A numerical model is a system of mathematical equations that depict the functioning of the system and its variables. A useful model must identify the factors that can influence the course of a particular decision and shows the impacts. A good model becomes a tool for decision making and predictions. A model helps in simulating the system for finding out some solutions . To simulate something physical, you will first need to create a mathematical model which represents that physical object. Models can take many forms including declarative, functional, constraint, spatial or multimodel.

1. MODELLING OF EARTH
SYSTEMS AND PROCESSES:
By
PROF.A.BALASUBRAMANIAN
Centre for Advanced Studies in Earth Science
UNIVERSITY OF MYSORE, INDIA

2. Decision making process needs
intelligent use of appropriate
tools.
The tools depend on the system
parameters, their relations and
expected solutions.

3. THE BEST TOOL IS
• A MODEL
• A MODEL IS A NUMERICAL
REPRESENTATION OF A REAL
SYSTEM CONSTRUCTED WITH
“SCALED-DOWN” PARAMETERS.
• IT IS A PROTOTYPE SYSTEM HAVING
ALL FEATURES OF THE REAL ONE.

4. TYPES OF MODELS:
• PHYSICAL MODELS- MATERIAL BASED
• OPTIMISATION MODELS- EQUATIONS
• EMPIRICAL MODELS -FORMULA
• ANALYTICAL MODELS- GRAPH
• STATISTICAL MODELS- FUNCTIONS
• NUMERICAL MODELS- EQUATIONS

5. NUMERICAL MODELS:
• A numerical model is a system of
mathematical equations that depict the
functioning of the system and its
variables.
• A useful model must identify the
factors that can influence the course of
a particular decision and shows the
impacts.

6. CHARACTERISTICS OF A
MODEL:
• A model must make certain assumptions about
the structure of the underlying problem
• A model will be a good replica of the system and
its parameters
• A good model becomes a tool for decision
making and predictions
• A model helps in simulating the system for
finding out some solutions

7. WHEN we DO NOT KNOW
THE ANSWER ?
• SIMULATE AND FIND OUT THE
ANSWER.
• TO SIMULATE A SYSTEM, THE FIRST
STEP IS TO DEVELOP A MODEL OF
THE SYSTEM AS A PROTOTYPE.
• CHECK ITS BEHAVIOUR,
SENSITIVITY, STABILITY AND
USEFULNESS.
• USE IT FOR SIMULATION.

8. MODELLING:
• Process of developing a model
• Understanding the system, its components,
processes, dependencies and role
• It involves 4 steps as:
• Formulation
• Approximating the system
• Transformation
• Calibration & testing
• Simulation

9. Types of simulations:
• Discrete event. The above sample path consisted of only
horizontal and vertical lines, as car arrivals and
departures occurred at distinct points of time, what we
refer to as events. Between two consecutive events,
nothing happens - the graph is horizontal. When the
number of events are finite, we call the simulation
"discrete event."
• In some systems the state changes all the time, not just
at the time of some discrete events. For example, the
water level in a reservoir with given in and outflows
may change all the time. In such cases "continuous
simulation" is more appropriate, although discrete
event simulation can serve as an approximation.

10. METHODOLOGY:
• CAUSE AND EFFECT
• TIME-ACTIVITY
• REACTION RATES
• INCREMENTAL ACTIVITY( RATE OF
CHANGE)
• TIME-DISTANCE-SPEED/ VELOCITY
• PROCESSES

11. Sub-fields:
• Within the overall task of simulation, there
are three primary sub-fields:
• model design,
• model execution and
• model analysis.

12. • To simulate something physical, you will
first need to create a mathematical model
which represents that physical object.
• Models can take many forms including
declarative, functional, constraint, spatial or
multimodel.

