# Mathematical and model analyses

a. General.

(1) Design. Design of a dewatering system re­quires the determination of the number, size, spacing, and penetration of wells or wellpoints and the rate at which water must be removed from the pervious strata to achieve the required groundwater lowering or pres­sure relief. The size and capacity of pumps and collec­tors also depend on the required discharge and draw­down. The fundamental relations between well and wellpoint discharge and corresponding drawdown are presented in paragraphs 4-2, 4-3, and 4-4. The equa­tions presented assume that the flow is laminar, the pervious stratum is homogeneous and isotropic, the water draining into the system is pumped out at a con­stant rate, and flow conditions have stabilized. Proce­dures for transferring an anisotropic aquifer, with re­spect to permeability, to an isotropic section are pre­sented in appendix E.

(2) Equations for flow and dmwdown to drainage slots and wells. The equations referenced in para­graphs 4-2, 4-3, and 4-4 are in two groups: flow and drawdown to slots (b below and fig. 4-1 through 4-9) and flow and drawdown to wells (c below and fig. 4-10 through 4-22). Equations for slots are applicable to

 4-1

 FLOW

 DRAWDOWN

 „ kDx, Q* — (H *he

 ARTESIAN FLOW

 FLOW

 kx / 2 ,2 Q = — (Н – h 2L о

 (3)

 (4)

 FROM FIG. 4.2

 GRAVITY FLOW

DRAWDOWN

L[D2-(ho + hs)2] L° ~2DH – D2 -(h + h f

0 s

COMBINED ARTESIAN-GRAVITY FLOW

(Modified from “Foundation Engineering,” G. A. Leonards, ed.,1962, McGraw-Hill Book Company.

Used with permission of McGraw-Hill Book Company.)

Figure 4-1. Flow and head for fully penetrating line slot; single-line source; artesian, gravity, and combined flows,

flow to trenches, French drains, and similar drainage systems. They may also be used where the drainage system consists of closely spaced wells or wellpoints. Assuming a well system equivalent to a slot usually simplifies the analysis; however, corrections must be made to consider that the drainage system consists of wells or wellpoints rather than the more efficient slot. These corrections are given with the well formulas dis­cussed in c below. When the well system cannot be simulated with a slot, well equations must be used. The figures in which equations for flow to slots and wells appear are indexed in table 4-1. The equations
for slots and wells do not consider the effects of hy­draulic head losses Hw in wells or wellpoints; proce­dures for accounting for these effects are presented separately.

(3) Radius of influence R. Equations for flow to drainage systems from a circular seepage source are based on the assumption that the system is centered on an island of radius R. Generally, R is the radius of influence that is defined as the radius of a circle be­yond which pumping of a dewatering system has no significant effect on the original groundwater level or piezometric surface. The value of R can be estimated

 _L H

 (Modified from "Foundation Engineering, " Q. A. Leonards, ed.,1962, McGraw-Hill Book Company. Used with permission of McGraw-Hill Book Company.)

 ea/d

 (b)

ARTESIAN FLOW

(Modified from “Foundation Engineering," G. A. Leonards, ed., 1962, McGraw-Hill Book Company. Used with permission of McGraw-Hill Book Company.)

Figure 4-3. Flow and head for partially penetrating line slot; single-line source; artesian, gravity, and combined flows.

FULLY PENETRATING SLOT

THE FLOW TO A FULLY PENETRATING SLOT FROM TWO LINE SOURCES, BpTH OF INFINITE LENGTH (AND PARALLEL), IS THE SUM OF THE FLOW FROM EACH SOURCE, WITH REGARD TO THE APPROPRIATE FLOW BOUNDARY CONDITIONS, AS DETERMINED FROM THE FLOW EQUATIONS IN FIG. 4-1. LIKEWISE, THE DRAWDOWN FROM EACH SOURCE CAN BE COMPUTED FROM THE DRAWDOWN EQUATIONS IN FIG. 4-1 AS IF ONLY ONE SOURCE EXISTED.

PARTIALLY PENETRATING SLOT

ARTESIAN FLOW

NOTE: WIDTH OF SLOT, b, ASSUMEO – 0.

t WITHIN THIS DISTANCE (1.3D) THE

PIEZOMETRIC SURFACE IS NONLINEAR DUE TO CONVERGING FLOW.

