Hydrogeology Concepts that help people memorize topics and formulae — Part I
Hydrogeology Topics and Formulae
Abstraction Rate
The aquifer above the stream of water is considered to fully penetrate through the long borehole.
The rate of flow going out of the borehole or abstraction rate is taken as Q, and the rate of stream of ground water or rate of stream flow depletion is taken as q.
Please watch the video on Deep water drilling.
The borehole is pumped by depleting the water from below. The stream depletion rate is given as:
Where tau is Square root of the product of:
tau = (t, T, L, S)
with coefficients = (1/2, 1/2, -1, -1/2)
termed, erfc = Error Function
t = Time, T = Transmissivity,
L = perpendicular distance of the borehole to the line of the river,
S = aquifer storage coefficient
Porosity
Consider metal spheres put inside a beaker of volume V. Then the Volume V can be written as:
V = Vs (Vol. of solid material)+ Vv (Vol. of voids)
Porosity and Void Ratio are related as:
Porosity is the fraction of given volume of material occupied by void space, or interstices.
Based on data contained in Freeze and Cherry 1979 and Back et. al. 1988, the hydraulic conductivity of a series of materials goes maximum up to 1 for Karstified Limestones and goes minimum up to 10^-13 for Glacial deposits of lacustrine silt and clay.
Hydraulic Conductivity and Intrinsic Permeability
Hydraulic conductivity refers to ease of movement of water through aquifer and porous material. It depends on physical properties of porous material and the properties of migrating fluid.
Intrinsic Permeability,
In here, Hydraulic Conductivity is a function of viscosity and specific weight; Intrinsic Permeability is dependent on the grain size in diameter (mm)
The Porosity here is a Volumetric Parameter and The Hydraulic Conductivity is a Flow Parameter
Hydraulic Conductivity impacts homogeneity and isotropy. Homogeneity refers to position independence and isotropy refers to directional independence.
Darcy’s Law and Darcy-Weisbach
Darcy’s Law: The flow rate is dependent on the cross-sectional area of the flow, hydraulic conductivity of the porous material, proportional to the difference in heads of water level and inversely proportional to the column length. The ratio is called as the hydraulic gradient.
Accurate calculation of groundwater flow and velocity is achieved using flow net analysis and groundwater modelling.
Fractured Rocks follow cubic equation: For the discrete fracture and dual-porosity models, the groundwater movement is dependent on the area which in turn is dependent on aperture of the fracture. It is also the square of the aperture that the hydraulic conductivity is dependent on which leads to Cubic Law of ground water flow:
Karst aquifers have two or three types of porosity:
— Intergranular pores in the rock matrix
— Common rock discontinuities such as fractures and bedding planes
— Solutionally enlarged voids such as channels and conduits
— Darcy’s Law Variations:
In materials with intergranular pores and in fractured porous materials, Darcy’s Law can successfully be applied lying within continuum scale. In a sub-continuum scale, where fracture dimensions change, we need to cater for the Cubic Law. In channels and conduits, we simulate the environment to reproduce the input and output responses. The empirical functions are developed based on field observations of flow.
Darcy-Weisbach: Average velocity in estimating peak discharge,
— Karst Variations:
In materials with Karst Variations: such as Field System, Single Fracture, Equivalent continuum and Dual Porosity, the representation of the hydraulic behavior of the system is complicated by difficulty in characterizing the hydraulic properties. One has to use Darcy’s Law and simulate the environment using field observations and field validation.
Average Velocity is also defined by discharge per unit area divided by porosity
Spring — Result of karst topography where there is a discharge from one end of the fissure to the other end which may open up as a spring.
Epikarstic Aquifer — Lies within the saturated zone of the superficial part of Karst that drains to the saturated zone and is recharged by infiltration.
Groundwater potential and Hydraulic Head
The fluid potential is equivalent to the sum of elevation head and the pressure head. The pressure head is written as the ratio of P and ρg. The formula is written as:
To be noted: The pressure head term is negative in the unsaturated zone. At the water level the water pressure is equal to zero (the atmospheric pressure).
