Fundamental Study on Fluid Flow and Mass Transport through Unsaturated Geologic Formation

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Fundamental Study on Fluid Flow and Mass
Transport through Unsaturated Geologic Formation

• Yuichi Niibori
• Dept. of Quantum Science Energy Engineering
• Tohoku University

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Contents

• Background
• Objective
• New Relative-Permeability-curves based on PDFs
• Hydraulic dispersion coefficient in unsaturated
  zone
• Summary

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Background

• To obtain more reliable estimate of fluid flow
  and mass transport in unsaturated condition
• A key issue in the safety assessment of the
  geologic disposal system.

Various models to understand transport
phenomena in unsaturated zone.
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Network model can focus on the fluid
flow-paths. no logic to determine
the network structures of fractures/cracks.
the degree of mixing at the connection
points between segments.
Continuous model (using Darcys law) can
image heterogeneous rock-body (matrix)
including the fractures/cracks. misleads us
about the travel time of nuclide through
unsaturated formation.
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What is the problem in the conventional,
continuous model?
The continuous model has squeezed the uncertainty
of unsaturated zone into the relative-permeability
-curves hydraulic dispersion coefficients.
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Objective

• To propose the two things
• New relative-permeability-curves to know
• the applicability of the continuous model
• Hydrodynamic dispersion-coefficient based on the
  fundamental experiments for unsaturated zone.

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New relative-permeability-curves
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Before New relative-permeability-curves
Darcys law of Two-Phase flow through porous
media The fluid flow velocity (m/s), vi, is
k absolute permeability(m2) kr the
correction value (-) of k
so-called, relative-permeability (0ltkrlt1)
the viscosity (Pa s) the pressure
gradient (Pa/m).
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Sw volume of water phase in unit pore volume
(so-called, water saturation). 0lt SW lt1, Sg1-SW
Coreys Eqs.
Here, Sw is normalized by Swr (the irreducible
water saturation) and Sgr (the water saturation
at irreducible gas saturation).
Sw(Sw-Swr)/(Sgr-Swr)
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Data from in-room experiments, Using core
sample or packed bed, whose size is less than 1
m. Controlling Sw to obtain these functions.
Practical analysis of Geologic formation
In the grid cell of the numerical analysis, the
saturation is not always distributed
uniformly. One grid cell must cover the space
larger than 10 meter.
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e.g., if the Sw-distribution in a grid-cell is
separated into Sw1 and Sw0, how do you decide
the value of kr ?
Swa(10)/20.5 krwa 0.54 0.0625
krw1 (at Sw1) , krw0 (at Sw0) ,
krwa (10)/20.5
kr-value depends on the distribution of Sw. In
this talk, such relative permeabilities with the
distribution (krwa and krga) are called
Apparent Relative-permeability.
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Definition of apparent relative permeability
Probability density function of Sw
Image of the saturation
Image of Sw-distribution
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However, no logic to determine a specific PDF.
Assuming various types of PDFs, e.g., the
Beta distribution, the triangle distribution, the
normal distribution, the Bernoulli trials,
the relation between kra and Swa was
examined. The results were (1) kra gt kr
described by Coreys eqs. (2) kra does not
depend on the type of PDF. The value
of kra depends on the three moment, i.e.,
the arithmetic mean of the saturation, Swa,
the standard deviation, , and the
skewness, .
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If the PDFs have the common values in Swa,
and SK, the values of kra almost agree.
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(1) Detail of kra gt kr (described by Coreys
eqs.)
Upper bounds of kra in case taking single peak or
two peak in PDF
the case of single peak.
the case of two peak.
If PDF has Single peak
Two peaks
Through the comparison of the data with these
upper bounds, in the experiment there was the
distribution of Sw taking single peak in PDF.
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(2) kra can be described by the three moments.
Bernoulli trials (discrete PDF)
These relations are not directly applicable to
the numerical analysis.
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Further, using index m and Swa,
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(m4 Coreys Eqs.) (m1 X-curve)
This model with m and Swa can express all values
of kra.
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Applicability
Tracer ammonia-water solution
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Analysis
NH3 gas
NH4OH solution
The mass and heat balance equations for the
two-phase flows were numerically solved, using
the proposed apparent relative permeability
curves.
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Parameters
The model parameters m and De mobtained
from the total flow rate QT. De obtained
through the other experiments
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Comparison
The tracer response strongly depends on the index
m.
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The Applicability of Continuous model with
Darcys law
If the Sw-distribution takes single peak, the
value of kra is limited. Its upper bound is
approximately described by the curve of m2.5.
Two-phase flow interrupting each other in the
same area.
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If the Sw-distribution takes two peaks, the kra
is close to the m1. The two-peaks (1ltmlt2.5)
suggest the existence of flow-paths isolated.
Such flow-paths can take their own tortuosity,
permeability or porosity. To obtain the
reliable e.g., the travel time, we have to treat
the continuous model carefully, and need to
detect the dominant flow-paths.
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Relative permeability-curves based on PDFs.
for the liquid phase for the gas phase
The index m represents the non-uniformity of
saturation. m4 ? homogeneous case(Coreys
equations), m1 ? Completely isolated case
of liquid and
gas flows 2.5ltmlt4, in case of Sw-distribution
taking single peak.
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Hydrodynamic dispersion-coefficient

