A Lecture Note INTRODUCTION TO ...

A Lecture Note INTRODUCTION TO GROUNDWATER MODELING (Pengantar Pemodelan Air Tanah)

Heru Hendrayana, 2012 [email protected]

According to Kresic (2007), groundwater modeling in some form is now a major part of projects dealing with groundwater development, protection, and remediation. As computer hardware and software continue to be improved and become more affordable, the role of models in highly qualitative earth sciences such as hydrogeology will continue to increase accordingly. Using mathematical equations, groundwater model describe the groundwater flow and transport processes using mathematical equations based on certain simplifying assumptions (Torak, 1992a,b; Cooley, 1992; Lin et al., 2003). These assumptions typically involve the direction of flow, geometry of the aquifer, the heterogeneity or anisotropy of sediments or bedrock within the aquifer, the contaminant transport mechanisms and chemical reactions. Groundwater model provides knowledge for programmers and users to understand and predict groundwater behavior. Because of the simplifying assumptions embedded in the mathematical equations and the many uncertainties in the values of data required by the model, a model must be viewed as an approximation and not an exact reconstruction of the real field conditions. Groundwater models, however, even as approximations are a useful investigation tool that groundwater hydrologists may use for a number of applications. I. Motive for Aquifer Modeling Based on Younger (2007), there are numerous reasons why the simulation of aquifer behavior is often desirable. Mostly, the motives are entirely practical, for instance:

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

Prediction the possible consequences of a proposed new abstraction (typically in terms of drawdown or changes in aquifer outflows).



Designing well fields for purposes of dewatering construction or mining sites.



Understanding the relationships between groundwater flow patterns and surface ecosystems.



Assessing the long-term safety of subsurface storage/disposal of radioactive wastes.



Predicting the possible impacts of climate change on groundwater resources.

Such applications of groundwater models are amongst the most valuable tools at the disposal of aquifer managers and are probably the single greatest application of advanced numerical analysis to the resolution of practical problems in any field of engineering [Younger, 2007]. In addition, simulation of groundwater systems is occasionally undertaken for purely scientific purposes, such as: 

Understanding how groundwater flow processes affect landform development, accumulation of ores, and other geochemical processes.



Simulating the coupled flow and mineral dissolution processes responsible for the formation of cave systems.



Reconstructing the behavior of aquifers during the cold periods of the Quaternary Era

II. Group of Groundwater Modeling According to Kresic (2007), groundwater modeling area divided into two main groups: (1) models of groundwater flow and (2) models of contaminant and solute transport. Some of the more common questions that fully developed and calibrated groundwater flow, and fate and transport models may help answer are: -

What is the safe (sustainable) yield of the aquifer portion targeted of groundwater development?

-

At what location and how many wells are needed to provide a desired flow area?

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-

What is the impact of current or planned groundwater extraction on the environment (e.g., on surface stream flows, wetlands)?

-

Is there a potential for saltwater intrusion from an increased groundwater withdrawal?

-

Where is the contaminant flowing to, and where is it coming from?

-

How long would it take the contaminant to reach potential receptors?

-

What would the contaminant concentration be once it reaches receptors?

Once of these questions area addressed by the models, many new ones may pop-up, which is exactly what the purpose of a well-documented and calibrated groundwater model should be answer all kinds of possible question related to groundwater flow, and fate and transport of contaminates. III Groundwater Flow Modeling Based on Mandle (2002), groundwater flow models are used to calculate the rate and direction of movement of groundwater through aquifers and confining units in the subsurface. These calculations are referred to as simulations. The simulation of groundwater flow requires a thorough understanding of the hydrogeologic characteristics of the site. The hydrogeologic investigation should include complete hydrogeological characteristics of aquifer such as: -

Subsurface extent and thickness of aquifers and confining units (hydrogeologic framework).

-

Hydrologic boundaries (also referred to as boundary conditions), which control the rate and direction of movement of groundwater.

-

Hydraulic properties of the aquifers and confining units.

-

A description of the horizontal and vertical distribution of hydraulic head throughout the modeled area for beginning (initial conditions), equilibrium (steady-state conditions) and transitional conditions when hydraulic head may vary with time (transient conditions).

-

Distribution and magnitude of groundwater recharge, pumping or injection of groundwater, leakage to or from surface-water bodies, etc. (sources or sinks, also referred to as stresses). These stresses may be constant (unvarying with time) or may change with time (transient).

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The outputs from the model simulations are the hydraulic head and groundwater flow rates which are in equilibrium with the hydrogeologic conditions (hydrogeologic framework, hydrologic boundaries, initial and transient conditions, hydraulic properties, and sources or sinks) defined for the modeled area. Through the process of model calibration and verification, which is discussed in later sections of this proposal, the values of the different hydrogeological conditions are varied to reduce any disparity between the model simulations and field data, and to improve the accuracy of the model. The model can also be used to simulate possible future changes to hydraulic head or groundwater flow rates because of future changes in stresses on the aquifer system. IV. Types of Groundwater Models Stated by Essink (2000), in general, three main classes of models can be distinguished: (i) a physical model or scale model, being scaled down duplicate of a full scale prototype; (ii) an analogue model, being a physical process which is translated to the hydrologic process involved such as model electric model; (iii) a mathematical model. However, nowadays, the third main class was described intensively; most models are of that kind. Various definitions of a mathematical model exist, as subject to the concept of the model and the field of application. Before going on, two commonly applied definitions are given: 

A mathematical model (Figure 1) is a model in which the behavior of the system is represented by a set of equations, perhaps together with logical statements, expressing relations between variables and parameters [Essink, 2000].



A mathematical model simulates groundwater flow indirectly by means of a governing equation thought to represent the physical processes that occur in the system, together with equations that describe heads or flows along the boundaries of the model [Anderson & Woessner, 1992].

