THE APPLICATION OF MATHEMATICAL MODELS IN MANAGEMENT OF AQUIFER
Groundwater Data Requirement and Analysis
1. Groundwater Data Requirement and Analysis
C. P. Kumar
Scientist ‘F’
National Institute of Hydrology
Roorkee – 247667 (India)
2. Outline
Introduction
Data Requirement for Groundwater Studies
Groundwater Data Acquisition
Processing of Groundwater Data
Interpolation of Hydrological Variables
Geostatistical Analysis using ArcGIS
Groundwater Data Management and Analysis Tools
4. Why should we devote resources for assessing
groundwater conditions?
Groundwater is a vital natural resource for our country…
• A major source of drinking water and irrigation water supply
• Groundwater baseflow sustains streamflow during low flow
periods
• Dependence on groundwater is rapidly increasing
• There’s a lot of stress on groundwater resource –
contamination, over-pumping…
5. Groundwater is an important, but often overlooked component of the
hydrologic cycle…
• Groundwater and surface water are in reality an interconnected
resource.
• Water management decisions that ignore the contributions of, or
impacts to, groundwater are not sustainable in the long run.
6. Accurate and reliable groundwater resource
information (including quality) is critical to planners
and decision-makers.
Huge investment in the areas of ground water
exploration, development and management at state
and national levels aims to meet the groundwater
requirement for drinking and irrigation and generates
enormous amount of data.
We need to focus on improved data management,
precise analysis and effective dissemination of data.
8. ALL GROUND-WATER HYDROLOGY WORK IS MODELING
A Model is a representation of a system.
Modeling begins when one formulates a concept of a
hydrologic system, continues with application of, for
example, Darcy's Law or the Theis equation to the
problem, and may culminate in a complex numerical
simulation.
9. The success of any groundwater study, to a large measure, depends
upon the availability and accuracy of measured/recorded data required
for that study.
Therefore, identifying the data needs and collection/monitoring of
required data form an integral part of any groundwater exercise.
The first phase of any groundwater study consists of collecting all
existing geological and hydrological data on the groundwater basin in
question.
Any groundwater balance or numerical model requires a set of
quantitative hydrogeological data that fall into two categories:
* Data that define the physical framework of the groundwater basin
* Data that describe its hydrological framework
10. Physical Framework
1. Topography
2. Geology
3. Types of aquifers
4. Aquifer thickness and lateral extent
5. Aquifer boundaries
6. Lithological variations within the aquifer
7. Aquifer characteristics
11. Hydrological Framework
1. Water table elevation
2. Type and extent of recharge areas
3. Rate of recharge
4. Type and extent of discharge areas
5. Rate of discharge
12. The data required for a groundwater flow modelling study under
physical framework are:
Geologic map and cross section or fence diagram showing the
areal and vertical extent and boundaries of the system.
Topographic map at a suitable scale showing all surface water
bodies and divides. Details of surface drainage system, springs,
wetlands and swamps should also be available on map.
Land use maps showing agricultural areas.
Contour maps showing the elevation of the base of the aquifers
and confining beds.
Isopach maps showing the thickness of aquifers and confining
beds.
Maps showing the extent and thickness of stream and lake
sediments.
These data are used for defining the geometry of the groundwater
domain under investigation, including the thickness and areal extent
of each hydrostratigraphic unit.
13. Under the hydrogeologic framework, the data requirements for a
groundwater flow modelling study are:
Water table and potentiometric maps for all aquifers.
Hydrographs of groundwater head and surface water levels.
Maps and cross sections showing the hydraulic conductivity and/or
transmissivity distribution.
Maps and cross sections showing the storage properties of the
aquifers and confining beds.
Spatial and temporal distribution of rates of evaporation,
groundwater recharge, groundwater pumping etc.
15. Some data may be obtained from existing reports of various
agencies/departments, but in most cases additional field work is
required.
The observed raw data obtained from the field may contain
inconsistencies and errors. Before proceeding with data
processing, it is essential to carry out data validation in order to
correct errors in recorded data and assess the reliability of a
record.
Amongst the hydrologic stresses including groundwater pumping,
evapotranspiration and recharge, groundwater pumpage is the
easiest to estimate.
Field information for estimating evapotranspiration is likely to be
sparse and can be estimated from information about the land use
and potential evapotranspiration values.
Recharge is one of the most difficult parameters to estimate.
16. Values of transmissivity and storage coefficient are usually
obtained from data generated during pumping tests and
subsequent data processing.