13. Let us try to
understand the
processes of
simulation !

14. What can we do with this
money?

15. SPEED, TIME OF START, MAINTENANCE
OF SPEED, OTHER FACTORS

16. •TIME
•WEIGHT
•AIR RESISTANCE ( UPWARD)
•ACC. DUE TO GRAVITY
•ALTITUDE

17. PLAYER- RUNNING SPEEDPLAYER- RUNNING SPEED
TRACKTRACK
ARRIVAL TIME OFARRIVAL TIME OF
PREVIOUS PLAYERPREVIOUS PLAYER
CUMULATIVE RUNNINGCUMULATIVE RUNNING
SCHEDULESCHEDULE
4 X 400 m RELAY

18. Traffic
simulators:
timer
timer
timer

19. Best shortest path:
VENUEVENUE
AIRPORTAIRPORT

20. WATER SUPPLY WELL:
SATURATED ZONESATURATED ZONE

21. WELL-FIELD PUMPING :WELL-FIELD PUMPING :
GROUNDWATER SYSTEM
UNDER STRESS

22. EARTH’S SYSTEMS:
• Earth is a dynamic evolving system
• Earth’ surface features undergo a lot of
changes due to natural or man-made
activities.
• There are endogenic( internal) and exogenic
processes which are active on earth
• Earth’s systems include atmosphere,
hydrosphere, lithosphere and biosphere

23. Earth system models:
• Atmospheric models
• hydrologic models
• hydrodynamic models- coastal
• geodynamic models- lithologic
• earth’s surface process models
• geotechnical models
• geo-environmental models

24. Atmospheric models:
• Aerodynamics
• Wind tunnels
• weather forecast
• particulate movements
• gaseous cycles
• Regional climatic models(RCM)

27. Hydrologic models:
• Surface water flow models
• river water quality models
• Groundwater flow models
• Solute transport models
• Radio-nucleide transport models
• aqueous geochemical models
• water management models

28. Hydrodynamic models:
• Coastal zone dynamics-
– littoral currents, long-shore currents
– sediment transport models
– depositional environment models
– hydraulic wave models

29. Geodynamic models:
• Solid earth simulators
• Earthquake simulation
• geo-thermal models
• geophysical models - gravity, magnetic,
paleo-magnetic
• tectonic models

34. Earth’s Surface process models:
• Drainage network simulators
• Soil erosion and soil loss
• coastal erosion
• siltation and sedimentation
• glacial erosion and melting
• floods

35. Erosional cycleErosional cycle

36. Geotechnical models:
• Landslides, land-subsidence
• stability of slopes,
• hill-slope processes
• Dams, Tunnel- seepage
• strength of earth’s materials
• reservoir regulations

37. Geo-environmental Models:
• Ecosystem models- estuarine, lake, rivers
• contaminant transport models
• Resource management models
• energy flow models
• bio-geochemical cycles
• biosphere-geosphere interaction models

38. OVERVIEW OF
NUMERICAL MODELS IN
HYDROLOGY

39. • deals with the occurrence, availability and
movement of water allover the globe above
and below the surface
• The presence or absence of water in earth
segments is controlled by several
hydrological processes and parameters
The Science of Hydrology

40. HYDROLOGICAL PARAMETERS
• vary with reference to both space and time
• it varies with reference to the age of
formations.
• The inter-relationship of several physico-
chemical, eco-biological and geo-
environmental matrices are governed by a
lot of well established mathematical
principles and numerical concepts.

41. HYDROLOGIC CYCLE:

42. Groundwater:
• Is present in small pore spaces of weathered
rocks & fractures of hard rocks, shallow or
deep, hills or lowlands

43. Groundwater is
prone to be affected by pollution

44. NOTABLE ASPECTS :
Natural flow patterns of surface and groundwater,
• Interdependence of several geochemical and
biochemical parameters in a pond, lake, river,
estuarine and other coastal ecosystems
• Chemistry of groundwater in different geological
media
• Trend of variations among the meteorological
elements and their impact on the local, regional
and global water resources
• Movement of pollutants through soil, water, and
other matrices

45. • Flow of water in pipes/rivers, porous
media, fresh and sheared rocks are
unique and governed by typical
PDEs.
• Hence, there are several possibilities
to evolve mathematical models for
analyzing these systems.

46. Mathematical models help in
order to
• evaluate the existing system ,
• generate new ideas,
• test new applications / approaches
• identify the problem areas and
• to reduce the cost of adhoc
experimentation.

47. • A digital model is a computer program that
numerically solves the rate equations by
stepping through a series of time-steps
during which the state variables are
updated.
• Several numerical models have been
developed throughout the world for
analyzing the dynamics of the water
resources environment.

48. FAMILIES OF MATHEMATICAL
MODELS IN HYDROLOGY
• Hydraulic and hydrologic models
• Water quality models
• Models of heat/solute transport in rivers
• Groundwater flow models
• Groundwater contaminant transport models
• Sea-water encroachment models and
• Geo-chemical models.

