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Prof.Dr.Ir. Sunjoto Dip.HE, DEA-Hydrology of Groundwater-Post Graduate Program JTSL-FT-UGM 1

GROUNDWATER FLOW ÉCOULEMENT SOUTERRAIN

By: Prof.Dr.Ir. Sunjoto Dip.HE,DEA

Lecture note:

Post Graduate Program Department of Civil and Environmental Engineering

Faculty of Engineering Gadjah Mada University

Yogyakarta, 2013


I. INTRODUCTION

1. Etymology

Hydrogeology (eng) Geohydrologie (fr) Geohidrologi (id)

Geohydrology (eng) Hydrogeologie (fr) Hidrogeologi (id)

2. Hydrology

a. Water cycle

Fig. 1.1. Hydrological cycle

THE WATER CYCLE

Water storage in ice and snow

Water storage in oceans

Evaporation

Groundwater discharge

Infiltration

Precipitation

Sublimation

Water storage in the atmosphere

Evapotranspiration

Spring Fresh water storage

Groundwater storage

Surface runoff

Snowmelt runoff to stream Condensation

SUN


b. Water Balance

Water balance on the ground surface is:

Fig 1.2. Water balance on the ground surface

Fig 1.3. Water balance of the storage

Acccording to Lee R. (1980): P + Ev annual 5 .105 km3/y, equal the depth 973

mm to cover the earth and needs 28 ceturies to evaporate by atmospheric

destilation.

I O ∆S

I - O = ∆S I : Inflow O : Outflow ∆S : Storage

P E

I

R P – E = R + I P : Precipitation E : Evapotranspiration R : Runoff I : Infiltration


c. Water Quantity in the Earth (Volume dimension x106 Km3 ) Table 1.1. Water distribution in the earth (Todd, 1970)

Items Volume x106 Percentage Ocean location

• Saline Water 1,320 Km3 97.300 % Continents location

• Lake fresh water 0.125 Km3 0.0090 % • Lake saline water 0.104 Km3 0.0080 % • Rivers 0.00125 Km3 0.0001 % • Soil moisture 0.067 Km3 0.0050 % • Groundwater (above – 4000 m) 8.350 Km3 0.6100 % • Eternal ice and snow 29.200 Km3 2.1400 %

Total volume 37.800 Km3 2.800 % Atmosphere location:

• Vapor 0.013 Km3 0.001 % Total water 1,360 Km3 100.000 % Table 1.2. Water distribution in the earth (Nace, 1971)

Items Volume x106 Percentage Saline water 1,370 Km3 94.000 % Ice & snow 30 Km3 2.000 % Vapor 0.010 % Groundwater 60 Km3 4.000 % Surface water 0.040 % Total water 100.000 %

Table 1.3. Water distribution in the earth (Huissman, 1978) Items Volume x106 Percentage

Free water, consist of: 1,370 Km3 • Saline water 97.200 % • Ice & snow 2.100 % • Vapor 0.001 % • Fresh water, consist of: 0.600 % Groundwater 98.80 % Surface water 1.20 %

Total water 100.000 %


Table 1.4. Water distribution in the earth (Baumgartner and Reichel, 1975) Items Volume Percentage

Solid 2.782 .107 Km3 2.010 % Liquid 1.356 .109 Km3 97.989 %

• Oceans 1.348 .109 Km3 97.390 % • Continent; groundwater 8.062 .106 Km3 0.583 % • Continent; surface water 2.250 .105 Km3 0.016 %

Vapor 1.300 .104 Km3 0.001 % Total (all forms) 1.384 .109 Km3 100.000 %

• Saline water 1.348 .109 Km3 97.938 % • Fresh water 3.602 .107 Km3 2.202 %

Table 1.5. Fresh water distribution in the earth (Baumgartner and Reichel, 1975) Items Volume Percentage

Solid 2.782 .107 Km3 77.23 % Liquid 8.187 .106 Km3 22.73 %

• Groundwater 7.996 .106 Km3 22.20 % • Soil moisture 6.123 .104 Km3 0.17 % • Lakes 1.261 .105 Km3 0.35 % • Rivers, organic 3.602 .103 Km3 0.01 %

Vapor 1.300 .104 Km3 0.04 % Total (all forms) 3.602 .107 Km3 100.00 %

Table 1.6. Annual average water balance components for the earth (Fig. 1.4)

Item Continent Ocean Earth Area (106 km2) Volume (103 km3)

• Precipitation • Evaporation • Discharge

Avererage depth (mm) • Precipitation • Evaporation • Discharge

148.90

+111 -71 -40

+745 -477 -269

361.10

+385 -425 +40

+1066 -1177 +111

510.00

+496 -496

0

+973 -973

0 Source: (Baumgartner & Reichel, 1975 in Lee R., 1980)


Fig. 1.4. Earth water balance components, in 103 km3 (Baumgartner & Reichel, 1975 in Lee R., 1980)

d. Management of Groundwater

1). Advantages and Disadvantages of Groundwater

Table 1.7. Conjunctive use of Surface and Groundwater Resources Advantages Disadvantages

1. Greater water conservation 2. Smaller surface storage 3. Smaller surface distribution system 4. Smaller drainage system 5. Reduced canal lining 6. Greater flood control 7. Ready integration with existing

development 8. Stage development facilitated 9. Smaller evapotranspiration losses 10. Greater control over flow 11. Improvement of power load 12. Less danger than dam failure 13. Reduction in weed seed distribution 14. Better timing of water distribution 15. Almost good quality of water resources

1. Less hydroelectric power 2. Greater power consumption 3. Decreased pumping efficiency 4. Greater water salination 5. More complex project operation 6. More difficult cost allocation 7. Artificial recharge is required 8. Danger of land subsidence

Source: Clendenen in Todd, 1980.

P=385

Q=40 P=111

E=425 Q=40

E=71

CONTINENT

OCEAN

ATMOSPHER

Water balance: P + E + Q = 0


Table 1.8. Advantages and Disadvantages of subsurface and Surface Reservoirs (USBR)

Subsurface Reservoirs Surface Reservoirs

Advantages 1. Many large-capacity site available 2. Slight to no evaporation loss 3. Require little land area 4. Slight to no danger of catastrophic

structural failure 5. Uniform water temperature 6. High biological purity 7. Safe from immediate radio active fallout 8. Serve as conveyance systems-canals or

pipeline across land of others unnecessary

Disadvantages

1. Water must be pumped 2. Storage and conveyance use only 3. Water maybe mineralized 4. Minor flood control value 5. Limited flow at any point 6. Power head usually not available 7. Difficult and costly to evaluate,

investigate and manage 8. Recharge opportunity usually dependent

of surplus of surface flows 9. Recharge water maybe require expensive

treatment 10. Continues expensive maintenance of

recharge area or wells

Disadvantages 1. Few new site available 2. High evaporation loss even in humid

climate 3. Require large land area 4. Ever-present danger of catastrophic

failure 5. Fluctuating water temperature 6. Easily contaminated 7. Easily contaminated radio active fallout 8. Water must be conveyed

Advantages 1. Water maybe available by gravity flow 2. Multiple use 3. Water generally of relatively low mineral

content 4. Maximum flood control value 5. Large flows 6. Power head available 7. Relatively to evaluate, investigate and

manage 8. Recharge dependent o annual

precipitation 9. No treatment require recharge of

recharge water 10. Little maintenance required of

facilities


Table 1.9. Attributes of Groundwater There is more ground water than surface water

Ground water is less expensive and economic resource.

Ground water is sustainable and reliable source of water supply.

Ground water is relatively less vulnerable to pollution

Ground water is usually of high bacteriological purity.

Ground water is free of pathogenic organisms.

Ground water needs little treatment before use.

Ground water has no turbidity and color.

Ground water has distinct health advantage as art alternative for lower sanitary

quality surface water.

Ground water is usually universally available.

Ground water resource can be instantly developed and used.

There is no conveyance losses in ground water based supplies.

Ground water has low vulnerability to drought.

Ground water is key to life in arid and semi-arid regions.

Ground water is source of dry weather flow in rivers and streams.

Source: http://www.tn.gov.in/dtp/rainwater.htm

e. Data collection 1). Topographic data 2). Geologic data 3). Hydrologic data

(a). Surface inflow and outflow (b). Imported and exported water (c). Precipitation (d). Consumptive use (e). Changes in surface storage (f). Changes in soil moisture (g). Changes in groundwater storage (h). Subsurface inflow and outflow

http://www.tn.gov.in/dtp/rainwater.htm�


3. History a. Dugwell

b. The simplest dug well is crude dug well where the people go down to draw a water directly. Then brick or masonry casing dug well which were build before century. The dug well with casing equipped by bucket, rope and wheel to draw water.

Fig. 1.5. A crude dug well in Shinyanga Region of Tanzania. (after DHV Con. Eng., in Todd, 1980) and Sketch of crude dug well cross section of step well.

Fig. 1.6. A traditional dug well and A modern domestic dug well with rock curb, concrete seal

and hand pump. (after Todd, 1980)


Fig 1.7. Communal dug well equipped by recharge systems surraunding the well.

Fig 1.8. Traditional step well in India it is called baollis or vavadi were built from 8th to 15th century (Source: Nainshree G. Sukhmani A. Design of Water Conservation System Through Rain Water Harvesting; An Excel Sheet Approach)


c. Qanat, Karez, Foggara

Qanat (arb), Karez (prs), Foggara (bbr-arb) is a system of water exploitation which

providing of irrigation water in Central East. Qanat is a method to get clean water by

digging horizontal gallery across the slope surface of ground till reach groundwater

table of the aquifer. From this aquifer water flow with smaller slope than original

slope of groundwater table of impervious canal go in the direction of irrigation area

(Fig. 1.9.). According to Todd (1980), the total gallery length of qanats in this area,

reach thousands of miles. Iran has the greatest concentration of qanats, here some

22,000 qanats are supplying 75% of all water used in the country. Lengths of qanats

extend up to 30 km but most are less than 5 km. The depth of qanats mother well is

normally less than 50 m but instances of depth exceeding 250 m. Discharges of

qanants vary seasonally with water table fluctuation and seldom exceed 100 m3/h.

The longest qanat near Zarand, Iran is 29 km with a mother well depth of 96 m with

966 shafts along its length and the total volume of material excavated is estimated

at 75,400 m3.

Fig. 1.9. Vertical cross section along a qanat, gallery and shaft (after Beaumont, in Todd,

1980)


d. Roman domestic water system. (http://staff.civil.uq.edu.au/h.chanson/rom_aq.html)

Roman aqueducts supplied waters to cities for public baths (thermes) and toilets

(latrines) (Hodge, 1992, Fabre et al. 2000), in addition of public fountains. They were

long subterranean conduits, following contours lines, with flat longitudinal slopes : i.e.,

1 to 3 m per km, even less at Nîmes (0.24 m/km). Numerous aqueducts were used for

centuries and some are still in use (e.g. Carthage, Mons). Their construction was a

huge task, often performed by the army under the guidance of military hydraulic

engineers. Their cost was extra-ordinary considering the real flow rate (i.e. less than

400 L/s) : about 1 to 3 millions sesterces per kilometre in average (Fevrier, 1979,

Leveau, 1991)

Several aqueducts (Fig.1.12.) were equipped with regulation basins installed along the

canal. For example, at Ars-sur-Moselle (Metz); at the Vallée de l'Eure, upstream of

Pont-du-Gard, at Lafoux along the Nîmes aqueduct; at Segovia upstream of the

aqueduct bridge. Most regulation basins were equipped with a series of gates and an

overflow system. Basic hydraulic considerastions imply that undershoot gates were

used to regulate the aqueduct flow while overshoot gates were used for the overflow

discharge (Chanson, 2000b). The aqueducts provided a constant flow of fresh water

into the bath houses. The best water supplied to the city was used for drinking

water. The baths were given the next best supply. Likewise, an equally impressive

system known as the Cloaca Maxima (literally, Latin for “giant sewer”) carried away

the used water filled with Rome’s trash, human and animal waste, and even bodies of

dead slaves. The sewer system flowed through and under much of Rome and still

drains rain water and debris from modern Rome. Hydraulic calculations were

conducted for two large regulation basins on the Gorze and Nîmes aqueducts. This

type of operation implied fine gate opening adjustment systems to enable precise

flow regulation.

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Fig 1.10. Roman water coveyance and water distribution system

Fig 1.11. Sketch of Roman city water system provider from ground water resources to the city.

Note: 1. Infiltration gallery/qanat 2. Steep chute in this case dropshafts 3. Settling tank 4. Tunnel and shafts 5. Covered trench

6. Aquaduct bridge 7. Siphon 8. Substruction 9. Arcade 10. Distribution basin 11. Water distribution (pipes)


Fig 1.12. Roman aquaduct

e. Springs

Spring is an outflow of ground water to the ground surface due to hydraulic head or

gravitational force (Fig. 1.13). This technique had been implanted since before

century like in Greek or Roman Kingdom. Spring water as a drinking water is usually be

conveyed by network of pipes or canals to the town.

Fig. 1.13. Diagrams that illustrating types of gravity springs. (a). Depression spring. (b).

Contact springs. (c). Fracture artesian spring. (d). Solution tabular spring and Schematic cross section illustrating unconfined and confined aquifer (after Bryan, in Todd, 1980)

f. Kaptering

Kaptering (ducth) is a building of spring catcher. The ancient kaptering in Indonesia

in Trowulan as capital of Majapahit Kingdom it was implemented since 12nd century


that on the site of spring was built a temple is now called Tikus Temple. Nowadays

from this temple still flowing water even though with small discharge and this

building installed by inflow-outflow and overflow system and conveyance pipes to

Segaran Pond with the area are more than 6 ha (Fig. 1.13).

This construction must keep that the spring have not the excess water pressure, it is

mean that the hydrostatic pressure must be equal or lower than before the

development. Much mistake de spring catcher development when designer aim to

increase the elevation of the natural elevation of the spring to get higher hydraulic

head. The problem will occur to the hydrostatic pressure of the soil or rock

surrounding the spring and when the pressure bigger than the carrying capacity of

soil it will create the leakage and finally the spring will be move to the other

direction.

Fig 1.14. Kaptering or spring water catcher of the Kingdom, recently it’s called Tikus Temple and Water pond with brick structure which is called Segaran Pond (pcp)


Fig. 1.15. Conveyance and distribution pipes to the housing in Trowulan as a capital of Majapahit Kingdom (Photo: Prof. Hardjoso P.)

Fig 1.16. Ancient fountains and dug well cased by bricks in the housing of the Kingdom (Photo: Prof. Hardjoso P.)

g. Crush Bore Well (Cable tool)

(http://www.welldrillingschool.com/courses/pdf/DrillingMethods.pdf)

Cable tool had its beginnings 4000 years ago in China. It was the earliest drilling

method and has been in continuous use for about 4000 years. The Chinese used tools


constructed of bamboo and well depths of 3000 ft. are recorded. However, wells of

this depth often took generations to complete. Cable tool rigs are sometimes called

pounders, percussion, spudder or walking beam rigs. They operate by repeatedly

lifting and dropping a heavy string of drilling tools into the borehole. The drill bits

breaks or crushes consolidate rock into small fragments. When drilling in

unconsolidated formations, the bit primarily loosens material.

Crush Bore Well is a well which is build to provide drinking water by crush or impact

of a sharp cylindrical metal using cable tool to rise on the certain height and then be

released and fall down to the ground and create a hole which reach ground water

table. In Egypt this system was implemented since 3000 BC, in Rome near the first

century and in a small town in south French Artois, which well had a hydraulic

pressure and it created an artesian well due to the water squirt out from the well .

For a cable tool drill to operate the drill string must have these four components:

• Drilling cable - lifts tools, turns tools, controls tool motion.

• Swivel socket - connects cable to tools, allows cable to unwind.

• Drill stem - provides weight, steadies and guides bit.

• Drill bit - penetrates formation, crushes and reams, mixes cuttings. Many cable

tool drillers now employ Tungsten Carbide studded bits to aid in hard rock

penetration.

h. Auger Drilling

Often used for site investigation, environmental and geotechnical drilling and

sampling, and boreholes for construction purposes, auger drilling can be an efficient

drilling method. The advantages of auger drilling include low operating costs, fast

penetration rates in suitable formations and no contamination of samples by fluids.

Augers come in continuous flight, short flight/plate augers and bucket augers.

Continuous flight augers driven by top head rotary machines (shown above) carry


their cuttings to the surface on helical flights. Continuous flight augers with hollow

stems are often used for sample recovery in environmental, geotechnical operations.

i. Rotary Drilling

Rotary bore well was implemented since 1890 in USA to draw gas and oil and the hole

reach 2,000 meter depth. Nowadays, the rotary bore well reach 7,000 meter depth.

Rotary drilling uses a sharp, rotating drill bit to dig down through the Earth's crust.

Much like a common hand held drill, the spinning of the drill bit allows for penetration

of even the hardest rock. The idea of using a rotary drill bit is not new.

Archeological records show that as early as 3000 B.C., the Egyptians may have been

using a similar technique. Leonardo Di Vinci, as early as 1500, developed a design for a

rotary drilling mechanism that bears much resemblance to technology used today.

Despite these precursors, rotary drilling did not rise in use or popularity until the

early 1900's. Although rotary drilling techniques had been patented as early as 1833,

most of these early attempts at rotary drilling consisted of little more than a mule,

attached to a drilling device, walking in a circle! It was the success of the efforts of

Captain Anthony Lucas and Patillo Higgins in drilling their 1901 'Spindletop' well in

Texas that catapulted rotary drilling to the forefront of petroleum drilling

technology.

While the concept for rotary drilling - using a sharp, spinning drill bit to delve into

rock - is quite simple, the actual mechanics of modern rigs are quite complicated. In

addition, technology advances so rapidly that new innovations are being introduced

constantly. The basic rotary drilling system consists of four groups of components.

The prime movers, hoisting equipment, rotating equipment, and circulating equipment

all combine to make rotary drilling possible.

j. Down the hole air hammer

To drill effectively in hard formations, rotary bits require very high pull down

pressures. These pressures may be beyond the design capabilities of small to medium


drill rigs. And, as was stated earlier, excessive pull down pressures may damage the

drill string and deflect the trueness of the hole. If the hard rock formation is near

the surface, even larger rigs have trouble with penetration as the weight of the drill

string is not relatively great when drilling is beginning. The down hole hammer is an

air activated percussive drilling bit which operates in the manner of the jack

hammer commonly seen in surface construction. Constructed from alloy steel with

heavy tungsten-carbide inserts that provide the cutting or chipping surfaces. These

inserts are subject to wear and may be replaced or reground improve penetration

rates. Corrosion (rust) is the DHH’s greatest enemy. It must be kept well lubricated

at all times. And it should be opened and inspected after every 100 hours of

continuous operation.

k. Jet Drilling

Drilling in unconsolidated formation with high water availability allows jet drilling to

be a viable drilling method. Often employed in drilling shallow irrigation wells, jet

drilling is achieved by water circulation down through the rods washing cuttings from

in front of the bit. The cutting flow up the annular space and in a settling pit so that

the water can be re-circulated. Jetting in semi consolidated formations may be

assisted by using a hammering technique to “chop” through hard bands. This

technique is a combination of jetting and percussion. A fish-tail type rotary bit may

be used and the pipe rotated to “cut the hole. All hydraulic (water based) drilling

requires that the hole be kept full off water until it is cased.


4. Qualitative Theory

a. Early Greek Philosophers

Homer, Thales (624-546 BC) and Plato (428-347 BC) hypothesized that springs were

formed by sea water conducted through subterranean channels below the mountains,

then purified and raised to the surface.

b. Aristoteles (384-322 BC

Water is every day carried up and is dissolved into vapor and rises to the upper

region, where it is condensed again by the cold and so returns to the earth.

c. Marcus Vitruvius (15 BC)

Theory of the hydrologic cycle, in which precipitation falling in the mountains

infiltrated the Earth's surface and led to streams and springs in the lowlands.

d. Early Roman Philosophers

Lucius Annaeus Seneca (1 BC – AD 65) and Pliny clarify theory of Aristoteles is

precipitation fall down in the mountain, a part of water infiltrate to the ground as a

storage water and then flow out as springs.

e. Bernard Palissy (1509-1589)

He described more clearly about hydrological cycle from evaporation in the sea till

water come back again to the sea in his book: Des eaux et fontaines.

f. Johannes Kepler (1571-1630)

The earth as a big monster whose suck water from the sea, be digested and flow out

as fresh water in springs.

g. Athanasius Kircher (1602-1680)

Interaction with magma heat which causes heated water to rise through fissures and

tidal and surface wind pressure on the ocean surface which forces ocean water into

undersea.


5. Quantitative Theory

a. Pierre Perrault (1608-1690)

He observed rainfall and stream flow in the Seine River basin, confirming Palissy's

hunch and thus began the study of modern scientific hydrology. He said that the

depth of precipitation in the Seine river, France was 520 mm/y

b. Edme Mariotte (1620-1684)

In his book Des mouvements des eaux Seine River: Discharge Q = 200.000 ft3/min,

local flow is 1/6 part, evaporation is 1/3 part and infiltration is 1/3 part.

c. Edmund Halley (1656–1742)

He developed the equation of balance : I – O = ∆S

d. Daniel Bernoulli (1700-1782)

He stated that, in a steady flow, the sum of all forms of mechanical energy in a fluid

along a streamline is the same at all points on that streamline.

e. Jean Leonard Marie Poiseuille (1797-1869).

The original derivation of the relations governing the laminar flow of water through a

capillary tube was made by him in the early of 19th century.

f. Reynold (1883)

The Reynolds number NR is a dimensionless number that gives a measure of

the ratio of inertial forces ρV2/L to viscous forces μV/L2 and consequently quantifies

the relative importance of these two types of forces for given flow conditions.

g. Henry Philibert Gaspard Darcy (June 10, 1803 – January 3, 1858)

On his books ‘Les fontaines publiques de Dijon (1856), he developed mathematical

equation for flow in porous media.

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h. Badon Gabon (1888) and Herzberg (1901)

They developed equilibrium theory of fresh water and saline water in the circular

island with porous soil.

i. Jules Dupuit (1863)

In his book: Estudes Thèoriques et Pratiques sur le mouvement des Eaux dans les

canaux dècouverts et à travers les terrains permèables, Dupuit developed the

formulas for groundwater flow from trench to trench with definite distance, radial

flow in unconfined and confined aquifer with definite distance.

j. Adolph Thiem (1870)

a German engineer who developed equation for the flow toward well and infiltration

galleries.

k. Gunther Thiem (1907)

In 1906, he continued Dupuit principle and his father research he developed steady

stage equation for the circular flow, using two test wells and drawdown data, and the

formula is nowaday called Dupuit-Thiem.

l. Lugeon (1930)

Lugeon developed the double packer bore hole inflow test made at constant head.

Lugeon is a measure of transmissivity in rocks, determined by pressurized injection

of water through a bore hole driven through the rock.

m. Theis (1936)

The Theis equation was developed to determine transmissivity of storage coefficient

by drawdown measuring at any given radius from the well in form exponential integral.