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System characteristics parameters

Input variables

Mathematical equations

Output variables

Condition: initial and boundary Figure 1: Part of Mathematical Model [Essink, 2000]

Based on Kresic (2007), a model simulates the areal and temporal properties of a system, or one of its parts, in either a physical (real) or mathematical (abstract) way. IV.1 Physical Model A physical model in hydrogeology can be consider as a tank filled with sand and saturated with water called “sandbox”, an equivalent to miniature aquifer of limited extent. This aquifer can be subject to miniature stresses such as pumping from a perforated tube placed into the sand thus representing a water well [Kresic, 2007]. IV.2 Mathematical Model According to Kresic (2007), models that use mathematical equations to describe elements of groundwater flow area called mathematical. Depending upon the natural of equation involve, these models can be: 

Empirical (experimental)



Probabilistic



Deterministic

IV.2.1

Empirical Models

These models are derived from experimental data that fitted to some mathematical function. Empirical models are limited in scope and are usually site or problem-specify. However, these models can be an important part of a more complex numeric modeling effort. For instant the behavior of a certain pollutant in porous Groundwater Modeling – Heru Hendrayana – 2012 – [email protected]

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media can be studied in the laboratory or in controlled field experiment, and the derived experimental parameters can then be used for developing numerical models of groundwater transport [Kresic, 2007]. IV.2.2 Probabilistic Models These models are based on laws of probability and statistics. They can have various forms and complexity starting with a simple probability distribution of a hydrogeological property of interest and ending with complicated stochastic, timedependent models. The main limitations for a wider use of probabilistic (stochastic) models in hydrogeology are that: (i) they required large data sets needed for parameter identification and (ii) they cannot be used to answer (predict) many of the most common questions from hydrogeologic practice such as effects of a future pumping, for example [Kresic, 2007]. IV.2.3 Deterministic Models These models assume that the stage or futures reactions of the system (aquifer) studied are predetermined by physical laws governing groundwater flow. Most problems in traditional hydrogeology are solved using deterministic models, which can be as simple as the Theis equation or as complicated as a multiphase flow through a multilayered, heterogeneous, anisotropic aquifer system. There are two large groups of deterministic models depending upon the type of mathematical equation involved: (a.) analytical methods and (b.) numerical methods [Kresic, 2007]. (a.) Analytical Methods Simply stated, analytical methods solve one equation of groundwater flow at a time and the result can be applied to one point or “line or point” in the analyzed field (aquifer). In the past, analytical methods were commonly used for the analysis of groundwater problems. An analytical solution of the partial differential equation was brought up for a particular problem with its corresponding initial and boundary conditions. For example, the piezometric head φ and the groundwater flow Q were computed by solving the equations directly and continuously in time and space.

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The advantage of analytical methods is that they can give a quick insight in the sensitivity of the solution for various physical parameters (such as transmissivity, storability). Moreover, they can serve as verification of solutions of more complex systems obtained by numerical methods. However, the application of analytical methods is limited as analytical solutions are only available for relatively simple and strongly schematised problems (e.g. homogeneous aquifers, 1D or 2D, steady state, interface between fresh and saline groundwater) [Kresic, 2007]. (b.) Numerical Methods If groundwater problems become more complex (e.g. inhomogeneous, anisotropic, transient, regional groundwater flow with changes of the properties of aquifers and semi-pervious layers, with wells, rivers, etc.), the system becomes too complicated for solutions obtained with analytical methods. In these cases, numerical methods have to be used. The introduction of micro-computers increased the application of numerical methods and these methods replaced almost completely the use of analytical methods [Kresic, 2007]. Referring to Kresic (2007), numerical methods described the entire flow field of interest at the same time, providing solutions for as many data points as specified by user. The area of interest is divided into many small areas (refer to as cells or elements) and a basic groundwater flow equation is solved for each cell usually considering its water balance (water input and output). The solution of a numerical model is the distribution of hydraulic heads at points representing individual cells. These points can be placed at the center of the cells, at interaction between adjacent cells, or elsewhere. The basic differential flow equation for each cell is replaced (approximated) by an algebraic equations so that the entire flow field is represented by x equation with x unknowns, where x is the number of cells. This system of algebraic equations is solved numerically, though an iterative process, thus the name numerical models. Since computers appeared on the scene, mathematical models either analytical models or numerical models gained ground than physical models. These analytical models, based on numerical techniques, apply infinite series of definite integrals to solve the solution. Numerical models, however, are directly based on computer codes. Groundwater Modeling – Heru Hendrayana – 2012 – [email protected]

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Groundwater Models Mathematical Models

Physical Models

Analytical Models Numerical Models

Analytical Models Numerical Models

Deterministic

Stochastic - Monte Carlo simulation - Random Walk Model

Direction computation

Inverse modeling - Optimalisation technique - Pumping tests

Saturated flow

Unsaturated flow

One fluid

Couple models

Two or more fluids

Only groundwater flow

Deformation models

Dissolved solute Head transport - salt-water: density, dependent - chemical reaction - contaminant transport

Porous media

Fractured rock - faults - joints Discrete fracture models

Steady-state

1D One aquifer

Dual porosity models

Transient

2D

3D

multilayered system

Figure 2: Classification of groundwater models [Hemker, 1994]

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At present, a large number of mathematical models is available, which are capable of handling many types of groundwater flow. Mathematical models specification of system geometry, boundary conditions and initial conditions for transient processes as shown in Figure 3 herein [Essink, 2000].

Figure 3: The schematisation of the calculation scheme, applied for mathematical modeling of groundwater [Essink, 2000] V. Finite Differences and Finite Elements Regarding to Essink (2000), the various methods of approximating is base on the differential flow equations and methods used for numerically solving the resulting system of algebraic equation, numeric models are divided into several methods. The two most widely applied methods are: 

Finite differences



Finite elements Both types of methods above have their advantage and disadvantage and for

certain problems one way are more appropriate than the other. Both the finite difference method and the finite element method are most widely used numerical techniques for solving mathematical models.

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The finite difference method is probably the oldest, most popular, and conceptually simplest of the numerical procedures governing groundwater behavior. The finite difference method consists of discretising the problem area into rectangular elements which are identified with discrete points or nodes. Meanwhile, the finite element method is a very well-known method to solve the governing partial differential equations. For the finite element method an integral approach (instead of a differential approach for the finite difference method) is applied. The domain is decomposed into a set of sub-regions, the so-called elements. The corners of the elements are called nodes. In groundwater problems, the polygonal shape of the element is almost always triangular (in two dimensions triangles), whereas occasionally more complex quadrilaterals are used. An irregular polygonal mesh allows the modeler to follow the natural shapes more accurately [Essink, 2000].