49. TYPES OF MODELS
•PREDICTION
•IDENTIFICATION
•MANAGEMENT

50. FLOW MODELS
PREDICTION
SINGLE PURPOSE MULTI PURPOSE
DEFORMATION
MODELS
MASS
TRANSPORT
MODELS
HEAT
TRANSPORT
MODELS

51. • Identification models help in estimating the
unknown system parameters, using the
known parameters
• Management models help in evolving
management plans based on simulation runs
with a precise model

52. HYDRAULIC MODELS
( FLOW IN OPEN CHANNELS AND RIVERS )
• quality of water flowing in any river
• (river water thermal dynamics), DO models
• Tidal Hydraulic models
• The hydraulic parameters which control the
quality of flowing water are the flow rate ,
flow type , suspended and bed load
sediments, channel slope, channel width
and depth of water column.

53. • The basic theory used to describe this flow
involves i) the equation of continuity and ii) the
equation of motion.
• These equations permit the evaluation of spatial
and temporal distribution of water flow rate,
velocity and depth by adopting the concept of
mass-balance.
• The application of these equations with
appropriate data could yield the changes and
dynamic characteristics of flow in channels.

54. HYDROLOGIC MODELS
• The hydrological processes of a river basin ,
movement of water or chemical in the soil
horizon and along the river channels are
controlled by many interdependent physical
phenomena in the earth’s spheres.
• River models, Ecosystem models- estuarine
models, snow melt models, Reservoir
models,

55. • There are two categories of hydrological
models as one dealing with the quantity of
surface flows and the other one deals with the
movement of chemicals and sediments along
with the flow of water.
• The models of the first category consider the
long term mean values of precipitation,
temperature, evaporation and runoff.
• They are mostly similar to time series
models, capable of forecasting the runoff
from rainfall.

56. SURFACE WATER
QUALITY MODELS
• Rivers are dynamic ecosystems where the
quality of water is also controlled by a large
number of variables like Biochemical
Oxygen Demand(BOD), Dissolved Oxygen
(DO), Total Dissolved Solids(TDS),
presence of bacteria, amount and nature of
industrial wastes and temperature(T).

57. GROUNDWATER FLOW
MODELS
• Groundwater models are simplified
representation of large and real
hydrogeologic systems like river basins or
watersheds.
• GWM is attempted to analyse the
mechanisms which control the occurrence
and movement of groundwater and to
evaluate the policies, actions and designs
which may affect the systems.

58. • These models are less complex prototypes of
complex hydrogeologic systems developed
using spatially varying aquifer parameters,
hydrologic properties, geologic boundary
conditions and positions of withdrawal wells
or recharging structures.
• These are designed to compute how
pumping or recharge might affect the local
or regional groundwater levels

59. • These models solve the basic partial
differential equations that govern the flow of
groundwater and solute transport through the
saturated and unsaturated porous medium.
• Models solve the equations analytically for
simple geometric problems and applies
numerical simulation to solve the equations
of more complex hydrogeological problems
involving aquifer heterogeneities,
anisotropic aquifer properties and
complicated boundary conditions

60. Categories:
• Flow and solute transport problems vary in
2 or 3 dimensions.
• The solution strategies may adopt
techniques like finite difference, finite
element and integrated finite difference
approaches.

61. FLOW MODELS
MULTI PHASE –
(IMMISIBLE))
SINGLE PHASE
(MISSIBLE)
LUMPED DISTRIBUTED
COMBINED
SUBSURFACE
—SURFACE
FLOW
UNSATURATED
FLOW
COMBINED
UNSATURATED
-SATURATED
FLOW
SATURATED-
FLOW
1-DIM
VERTICAL
2-DIM
HORIZ /
VERTICAL
FULLY
3-DIM

62. SATURATED FLOW
HYDRAULIC (DUPUIT’S
APPROXIMATION)
HYDRODYNAMIC
SINGLE
AQUIFER
SINGLE
AQUIFER
2 DIM
VERTICAL
FULLY
3 DIM
AX1
SYMMETRIC

63. Groundwater modelling requires
• the following domain specific information:
• physical units,
• hydrologic conditions,
• aquifer parameters ,
• time varying inputs and
• boundary conditions.

