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Two-phase flow in heterogeneous porous media III: Laboratory experiments for flow parallel to a stratified system

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Abstract

Two-phase flow in stratified porous media is a problem of central importance in the study of oil recovery processes. In general, these flows are parallel to the stratifications, and it is this type of flow that we have investigated experimentally and theoretically in this study. The experiments were performed with a two-layer model of a stratified porous medium. The individual strata were composed of Aerolith-10, an artificial: sintered porous medium, and Berea sandstone, a natural porous medium reputed to be relatively homogeneous. Waterflooding experiments were performed in which the saturation field was measured by gamma-ray absorption. Data were obtained at 150 points distributed evenly over a flow domain of 0.1 × 0.6 m. The slabs of Aerolith-10 and Berea sandstone were of equal thickness, i.e. 5 centimeters thick. An intensive experimental study was carried out in order to accurately characterize the individual strata; however, this effort was hampered by both local heterogeneities and large-scale heterogeneities.

The theoretical analysis of the waterflooding experiments was based on the method of large-scale averaging and the large-scale closure problem. The latter provides a precise method of discussing the crossflow phenomena, and it illustrates exactly how the crossflow influences the theoretical prediction of the large-scale permeability tensor. The theoretical analysis was restricted to the quasi-static theory of Quintard and Whitaker (1988), however, the dynamic effects described in Part I (Quintard and Whitaker 1990a) are discussed in terms of their influence on the crossflow.

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Abbreviations

A ωη :

interfacial area between the ω-region and the η-region contained within V∞, m2

a β :

vector that maps \((\partial \{ \varepsilon _\beta \} */\partial t)\) onto \(\hat v_\beta\), m

b β :

vector that maps \((\nabla \{ \left\langle {p_\beta } \right\rangle ^\beta \} ^\beta - p_\beta g)\) onto \(\hat p_\beta\), m

b βω :

vector that maps \((\nabla \{ \left\langle {p_\beta } \right\rangle ^\beta \} ^\beta - p_\beta g)\) onto \(\hat p_{\beta \omega }\), m

B β :

second order tensor that maps \((\nabla \{ \left\langle {p_\beta } \right\rangle ^\beta \} ^\beta - p_\beta g)\) onto \(\hat v_\beta\), m2

C β :

second order tensor that maps \((\partial \{ \varepsilon _\beta \} */\partial t)\) onto \(\hat v_\beta\), m2

E :

energy of the gamma emitter, keV

f β :

fractional flow of the β-phase

g:

gravitational vector, m/s2

h :

characteristic length of the large-scale averaging volume, m

H :

height of the stratified porous medium \(l_\omega + l_\eta\), m

i :

unit base vector in the x-direction

K :

local volume-averaged single-phase permeability, m2

\(\hat K\) :

K - {K}, large-scale spatial deviation permeability

{ K}:

large-scale volume-averaged single-phase permeability, m2

K * :

large-scale single-phase permeability, m2

K ** :

equivalent large-scale single-phase permeability, m2

K βω :

local volume-averaged β-phase permeability in the ω-region, m2

K βη :

local volume-averaged β-phase permeability in the η-region, m2

\(\hat K_\beta\) :

K β - {K β }β, large-scale spatial deviation for the β-phase permeability, m2

K * β :

large-scale permeability for the β-phase, m2

l :

thickness of the porous medium, m

l ω :

characteristic length for the ω-region, m

l η :

characteristic length for the η-region, m

L :

length of the experimental porous medium, m

ℒ:

characteristic length for large-scale averaged quantities, m

n ω :

outward unit normal vector for the ω-region

n η :

outward unit normal vector for the η-region

n ωη :

unit normal vector pointing from the ω-region toward the η-region (n ωη = - n ηω )

N :

number of photons

p β :

pressure in the β-phase, N/m2

p γ0 :

reference pressure in the γ-phase, N/m2

\(\langle p\beta \rangle ^\beta\) :

local volume-averaged intrinsic phase average pressure in the β-phase, N/m2

\(\{ \langle p\beta \rangle ^\beta \}\) :

large-scale volume-averaged pressure of the β-phase, N/m2

\(\{ \langle p\beta \rangle ^\beta \} ^\beta\) :

large-scale intrinsic phase average pressure in the capillary region of the β-phase, N/m2

