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In this article we will discuss about:- 1. Introduction to Ground Water Basin Management 2. Conjunctive Use of Ground Water 3. Mathematical Modelling of a Dual Aquifer System 4. Mathematical Model for a Basin 5. Finite Element Method.

**Introd****uction to Ground Water Basin Management: **

**For optimum development of water resources of any basin and their management, the step by step studies to be made and data to be collected are given in the following: **

(a) Identify the basin boundary, the main river and its tributaries and other physiographic features.

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(b) Divide into sub-basins of controllable size depending on factors like steep hill slopes, forest areas, irrigated and unirrigated lands, fallow areas, etc.

(c) Establish a hydro meteorological set-up for each sub-basin.

(d) Select a convenient base period for the hydrologic equation.

The hydrologic equation simply states that all water entering a river basin or sub-basin during any period of time should either go into storage within its boundaries or leave the basin during the same period and a water balance is obtained. The different components of inflow to and outflow from the basin and the method of estimation of each item is given in Table 16.1.

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**A base period is selected so as to: **

(i) Allow direct determination of as many items in the equation as possible.

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(ii) Cover a length of time during which the investigator has reasonable confidence that the items used reflect truly average conditions.

(iii) Eliminate some items that are negligibly small during the period selected.

If the base period does not represent long time mean climatological conditions or the proper stage, present or future, of water use for which an answer is being sought, adjustments have to be made.

(e) Apply the water balance equation for the basin for the period selected as in item (d) above. The components of water balance equation and methods of their estimation are given in Table 16.1.

(f) Determine average depth of rainfall over the basin or sub-basin by Thiessen polygon or isohyetal methods.

(g) Draw stage and discharge hydrographs at control points.

(h) Draw cropping pattern maps for different seasons and estimate evapotranspiration. Evapotranspiration figures for different crops may be obtained either by field experiments or by climatological data. Even open-pan evaporation data are helpful in this regard.

(i) Determine evaporation from soil and water surfaces.

(j) Draw soil map of the basin or sub-basin and conduct infiltration studies in different soils and irrigated and unirrigated lands.

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(k) Determine isobath and ground water level contours monthly or at least for two different seasons, i.e., driest (LWL) and monsoon (HWL) seasons.

(I) Correlate hydrographs of monthly rainfall, river stage and discharge and ground water levels.

(m) Change in ground water storage = change in GWL × involved area of the aquifer × specific yield in the case of water table aquifer or storage coefficient for confined aquifer.

(n) Determine specific yield in the laboratory and storage coefficient by pumping tests in the field.

(o) Directions of flow can be determined from water table (or piezometric surface in the case of confined aquifers) contours.

(p) Arrive by water balance the monthly ground water accretion.

(q) Construct plots of monthly rainfall, ground water levels and cumulative change in ground water storage for period selected.

(r) The usable capacity of the ground water reservoir can be developed by planned extractions of ground water during periods of low precipitation while subsequent replenishment can be made during periods of surplus surface supply.

(s) The ground water investigation team is mainly concerned with location of sites of discharge (borewells) and recharge and design of suitable recharge facilities; and also to prevent sea water intrusion in coastal aquifers.

(t) Aerial and infra-red photography, electrical resistivity surveys and well logging techniques can provide valuable information in regard to item(s) above. The US Geological Survey, by the use of an infra-red scanner, has published an atlas of Hawaii’s coastal areas, pinpointing the location of underground fresh water flows.

Thus with the hydrometeorological data combined with geophysical and hydrogeological investigations, test drilling and pumping tests, it is possible to develop and manage the ground water resources of a basin.

For example- in Israel the success ratio in ground water drilling has been raised from 35 to 85% with the help of geophysical surveys.

Conjunctive utilisation of river-aquifer systems, mathematical and economic models of alternative well field configurations, pumping patterns, cropping patterns and irrigation sequences must be studied in detail before arriving at a final design.

Methods of systems analysis techniques and computer applications provide a better insight for the problems of ground water management. An excellent example of the application of the computer is the study of ground water conditions in southern California.

**Conjunctive Use of Ground Water****: **

Optimum development of water resources can be achieved by the conjunctive use of surface and ground waters. Ground water recharge occurs in nature by seepage from canals and reservoirs and return flow from irrigation. It can be augmented by artificial methods such spreading of storm water in ponds or basins, recharge wells, pits and shafts.

The usable capacity of the ground water reservoir can be developed by planned extractions of ground water during periods of low precipitation while subsequent replenishment can be made during periods of surplus surface supply.

Such a coordinated operation of surface and ground water supplies is possible if there is sufficient ground water storage to meet the requirements for regulation of local and imported water supplies and if the aquifers possess sufficient transmissibility to permit the movement of recharged water to the area of extraction. Also the underground storage is devoid of losses due to evaporation, quality deterioration due to pollution, etc.

Their reduction from danger of destruction of reservoir structures and wide dispersion of outlet facilities in earthquake areas, in places liable for atomic attacks, make ground water basins of inestimable value as an emergency supply.