(o)

FLOW

t ORAWOOWN WHENyC 1.30 CAN BE ESTIMATEO BYDRAWING A FREEHAND CURVE FROM h TANGENTTOTHE SLOPE OF THE LINEARPARTATySl,3D. *

G R Avityflow

FLOW

APPROXIMATELY, BUT SOMEWHAT LESS THAN, TWICE THAT COMPUTED FROM A SINGLE SOURCE, EQ3, FIG. 4-3,

DRAWDOWN

APPROXIMATELY THAT COMPUTED FROM A SINGLE SOURCE, EQ 4, FIG. 4-1.

(Modified from “Foundation Engineering, " G. A, Leonards, ed., 1962, McGraw-Hill

Book Company. Used with permission of McGraw-Hill Book Company.)

 A FREQUENTLY ENCOUNTERED DEWATER ING SY ST E M IS ONE WITH TWO LINES OF PARTIAI I Y PENETRATING WELL POINTS ALONG EACH SIDE OF A LONG EXCAVATION, WHERE THE FLOW CAN BE ASSUMED TO ORIGINATE FROM TWO EQUIDISTANT LINE SOURCES.

 ct 0.6

 4 6 ^/h0 (c)

 0.05 0.10 0.15 Ь/н

 0 2

 8 10

 0

 (b)

 M)

FLOW

FLOW TO EACH SLOT APPROXIMATELY THAT ONE SLOT WITH ONE LINE SOURCE, EQ 3, FIG. 4-3.

WHERE C, AND Cj ARE OBTAINED FROM FIG.(C) AND (d) ABOVE

GRAVITY FLOW

t MAXIMUM RESIDUAL HEAO MIDWAY BETWEEN THE TWO SLOTS

(Modified from “Foundation Engineering," G. A. Leonards, ed., 1962, McGraw-Hill Book Company. Used with permission of McGraw-Hill Book Company.)

 FLOW Q w OR DRAWDOWN (h – K#) can be ESTIMATED FROM

 Qw • (h * he) kD £

 WHERE ^ = SHAPE FACTOR FOR ARTESIAN F L OW OBTAINED FROM (c)

 W/D, PERCENT t IF R IS OBTAINED FROM FIG. 4-23, S U В S T I T U T E h FOR h

 Kc CANBE OBTAINED FROM PLOTS IN FIG. 4-7

(b)

U. S. Army Corps of Engineers

Figure 4-6. Flow and head for fully andpartiullypenetratingcircular slots; circular source ; artesian flow

from the equation and plots in figure 4-23. Where there is little or no recharge to an aquifer, the radius of influence will become greater with pumping time and with increased drawdown in the area being clewaterecl. Generally, R is greater for coarse, very pervious sands than for finer soils. If the value of R is large relative to the size of the excavation, a reasonably good approxi­mation of R will serve adequately for design because flow and drawdown for such a condition are not espe­cially sensitive to the actual value of R. As it is usually impossible to determine R accurately, the value should
be select-d conservatively from pumping test data or, if necessary, from figure 4-23.

(4) Wetted screen. There should always be suffi­cient well and screen length below the required draw – dom in a well in the formation being dewatered so that the design or required pumping rate does not produce a gradient at the interface of the formation and the well filter (or screen) or at the screen and filter that starts to cause the flow to become turbulent. There­fore, the design of a clewatering system should always be checked to see that the well or wellpoints have ade-

 -U ■ oo

 c CO

 4! О о ч тз и

 о И) и 3 ста & Ф Ф 3

 NOTE: SEE FIG. 4-6 FOR EXPLANATION OF TERMS. (H – he) AND h0 ARE COMPUTED FROM EQ 1 (FIG. 4-6). HEAD AT CENTER (h£) = hg + (hc – h )

FLOW. QT, OR DRAWDOWN, H – he. CAN BE E5TI MATED FROM

QT = (H – he)kD \$ (1)

WHEREJ)I5 OBTAINED FROM PLOTS SHOWN BELOW AND PERCENT PENETRATION =WDx100

NOTE HEAD ALONG LINE A-A WITHIN THE ARRAY, h, ISOBTAINED FROM FIG 4-9

 b’RT

 b/R + (d)

 R iS OBTAINED FROM FIG. 4-23.

