AHMADI-SENICHAULD, Azita

Colombano, Stéfan

Davarzani, Hossein

Omirbekov, Sagyn

English

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Derivation of the Darcy-scale filtration equation for
foam flow using the volume averaging technique
Sagyn Omirbekov, Hossein Davarzani, Azita Ahmadi-Senichauld, Stéfan
Colombano
To cite this version:
Sagyn Omirbekov, Hossein Davarzani, Azita Ahmadi-Senichauld, Stéfan Colombano. Derivation of the
Darcy-scale filtration equation for foam flow using the volume averaging technique. 14èmes Journées
d’Etude des Milieux Poreux - JEMP 2018, Oct 2018, Nantes, France. 2018. ￿hal-01830155￿
______________________________________________________________________________________________________ 
FICPM - Nantes, France – October 8-10, 2018. 
 
Derivation of the Darcy-scale filtration equation for foam flow using 
the volume averaging technique 
 
S. Omirbekova,b, H. Davarzania, A. Ahmadi-Senichaultb, S. Colombanoa 
aBRGM, 3 Avenue Claude Guillemin, Orleans, 45100 
bInstitut de Mécanique et Ingénierie (I2M, TREFLE), Arts et Métiers ParisTech-Talence , 33405 
 
Keywords: foam flow, porous media, upscaling, volume averaging technique, non-Newtonian fluid, Herschel-
Bulkley model 
1 Introduction  
Physically, foam is represented as a two-phase system, where gas bubbles are dispersed in a liquid phase and 
are separated by thin borders, which are called lamellae. When the bubble-size is much smaller than the 
characteristic pore size, foam in porous media can be considered as a single non-Newtonian fluid phase, as is 
the case for bulk foam. The rheology of foam is complex and depends on several characteristics as bubble 
size, quality, stability, etc. To avoid all these complexities and simulate foam transport in porous media, the 
“population balance” approaches were suggested by several authors [3]. Generally, there are two main 
approaches of foam modelling: Local-Equilibrium models (empirical, semi-empirical) and Bubble Population-
Balance models (mechanistic). In addition, there are some other modelling approaches, like percolation theory, 
fractional flow theory, catastrophic theory, and filtration theory. Both Local-equilibrium and Bubble 
Population-Balance models are designed by a modification of Darcy’s law parameters such as gas relative 
permeability, as well as gas viscosity. Application of these approaches needs some general assumptions. 
Mathematically, the relative permeability and the viscosity of foam in porous media are grouped together and 
mentioned as gas phase mobility in Darcy law, but physically they are separable. 
The main objective of this study is to derive a 3D Darcy-scale momentum equation for liquid foam (surfactant, 
water and air) flow using the volume averaging method, which allows modelling foam injection process at the 
local or column scale experiments. In particular, the impact of the scale on pressure gradient threshold above 
which the foam flows will be determined. 
2 Upscaling of foam flow in porous media   
The upscaling process is a very important procedure in which the static and dynamic characteristics of a pore-
scale model should be taken into account in the local-scale model. Certainly, the direct modelling of fluid 
flow at the pore scale in a porous medium sample is practically impossible due to the complexity of the pore 
structure and the huge demand for computing resources. Therefore, it is necessary to derive a local scale 
model. Upscaling and modelling the flow of non-Newtonian fluids through porous media are involved in 
many problems substantial to science and industry. Appreciable examples are in biological systems, oil 
recovery operations, food processing, cosmetics, textile, paper industries and also in soil remediation 
technologies. For Newtonian fluids, the flow through porous media is modelled by Darcy’s law, which has 
been theoretically derived from the Stokes equation via homogenization [1] and the volume averaging 
methods [4]. However, the filtration laws for non-Newtonian fluids are more complicated due to their 
complex rheology and the complex microstructure of porous media. In our study, we investigate the pore to 
Darcy scale upscaling of the flow of foam as a non-Newtonian, one phase fluid, using the volume averaging 
method. The apparent viscosity of bulk foam was first measured with a rotational rheometer (Haake Mars 60, 
ThermoFisher Scientific Company). As shown in Figure 1a, the apparent viscosity of foam fits well the 
Herschel-Bulkley model for 𝜏 ≥ 𝜏0, 
 
 𝜏 = 𝜏0 + 𝑘?̇?
𝑛 (1) 
where 𝜏 [Pa] is the shear stress, 𝜏0 [Pa] is the yield stress, ?̇? [s-1] is the shear rate, 𝑘 [Pa sn] is the consistency 
index, and n is the flow index [2]. 
______________________________________________________________________________________________________ 
FICPM - Nantes, France – October 8-10, 2018. 
 