Figure 4: The finite difference and finite element methods [Essink, 2000] In the case of finite difference models, the elements have to be rectangular, whereas in case of finite element models, not only rectangular but also triangular elements may be used (Figure 4). The computations result in values at nodal points. This means that the piezometric head is determined for a certain area around these points. So the piezometric head is an average for the so-called influence area. Figure 5 shows influence areas for triangular and rectangular elements. In areas of interest a more dense grid may be required. In the finite difference method, fluxes through a boundary are inserted over the area of the elements, whereas in the finite element method, boundary fluxes are inserted in the node [Essink, 2000].

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Figure 5: The influence area [Essink, 2000] VI. Transient and Steady State Regarding to Pollock (1994), the option of steady-state or transient flow is designated by the value of the number of stress periods (NPER) that specified in the main data. For a steady state simulation, set NPER=0. For a transient simulation, set NPER equal to the number of stress periods in the MODFLOW simulation. Meanwhile stated by National Academy of Sciences (1990), another valuable assumption in the application of saturated continuum flow models is that of steady state, i.e., that condition remains constant over time. Because the stresses that drive ground water flow often vary only slowly in time-much more slowly than the system requires to respond-steady state assumptions are often justified. However, there are situations where transients must not be ignored. For example, the ignoring seasonal periodicities in ground water flow direction can lead to otherwise unexpected dispersion during transport. Pulsed-pumping remedial schemes, which are becoming more common, also may demand explicit consideration of transient effects if accurate prediction of contaminant breakthrough is required.

VII. Aquifers Properties and Fundamental of Groundwater Flow VII.1. Henry Darcy and his Law “Father of Hydrogeology”: Henry Darcy (1803–1858), was instrumental in the derivation of one of a very important formula for describing frictional head losses during flow through pipes (the Darcy–Weisbach Formula, which remains in widespread use today). Without even trying to investigate the movement of natural groundwater, Henry Darcy managed to

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derive a fundamental law which has been used ever since for the quantification of flows in aquifers [Younger, 2007]. Stated by Younger (2007), Darcy (1856) did experiments on flow through a cylinder of saturated sand (homogeneous and isotropic porous medium) (Figure 6). He found relations between different parameters which influence the flow of water. The rate of flow appears to be proportional directly to head loss and inversely to the length of the flow path, with a constant proportionally factor, see the equations I.1 to I.3. i

h1  h2 h  L L

v  k .i  k .

h1  h2 h  k. L L

Q  k .i. A  k .  q  k .

h1  h2 h .A  k. .A L L

h l

(I. 1) (I. 2) (I. 3) (I. 4)

Where i = hydraulic gradient k = proportionally factor or hydraulic conductivity/permeability (L/T) h1 & h2 = heads (L) v = velocity of flow through the aquifer (L/T) A = cross-sectional area of diameter cylinder (L2) Q = volume rate of flow of ground water (L3/T) L = length of between hydraulic heard (T) q = darcian specific discharge (L/T)

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Figure 6: Darcy's experiment [Younger, 2007] Darcy’s law is only valid in case of laminar flow: viz. at relative low velocities when water particles move more or less parallel to each other. In quantified terms, Darcy’s law is valid as long as the so-called Reynolds number Re does not exceed some value between 1 and 10. Re 

p.q.R



 1  10

(I. 5)

Where p = mass density of water μ = dynamic viscosity (M/ L/T) R = hydraulic radius of the pore (L) VII.2. Groundwater Head The understanding flow of groundwater is an important process; however the actual groundwater flows cannot be measured directly. Consequently, an alternative method of identifying groundwater conditions is required and this is provided by the groundwater head (or groundwater potential). The groundwater head at a location in an aquifer is the height to which water will rise in a piezometer (or observation well). So that the conditions at a specific location in an aquifer can be identified, the open

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section of the piezometer which monitors the conditions should extend for no more than a metre [Younger, 2007]. In natural of groundwater environment, groundwater flows from a higher to a lower groundwater head (Figure 7).

Figure 7: Direction flow based on hydraulic head [Younger, 2007]

Figure 8: Definition of the piezometric head [Essink, 2000] In case the atmospheric pressure equals zero (Figure 8), the relation between the pressure and the so-called piezometric head is as follows: Groundwater Modeling – Heru Hendrayana – 2012 – [email protected]

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

p z i .g

(I. 6)

where



 piezometric head ( L). Also called head or piezometric level

p  pressure head ( L) i .g z

 elevation with respect to the referencelevel ( L)

VII.3. Permeability Based on Kresic (2007), a permeable or previous rock, with respect to groundwater, is one having a texture that permits water to move through if freely under the influence of gravity and hydraulic gradients commonly found in aquifers. Such a rock has communicating interstices (voids) of sufficient size to allow free groundwater flow. An impermeable or impervious rock, one the other hand, does not permit movement of water under common hydraulic heads. Such a rock may have subcapillary interstices or isolated interstices of larger size. It may possible also have small communicating capillary interstices. A rock that is impermeable under common conditions may not be impermeable to water under a hydrostatic pressure in excess of those found in subsurface. It may also allow water to move through it under the influence of other forces, such as molecular attraction. VII.4. Hydraulic Conductivity (k) Based on Younger (2007), hydraulic conductivity is one of the most crucial physical properties studied by hydrogeologists. It is helpful to consider hydraulic conductivity as “the permeability of a given rock with respect to fresh groundwater”. This is because the value of K found in Darcy’s experiments relates specifically to freshwater, not too dense, saline waters or other fluids. As such, hydraulic conductivity is a function of both the properties of the rock and the properties of the water (especially its kinematic viscosity and density). Hydraulic conductivity is a remarkable physical property in its own right, for it is known to vary over more than 13 orders of magnitude from less than 10− 8 m/day to more than 105 m/day), which is a far wider range than is exhibited by most other physical properties.