Large ground water reservoirs thus developed not only meet the deficiencies of the surface supplies in seasons of drought but also supplement them to a large extent. These conjunctive operations result in a more economic yield as they provide more water at a lower average cost. Tubewell schemes can be integrated with the canal irrigation scheme by suitably spacing them along the drainage lines in the distribution area.

**The benefits accruing from the conjunctive use of waters are: **

(i) A large sub-surface storage at a relatively lower cost and safe against any risk of dam failures.

(ii) Provides water supplies during a series of drought years while a surface storage can at the most tide over one such year.

(iii) Efficient water use from well-spaced wells due to smaller surface distribution system than a canal irrigation scheme.

(iv) Water table can be controlled by pumping from wells and prevent water logging in canal irrigated areas and reduce land subsidence due to reduced ground water levels particularly in confined aquifers.

(v) Both water conservation and flood protection can be achieved simultaneously.

(vi) A sub-surface scheme can be developed in a shorter period while it takes 10-15 years for the completion of a big surface water project.

(vii) No evaporation and percolation losses, thus obviating the construction of expensive storm and seepage drains.

(viii) In project under conjunctive use of waters, tubewell loads can be reduced by releasing surface water for irrigation during periods of peak power demand thus resulting in lower power costs.

(ix) Crop water requirements can be ensured right through the year using surface water during the monsoons and ground water supplies when the surface water is not available.

(x) Ground water and surface water can be mixed in proper proportions to obtain a desired water quality for irrigating certain crop types (particularly when the ground water has a higher salt concentration); when the ground water has a higher salt concentration only certain salt tolerant crops can be grown.

(xi) Integration of the two types of schemes can be obtained with the existing water resources without loss of earlier investment.

Figure 16.3. broadly depicts various aspects of system approach for planning conjunctive use of surface and ground water resources.

**Mathematical Modelling of a Dual Aquifer System****: **

A typical grid network used in mathematical modelling of a dual aquifer system concept in alluvial valleys is shown in Fig. 16.4. A multi-layered mathematical model of the Gujarat area was constructed and tested by UNDP. Results have provided guidelines for optimum development of ground water resources with due considerations and artificial recharge.

Digital computer models are currently being used as a management tool in the water resources field whereby the actual field situation is recreated mathematically and proposed changes made after studies on computer outputs before implementation in the field.

**Mathematical Model for a Basin****: **

If Q is the net Inflow-Outflow to the system per unit area (draft, subsurface flow, recharge due to precipitation, etc.), Eq. 4.65 can be written in the form,

For two dimensional flow, h being the height of water table above a datum.

In the Tyson-Weber scheme, Eq. (16.1) is replaced by an equivalent system of difference-differential equations, the simultaneous solution of which gives the wanted function ^{‘}h’ at a finite number of node points lying within the boundaries of the aquifer. The basin is subdivided into a number of polygonal areas, each having a node point, which is the control point for the polygon, and all inputs and outputs to the polygon are assumed concentrated at the node. For a polygon associated with a typical node B in Fig. 16.5, the difference-differential equation can be written as-

Where i,

B = adjacent continuous nodes and node in question

A_{B} = area of polygon associated with node B

W_{iB} = length of perpendicular bisector associated with nodes i and B

T_{iB} = transmissibility at midpoint between nodes i and B

L_{iB} = distance between nodes i and B

Y_{iB} = conductance of path between nodes i and B

S_{B} = storage coefficient of polygonal zone associated with node B

Q_{B} = net volumetric flow rate per unit area at node B

h_{i}, h_{B}= water table elevations at nodes i and B

Superscripts j, j + 1 represent continuous points along the time axis, i.e., t^{J+1} = t^{J} + ∆t.

Equation (16.2) states that-

If the polygon borders the basin boundaries, any subsurface flow crossing the boundary is usually included in A_{B}Q_{B}.

The system of Eq. (16.2) can be solved on a digital computer. Initial value of water table elevations is impressed at each node. The flows, subsurface storage and extraction are balanced at each node by setting their sum equal to the residual term for any time step. Water table elevation at the node is then adjusted by the magnitude of the residual attenuated by a relaxation coefficient given by-

When all the water table elevations have been adjusted, the sum of nodel residuals is formed and compared with an error criteria. The calculations are repeated till the sum of these residuals become ≤ ԑ the permissible error. At this stage the calculations for that time step are complete. These values then become the initial water level elevations for the next step in time. By comparison of the water table elevations thus computed with the historical records, the model is verified. Then the model can be used to predict the future response of the basin for varying inputs, extractions, etc.

**The case history of a mathematical model for the Varuna Basin, U.P. to determine the average annual recharge and water balance available for the year 1972-73 for which water table elevation records were available, is given in the following: **

**Ground Water Balance Study of the Varuna Basin during 1972-73 (Fig. 16.6): **

The hydrologic balance equation for ground water, considering long term averages can be written as-

Where,

R = Recharge into ground water due to rainfall; irrigation water percolating down and seepage from canals.