For modelling at a local scale, values of hydraulic conductivity may
be determined by pumping tests if volume-averaged values are
required.
In the field, in-situ hydraulic conductivity may be measured by
Guelph Permeameter.
For unconsolidated sand-size sediment, hydraulic conductivity may
be obtained from laboratory permeability tests using permeameters.
Laboratory analyses of core samples tend to give lower values of
hydraulic conductivity than are measured in the field.
17. Monitoring of Groundwater Levels
A network of observation wells and/or piezometers are established
to obtain data on the -
o Depth and configuration of the water table
o Direction of groundwater movement
o Location of recharge and discharge areas
In any drainage investigation, the highest and the lowest water
table positions, as well as the mean water table during a
hydrological year are important.
For this reason, water level measurements should be made at
frequent intervals. The interval between readings should
preferably not exceed one month.
All measurements in a study area should, as far as possible, be
made on the same day because this gives a complete picture of
the water table.
18. Monitoring of Groundwater Quality
The objectives of the water quality monitoring network are to:
o Detect water quality changes with time
o Identify potential areas that show rising trend
o Detect potential pollution sources
o Study the impact of land use and industrialization on groundwater
quality
Substantial costs are incurred to obtain and analyze samples.
Field costs for drilling, installing, and sampling monitoring wells
and laboratory costs for analyzing samples are not trivial.
Comprehensive data analysis and evaluation by a
knowledgeable professional should be the final quality assurance
step .
19. The frequency of sampling required in a ground-water-quality
monitoring program is dictated by the expected rate of change in
the concentrations of chemical constituents.
For monitoring concentrations of major ions and nutrients, and
values of physical properties of ground water, twice yearly
sampling should be sufficient.
More frequent sampling should be considered if the types and
conditions of any upgradient sources of these compounds are
changing.
Monitoring of ground-water quality should be a long-term activity.
21. Processing of Groundwater Data
Before any conclusions can be drawn about the cause, extent, and
severity of an area’s groundwater related problems, the raw
groundwater data on water levels and water quality have to be
processed.
This data then have to be related to the geology and hydrogeology of
the area. The results, presented in graphs, maps, and cross-sections,
will enable a diagnosis of the problems.
The following graphs and maps have to be prepared that are
discussed hereunder:
o Groundwater hydrographs
o Water table-contour map
o Depth-to-water table map
o Water table-fluctuation map
o Head-differences map
o Groundwater-quality map
A proper interpretation of groundwater data, hydrographs, and maps
requires a coordinate study of a region’s geology, soils, topography,
climate, hydrology, land use, and vegetation.
22. Groundwater Hydrographs
When the amount of groundwater in storage increases, the water
table rises; when it decreases, the water table falls. This response of
the water table to changes in storage can be plotted in a
hydrograph.
Groundwater hydrographs show the water-level readings, converted
to water levels below ground surface, against their corresponding
time.
A hydrograph should be plotted for each observation well or
piezometer. It is important to know the rate of rise of the water table,
and even more important, that of its fall.
July Aug Sept Oct Nov Dec Jan Feb Mar Apr May June
23. Groundwater hydrographs also offer a means of estimating the annual
groundwater recharge from rainfall. This, however, requires several years
of records on rainfall and water tables.
An average relationship between the two can be established by plotting
the annual rise in water table against the annual rainfall.
Extending the straight line until it intersects the abscissa gives the amount
of rainfall below which there is no recharge of the groundwater. Any
quantity less then this amount is lost by surface runoff and
evapotranspiration.
24. Water Table – Contour Map
A water table - contour map shows the elevation and
configuration of the water table on a certain date.
To draw the water table-contour lines, we have to interpolate
the water levels between the observation points, using the
linear interpolation method.
A proper contour interval should be chosen, depending on the
slope of the water table. For a flat water table, 0.25 to 0.50 m
may suit; in steep water table areas, intervals of 1 to 5 m or
even more may be needed to avoid overcrowding the map with
contour lines.
A water table-contour map is an important tool in groundwater
investigations because, from it, one can derive the gradient of
the water table (dh/dx) and the direction of groundwater flow,
which is perpendicular to the water table-contour lines.
25. The topographic base map should contain contour lines of
the land surface and should show all natural drainage
channels and open water bodies.
For the given date, the water levels of these surface waters
should also be plotted on the map. Only with these data
and data on the land surface elevation can water table
contour lines be drawn correctly.