U. S. Army Corps of Engineers

Figure 4-8. Flow and drawdown at slotfor fully and partially penetrating rectangular slots; circular source; artesian flow.

quate “wetted screen length hws or submergence to pass the maximum computed flow. The limiting flow qc into a filter or well screen is approximately equal to

2тіг*/Е 7.48 gallons per minute clc = і Qy x per foot of filter or screen

where

rw = radius of filter or screen к = coefficient of permeability of filter or aquifer sand, feet per minute

(5) Hydraulic head loss Hw. The equations in fig­
ures 4-1 through 4-22 do not consider hydraulic head losses that occur in the filter, screen, collector pipes, etc. These losses cannot be neglected, however, and must be accounted for separately. The hydraulic head loss through a filter and screen will depend upon the diameter of the screen, slot width, and opening per foot of screen, permeability and thickness of the filter; any clogging of the filter or screen by incrustation, drilling fluid, or bacteria; migration of soil or sand par­ticles into the filter; and rate of flow per foot of screen. Graphs for estimating hydraulic head losses in pipes, wells, and screens are shown in figures 4-24 and 4-25.

 NOTE: HEAD, h. ALONG LINE A-A IN FIG. 4-80 CAN BE OBTAINED FROM CURVES ABOVE. P

h= h + (h — h )

p e p e

U. S. Army Corps of Engineers

HYDRAULIC HEAD LOSS, IS OBTAINED FROM FIG. 4-2 4

RADIUS о f INFLUENCE, R, IS OBTAINED FROM FIG. 4-23

(a)

2ffkD (H – h)

DRAWDOWN, H ■

WHERE G IS EQUAL TO THE RATIOOF FLOW FRO,., A PARTIALLY PENETRATING WELL, Qwp, TO THAT FOR A FULLY P enetrAting WELL fO r THE SAME DRAWDOWN, H-hw, AT THEPERIPHERY OF THEWELLS.

DRAWDOWN

2. COMPUTE l-l-h w FROM EQ 2 FOR A FULLY PENE­TRATING. w E L L FORA DISCHARGE OF Qwp (20N (O).

3. PLOT DRAWDOWN FOR FULLY PENETRATING WELL VS

-POINT В IN I L L U S T R A T I О N – FOR THEPARTIALLY PENE-

THEDRAWDOWNCURVE FORA PARTIALLY

(Modified from “Foundation EngineeringG. A. Leonards, ed., 1962, McGraw,-Hill Book Company.

Used with permission of McGraw-Hill Book Company.)

 FLOW. Q OR DRAWDOWN. H2> h2 neglectInG height of F R Ее discharge, h’ (condition (0))< w ‘ *

 7гк(н2- h2) °w In (R/r)

 (l)

 FLOW, n… TAKING h’ INTO ACCOUNT (b) CAN BE ESTIMATED ACCURATELY FROM EO 2 USING W ’ HEIGHT OF WATER, t t S (S =0 FOR FULLY PENETRATING WELL), FOR THE TERM hw.

 FULLY OR PARTIALLY PENETRATING WELL

 із)

 (4)

 (6 (7) (7) (9)

 FOR 0.3

 p = 0.13 In R/r p = c + Лс

 Ac of-

(Modified from “Foundation Engineering, " C. A. Leonards, ed., 1962, McGraw-FIill Book Company. Used with permission of McGraw-Hill Book Company.)

FLOW, Q CAN BE COMPUTED FROM

ffk (2DH – d2 –

w" In (R/r )

4 w’

DRAWDOWN, H – h, CAN BE COMPUTED AT ANY DISTANCE FROM

R, DISTANCE FROM WELL AT WHICH FLOW CHANGES FROM GRAVITY TO ARTESIAN CAN BE COMPUTED FROM

R IS DETERMINED CROMFIG. 4-23.