Due to the equations are difficult to compute so the graphic solutions are needed.

http://en.wikipedia.org/w/index.php?title=Lugeon&action=edit�


n. Expansion of Theis

Cooper-Jacob simplified the Theis formula by negligible after the first two terms.

The same manner it was expanded to by Chow (1952) and Todd (1980) but all

together still need graphic solution.

o. Forchheimer (1930)

He developed the flow equation of steady state radial flow in borehole using new

parameter is ‘shape factor’ and neglected data of observation well.

p. Expansion of Forchheimer

Development of formulas of shape factors by Samsioe (1931), Dachler (1936), Taylor

(1948), Hvorslev (1951), Aravin (1965), Wilkinson (1968), Al-Dahir & Morgenstern

(1969), Luthian & Kirkham (1949), Kirkham & van Bavel (1948), Raymond & Azzouz

(1969), Smiles & Young (1965) and Sunjoto (1988-2008).

q. Taylor (1940)

Certain guiding principles are necessary such as the requirement that the formation

of the flownet is only proper when it is composed of ‘curvilinear squares’.

r. Sunjoto (1988)

Base on Forchheimer (1930) principle, Sunjoto (1988) developed an unsteady state

radial flow equation for well which was derived by integration solution and shape

factors of the tip of the well. In 2008 he developed too the formula of unsteady

state condition of recharge trench and its shape factors.

6. Interest of Research

a. Russian ⇒ Groundwater in ice region

b. Dutch ⇒ Groundwater in sand dunes

c. Japanese ⇒ Hot groundwater

d. Indonesian ⇒ Recharge Systems


7. Dimension and Unit

a. Georgy System (mks)

Table 1.8. Dimension and Unit Description Dimension Unit

mass length time

m l t

gram meter second

Force

Energy

Power

Pressure

mlt-2

ml2t-2

ml2t-3

ml-1t-2

N (Newton) = kgm.s-2

J (Joule) = N.m

W (Watt) = N.m.s-1

N.m-2

b. Metric prefixes

Table 1.9. Metric prefices Prefix Symbol Factor Prefix Symbol Factor

tera T 1012 centi c 10-2

giga G 109 milli m 10-3

mega M 106 micro µ 10-6

kilo k 103 nano n 10-9

hecto h 102 pico p 10-12

deca da 101 femto f 10-15

deci d 10-1 atto a 10-18


c. Conversion of unit

Table 1.10. Conversion Description Unit mks Note Force

Energy

Power

1 kg

1 kg.m

1 kg.ms-1

g.N

g.J

g.W

1 N = 105 dynes

g = 9.78 m.s-2 = 32.3 ft.s-2

1 HP = 75.g.W = 734 W

d. Metric-English equivalents

Table 1.11. Metric-English equivqlent 1). Length

1 cm = 0.3937 in

1 m = 3.281 ft

1 km = 0.6214 mi

2). Area

1 cm2 = 0.1550 in2

1 m2 = 10.76 ft2

1 ha = 2.471 acre

1 km2 = 0.3861 mi2

3). Volume

1 cm3 = 0.06102 in3

1 l = 0.2642 gal = 0.03531 ft3

1m3 = 264.2 gal = 35.31 ft3

= 8.106 .10-4 acre.ft

4). Mass

1 g = 2.205 .10-3 lb (mass)

1 kg = 2.205 lb (mass)

= 9.842 .10-4 long ton

5). Velocity

1 m/s = 3.281 ft/s

= 2.237 mi/hr

1 km/hr = 0.9113 ft/s

= 0.6214 mi/hr

6). Temperature o C = K – 273.15

= (o F – 32)/1.8

7). Pressure

1 Pa = 9.8692 .10-6 atm

= 10-5 bar

= 10-2 millibar

= 10 dyne/cm2

= 3.346 .10-4 ft H2O (4o C)

= 2.953 .10-4 in Hg ( 0o C)

= 0.0075 mm Hg

= 0.1020 kg (force)/m2

= 0.02089 lb (force)/ft2


8). Flow rate

1 l/s = 15.85 gpm

= 0.02282 mgd = 0.03531 cfs

1 m3/s = 1.585 .104 gpm

= 22.82 mgd = 35.31 cfs

1 m3/d = 0.1834 gpm

= 2.642 .10-4 mgd = 4.087 .10-4 cfs

9). Force

1 N = 105 dyne

= 0.1020 kg (force)

= 0.2248 lb (force)

10). Power

1 W = 9.478 .10-4 BTU/s

= 0.2388 cal/s

= 0.7376 ft.lb (force)/s

11). Water quality

1 mg/l = 1 ppm = 0.0584 grain/gal

12). Hydraulic conductivity

1 m/d = 24.54 gpd/ft2

= 1.198 darcy (water 20o C)

1 cm/s = 2.121 .104 gpd/ft2

= 1035 darcy (water 20o C)

13). Viscosity

1 Pa.s = 103 centistoke= 10 poise

= 0.02089 lb (force).s/ft2

1 m2/s = 106 centistoke = 10.76 ft2/s

14). Gravitational acceleration, g

9.807 m/s2 = 32.2 ft/s2 (std., free fall)

15). Heat

1 J/m2 = 8.806 .10-5 BTU/ft2

= 2.390 .10-5 cal/cm2

1 J/kg = 4.299 .10-4 BTU/lb (mass)

= 2.388 .10-4 cal/g

16). Density of water, ρ

1000 kgmass/m3 = 1.94 slugs/ft3

(when 50o F/10o C)

17). Specific weight of water, γ

9.807 .103 N/m3 = 62.4 lb/ft3 (50oF/10oC)

18). Dynamic viscosity of water, µ

1.30 .10-3 Pa.s=2.73 .10-5lb.s/ft2(50o/10oC)

10-3 Pa.s = 2.05 .10-5 lb.s/ft2 (68o F/20o C)

19). Kinematic viscosity of water, ν

1.30.10-6m2/s=1.41 .10=5 ft2/s(50o F/10oC)

10-6 m2/s = 1.06 .10-5 ft2/s (68o F/20o C)

20). Atmospheric pressure, p (std)

1.013 .105 Pa = 14.70 psia

21). Energy

1 J = 9.478 .10-4 BTU

= 0.2388 cal

= 0.7376 ft.lb (force)

= 2.788 .10-7 kw.hr


e. Legends

1). Density

• Symbol : ρ

• Dimension : ml-3

• Unit : kgmass.m-3 or slug.ft-3

Detail:

• 1 slug = 14.60 kgmass

• 1 feet = 0.305 m

• 1 slug.ft-3 = 514.580 kgmass.m-3

In practical use:

• ρpure water = 1,000 kgmass.m-3 = 1.94 slug.ft-3

• ρsea water = 1,026 kgmass.m-3 = 1.99 slug.ft-3

Table 1.12. Density of pure water in kgmass.m-3 dependent temperature to C t ρ t ρ t ρ t ρ 0

2

4

6

8

999.8679

999.9267

1000.0000

999.9081

999.8762

10

12

14

16

18

999.7277

999.5247

999.2712

998.9701

998.6232

20

22

24

26

28

998.2323

997.7993

997.3256

996.8128

996.2623

30

32

34

36

38

995.6756

995.0542

994.3991

993.7110

992.9936

2). Specific weight

• Symbol : γ ⇒ γ = ρ.g

• Dimension : ml-2t-2

• Unit : N.m-3 atau lbs.ft-3


3). Specific Gravity

• Symbol : s ⇒ s = ρ/ρw = γ/γw

• Dimension : -

• Unit : -

4). Viscosity

(a). Dynamic viscosity

• Symbol : µ

• Dimension : ml-1t-1

• Unit : N.s.m-2

• 1 N.s.m-2 = 10 poise; 478 poise = 1 lbs.ft-2

Table 1.13. Dynamic viscosity of water in 10-2 poisses dependent temperature to C t µ t µ t µ t µ

0

2

4

6

8

1.7921

1.6728

1.5674

1.4728

1.3860

10

12

14

16

18

1.3077

1.2363

1.1709

1.1111

1.0559

20

22

24

26

28

1.0050

0.9579

0.9142

0.8737

0.8360

30

32

34

36

38

0.8007

0.7679

0.7371

0.7085

0.6814

(b). Cinematic viscocity

• Symbol : υ

• Dimension : l2t-1

• Unit : m2s-1 or stokes

• 1 m2s-1 = 10-4 stokes

• 1 ft2s-1 = 929 stokes

• υ = µ /ρ


5). Surface Tension

• Symbol : σ

• Dimension : mt-2

• Unit : N.m-1

• σwater/air = 0.074 N.m-1

Table 1.14. Relationship of ρ, σ and υ of water t = 10o C; p = atm t = 60o F; p = atm

Water Air Unit Water Air Unit

ρ

µ

υ

1000

1.3 .10-2

1.3 .10-6

1.37

1.8 .10-4

1.3 .10-5

kgmass.m-3

poise

m2s-1

1.94

2.3 .10-5

1.2 .10-5

2.37 .10-3

3.7 .10-7

1.6 .10-4

slug.ft-3

lbs.s.ft-2

ft2s-1

6). Specific Surface (Am)

Suface specific is the total area of particle (m2) per unit mass (m2/g)

This value depend on the forme of particle and has big role on the phenomena the

liquid-solid suface, especially for the absorbstion and swelling.

For example one sphere with d diameter and ρs= 2.7 g/cm3 of specific mass and the

thikness of particle is e so the sufarce specific is:

𝐴𝐴𝑚𝑚 =6𝜋𝜋𝑑𝑑2. 10−4

4.𝜋𝜋 𝑑𝑑2. 𝑒𝑒. 2.7

4

=2. 10−4

𝑒𝑒. 2.7=

0.75. 10−4

𝑒𝑒 𝑐𝑐𝑚𝑚 𝑚𝑚2 𝑔𝑔⁄

When this formula is implemented for the plaquette of monmorollonite so

𝐴𝐴𝑚𝑚 =0.75. 10−4

10−7 𝑚𝑚2 𝑔𝑔⁄ = 750𝑚𝑚2 𝑔𝑔⁄

Prof.Dr.Ir. Sunjoto Dip.HE, DEA-Subsurface Hydrology-Post Graduate Program JTSL-FT-UGM=2012

30

II. GENERAL DESCRIPTION

1. Terminology

a. Aquifer

The origin of aqua is water and ferre is ’contain’.

b. Aquiclude

The origin of claudere is ‘to shut’.

c. Aquifuge

The origin of fugere is ‘to expel’.

d. Aquitard

The origin of tard is ’late’.

2. Vertical Distribution

Fig. 2.1. Diagram of zones in permeable soil

Ground surface

Soil water zone

Intermediate vadoze

zone

Capillary zone

Saturated zone

ZONE OF AERATION

ZONE OF SATURATION

VADOZE WATER

GROUND / PHREATIC WATER

Groundwater table

Impermeable

P e r m e a b l e


31

a. Zone of Aeration

This zone divided into:

• Soil water zone

• Intermediate vadose zone

• Capillary zone

2𝜋𝜋𝜋𝜋𝜋𝜋𝑐𝑐𝜋𝜋𝜋𝜋𝜋𝜋 = ℎ𝑐𝑐𝜋𝜋𝜋𝜋2𝛾𝛾

ℎ𝑐𝑐 =2𝜋𝜋𝑐𝑐𝜋𝜋𝜋𝜋𝜋𝜋𝜋𝜋𝛾𝛾 (2.1)

Fig. 2.2. Schematic of capillary rise

hc

2r

λ

ℎ𝑐𝑐 =0.15𝜋𝜋

hc : height of capillary zone τ : surface tension (dynes/cm) γ : specific weight of water r : radius of tube λ : contact angle of water and wall When pure water in clean glass, λ = 0 and temperature at 20o C so value of τs = 75 dyne/cm = 0.076 g/cm and,


32

Table 2.1. Capillary rise in samples of unconsolidated materials (after Lohman in Todd, 1980)

Soils Type Grain size (mm) Height of capillary (cm)

Fine gravel

Very coarse sand

Coarse sand

Medium sand

Fine sand

Silt

Silt

5 - 2

2 - 1

1 – 0.5

0.5 – 0.2

0.2 – 0.1

0.1 – 0.05

0.05 – 0.002

2.50

6.50

1.50

24.60

42.80

105.50

200.00

Table 2.3. Capillary rice of some soils type (Murthy, 1977) Soils Type Size of particles (mm) Capillary rise (cm)

Sand, coarse

Sand, medium

Sand, fine

Silt

Clay, coarse

Clay, colloid

2.00 - 0,60

0.60 – 0.20

0.20 – 0.06

0.06 – 0.002

0.002 – 0.0002

< 0.0002

1.50 – 5

5 – 15

15 - 50

50 - 1,500

1,500 – 15,000

>15,000


33

b. Zone of Saturation

1). Specific retention (Sr)

Sr = Wr / V (2.2)

Wr : the rest water volume after drainage

V : total volume of soil

2). Specific yield (Sy)

Sy = Wy / V (2.3)

Wy : volume of water which be drained

α = Sr + Sy (2.4)

c. Solid Liquid and Air System

• Solid phase : geometricly difficult be soluble

• Liquid phase : solution → organic & unorganic

• Air phase : vapor

Fig. 2.3. Diagram of solid, water and air relationship

V

Vv

Va

Vw

Vs

Wa

Ww

Ws

1

air

water

solid


34

1). Void ratio (e)

The ratio of the volume of voids (Vv) to the volume of solids (Vs), is defined as

void ratio, and:

𝑒𝑒 =𝑉𝑉𝑒𝑒𝑉𝑉𝜋𝜋

(2.5)

2). Porosity (n)

The ratio of the volume of voids (Vv) to the total volume (V), is defined as

porosity, so:

𝑛𝑛 =𝑉𝑉𝑣𝑣𝑉𝑉

× 100% (2.6)

3). Degree of saturation (S)

The ratio of volume of water (Vw) to the volume of voids (Vv) sis defined as

degree of saturation so:

𝑆𝑆 =𝑉𝑉𝑤𝑤𝑉𝑉𝑣𝑣

× 100% (2.7)

4). Water content (w)

The ratio of weight of water (Ww) in the voids to the weight of solids so:

𝑤𝑤 =𝑊𝑊𝑤𝑤

𝑊𝑊𝜋𝜋× 100% (2.8)

5). Unit Weight

a). Unit weight of water (γw)

The ratio of weight of water to the volume of water in the same temperature

(γw) and (γo) is designated as unit weight of water at 4o C.

γ𝑤𝑤 = 1 𝑔𝑔𝑓𝑓𝜋𝜋𝜋𝜋𝑐𝑐𝑒𝑒

𝑐𝑐𝑚𝑚3� = 1 𝑘𝑘𝑔𝑔𝑓𝑓𝜋𝜋𝜋𝜋𝑐𝑐𝑒𝑒 𝑑𝑑𝑚𝑚3� = 1 𝑡𝑡𝑓𝑓𝜋𝜋𝜋𝜋𝑐𝑐𝑒𝑒 𝑚𝑚3� = 1000 𝑘𝑘𝑔𝑔𝑓𝑓𝜋𝜋𝜋𝜋𝑐𝑐𝑒𝑒 𝑚𝑚3�


35

b). Total unit weight of soil mass (γt)

The ratio of the weight of the mass (W) to the volume of the mass (V) so:

γ𝑡𝑡 =𝑊𝑊𝑉𝑉

(2.9)

c). Dry unit weight mass (γd)

The ratio of the weight of solids (Ws) to the total volume (V)

γ𝑑𝑑 =𝑊𝑊𝜋𝜋

𝑉𝑉 (2.10)

d). Ratio of the saturated weight of the mass (γsat)

Saturated unit weight soil mass (when S = 100%) to the total volume (V).

γ𝜋𝜋𝑠𝑠𝑡𝑡 =𝑊𝑊𝑉𝑉

(2.11)

e). Unit weight of solid (γs)

The ratio of the weight of solids (Ws) to the volume of solids (Vs)

γ𝜋𝜋 =𝑊𝑊𝜋𝜋

𝑉𝑉𝜋𝜋 (2.12)

f). Specific gravity (Gm)

Specific gravity of a substance is the ratio of its weight in air to the weight of

an equal volume of water at reference temperature 4o C.

• The specific gravity of mass of soil including air, water and solid:

𝐺𝐺𝑚𝑚 =γ𝑡𝑡γ𝜋𝜋

=𝑊𝑊𝑉𝑉γ𝜋𝜋

= γ𝜋𝜋 = 𝐺𝐺 (2.13)

• The specific gravity of mass of soil excluding air, water and solid:

𝐺𝐺 =γ𝜋𝜋γ𝜋𝜋

=𝑊𝑊𝑉𝑉𝜋𝜋γ𝜋𝜋

= γ𝜋𝜋 =𝑊𝑊𝜋𝜋

𝑉𝑉𝜋𝜋 (2.14)


36

3. Type of Aquifer

gs

gwt’

gwt H

e. Suspended aquifer

Note: gs : ground surface ps : piezometric surface gwt : groundwater table gwt’ : groundwater table of perched water D : thickness of aquifer H : depth of groundwater K : coefficient of permeability

Note: Compare to Todd (1980) page 44 about leaky aquifer, which the elevation of gwt is higher than ps.

Fig. 2.4. Types of aquifers

gs

gwt

K=0

gs

gwt

K D H

ps

D=H K

ps K1<K

c. Confined aquifer d. Semi confined/leaky aquifer

gs

gwt = ps K

H

gs

gwt = ps

K

K1<K

D H

a. Unconfined aquifer b. Semi unconfined aquifer


37

III. BASIC PARAMETERS

1. Law of Groundwater Flow

a. Poiseuille’s Law

𝑣𝑣𝑠𝑠 =𝛾𝛾𝑤𝑤𝑅𝑅2𝑖𝑖

8µ 𝜋𝜋𝜋𝜋 𝑄𝑄𝑠𝑠 = 𝑍𝑍𝑖𝑖𝐴𝐴 (3.1)

where va : average velocity γw : unit weight of water R : radius of tube µ : viscosity of fluid i : hydraulic gradient A : area Qa : average discharge Z = γw.R2/8μ

This equation is the proof of Poiseuille’s Law which states that the velocity in laminar

flow is proportional to the first power of the hydraulic gradient i.

b. Darcy’s Law (1856),

1). Equation

𝑉𝑉 = 𝐾𝐾𝑖𝑖 ⇒ 𝑖𝑖 = ℎ𝑙𝑙

(3.2)

General equation can be written as a vector form:

𝑉𝑉 = 𝐾𝐾𝐾𝐾𝐾𝐾

Substitute to the Laplace Equation:

𝑉𝑉 = −𝐾𝐾 �𝛿𝛿ℎ𝑖𝑖𝛿𝛿𝛿𝛿

𝑖𝑖 +𝛿𝛿ℎ𝑗𝑗𝛿𝛿𝛿𝛿

𝑗𝑗 +𝛿𝛿ℎ𝑘𝑘𝛿𝛿𝛿𝛿

𝑘𝑘� (3.3)

Consider on x direction only so:

𝛿𝛿ℎ𝑗𝑗𝛿𝛿𝛿𝛿

= 0 & 𝛿𝛿ℎ𝑘𝑘𝛿𝛿𝛿𝛿

= 0


38

The equation becomes:

𝑉𝑉 = −𝐾𝐾 �𝛿𝛿ℎ𝑖𝑖𝛿𝛿𝛿𝛿

�

𝑉𝑉 = −𝐾𝐾𝑑𝑑𝑑𝑑𝛿𝛿

�−𝑠𝑠𝑏𝑏𝛿𝛿 + 𝑠𝑠�

𝑽𝑽 = +𝑲𝑲𝒂𝒂𝒃𝒃

𝒐𝒐𝒐𝒐 𝑽𝑽 = +𝑲𝑲𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅

𝒐𝒐𝒐𝒐 𝑽𝑽 = +𝑲𝑲𝒅𝒅𝒅𝒅

𝒐𝒐𝒐𝒐 𝑽𝑽 = 𝑲𝑲𝑲𝑲 (3.4)

The essential point of above equation is that the flow through the soils is also

proportional to the first power of the hydraulic gradient i as propounded by

Posseuille’s Law. And the discharge is by Darcy’s equation is:

𝑄𝑄 = 𝐾𝐾𝑑𝑑ℎ𝑑𝑑𝑙𝑙𝐴𝐴 𝜋𝜋𝜋𝜋 𝑄𝑄 = 𝐾𝐾𝑖𝑖𝐴𝐴 (3.4𝑠𝑠)

where,

Q : discharge K : coefficient of permeability A : section area of aquifer dh : difference water elevation dl : length of aquifer i = dh/dl

c. Based on Dupuit (1863), according to Castany (1967): 𝑉𝑉 = 𝐾𝐾. 𝑖𝑖 ⇒ i = sin ⇒ 𝛼𝛼 𝜋𝜋𝑖𝑖𝑛𝑛𝛼𝛼 =

𝑑𝑑𝛿𝛿

�𝑑𝑑𝛿𝛿2 + 𝑑𝑑𝛿𝛿2

𝑉𝑉 = 𝐾𝐾𝑑𝑑𝛿𝛿

�𝑑𝑑𝛿𝛿2 + 𝑑𝑑𝛿𝛿2 =

𝐾𝐾

�1 + �𝑑𝑑𝛿𝛿𝑑𝑑𝛿𝛿�2

𝑑𝑑𝛿𝛿𝑑𝑑𝛿𝛿

dx

dy

𝛼𝛼

�𝑑𝑑𝛿𝛿2 + 𝑑𝑑𝛿𝛿2

𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴: 𝑖𝑖 =𝑑𝑑𝛿𝛿

�𝑑𝑑𝛿𝛿2 + 𝑑𝑑𝛿𝛿2∶𝑑𝑑𝛿𝛿𝑑𝑑𝛿𝛿

=1

�1 + �𝑑𝑑𝛿𝛿𝑑𝑑𝛿𝛿�2

𝑑𝑑𝛿𝛿𝑑𝑑𝛿𝛿


39

Due to the assumption of vertical velocity is so small, (dy/dx)2 can be neglected so :

�1 + �𝑑𝑑𝛿𝛿𝑑𝑑𝛿𝛿�

2

= 1 ⇒ 𝑉𝑉 = 𝐾𝐾𝑑𝑑𝛿𝛿𝑑𝑑𝛿𝛿

(3.5)

2). Similar equations

Fourier’s Law on heat transfer {Jean Baptiste Joseph Fourier (1768 – 1830)}:

𝑯𝑯 = 𝒌𝒌𝑴𝑴𝒅𝒅𝑻𝑻𝒅𝒅𝒅𝒅

𝒐𝒐𝒐𝒐 𝑯𝑯 = 𝒌𝒌𝑲𝑲𝑴𝑴 (3.6)

where,

H : rate of heat flow k : thermal conductivity A : cross section area dT : temperature difference dx : thickness i = dT/dx

Ohm’s Law on electrical current flow {George Simon Ohm (1787 - 1854)}:

𝑴𝑴 = 𝑪𝑪𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅

𝒂𝒂 𝒐𝒐𝒐𝒐 𝑴𝑴 = 𝑪𝑪𝑲𝑲𝒂𝒂 (3.7)

where,

I : current C : coefficient of conductivity a : sectional area of conductor dv : drop in voltage dl : length of conductor i : dv/dl


40

3). Validity of Darcy’ Law

𝑴𝑴𝑹𝑹 =𝝆𝝆𝝆𝝆𝝆𝝆𝝁𝝁

(3.8)

It can be written in other equation as:

𝑴𝑴𝑹𝑹 =γ𝝆𝝆𝝆𝝆𝝁𝝁𝝁𝝁

(3.9)

where, NR : Reynold’s Number D : diameter of pipe ρ : density of water ν : flow velocity µ : viscosity of fluid γ : unit weight of fluid g : acceleration of gravity

Experiments show that Darcy’s law is valid for NR < 1 and does not depart seriously up

to NR = 10, and this value represents an upper limit to the validity of Darcy’s law

(Todd, 1980).