64. Fundamental equations:
• Two-dimensional case:
• d (Tx - dh) + d (Ty dh) = S dh + w(x,y,t)
• dx dx dy dy dy
•
• Three-dimensional case:
• d (Tx - dh) + d (Ty dh) + d (Tz dh) = S dh + w(x,y,z, t)
• dx dx dy dy dz dz dy

65. APPROXIMATION:
• Finite Difference method
• Finite element method
• Integrated finite difference method
• Boundary integral method
• Random walk method
• Method of characteristics using FD/FE

74. Solution strategies:
• Implicit
• Explicit
• Alternating direction explicit
• Alternating direction implicit
• Iterative alternating direction implicit
• SOR

75. Aquifer types and issues:
• Homogeneous, Heterogeneous, Single layer, multi-
layered, confined , unconfined, leaky, Lateral and
vertical boundaries
• Recharge / barrier / closed / fixed
• Flow- steady state, transient, radial, linear, bilinear
• Pumping schedules, no, loc, rates ?
• Fully/partially penetrating, small/large dia
• single /double porosity( Barrenblatt’s), fractured/
sheared, Arbitrary fracture networks, subsurface
barriers/ dykes

76. Macros and Micros:
• Basin models
• Watershed models
• Micro-watershed models
• Home aquifer system models

77. Recent Concepts:
• Dual porosity medium models
• Discrete Fracture Network Models
(Geometry-based, orientation of fractures)
• Parallel plate flow models ( single fracture)
• Simulation models for soil water, well head
protection
• Animations

78. CONTAMINANT
TRANSPORT MODELS
• analyse the movement , mixing and
chemical reactions of various anthropogenic
pollutants entering into the groundwater
system.

79. The three major processes are:
• Movement due to groundwater
flow(advection or convection),
• Mixing of groundwater and an
effluent(hydrodynamic dispersion)
• Chemical reaction as ; (a)
conservative transport(models which
do not consider the chemical reactions
(b) non-conservative transport
(models which consider the chemical
reactions).

80. MASS TRANSPORT
MODELS
LUMPED PARAMETER DISTRIBUTED
PARAMETER
UNCOUPLED COUPLED
CONSERVATIVE
TRANSPORT
NON-
CONSERVATIVE
TRANSPORT
RANDOM WALK
MODELS

81. NON-CONSERVATIVE
TRANSPORT
ABIOTIC
PROCESSES
BIOTIC
PROCESSES

82. HEAT TRANSPORT MODELS
UNCOUPLED COUPLED
SINGLE PHASE MULTI PHASE
AQUIFER THERMAL ENERGY STORAGE MODELS
RADIACTIVE WASTE ENERGY DISSIPATION

83. SEA-WATER ENCROACHMENT
MODELS
• In coastal areas, freshwater will overlie the
saltwater because of the difference in
density.
• A boundary surface between these two
fluids is known as saltwater-freshwater
interface. Both these fluids will be normally
in a state of equilibrium.

84. • Due to pumping of fresh groundwater the
hydrodynamic balance will be disturbed and the
saltwater moves inland until a new equilibrium
is established.
• Conversely an increase in freshwater flow in
the aquifer flow will push the interface
seawards.
• This movement is controlled by density of
freshwater & saltwater, rate of recharge and
discharge, extent of aquifer media, their
hydrologic parameters like porosity,
permeability, thickness and dispersion
characteristics.

85. Geochemical Models:
• interpret and predict the
chemical reactions of minerals,
gases and organic matter with
aqueous solutions in real or
hypothetical water-rock
systems .

86. Geochemical models aid to
• identify geochemical processes that
regulate the concentration of dissolved
constituents and
• quantify the effects of temperature,
speciation, sorption and solubility on the
concentrations of dissolved constituents.
• Ion activity of water , solubility of mineral
species and saturation indices

87. • a thorough understanding of the system,
determination of the system parameters,
equations,
• solution strategies,
• computer coding,
• database creation,
• calibration, sensitivity analysis, simulation
and applications.
Any Modelling application
requires

88. WHERE TO BROWSE?
• www.ubmail.ubalt.edu/~harsham/simulatio
n/sim.htm
• statistics & probability for simulation
• descriptive simulation modeling
• sensitivity, optimization
• meta-modeling and goal seeking
• what-if analysis & decision making
resources

89. Thank You !