SHORT COURSE ON
MODELING
FLOW AND SOLUTE TRANSPORT
IN THE SUBSURFACE
by
JACOB BEAR
Professor Emeritus, Technion—Israel Institute of Technology,
Haifa, Israel
Lectures presented at the Instituto de Geologia, UNAM,
Mexico City, Mexico, December 6--8, 2003
Copyright © 2002 by Jacob Bear, Haifa Israel. All Rights Reserved.
To use, copy, modify, and distribute these documents for any purpose
is prohibited, except by written permission from Jacob Bear.
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OBJECTIVES OF LECTURES
To discuss the role of models in the decision making process for
ground water management, including of the subsurface, and to
subsurface to describe the modeling process.
To describe the mechanisms that govern the movement and storage of
fluids in the subsurface.
To describe the mechanisms that govern the movement, accumulation,
and transformations of contaminants in the subsurface.
To discuss models of sea water intrusion into coastal aquifers.
To construct conceptual and well-posed mathematical models of
flow and contaminant transport of in the saturated and unsaturated zones,
taking into account modeling objectives and the relevant processes.
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PROGRAM OF LECTURES:
Models. The modeling process. Conceptual and mathematical modeling.
The continuum approach to the modeling of transport phenomena in
porous media. Aquifers. Essentially horizontal flow in aquifers.
The motion equation (Darcy's law) for saturated flow, and its
extensions.. Aquifer transmissivity. Dupuit assumption for flow in a
phreatic aquifer.
Specific storativity. Balance equation for 3-d saturated flow.
boundary conditions. Complete model of 3-d saturated flow.
Aquifer storativity. Balance equation for 2-d horizontal flow
in confined, phreatic, and leaky aquifers. Boundary conditions.
Complete models of 2-d flow in aquifers.
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The unsaturated zone. Two and three fluid phases. Capillary
pressure. Retention curves for two and three fluid phases. Motion
equations.
Balance equations for multiphase flow in the vadose zone. Boundary
conditions. Complete, well-posed flow models for multiphase flow.
Modeling subsurface contamination. Sources of subsurface
contamination. Movement of a contaminant in single and multiphase
flow systems. Advective, diffusive, and dispersive fluxes.
Adsorption. Volatilization. Chemical reactions. Chemical equilibrium.
Balance equations. Boundary conditions. Complete well-posed models
of flow and contaminant transport.
Modeling sea water intrusion into coastal aquifers.
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THE SUBSURFACE?
The saturated zone---aquifers, groundwater.
The unsaturated zone (= vadose zone).
Why the interest in THE UNSATURATED ZONE?
Hydrologist or geohydrologist, interested in aquifers, have regarded
the unsaturated zone only as the passage for water from ground surfac
to an aquifer, with a year or a season as the time unit of interest.
The current interest in the unsaturated zone is due to the rising interest
in ground-water contamination..
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Hydrologist or geohydrologist, interested in aquifers, have
regarded the unsaturated zone only as the passage for water
from ground surface to an aquifer, with a year or a season as the time
unit of interest.
The current interest in the unsaturated zone is due to the rising interest
in ground-water contamination by pollutants originating at ground
surface. The vadose zone is acts also as chemical-biological reactor.
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WHAT IS A MODEL?
A model may defined as a selected simplified version of a real
system and phenomena that take place within it, which approximately
simulates the system's excitation - response relations that are of interest
WHY DO WE NEED MODELS?
Our approach here is the need to solve problems of practical interest.
For example, the need to remove pollutants from the unsaturated zone
in order to prevent or reduce the contamination of groundwater in an
underlying aquifer.
Good management requires a tool for predicting the consequences
of implementing proposed decisions. This tool is the MODEL of the
investigated system.
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Problem
Objectives
Proposed Alternative solutions
Model Predictions of System’s
Response to Alternatives
Evaluate Objective Function
Select Preferred Alternative
Implement Alternative
Monitoring
Role of Modeling in the
Decision Making process
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Thus, the basic role of a model is to predict the future
behavior of a system. However, this information can be used to:
To predict the system's future behavior in response to excitations,
i.e., the implementation of decisions.
To provide information required in order to comply with
regulations.
To obtain a better understanding of the system from the geological,
hydrological, and chemical points of view.
To provide information for the design of observation networks, by
anticipating the system's future behavior.
To provide information for the design of field experiments.
The mathematical model is used only as a compact way of describing
the physical/chemical/biological phenomena that are relevant to the
considered problem.