• based on the fundamental experiments for
  unsaturated zone.

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Schematic diagram of experimental apparatus
Acrylic cylinder Length 500 mm InsideF
41mm Glass beads with F1 mm k 6.510-10 m2 ,e
0.37
/ Deionized water / Nitrogen gas saturated with
water vapor (The hydraulic head at inlet was
fixed.) / Tracer experiments 5.0 mL of KCl
solution (0.1 M) or 8.0 mL of an
ammonia-water solution (about 28 wt
NH4OH)
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Measurement of Sw and Tracer Injection
Chen et al. (1996) showed that Sw became
approximately uniform in the homogeneous bed as
the continuous flow of water and gas became
steady. The time required to obtain the steady
state mainly depends on permeability of
column. The packed bed was disconnected once
in order to determine the Sw by weighting the
column. After measuring the Sw and continuing
the injection of the water and gas for 1 hour,
either tracer was injected with a syringe within
a few seconds.
(Potassium chloride or
ammonia-water solution)
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Relationship between the gas flow rate and water
saturation in a steady state.
The Sw in the bed was held within the range of
0.41-0.85 by controlling the flow rate of the
gas phase.
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Schematic representations of the three different
flow conditions through the bed.
(a) Saturated flow
(b) Unsaturated one-phase flow
(c) Unsaturated two-phase flow
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(Saturated flow)
(Unsaturated flow)
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Pe0.8Sw1.2
(Saturated)
(Unsaturated one-phase flow)
Pe0.9Sw3.1
(Unsaturated two-phase flow)
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NH3 Gas tracer response
A-D eqs. for w- g- phases Transfer of
tracer from w- to g-phase The proposed
empirical eqs.
The tracer remaining in the water phase
Using the relations of Pe with Sw, the numerical
model well expresses the experimental responses
for the water phase and the gas phase with no
fitting parameter.
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Data of beads size 0.2 cm Same tendency as 0.1
cm Another flow pattern
Bubble flow
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Mechanisms of generating bubble flow
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Summary

• New relative-permeability-curves
• based on PDFs of Sw.
• Hydrodynamic dispersion-coefficient based on the
  fundamental experiments for unsaturated zone.

m represents the non-uniformity of saturation.
2.5ltmlt4, in case of Sw-distribution taking single
peak.
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Definition of apparent relative permeability
Probability density function of Sw
Pipe model
seems
e.x., 2-D area
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However, two-dimensional effect should be
considered.
Sw11, krw,11
Sw1m, krw,1m
Swn1, krw,n1
Swnm, krw,nm
Swij, krw,ij
kT0.5 (ka)
kT0 (?ka)
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