4. The development of the post post Darcy

After Darcy developed his fundamental formula on his book Les fountains publiques de Dijon (1856), many researchers developed other formulas based on his, except Forhheimer (1930). The Darcy’s formula in vector form is very advance and easy to developed basic formula to explain for many condition of flow for instance together with Laplace equation or for mono-phase or bi-phase flow, but it is impossible to use it to compute the design of groundwater flow. Forchheimer (1930) simplified Darcy’s formula and introduce new parameter is shape factor (F) and this formula easily to use to compute the design of groundwater flow. Sunjoto (1988), developed new unsteady flow condition formula based on Forchheimer’s formula which is a steady state condition. Beside the unsteady state formula, Sunjoto developed more than 20 formulas of shape factors.


41

Figure 3.1. Diagram of development of groundwater science

DUPUIT (1863)

FORCHHEIMER (1930)

LUGEON (1930)

Sichardt Cambefort Choultse Koussakine Castany Kozen Bogomolov

DARCY (1856)

POISEUILLE (1797-1869) Qa=Z.i.A

FOURIER (1768-1830) H= K.i.A

OHM (1789 -1854 I=C.i..a

THEIS (1936)

Samsioe (1931), Dahler (1936), Taylor (1948), Hvorslev (1951), Aravin (1965), Wilkinson (1968), Al-Dahir & Morgenstern (1969), Luthian & Kirkham (1949), Kirkham & van Bavel (1948), Raymond & Azzouz (1969), Smiles & Young (),Sunjoto (1988; 2002)

Cooper-Jacob (1946) Chow (1952) Todd (1980)

Ehrenberger (1928), Vodgeo Institut (1954), Iokutaro Kano (1939), Vibert (1949), Castany (1967)

Mikel & Klaer (1956), Spiridonoff & Hantush (1964), Nasjono (2002), Das, Saha, Rao & Uththmanthan (2009)

F

Q, K Castany (1967) Murthy (1977) Suharjadi

S & T

SUNJOTO (1988-2010)

H, Q, K

Q

Q, K K Q, K, s

S & T

h’

Ri

Note: V : velocity Q : discharge h’ : drawdown correction S : K : permeability F : shape factor s : drawdown T : I : hydraulic head H : hydraulic head Ri : radius of depletion

V= K.i.A

Glover (1966)

s


42

2. Permeability of soils

a. Factors that affect permeability • Void ratio • Grain size • Temperature • Structure and stratification Interrelated of grain size and void ratio will affect permeability of soils. Smaller grain size, smaller void ratio which leads to reduce size of flow channels and lower permeability.

1). Void ratio The ratio of the volume of voids (Vv) to the volume of solids (Vs), is defined as

void ratio, and:

𝑒𝑒 =𝑉𝑉𝑒𝑒𝑉𝑉𝜋𝜋

𝐴𝐴𝑣𝑣 = 𝐴𝐴.𝑒𝑒

1 + 𝑒𝑒 (3.10)

The relationship between real pore channels to the idealized pore channel is:

𝐿𝐿 × 𝑠𝑠′ = 𝐿𝐿′ × 𝑠𝑠 (3.11) where,

L : length of idealized channel a : area of idealized channel L’ : length of real channel a’ : area of real channel

2). Grain size

If the cross section of a tube is circular, the flow in the tube as per Poiseuille’s Law is:

𝑞𝑞 =γ𝑤𝑤𝑅𝑅

2

8µ𝑠𝑠𝑖𝑖 (3.12)

The average velocity flow in the tube:


43

𝑣𝑣𝑠𝑠 =𝑞𝑞𝑠𝑠

=γ𝑤𝑤𝑅𝑅

2𝑖𝑖8µ

=γ𝑤𝑤𝑑𝑑

2

32µ. 𝑖𝑖 (3.13)

3). Temperature

The coefficient of permeability K is product of k’ which is dependent on

temperature and a function of the void ratio e, and the value of k’ is expressed:

𝑘𝑘′ =𝑠𝑠1

,

16𝜋𝜋𝜋𝜋22 .γ𝑤𝑤µ

=𝜋𝜋γ𝑤𝑤µ

(314)

Where, C is constant which is independent of temperature and the expression of K may now be as below and K varies as γw/µ.

𝐾𝐾 = 𝜋𝜋.𝐹𝐹. (𝑒𝑒).γ𝑤𝑤µ

(3.15)

4). Structure and stratification

Fig 3.2. Diagram of soil layers structure

a). Flow in the Horizontal Direction

Q = V.A = V. Z = K.i.Z

Q = (V1.Z1 + V2.Z2 + ………+ Vn-1.Zn-1 + Vn.Zn)

Q = (K1.i.Z1 + K2.i.Z2 + … + Kn-1.i.Zn-1 + Kn.i.Zn)

𝑲𝑲𝒅𝒅 =𝟏𝟏𝒁𝒁

(𝑲𝑲𝟏𝟏𝒁𝒁𝟏𝟏 + 𝑲𝑲𝟐𝟐𝒁𝒁𝟐𝟐 + ⋯+ 𝑲𝑲𝒏𝒏𝒁𝒁𝒏𝒏) (3.16)

K1

K2

Kn-1

Kn

Z1

Z2

Zn-1

Zn

Z

Kv ▼

Kh ►

V1.i.K1

V2.i.K2

Vn-1.i.Kn-i

Vn.i.Kn


44

b). Flow in the Vertical Direction

The hydraulic gradient is h/Z and:

𝑉𝑉 = 𝐾𝐾𝑣𝑣ℎ𝑍𝑍

= 𝐾𝐾1𝑖𝑖1 = 𝐾𝐾2𝑖𝑖2 = … … … …𝐾𝐾𝑛𝑛𝑖𝑖𝑛𝑛

If h1, h2 ………hn are the loss of heads in each of the layers, therefore:

H = h1 + h2 + …………hn

or, H = Z1h1 + Z2H2+ ……..ZnHn

Substitution:

𝑲𝑲𝒅𝒅 =𝒁𝒁

𝒁𝒁𝟏𝟏𝑲𝑲𝟏𝟏

+ 𝒁𝒁𝟐𝟐𝑲𝑲𝟐𝟐

+ ⋯+ 𝒁𝒁𝒏𝒏𝑲𝑲𝒏𝒏

(3.17)

b. Method of Determination

1). Laboratory Method

a). Constant head permeability method

The coefficient of permeability K is computed:

𝑸𝑸 = 𝑲𝑲𝒅𝒅𝑳𝑳𝑴𝑴 𝒕𝒕 (3.18)

𝑲𝑲 =𝑸𝑸𝑳𝑳𝒅𝒅 𝑴𝑴 𝒕𝒕

(3.19)

b). Falling head permeability method

The coefficient of permeability K can be determined on the basis of drop in head

(ho- h1 ) and the elapse time (t1- to).

𝑑𝑑𝑄𝑄 = 𝐾𝐾𝑖𝑖𝐴𝐴 𝑑𝑑𝑡𝑡 = 𝐾𝐾.ℎ𝐿𝐿

.𝐴𝐴 𝑑𝑑𝑡𝑡 (3.20)

𝑲𝑲 =𝒂𝒂𝑳𝑳

𝑴𝑴(𝒕𝒕𝟏𝟏 − 𝒕𝒕𝒐𝒐) 𝒅𝒅𝒏𝒏 �𝒅𝒅𝒐𝒐𝒅𝒅𝟏𝟏� (3.21)

when A = a the equation be:


45

𝑲𝑲 =𝑳𝑳

(𝒕𝒕𝟏𝟏 − 𝒕𝒕𝒐𝒐) 𝒅𝒅𝒏𝒏 �𝒅𝒅𝒐𝒐𝒅𝒅𝟏𝟏� (3.22)

where:

K : coefficient of permeability L : length of sample A : cross section area of sample a : cross section area stand pipe ho h1 : head of water in observation well 1 and 2 respectively to t1 : duration of flow in observation well 1 and 2 respectively

c). Computation from consolidation test data

In the case of materials of very low permeability with K less than 10-6 cm/s

consolidation test apparatus with permeability attachment may be used. The

coefficient of permeability K of sample can be computed from equation:

𝐾𝐾 =𝑄𝑄𝐿𝐿ℎ.𝐴𝐴. 𝑡𝑡

(3.23)

where, K : coefficient of permeability L : length of sample A : cross section area of sample Q : discharge in certain time t h : average head t : duration of flow

d). Computation from grain size distribution

On the basis of Poiseuille’s Law the coefficient of permeability can be computed:

𝐾𝐾 = 𝜋𝜋𝐶𝐶2 (3.41)

According to Allen Hazen (1911) in Murthy (1977) the empirical equation can be

computed as:

𝐾𝐾 = 𝜋𝜋𝐶𝐶102 (3.24)

where, K : coefficient of permeability (cm/s) C : a factor (100 <C< 150) D10 : effective size of grain (cm)


46

e). Computation from horizontal capillary test

This method base on the Darcy’s Law and compute the K are sometimes used

where the soil permeability fall within the range of 10-3 to 10-6 cm/s but this

method not very accurate (Murthy, 1974).

2). Field Methods

The field method of permeability test of soils usually carried out by pumping test

or bore hole test. That is why the parameters of testing are similar to the

parameters of radial flow for instance discharge, coefficient of permeability and

some of them is shape factors, so this matter will be discused in the next section

on paragraph V.


47

IV. RADIAL FLOW

Assumptions for the equations are (Dupuit-Thiem):

The soils surrounding the well is assumed homogeneous

The flow towards the well is assumed as steady, laminar, radial and horizontal

The horizontal velocity is independent of depth

The ground water table is assumed as horizontal in all direction

The hydraulic gradient at any point on the drawdown is equal to the slope of

the tangent at the point. According to Castany G. (1967) that value is sinus at

the point.

1. Unconfined aquifer

a. Dupuit (1863)

Fig. 4.1. Circular unconfined aquifer

Let h be the depth of water at radial distance r. The area of the vertical cylindrical

surface of radius r and depth h through which water flow is:

A = 2πrh (4.1)

rw r

R

hw

h H


48

The hydraulic gradient is:

𝑖𝑖 =𝑑𝑑ℎ𝑑𝑑𝜋𝜋

(4.2)

Discharge of inflow when the water levels in the well remain stationary (Darcy’s Law) V = Ki (4.3) Q = KiA (4.4) Substituting for Eqn (4.1) and (4.2) for (4.3), the rate inflow across the cylindrical surface is:

𝑄𝑄 = 𝐾𝐾𝑑𝑑ℎ𝑑𝑑𝜋𝜋

2π𝜋𝜋ℎ (4.5)

The equation for discharge outflow from pumping is:

𝑸𝑸 = 𝝅𝝅𝑲𝑲(𝑯𝑯𝟐𝟐 − 𝒅𝒅𝒘𝒘𝟐𝟐 )

𝒅𝒅𝒏𝒏 � 𝑹𝑹𝒐𝒐𝒘𝒘�

(4.6)

The equation for permeability of soil is:

𝑲𝑲 =𝑸𝑸

𝝅𝝅(𝑯𝑯𝟐𝟐 − 𝒅𝒅𝒘𝒘𝟐𝟐 ) 𝒅𝒅𝒏𝒏 �𝑹𝑹𝒐𝒐𝒘𝒘� (4.7)

where, H : depth of water outside of aquifer layer hw : depth of water at face of pumping well R : radius of outside of aquifer layer rw : radius of pumped well


49

b. Dupuit-Thiem

1). According to UNESCO (1967),

G. Thiem (1906) based on Dupuit and Darcy principle developed a formula

of pumping and the formula is called Dupuit-Thiem.

Let h be the depth of water at radial distance r (Fig. 4.2.). The area of the vertical cylindrical surface of radius r and depth h through which water flow is:

Fig. 4.2. Pumping in unconfined aquifer

Area of cylinder of piezometric h and radius r: A = 2πrh

The hydraulic gradient is: 𝑖𝑖 =𝑑𝑑ℎ𝑑𝑑𝜋𝜋

Darcy’s Law: V = Ki and Q = KiA Substituting, so the rate inflow across the cylindrical surface is:

𝑄𝑄 = 𝐾𝐾𝑑𝑑ℎ𝑑𝑑𝜋𝜋

2𝜋𝜋𝜋𝜋ℎ (4.8)

Rearranging the terms, so:

r1 r

r2

h1 h h2


50

𝑑𝑑𝜋𝜋𝜋𝜋

=2𝜋𝜋𝑘𝑘𝑄𝑄

ℎ𝑑𝑑ℎ

The equation for permeability of soil is:

𝑲𝑲 =𝑸𝑸

𝝅𝝅�𝒅𝒅𝟐𝟐𝟐𝟐 − 𝒅𝒅𝟏𝟏𝟐𝟐�𝒅𝒅𝒏𝒏 �

𝒐𝒐𝟐𝟐𝒐𝒐𝟏𝟏� (4.9)

The equation for discharge outflow from pumping is (Fig, 4.2):

Dupuit-Thiem Formula for the full penetration well in free aquifer:

𝑸𝑸 = 𝝅𝝅𝑲𝑲𝒅𝒅𝟐𝟐𝟐𝟐 − 𝒅𝒅𝟏𝟏𝟐𝟐

𝒅𝒅𝒏𝒏 �𝒐𝒐𝟐𝟐𝒐𝒐𝟏𝟏�

(4.10)

where,

Q : discharge of pumping K : coefficient of permeability D : thickness of aquifer layer r1 r2 : distance from well to observation well 1 and 2 respectively h1 h2 : head of water in observation well 1 and 2 respectively

2). According to Castany (1967)

G. Thiem (1906) based on Dupuit principle developed a formula of pumping in unconfined aquifer and the formula is called Dupuit-Thiem (Fig. 4.3.).

Darcy’s law:

𝑄𝑄 = 2𝐾𝐾𝜋𝜋ℎ𝑑𝑑𝜋𝜋𝑑𝑑ℎ

(4.11)

dr/dh = tgα (4.12)

𝑄𝑄 = 2𝜋𝜋𝐾𝐾𝜋𝜋ℎ. tgα (4.13)


51


tgα =∆1 − ∆2

r2 − r1 (4.14)

For first permanent regime:

𝑄𝑄 = 2𝜋𝜋𝐾𝐾𝜋𝜋1ℎ1. tgα (4.15) For second permanent regime:

𝑄𝑄 = 2𝜋𝜋𝐾𝐾𝜋𝜋1ℎ11. tgα1 (4.16)

Dupuit-Thiem equation for the full penetration well in free aquifer:

𝑸𝑸 = 𝝅𝝅𝑲𝑲(𝐡𝐡𝟏𝟏 + 𝐡𝐡𝟐𝟐)(∆𝟏𝟏 − ∆𝟐𝟐)


(4.17)

𝑲𝑲 =𝑸𝑸

𝝅𝝅(𝐡𝐡𝟏𝟏 + 𝐡𝐡𝟐𝟐)(∆𝟏𝟏 − ∆𝟐𝟐) 𝒅𝒅𝒏𝒏 �𝒐𝒐𝟐𝟐𝒐𝒐𝟏𝟏� (4.18)

where:

Q : discharge of pumping K : coefficient of permeability r1 r2 : distance from well to observation well 1 and 2 respectively ∆1 ∆2 : drawdown in observation well 1 and 2 respectively

r1 r2

h1

h2

∆2 ∆1

∆w

hw

rw

Ri

H


52

3). According to Murthy V.N.S. (1977) Murthy developed the formula for unconfined aquifer by other parameters

and can be found as (Fig.5.3.):

𝑸𝑸 =𝝅𝝅𝑲𝑲(𝑯𝑯𝟐𝟐 − 𝒅𝒅𝒘𝒘𝟐𝟐 )

𝒅𝒅𝒏𝒏 �𝑹𝑹𝑲𝑲𝒐𝒐𝒘𝒘�

(4.19)

𝑲𝑲 =𝑸𝑸

𝝅𝝅(𝑯𝑯𝟐𝟐 − 𝒅𝒅𝒘𝒘𝟐𝟐 ) 𝒅𝒅𝒏𝒏 �𝑹𝑹𝑲𝑲𝒐𝒐𝒘𝒘� (4.20)

If we write hw = (H - ∆w) where ∆w is the depth of maximum drawdown in the test well or pumped well so (Castany, 1967):

𝐾𝐾 − ℎ𝑤𝑤 = ∆𝑊𝑊

𝑸𝑸 =𝝅𝝅∆𝒘𝒘𝑲𝑲(𝟐𝟐𝑯𝑯 − ∆𝒘𝒘)


(4.21)

𝑲𝑲 =𝑸𝑸

𝝅𝝅∆𝒘𝒘(𝟐𝟐𝑯𝑯 − ∆𝒘𝒘) 𝒅𝒅𝒏𝒏 �𝑹𝑹𝑲𝑲𝒐𝒐𝒘𝒘� (4.22)

where:

Q : discharge of pumping K : coefficient of permeability Ri : radius of influence rw : radius of pumped well H : depth of water before pumping ∆w : maximum drawdown (on well)


53

2. Confined aquifer a. Dupuit (1863)


𝑉𝑉 = 𝐾𝐾𝑖𝑖

𝑄𝑄 = 𝑉𝑉.𝐴𝐴 = 𝐾𝐾𝑑𝑑𝛿𝛿𝑑𝑑𝛿𝛿

2𝜋𝜋𝛿𝛿𝐶𝐶 𝑑𝑑𝛿𝛿𝛿𝛿𝑄𝑄 = 2𝜋𝜋𝐾𝐾𝐶𝐶𝛿𝛿

𝑄𝑄𝑙𝑙𝑛𝑛𝛿𝛿]𝜋𝜋𝑤𝑤

𝑅𝑅 = 2𝜋𝜋𝐾𝐾𝐶𝐶𝛿𝛿]ℎ𝑤𝑤𝐾𝐾

Dupuit (1863) formula for full penetration well on confined aquifer: 𝑸𝑸 = 𝟐𝟐𝝅𝝅𝑲𝑲𝝆𝝆

𝑯𝑯 − 𝒅𝒅𝒘𝒘

𝒅𝒅𝒏𝒏 � 𝑹𝑹𝒐𝒐𝒘𝒘�

(4.23)

𝑲𝑲 =𝑸𝑸

𝟐𝟐𝝅𝝅𝝆𝝆(𝑯𝑯− 𝒅𝒅𝒘𝒘) 𝒅𝒅𝒏𝒏 �𝑹𝑹𝒐𝒐𝒘𝒘� (4.24)

where,

Q : discharge of pumping K : coefficient of permeability D : thickness of aquifer R : radius of influence rw : radius of pumped well H : depth of water outside of aquifer layer hw : depth of water at face of pumping well

hw H

D

rw R


54

b. Dupuit-Thiem (1906)

1). According to UNESCO (1967)

Fig. 4.5. Circular unconfined embankment

𝑉𝑉 = 𝐾𝐾𝑖𝑖 Dupuit-Thiem formula for full penetration well on confined aquifer:

𝑸𝑸 = 𝟐𝟐𝝅𝝅𝑲𝑲𝝆𝝆𝒅𝒅𝟐𝟐 − 𝒅𝒅𝟏𝟏𝒅𝒅𝒏𝒏 �𝒐𝒐𝟐𝟐𝒐𝒐𝟏𝟏

� (4.25)

𝑲𝑲 =𝑸𝑸

𝟐𝟐𝝅𝝅𝝆𝝆(𝒅𝒅𝟐𝟐 − 𝒅𝒅𝟏𝟏) 𝒅𝒅𝒏𝒏 �𝒐𝒐𝟐𝟐𝒐𝒐𝟏𝟏� (4.26)

where,

Q : discharge of pumping K : coefficient of permeability D : thickness of aquifer r1 r2 : distance from well to observation well 1 and 2 respectively h1 h2 : head of water in observation well 1 and 2 respectively

h1 h2 D

r1 r2


55

2). According to Castany (1967)


Dupuit-Thiem equation for the full penetration well in confined aquifer:

𝑸𝑸 = 𝟐𝟐𝝅𝝅𝑲𝑲𝝆𝝆(∆𝟏𝟏 − ∆𝟐𝟐)


(4.27)

𝑲𝑲 =𝑸𝑸

𝟐𝟐𝝅𝝅𝝆𝝆(∆𝟏𝟏 − ∆𝟐𝟐) 𝒅𝒅𝒏𝒏 �𝒐𝒐𝟐𝟐𝒐𝒐𝟏𝟏� (4.28)

where:

Q : discharge of pumping K : coefficient of permeability D : thickness of aquifer layer r1 r2 : distance from well to observation well 1 and 2 respectively ∆1 ∆2 : drawdown in observation well 1 and 2 respectively

3. Alternate equations of the Dupuit-Thiem principle for radial flow are:

1). Pumping in circular aquifer

a). Unconfined aquifer:

o Without observation well and with piezometric head data:

r1

r2

h2

∆2 ∆1

h1

D


56

𝑲𝑲 =𝑸𝑸

𝝅𝝅(𝑯𝑯𝟐𝟐 − 𝒅𝒅𝒘𝒘𝟐𝟐 ) 𝒅𝒅𝒏𝒏 �𝑹𝑹𝒐𝒐𝒘𝒘� (4.29)

o Without observation well and with drawdown data:

𝑲𝑲 =𝑸𝑸

𝝅𝝅∆𝒘𝒘(𝟐𝟐𝑯𝑯 − ∆𝒘𝒘) 𝒅𝒅𝒏𝒏 �𝑹𝑹𝒐𝒐𝒘𝒘� (4.30)

b). Confined aquifer:


𝑲𝑲 =𝑸𝑸

𝟐𝟐𝝅𝝅𝝆𝝆(𝑯𝑯− 𝒅𝒅𝒘𝒘) 𝒅𝒅𝒏𝒏 �𝑹𝑹𝒐𝒐𝒘𝒘� (4.31)

2). Pumping in unlimited aquifer

a). Unconfined aquifer:


𝑲𝑲 =𝑸𝑸

𝝅𝝅(𝑯𝑯𝟐𝟐 − 𝒅𝒅𝒘𝒘𝟐𝟐 ) 𝒅𝒅𝒏𝒏 �𝑹𝑹𝑲𝑲𝒐𝒐𝒘𝒘� (4.32)

o Without observation well and with drawdown data:

𝑲𝑲 =𝑸𝑸

𝝅𝝅∆𝒘𝒘(𝟐𝟐𝑯𝑯 − ∆𝒘𝒘) 𝒅𝒅𝒏𝒏 �𝑹𝑹𝑲𝑲𝒐𝒐𝒘𝒘� (4.33)

o With one observation well and with piezometric head data:

𝑲𝑲 =𝑸𝑸

𝝅𝝅�𝒅𝒅𝟏𝟏𝟐𝟐 − 𝒅𝒅𝒘𝒘𝟐𝟐 �𝒅𝒅𝒏𝒏 �

𝒐𝒐𝟏𝟏𝒐𝒐𝒘𝒘� (4.34)

o With one observation well and with drawdown data:


57

𝑲𝑲 =𝑸𝑸

𝝅𝝅∆𝒘𝒘(𝟐𝟐𝒅𝒅𝟏𝟏 − ∆𝒘𝒘) 𝒅𝒅𝒏𝒏 �𝒐𝒐𝟏𝟏𝒐𝒐𝒘𝒘� (4.35)

𝑲𝑲 =𝑸𝑸

𝝅𝝅(𝒅𝒅𝒘𝒘 + 𝒅𝒅𝟏𝟏)(∆𝒘𝒘 − ∆𝟏𝟏) 𝒅𝒅𝒏𝒏 �𝒐𝒐𝟏𝟏𝒐𝒐𝒘𝒘� (4.36)

o With two observation wells data and piezometric head data:

𝑲𝑲 =𝑸𝑸


𝒐𝒐𝟐𝟐𝒐𝒐𝟏𝟏� (4.37)

o With two observation wells and drawdown data:

𝑲𝑲 =𝑸𝑸

𝝅𝝅(𝒅𝒅𝟏𝟏 + 𝒅𝒅𝟐𝟐)(∆𝟏𝟏 − ∆𝟐𝟐) 𝒅𝒅𝒏𝒏 �𝒐𝒐𝟐𝟐𝒐𝒐𝟏𝟏� (4.38)

b). Confined aquifer:


𝑲𝑲 =𝑸𝑸

𝟐𝟐𝝅𝝅𝝆𝝆(𝑯𝑯− 𝒅𝒅𝒘𝒘) 𝒅𝒅𝒏𝒏 �𝑹𝑹𝑲𝑲𝒐𝒐𝒘𝒘� 𝜋𝜋𝜋𝜋 𝑲𝑲 =

𝑸𝑸𝟐𝟐𝝅𝝅𝝆𝝆.∆𝐰𝐰

𝒅𝒅𝒏𝒏 �𝑹𝑹𝑲𝑲𝒐𝒐𝒘𝒘� (4.39)

o With one observation well and with piezometric head data:

𝑲𝑲 =𝑸𝑸

𝟐𝟐𝝅𝝅𝝆𝝆(𝒅𝒅𝟏𝟏 − 𝒅𝒅𝒘𝒘) 𝒅𝒅𝒏𝒏 �𝒐𝒐𝟏𝟏𝒐𝒐𝒘𝒘� 4.40)

o With one observation well and with drawdown data:

𝑲𝑲 =𝑸𝑸

𝟐𝟐𝝅𝝅𝝆𝝆(∆𝐰𝐰 − ∆𝟏𝟏) 𝒅𝒅𝒏𝒏 �𝒐𝒐𝟏𝟏𝒐𝒐𝒘𝒘� (4.41)

o With two observations well and piezometric head data:

𝑲𝑲 =𝑸𝑸


o With two observations well and drawdown data:


58

𝑲𝑲 =𝑸𝑸

𝟐𝟐𝝅𝝅𝝆𝝆(∆𝟏𝟏 − ∆𝟐𝟐) 𝒅𝒅𝒏𝒏 �𝒐𝒐𝟐𝟐𝒐𝒐𝟏𝟏� (4.43)

𝑸𝑸 = 𝟐𝟐𝝅𝝅𝑲𝑲𝝆𝝆(∆𝟏𝟏 − ∆𝟐𝟐)


(4.44)

4. Field test of soils permeability

a). Pumping test

The pumping test method is equal to the method of computing discharge from the

well using equation of Dupuit or Dupuit-Thiem for confined and unconfined aquifer

as mentioned in above article.

b). Casing Bore hole test

1). Murthy (1977)

According to Murthy (1977), hydraulic gradient of the some conditions are:

(a). Without pressure and end casing above groundwater table

H = hw (4.45)

(b). Without pressure and end casing below groundwater table

H = hw’ (4.46)

(c). With pressure and end casing above groundwater table

H = hw + hp (4.47)

(d). With pressure and end casing below groundwater table

H = hw’ + hp (4.48)


59

Fig 4.7. Bore hole in some conditions

The coefficient of permeability is calculated by making use of formula:

𝐾𝐾 =0.18 𝑄𝑄𝜋𝜋 𝐾𝐾

(4.49)

where: Q : discharge (L3/T) K : coefficient of permeability (L/T) H : hydraulic head (L) Fig. 3.2.

Note: Compare to Forchheimer (1930) that Q= FKH and to Harza (1935), Taylor (1948) and

Hvorslev (1951) that F = 5,5 r. And Sunjoto (2002) developed the formula for the

same condition that F = 2πr.

2). Forchheimer (1930)

Forchheimer (1930) proposed to find a coefficient of permeability (K) by bore

hole with certain diameter and depth.

𝑲𝑲 =𝝅𝝅𝑹𝑹𝟐𝟐

𝑭𝑭(𝒕𝒕𝟐𝟐 − 𝒕𝒕𝟏𝟏) 𝒅𝒅𝒏𝒏𝒅𝒅𝟐𝟐𝒅𝒅𝟏𝟏

(4.50)

Q Q Q & hp Q & hp

hw

hw’

hw

hw’

(2). H=hw‘ (3). H=hw+ hp (4). H=hw’+ hp

Hb

Hg


60

where: K : coefficient of permeability (L/T) R : radius of well (L) F : shape factor (L) (F = 4 R, Forchheimer, 1930) t1 t2 : time of the measurement respectively (T) h1 h2 : height of water of the measurement respectively (L)

As : cross section area of well (L2 , As = π R2)

c). Partial permeable casing bore hole test

1). Suharyadi (1984)

There are two conditions of hydraulic head (Fig. 3.3) as:

• The hole is submerged in groundwater:

H = difference of groundwater table to the water elevation test

• The hole above the groundwater table:

H = Depth of water test on the hole minus half of permeable hole length

Fig. 4.8 Hydraulic head dimension on bore hole test according to Suharyadi (1984)

Q

Q

(2). The hole test above ground water table

L L

2R 2R

gwt Hw

Hw

(1). The hole test below ground water table

gwt

(H=Hw) H=Hc+1/2L


61

The coefficient of permeability can be computed by:

𝐾𝐾 =2.30 𝑄𝑄2𝜋𝜋𝐿𝐿𝐾𝐾

𝑙𝑙𝜋𝜋𝑔𝑔𝐿𝐿𝑅𝑅

=𝑄𝑄

2𝜋𝜋𝐿𝐿𝐾𝐾𝑙𝑙𝑛𝑛𝐿𝐿𝑅𝑅

(4.51)

where,

K : coefficient of permeability L : length of permeable part H : Hydraulic head (L ≥ R) R : radius of casing

d). Uncasing bore hole test

1). Pecker test

• Suharyadi (1984)

𝑲𝑲 =𝟐𝟐.𝟑𝟑𝟑𝟑 𝑸𝑸𝟐𝟐𝝅𝝅𝑳𝑳𝑯𝑯

𝒅𝒅𝒐𝒐𝝁𝝁𝑳𝑳𝑹𝑹

=𝑸𝑸

𝟐𝟐𝝅𝝅𝑳𝑳𝑯𝑯𝒅𝒅𝒏𝒏

𝑳𝑳𝑹𝑹

(4.52)

𝑯𝑯 = 𝑯𝑯𝟏𝟏 + 𝑯𝑯𝟐𝟐 (4.53)

Fig. 4.9. Hydraulic head dimension on packer test (after Suharyadi, 1984)

Q and H2 Q and H2 Q and H2 Q and H2

(b). One pecker test which zone test is above groundwater table

(c). Two peckers test which zone test is submerged

(d). Two peckers test which zone test is above groundwater table

L

2R

L L

2R 2R 2R

(a). One pecker test which zone test is submerged

gwt

gwt

gwt

gwt

H1

H1

1/2L

H1 H1

1/2L L


62

2). Boast and Kirkham (in Todd, 1980)

𝑲𝑲 =𝑪𝑪𝟖𝟖𝟖𝟖𝟖𝟖

.𝒅𝒅𝒅𝒅𝒅𝒅𝒕𝒕

(4.54)

Fig. 4.10. Diagram of auger hole and dimensions for determining coefficient of permeability (after Boast and Kirkham, in Todd, 1980)

3). Sunjoto (1988)

𝐾𝐾 =𝑄𝑄𝐹𝐹𝐾𝐾

�1 − 𝑒𝑒𝛿𝛿𝑒𝑒 �−𝐹𝐹𝐾𝐾𝐹𝐹𝜋𝜋𝑅𝑅2 �� (4.55)

where: H : depth of hollow well (L) F : shape factor (L) K : coefficient of permeability (L/T) Q : inflow discharge (L3/T), dan Q = C I A C : runoff coefficient of roof ( ) I : precipitation intensity (L/T) A : roof area (L2)

Note:

• When steady flow condition (8.53) become F =Q/KH • The solution of this equation by trial and error.

Lw y

2rw H


63

Table 4.1. Value of C after Boast and Kirkham (in Todd, 1980) Lw/ rw

y/ Lw

(H-Lw)/Lw for Impermeable Layer

H-Lw (H-Lw)/Lw for Infinitely Impermeable Layer

0 0.05 0.1 0.2 0.5 1 2 5 ∞ 5 2 1 0.5

1 1.00 447 423 404 375 323 286 264 255 254 252 241 213 166 0.75 469 450 434 408 360 324 303 292 291 289 278 248 198 0.50 555 537 522 497 449 411 386 380 379 377 359 324 264

2 1.00 186 176 167 154 134 123 118 116 115 115 113 106 91

0.75 196 187 180 168 149 138 133 131 131 130 128 121 106 0.50 234 225 218 207 188 175 169 167 167 166 164 156 139

5 1.00 51.9 48.6 46.2 42.8 38.7 36.9 36.1 35.8 35.5 34.6 32.4

0.75 54.8 52.0 49.9 46.8 42.8 41.0 40.2 40.0 39.6 38.6 36.3 0.50 66.1 63.4 61.3 58.1 53.9 51.9 51.0 50.7 40.3 49.2 466

10 1.00 18.1 16.9 16.1 15.1 14.1 13.6 13.4 13.4 13.3 13.1 12.6

0.75 19.1 18.1 17.4 16.5 15.5 15.0 14.8 14.8 14.7 14.5 14.0 0.50 23.3 22.3 21.5 20.6 19.5 19.0 18.8 18.7 18.6 18.4 17.8

20 1.00 59.1 55.3 53.0 50.6 48.1 47.0 46.6 46.4 46.2 45.8 44.6

0.75 62.7 59.4 57.3 55.0 52.5 51.5 51.0 50.8 50.7 50.2 48.9 0.50 76.7 73.4 71.2 68.8 66.0 64.8 64.3 64.1 63.9 63.4 61.9

50 1.00 1.25 1.28 1.14 1.11 1.07 1.05 1.04 1.03 1.02

0.75 1.33 1.27 1.23 1.20 1.16 1.14 1.13 1.12 1.11 0.50 1.64 1.57 1.54 1.50 1.46 1.44 1.43 1.42 1.39

100 1.00 0.37 0.35 0.34 0.34 0.33 0.32 0.32 0.32 0.31

0.75 0.40 0.38 0.37 0.36 0.35 0.35 0.35 0.34 0.34 0.50 0.49 0.47 0.46 0.45 0.44 0.44 0.44 0.43 0.43

Table 4.2. Coefficient of Permeability of some Soils (Casagrande and Fadum)

K (cm/sec) Soils type Drainage Condition

Recommended method of determining K

101 - 102 Clean gravels Good Pumping Test

101 Clean sand Good Constant head or Pumping test

10-1 – 10-4 Clean sand and gravel

mixtures

Good Constant head, Falling head

or Pumping test

10-5 Very fine sand Poor Falling head

10-6 Silt Poor Falling head

10-7 – 10-9 Clay soils Practically impervious

Consolidation test


64

e). Lugeon Test

Maurice Lugeon (July 10, 1870 - October 23, 1953) was a Swiss geologist, and

the pioneer of nape tectonics. He was a pupil of Eugène Renevier. The Lugeon test,

extensively used in Europe, is a special case of double packer bore hole inflow test

made at constant head.

Lugeon is a measure of transmissivity in rocks, determined by pressurized

injection of water through a bore hole driven through the rock.

o One Lugeon (LU) is equal to one liter of water per minute injected into 1 meter

length of borehole at an injection pressure of 10 bars.

o 1 Lugeon Unit = a water take of 1 liter per meter per minute at a pressure of 10

bars.

o Lugeon value : water take (liter/m/min) x 10 bars/test pressure (in bars)

The Lugeon unit is not strictly a measure of hydraulic conductivity but it is a good

approximation for grouting purposes and 1 Lugeon is approximately equivalent to

1x10-5 cm/s or 1x10-7 m/s.

The three successive test runs, each of 5 minutes duration enable a rough

assessment of the water behavior.

Uncertainaty of Lugeon Unit Theory:

• The test carried out 5 minute only, so you don’t know is the flow in stady

state or unsteady state flow condition

• The layer of the soil or rock have no to be taken as consideration

• The diameter of hol is not be taken as consideration.

http://en.wikipedia.org/wiki/July_10�

http://en.wikipedia.org/wiki/1870�

http://en.wikipedia.org/wiki/October_23�

http://en.wikipedia.org/wiki/1953�

http://en.wikipedia.org/wiki/Swiss�

http://en.wikipedia.org/w/index.php?title=Nappe_tectonics&action=edit�

http://en.wikipedia.org/wiki/Eug%C3%A8ne_Renevier�

http://en.wikipedia.org/w/index.php?title=Lugeon&action=edit�


65

Analysis:

This test will be analyzed by principle of:

• Forhheimer (1930) for steady flow condition

𝑸𝑸 = 𝑭𝑭𝑲𝑲𝑯𝑯 ⇒ 𝑲𝑲 =𝑸𝑸𝑭𝑭𝑯𝑯

• Sunjoto (2010) for un steady flow condition



• Sunjoto (2010) for the shape factor of each condition (Fig.4.11.)

Fig. 4.11. Schematic of condition of well and packers location.

To compute the value of Shape Factor, Sunjoto (2010) proposed formula for

three conditions of well as:

• Condition of (a) well Fig. 4.11.(a):

𝐹𝐹𝑠𝑠 =2π𝐿𝐿

𝑙𝑙𝑛𝑛 �𝑠𝑠(𝐿𝐿 + 2𝑅𝑅)𝑏𝑏𝑅𝑅 + ��𝑠𝑠𝐿𝐿𝑏𝑏𝑅𝑅�

2+ 1�

(4.56)

Q in 10 bar Q in 10 bar Q in 10 bar Q in 10 bar

(2). Condition of well b. (3). Condition of well b. (4). Condition of well c.

L

2R

L L L

2R 2R 2R

(1). Condition of well a.


66

• Condition of well (b) Fig. 4.11.(b):

𝐹𝐹𝑏𝑏 =2π𝐿𝐿

𝑙𝑙𝑛𝑛 �𝑐𝑐(𝐿𝐿 + 2𝑅𝑅)𝑑𝑑𝑅𝑅 + ��𝑐𝑐𝐿𝐿𝑑𝑑𝑅𝑅�

2+ 1�

(4.57)

• Condition of well (c) Fig. 4.11.(c):

𝐹𝐹𝑐𝑐 =2π𝐿𝐿

𝑙𝑙𝑛𝑛 �𝑒𝑒(𝐿𝐿 + 2𝑅𝑅)𝑓𝑓𝑅𝑅 + ��𝑒𝑒𝐿𝐿𝑓𝑓𝑅𝑅�

2+ 1�

(4.58)

The coefficient of a,b,c,d,e and f must be definite by pumping test. 3). Special case of confined aquifer

According to Murthy (1977), figure below shows a confined aquifer with the test well and two observation wells. The elevation of water in the observation wells rises above the top of the aquifer due to artesian pressure. When pumping at steady flow condition from artesian well two cases might found they are:

Case 1: The water level in the test well might remain above the roof level (hw > D)

Case 2: The water level in the test well might fall below the roof level (hw < D)


rw r1

Ri

D hw

h1 h

r

H

Case 1

Case 2


67

Case 1: (hw > D)

𝑸𝑸 = 𝟐𝟐𝝅𝝅𝑲𝑲𝝆𝝆𝒅𝒅𝟐𝟐 − 𝒅𝒅𝟏𝟏𝒅𝒅𝒏𝒏 �𝒐𝒐𝟐𝟐𝒐𝒐𝟏𝟏

� (4.59)

𝑲𝑲 =𝑸𝑸


This equation is like mention above.

Case 2: (hw < D)

𝑸𝑸 =𝝅𝝅𝑲𝑲(𝟐𝟐𝑯𝑯𝝆𝝆 − 𝝆𝝆𝟐𝟐 − 𝒅𝒅𝒘𝒘𝟐𝟐 )


(4.61)

𝑲𝑲 =𝑸𝑸

𝝅𝝅(𝟐𝟐𝑯𝑯𝝆𝝆 − 𝝆𝝆𝟐𝟐 − 𝒅𝒅𝒘𝒘𝟐𝟐 ) 𝒅𝒅𝒏𝒏 �𝑹𝑹𝑲𝑲𝒐𝒐𝒘𝒘� (4.62)

4. Correction to flow line


a. Castany (1967) implemented Dupuit (1868) equation:

For the lateral flow:

𝑄𝑄 = 𝐾𝐾𝐾𝐾2 − (ℎ + ℎ′)2

2𝑅𝑅

Real curve

Theoretic curve

h h+h’

H


68

𝒅𝒅𝟐𝟐 − (𝒅𝒅 − 𝒅𝒅′)𝟐𝟐 = [𝑯𝑯𝟐𝟐 − (𝒅𝒅 − 𝒅𝒅′)𝟐𝟐]𝒅𝒅𝑹𝑹

(4.63) For the free aquifer and parallel flow:

𝑄𝑄 = 𝜋𝜋𝐾𝐾𝐾𝐾2 − (ℎ + ℎ′)2

𝑙𝑙𝑛𝑛 �𝑅𝑅𝜋𝜋 �

𝒅𝒅𝟐𝟐 − (𝒅𝒅 − 𝒅𝒅′)𝟐𝟐 = [𝑯𝑯𝟐𝟐 − (𝒅𝒅 − 𝒅𝒅′)𝟐𝟐]𝒅𝒅𝒏𝒏 �𝒅𝒅𝒐𝒐�

𝒅𝒅𝒏𝒏 �𝑹𝑹𝒐𝒐� (4.64)

b. Ehrenberger (1928)

𝒅𝒅′ = 𝟑𝟑,𝟓𝟓(𝑯𝑯 −𝒅𝒅)𝟐𝟐

𝑯𝑯 (4.65)

a. Vodgeo Institut (1954)

𝒅𝒅′ = 𝟑𝟑,𝟓𝟓(𝑯𝑯− 𝒅𝒅)𝟐𝟐,𝟐𝟐 (4.66)

b. Iokutaro Kano (1939)

𝒅𝒅′ = 𝑪𝑪𝑸𝑸𝒅𝒅 (4.67)

0,324 < C < 1,60

c. Vibert (1949)

𝒅𝒅′ = 𝟑𝟑,𝟓𝟓 ��𝑹𝑹𝟐𝟐 + 𝟖𝟖𝑯𝑯𝟐𝟐 −𝑹𝑹� (4.68)

5. Radius of depletion

According to many researchers, the radius of depletion depends on the depression cone because the drawdown of pumping:


69

a. W.Sichardt (in Castany, 1967)

𝑹𝑹𝑲𝑲 = 𝟑𝟑𝟑𝟑𝟑𝟑𝟑𝟑(𝑯𝑯− 𝒅𝒅)√𝑲𝑲 (4.69) where,

Ri : radius of depletion (m) H – h : drawdown (m) K : permeability (m/s)

b. H.Cambefort (in Castany, 1967)

𝑹𝑹𝑲𝑲 = 𝟓𝟓𝟓𝟓𝟑𝟑�𝑯𝑯𝑲𝑲𝑲𝑲𝟖𝟖 (4.70)

where,

Ri : radius of depletion (m) H : drawdown (m) Ki : permeability (m/s)

c. I. Choultse (in Castany, 1967)

𝑹𝑹𝑲𝑲 = �𝟖𝟖𝑯𝑯𝑲𝑲𝑻𝑻𝒎𝒎𝒆𝒆

(4.71)

where,

me : porosity of soil T : duration of pumping (s or h) H : drawdown (m) K : permeability (m/s or m/h) Ri : radius of depletion (m)


70

d. I.P. Koussakine (in Castany, 1967)

𝑹𝑹𝑲𝑲 = 𝟖𝟖𝟒𝟒�𝟖𝟖𝑯𝑯𝑲𝑲𝑻𝑻𝒎𝒎𝒆𝒆

(4.72)

where,

K : permeability (m/s) T : duration of pumping (hour)

e. Dupuit 1). Lateral flow :

1). Dupuit (in Castany, 1967)

𝑹𝑹𝑲𝑲 = 𝑲𝑲𝑳𝑳𝑯𝑯𝟏𝟏𝟐𝟐 −𝑯𝑯𝟐𝟐

𝟐𝟐

𝟐𝟐𝑸𝑸 (4.73)

2). Castany (1967)

𝑹𝑹𝑲𝑲 = 𝑲𝑲𝑳𝑳𝑯𝑯𝟏𝟏𝟐𝟐 −𝑯𝑯𝟐𝟐

𝟐𝟐

𝑸𝑸 (4.74)

2). Radial flow (in Castany, 1967):

Using Darcy’s Law, Castany (1967) proposed an equation:

𝑙𝑙𝑛𝑛𝑹𝑹𝑲𝑲 =𝑄𝑄

𝜋𝜋𝐾𝐾(𝐾𝐾2 − ℎ2) + 𝑙𝑙𝑛𝑛𝜋𝜋 (4.75)

Sunjoto tried to improve above formula as:

𝒅𝒅𝒏𝒏𝑹𝑹𝑲𝑲 =𝝅𝝅𝑲𝑲(𝑯𝑯𝟐𝟐 − 𝒅𝒅𝟐𝟐)

𝑸𝑸+ 𝒅𝒅𝒏𝒏𝒐𝒐

𝑹𝑹𝑲𝑲𝜋𝜋 = 𝑒𝑒𝛿𝛿𝑒𝑒�

𝝅𝝅𝑲𝑲(𝑯𝑯𝟐𝟐 − 𝒅𝒅𝟐𝟐)𝑸𝑸

�

𝑹𝑹𝑲𝑲 = 𝒐𝒐.𝒆𝒆𝒅𝒅𝒆𝒆 �𝝅𝝅𝑲𝑲(𝑯𝑯𝟐𝟐 − 𝒅𝒅𝟐𝟐)

𝑸𝑸� (4.76)


71

where,

Ri : radius of depletion (m) r : radius of observation well location (m) Q : discharge (m3/h) H : drawdown (m) K : permeability (m/h) h : height of water on observation well (m)

f. Some authors (in Castany, 1967)

𝑹𝑹𝑲𝑲 = �𝑸𝑸𝝅𝝅𝑴𝑴 (4.77)

where,

Ri : radius of influence (L) Q : rate of pumping (L/T3) I : precipitation intensity (debit/L2/T)

g. Kozen (in Bogomolov et Silin-Bektchoutine (1955)

𝑹𝑹𝑲𝑲 = �𝟏𝟏𝟐𝟐𝑻𝑻𝝁𝝁

�𝒒𝒒𝒌𝒌𝝅𝝅 (4.78)


72

h. G.V. Bogomolov (in Castany, 1967)

Table 4.3. Coefficient of permeability and Radius of depletion Aquifer material Granulometric

fraction (mm)

Coefficient of Permeability

(m/day)

Well discharge (m3/hour)

Radius of Depletion

(m) Clay sand 0,01-0,05 0,500-1,000 0,100-0,300 65 Fine sand 0,01-0,05 1,500-5,000 0,200-0,400 65 Clay sand in small grains

0,10-0,25 10,00-15,00 0,500-0,800 75

Sand in small grains 0,10-0,25 20,00-25,00 0,800-1,700 75 Clay sand in medium grains

0,25-0,50 20,00-25,00 1,600-10,00 100

Sand in medium grains 0,25-0,50 35,00-50,00 15,00-20,00 100 Clay sand in big grains 0,50-1,00 35,00-40,00 20,00-25,00 100 Sand in big grains 0,50-1,00 60,00-75,00 40,00-50,00 125 Gravels - 100,0-125,0 75,00-100,0 150 Note: drawdown 5-6 meter


73

V. FRESH AND SALINE WATER BALANCE 1. Basic equation

Badon Ghyben (1888) and Herzberg (1901),

Fig. 5.1. Schematic of cross section circular homogenous, isotropic and porous island.