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MODELING PROCESS
STEP 1: Development of a conceptual model
The real system and its behavior may be very complicated.
Simplifications are introduced in the form of a
SET OF ASSUMPTIONS that expresses our understanding of
the nature of the system and its behavior.
Because the model is a simplified version of the real system, no
unique model exists to describe it.
SET OF ASSUMPTIONS (In words)
= CONCEPTUAL MODEL
LET'S ASSUME THAT..
•Dimensionality of the model (1-, 2-, or 3-dimensions).
• Steady or unsteady behavior.
• Kinds of soil and rock materials.
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CONCEPTUAL MODEL (cont.)
More (possible) assumptions:
•Homogeneity (or inhomogeneity), isotropy (or anisotropy),
and deformability of these materials.
•Number and kinds of fluid phases.
•Relevant material properties of the fluid phases (density, viscosity..)
• Relevant chemical components.
•Relevant transport mechanisms.
•Possibility of phase and exchange between adjacent phases.
•Flow regimes of the fluids (e.g., laminar or not).
•Existence, of isothermal or nonisothermal conditions.
•Presence or absence of assumed sharp macroscopic fluid-fluid
boundaries, such as a phreatic surface.
•Sources and sinks of fluids and pollutants within the domain.
•Relevant, chemical, physical,, and biological processes that take
place in the domain.
•Initial conditions and Conditions on the domain's boundaries.
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STEP 2: Development of a mathematical model
The conceptual model is translated into a mathematical model. A
continuum model is usually employed.
. MATHEMATICAL MODEL:
THE
•Definition of boundary surfaces.
•Equations that express balances of the considered extensive
quantities (mass, energy of phases and components).
•Flux equations that relate the fluxes of the considered extensive
quantities to the relevant state variables.
•Constitutive equations, that define the behavior of phases and
components.
•Sources and sinks of the extensive quantities.
•Initial conditions that describe the known state of the considered
system at some initial time.
•Boundary conditions that describe the interaction of the system with
its environment.
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In CONTINUUM MODELS:
•Balances are stated for “a point within the domain”.
•State variables are stated at “every point”.
EVERY point?
In NUMERICAL MODELS:
•State variables are defined at nodes or as averages over small cells.
Coefficients appear in models. Where do they come from?
Coefficients are born in the passage from the microscopic scale
to the macroscopic model by the process of averaging. Examples:
permeability, moisture diffusivity, and dispersivity.
STEP 3: Development of a numerical model and code
The preferred method of solution is the analytical one. However,
in most practical cases, this approach is not feasible.
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STEP 4: Code verification.
A new code must be verified.
VERIFICATION means that the code does what it proclaims to
do---to solve the mathematical model.
Verification involves comparing solutions obtained by using the code
with analytical ones---if available (usually for simplified domain
geometries, homogeneous, etc.)
STEP 5: Model validation.
A model must be validated for a particular problem (and site), i.e.,
making sure that it indeed describes a considered process.
Validation---only by an experiment, preferably for the actual site.
If model validation cannot be implemented, it is often
combined with model calibration.
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STEP 6: Model calibration and parameter estimation.
The identification problem, the inverse problem, parameter estimation
problem---the activity of determining the coefficients.
Model calibration---the activity that combines model validation at
a specific site and parameter estimation.
Use of historical data --- a period with
(a) initial conditions, (b) excitations, (c) observations of the response.
Use this information to compare observed and modeled responses.
The sought values of the coefficients are those that would make
the two sets of values closest ( `best fit‘).
The inverse problem is usually not a well-posed one, that results in a
unique solution.
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STEP 7: Model applications.
Computer runs are conducted to provide the required
information
Step 8: Analysis of uncertainty and stochastic modeling.
SOURCES OF MODEL UNCERTAINTY:
Is the selected conceptual model appropriate for the problem?
Are values of the various coefficients correct?
Errors may result from errors in observed data.
Insufficient data about heterogeneity of the domain.
Are selected boundaries appropriate?
Are conditions assumed to prevail on domain boundaries correct?
Consequence--STOCHASTIC MODELS: information appears as
probability distributions of values, rather than as deterministic ones.
Special/main interest---HETEROGENEITY OF DOMAIN.
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Coping with heterogeneity by
MONTE CARLO SIMULATIONS.
Spatial structure in heterogeneous soils (mean, variance about
mean, etc.).
Create various realizations of soil properties and features.
It is hoped that if the statistical representation is a good
approximation, then Monte Carlo analyses can determine average
behavior that would occur over many independent realizations.