26. For a proper interpretation of a water table-contour map, one has to
consider not only the topography, natural drainage pattern, and
local recharge and discharge patterns, but also the subsurface
geology.
More specifically, one should know the spatial distribution of
permeable and less permeable layers below the water table.
For instance, a clay lens impedes the downward flow of excess
irrigation water or, if the area is not irrigated, the downward flow of
excess rainfall. A groundwater mound will form above such a
horizontal barrier.
27. Depth-to-Water Table Map
A depth-to-water table map shows the spatial distribution of the
depth of the water table below the land surface. A suitable contour
interval may be 50 cm.
The regions of map where the groundwater level is between 0-2 m
depicts the area having drainage problems.
Based on measurement results for a year, the map drawn using
the lowest water table levels indicates to which extent the
groundwater falls in a year.
The section where the water table level is between 0-1 m
determines the areas in which groundwater exists in the root-zone
throughout a year.
The depth and shape of the first impermeable layer below the
water table strongly affect the height of the water table.
28. Water Table - Fluctuation Map
A water table - fluctuation map is a map that shows the
magnitude and spatial distribution of the change in water table
over a period (e.g. a season or a whole hydrological year).
A water table-fluctuation map is a useful tool in the interpretation
of drainage problems in areas with large water table fluctuations.
The change in water table in fine-textured soils will differ from
that in coarse-textured soils, for the same recharge or discharge.
29. Head - Differences Map
A head-differences map is a map that shows the magnitude and
spatial distribution of the differences in hydraulic head between
two different soil layers.
We calculate the difference in water level between the two
piezometers, and plot the result on a map. After choosing a
proper contour interval (e.g. 0.10 or 0.20 m), we draw lines of
equal head difference.
The map is a useful tool in estimating upward or downward
seepage.
30. Groundwater - Quality Maps
A groundwater-quality map (for example, electrical-conductivity
map) is a map that shows the magnitude and spatial variation in
the salinity of the groundwater.
The EC values of all representative wells (or piezometers) are
used for this purpose.
Groundwater salinity varies not only horizontally but also
vertically. It is therefore advisable to prepare an electrical-
conductivity map not only for the shallow groundwater but also
for the deep groundwater.
In electrical conductivity maps, critical groundwater salinity is
taken as 5000 micromhos/cm, although it changes according
to species of the crop to be grown.
31. By plotting all the EC values on a map, lines of equal electrical
conductivity (equal salinity) can be drawn. Preferably the
following limits should be taken: less than 100 micromhos/cm,
100 to 250; 250-750; 750 to 2500; 2500 to 5000; and more
than 5000. Other limits may, of course, be chosen, depending
on the salinity found in the waters.
Other types of groundwater-quality maps can be prepared by
plotting different quality parameters (e.g. Sodium Adsorption
Ratio (SAR) values).
The groundwater in the lower portions of coastal and delta
plains may be brackish to extremely salty, because of sea-
water encroachment.
In the arid and semi-arid zones, shallow water table areas may
contain very salty groundwater because of high rates of
evaporation. Irrigation in such areas may contribute to the
salinity of the shallow groundwater through the dissolution of
salts accumulated in the soil layers.
32. Interpretation of Hydraulic Head and Groundwater Conditions
Measurements of hydraulic head, normally achieved by the
installation of a piezometer or well point, are useful for determining
the directions of groundwater flow in an aquifer system.
In the above figure, three piezometers installed to the same depth enable the
determination of the direction of groundwater flow and, with the application of
Darcy’s law, the calculation of the horizontal component of flow.
33. In the above figure, two examples of piezometer nests are shown that allow the
measurement of hydraulic head and the direction of groundwater flow in the
vertical direction to be determined either at different levels in the same aquifer
formation or in different formations.
35. A fundamental problem of Hydrology is that our
models of hydrological variables assume continuity in
space (and time), while observations are done at
points.
The elementary task is to estimate a value at a given
location, using the existing observations.
36. Hydrological data have variability in space and time.
• Spatial variability is observed by a sufficient number of
stations
• Time variability is observed by recording time series
• Spatial variability can be in different range of values or
in different temporal behaviour
A continuous field v = v(x,y,z,t) is to be estimated from
discrete values vi = v(xi,yi,zi,ti)
37. Global estimation: characteristic value for area
Point estimation: estimation at a point P = P(x,y)
We need data AND a conceptual model, how these
data are related, (i.e. a conceptual model of the
process)
If the process is well defined, only few data are
needed to construct the model
38. Example
A groundwater table in a
confined, homogeneous, Q ⎛ r2 ⎞
h( r2 ) − h( r1 ) = ln⎜ ⎟
isotropic aquifer under steady 2πT ⎜ r1 ⎟
⎝ ⎠
state discharge from a well is
described by the Thiem well
formula.