∆𝒅𝒅 = 𝒅𝒅𝒔𝒔 �𝝆𝝆𝒔𝒔 − 𝝆𝝆𝒇𝒇𝝆𝝆𝒇𝒇

� (5.1)

Normal condition: Sea water ρs = 1.025 tmass/m3 = 1,025 kgmass/m

3

} so: ∆𝒅𝒅 = 𝟏𝟏𝟖𝟖𝟑𝟑𝒅𝒅𝒔𝒔�

Fresh water ρf = 1.00 tmass/m3 = 1,000 kgmass/m3

hf hs

A

Δh

precipitation

ground surface

groundwater surface

sea level

fresh water boundary area of saline water and fresh water

saline water


74

2. Shape of the Fresh-Salt Water interface

Fig. 5.2. Flow pattern of fresh water in an unconfined coastal aquifer The exact shape of the interface is (Glover in Todd, 1927):

𝛿𝛿2 =2𝜌𝜌𝑞𝑞𝛿𝛿∆𝜌𝜌𝐾𝐾

+ �𝜌𝜌𝑞𝑞∆𝜌𝜌𝐾𝐾

�2

(5.2)

The corresponding shape for the water table is given by:

ℎ𝑓𝑓 = �2∆𝜌𝜌𝑞𝑞𝛿𝛿

(𝜌𝜌 + ∆𝜌𝜌)𝐾𝐾�

1 2⁄

(5.3)

The width xo of the submarine zone through which fresh water discharges into the sea can be obtained for z=0,

𝛿𝛿𝜋𝜋 = −𝜌𝜌𝑞𝑞

2∆𝜌𝜌𝐾𝐾 (5.4)

Sea

Saline water

Fresh water

Ground surface

Water table

Interface

xo

zo


75

The depth of the interface beneath the shoreline zo, occurs where x = 0 so that: 𝛿𝛿𝜋𝜋= −

𝜌𝜌𝑞𝑞∆𝜌𝜌𝐾𝐾

(5.5)

3. Upconing Upconing is phenomenon that occurs when an aquifer contains an underlying of

saline water and is pumped by a well penetrating only the upper freshwater

portion of the aquifer, a local rise of the interface bellow the well occurs.


76

Fig. 5.3. Diagram of upconing of underlying saline water to a pumping well (after Schmorak and Mercado ini Todd, 1980)

According to Todd (1980) using Dupuit assumption and Ghyben-Herzberg relation, the upconing is:

𝒛𝒛 =𝑸𝑸

𝟐𝟐𝝅𝝅𝒅𝒅𝑲𝑲(∆𝝆𝝆 𝝆𝝆𝒔𝒔⁄ ) (5.6)

Comment:

Compare 2πd of this equation to the shape factor of Sunjoto (2002) F = 2πR

Base on Forchheimer (1930) principle, Sunjoto proposes that the upconing is:

𝛿𝛿 =𝑄𝑄

𝐹𝐹𝐾𝐾 �𝝆𝝆𝒔𝒔 − 𝝆𝝆𝒇𝒇𝝆𝝆𝒇𝒇

� (5.7)

Usually: o Sea water ρs = 1,000 kgmass/m3 = 1.00 tmass/m3 o Fresh water ρf = 1,000 kgmass/m3 = 1.00 tmass/m3

And for the security take z/d < 0.50


77

4. Drawdown versus Built up a. Theory of Dupuit-Thiem

Fig.5.4. Schematic of pumping

• Discharge (Dupuit-Thiem) base on Darcy’s Law:

𝑲𝑲 =𝑸𝑸


𝒐𝒐𝟐𝟐𝒐𝒐𝟏𝟏� (5.8)

Problem: Solution of this equation needed minimum two dependent unknown (h2 & r2) so this formula is difficult for predicting computation.

• From the above legends and schematic (Fig. 6.3) so the Power:

𝑴𝑴 =𝑸𝑸γ(𝑯𝑯+ 𝑺𝑺)

𝜼𝜼 (5.9)

pump axis level

gs H

S

Q

gwl

r1 r2

h1 h

𝑴𝑴 =𝑸𝑸γ(𝑯𝑯+ 𝑺𝑺)

𝜼𝜼

𝑲𝑲 =𝑸𝑸

𝝅𝝅(𝒅𝒅𝟐𝟐𝟐𝟐 − 𝒅𝒅𝟏𝟏𝟐𝟐)𝒅𝒅𝒏𝒏 �

𝒐𝒐𝟐𝟐𝒐𝒐𝟏𝟏�

Drawdown due to pumping

where, P : power (kN.m/s = kW) Q : discharge (m3/s) γ : specific weight of water (9.81 kN/m3) H : gap of groundwater level to pump axis (m) S : drawdown (m)

η : pump efficiency K : coefficient of permeability (m/s) h1 : piezometric of observation well 1 h2 : piezometric of observation well 2

r1 : radius of observation well 1 r2 : radius of observation well 2


78

b. Theory of Forhheimer (1930)

Fig.5.5. Theory of Forchheimer (1936)

According to Forchheimer (1930) discharge (Q) on the hole with casing is hydraulic head (H) multiplied by coefficient of permeability (K) multiplied by shape factor (F), and for the hole with casing F = 4 R.

On his auger test with Q = 0, or water was poured instantly and then be measured the relationship between duration (t) and height of water on hole (h), he derived mathematically the equation to compute coefficient of permeability:



(5.10)

where, K : coefficient of permeability R : radius of hole F : shape factor (F=4R) h1 : depth of water in the beginning h2 : depth of water in the end t1 : time in the beginning t2 : time in the end

𝑸𝑸 = 𝑭𝑭𝑲𝑲𝑯𝑯



t2

t1 h1

h2

2R


79

c. Theory of Sunjoto (1988)

Fig.5.6. Theory of recharge well and anti-drawdown (Sunjoto, 1988)

1). Discharge

Base on the steady flow condition theory of Forchheimer (1930), Sunjoto (1988) developed the equation of discharge through the hole with continue discharge flow to the hole which was derived mathematically by integration and the result is unsteady flow condition:

• Forchheimer (1936) formula:

𝑸𝑸 = 𝑭𝑭𝑲𝑲𝑯𝑯 (5.11)

• Sunjoto (1988) formula:

𝑸𝑸 =𝑭𝑭𝑲𝑲𝑯𝑯

�𝟏𝟏 − 𝒆𝒆𝒅𝒅𝒆𝒆 �−𝑭𝑭𝑲𝑲𝑻𝑻𝝅𝝅𝑹𝑹𝟐𝟐 �� 𝒐𝒐𝒐𝒐 𝑯𝑯 =

𝑸𝑸𝑭𝑭𝑲𝑲

�𝟏𝟏 − 𝒆𝒆𝒅𝒅𝒆𝒆�−𝑭𝑭𝑲𝑲𝑻𝑻𝝅𝝅𝑹𝑹𝟐𝟐

�� (5.12)

This formula (6.14) when duration T is infinite so the equation will become Q = FKH

(see Fig. 6.5)

H

T

Q/FKK

𝑯𝑯 =𝑸𝑸𝑭𝑭𝑲𝑲

�𝟏𝟏− 𝒆𝒆𝒅𝒅𝒆𝒆�−𝑭𝑭𝑲𝑲𝑻𝑻𝝅𝝅𝑹𝑹𝟐𝟐 ��

0

Built up due to recharging

Q

K

H 𝑸𝑸 = 𝑭𝑭𝑲𝑲𝑯𝑯

Relationship between H an T


80

2). Drawdown - Built up value

Drawdown due to pumping (S) will occur in discharge system by pumping (Fig. 6.3) and the reverse side the built up (anti-drawdown) due to recharging (H) will occur (Fig. 6.5) for the recharge system. For the equal condition and equal parameters the both value drawdown and anti-drawdown are equal with opposite direction.

a). Steady flow condition 𝑺𝑺 = −𝑯𝑯

=−𝑸𝑸𝑭𝑭𝑲𝑲

(5.13)

b). Unsteady flow condition 𝑺𝑺 = −𝑯𝑯

=−𝑸𝑸𝑭𝑭𝑲𝑲

�𝟏𝟏

− 𝒆𝒆𝒅𝒅𝒆𝒆�−𝑭𝑭𝑲𝑲𝑻𝑻𝝅𝝅𝑹𝑹𝟐𝟐

�� (5.14)

(negative sign means that the direction is opposite and in this case downward)

where, S : drawdown (m) H : depth of water on the hole/well (m) Q : discharge through the well (m3/s) F : shape factor (m) K : coefficient of permeability (m/s) T : duration of flow (s) R : radius of pipe/well (m)


81

EXAMPLE: Pumping system with discharge Q = 0.1667 m3/s, distance between pumping axis to the groundwater level H = 6.50 m, coefficient of permeability K = 0.00047 m/s, length of screen casing or perforated pipe L = 18 m and diameter of casing is 45 cm, fresh water: ρf = 1,000 kg/m3 or γf = 9.81 kN/m3 and saline water: ρs = 1,025 kg/m3 or γs = 10.552 kN/m3. Tip of the well in -28 m and the pumps are installed on the sandy costal which beneath of the pump in -160.00 m laid the boundary of fresh and saline water. Compute: Power needed and how is the pumping system related to salt water intrusion.

Fig.5.7. Pumping data

• Shape factor installed:

𝐹𝐹 =2 × π × 18 + 2 × π × 0.225 × 𝑙𝑙𝑛𝑛2

𝑙𝑙𝑛𝑛 �18 + 2 × 0.2252 × 0.225 + �� 18

2 × 0.225�2

+ 1�= 25.95 𝑚𝑚

K=4.70*10-4

S

5.00 m

Q=0.1667 m3/s

6.50 m

23.00 m

18.00 m

+1.5

-5.00

-28.00


82

• The drawdown of 1 pump installed:

𝑆𝑆 =𝑄𝑄𝐹𝐹𝐾𝐾

⇒ 𝑆𝑆 =0.1667

25.95 × 0.00047= 13.667 𝑚𝑚

To decrease of drawdown value S is by increasing value of F value, in this case be

installed 4 wells with same dimension and each well equipped by P = 4.30 KW.

• The drawdown of 4 pumps installed:

𝑆𝑆 =0.1667

4 × 25.95 × 0.00047= 𝟑𝟑. 𝟖𝟖𝟏𝟏 𝑚𝑚

The pumps are installed on the sandy costal which beneath of them laid down the boundary of fresh and saline water in –200,00 m.

Upconing:

According to Sunjoto Eq.(6.9) is:

𝛿𝛿 =3.41

1,025 − 1,0001,000

= 136,40 𝑚𝑚

• Power needed:

P = 0.1667 m3/s x 9.81 kN/m3 x (6.50+3.41) m/ 0.60 = 27 kN.m/s = 27 kW

• Conclusion:

The level of boundary will move upward to –200 + 136.40 = –63.60 m and due to the tip of the well level is –28 m so the saline water will not flow into tip of pipe so there is not sea water intrusion.

• Recommendation:

To avoid saline water intrusion to the pump so the shape factor Fd should be increased by enlarging the diameter of well or/and adding the length of porous well.


83

5. Saline water pumping

Since the last three decades, the cultivation of fish in coastal area speedy increase due to the demand of fish consumption increases. The fishpond in fresh water and brackish water had been developed largely in Indonesia and then the fish cultivation in seawater is now it’s beginning to be developed. A seawater fishpond in sandy coastal area which was equipped by geo-membrane had been developed in Yogyakarta Special Province with 7.20 ha area, 60 cm depth. One third of water should be replaced by seawater. The needed pumping system for hydraulic head H = 7.50 m and coefficient of permeability K = 0.00047 m/s and saline water: ρs = 1,025 kg/m3 or γs = 10.552 kN/m3. This fishpond was installed 4 types of pumping system and one system still under design. The problem is that the discharge of pumping only less than half of the design discharge even though the power was doubled.

• Volume of pond: Vp = 72,000 m2 x 0.60 m = 43,200 m3

• Daily seawater volume needed: Vn = 33 % x 43,200 m3 = 14,400 m3

• Daily seawater discharge needed: Qn = 14,400/24/3,600 = 0.1667 m3/s ≈ 10 m3/mnt

• Power needed (without drawdown occurs): Pn = Q γ H / η kNm/s Pn = 0.1667 m3/s x 10.552 kN/m3x 7,50 m/ 0.60 = 21.99 kN.m/s = 21.99 kW Analysis: According to Forchheimer (1930) that radial flow in porous media, discharge (Q) is equal to shape factor (F) multiplied by coefficient of permeability (K) multiplied by hydraulic head (h).

𝑄𝑄 = 𝐹𝐹𝐾𝐾ℎ (5.15)

Pumping power is discharge multiplied by specific weight of water multiplied by hydraulic head divided by efficiency of pump system.

𝑃𝑃 =𝑄𝑄γ𝐾𝐾𝜂𝜂

(5.16)

According to Sunjoto (2008), when drawdown of pumping is equal to hydraulic head the equation becomes:

𝐹𝐹 =𝑄𝑄2γ𝜂𝜂𝑃𝑃𝐾𝐾

(5.17)


84

where, Q : discharge (m3/s)

F : shape factor of well (m) K : coefficient of permeability (m/s) H : hydraulic head of pumping (m) P : power (kN.m/s) γ : specific weight of water (kN/m3)

η : pump efficiency

Due to there is not data of saline water and fresh water boundary so it was decided that the value of drawdown should be big enough to achieve the high upconing and it will get get saline water discharge, In this case the drawdown was decided equal to hydraulic gradient and shape factor needed can be computed by (5.17):

𝐹𝐹𝑛𝑛 =0,16672 × 1.025

0,60 × 2.135,85 × 0,00047= 47,29 𝑚𝑚

a. Lying pipes This pumping system consists of four pipes of 20 cm diameter non-perforated and the tip of pipes was covered by screen filter. The pipes were lied down about 1 m under the ground (sand) surface and always sink under low sea water surface to achieve the discharge water free from predators. The installed shape factors is (Sunjoto, 2002):

F = 2 π R (5.18)

where, F : shape fator of pipe (m) R : radius of pipe (m) Computed by (5), the installed shape factor for the 4 pipes is (5.18): Fi = 4 x 2 x π x 0,10 = 2,51 m

• This system was not installed the pump due to the current of the sea is big enough to destroy the lied pipes.


85

Fig.5.8. Lying pipes b. Cubical Water Intake

This system consist of hollow 6 m sides cubical concrete structure and the base of cube without concrete slab lied down on the costal sand and sink always under lowest sea level. The aim of this system is keeping of 2 pumps from fast current and high wave. Inside of the cube was installed two cylinder concrete of 60 cm diameter where the tip of suction pumps take a water. So the shape factor of this install system is (5.18):

Fi = 2 x 2 x π x 0,30 = 3,77 m This system was installed 2 pumps of 1x3.00 KW and 1x4.00 KW

Fig.5.9. Cubical Water Intake c. Impermeable Deep well This system consists of 2 steel non perforated pipes of 45 cm diameter with length 60 m and the installed shape factor can be computed by (5.18): F = 2 x 2 x π x R = 2 x 2 x π x 0,225 = 2,827 m This system was installed 2 pumps of 16.00 KW

6.00

Indian Ocean

Q

4 Φ 0,20 m

Q

Indian Ocean


86

Fig.5.10. Deep Well

d. Perforated swallow pipes This system consists of 6 meter perforated pipes 30 cm diameter was installed in

costal sandy area and according to Sunjoto (2002) the shape factor is:

𝐹𝐹 =2π𝐿𝐿 + 2π𝑅𝑅𝑙𝑙𝑛𝑛2

𝑙𝑙𝑛𝑛 �𝐿𝐿 + 2𝑅𝑅2𝑅𝑅 + �� 𝐿𝐿2𝑅𝑅�

2+ 1�

(5.19)

where, F : shape factor of pipe (m) R : radius of pipe (m)

L : porous length (m) So shape factor (5.19):

𝐹𝐹 =2 × π × 6 + 2 × π × 0,15 × 𝑙𝑙𝑛𝑛2

𝑙𝑙𝑛𝑛 �6 + 2 × 0,152 × 0,15 + �� 6

2 × 0,15�2

+ 1�

= 10,326 𝑚𝑚

• This system was installed 1 pumps of 1x3.00 KW

Q

60 m

Indian Ocean


87

Fig.5.11. Swallow Porous Pipes

Analysis a. Installations

Acctually there were 4 types of pumping systems were built in this project but the Lying Pipes was broken down by the current and the wave of the ocean and the pump was not installed so its rest 3 pumping systems operate with the conditions:

1). Total installed power P = 0 + (3,00 + 4,50) + (16.00 + 16.00) + 3,00 = 42,50 KW

Design power was 21,99 KW

2). Total installed shape factor:

F = 0 + 3,770 + 2,827 + 10,326 = 16,923 m Needed shape factor is 47,29 m.

3). Total real discharge:

Q = Q1 + Q2 + Q3 + Q4

Q = 0 + (0,18 + 0,27) + (1,80 + 1,80) + 0,18 = 4,23 m3/mnt

Design discharge was 10 m3/mnt. b. Shape factor point of view 1). Cubic Water Intake

When this system without 60 cm cylinder concrete, it will get bigger shape factor as:

𝑓𝑓 = 4√𝑏𝑏 × 𝑏𝑏 (5.20)

6,00

Q

Indian Ocean


88

𝑓𝑓 = 4√6 × 6 = 24 𝑚𝑚, 𝑐𝑐𝜋𝜋𝑚𝑚𝑒𝑒𝑠𝑠𝜋𝜋𝑒𝑒 𝑡𝑡𝜋𝜋 𝑡𝑡ℎ𝑒𝑒 𝑒𝑒𝛿𝛿𝑖𝑖𝛿𝛿𝑡𝑡𝑖𝑖𝑛𝑛𝑔𝑔 𝑓𝑓 = 3,77 𝑚𝑚

To get shape factor F = 47,29 m you can build Cubical Water Intake Pumping System with dimension:

• When Cylinder form so the radius is:

R = 47,29 / 2 π = 7,50 m.

• When Rectangular form the sides are (5.20):

𝑓𝑓 = 4√𝑏𝑏 × 𝑏𝑏 = 47,29 ⇒ 𝑏𝑏 = 11,83 𝑚𝑚 ∞ 12 𝑚𝑚

To provide the discharge of the project 10 m3/mnt it can be built only one Cubic Water Intake Pumping System with dimension radius 7.50 m for the Cylinder form or Rectangular form with the sides 12 m, equiped by 5 x 4,50 KW pumps.

2). Deep well To provide the discharge of the project 10 m3/mnt it can be built only 3 Deep

Wells equiped by 16 m perforated pipes and the shape factor (6.22):

𝐹𝐹 = 2 ×2 × π × 16 + 2 × π × 0,225 × 𝑙𝑙𝑛𝑛2

𝑙𝑙𝑛𝑛 �16 + 2 × 0,2252 × 0,225 + �� 16

2 × 0,225�2

+ 1�

= 2 × 23,726 = 47,452 𝑚𝑚

To provide the discharge demand of the project 10 m3/mnt it can be built only 2 Deep Wells with 16 m perforated pipe each, equiped by 2 x 12 KW pumps.

3). Perforated swallow well

To provide the discharge of the project 10 m3/mnt it can be built only 5 Perforated Swallow Well Systems due to total shape factor is 5 x 10,326 = 51.63 m > 47,26 m with 5 x 4.50 Kw Pumps.


89

c. Horizontal perforated pipes (Imron Rosyadi, 2004)

According to Imron Rosyadi (2004) in his Master Thesis that the best solution is 3 m diameter concrete cylinder with height of 13 m shoud be sunk 8 m on the sand and equiped 5 perforated pipes 4 m length and 10 cm diameter (Fig.5.12)

Fig.5.12. Horizontal perforated pipes

1). Shape factor of concrete cylinder is (6.21):

F1 = 2 x π x 1,50 = 9,42 m

Indian Ocean

13.00

Q

3.0

4.00

Φ 10 cm


90

2). Shape factor of perforated pipes is (6.22):

𝐹𝐹2 = 5 ×2 × π × 4 + 2 × π × 0,05 × 𝑙𝑙𝑛𝑛2

𝑙𝑙𝑛𝑛 �4 + 2 × 0,052 × 0,05 + �� 4

2 × 0,05�2

+ 1�

= 5 × 5,769 = 28,845 𝑚𝑚

Total shape factor of concrete cylinder and horizontal perforated pipes is: F = F1 + F2 = 9,42 + 28, 845 = 38,265 m < 47,29 m.

Conclusion: The all designs never considerated shape factor of tip of well therefore the power was doubled but the discharge was only less than half of the designed value.

e. Horizontal Perforated Pipe (HPP)

HPP is perforated pipe which are installed horizontally to get bigger discharge or

recharge of the well. For discharge well the hydraulic head is the drawdown of

pumping and for recharge well the hydraulic head is the difference of groundwater

elevation on the well before and after pumping.