Transmissivity, Storativity, Compressibility and Specific Yield
Transmissivity: The rate at which water content is transmitted from one end to another end separated by depth b, under unit hydraulic head.
Storativity: Within a depth of b, the amount of water (in volume) that is contained per unit surface area of aquifer with the decline along the hydraulic head component.
Hydraulic Diffusivity: Ratio of hydraulic conductivity and specific storage or alternatively termed as ratio of Transmissivity and Storativity.
Compressibility of Water and Compressibility of Aquifer: is termed as volumetric strain per the induced pore water-pressure change and the volumetric strain per the change in effective stress.
The central Valley of California observes subsidence of aquifer over years and it is also noted that when water in pumped up the aquifer subsides by an amount due to groundwater extraction. Areas of greater subsidence have been controlled by importing surface water.
Equations of Groundwater Flow
In steady-state saturated flow, the sum of partial derivatives of fluid mass flow into and out of a control volume (which is the product of fluid density multiplied by specific discharge across a unit cross-sectional area) is zero.
Using Darcy’s Law, one can show the specific discharge is the product of hydraulic conductivity and hydraulic gradient.
Hence, for an isotropic and homogeneous porous material such as the Geosynthetics
K(x, y, z) = constant
The equation becomes:
Transient Saturated: In transient flow where there the time dimension is equated to the spatial dimension, the mass rate of water produced by expansion of the water under a change in its density, controlled by compressibility and mass rate of water produced by compaction of the porous material (ρn) is equated to the rate of discharge of the control volume pq.
Transient Unsaturated: In transient unsaturated flow, the porosity is replaced by moisture content θ.
Using h = z +ψ,
Richards Equation becomes:
In Compaction driven fluid-flow, the pore fluid pressure increases and it drives fluid flow with the increasing weight of overburden. Using Terzaghi, the mechanical compaction of a pore network requires an increase of the effective stress. Also it is found from Bethke and Corbet 1988 that porosity is dependent on compressibility and effective stress by an exponential relation.
MODFLOW
Example Project:
This is a docker box which consists of Fortran and MODFLOW6 implementation in Linux. This is compiled according to the USGS website implementation.
Compiling Sample Fortran Code:
This is a sample hello world code on Fortran. The shell code given below compiles Fortran.
gfortran -o hello_world hello_world.f90
ModFlow Contour Lines:
More Details on this article:
Groundwater flow patterns
Flow net analysis is used to model ground water flow lines in the aquifers, aquitards and aquicludes. Freeze and Witherspoon developed mathematical approach to simulate the effects of topography and geology on the nature of regional groundwater flow patterns.
3 Types of Groundwater flow systems are: Local, Intermediate and Regional. They are caused due to a change in ratio of depth to lateral extent of the entire aquifer system. Water quality improves with greater aquifer depth sequentially layer by layer. Due to high sodium concentration, it is generally unsuitable for irrigation.
Variable-density driven fluid flow
Hydrostatic and Lithostatic flow: is dependent on permeability and compressibility of soil below ground.
Topography and Compaction driven flow: is dependent on Groundwater circulation pattern.
The layers of the Gulf of Mexico are: Quaternary, Pliocene, Miocene, Frio, Vicksburg, Upper Claiborne, Lower Claiborne, Lower Wilcox, Paleocene, Upper Cretaceous, Jurassic Lower Cretaceous
Gulf of Mexico basin is an example of actively subsiding sedimentary basin. Free-convection happens when the permeability of the aquifer, the density contrast between the seawater and the groundwater in the aquifer, and the thickness of the aquifer are large enough.
Rayleigh Number, Ra is given as:
In the analysis of hot springs over Borax Lake, thermal imaging led to significant discoveries. The fault lines, are connected indicating where one fault strand ends the fault displacement is transferred to a second fault trace. Also, along the fault trace the spring and ground temperatures are highly variable.
Higher surface temperatures indicate higher flow rate. Near boiling temperatures of springs sufficed with oxygen-isotope data indicate that recharge to the springs are meteoric.
Thermal convection is likely to be the primary mechanism to circulate groundwater from and towards the surface accompanied by more shallow topography-driven flow.