 
If 𝜏0 is zero, the model alters to the power law model, which is first investigated here. Application of the 
volume averaging method to the pore scale equations, without yield stress, leads to the following macroscopic 
equation: 
 
 
〈𝐯𝛽〉 = −
𝐊𝑎𝑝𝑝 ∙ (𝛁〈𝑝𝛽〉
𝛽 − 𝜌𝛽𝐠)
(𝑘‖〈𝐯𝛽〉‖
𝑛−1
)
 (2) 
 
Equation 2 is a generalized Darcy’s law for the 𝛽-phase considered as a power law fluid,  where 〈𝐯𝛽〉 is the 
superficial average velocity, ‖〈𝐯𝛽〉‖ is the magnitude of superficial average velocity, 〈𝑝𝛽〉
𝛽  is the intrinsic 
average pressure, 𝜌𝛽 is the density, g is the gravity acceleration, and 𝐊𝑎𝑝𝑝 is the apparent permeability which 
depends on the geometry and the direction of the flow velocity. The apparent permeability in longitudinal 
direction was obtained by solving the associated closure problem in a simple isotropic cylindrical geometry. 
The variation of the apparent permeability versus porosity for a Newtonian and a non-Newtonian fluids are 
presented in Figure 1b. We found that the apparent permeability of non-Newtonian fluid is also increased 
with increasing the porosity, as a Newtonian fluid. The values of apparent permeability for non-Newtonian 
fluid are about two orders of magnitude smaller than values for Newtonian fluid.  
 
(a) (b) 
  
 
Figure 1: (a) Non-Newtonian foam behaviour (10*CMC solution of Triton X-100, Foam quality 85%, T=20 °C) and 
(b) Apparent permeability of Newtonian and non-Newtonian fluids. 
3 Perspectives  
Currently, we have almost developed a macroscopic model for power-law, non-Newtonian fluid flow in 
porous media. Besides, we are working in order to take into account the yield stress term in the up-scaled 
equations. In addition, validation of the macroscopic model by comparison with pore-scale, direct numerical 
simulation and also the experiments will be presented during the conference. 
 
References 
[1] J.L. Auriault. Heterogeneous medium. Is an equivalent macroscopic description possible? International Journal of Engineering Science, 
29(7):785 – 795, 1991. 
[2] R. Hohler and S. Cohen-Addad. Rheology of liquid foam. Journal of Physics: Condensed Matter, 17(41), 2005. 
[3] K. Ma, G. Ren, K. Mateen, D. Morel, and P. Cordelier. Modeling techniques for foam flow in porous media. Society of Petroleum 
Engineers, 20, June 2015. 
[4] S. Whitaker. Flow in porous media I: A theoretical derivation of Darcy’s law. Transport in Porous Media, 1(1):3–25, March 1986. 
0
2
4
6
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14
16
18
20
0.01 0.1 1 10 100
τ
[P
a]
𝛾 ̇ [s-1] 
Experiment results
Herschel-Bulkley model  
τ_0=5.1  k=0.75  n=0.62
1.E-11
1.E-10
1.E-09
1.E-08
1.E-07
0 0.2 0.4 0.6 0.8 1
A
p
p
ar
en
t 
P
e
rm
ea
b
ili
ty
 [
m
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]
Porosity [-]
K1 Newtonian fluid
K2 Non-newtonian
fluid; n=0.62


Derivation of the Darcy-scale filtration equation for foam flow using the volume averaging technique

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Derivation of the Darcy-scale filtration equation for foam flow using the volume averaging technique

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