I_{Q} = Inflow into the basin from other basins

S_{1} = Influent seepage from streams

O_{Q} = Outflow from basin to other basins

S_{E} = Effluent seepage from streams

E_{t} = Evapotranspiration from the region in direct contact with the aquifer

D_{Gw} = Draft from ground water.

GWS = Change in ground water storage of the aquifer

**Area of basin = 258250 ha; a.a.r. = 79.4 cm. Applying Eq. (16.4) to the Varuna basin, U.P. for the period of June to October 1972:**

Which corresponds to 32% of the a.a.r of 79.4 cm (205,000 ha-m); this is the same as that given by the empirical Amritsar formula, i.e., R_{r} = 31.2%.

**Applying this water balance equation, Eq. (16.4), for the period of November 1972 to May 1973: **

**Development of the Mathematical Model:**

(i) The formation constants S and T were determined by the pump test data using the Boulton’s method (for an aquifer with delayed yield): S = 0.0963; T = 1240 m^{2}/day, these values were used for the entire basin.

(ii) The entire basin was divided into 25 Thiessen polygons, Fig 16.6, such that the water levels at the nodes were known for some period: The six rain gauge stations in the area are shown in Fig. 16.6 and Thiessen polygons were drawn for these stations also.

(iii) The canal command area was delineated.

(iv) For each polygon A_{B}, W_{iB}, L_{jB} were measured and Y_{iB} determined using T = 1,240 m^{2}/day.

(v) The inputs to various nodes were due to precipitation and canal input. The precipitation at each rain gauge station being known for different months, the average value for each polygon associated with any node was worked out on the basis of proportion of the area of that node under the influence of a given rain gauge station (with the help of Thiessen polygons for rainfall). The canal input was assumed zero for polygons other than those falling in the canal command. The input was distributed uniformly over the total area of those polygons. The irrigation efficiency was assumed as 60%.

(vi) The tubewell draft (output) was distributed uniformly over the area of all the polygons except those falling in the canal command. The irrigation efficiency was assumed as 65% and conveyance losses as 20%.

(vii) The forest land and waterlogged area was assumed as 1.25% of the total for computation of evapotranspiration.

(viii) The water levels at node points were known for the years 1972 and 1973.

(ix) The water levels in January 1972 were taken as the initial values. The time step taken was one month. For each node the input was determined for the month of January. This included the recharge due to rainfall and canal input, etc. The recharge due to rainfall (R_{r}) was taken R × C where C is a fraction and R the rainfall. The extractions from the node were the tubwell draft, evapotranspiration, etc. The algebraic sum of these replenishments and extractions gave the net recharge A_{B}Q_{B} for the polygonal area. The storage factor A_{B}S_{B} for the polygon was calculated. The water levels at the beginning of February 1972 were calculated adopting the scheme outlined in Eq. (16.2) and (16.3).

(x) The computations of water levels were carried out and compared with the available historic data. If they did not agree within a permissible error (Ɛ_{1}), a new value of C was chosen and the computations repeated. In this manner, the computations were carried out on a digital computer for all nodes for the entire period of 2 years. Figure 16.7 shows the flow diagram used for the computations. The value of C differed from node to node and from month to month (deviations = ± 30% of the average value of 32%).

(xi) Thus, a mathematical model, based on Tyson-Weber scheme of polygonal areas was prepared for the basin. The recharge for each polygon was worked out using the water table elevation data for two years. The model, with the recharge values for individual polygons as worked out can be used for predictions of future response of the basin under different hydrologic conditions being imposed.

(xii) The model can be refined with some more data on the variation of S and T values and verification with historical data of a longer duration. The refined model can be used for optimum utilisation of the water resources of the basin.

**Finite Element Method****: **

In this Method, the solution to flow system through a basin is obtained through an equivalent variational functional, rather than through a finite difference solution of the differential flow equation. With the finite element technique, the solution of the differential flow equation, with a source or sink term Q added to its left side, is obtained by finding a solution for the head h that minimises an equivalent variational functional of the form.

To find the solution, the flow domain (D) is divided into a number of small areas or finite elements which are of triangular and quadrilateral pattern for two-dimensional systems, Fig. 16.8 and tetrahedral or parallelepiped for three-dimensional systems. The elements are of irregular pattern to facilitate representation of irregular boundaries and are smallest where the flow is concentrated. For the solution of each step, the parameters, K, S and Q are kept constant for a given element, but they may vary between elements. To minimize Eq. (16.5) with respect to head h, the differential ∂F/∂h is evaluated for each node and equated to zero; this results in a number of simultaneous equations which are readily solved by computer.

One of the methods of solution of the system of simultaneous equations is through the generation of the system matrix of block tri-diagonal form by a partitioning scheme. Since the matrices of only one block are handled at a time, the requirement of the computer core storage is considerably cut down, making it possible to solve the problems on small and medium sized computers.

The choice between finite-element and finite-difference methods will depend on the complexity of the flow system, computer time required, individual preference and experience, etc.