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Besides their utility in allowing quantification of groundwater flow rates using Darcy’s Law, hydraulic conductivity values are also very useful as criteria with which to compare the water bearing capabilities of different rocks. This not only allows us to identify the more and less permeable parts of a given aquifer, but also allows us to compare values obtained for a particular aquifer with those obtained for other aquifers of similar rock type [Younger, 2007]. Such ranges of values of hydraulic conductivity which are commonly encountered in various rock types are summarized in Table 1 below. The darker the shading, the more common are values in that range for the rocks indicated. Table 1: Ranges of hydraulic conductivities encountered in various rock types (Younger, 2007)

VII.4.1. Slug Tests- Hvorslev Method One of the effective method to calculate the hydraulic conductivity is Slug Tests Method. This method is applicable to determine the horizontal hydraulic conductivity of distinct geologic horizons under in-situ conditions. Slug tests involve a short-term introduction or removal of water via a well or boring into (or out of) a subsurface interval of sediment, soil, or fractured rock. Monitoring the water level rise or fall as it Groundwater Modeling – Heru Hendrayana – 2012 – [email protected]

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returns to quasi-equilibrium conditions produces data based on which numerous researchers have developed method to determine hydraulic conductivity and transitivity. Each slug test method was developed in response to a particular subsurface condition, but the whole, each method is related in some way to some extent to the other methods. One method was developed to accommodate certain features that previous methods either overlooked or ignored. Based on Fowler & Klein (1990), the Slug tests method are included: (1). Hvorslev Method, 1951; (1). CooperBredehoeft-Papadopulos Method, 1967; (3). Ferris-Knowles Method, 1963;

(4).

Theis (Modified) Method, 1935; (5). Bureau of Reclamation Method, 1960; (6). Nguyen-Pinder Method, 1984; (7). Bouwer-Rice Method, 1976. Likewise, one of the most applicable is Hvorslev Method, 1951 because this test method is commonly use in the field since it is suitable for each type of aquifers and tend to be more free and easy way of calculation. The parameters necessary recorded at the observation wells such as radius of well casing, radius of well screen, length of well screen, initial water level prior to removal of slug, and water level at time of recorded equaled to zero.

Figure 9: Parameters recorded of Slug test method in a dug well

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VII.5. Transmissivity (T) Although hydraulic conductivity is an extremely important characteristic of an aquifer, it will only be effective in contributing to the transmission of large quantities of water where it is developed in an aquifer of substantial saturated thickness. For many practical purposes therefore, it is precisely the combination of hydraulic conductivity and saturated thickness, which is needed to know. This combination of hydraulic conductivity and saturated thickness is captured in the property transmissivity, which in strict terms is defined as the integration of the values of (horizontal) hydraulic conductivity between the base and top of the aquifer [Younger, 2007].

Opening B, 1m wide and aquifer height b Opening A, 1m square

Figure 10: Illustration of the concept of transmissivity [Delleur, 1999] In quantitative terms, the transmissivity (T) is product of hydraulic conductivity (K) of aquifer material and the saturated thickness of the aquifer (b). T  K .b

(I. 7)

It has unites of squared length over time (e.g.,m 2/d or ft2/d). In practice terms, the transmissivity equals the horizontal groundwater flow rate through a vertical strip

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of aquifer one unit wide. The larger the transmissivity, the larger the hydraulic conductivity and the aquifer thickness. VII.6. Porosity and Effective Porosity Referring to Naven (2009), porosity (n) is defined as the percentage of voids (empty space occupied by water or air) in the total volume of rock, which includes both solids and voids: n

VV .100 V

(I. 8)

where Vv = volume of all rock voids and V = total volume of rock (in geologic terms, rock refers to all the following: soils, unconsolidated and consolidated sediments, and any type of rock in general). Assuming the specific gravity of water equals unity, total porosity, as a percentage, can be expressed in four different ways [Lohman, 1972]. n

Vi Vw V  Vm V    1  m .100% V V V V

(I. 9)

where n  porosity, in percent per volume VV  volume of rock voids V

 total volume

Vi  volume of all int erstices  voids  Vm  aggregate volume of min eral  solid  particles Vw  volume of water in saturated sample Porosity can also be expressed as:

n

m  d  1  d .100% m m

(I. 10)

where ρm = average density of mineral particles (grain density) and ρd = density of dry sample (bulk density). The shape, amount, distribution, and interconnectivity of voids influence the permeability of rocks. Voids, on the other hand, depend on the depositional mechanisms of unconsolidated and consolidated sedimentary rocks, and on various other geologic processes that affect all rocks during and after their formation. Primary porosity is the porosity formed during the formation of rock itself, such as voids Groundwater Modeling – Heru Hendrayana – 2012 – [email protected]

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between the grains of sand, voids between minerals in hard (consolidated) rocks, or bedding planes of sedimentary rocks. Secondary porosity is created after the rock formation mainly due to tectonic forces (faulting and folding), which create microand macrofissures, fractures, faults, and fault zones in solid rocks. Both the primary and secondary porosities can be successively altered multiple times, thus completely changing the original nature of the rock porosity. These changes may result in porosity decrease, increase, or altering of the degree of void interconnectivity without a significant change in the overall void volume, [Patel & Shah, 2008]. A related parameter is termed the void ratio, designated as e, and stated as: e

Vv Vs

(I. 11)

expressed as a fraction, where Vs is the solid volume. As total volume is the sum of the void and solid volume, the following relationships can be derived: e

n 1 n

or n 

e 1 e

(I. 12)

Porosity of sedimentary rock will depend not only on particle shape and arrangement, but on a host of diagenetic features that have affected the rock since deposition. Porosity can range from zero or near zero to more than 60% as shown in Table 2. Table 2: Range in value of porosity (Domenico & Schwartz, 1997) Material SEDIMENTARY Gravel, coarse Gravel, fine Sand, coarse Sand, fine Silt Clay SEDIMENTARY ROCKS Sandstone Siltstone Limestone, dolomite Karst limestone Shale CRYSTALLINE ROCKS Fractured crystalline rocks

Porosity (%) 24-36 25-38 31-46 26-53 34-61 34-60 5-30 21-41 0-40 0-40 0-10 0-10

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Dense crystalline rocks Basalt Weathered granite Weathered gabbro