EQUATIONS 1 AND 2 ARE BASED ON THE ASSUMPTION THAT THE HEAD h AT THE WELL IS AT

w

THE SAME ELEVATION AS THE WATER SURFACE IN THE WELL. THIS WILL NOT BE TRUE WHERE THE DRAWDOWN IS RELATIVELY LARGE. IN THE LATTER CASE, THE HEAD AT AND IN THE CLOSE VICINITY OF THE WELL CAN BE COMPUTED FROM EQ 4 THROUGH 9 (FIG. 4-111. IN THESE EQUATIONS THE VALUE OF Q

USED IS THAT COMPUTED FROM EQ 1, ASSUMING hw EQUAL TO THE HEIGHT OF WATER IN THE WELL,

AND THE VALUE OF R COMPUTED FROM EQ 3 IS USED IN LIEU OF R.

(Modified from “Foundation Engineering,” G. A. Leonards, ed., 1962, McGraw-Hill Book Company. Used with permission of McGraw-Hill Book Company.)

Hw IS OBTAINED FROM FIG. 4-24

(Ы ARTESIAN FLOW ( c) GRAVITY FLOW

ARTESIAN FLOW

DRAWDOWN (H – hp) AT ANY POINT P

t DRAWDOWN FACTORS, F, FOR SEVERAL COMMON WELL ARRAYS ARE GIVEN IN FIG. 4-14 t FOR RELATIVELY SMALL DEWATERING SYSTEMS AND WHERE NO UNUSUAL BOUNDARY C ONDITIONS EXIST, THE RADIUS OF INFLUENCE FOR ALL WELLS CAN BE ASSUMED CONSTANT AS IN (a) ABOVE. SEE FIG. 4-23 FOR DETERMINING THE VALUE OF R.

(Modified from “Foundation Engineering," G. A. Leonards, ed., 1962, McGraw-Hill Book Company. Used with permission of McGraw-Hill Book Company.)

ALL WELLS ARE FULLY PENETRATING WITH A CIRCULAR SOURCE. THE FLOW, Ow, FROM ALL WELLS IS EQUAL. F = DRAWDOWN FACTOR FORANYWELL IN THE ARRAY. F = DRAWDOWN FACTOR FOR CENTER OF THEARRAY.

ARRAY 1. CIRCULAR ARRAY OF EQUALLY SPACED WELLS

DRAWDOWNAT POINTS p And c FOR ARTESIAN FLOW CANBE COMPUTED FRO M

WHERE |= WELL NUMBER AS SHOWN IN THE ARRAY ABOVE.

NOTE THAT THE LOCATION OF M IS MIDWAY BETWEEN THE LINES OF WELLS AND CENTERED BETWEEN

THE END TWO WELLS OF THE LINE. THIS POINT CORRESPONDS TO THE LOCATION OF THE MINIMUM DRAW­DOWN WITHIN THE ARRAY.

VALUES DETERMINED FOR F, F.AND F ARE SUBSTITUTED FOR F IN EQ 1 AN D 3 (FIG.4-13)To COMPUTE

w c m

DRAWDOWNAT THE RESPECTIVE POINTS.

(Modified from “Foundation Engineering, ” G. A. Leonards, ed., 1962, McGraw-Hill Book Company. Used with permission of McGrow – Hill Book Company.)

FULLY PENETRATING WELL

 CIRCULAR ARRAY OF П NUMBER OF EQUALLY SPACED WELLS

(H-h ) \$

e y

2/ГГ

 DRAWDOWN MIDWAY BETWEEN WELLS HEAD INCREASE IN CENTER OF A RING OF WELLS, Ah IS EQUAL D TO Ah and can be computed FROM EQ 1.drawdown at the w CENTER OF THE RING OF WELLS, H — h D, IS EQUAL TO H – h -^A h ^

OR H-h AND, CONSEQUENTLY, CAN BE COMPUTED FROM EQ 1 ( FIG. 4-6). e

FOR EQ 1 THROUGH 4:

FLOWS FROM ALL WELLS ARE EQUAL

SHAPE FACTOR § IS OBTAINED FROM FIG 4-6c.

COEFFICIENT OF PERMEABILITY

all Other terms are e XP LaIned In a, b, and c

U. S. Army Corps of Engineers

Figure 4-15. Flow and drawdown for fully penetrating circular well arrays; circular source; artesian flow