Theoretically, the observation
of two groundwater heads at
different distances from the
well is sufficient to reconstruct
the complete groundwater
surface.
39. Hydrological variables are random and uncertain
Geostatistical Methods
Mostly 2D consideration v = v(x,y,t)
40. Regionalisation and Interpolation
Regionalisation: Identification of the spatial
distribution of a function g, depending on local
information as well as by transfer of information from
other regions by transfer functions.
Regionalisation therefore means to describe spatial
variability (or homogeneity) of
• Model parameters
• Input variables
• Boundary conditions and coefficients
41. Regionalisation includes the following tasks:
• Representation of fields of hydrological parameters and
data (contour maps)
• Smoothing spatial fields
• Identification of homogeneous zones
• Interpolation from point data
• Transfer of point information from one region to others
• Adaptation of model parameters for the transfer from
point to area
42. Interpolation
Given z = z(x,y) at some points we want to estimate z0
at (x0, y0)
z
z2
z0 ?
z3
z1
y
(x3,y3)
(x0,y0)
(x2,y2)
(x1,y1)
x
43. Weighted linear combination -
n
z = z0 =
ˆ ˆ ∑w z
i =1
i i
The methods differ in the way how they establish the
weights.
44. Global and Local Interpolation
An interpolation method is working globally, if all data
points are evaluated in the interpolation.
Local interpolation techniques use only data points in
a certain neighbourhood of the
estimated point.
y
y0 z0
r
x0 x
45. Deterministic or Statistical Interpolation
Deterministic methods attempt to fit a surface of given
or assumed type to the given data points
• Exact
• Smoothing
Statistical (stochastic) methods
46. Choice of Interpolation Method
Depends primarily on the nature of the variable and
its spatial variation.
Examples: Rainfall, groundwater, soil physical
properties, topography
47. Example: Groundwater Data
Groundwater tables have smooth surface, but trend!
Hydrogeological information is highly random, has
faults, few points with “good” data
48. Deterministic Interpolation Methods
Polynomials
Spatial join (point in polygon)
Thiessen polygons
TIN and linear interpolation
Spline
Inverse Distance Weighting (IDW)
49. Polynomials
n n
fs (x, y) = ∑ ∑
n
•General: f s ( x) = ∑ ci x i
cij xi y j
i =0 i =0 j =0
•Plane: f s ( x, y ) = c0 + c1 x + c2 y
•Second f s ( x, y ) = c0 + c1 x + c2 y + c3 x 2 + c4 xy + c5 y 2
Order:
n(n + 3)
nk = +1 f(x)
•Number of 2
coefficients
•Over- and
undershoots
x1 x2 x3 x4 x
53. TIN and Linear Interpolation
Surface is approximated by facets of plane triangles
Continuous surface, but discontinuous 1st derivative
42.0
#
45.0
36.0 #
#
50.0
#
#55.0
74.0 #
#82.0
#
# 70.0
65.0
54. Splines
Spline estimates values using a mathematical
function that minimizes overall surface curvature,
resulting in a smooth surface that passes exactly
through the input points.
Conceptually, it is like bending a sheet of rubber to
pass through the points while minimizing the total
curvature of the surface.
55. Inverse Distance Weighting (IDW)
N
z ( xi , yi )
∑ hβ
i =1
z ( x0 , y0 ) =
ˆ i ,0
hi , 0 = d i2, 0 + δ 2
N
1
∑ hβ
i =1 i , 0
Default method in many software packages β = 2
Controlled by exponent β
56. Stochastic (Geostatistical) Interpolation
Analysis of the spatial correlation in the random
component of a variable
Optimum determination of weights for interpolation
57. Semi-variance
Regionalized variable theory uses a related
property called the semi-variance to express
the degree of relationship between points on a
surface.
The semi-variance is simply half the
variance of the differences between all
possible points spaced a constant distance apart.
(Semi-variance is a measure of the degree of
spatial dependence between samples)
58. Semi-variance :The magnitude of the semi-
variance between points depends on the
distance between the points. A smaller distance
yields a smaller semi-variance and a larger
distance results in a larger semi-variance.