0-5 3-35 34-57 42-45

VII.7. The Storage Properties of Porous Media VII.7.1. Storage (Storativity) Referring to Kresic (2007), the storativity of an aquifer is the volume of water it releases from or takes into or storage due to change in the component of hydraulic head normal to that surface. VII.7.2. Specific Storage Based on Domenico & Schwartz (1997), the specific storage was defined as a proportionality constant relating the volumetric changes in fluid volume per unit volume to the time rate of change in hydraulic head. The amount of water obtained per unit volume drained is rather substantial and is obviously equal to the volume of pore space actually drained. However, the physical mechanism that releases or stores water in the storage is not the same for unconfined and confined aquifers (Figure 11). Unit area Unit declines

in heads

Confined aquifer

Confined aquifer

Volume of water released from storage

Figure 11: Schematic show storativity in confined and unconfined aquifers [Domenico & Schwartz, 1997] Groundwater Modeling – Heru Hendrayana – 2012 – [email protected]

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VII.7.3. Specific Retention The specific retention Sr of a soil or rock is the ratio of the volume of water it will retain after saturation against the force of gravity to its own volume. Thus, Sr 

wr Vt

(II. 13)

where wr is the volume occupied by retained water and Vt is the bulk volume of the soil or rock. VII.7.4. Specific Yield Based on Todd (1990), a specific yield Sy of a soil or rock is the ratio of the volume of water that, after saturation, can be drained by gravity to its on volume. Therefore,

Sy 

wy Vt

(I. 14)

where wy is the volume of water drained. Values of Sr and Sy can also be expressed as percentages. Regarding to Kresic (2007), specific yield has been called the “effective porosity” or “practical porosity” because it represents the pore space that will surrender water to wells and is there effective in furnishing water supplies. Therefore,   S r S y

(I. 15)

where α presented the effective porosity. Figure 12 below illustrated the effective porosity of unconfined aquifer in function to specific retention and specific yield.

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Figure 12: Sketch illustrated the effective porosity of unconfined aquifer [Younger, 2007]

Regarding to Todd (1980), the values of specific yield depend on grain size, shape and distribution of pores, compaction of the stratum, and time of drainage. Representative specific yield for various geologic materials are listed in Table II. 3; individual values for a soil or rock can very considerably from these value. It should be noted that fine-grained materials yields little water, whereas coarse-grained materials permit a substantial release of water and hence serve as aquifers. In general, specific yields for thick unconsolidated formations tend to fall in the range of 7 to 15 percent, because the mixture of grain sizes presented in the various strata; furthermore, thy normally decrease with depth due to compaction. Table 3: Values of Specific Yields for Various Geologic Materials (Johnson, 1967) Material

Specific Yield (%)

Gravel, coarse Gravel, medium Gravel, fine Sand, coarse Sand, medium Sand, fine Silt Clay Sandstone, fine-grained

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23 24 25 27 28 23 8 3 21

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Sandstone, medium-grained Limestone Dune sand Loess Peat Schist Siltstone Till, predominantly silt Till, predominantly sand Till, predominantly gravel Tuff

27 14 38 18 44 26 12 6 16 16 21

VII.8. Inflows and Outflows To understand the groundwater flow within the aquifer required information about inflows and outflow. Based on Rushton and Kruseman (1979), the term recharge is used for the inflow to an aquifer system arising from precipitation, return flow from irrigation and flows from various surface water bodies such as rivers, canals and lakes. Regarding to Rushton and Kruseman (1979), the magnitude of the recharge is likely to change significantly with time. Anyway, outflows from the aquifer system can be divided into natural outflows and man-made outflows. -

Natural outflows occur when water leaves the aquifer at springs or into rivers. Other natural outflows include low-lying areas which act as a sink to groundwater systems; this form of outflow may be associated with areas of evapotranspiration especially from deep-rooting vegetation. These low-lying areas often form wetlands which have a high ecological value. One further natural outflow occurs when water flows into other aquifers.

-

Man-made outflows occur when water is pumped from wells and boreholes. Since the velocities in the vicinity of the pumped borehole are far higher than the natural groundwater velocities, there is a risk of deterioration of the aquifer in the vicinity of the well or borehole and a deterioration of the borehole structure. Horizontal wells or adits provide alternative means of collecting water from an aquifer; this approach is especially suitable for shallow aquifers or for aquifers with thin lenses of good quality water.

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VII.9. Anisotropy and Heterogeneity Regarding to Delleur (1999), isotropic referred to the unique value of aquifer parameters directionally at a point. However, in the geological material is very rarely homogeneous in all direction. Hence, the permeability, the hydraulic conductivity and transmissivity, are rarely equal in all directions at any point in the porous medium. If the parameters differ in value directionally at a point, the medium is then said to be anisotropic. If the condition of directional equality of properties is the same from point to point anywhere in the medium, the medium is termed homogeneous. If the condition of either isotropy or anisotropy varies from point to point, the medium is then said to be heterogeneous. Figure 3 demonstrate these four possible descriptions of a medium (1) isotropic and homogeneous, (2) anisotropic and homogeneous, (3) isotropic and heterogeneous, and (4) anisotropic and heterogeneous [Delleur, 1999].

Figure 13: Combinations of isotropy, anisotropy, homogeneity, and heterogeneity [Delleur, 1999]

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The average hydraulic conductivity (Kp) perpendicular to a layered sequence of m beds, each of which is either isotropic or anisotropic, can be determined using a weighted harmonic average as KP 

d d 1 K m pm m

(I. 16)

Where d = total thickness dm = thickness of each layer Kpm = perpendicular hydraulic conductivity of each bed VIII. Groundwater Flow and Solute Transport Modeling Methodology According Essink (2000), groundwater modeling is very tempting to pass over some steps as describes following: VIII.1 Define purpose of the modeling effort Based on Essink (2000), beefore going on, it is essential to identify the purpose of the modeling effort. Therefore, some of the following questions are needed. -

What is the application of the model (from a scientific, engineering or management point of view)?

-

What is the new knowledge can be gained from the model?

-

What is the question of problems that the model can be answer?

-

Is a modeling exercise the best way to answer the question?

-

Do we really need a mathematical model? Can an analytical model provide the answer or must a numerical model be constructed?

The responses of these questions will lead the modeler in determining the modeling effort: analytical or numerical, lumped or distributed, transient or steady state, etc [Essink, 2000].