Effective properties (from many realizations) can be used to
capture the mean behavior in practical modeling efforts, while the
variance estimates may used to calibrate or define the expected
scatter in predictions. in practical modeling efforts.
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STEP 9: Summary, conclusions, and reporting.
MODEL USE
• Most natural porous domains of interest are highly heterogeneous.
• Insufficient data for model calibration.
• Uncertainty about model boundaries and boundary conditions.
• Insufficient knowledge for modeling the complex case of multiple
multi-component phases, possibly under non-isothermal conditions.
• Model validation not always available.
WHAT, then, IS THE USE OF THE MODEL?
Decisions will be made anyway. Hence,….
WE HAVE NO BETTER ALTERNATIVE.
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However, model use should be extended:
•
•
•
•
•
Predicting system's behavior.
Enhancing understanding and organizing it.
Gaining insight into the roles of various processes.
Performing sensitivity analysis to indicate significant features
Guiding the acquisition of field data. Design observation networks
and design of experiments.
• Designing of early-warning networks.
Models can aid in making informed decisions, even in the absence of
model validation and parameter identification in the strict sense of
these terms.
NEXT: THE CONTINUUM APPROACH
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The most common conceptual model
CONTINUUM APPROACH to PHENOMENA of
TRANSPORT in POROUS MEDIA
What is a transport problem? Transport of what?
What is a CONTINUUM?
Why do we need a continuum approach to describe phenomena of
transport in porous media?
What is A POROUS MEDIUM? (plural: porous media !)
TRANSPORT mean the movement, accumulation, and
transformation of extensive quantities of PHASES and COMPONENTS
A domain is said to behave as a continuum, if every state variable is
defined for EVERY point within it.
A phase is regarded as continuum that fills up the entire spatial
domain occupied by it. Similarly, a component is regarded as a
continuum that fills up the entire domain occupied by the phase.
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In a phase continuum, state variables and phase coefficients
are obtained by averaging the relevant behavior of matter at the
molecular level over a REPRESENTATIVE ELEMENTARY
VOLUME around every point within the phase.
Why do we need a continuum approach for a porous medium domain?
Why not apply the theory of fluid mechanics in order to determine
velocity, pressure, etc., at every point within the void?
IN PRINCIPLE, we could. We could write ALMOST a complete model
for any transport phenomenon in a porous medium domain,
considering the behavior at every point within the fluid.
HOWEVER, we do not know the
boundaries of the void space!
(or of other phases).
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Furthermore….Do we really want to know what happens
at points WITHIN a phase?
To circumvent the need to specify the exact configuration of interphase
boundaries we introduce the continuum approach.
The underlying idea---AVERAGING to obtain a continuum description.
Advantage: Variables are measurable differentiable quantities.
Disadvantage: Loss of information (which we do not have, anyway).
.
MICROSCOPIC vs. MACROSCOPIC quantities.
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LEVELS of OBSERVING/DESCRIBING PHENOMENA:
Molecular

Microscopic
Macroscopic
.
P
Megascopic
What is the meaning of FLUID DENSITY    ( P)
MICROSCOPIC QUANTITY ---averaging over molecular level.
MACROSCOPIC LEVEL– averaging over what volume?
WHAT IS A POROUS MEDIUM?
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Consider a series of volumes DU1<DU2<DU3<…
Determine mass per unit volume.
Fluid
Continuum
.
Domain of
Molecular effects
Mass per
Unit vol.

Domain of
fluid continuum
Inhomogeneous
Homogeneous
Volume of REV
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EXAMPLES OF POROUS MEDIA:
Soil, sand, fissured rocks, sandstone, and karstic limestone.
. COMMON FEATURE:
SOLID MATRIX
VOID SPACE occupied by one or more fluid phases.
The void space may be interconnected or not. Here we shall
consider porous media in which both the void space and the
solid matrix are interconnected.
Another common feature:
both the solid matrix and void space are
distributed throughout the porous medium domain.
HOW DO WE CHECK?----- BY TAKING SAMPLES.
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LET US TAKE SAMPLES
Some samples have only void space.
Some samples have only solid matrix
Some samples have both solid and void space.
How large should the sample be to represent a “point”?
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Consider a sequence of volumes centered at point:
Determine porosity at the point.
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REV = Representative Elementary
Volume.
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DEFINITION OF A POROUS MEDIUM:
Combine:
The CONTINUUM APPROACH.
The concept of REV.