Gambar 5.13. Cross section of horizontal perforated pipes


91

Data: Coefficient of permeability K = 10-3 m/s Length of HPP L = 4 m Radius of HPP r = 0.15 m Radius of well R = 2 m Number of pipe n = 8 pcs Diameter of pipe pore f = 0.003 m Pores distance 0.15 m Axis of HPP elevation: – 9.50 m Groundwater elevation above HPP: -6.50 m Ground surface elevation: 0.00 m Some of the methods of computation are:

a. Mikel & Klaer’s Methode (1956)

𝑸𝑸 = 𝒏𝒏 ∗ 𝝅𝝅 ∗ 𝑳𝑳𝟐𝟐 ∗ 𝑾𝑾 (6.24)

where, Q : discharge (m3/s) n : number of pipe L : length of pipe (m) W : flow velocity (m/s)

𝑊𝑊 = 𝐾𝐾 ∗ 𝑖𝑖 = 𝐾𝐾 ∗ℎ𝑙𝑙

= 10−3 ∗33

= 0.001 𝑚𝑚 𝜋𝜋⁄

Discharge of 8 pore pipes:

𝑄𝑄 = 8 × 𝜋𝜋 × 42 × 0,001 = 0.402285 𝑚𝑚3 𝜋𝜋⁄

b. Spiridonoff & Hantush’s Method (1964) 𝑸𝑸 = 𝒏𝒏 ∗ 𝑺𝑺𝒅𝒅 ∗ 𝑴𝑴𝒇𝒇 ∗ 𝒅𝒅 (6.25)

where, Q : discharge (m3/s) Sv : specific yield aquifer of sand and gravel (Sv = 20 %) Af : total area of pore hole of each pipe (m2) h : distance between axis of pipe to groundwater level (m) D : diameter of pipe (m)


92

Total area of pore holes of each pipe:

𝐴𝐴𝑓𝑓 =14∗ 𝜋𝜋 ∗ 𝐶𝐶2 ∗ 𝑛𝑛𝑓𝑓 =

14

× 𝜋𝜋 × 0.0032 × 162 = 0.114557 𝑚𝑚2

Distance between axis of HPP to groundwater level: h= 3 m Discharge of 8 pipes: 𝑄𝑄 = 8 × 0.20 × 0.114557 × 3 = 0.549874 𝑚𝑚3 𝜋𝜋⁄ c. Nasjono’s Method (2002)

𝑸𝑸 = 𝟏𝟏𝟖𝟖𝟖𝟖.𝟖𝟖𝟏𝟏 �𝑴𝑴𝒇𝒇𝑳𝑳 ∗ 𝝆𝝆

∗𝒅𝒅𝒅𝒅�𝟑𝟑.𝟐𝟐𝟑𝟑𝟖𝟖𝟖𝟖

∗ 𝑲𝑲 ∗ 𝑳𝑳 ∗ 𝝆𝝆 (6.26)

where,

Q : discharge (m3/s) Af : total area of pore hole of each pipe (m2) K : coefficient of permeability (m/s) L : length of pipe (m) D : diameter of pipe (m) h : distance between axis of pipe to groundwater level (m) l : distance of flow (m)

Discharge of 8 pore pipes:

𝑄𝑄 = 148.41 × �0.114557

4 × 0.3×

33�

0.2366

× 10−3 × 32 × 0.30 = 0.102159 𝑚𝑚3 𝜋𝜋⁄

d. Das, Saha, Rao dan Uththmanthan’s Method (2009)

• The assumption of pores clogging is 50% • The assumption Af is 20% the surface area of pipe

𝑸𝑸 = 𝒏𝒏�𝝅𝝅 ∗ 𝑳𝑳 ∗ 𝝆𝝆 ∗ 𝑴𝑴𝒇𝒇� ∗ 𝟓𝟓𝟑𝟑% ∗ 𝑽𝑽 (6.27)


93

where,

Q : discharge (m3/s) L : length of pipe (m) D : diameter of pipe (m) Af : total area of pore hole of each pipe (m2) ⇒ = 20 % area of pipe V : flow velocity in the pipe (m/s) ⇒ V = 0.50 cm/s=0.005 m/s

Discharge of 8 pipes: 𝑄𝑄 = 8 × (𝜋𝜋 × 4 × 0.30 × 20%) × 50% × 0.005 = 0.150857 𝑚𝑚3 𝜋𝜋⁄ e. Sunjoto’s Method (1988; 2002)

This method describes that when the condition is steady flow so the formula is

Forhheimer (1930) but that when the condition is unsteady flow so the formula is

Sunjoto (1988) as follows:

• When steady flow condition the discharge (Forhheimer,1930) is:

𝑸𝑸 = 𝑭𝑭 ∗ 𝑲𝑲 ∗ 𝑯𝑯 (6.28)

where, Q : discharge (m3/s) F : shape factor of pipe or well (m) ⇒ Table 8.1. K : coefficient of permeability (m/s) H : hydraulic head (m) R : radius of well or pipe (m) T : duration of flow (s)

• Total length of HPP is L = 4 x 8 = 32 m • The assumption that hydraulic head is H = 3 m • Diameter of well is D = 4 m or radius R = 2 m • The porosity and coefficient of permeability of pipe pore is bigger than the

soil and permeability of porous wall of well is bigger to the permeability of soil.

All the methods are computed in steady flow condition using the above data so:


94

1). Discharge through 8 pipes when base and wall of well are impermeable:

Shape factor for 8 pipe pores is (Sunjoto, 2002):

𝐹𝐹6𝑏𝑏 =2π𝐿𝐿 + 2π𝑅𝑅𝑙𝑙𝑛𝑛2


2+ 1�

(6.29)

𝐹𝐹6𝑏𝑏 =2 × π × 32 + 2 × 𝜋𝜋 × 0.15 × 𝑙𝑙𝑛𝑛2

𝑙𝑙𝑛𝑛 �32 + 2 × 0,152 × 0,15 + �� 32

2 × 0,15�2

+ 1�

= 37.58047 𝑚𝑚

• Discharge through 8 pipes:

𝑄𝑄1 = 𝐹𝐹 ∗ 𝐾𝐾 ∗ 𝐾𝐾 = 37.58047 × 10−3 × 3 = 0.112743 𝑚𝑚3 𝜋𝜋⁄

2). Discharge when 8 pipe pores and base of well are permeable but wall of well is

impermeable:

Shape factor for well when the base is permeable (Sunjoto, 2002) is:

𝐹𝐹4𝑏𝑏 = 2 ∗ π ∗ 𝑅𝑅 = 2 × π × 2 = 12.566371 𝑚𝑚

• Discharge through the well base:

𝑄𝑄 = 𝐹𝐹 ∗ 𝐾𝐾 ∗ 𝐾𝐾 = 𝑄𝑄 = 12.566371 × 10−3 × 3 = 0.037699 𝑚𝑚3 𝜋𝜋⁄

Total discharge 1). + 2). is:

𝑄𝑄2 = (0.112474 + 0.037699) = 0.150441 𝑚𝑚3 𝜋𝜋⁄

3). Discharge when 8 pipe pores of base and wall of well are permeable:

Shape factor for well with permeable perimeter with L = 4 m and radius R = 2 m

(Sunjoto, 2002) is:

𝐹𝐹6𝑏𝑏 =2π𝐿𝐿 + 2π𝜋𝜋𝑙𝑙𝑛𝑛2


2+ 1�

=2 × π × 4 + 2 × 𝜋𝜋 × 0.15 × 𝑙𝑙𝑛𝑛2

𝑙𝑙𝑛𝑛 �4 + 2 × 22 × 2 + �� 4

2 × 2�2

+ 1�= 20.99929 𝑚𝑚3 𝜋𝜋⁄


95

• Discharge through the perimeter of well:

𝑄𝑄 = 𝐹𝐹 ∗ 𝐾𝐾 ∗ 𝐾𝐾 = 𝑄𝑄 = 20.99929 × 10−3 × 3 = 0.062999 𝑚𝑚3 𝜋𝜋⁄

Total discharge 1). + 3). is: 𝑄𝑄3 = (0.112474 + 0.062999) = 0.175426 𝑚𝑚3 𝜋𝜋⁄

f. Sriyono’s Method (2011)

Sriyono developed the formulas for the 1, 2 and 3 only number of horizontal pipes by

hydraulic modeling research as follows:

For number of pipe n = 1: 𝑄𝑄𝐾𝐾𝐿𝐿𝐶𝐶

= 54.600 �𝐴𝐴𝑓𝑓 .ℎ𝐿𝐿.𝐶𝐶. 𝑙𝑙

�−.427

(6.30)

For number of pipe n = 2: 𝑄𝑄𝐾𝐾𝐿𝐿𝐶𝐶

= 80.354 �𝐴𝐴𝑓𝑓 .ℎ𝐿𝐿.𝐶𝐶. 𝑙𝑙

�−.385

(6.30)

For number of pipe n = 3:

𝑄𝑄𝐾𝐾𝐿𝐿𝐶𝐶

= 103.936 �𝐴𝐴𝑓𝑓 . ℎ𝐿𝐿.𝐶𝐶. 𝑙𝑙

�−.354

(6.30)

where, Q : discharge (m3/s) Af : total area of pore hole of each pipe (m2) K : coefficient of permeability (m/s) L : length of pipe (m) D : diameter of pipe (m) h : distance between axis of pipe to groundwater level (m) l : distance of flow (m)


96

Table 5.1. Comparison of the result No. Method Discharges (m3/s)

1 Mikel & Klaer (1956) 0.402285

2 Spiridonoff & Hantush (1964) 0.549874

3 Nasjono (2002) 0.102159

4 Das, Saha, Rao & Uththmanthan (2009)

0.150857

5 Sunjoto (1988; 2002) 1) 2) 3) 𝑄𝑄1 = 0.112474 𝑄𝑄2 = 0.150441 𝑄𝑄3 = 0.175426

6 Sriyono n=1 n=2 n=3

0.01290 0.02094 0.02898


97

VI. UNSTEADY FLOW

1. Theis (1935)

The assumptions made in applying these equations to solution of aquifer problems are:

• The system is infinite • The aquifer is homogenous, isotropic and uniform thickness • Prior to removal or addition of water the piezometric is horizontal • The pumping is at constant rate • The pumped well penetrates the aquifer • Water removed from storage is discharged immediately

Theis (1906) used the exponential integral solution to analyze unsteady flow in the

following term:

𝜋𝜋 =𝑄𝑄

4𝝅𝝅𝐹𝐹�𝑒𝑒−𝑢𝑢𝑑𝑑𝑢𝑢𝑢𝑢

∞

𝑢𝑢

−𝑄𝑄

4𝝅𝝅𝐹𝐹𝑊𝑊(𝑢𝑢) (6.1)

The integral is a function of lower limit u and is known as an exponential integral. It

can be expanded as a convergent series so that Eq. 6.1. becomes:

𝒔𝒔 =𝑸𝑸𝟖𝟖𝝅𝝅𝑻𝑻

�−𝟑𝟑.𝟓𝟓𝟒𝟒𝟒𝟒𝟐𝟐 − 𝒅𝒅𝒏𝒏 𝒖𝒖 + 𝒖𝒖 − 𝒖𝒖𝟐𝟐

𝟐𝟐.𝟐𝟐!+𝒖𝒖𝟑𝟑

. 𝟑𝟑! −

𝒖𝒖𝟖𝟖

𝟖𝟖.𝟖𝟖!∗ … � (6.2)

where,

𝑢𝑢 =𝜋𝜋2𝑆𝑆4𝐹𝐹𝑡𝑡

(6.3)

The storage coefficient is

𝑆𝑆 =4𝐹𝐹𝑢𝑢𝜋𝜋2 𝑡𝑡⁄

(6.4)

The exponential integral W(u) = -Ei(-u) can be represented by the series below and

the values is tabulated in Table 6.1.


98

𝑾𝑾(𝒖𝒖) = −𝟑𝟑. 𝟓𝟓𝟒𝟒𝟒𝟒𝟐𝟐 − 𝒅𝒅𝒏𝒏 𝒖𝒖 + 𝒖𝒖− 𝒖𝒖𝟐𝟐

𝟐𝟐.𝟐𝟐!+𝒖𝒖𝟑𝟑

.𝟑𝟑! −

𝒖𝒖𝟖𝟖

𝟖𝟖.𝟖𝟖!∗ … (6.5)

Table 6.1. Values of W(u) for Values of u

u 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 ×1 0.219 0.049 0.013 0.0038 0.0011 0.00036 0.00012 0.000038 0.000012

×10-1 1.82 1.22 0.91 0.70 0.56 0.45 0.37 0.31 0.26

×10-2 4.04 3.35 2.96 2.68 2.47 2.30 2.15 2.03 1.92

×10-3 6.33 5.64 5.23 4.95 4.73 4.54 4.39 4.26 4.14

×10-4 8.63 7.94 7.53 7.25 7.02 6.84 6.69 6.55 6.44

×10-5 10.94 10.24 9.84 9.55 9.33 9.14 8.99 8.86 8.74

×10-6 13.24 12.55 12.14 11.85 11.63 11.45 11.29 11.16 11.04

×10-7 15.54 14.85 14.44 14.15 13.93 13.75 13.60 13.46 13.34

×10-8 17.84 17.15 16.74 16.46 16.23 16.05 15.90 15.76 15.65

×10-9 20.15 19.45 19.05 18.76 18.54 18.35 18.20 18.07 17.95

×10-10 22.45 21.76 21.76 21.06 20.84 20.66 20.50 20.37 20.25

×10-11 24.75 24.06 24.06 23.36 23.14 22.96 22.81 22.67 22.55

×10-12 27.05 26.36 26.36 25.67 25.44 25.26 25.11 24..97 24.86

×10-13 29.36 28.66 28.66 27.97 27.75 27.56 27.41 27.28 27.16

×10-14 31.66 30.97 30.56 30.27 30.05 29.87 29.71 29.58 29.46

×10-15 33.96 33.27 32.86 32.58 32.35 32.17 32.02 31.88 31.76

Example:

Pumping in confined aquifer, with full penetration and a discharge 2500 m3/d.

Observation well 60 m away from the well. Data found of drawdown in function of

duration of pumping and value of r2/t is tabulated in Table 6.2.:


99

Table 6.2. Pumping test data

t (min)

s (m)

r2/t m2/min

t (min)

s (m)

r2/t m2/min

t (min)

s (m)

r2/t m2/min

0 0 ∞ 8 0,53 450 60 0,90 60 1 0,20 3600 10 0,57 360 80 0,93 45

1,5 0,27 2400 12 0,60 300 100 0,96 36 2 0,30 1800 14 0,63 257 120 1,00 30

2,5 0,34 1440 18 0,67 200 150 1,04 24 3 0,37 1200 24 0,72 150 180 1,07 20 4 0,41 900 30 0,76 120 210 1,10 17 5 0,45 720 40 0,81 90 240 1,12 15 6 0,48 600 50 0,85 72 - -

Solution:

Values of s and r2/t are plotted on logarithmic paper and values of W(u) and u from

Table. 6.1. are plotted on another sheet of logarithmic paper and curve is drawn

through the points. The two sheets are superposed and shifted with coordinate axe

parallel until the observational point coincide with the curve as shown in Fig. 6.1.

convenient match point is selected with W(u) = 1.00 and u = 1 x 10-2, so that s = 0.18 m

and r2/t = 150 m3/min = 216,000 m3/d. Thus, from equation:

𝐹𝐹 =𝑄𝑄

4𝜋𝜋𝑆𝑆𝑊𝑊(𝑢𝑢) =

2500(1.00)4𝜋𝜋(0.18) = 1110 𝑚𝑚2 𝑑𝑑⁄

𝑆𝑆 =4𝐹𝐹𝑢𝑢𝜋𝜋2 𝑡𝑡⁄

=4(1110)(1 x 10−2)

216,000= 0.000206


100

Fig. 6.1. Theis method of superposition for solution of the non equilibrium equation

1. Cooper-Jacob (1946)

The expansions of Theis (1935) were carried out by Cooper-Jacob (1946), Chow

(1953), Todd (1980). The third method are developed as similar method to Theis

that’s are developing exponential integration formula which are difficult to compute,

using pumping data, then plotting the curve and fitting the curves. Glover (1966)

developed the similar exponential integration formula but his formula supported by

table. Due to Glover uses the parameters of computation of pumping method which it

similar to parameters of formula developed by Sunjoto (1988) so those data its can

be computed by both methods that are Glover and Sunjoto.


101

Cooper-Jacob noted that for small value of r and large value of t, u is small so that

the series terms of Theis formula become negligible after the first two terms then

the drawdown can be expressed by the asymptote:

𝜋𝜋 =𝑄𝑄

4𝜋𝜋𝐹𝐹�−0.5772 − 𝑙𝑙𝑛𝑛

𝜋𝜋2𝑆𝑆4𝐹𝐹𝑡𝑡

� (6.6)

Rewriting and changing to decimal logarithms, this reduce to:

𝜋𝜋 =2.30𝑄𝑄4𝜋𝜋𝐹𝐹

𝑙𝑙𝜋𝜋𝑔𝑔2.25𝐹𝐹𝑡𝑡𝜋𝜋2𝑆𝑆

(6.7)

Therefore, a plot of drawdown s versus the logarithms of t shows a straight line.

Projecting this line to s = 0, where t = to (Fig. 6.2)

0 =2.30𝑄𝑄4𝜋𝜋𝐹𝐹

𝑙𝑙𝜋𝜋𝑔𝑔2.25𝐹𝐹𝑡𝑡𝜋𝜋𝜋𝜋2𝑆𝑆

(6.8)

Fig. 6.2. Cooper-Jacob method for solution of the non equilibrium equation

and it follows:

2.25𝐹𝐹𝑡𝑡𝜋𝜋𝜋𝜋2𝑆𝑆

= 1 (6.9)


102

resulting in:

𝑆𝑆 =2.25𝐹𝐹𝑡𝑡𝜋𝜋𝜋𝜋2 (6.10)

And value for T can be obtained by noting that if t/to = 10, then log t/to = 1, there

for replacing s by Δs, where Δs is the drawdown difference per log cycle of t and

equation becomes:

𝐹𝐹 =2.30𝑄𝑄4𝜋𝜋𝜋𝜋𝜋𝜋

The straight line approximation for this method should be restricted to small values of u (u < 0.01) to avoid large errors.

EXAMPLE:

From pumping test data Table 6.2, s and t plotted on semilogathmic paper, as shown in

Fig. 6.2. A straight line is fitted through the points, and ∆s = 0.40 m and to = 0.39

min = 2.70 .10-4 day are read.

Then,

𝐹𝐹 =2.30𝑄𝑄4𝜋𝜋𝜋𝜋𝜋𝜋

=2.30(2500)

4𝜋𝜋(0.40) = 1090 𝑚𝑚2 𝑑𝑑⁄

and,

𝑆𝑆 =2.25𝐹𝐹𝑡𝑡𝜋𝜋𝜋𝜋2 =

2.25(1090)(2.70 . 10−4)(60)2

2. Chow (1952)

He introduced a method of solution with the advantages of avoiding curve fitting and

being unrestricted in application. The observational data are plotted on

semilogarithmic paper in the same manner as for the Cooper-Jacob method. On the

plotted curve, choose an arbitrary point and note the coordinates, t and s. Next, draw


103

a tangent to the curve at the chosen point and determine the drawdown difference

∆s, in feet, per log cycle of time. Then compute F(u) from: 𝜋𝜋∆𝜋𝜋

= 𝐹𝐹(𝑢𝑢) (6.11)

or,

𝜋𝜋∆𝜋𝜋

=𝑊𝑊(𝑢𝑢)𝑒𝑒𝑢𝑢

2.30 (6.12)

and find corresponding values of W(u) and u from Fig. 6.3. and finally compute the

formation constants T , s and r2/t of Theis equation.

Fig. 6.3. Relation among F(u), W(u) and u (After Chow 1952, in Todd, 1980)

EXAMPLE:

In Fig. 6.4. data are plotted from Table 6.2. and point A is selected on the curve

where t = 6 min = 4.20 .10-3 day and s = 0.47 m. A tangent is constructed as shown;

the drawdown difference per log cycle of time is ∆s = 3.80 m. Then F(u) = 0.47/0.38

= 1.24, and from Fig. 6.4. W(u) = 2.75 and u = 0.038.


104

Hence,

𝐹𝐹 =𝑄𝑄

4𝜋𝜋𝑆𝑆𝑊𝑊(𝑢𝑢) =

2500(2.75)4𝜋𝜋(0.47) = 1160 𝑚𝑚2 𝑑𝑑⁄

𝑆𝑆 =4𝐹𝐹𝑡𝑡𝑢𝑢𝜋𝜋2 =

4(1160)(4.2 x 10−3)(0.038)(60)2 = 0.000206

Fig. 6.4. Chow method for solution of the non equilibrium equation

3. Recovery Test (Todd, 1980)

At the end of a pumping test, when pumping is stopped, the water levels in pumping

observation wells will begin rice. This is referred to as the recovery of groundwater

levels, while measurements of drawdown below the original static water level during

the recovery period are known as residual drawdown. (See Fig. 6.5). It should be

noted that measurement of the recovery within a pumped well provide an estimate of

transmissivity even without an observation well and no comparable value of S can be

determined by this recovery test method.


105

The rate f recharge Q to the well during recovery is assumed constant and equal to

the mean pumping rate. The drawdown after pumping shut down will be identically the

same as if the discharge had been continued and hypothetical recharge well with the

same flow were superposed on the discharging well at the instant the discharge is

shut down.