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VIII.2. Conceptualisation of a mathematical model Developing a modeling concept is the most important part of model effort. It requires a thorough understanding of hydrogeology, hydrology, and dynamics of groundwater flow in and around the area of interest. The conceptualization of the model consists of two modules: (1) a schematization of the hydrological problem and (2) a concept of the mathematical model [Essink, 2000]. VIII.2.1. A schematization of the hydrological problem In general, schematisations in groundwater problems are focused the composition of the subsoil (layered system, number of aquifers); the type of groundwater flow (steady state, 1D or 2D); the properties of groundwater (density, temperature, fresh-saline interface, fresh/saline, dissolved solutes); the boundaries of the study area (location of the boundary, type of boundary condition); and the use of averaged values (piezometric head, polder level, thickness of layers, porosity, groundwater extraction) [Essink, 2000]. VIII.2.2. Concept of the mathematical model Based on the schematisation of the hydrologic problem, the concept of the mathematical model is built. The purpose to building a concept is to simplify the field problem in order to make the schematisation suitable for numerical modeling. In other words, in order to simplify the system that interested in to a large extent. In addition, the building of a concept organizes the associated field data so that the hydrologic system can be analyzed more easily. A concept is set up to define system characteristics, processes and interactions [Essink, 2000]. VIII.3. Select of the computer code Regarding to Essink (2000), since a very wide range of computer codes exists for application of different problems, it is needed to select the best code which is the most available to the interest area. When choosing a code from the selection available, the following points should be considered: -

What code is best in solving your particular problem?

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-

What are the data requirements for both code and problem?

-

What computer hardware and supporting staff are required?

-

How much will the computer code cost?

-

How accurate will the code be in representing the real world?

VIII.4. Design Model The concept of the mathematical model is transformed to a form suitable for numerical modeling by converting the concept of the specific hydrologic problem to a model which can be implemented in the chosen computer code. This step includes the design of the domain partition, the selection of the length of the time steps (when transient), the setting of the boundary and initial conditions, and the selection of the initial values for system parameters and hydrologic stresses [Essink, 2000].. VIII.4.1. Grid Design Model In a numerical model, the continuous space domain of the hydrological problem is replaced by a discretised domain, the so-called grid. The concept, the selected code and the model scale determine the overall dimensions of the elements (also called blocks or grid cells) in the grid. There are numerous types of elements, see figure 3.6. The two most commonly used grids, applied in mathematical models, are based on the finite difference method and the finite element method [Essink, 2000]. o Model Grid Orientation The model grid orientation is dependent on aquifer’s shape and degree of anisotropy. Based on Kresic (2007), when the aquifer is isotropic, i.e., when its hydraulic conductivity is the same in all directions, orientation of the grid is not critical. In such cases it is still recommended to orient the grid so that the number of inactive cells is minimized. Figure 14 (a) shows an isotropic system of irregular shape without any dominant direction. Here a common south-north orientation seems the most “natural”: columns and rows are aligned with x-and y- axes, respectively. Figure 14 (b) shows an isotropic system that is elongated in one direction. In this case it is recommended that one of the model axes (either row or column) be aligned with the predominant direction. This reduces the number of inactive cells and the size of the model. When the aquifer is anisotropic, the model

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coordinate i.e., directions of maximum and minimum hydraulic conductivities are always perpendicular to each other.

Figure 14: Orientation is dependent on aquifer’s shape and degree of anisotropy [Kresic, 2007]

o Choosing the Model-Grid Cell Size

Stated by Essink (2000), Selecting the model grid is comparable to the selecting a net for fishing; the openings in the net must match the size of the “fish” (heterogeneities and predictive details) to be captured. In choosing a model grid size, the following factors should be considered: - Degree of heterogeneity in hydraulic or transport parameters, and in boundary conditions. - Model domain size. - Predictive resolution required to meet modeling objectives. - Restrictions imposed by computational resources. Similar constraints apply to vertical discretization, with the added consideration of stratification due to density effects, recharge, and shallow or deep sources or sink of water or contaminant. In general, the accuracy of the predicted results improve with finer calculation meshes, but computational time and space requirements increase correspondingly .

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o Selecting Time Step Size Two kinds of time interval are used in models: stress periods (during which boundary conditions are constant and between which boundary conditions vary) and time steps (during which model calculations are made). Time steps are required for transient calculation. Factors effecting choice of time step include stability consideration, numerical dispersion in transport calculation, time variation of boundary conditions and time-related modeling objectives. In general the small the time step, the more accurate are the predicted results. Too small a time step results in excessive computation time, while too large a time step results in an excessive number of iterations required to reach a mass-balanced solution, and possibly numerical dispersion or instability [Essink, 2000]. VIII.4.2. Model Boundary Conditions Based on Essink (2000), a model boundary is the interface between the model calculation domain and the surrounding environment. Boundaries occur at the edges of the model domain and at other points where external influences are represented, such as rivers, wells, leaky impoundments, or chemical spills and so forth. Boundary conditions are expressions of the effect of the external world on the model domain, and they required to complete the description of a flow or transport problem. The mathematical expression of boundary condition is required for a well-posed problem. Likewise, boundary conditions are mathematical statements at the boundary of the problem domain. A correct selection of boundary conditions is a critical step in the model design, as a wrong boundary may lead to serious errors in the results. Mathematically, the boundaries are divided in three types: VIII.4.2.1. Dirichlet Condition (specific head boundary) Specific head boundary describing specified head boundaries for which a head is given. Examples of specified head boundaries are: the water level at a lake or at the sea. A specified head boundary represents an inexhaustible supply of water. For example, water is pulled from or discharged in the boundary without changing the head at the boundary. In some situations, this is probably an unrealistic approximation of the response of the system. The equation of specific head boundary expressed by:

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( x , y , x ,t )  cons tan t ,t

(I. 17)

VIII.4.2.2. Neumann Condition (specific flow boundary) Specific flow boundary is:

q( x , y , z .t )   .

  qcons tan t ,t x

(I. 18)

This type of boundary describing specified flow boundaries for which a flow (the derivative of head) is given across the boundary. Examples of specified flow boundaries are: natural groundwater recharge in an aquifer (areal recharge); groundwater injection or extraction wells; groundwater springflow or underflow; seepage to a hydrologic system. A special Neumann condition is the no-flow boundary condition. A no-flow boundary condition is set by specifying the flux to be zero. Examples of no-flow boundaries are: the groundwater divide in a catchment area; a streamline (a cross-section perpendicular to the contour lines of the piezometric head may also be considered as a no-flow boundary for groundwater problems); a freshsaline interface in a coastal aquifer (interface is a streamline boundary); and an impermeable fault zone. The no-flow boundary condition is simulated in a (blockcentered) finite difference grid by assigning zeros to the transmissivities or the hydraulic conductivities in the inactive elements just outside the boundary. In a finite element grid, the no-flow boundary condition is simulated by simply setting the flux in the node equal to zero [Essink, 2000]. VIII.4.2.3. Cauchy Condition (head-dependent flow boundary) This boundary describing the head-dependent flow for which flux across a boundary is calculated, given a value of the boundary head. The condition of hearddependent flow boundary is:

