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MODELINGFLOW AND SOLUTE TRANSPORT

IN THE SUBSURFACEby

JACOB BEAR

SHORT COURSE ON

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.

Lectures presented at the Instituto de Geologia, UNAM,Mexico City, Mexico, December 6--8, 2003

Professor Emeritus, Technion—Israel Institute of Technology,Haifa, Israel

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To discuss the role of models in the decision making process for ground water management, including of the subsurface, and tosubsurface 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.

OBJECTIVES OF LECTURES

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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 aphreatic 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 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..

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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

Model Predictions of System’sResponse to Alternatives

Proposed Alternative solutions

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 PROCESSSTEP 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 ofthe 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 MODELLET'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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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.

CONCEPTUAL MODEL (cont.)

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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.

THE MATHEMATICAL MODEL:•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”.

In NUMERICAL MODELS:

•State variables are defined at nodes or as averages over small cells.

EVERY point?

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 codeThe 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 ata 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 aunique 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 modelCONTINUUM 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 modelfor 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:MolecularMicroscopicMacroscopicMegascopic

What is the meaning of FLUID DENSITY

MICROSCOPIC QUANTITY ---averaging over molecular level.

MACROSCOPIC LEVEL– averaging over what volume?

WHAT IS A POROUS MEDIUM?

( )P

P

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FluidContinuum

Consider a series of volumes U1<U2<U3<…Determine mass per unit volume.

Inhomogeneous

Domain of fluid continuum

Domain of Molecular effects

Homogeneous

Mass perUnit vol.

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 shallconsider porous media in which both the void space and the solid matrix are interconnected.

Another common feature:

both the solid matrix and void space aredistributed 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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An REV is

(1) The smallest sample that wherever taken in a given domainalways contains both solid and void space.

(2) An average over it should be independent of the size of theREV.

REV = Representative Elementary Volume.

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A POROUS MEDIUM is defined as a portion of space occupiedby a number of phases, at least one of which is a solid, for whichan REV can be found. If an REV cannot be found, the considereddomain cannot qualify as a porous medium.

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 quantityof a phase, or component of a phase.

OVERLAPPING CONTINUA

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E = amount of an extensive quantity, E, in a phase denoted by . e = the density of E per unit volume of the -phase.

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.

0 ( , )0

1( , ) ( ', , ) ( ', )

( , ) te t e t d t

t

xx x x x

x oUoo o

UU

INTRINSIC PHASE AVERAGE

= volume of -phase within = a point in the REV centered at x.

0 ( , )t xU

X’

Uo

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0 ( , )0

1( , ) ( ', , ) ( ', )

( , ) te t e t d t

t x

x x x xx oU

oo o

UU

PHASE AVERAGE

with e e

0UU

= volumetric fraction of the -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 determinedEXPERIMENTALLY.

…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 ofinterest

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 levelWe MEASURE, PREDICT..at macroscopic level.

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…and from now on, we’llbe working in a continuum,UNLESS,…

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DISTRIBUTION OF SUBSURFACE WATER

Subsurface moisture zones:Unsaturated zone: (soil water zone, intermediate zone, capillary fringe.

Saturated zone

VADOSE ZONE

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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.

Aquitard Aquifer

Leakage

Leakage

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PHREATIC SURFACE

Real moisture distribution.

As an approximation of moisture distribution: Capillary fringe.

Ground surface

Approximate

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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 (verticalequal piezometric head surfaces).

End of part 1.

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MODELINGFLOW AND SOLUTE TRANSPORT

IN THE SUBSURFACEby

JACOB BEAR

WORKSHOP ON

Copyright © 2002 by Jacob Bear, Haifa Israel. All Rights Reserved.

The Second International conference on Salt Water Intrusion and Coastal Aquifers--Monitoring, Modeling and Management

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MODELINGFLOW AND CONTAMINANT TRANSPORT

IN THE SUBSURFACE

by

JACOB BEAR

These notes were especially prepared for the lectures presented at the pre-conferenceworkshop, 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.

COPYRIGHT NOTICE:

The Second International conference on Salt Water Intrusion and Coastal Aquifers--Monitoring, Modeling and Management