59. Calculating the Semi-variance (Regularly Spaced Points)
Consider regularly spaced points distance (d) apart, the
semi-variance can be estimated for distances that are
multiple of (d) (Simple form):
1 Nh
γ ( h) = ∑ (z − z ) i+h
2
2N
i
i =1
h
60. Semi-variance
1 Nh
γ ( h) = ∑ (z − z ) i+h
2
2N
i
i =1
h
Zi is the measurement of a regionalized variable
taken at location i ,
Zi+h is another measurement taken h intervals
away
Nh is number of points
61. Semi-variogram
The plot of the semi-variances as a function of
distance from a point is referred to as a semi-
variogram or variogram.
62. Semi-variogram
The semi-variance at a distance d = 0 should be zero,
because there are no differences between points that are
compared to themselves.
However, as points are compared to increasingly distant
points, the semi-variance increases.
63. Semi-variogram
The range is the greatest distance over which the value at a
point on the surface is related to the value at another point.
The range defines the maximum neighborhood over which
control points should be selected to estimate a grid node.
64. Characteristics of the Semi-variogram
(and therefore of the data)
Nugget: variance at zero distance which should be zero but isn’t…
Range: Distance at which max. variance is reached (data considered decorrelated)
Sill: Level of max. variability
Anisotropy might thus be manifested by varying range with direction (constant sill; so-
called geometric anisotropy). This is observed with the elevation data.
65. 1
Experimental semi-
variogram
γ * (h) = ∑=(hZ (u i ) − Z (u j )) 2
2 N ( h ) ui −u j
Things nearby tend to
be more similar than
things that are farther
apart
350 181
181 186
183
222
200
153
180
300 124 201
181
126 159 177
250 167
200
Variogram
80 84
86 106
150 2450
84
38
100
50
0
0 200 400 600 800 1000 1200 1400 1600 1800
Lag Distance
66. Theoretical semi-variogram: fit function through
empirical semi-variogram
350
300
250
Variogram
200
150
100
50
0
0 200 400 600 800 1000 1200 1400 1600 1800
Lag Distance
67. Variogram - Spherical
It is a ‘model’ semi-variogram and is usually called the
spherical model.
a is called the range of influence of a sample.
C is called the sill of the semi-variogram.
⎧ ⎛ 3 h 1 h3 ⎞
⎪C ⎜ − ⎟ where h ≤ a
γ ( h) = ⎨ ⎜ 2 a 2 a 3 ⎟
⎝ ⎠
⎪C where h ≥ a
⎩
68. Variogram - Exponential
γ ( h) = C 1 − e ( −h a
)
Spherical and Exponential with the Spherical and Exponential with the same
same range and sill sill and the same initial slope
69. Semi-variogram from ArcGIS
This is an example of
a variogram produced
using ArcGIS's
Geostatistical Analyst.
70. Interpolation by Kriging
Kriging is named after the South African
engineer, D. G. Krige, who first developed the
method.
Kriging uses the semi-variogram, in calculating
estimates of the surface at the grid nodes.
71. Interpolation by Kriging
Kriging goes through a two-step process:
1. Variograms and covariance functions are created to
estimate the statistical dependence (called spatial
autocorrelation) values, which depends on the model
of auto-correlation (fitting a model),
2. Prediction of unknown values
75. A linkage between GIS and spatial data analysis is considered to
be an important aspect to explore and analyze spatial relationships.
The GIS methodology for the spatial analysis of the groundwater
levels involves the following steps:
(a) Exploratory spatial data analysis (ESDA) using ArcGIS software for
the data (e.g. groundwater level) to study the following:
Data distribution
Global and local outliers
Trend analysis
(b) Spatial interpolation for data using ArcGIS software, while kriging is
applied by involving the following procedures:
Semivariogram and covariance modelling
Model validation using cross validation
Surfaces generation of the groundwater level data
78. Following are some of the software packages used for groundwater data
management and analysis.
AquaChem: AquaChem is an integrated software package developed
specifically for graphical and numerical analysis of geochemical data sets.
AquaChem also includes a direct link to the popular PHREEQC program for
geochemical modeling.
AquiferTest Pro: Graphical analysis and reporting of pumping test and slug
test data.
EnviroInsite: EnviroInsite is a desktop, groundwater visualization package
for analysis and communication of spatial and temporal trends in multi-
analyte, environmental groundwater data.
GW Contour: Data interpolation and contouring program for groundwater
professionals that also incorporates mapping velocity vectors and particle
tracks.
HydroGeo Analyst: Groundwater and borehole data management and
visualization technology.
RockWorks: Geological data management, analysis & visualization.