.

    cons tan t x

(I. 19)

This condition is also called the mixed boundary condition, as it relates boundary heads to boundary flows. It is dependent on the difference between a specified head, supplied by the user, on one side of the boundary and the model calculated head at the other side. Examples of head-dependent flow boundaries are:

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leakage to or from a river, lake or reservoir, evapotranspiration (flux across the boundary is proportional to the depth of the water table below the land surface) [Essink, 2000]. VIII.4.3. Initial Conditions When the problem is transient, an initial condition is necessary at the beginning of the simulation everywhere in the hydrologic system. Due to the parameters and hydrologic stresses inserted in the model are consistent with the generated heads and not with the field-measured heads during the early time steps of the simulation, it appears to be a standard practice to apply the steady state initial condition which is generated with the calibrated model (by setting the storage equal to zero or by setting the time step to a very large value) instead of the initial condition which is obtained with field-measured head values. Furthermore, the alternative in selecting a starting variable distribution is to use an arbitrarily defined variable distribution and then run the transient model until it matches field-measured variables. Then, these new calibrated variables are used as starting conditions in predictive simulations [Essink, 2000]. The mathematical expression for this condition is:

h  h ( x, y, z,0)

(I. 20)

where h (x,y,x,0) is the initial distribution function. Based on Essink (2000), the initial condition of groundwater flow can be given in three features (Figure15): (1) the static steady state condition in which the head is constant throughout the problem domain and in which there is no flow is the system (e.g. used for drawdown simulations in response to pumping); (2) the dynamic average steady state condition in which the head varies spatially and flow into the system equals flow out the system (this condition is used most frequently); and (3) the dynamic cyclic steady state condition in which the head varies in both space and time (a set of heads represent cyclic water level fluctuations, e.g. monthly head fluctuations or monthly average recharge rates).

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Figure 15: Three types of initial conditions for one-dimensional groundwater flow [Essink, 2000] VII.4.4. Preliminary selection of parameters and hydrologic stresses In this phase, the physiographic characteristics of the hydrologic system (e.g. subsoil parameters as porosity and hydraulic conductivity) and hydrologic stresses (e.g. sources and sinks as injection and pumping well rates; flux across a water table as natural groundwater recharge and leakage through a resistance layer) have to be discretised for the input data file of the model. Moreover, mostly numerous other model parameters, such as dummy parameters which set the printing options, must also be inserted in the input data file [Essink, 2000].

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VIII.5. Calibration According to Essink (2000), calibration of a model is one of the most important steps in the application of models. Calibration is adjustments of hydrological models to the parameters in order to tune the model to mach model output with measured data. A numerical model, which is applied to simulate hydrologic processes, must be validated with available data in order to prove its predictive capability, accuracy and reliability. A valid model is an unattainable goal of model validation. The parameters are adjusted within a predetermined range of uncertainty until the model produces results that approximate the set of field measurements selected as calibration targets. A calibration target is defined as a calibration value and its associated error. The effects of uncertainty in hydrologic parameters, hydrologic stresses, and possibly boundary and initial conditions are tested [Essink, 2000]. Model calibration can be performed to steady state or transient data sets. It is common practice to begin the calibration of a transient hydrologic process with a steady state data set and then to continue the calibration under the transient conditions. The selection of a proper steady state data set can be difficult, especially when the hydrologic process to be modeled is really a transient one. For example, it is complicated to define a steady state water level when seasonal fluctuations in water level are large [Essink, 2000]. Though calibration procedures vary from model to model, general alternatives can be listed:

 Trial-and-error calibration In this alternative is the input all the parameters based on physical observation, and provides estimates of the unknown parameters as a first trial. As such, the adjustment of parameters is manual. The model is run and the computed output is compared to the measured output from the prototype (Figure 16). The comparison is done by means of visual pattern recognition of the measured and computed flow hydrographs or solute distributions, or it is based on some mathematical criterion. Based on this comparison, adjustments are made to one or more of the trial parameters to improve the fit between

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measured and computed output. The trial runs of the model are repeated until some kind of required accuracy or calibration target is achieved. Tens to hundreds of runs are typically needed to achieve calibration. Parameters which are known with a high degree of certainty should only be modified sightly or not at all during the calibration procedure.

Figure 16: Procedure of the trial-and-error calibration [Essink, 2000]  Automated Parameter Estimation Codes In this alternative, also called the inverse problem, the model itself contains internal programming which will adjust the trial parameters in a systematic step by step manner until the goodness of fit criterion is satisfied. In this way, the model will automatically calibrate itself and carry out the necessary number of trial runs until the best set of parameters is achieved [Essink, 2000].  Combined Both Alternatives In this alternative, first a trail-and-error manual adjustment of parameters is carried out until the model is almost calibrated, then to introduce the automatic search technique to refine the goodness of fit. A model calibrated with the automated technique is not necessarily superior to a model calibrated with the trial-and-error method. Points in favor of the automated calibration codes is that they are objective compared to the trial-and-error method, they provide information on uncertainty in the calibrated parameters and they may speed in the time-consuming and criticized because of problems of non-uniqueness

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(e.g. due to the absence of prior information on transsmisivities in groundwater problems) and instability [Essink, 2000]. VIII.5.1 Evaluating the Calibration Regarding to Essink (2000).The results of the calibration should be evaluated both qualitatively and quantitatively. Whether or not the fit between model and reality is good is a subjective judgment. Traditionally, two methods are used to evaluate the calibration: (a) qualitatively, by comparison of contour maps of measured and computed parameters, which provides only a qualitative measure of the similarity between the patterns; and (b) quantitatively, by a scatterplot of measured and computed parameters, where the deviation of points from the straight line should be randomly distributed. Three ways of expressing the average difference between measured and computed parameters are normally used to quantify the average error in the calibration: a.

