Mathematical simulation models and modeling with software and calculators
The home page of this website is waterlog.info.
Totally free download.

Mathematical models are a description of natural phenomena in mathematical terms. They can be descriptive or meant for simulation to predict the occurrence in the future or in different conditions.
The models can be statistical using existing data, or simulative, that is based on equations developed for physical or chemical relations between magnitudes.

1. Statistical models

Statistical modeling is divided into two groups: regression analysis and frequency analysis.

1a. Regression analysis

Regression analysis is meant to detect the mathematical relation between two or more variables. Variables are the different outcomes of the measurement of a phenomenon. If one wishes to detect the influence of one variable on the other, then there must be an independent (or explanatory or predictive) variable, which is normally taken as the X-variable, and a depentent (or response) variable that reacts on the value of X. It is usually indicated as the Y-variable.
The relation between Y and X can be linear (a straight line) or curved (non-linear). A curved relation can sometimes be described by two different straight lines in two different segments in which the X-variable is divided. This is called segmented regression.
          A computer program (software, app, calculator) for segmented regression can be found on the SegReg page. Here on can see the various combinations that can occur with segmented regression. Apart from the standard version there is also a version that gives the option to perform non-linear regression like quadratic (parabolic) regression or S-curve regression.
          If one is intested in a segement where the X-value has no effect on Y and a segment where the effect is clearly present then on can resort to the PartReg page.

1b. Frequency analysis

Frequency analysis is done to detect how often a certain event occurs. The results can be presented in a histogram. A histogram may show that extreme values occur seldom. It is possible to fit a probabilty distribution to the data so that the frequency of events is given in a mathematical expression, which can be called frequency model.
          Software and calculator for distribution fitting can be freely downloaded from the CumFreq page where the computer application is found. It uses 20 different proability distributions with different skewnesses (skew to left, skew to right, or symmetrical).

1c. Normal distribution calculator

The normal probability distribution consists of magnitudes and frequencies/probabilities. The cumulative probability cannot be calculated analytically but it needs a numerical method.
          Software and calculator for the normal distribution can be freely downloaded from the NormDis page where the computer application is found.

1d. F-test calculator

Fisher's F-test is used in variance analysis (Anova) to determine the significance of the difference between the deviations of data from models and to find the model with the best fit.
          Software and calculator for F-distribution can be freely downloaded from the F-test page where the computer application is found.

1e. T-test calculator

Student's T-test is used to determine the significance of the difference between mean values of data series.
          Software and calculator for F-distribution can be freely downloaded from the T-tester page where the computer application is found.


2. Simulation models

The simulation models provided freely on this website are divided into three groups: hydrologic models, soil salinity models, and drainage system models.

Hydrological modelling can be done for surface hydrology with respect to runoff of rainfall and conversion into streamflow in rivers, groundwater flow or groundwater hydraulics of the aquifer whereby the availability of groundwater for domestic or agricultural use (irrigation) is assessed

Soil salinity modeling is done to assist with the solution of soil salinization problems that frequently happen in irrigated land. The mathematical techniques can focuss on long term soil conservation or on short term leaching practices for reclamation and improvement. In addition, the influence of the aquifer behaviour need sometimes to be taken into account to check the displacement and flow of groundwater from the higher areas to the lower lands that may cause waterlogging and salinity problems.

Drainage system modeling is necessary to design drainage systems for the alleviation of waterlogging problems and high, elevated, watertables. The hydraulic conductivity of the soil plays here an inportant role. The drainage can de done by ditches, horizontal undergound burried pipedrains, or by vertical tubewells equipped with pumps. It is important to know whether te aquifer is unconfined or semi-confined.


2A. Hydrological models

2Aa Surface hydrology

Surface hydrology concerns the transformation of pricipitation into surface drainage and stream flow. The conversion process can be simulated using a reservoir model in which the discharge is proportional to the storage of water. One can also use a linear reservoir or a cascade of reservoirs. The software model can be downloaded freely from the Rain-Off page.

2Ab Groundwater hydrology

The groundwater flow can be calculated in a polygonal network where the water flows from one polygon to the other. When the water level or the artesian pressure in a polygon is high, the flow will be away from it. A computer program for a mathematical groundwater model can be seen as software in the SahysMod page. In fact, SahysMod also deals with salt balances, see below.


2B. Soil salinity models

2Ba Short term leaching

For leaching of the soil and the calculation of salt balances from day to day on may use the LeachMod model that takes into account leaching efficiency and capillary rise as wel as the salinity of the irrigation water.

2Bb Long term soil conservation

For annual or perannual stability of the soil salinity and the depth of the water table in irrigated land to obtain a long term equilibrium at a safe level the use of the SaltMod program is recommended. It handles both the rootzone and the aquifer.

2Bc Groundwater interaction

For the influence of groundwater flow on the salinity of the soil and depth of the water table, on can make use of a calculation application that employs SaltMod in a number of polygons whose aquifers are interconnected. Upward seepage of groundwater can thus be detected. The corresponding model can be seen at the Sahys-Mod page.


3. Drainage models

3a. Horizontal subsurface drainage

Horizontal drainage is drainage by ditches or burried subsurface drain pipes. The model is based on the drain spacing equation of Hooghoudt adjusted to sloping land, entrance resistance and energy balance. The software, which is an application of a computer program, is free for download, see the En-Drain page

3b. Vertical drainage by wells

Vertical drainage can be done placing tube wells down into the underground while istalling pumps to move the water up and away to the outlet. The groundwater flow is radial to the wells and needs a well spacing equation for evaluation. The dimension of the well field can be computed by the app Well-Drain