Fig. 6.5. Drawdown and recovery curves in an observation well near pumping well

Using Theis principle that the residual drawdown s’ can be given as,

𝑆𝑆 =𝑄𝑄

4𝜋𝜋𝐹𝐹[𝑊𝑊(𝑢𝑢) −𝑊𝑊(𝑢𝑢′)] (6.13)

where,

𝑢𝑢 =𝜋𝜋2𝑆𝑆4𝐹𝐹𝑡𝑡

𝑠𝑠𝑛𝑛𝑑𝑑 𝑢𝑢′ =𝜋𝜋2𝑆𝑆4𝐹𝐹𝑡𝑡′

(6.14)


106

and t and t’ are defined in Fig. 9.5. and for small r , large t’ the well functions can be

approximated by the equations:

𝒔𝒔′ =𝟐𝟐.𝟑𝟑𝟑𝟑𝑸𝑸𝟖𝟖𝝅𝝅𝑻𝑻

𝒅𝒅𝒐𝒐𝝁𝝁𝒕𝒕𝒕𝒕′

(6.15)

And the transmissivity becomes:

𝐹𝐹 =2.30𝑄𝑄4𝜋𝜋𝜋𝜋𝜋𝜋′

(6.16)

EXAMPLE:

A well pumping at an uniform rate 2500 m3/d was shut down after 240 min and

measurements were made in an observation well of s’ and t’ and computation of values

of t/t’ tabulated in Table. 6.3, and then plotted versus s’ on semilogarithmic paper

(Fig. 6.6 ). A straight line is fitted through the points and ∆s’ = 0.40 m is determined,

then:

𝐹𝐹 =2.30𝑄𝑄4𝜋𝜋𝜋𝜋𝜋𝜋′

=2.30(2500)

4𝜋𝜋(0.40) = 1140 𝑚𝑚3 𝑑𝑑⁄

Fig. 6.6. Recovery test method for solution of the non equilibrium equation


107

Table 6.3. Recovery test data, pump shut down at 240 min (after Todd, 1980)

t’ (min) t (min) t/t’ S’ (m)

1 241 241 0.89

2 242 121 0.81

3 243 81 0.76

5 245 49 0.68

7 247 35 0.64

10 250 25 0.56

15 255 17 0.49

20 260 13 0.55

30 270 9 0.38

40 280 7 0.34

60 300 5 0.28

80 320 4 0.24

100 340 3.4 0.21

140 380 2,7 0.17

180 420 2.3 0.14

4. Glover (1966)

a. General formulation

The flow Q through a unit width and the height h at the distance x from the origin is:

𝑄𝑄 = 𝐾𝐾ℎ𝜕𝜕ℎ𝜕𝜕𝛿𝛿

(6.17)

The continuity condition is:

𝜕𝜕𝑄𝑄𝜕𝜕𝛿𝛿

𝑑𝑑𝛿𝛿 𝑑𝑑𝑡𝑡 = 𝑉𝑉𝜕𝜕ℎ𝜕𝜕𝑡𝑡𝑑𝑑𝑡𝑡 𝑑𝑑𝛿𝛿

By substitution and arrangement

𝐾𝐾𝜕𝜕𝜕𝜕𝛿𝛿

�ℎ𝜕𝜕ℎ𝜕𝜕𝛿𝛿� = 𝑉𝑉

𝜕𝜕ℎ𝜕𝜕𝑡𝑡


108

If, as an approximation the quantity ℎ 𝜕𝜕ℎ𝜕𝜕𝛿𝛿

replaced by 𝐶𝐶 𝜕𝜕ℎ𝜕𝜕𝛿𝛿

and

𝛼𝛼 =𝐾𝐾𝐶𝐶𝑉𝑉

(6.18)

the above relation reduce to

𝛼𝛼𝜕𝜕2ℎ𝜕𝜕𝛿𝛿2 =


(6.19)

If y represents a coordinate whose direction is horizontal and normal to that of x and

if there are gradients 𝜕𝜕ℎ𝜕𝜕𝛿𝛿

, the above relation takes the form

𝛼𝛼 �𝜕𝜕2𝜑𝜑𝜕𝜕𝛿𝛿2 +

𝜕𝜕2𝜑𝜑𝜕𝜕𝛿𝛿2� =


(6.20)

In radial symmetrical cases, the differential equation takes for

𝛼𝛼 �𝜕𝜕2𝜋𝜋𝜕𝜕𝜋𝜋

+1𝜋𝜋𝜕𝜕𝜋𝜋𝜕𝜕𝜋𝜋� =

𝜕𝜕𝜋𝜋𝜕𝜕𝑡𝑡

The Laplace formulation with the condition that the flow into the element of volume

must equal the flow out of it, is

𝐾𝐾 �𝜕𝜕2ℎ𝜕𝜕𝛿𝛿2 𝑑𝑑𝛿𝛿 𝑑𝑑𝛿𝛿 𝑑𝑑𝛿𝛿 +

𝜕𝜕2ℎ𝜕𝜕𝛿𝛿2 𝑑𝑑𝛿𝛿 𝑑𝑑𝛿𝛿 𝑑𝑑𝛿𝛿 +

𝜕𝜕2ℎ𝜕𝜕𝛿𝛿2 𝑑𝑑𝛿𝛿 𝑑𝑑𝛿𝛿 𝑑𝑑𝛿𝛿� = 0 (6.21)

Or

𝜕𝜕2ℎ𝜕𝜕𝛿𝛿2 +

𝜕𝜕2ℎ𝜕𝜕𝛿𝛿2 +

𝜕𝜕2ℎ𝜕𝜕𝛿𝛿2 = 0 (6.22)

If the flow is radial symmetrical this continuity equation takes the form

𝜕𝜕2ℎ𝜕𝜕𝜋𝜋2 +

1𝜋𝜋𝜕𝜕ℎ𝜕𝜕𝜋𝜋

+𝜕𝜕2ℎ𝜕𝜕𝛿𝛿2 = 0 (6.23)

b. Pump well

1). Confined aquifer

The case of a well in confined aquifer may be met in an artesian area where the

pressure has declined to the point where pump must be used. The aquifer of


109

permeability K and thickness D is confined above and below between impermeable

formation, with the discharge of pump Q, the condition of continuity is

𝛼𝛼 �𝜕𝜕2𝜋𝜋𝜕𝜕𝜋𝜋2 +

1𝜋𝜋𝜕𝜕𝜋𝜋𝜕𝜕𝜋𝜋� =

𝜕𝜕𝜋𝜋𝜕𝜕𝑡𝑡

(6.24)

A solution which satisfies the continuity requirement and the conditions

s = 0 when t = 0 for r>1

s→0 when r → ∞

is:

𝜋𝜋

=𝑄𝑄

2𝜋𝜋𝑘𝑘𝐶𝐶�𝑒𝑒−𝑢𝑢2

𝑢𝑢

∞

𝜋𝜋√4𝛼𝛼𝑡𝑡

𝑑𝑑𝑢𝑢 (6.25)

where:


Q : discharge of pumping (ft3/s) k : coefficient of permeability (ft/s) D : thickness of aquifer (ft) t : duration of flow (s) s : drawdown (ft)

Above equation (6.25) is a form of the exponential integral and values of this function have been tabulated. In term of the exponential integral function its value is

�𝑒𝑒−𝑢𝑢2

𝑢𝑢

∞

𝜋𝜋√4𝛼𝛼𝑡𝑡

𝑑𝑑𝑢𝑢 = −12𝐸𝐸𝑖𝑖 �−

𝜋𝜋2

4 α t� (6.26)

Value of ∫ 𝑒𝑒−𝑢𝑢2

𝑢𝑢∞𝜋𝜋

√4𝛼𝛼𝑡𝑡𝑑𝑑𝑢𝑢 can be obtained from the Table 6.4. (Glover, 1966) or they can

be computed from the series


110

�𝑒𝑒−𝑢𝑢2

𝑢𝑢𝑑𝑑𝑢𝑢

∞

𝛿𝛿

= −0,288608 − 𝑙𝑙𝑛𝑛𝛿𝛿 +𝛿𝛿2

2−𝛿𝛿4

2! 4+𝛿𝛿6

3! 6… … … … ( 6.27)

When used for finding values to use in equation, (6.25) 𝛿𝛿 = 𝜋𝜋√4𝛼𝛼𝑡𝑡

. This integral can also

be evaluated by use of the tabulated exponential integral as − 12𝐸𝐸𝑖𝑖 �−

𝜋𝜋2

4 α t� as noted

previously.

2). Unconfined aquifer

A well that is to be pumped from unconfined aquifer occurs commonly. The aquifer rests on an impermeable bed and the saturated portion of aquifer terminate at the top in the water table. According to Glover (1966), a moment’s consideration will show that equation (13) can be used to provide an approximate treatment for this case if the drawdown s is everywhere small compare to D, this is the customary treatment for the water table case.

Example:

• Radius of well R = 1 ft (r1 = rw) • Coeffisient of permeability = 0,002 ft/s • Discharge Q = 500 gal/mnt • Thikness of aquifer D = 70 ft • Void ratio V = 0.20 • Duration of pumping T = 72 hrs

Cari drawdown pada radius 1 ft (s1), 50 ft (s50) dan 100 ft (s100):


111

Fig.6.7. Sketch of data condition of pumping

Solution:

Conversion from galon/mnt to ft3/s is:

𝑄𝑄 =500

448,8= 1,1141 𝑓𝑓𝑡𝑡

3𝜋𝜋�


=0,002 × 70

0,20= 0,70 𝑓𝑓𝑡𝑡

2𝜋𝜋�

t = 72 x 3600 = 259200 s so:

𝑄𝑄2𝜋𝜋𝐾𝐾𝐶𝐶

=1,1141

2 × 𝜋𝜋 × 0,002 × 70=

1,11410,87965

= 1,2665 𝑓𝑓𝑡𝑡

√4 α t = �4 × 0,70 × 259200 = √725760 = 851,90 𝑓𝑓𝑡𝑡

𝜋𝜋1√4𝛼𝛼𝑡𝑡

=1

851,9= 0,00174

𝜋𝜋50

√4𝛼𝛼𝑡𝑡=

50851,9

= 0,0587

D=70 ft h100 h50 h1

Q

s50

50

100


112

𝜋𝜋100

√4𝛼𝛼𝑡𝑡=

100851,9

= 0,1174

From Table 6.4. Can be found value of,

𝑌𝑌 = �𝑒𝑒−𝑢𝑢2

𝑢𝑢𝑑𝑑𝑢𝑢

∞

√4𝛼𝛼𝑡𝑡

The value of X is:

• 𝑊𝑊ℎ𝑒𝑒𝑛𝑛 𝜋𝜋 = 1 𝑓𝑓𝑡𝑡 ⇒ 𝜋𝜋1

√4𝛼𝛼𝑡𝑡= 0,00174 𝑓𝑓𝜋𝜋𝜋𝜋𝑚𝑚 𝐹𝐹𝑠𝑠𝑏𝑏𝑙𝑙𝑒𝑒 .⇒ 𝑌𝑌1 = 6,5328





The value of drawdown in distance r from the well is:

𝜋𝜋𝜋𝜋 =𝑄𝑄

2𝜋𝜋𝐾𝐾𝐶𝐶× 𝑌𝑌𝜋𝜋

So the drawdown in distance from the well r is:

• 𝜋𝜋1 = 1 𝑓𝑓𝑡𝑡 ⇒ 𝜋𝜋1 = 1,2665 × (6,5328) = 8,27 𝑓𝑓𝑡𝑡 𝜋𝜋𝜋𝜋 ℎ1 = 61,73 𝑓𝑓𝑡𝑡



The verification is carried out based on the dept of water on r1 is h1 = 61,73 ft as

known value, using Dupuit-Thiem formula Equation (7.32) will be computed r50 and r100

as follows:

• Drawdown in r50 = 50 ft:

𝑄𝑄 =𝜋𝜋 × 0.002(ℎ50

2 − ℎ12)

𝑙𝑙𝑛𝑛�𝜋𝜋50 𝜋𝜋1� � ⇒ 1,1141 =

𝜋𝜋 × 0.002(ℎ502 − 61,732)

𝑙𝑙𝑛𝑛�501� �

⇒ ℎ50 = 67,11 𝑓𝑓𝑡𝑡

⇒ s50 = 70 – 67,11 = 2,89 ft


113

• Drawdown in r100 = 100 ft:

𝑄𝑄 =𝜋𝜋 × 0.002(ℎ100

2 − ℎ12)

𝑙𝑙𝑛𝑛�𝜋𝜋100 𝜋𝜋1� � ⇒ 1,1141 =

𝜋𝜋 × 0.002(ℎ1002 − 61,732)

𝑙𝑙𝑛𝑛�1001� �

⇒ ℎ100 = 68,02𝑓𝑓𝑡𝑡

⇒ s100 = 70 – 68,02 = 1,98 ft


114

Table 6.3. Value of integral 𝑌𝑌 = ∫ 𝑒𝑒−𝑢𝑢2

𝑢𝑢𝑑𝑑𝑢𝑢∞

√4𝛼𝛼𝑡𝑡 for given values of parameters 𝛿𝛿 = 𝜋𝜋√4𝛼𝛼𝑡𝑡

Continued


115




Continued


116




Continued


117




Continued


118

5. Sunjoto Sunjoto developed his unsteady flow condition method based on the steady flow

condition theory of Forchheimer (1930). In this case, Glover (1966) was using

parameters data are ; ‘Radius of well, Coeffisient of permeability, Discharge,

Thikness of aquifer, Void ratio and Duration of pumping’ that those parameters are

needed by Sunjoto’s theory except Void ratio, due to the this value influences already

the value of Coefficient of permeability or in other word that the Coeffisient of

permeability is function of Void ratio.

For the computation Sunjoto needs the new parameter is shape factors that it can be

computed by above data as:

• Shape factor of the tip of full penetration well of the confined aquifer with

the piezometric above the water table is:

𝐹𝐹 =2π𝐶𝐶

𝑙𝑙𝑛𝑛 �2(𝐶𝐶 + 2𝑅𝑅)𝑅𝑅 + ��2𝐶𝐶

𝑅𝑅 �2

+ 1� (6.28)

• Shape factor of the tip of well of the unconfined aquifer is:

𝐹𝐹 =2π𝐿𝐿

𝑙𝑙𝑛𝑛 �(𝐿𝐿 + 2𝑅𝑅)𝑅𝑅 + ��𝐿𝐿𝑅𝑅�

2+ 1�

(6.29)

where:

F : shape factor of well (L) R : radius of well (L) L : length of porous casing (L) D : thickness of the aquifer (L)

This computation needs trial and error or iteration by computer by taking some value of drawdown (s1), then you get (h1) and compute the value of shape factor (F) 0f well.

Then compute drawdown using Sunjoto (1988) formula based of the recharge well with H is ‘built up’ as:


119



Based on the the above formula can be used the pumping well formula with drawdown (s1) as ‘drawdown’ as:

𝜋𝜋 =𝑄𝑄𝐹𝐹𝐾𝐾

�1− 𝑒𝑒𝛿𝛿𝑒𝑒 �−𝐹𝐹𝐾𝐾𝐹𝐹𝜋𝜋𝑅𝑅2 �� (6.31)

where:

H : built up (L) s : drawdown (L) Q : recharge or discharge (L3/T) F : shape factor of well (L) K : coeffisient of permeability (L/T) T : duration of flow (T) R : radius of well (L)

Then compute the value of drawdown in the distance r2 as unknown with data of r1 using Dupuit-Thiem theory as follows:

𝑄𝑄 =𝜋𝜋𝐾𝐾(ℎ1

2 − ℎ22)

𝑙𝑙𝑛𝑛�𝜋𝜋1 𝜋𝜋2� � (6.32)

Example:

With the data above it can be compute as follows:

1). First step

Take some value of drawdown, for instantce (s1)= 6 ft so L = 70-6 = 64 ft

𝐹𝐹 =2 × π × 64

𝑙𝑙𝑛𝑛 �(64 + 2 × 1)1 + ��64

1 �2

+ 1�= 82,6135 𝑓𝑓𝑡𝑡

Subtitute the above value of F to Equation (7.31):

𝜋𝜋 =1,1141

82,6135 × 0,002�1 − 𝑒𝑒𝛿𝛿𝑒𝑒 �

−82,6135 × 0,002 × 259200𝜋𝜋 × 12 �� = 6,74

So height of water in r1 is h1 = 70 – 6,74 = 63, 36 ft


120

2). Second step

Take value of drawdown (s1) = 6,50 ft so L = 70-6,50 = 63,50 ft and compute shape factor as follows:

𝐹𝐹 =2 × π × 63,5

𝑙𝑙𝑛𝑛 �(63,5 + 2 × 1)1 + ��63,5

1 �2

+ 1�= 82,0983

Subtitute above shape factor F to Equation (7.31):

𝜋𝜋 =1,1141

82,0983 × 0,002�1 − 𝑒𝑒𝛿𝛿𝑒𝑒 �

−82,0983 × 0,002 × 259200𝜋𝜋 × 12 �� = 6,78

So height of water in r1 is h1 = 70 – 6,78 = 63,22 ft

3). Third step


𝐹𝐹 =2 × π × 63,2

𝑙𝑙𝑛𝑛 �(63,2 + 2 × 1)1 + ��63,2

1 �2

+ 1�= 81,7889

Subtitute F to Equation (7.31):

𝜋𝜋 =1,1141

81,7889 × 0,002�1 − 𝑒𝑒𝛿𝛿𝑒𝑒 �

−81,7889 × 0,002 × 259200𝜋𝜋 × 12 �� = 6,81

So height of water in r1 is h1 = 70 – 6,81 = 63,19 ft

4). Fourth step


𝐹𝐹 =2 × π × 63,19

𝑙𝑙𝑛𝑛 �(63,19 + 2 × 1)1 + ��63,19

1 �2

+ 1�= 81,7856

Subtitute F to Equation (7.31):


121

𝜋𝜋 =1,1141

81,7856 × 0,002�1 − 𝑒𝑒𝛿𝛿𝑒𝑒 �

−81,7856 × 0,002 × 259200𝜋𝜋 × 12 �� = 6,81

Due to the both value of s are already the equal value is 6,81 or value of your input data of drawdown is equal to the result of computation so it means that the final result of drawdown is 6,81 ft.

Then compute the value of drawdown in in the distance r50 and r100 using Equation (6.32) as follows:

• Drawdown in r = 50 ft:

𝑄𝑄 =𝜋𝜋 × 0.002(ℎ50

2 − ℎ12)

𝑙𝑙𝑛𝑛�𝜋𝜋50 𝜋𝜋1� � ⇒ 1,1141 =

𝜋𝜋 × 0.002(ℎ502 − 63,192)

𝑙𝑙𝑛𝑛�501� �

⇒ ℎ50 = 68,46 𝑓𝑓𝑡𝑡

Drawdown s50 = 70 – 68,46 = 1,54 ft

• Drawdown in r = 100 ft:

𝑄𝑄 =𝜋𝜋 × 0.002(ℎ100

2 − ℎ12)

𝑙𝑙𝑛𝑛�𝜋𝜋100 𝜋𝜋1� � ⇒ 1,1141 =

𝜋𝜋 × 0.002(ℎ1002 − 63,192)

𝑙𝑙𝑛𝑛�1001� �

⇒ ℎ100 = 69,35 𝑓𝑓𝑡𝑡

Drawdown s100 = 70 – 69,35 =0,65 ft

Based on the same data the drawdown using Sunjoto’s method are: r1 = 1 ft --> s1 = 6,81 ft r 50 = 50 ft --> s50 = 70 - 68,80 = 1,54 ft r100 = 100 ft --> s100 = 70 - 69,70 = 0,65 ft Table 6.4 Recapitulation of drawdown values of Glover’s and Sunjoto’s method. Drawdown (s) Radius (r)

Glover (ft ) Sunjoto

(ft) Computed by

Glover Method Computed by Dupuit-

Thiem Method 1 ft

50 ft 100 ft

8,27 )* 3,23 )* 2,36 )*

8, 27 )* 2,89 )** 1,98 )**

6, 81 )*** 1, 54 )** 0,65 )**

Note: )* computed by Glover’s method. )** computed by Dupuit-Thiem method. )*** computed by Sunjoto’s method.


122

VII. DISCHARGE AND RECHARGE SYSTEM

1. Well

Using Forchheimer (1930) principle which form is steady state flow condition,

Sunjoto (1988) developed an unsteady state radial flow formula for well which

was derived by integration solution. His formula computes a dimension of

recharge well, which catch rainwater to infiltrate to the ground to increase

groundwater storage.

• Hollow well

𝑯𝑯 =𝑸𝑸𝑭𝑭𝑲𝑲

�𝟏𝟏 − 𝒆𝒆𝒅𝒅𝒆𝒆�−𝑭𝑭𝑲𝑲𝑻𝑻π𝑹𝑹𝟐𝟐

�� (7.1)

• Filled material well

𝑯𝑯′ =

𝑸𝑸𝑭𝑭𝑲𝑲

�𝟏𝟏 − 𝒆𝒆𝒅𝒅𝒆𝒆 �−𝑭𝑭𝑲𝑲𝑻𝑻nπ𝑹𝑹𝟐𝟐

�� (7.2)

where:

H : depth of hollow well (L) H’ : depth of filled material well (L) F : shape factor (L) K : coefficient of permeability (L/T) T : dominant duration of precipitation (T) R : radius of well (L) Q : inflow discharge (L3/T), dan Q = C I A C : runoff coefficient of roof ( ) I : precipitation intensity (L/T) A : roof area (L2) n : porosity of filled material ( )


123

Formula Development of Shape Factors a. Ellipse

Basic equation of ellipse (LaRue et Risi, 1960. Mathématiques Intermediaires):

𝛿𝛿2

𝑠𝑠2 +𝛿𝛿2

𝑏𝑏2 = 1 Theoreme:

𝛿𝛿2 = 𝛿𝛿2 + 𝛿𝛿 2 (7.3)

Fig. 1.1. Ellipse

b. Basic equation of radial flow

Fig. 7.2. Cross section of aquifer between two impermeable layers Boundary condition: Y = Ho → x = Ro Y = H1 → x = R

R

H

Ho H1

L’ R0

dh

dr

ae

b

a

e : excentrisity of ellipse and e < 1 a and b positive

z

x

y


124

Darcy’s Law (1856)

𝑉𝑉 = 𝐾𝐾𝑖𝑖 𝑠𝑠𝑛𝑛𝑑𝑑 𝑖𝑖 =𝑑𝑑ℎ𝑑𝑑𝜋𝜋

𝑄𝑄 = 𝑉𝑉𝐴𝐴 𝑠𝑠𝑛𝑛𝑑𝑑 𝑉𝑉 = 2𝜋𝜋𝛿𝛿𝐿𝐿′

𝑆𝑆𝜋𝜋 ∶ 𝑄𝑄 =2𝜋𝜋𝐿𝐿′𝐾𝐾𝐾𝐾

𝑙𝑙𝑛𝑛 𝑅𝑅𝜋𝜋𝑅𝑅 (7.4)

c. Well condition 5b of Dachler.

According to Dachler (1936), the direction of equipotential to the permeable casing

will be an ellipses form and the stream lines which are perpendicular to them are flow

lines which hyperbolic form, and from his equation can be concluded that no water

flow through the base of the well (Fig. 8.5.).

When h = H and a = ∞ the equation will be: 𝐾𝐾(𝐾𝐾 − ℎ) = 𝑄𝑄

2π𝑡𝑡𝐴𝐴𝜋𝜋𝑆𝑆ℎℎℎℎℎℎℎℎℎ�𝑡𝑡

𝑠𝑠�ℎ (7.5)

𝐾𝐾(𝐾𝐾 − ℎ) =𝑄𝑄

2π𝑡𝑡𝑙𝑙𝑛𝑛�

𝑡𝑡𝑠𝑠

+ �1 + �𝑡𝑡𝑠𝑠�

2�

𝑄𝑄 =2π𝑡𝑡𝐾𝐾(𝐾𝐾 − ℎ)

𝑙𝑙𝑛𝑛 �𝑡𝑡𝑠𝑠 + �1 + �𝑡𝑡𝑠𝑠�2�

When t = L, a = R so:

𝑭𝑭 =𝟐𝟐π𝑳𝑳

𝒅𝒅𝒏𝒏 �𝑳𝑳𝑹𝑹 + �𝟏𝟏 + �𝑳𝑳𝑹𝑹�𝟐𝟐�

(7.6)


125

Fig. 7.3. Cross section of aquifer under impermeable layer (Dachler, 1936)

d. Well condition 5b of Sunjoto (1989)

Assumption I: 𝐿𝐿′ = 𝐿𝐿 + 𝑅𝑅𝑙𝑙𝑛𝑛2 Assumption II: a = Ro – 2(½ L + R); b = L; c = R Explication of assumption I.