The mean error (ME): which is the mean difference between measured (pmeasured) and computed (pcomputed) parameters, such as piezometric heads: 1 n ME  . ( Pmeasured  Pcomputed )i n i 1

(I. 21)

Where n = number of calibration values As both negative and positive differences are incorporated in the calculation, they may cancel out the error. As such, a small error may not indicate a good calibration, and this way of quantifying the error should be used with care. b. The mean absolute errors (MAE): which is the mean of the absolute value of the difference between measured and computed parameters: 1 n MAE  . Pmeasured  Pcomputed )i n i 1

(I. 22)

This error clears away the difficulty in item 1.

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c.

The root mean squared error (RMS), standard deviation or standard error of estimate (SE): which is the average of the squared differences between measured and computed parameters:

1 n  RMS   . ( Pmeasured  Pcomputed )i2   n i 1 

0.5

(I. 23)

This error is usually thought to be the best measure of error if errors are normally distributed. VIII.5.2. Error criterion When iteration is involved to solve the mathematical equations, an error criterion or convergence criterion can be used to judge whether or not the solution converges. Iteration stops when the change in e.g. fluxes, water balance, heads or solutes between two successive iterations is less than the error criterion. As a rule of thumb, the error criterion should be one or two orders of magnitude smaller than the level of accuracy desired in the results. The residual error of the iteration should progressively decrease during the solution of the mathematical equations. The check on the water balance can be very useful in designing the model. For example, when the fluxes to or from the model are unreasonable high or low, then the inserted transmissivity file may be wrong, whereas unreasonable high or low volumes of water entering or leaving the storage may indicate a wrong storage parameter. As a rule of thumb, the ideal error in the water balance for numerical modeling should be less than 0.1 %, whereas an error around 1 % is usually considered acceptable [Essink, 2000]. VIII.5.3. First model execution In general, the making of the first error-free output will probably require many additional hours, at least, more than expected. The whole phase of the first model execution includes the preparation of the input data file, the entry of input data file into computer lines, the execution of the model (the so-called run), and the interpretation of the results [Essink, 2000].

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VIII.5.4. Sensitivity analysis According to Essink (2000), sensitivity analysis is used to demonstrate the model response to variation in uncertain input parameters. In generally, given that the calibration may be non-unique, hence we have no guarantee that the predictive model will produce accurate results when the model is stressed differently from the calibrated conditions. Moreover, calibration is difficult as values for hydrologic parameters, stresses and boundary conditions are typically known at only a few nodes and are associated with uncertainty. In addition, there is even uncertainty about the geometry of the hydrologic system (e.g. lithology and stratigraphy) that is trying to analyze. In order to reduce the uncertainty, it is essential to analysis sensitivity. The purpose of a sensitivity analysis is to quantify the uncertainty in the calibrated model. A sensitivity analysis is typically performed by changing the value of one parameter at a time. The widest range of plausible solutions can also be examined by changing two or more parameters. The procedure of calculating sensitivities can also be automated or can be done by stochastic modeling. During the sensitivity analysis, calibrated values of the most important hydrological parameters, such as transmissivities, are systematically changed within a (previous established) plausible range, e.g. by means of a coefficient of variation (standard deviation divided by the expected value). The sensitivity analysis of a parameter has its effect on relevant variables of the hydrologic process, such as head, solute concentration, sources of water to a pumping well, etc. VIII.6. Model verification In order to improve the confidence in the calibration of the model, the model has to be tested by using a second independent set of data. This is called the model verification or model validation. A model is verified when the verification targets are matched without changing the calibrated parameters. If it is necessary to adjust parameters during the model verification because the verification targets are not matched, the verification becomes a second calibration and other independent data sets should be needed until the verification of the model is performed [Essink, 2000]. VIII.7. Simulation Whereas the objective of calibration was to demonstrate that the calibrated model can reproduce measured hydrologic processes, the ultimate modeling objective Groundwater Modeling – Heru Hendrayana – 2012 – [email protected]

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is to produce a model that can accurately simulate or predict (future) conditions for which no data is available [Essink, 2000]. VIII.7.1. Short-term forecasting and prediction In this category can often rely on the laws of fluid mechanics and hydraulics. At the most, interpolation and short extrapolation occurs. The availability of test data provides a relatively good safeguard against misconceptions [Essink, 2000]. VIII.7.2. Hydrologic simulation In this category, at least an indirect testing on analogous empirical data is often possible. It is mentioned that a predictive simulation should not be extended into the future more than twice the period for which the calibration data are available [Essink, 2000]. VIII.7.3. Long-term forecasting and prediction In this third category, the possibility of testing is nonexistent. Some environmental problems require a length of time of many years, perhaps as many as 10,000 years. For example, as groundwater flow and solute transport are slow processes, a long simulation time of several centuries for large-scale coastal groundwater flow systems is not rare.

VIII.8. Presentation of results In this (final) phase, a report has to be completed. A good report is essential to the effective completion of the modeling study. A modeling report should contain the following elements: -

purpose of the model

-

formulation of the concept of the model

-

information about the computer code

-

model design

-

calibration and model verification

-

sensitivity analysis

-

results of the predictive simulations

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Mostly, results from model simulations are used by decision makers to plan the future changes in hydrologic processes. Therefore, the results of simulations should (somewhat) be adapted to the imaginative powers of the decision maker [Essink, 2000].

VIII.9. Postaudit Based on Essink (2000), postaudits is the verifications of the complete modeling result examine the accuracy of a prediction by a model which was executed a considerable number of years ago. Postaudits can help us to know how good can models predict the future (e.g. at least 10 years to allow adequate time for the hydrologic system to move far from the calibrated solution). Figure 17 below illustrated the steps of groundwater model methodology and application proposed by Anderson & Woessner (1992).

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Define purpose Field data

Conceptualization of mathematical model

Analytical solutions

Numerical formulation Computer program Code verification Verified?

CODE SELECTION

No

Yes

Model design

Field data

Calibration

Sensitivity analysis

Analytical solutions Model verification

Simulation

Sensitivity analysis

Presentation of result

Field data

Postaudits

Figure 17: Steps of groundwater model application [adapted from Anderson & Woessner, 1992]

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