The field observations using the one dimensional heat flow model gives information on fault permeability distribution. A more detailed model with the description of flow driven by thermal convection with the variability of groundwater discharge rates as well as surface temperatures is also statistically equivalent to the field observations.
Salinity of water determines its stages of development in the coastal regions as well as aquifers. Hence the age can be determined by the presence of isotope. The process of salinizing aquifers, seawater and groundwater has been explained below.
Free convection would be the cause for salinizing aquifers where the seawater transgresses over an aquifer. When free-convection is not possible, the groundwater gets salinized by the process of molecular diffusion.
Subsidence in sedimentary basins, mountain building, sea-level fluctuation and glaciation can have profound impacts on the rates and direction of groundwater movement.
It is important to be noted that geological processes happen in millennia and longer and groundwater processes happen in a slow speed in response to boundary conditions.
Law of Mass Action
The relation between reactants and the products when the chemical reaction is at the equilibrium.
Adsorption occurs at the groundwater level and ion-exchange leads to the migration of heavy metals and polar organic chemicals. The process of ion-exchange can lead to changes in hydraulic conductivity of natural materials. For example, the Ca2+ ion is more strongly adsorbed than the Na+ ion by the negatively charged material such as clay and the amount of water surrounding the ions will thus be reduced. The energy absorption sequence is: Ca2+ > Mg2+ > K+ > Na+
Graphical Representation of hydrochemical data (Piper Plot)
piperPlot: Piper Diagram in USGS-R/smwrGraphs: Graphing Functions (rdrr.io)
A Piper Excel File in Google Drive:
Piper Diagram Plot using Microsoft R Open
# import library smwrGraphs, install the library using remotes
library("smwrGraphs")# read csv downloaded from the google drive excel file
df = read.csv("piper-Diagram.csv")# plot piper plot
piperPlot(
xCat = df['Ca'], yCat = df['Mg'],
zCat = df['Na...K'], xAn = df['Cl...other'],
yAn = df['Carbonate...Bicarbonate'], zAn = df['Sulfate']
)
Redox reactions have a controlling influence on the solubility and transport of some minor elements in groundwater such as Fe and Mn and also on redox sensitive species such as NO3 - and SO4 2-. Issues that are relevant to redox reactions are: (1) Groundwater Quality, (2) Landfill Leachate Plumes Attenuation and (3) Site Remediation.
Sequence of Redox Processes in a Closed System:
A Closed System describes the groundwater containing dissolved oxidized species and also excess dissolved organic carbon (DOC). A Closed System is closed to the input of further oxidants or oxidized species.
- Aerobic Respiration
- Denitrification
- Mn(IV) Reduction
- Fe(III) Reduction
- Sulfate Reduction
- Methane Reduction
- Nitrogen fixation
Sequence of Redox Processes in an Open System:
In an Open System, excess dissolved oxygen is present reacts with reduced species such as HS- and NH4 + (eg:- landfill leachate + groundwater).
- Aerobic Respiration
- Sulfide Oxidation
- Fe(II) Oxidation
- Nitrification
- Mn(II) Oxidation
Stable Isotope Chemistry
Concentrations of dissolved atmospheric noble gases as well as stable isotopes of water (16O, 18O, 1H, 2H) are found in groundwater which provide the information on aquifer evolution.
The isotopic abundances expressed in parts per thousand, 𝛿, gives us whether a rare, heavy isotope is present in the mixture by getting to know the plot of the rate in 𝛿.
In liquid state, i.e. water, the ratio of concentrations of heavy to light isotope is expressed as,
In vapor state the ratio of concentrations of heavy to light isotope is expressed as,
Fractionation Factor is expressed as the ratio of phase A to phase B:
Fractionation Factor (Liquid to Water Vapor):
Relation between isotopic precipitation and Tropical Storms:
The process of evaporation and condensation is repeated many times due to the water vapor moving inland across continental areas. The data on isotopic abundance on ground water indicate meteoric presence as well as is used to determine the age of the ground water hydrology process. Tropical Storms deplete isotopic abundance ratio whereas the absence of it improves the isotopic abundance ratio.
Part 2 with details of Hydrogeology topics and formulae will be published as soon as possible