\(\hat p_\beta\) :

\(\langle p\beta \rangle ^\beta\)- \(\{ \langle p\beta \rangle ^\beta \}\), large-scale spatial deviation for the β-phase pressure, N/m2

pc :

\(\langle p_\gamma \rangle ^\gamma - \langle p_\beta \rangle ^\beta\), capillary pressure, N/m2

p cω :

capillary pressure in the ω-region, N/m2

p η :

capillary pressure in the η-region, N/m2

{p c }c :

large-scale capillary pressure, N/m2

q β :

β-phase velocity at the entrance of the porous medium, m/s

q γ :

γ-phase velocity at the entrance of the porous medium, m/s

Swi :

irreducible water saturation

S β :

∈ β /∈, local volume-averaged saturation for the β-phase

S βi :

initial saturation for the β-phase

S γr :

residual saturation for the γ-phase

S * β :

{∈ β }*/∈}*, large-scale average saturation for the β-phase

S βω :

saturation for the β-phase in the ω-region

S βη :

saturation for the β-phase in the η-region

t :

time, s

v β :

β-phase velocity vector, m/s

〈v β 〉:

local volume-averaged phase average velocity for the β-phase, m/s

{〈v β 〉}:

large-scale averaged velocity for the β-phase, m/s

〈v β 〉 ω :

local volume-averaged phase average velocity for the β-phase in the ω-region, m/s

〈v β 〉 ω :

local volume-averaged phase average velocity for the β-phase in the η-region, m/s

\(\hat v_\beta\) :

〈v β 〉-{〈v β 〉}β, large-scale spatial deviation for the β-phase velocity, m/s

\(\hat v_{\beta \omega }\) :

〈v β 〉 ω -{〈v β 〉}β, large-scale spatial deviation for the β-phase velocity in the β-region, m/s

\(\hat v_{\beta \eta }\) :

〈v β 〉 η -{〈v β 〉}β, large-scale spatial deviation for the β-phase velocity in the η-region, m/s

V ∞ :

large-scale averaging volume, m3

y :

position vector relative to the centroid of the large-scale averaging volume, m

{y}c :

large-scale average of y over the capillary region, m

∞:

local porosity

∞ ω :

local porosity in the ω-region

∞ η :

local porosity in the η-region

∞ β :

local volume fraction for the β-phase

∞ βω :

local volume fraction for the β-phase in the ω-region

∞ βη :

local volume fraction for the β-phase in the η-region

{∈}* :

{∈ β }*+{∈ γ }*, large-scale spatial average volume fraction

{∈ β }* :

large-scale spatial average volume fraction for the β-phase

ρ β :

mass density of the β-phase, kg/m3

ρ γ :

mass density of the γ-phase, kg/m3

μ β :

viscosity of the β-phase, N s/m2

μ γ :

viscosity of the γ-phase, Ns/m2

φ β :

V ω /V ∞ , volume fraction of the ω-region (ω β +ω η =1)

φ η :

V η /V ∞ , volume fraction of the η-region (ω β +ω η =1)

ζ:

attenuation coefficient to gamma-rays, m-1

Ω β :

\(\nabla \{ \langle p_\beta \rangle ^\beta \} ^\beta - p_\beta g), N/m^3\)

Ω γ :

\(\nabla \{ \langle p_\gamma \rangle ^\gamma \} ^\gamma - p_\gamma g), N/m^3\)

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Authors and Affiliations

  1. Laboratoire Energétique et Phénomènes de Transfert-ENSAM Unité associée au CNRS 873, Esplanade des Arts et Métiers, 33405, Talence Cédex, France

    Henri Bertin & Michel Quintard

  2. Elf Aquitaine, 64000, Pau, France

    Vincent Corpel

  3. Department of Chemical Engineering, University of California, 95616, Davis, CA, U.S.A.

    Stephen Whitaker

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  1. Henri Bertin
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  2. Michel Quintard
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  3. Vincent Corpel
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  4. Stephen Whitaker
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Bertin, H., Quintard, M., Corpel, V. et al. Two-phase flow in heterogeneous porous media III: Laboratory experiments for flow parallel to a stratified system. Transp Porous Med 5, 543–590 (1990). https://doi.org/10.1007/BF00203329

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