AVERAGING OVER AN REV.
MACROSCOPIC LEVEL---averaged (over an REV)
behavior at a point in a porous medium domain.
A continuum for every phase, every extensive quantity
of a phase, or component of a phase.
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INTRINSIC PHASE AVERAGE
Ea = amount of an extensive quantity, E, in a phase denoted by a.
ea = the density of E per unit volume of the a -phase.
a.
1
ea (x, t ) 
U0a (x, t ) oo o

U0a ( x ,t )
ea (x ', t , x) d Ua (x ', t ) o
U0a (x, t ) = volume of a-phase within
X’
= a point in the REV centered at x.
U
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o
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PHASE AVERAGE
1
ea (x, t ) 
U0 (x, t ) oo
o
U0a ( x ,t )
ea ( x ', t , x) d Ua ( x ', t ) o
ea  a ea
with
U0 a
a 
Ua

a
= volumetric fraction of the a -phase within
.
The kind to be used depends on the way the averaged quantity
is measured.
SIZE OF REV?
l = characteristic dimension of an REV.
d = length characterizing the microscopic structure of the void
space (say, grain size, or the HYDRAULIC RADIUS = reciprocal
of the specific surface area of the solids within an REV).
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A necessary condition for obtaining non-random estimates of
the geometrical characteristics of the void space at any point is
l >> d.
For upper limit:
l << lmax
Lmax = distance beyond which the spatial distribution of the relevant
macroscopic coefficients that characterize the configuration of the
void space, or of a phase, deviates from the linear one by more than
some acceptable value.
Also, for upper limit:
l << L.
L= characteristic length of domain.
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How are COEFFICIENTS Created?
In the macroscopic model, the effects of the shape of interphase
boundaries, within the REV, appears in the form of COEFFICIENTS.
The numerical values of these coefficients must be determined
EXPERIMENTALLY.
…so we have a A POROUS MEDIUM DOMAIN……..
HOMOGENEITY A porous medium domain is said to be
homogeneous if its properties are the same AT ALL ITS POINTS.
SCALE OF HETEROGENEITY depends on the size of domain of
interest
ANISOTROPY: A porous medium domain is said to be anisotropic,
if its properties AT A POINT vary with direction.
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We see: Phases, components,
inter-phase boundaries…
We UNDERSTAND: at microscopic level
We MEASURE, PREDICT..at macroscopic level.
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…and from now on, we’ll
be working in a continuum,
UNLESS,… JB/SWICA 36
DISTRIBUTION OF SUBSURFACE WATER
Subsurface moisture zones:
Unsaturated zone:
Saturated zone
(soil water zone,
intermediate zone,
capillary fringe.
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AQUIFER: A geological formation which
(a) contains water, and
(b) transmits water, in significant quantities.
"SIGNIFICANT"?
Aquifer properties will describe the ability to
transmit and store water.
AQUITARD: A relatively thin formation, underlying and/or
overlying an aquifer, which may contain water, but permits
only a small leakage.
Leakage
Aquitard
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Aquifer
Leakage
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PHREATIC SURFACE
Ground surface
Real moisture distribution.
Approximate
As an approximation of moisture distribution: Capillary fringe.
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PRESSURE DISTRIBUTION:
Above water table: pressure in the water is less than atmospheric
Below water table: pressure in the water is above atmospheric
For horizontal flow: Hydrostatic pressure distribution (vertical
equal piezometric head surfaces).
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WORKSHOP ON
MODELING
FLOW AND SOLUTE TRANSPORT
IN THE SUBSURFACE
by
JACOB BEAR
The Second International conference on
Salt Water Intrusion and Coastal Aquifers
--Monitoring, Modeling and Management
Copyright © 2002 by Jacob Bear, Haifa Israel.
All Rights Reserved.
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The Second International conference on
Salt Water Intrusion and Coastal Aquifers
--Monitoring, Modeling and Management
COPYRIGHT NOTICE:
MODELING
FLOW AND CONTAMINANT TRANSPORT
IN THE SUBSURFACE
by
JACOB BEAR
These notes were especially prepared for the lectures presented at the pre-conference
workshop, Conference on Sea Water Intrusion in Coastal Aquifers, held at Essouira,
Morroco, April 23-May 25, 2001.
This material/notes is
Copyright © 2000 by Jacob Bear, Haifa, Israel.
All Rights Reserved. To use, copy, modify, and distribute these documents
for any purpose is prohibited, except by written permission from Jacob Bear.
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