solid surfaces
In practice, it turns out that the dissipative viscous term is neglegible compared with
other terms
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Darcy is not natural porous media only
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Ex: Viscous flow between parallel plates
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Ex: Viscous flow through a capillary tube
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Thank you for your attention.
Question?