• The fact that there is a flow of water though the base of well so it must be taken consideration.

• Area of base of well is equal to the area of the wall which length ½ R but due to the hydraulic gradient on the base of well is bigger than on the wall so we take value 2/3 R as an addition of length of permeable well.

• Finally on the detail computation it found that addition of length of permeable wall is not 2/3 R but R. ln2:


126

Fig. 7.4. Cross section of real and theoritic aquifer

𝑅𝑅𝜋𝜋 − 2 �12𝐿𝐿 + 𝑅𝑅� = �𝐿𝐿2 + 𝑅𝑅2

𝑹𝑹𝒐𝒐 = (𝑳𝑳 + 𝟐𝟐𝑹𝑹) + √𝑳𝑳𝟐𝟐 + 𝑹𝑹𝟐𝟐 (7.7)

Substitution: (8.7) → (8.4)

𝑄𝑄 =2𝜋𝜋(𝐿𝐿 + 𝑅𝑅𝑙𝑙𝑛𝑛2)𝐾𝐾𝐾𝐾

𝑙𝑙𝑛𝑛 (𝐿𝐿 + 2𝑅𝑅) + √𝐿𝐿2 + 𝑅𝑅2

𝑅𝑅

𝑄𝑄 =2𝜋𝜋(𝐿𝐿 + 𝑅𝑅𝑙𝑙𝑛𝑛2)𝐾𝐾𝐾𝐾

𝑙𝑙𝑛𝑛 �𝐿𝐿 + 2𝑅𝑅𝑅𝑅 + �1 + �𝐿𝐿 𝑅𝑅� �

2�

𝑭𝑭𝟓𝟓𝒃𝒃 =𝟐𝟐π𝑳𝑳+ 𝟐𝟐π𝑹𝑹𝒅𝒅𝒏𝒏𝟐𝟐

𝒅𝒅𝒏𝒏 �𝑳𝑳 + 𝟐𝟐𝑹𝑹𝑹𝑹 + ��𝑳𝑳𝑹𝑹�

𝟐𝟐+ 𝟏𝟏�

(7.8)

• When R = 1, L = 0 and 𝐿𝐿′ = 𝐿𝐿 + 𝑅𝑅𝑙𝑙𝑛𝑛2 so F5b = 3,964 R and this value approach

99% of F3b = 4R (Forchheimer, 1930)

L L

R R

L’=L+Rln2

a. Real b. Theoritic


127

Table 7.1. Assumption I, between real and theorem condition on the tip of well Description

Dachler (1936)

Length of permeable wall

Sunjoto (1989; 2010)

Length of permeable wall

Real Function Real Function

Condition 1

-

-

L

L

Condition 5b

L

L

L

L+ R.ln2

Condition 6b

L

L

L

L+ R.ln2

Table 7.2. Assumption II between real and theorem condition on the tip of well

Description

Dachler (1936)

Sunjoto (1989; 2010)

Condition 1

-

a = Ro – 4(½ L + R)

b = 2L c = R

Condition 5b

a = Ro – L

b = L c = R

a = Ro – 2(½ L + R)

b = L c = R

Condition 6b

a = Ro – ½ L

b = ½ L c = R

a = Ro – (½ L + R)

b = ½ L c = R


128

Table 7.3. Flowchart of formula derivation No Condition Shape Factor Reverences

F when L=0

3b

𝐹𝐹3𝑏𝑏 = 4𝑅𝑅

Forchheimer (1930)

Dachler (1936) Aravin (1965)

4,000

5b

𝑭𝑭𝟓𝟓𝒃𝒃 =𝟐𝟐π𝑳𝑳

𝒅𝒅𝒏𝒏 �𝑳𝑳𝑹𝑹 + ��𝑳𝑳𝑹𝑹�𝟐𝟐

+ 𝟏𝟏�

Dachler (1936)

0/0


𝑙𝑙𝑛𝑛 �𝐿𝐿 + 2𝑅𝑅𝑅𝑅 + ��𝐿𝐿𝑅𝑅�

2+ 1�

Sunjoto (2002)

3,964

4b

𝐹𝐹4𝑏𝑏 = 5.50𝑅𝑅

Harza (1935) Taylor (1948)

Hvorslev (1951)

5,50

𝑭𝑭𝟖𝟖 𝒃𝒃 = 𝟐𝟐𝝅𝝅𝑹𝑹

Sunjoto (2002)

6,283

6b

𝑭𝑭𝟖𝟖𝒃𝒃 =𝟐𝟐π𝑳𝑳

𝒅𝒅𝒏𝒏 � 𝑳𝑳𝟐𝟐𝑹𝑹 + �� 𝑳𝑳𝟐𝟐𝑹𝑹�𝟐𝟐

+ 𝟏𝟏�

Dachler (1936)

0/0



2+ 1�

Sunjoto (2002)

6,283

Note: The flowchart of thinking • Formula F3b was derived mathematically like F2a and F3a. • Based on F3b, be derived the first F5b then the second F6b finally the third F4b.

1 2

3

4

5


129

2. Trench When the groundwater surface high so the efficiency of recharge well will be

decrese. The structure mus be developed horizontally and it is called Recharge

Trench. The design is to compute the length (B) of trench with know width (b)

and depth (H).

Fig. 7.5.. Sketch of water balance on the trench

Volume of storage of trench is the difference of input flow and water to

infiltrate on the trench.

𝑑𝑑𝑉𝑉𝜋𝜋𝑙𝑙𝑡𝑡 = 𝐴𝐴𝜋𝜋𝑑𝑑ℎ

𝑑𝑑𝑉𝑉𝜋𝜋𝑙𝑙𝑡𝑡 = (𝑄𝑄 − 𝑄𝑄0)𝑑𝑑𝑡𝑡 = (𝑄𝑄 − 𝐹𝐹𝐾𝐾ℎ)𝑑𝑑𝑡𝑡

h2 H

h h1

dh

t2 T

B

t

Qo=FK

b

t1

dt

Y

X

Qi = Q


130

where, Qo : outflow discharge Q : inflow discharge As : cross section area of casing

h : depth of water t : duration of flow F : shape factor of casing K : coefficient of permeability

The above equation is solved by integration :

𝐴𝐴𝜋𝜋𝑑𝑑ℎ = (𝑄𝑄 − 𝐹𝐹𝐾𝐾ℎ)𝑑𝑑𝑡𝑡 ⇒ 𝑑𝑑𝑡𝑡 =𝐴𝐴𝜋𝜋𝑑𝑑ℎ

𝑄𝑄 − 𝐹𝐹𝐾𝐾ℎ ÷

𝐹𝐹𝐾𝐾𝐹𝐹𝐾𝐾

Finally the result are::

1). Hollow trench (Sunjoto, 2008)

𝑩𝑩 =−𝒇𝒇𝑲𝑲𝑻𝑻

𝒃𝒃�𝒅𝒅𝒏𝒏 �𝟏𝟏 − 𝒇𝒇𝑲𝑲𝑯𝑯𝑸𝑸 ��

(𝟒𝟒.𝟗𝟗)

2). Material filled trench (Sunjoto, 2008)

𝑩𝑩′ =−𝒇𝒇𝑲𝑲𝑻𝑻

𝒏𝒏𝒃𝒃 �𝒅𝒅𝒏𝒏 �𝟏𝟏 − 𝒇𝒇𝑲𝑲𝑯𝑯𝑸𝑸 ��

(𝟒𝟒. 𝟏𝟏𝟑𝟑)

where, B : length of trench (L) B’ : length of trench material filled (L) b : width of trench (L) f : shape factor of trench (L) ⇒ Tabel 28. K : coefficient of permeability (L/T) H : depth of water on trench (L) T : dominant duration of precipitation (T) Q : inflow discharge (L3/T) and Q = CIA C : runoff coefficient of roof (-) I : precipitation intensity (L/T) A : area of roof (L2) n : porosity of material filled


131

Shape factor of trench (f) is developed from well shapoe factor as follows (Sunjoto, 2008):

(a). Shape factor of trench is well shape factor multiplied by ‘shape coefficient’ (SC). (b). Shape coefficient is ‘perimeter coefficient’ multiplied by ‘area coefficient’ (c). ‘Perimeter coefficient’ circular form to square form is perimeter of square (4b) divided by perimeter of circle (2πR) or equal to ( )Rb π2/4 . (d). ‘Area coefficient’ from square form to rectangular form is root of the rectangular area devided by square area or equal to ( 2/)( bbB ). (e). Finally value of ‘shape coefficient’ (SC) from circle form to the rectangular form is equal to: 4𝑏𝑏 (2𝜋𝜋𝑅𝑅)⁄ × �(𝑏𝑏.𝐵𝐵) 𝑏𝑏2⁄ = �2√𝑏𝑏.𝐵𝐵� (𝜋𝜋𝑅𝑅)�

𝑺𝑺𝒐𝒐 ∶ 𝑺𝑺𝑪𝑪 =𝟐𝟐√𝒃𝒃.𝑩𝑩𝝅𝝅𝑹𝑹

𝒅𝒅𝒂𝒂𝒏𝒏 𝒇𝒇𝑲𝑲 = 𝑭𝑭𝑲𝑲 ×𝟐𝟐√𝒃𝒃.𝑩𝑩𝝅𝝅𝑹𝑹

(𝟒𝟒.𝟏𝟏𝟏𝟏)

where:

fi : trench shape factor in i condition of tip of trench Fi : well shape factor in i condition of tip of well


132

Tabel 28. Shape factor of trenches (Sunjoto, 2008) No Condition Shape factor of trenchs (f)

1

𝒇𝒇𝟏𝟏 =𝟖𝟖𝑳𝑳

𝒅𝒅𝒏𝒏 �𝑳𝑳 + 𝟖𝟖√𝒃𝒃𝑩𝑩√𝒃𝒃𝑩𝑩

+ �� 𝑳𝑳√𝒃𝒃𝑩𝑩

�𝟐𝟐

+ 𝟏𝟏�

2

𝒇𝒇𝟐𝟐𝒂𝒂 = 𝟖𝟖√𝒃𝒃𝑩𝑩

𝒇𝒇𝟐𝟐𝒃𝒃 =𝟑𝟑𝟖𝟖𝝅𝝅√𝒃𝒃𝑩𝑩

3

𝒇𝒇𝟑𝟑𝒂𝒂 = 𝟖𝟖√𝒃𝒃𝑩𝑩

𝒇𝒇𝟑𝟑𝒃𝒃 =𝟖𝟖𝝅𝝅√𝒃𝒃𝑩𝑩

4

𝒇𝒇𝟖𝟖𝒂𝒂 = 𝟐𝟐π √𝒃𝒃𝑩𝑩

𝒇𝒇𝟖𝟖𝒃𝒃 = 𝟖𝟖√𝒃𝒃𝑩𝑩

5

𝒇𝒇𝟓𝟓𝒂𝒂 =𝟖𝟖𝑳𝑳 + 𝟐𝟐π√𝒃𝒃𝑩𝑩𝒅𝒅𝒏𝒏𝟐𝟐

𝒅𝒅𝒏𝒏�𝑳𝑳 + 𝟖𝟖√𝒃𝒃𝑩𝑩𝟐𝟐√𝒃𝒃𝑩𝑩

+ �� 𝑳𝑳𝟐𝟐√𝒃𝒃𝑩𝑩

�𝟐𝟐

+ 𝟏𝟏�

𝒇𝒇𝟓𝟓𝒃𝒃 =𝟖𝟖𝑳𝑳 + 𝟖𝟖√𝒃𝒃𝑩𝑩𝒅𝒅𝒏𝒏𝟐𝟐

𝒅𝒅𝒏𝒏 �𝑳𝑳 + 𝟖𝟖√𝒃𝒃𝑩𝑩𝟐𝟐√𝒃𝒃𝑩𝑩

+ �� 𝑳𝑳𝟐𝟐√𝒃𝒃𝑩𝑩

�𝟐𝟐

+ 𝟏𝟏�

𝒇𝒇𝟖𝟖𝒂𝒂 =𝟖𝟖𝑳𝑳 + 𝟐𝟐π√𝒃𝒃𝑩𝑩𝒅𝒅𝒏𝒏𝟐𝟐

𝒅𝒅𝒏𝒏�𝑳𝑳 + 𝟖𝟖√𝒃𝒃𝑩𝑩𝟖𝟖√𝒃𝒃𝑩𝑩

+ �� 𝑳𝑳𝟖𝟖√𝒃𝒃𝑩𝑩

�𝟐𝟐

+ 𝟏𝟏�

b

b

b

b

b

b

b

b

b

L

b

b

L

L

L


133

6

𝒇𝒇𝟖𝟖𝒃𝒃 =𝟖𝟖𝑳𝑳 + 𝟖𝟖√𝒃𝒃𝑩𝑩𝒅𝒅𝒏𝒏𝟐𝟐

𝒅𝒅𝒏𝒏 �𝑳𝑳 + 𝟖𝟖√𝒃𝒃𝑩𝑩𝟖𝟖√𝒃𝒃𝑩𝑩

+ �� 𝑳𝑳𝟖𝟖√𝒃𝒃𝑩𝑩

�𝟐𝟐

+ 𝟏𝟏�

7

𝒇𝒇𝟒𝟒𝒂𝒂 =𝟖𝟖𝑯𝑯 + 𝟐𝟐π√𝒃𝒃𝑩𝑩𝒅𝒅𝒏𝒏𝟐𝟐

𝒅𝒅𝒏𝒏�𝑯𝑯 + 𝟖𝟖√𝒃𝒃𝑩𝑩𝟖𝟖√𝒃𝒃𝑩𝑩

+ �� 𝑯𝑯𝟖𝟖√𝒃𝒃𝑩𝑩

�𝟐𝟐

+ 𝟏𝟏�

𝒇𝒇𝟒𝟒𝒃𝒃 =𝟖𝟖𝑯𝑯 + 𝟖𝟖√𝒃𝒃𝑩𝑩𝒅𝒅𝒏𝒏𝟐𝟐

𝒅𝒅𝒏𝒏�𝑯𝑯 + 𝟖𝟖√𝒃𝒃𝑩𝑩𝟖𝟖√𝒃𝒃𝑩𝑩

+ �� 𝑯𝑯𝟖𝟖√𝒃𝒃𝑩𝑩

�𝟐𝟐

+ 𝟏𝟏�

3. Dewatering

a. Pump

The power of pump required can be calculated using the formula as follows:

ηρHQP = (7.12)

where, Q : discharge (m3/s) P : power of pump (kg m/s) H : hydraulic head (m) ρ : unit weight of water (1000 kg/m3)

η : efficiency of pump

b. Water losses

The following is an example of a building site where the digging dimension is

100 x 100 m2, the lowering of elevation of unconfined aquifer surface is 8 m (from -

3.00 m to -11.00 m), the coefficient of soil permeability is 5,10-5 m/s and the duration

of continuous pumping is 7 months, that is the time used to construct the lower parts

of a building. This duration consists of 1-month continuous pumping in unsteady flow

condition and 6-month continuous pumping in steady flow condition. For pumping in the

b

b

b

L

H

H


134

well when the flow is in a steady flow condition can be calculated using the formula as

follows (Forchheimer, 1930):

Q = F K H (7.13)

where, Q : discharge (m3/s) F : shape factor of well (m) K : coefficient of permeability (m/s) H : depth of water (m) Hvorslev (1951) had developed the following shape factor of well formula with

R is radius of well, K is coefficient of soil permeability and Kv is vertical coefficient

of soil permeability as well as h length of under part casing and well laid on porous

layer:

vKK

RhRF××+

=

π111

5.5 (7.14)

According to Hvorslev (1951) the value 5.5 R is average amount from the three

researchers were Harza (1935), Taylor (1948) and Hvorslev (1951). On analytical

study, Sunjoto (2002) found that value is 2 π R and Eq. (4) becomes:

vKK

RhRF××+

=

π

π111

2 (7.15)

where, F : shape factor of well (m) R : radius of well (m) h : depth of sheet pile (m) K : coefficient of permeability (m/s) Kv : vertical coefficient of permeability (m/s)


135

By analogy with the Eq. (5), the shape factor of rectangular cross section hole

of dewatering with sheet pile as deep as h below the digging area and assuming that

the soil is homogenous and isotropic (K = Kv), a proposed equation for rectangular

form can be formulated as follows:

bBh

bBf×+

=

π111

4 (7.16)

where, f : shape factor of rectangle (m) h : depth of sheet pile (m) B : length (m) b : width (m) K : coefficient of permeability (m/s)

Base on the principle of Forchheimer (1930), Sunjoto (1988) developed a

formula of pumping or recharging on the well in unsteady flow condition with equation

as follows:

−−= 2exp1

R

FKTFKQ

Hπ

(7.17)

where, H : depth of water on well (m) F : shape factor of well (m) K : coefficient of permeability (m/s) T : duration of flow (s) R : radius of well (m) Q : discharge (m3/s), Q = CIA C : runoff coefficient I : precipitation intencity (m/s) A : area of roof (m2) For the pumping in dewatering area with rectangular cross section can be found

by modifying the circle area π R2 into a square area bB, the shape factor of circle or


136

well (F) becomes the shape factor of rectangle (f), therefore, the equation (7.77) can

be changed into:

−−

=

bBfKT

fKHQexp1

(7.18)

where: Q : pumping discharge (m3/s) B : length (m) b : width (m) f : shape factor of rectangular (m) K : coefficient of permeability (m/s) T : duration of pumping (s) H : hydraulic head (m) c. Computation

With K = Kv = 5,10-5 m/s, B = b = 100 m, H = 8 m, h = 8 m and the sketch as of

Fig. 1, the discharge, water losses, power of pump can be calculated using the above

formulas as follows:

1). Influence of drawdown

With the drawdown is 8 m and the coefficient of soil permeability is 5.10-5 m/s,

the radius of influence by substituting to Eq. (1) is as follows:

mxxL 17010,583000 5 == −

In this case the water table will lower, starting from the biggest drawdown at

the edge of the digging area or sheet pile as far as 170 m, around the digging area.

2). Shape factor

By using the proposed equation Eq. (6), value of shape factor can be calculated:

472.312

1001008111

1001004=

××+

×=

π

f m

3). Unsteady flow pumping


137

Unsteady flow pumping is a pumping to lower the water table from elevation of

-3.00 to -11.00 or hydraulic head H = 8 m. The discharge in unsteady flow condition

within the planned time of one month can be calculated using Eq. (8):

×

××××−−

××=

−

−

1001003024600,310,5472.312exp1

810,5472.3125

5

Q = 0.1272 m3/s

4). Steady flow pumping

Steady flow pumping is a pumping to maintain water surface after the elevation

has reached -11.00 m or the hydraulic head H = 11 m. The pumping duration is 6

months according to the carrying out basement construction duration and using Eq.

(3), the discharge is:

Q = 312.472 x 5,10-5 x 11 = 0.1719 m3/s

5). Volume of water losses

During the underground construction, the volume of water losses (V) is:

a) Volume of water losses in one-month pumping:

Vuf = 30 x 24 x 3,600 x 0.1272 = 458,164 m3

b) Volume of water losses in six-month pumping:

Vsf = 180 x 24 x 3,600 x 0.1719 = 2,673,389 m3

Thus, total volume of water losses during construction with dewatering is:

V = Vuf + Vsf = 458,164 + 1,944,000 = 3,131,553 m3

The number is resulted from the pumping volume of the one-month unsteady

flow condition added with the volume of six-month of steady flow condition. The

number, however, does not calculate water addition caused by rain, which might fall,

during the construction period.

6). Equivalence with domestic water consumption

When compared with the domestic water consumption of average of Indonesian

people consuming 100 l/day/cpt, the volume of water losses equals with:


138

Number of people = (3,131,553 /7/30/0,100) = 149,121 cpt

Thus, the water losses caused by dewatering equals to the domestic water

consumption for 149,121 people for 7 months.

7). Pump capacity in unsteady flow condition

The running pump capacity required can be calculated using Eq. (2) where Q =

0.1272 m3/s and H = 11 m that is, from -11.00 to + 0.00 m and pump efficiency η = 0.60

:

=××

=60.0

1110001272.0P 2,332 kgm/s = 23.11 KW

8). Pump capacity in steady flow condition

The running pump capacity required can be calculated using Eq. (2) where Q =

0.1768 m3/s and H = 11 m, that is, from -11.00 to + 0.00 m.

60.0

1110001719.0 ××=P = 3,152 kgm/s = 31.26 KW

In the implementation, several pumps with the determined total capacity are placed

around the edge of the digging area near the sheet pile.

B=b=100 m

-10.00

-18.00

Ground surface: gs1 -0.00 Groundwater surface : gws1 -3.00

sheetpile

gs2 -10.00 m

K

h = 8 m

Figure. 7.6. Sketch of dewatering site cross section (non scale).

gws2 -11.00 m


139

REFERENCES Chow, V.T. 1952. On the determination of transmissibility and storage coefficients from pumping test data , Chow, V.T. 1964. Handbook of Applied Hydrology. New York, McGraw Hill Book Co. Cooper H.H,Jr. and Jacob C.E. 1946. A generalized graphical method for evaluating formations constants

and summarizing well-field history, Trans. Amer. Geophysical Union, v.27, pp. 526-534. Glover R.E.1966. Groundwater movement, U.S. Bureau of Reclamation Engineering Monograph no 31,

Denver,76. Lee, Richard. 1980. Forest Hydrology, translated by Subagio Sentot, Gadjah Mada Press, Yogyakarta LinsleynR.K., M.A. Kohler J.I.H. Paulhus. 1975. Hydrology for Engineers. New York, McGraw Hill Book

Co. Murthy V.N.S. 1977. Soil Mechanic and Foundation Engineering, Delhi (2nd ed.) Porchet M., 1931. Hydrodinamique des puits. Ann. Du Genie Rural fasc.6 Todd, D.K. 1980. Groundwater Hydrology, John Wiley & Sons. Trans. Amer. Geophysical Union, v.33, pp.

397-404. Theis C.V. 1935. He relation between the lowering of piezometric surface and the rate and duration of

discharge of well using groundwater storage, Trans. Amer. Geophysical Union, v.16, pp. 519-524 Sriyono E, 2011. Debit Aliran Air Tabah Melalui Pipa Berpori Sistem Sumur Kolektor Berjari, Jurnal Teknik

Universitas Jana Badra Yogyakarta, Vol. 1 No. 2, Oktober 2011 Suharyadi. 1984. Geohidrologi (Ilmu Air Tanah) Lecture none, Jurusan Teknik Geologi Fakultas Teknik

Universitas Gadjah Mada, Yogyakarta UNESCO, 1967. 1967. Methods and Techniques of Groundwater Investigation and Development, Water

Resources Series No. 33, New York.

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