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Theories of seepage into auger holes in homogeneous anisotropic soil
Journal of ELSEVIER
Journal of Hydrology 167 (1995) 1-22
Hydrology
[3]
Theories of seepage into auger holes in homogeneous anisotropic soil G a u t a m B a r u a * , K . N . Tiwari Department of Agricultural and Food Engineering, LL T. Kharagpur, West Bengal, India
Received 6 December 1993;revision accepted 26 September 1994
Abstract Steady-state theories have been developed for flow into an auger hole in a homogeneous and anisotropic soil medium when (i) there is an impermeable layer at the bottom of the auger hole and (ii) there is an impermeable barrier at a finite distance below the bottom of the auger hole. The proposed theories can be modified easily for an isotropic soil condition and the partially penetrating auger hole theory presented here is simpler than the existing theories (Kirkham, 1958; Boast and Kirkham, 1971) for the same soil condition. Furthermore, the theories can be applied directly, i.e. without transformation of coordinate axes, to determine the directional conductivities of an anisotropic medium (and hence also the hydraulic conductivity of an isotropic medium since an anisotropic medium can be transformed into an isotropic one simply by treating the horizontal hydraulic conductivity of the medium as equal to its vertical hydraulic conductivity) from auger hole experimental data below a water table.
1. Introduction The auger hole method for determining the saturated hydraulic conductivity of soil below a water table has been described by Van Bavel and Kirkham (1948), Luthin (1957), Bouwer and Jackson (1974) and Kessler and Oosterbaan (1974). A steadystate theory of flow into an auger hole fully penetrating a homogeneous and isotropic porous medium of infinite horizontal extent was given by Kirkham and Van Bavel (1948). Kirkham (1958) gave an exact theory of seepage into an auger hole partly penetrating a porous medium bounded above by a water table and below by an impermeable layer. Boast and Kirkham (1971) developed exact theories of flow into a partially penetrating auger hole both for situations when the bottom boundary
* Corresponding author. 0022-1694/95/$09.50 © 1995 - Elsevier Science B.V. All rights reserved SSDI 0022-1694(94)02629-7
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G. Barua, K.N. Tiwari / Journal of Hydrology 167 (1995) 1-22
Notation
A~,,Bm,B, h HI tt3
It(.) l~(.) K
Ko(.) K~(.) K, K~ (K~)2 rn~n
M N
Qb Qf Qp r z
constants with m = 1,2, 3,... to and n = 1,2, 3,... ec radius of auger hole (L) depth to the impermeable layer from the water table (L) water level depth in auger hole (L) depth of penetration of the partially penetrating auger hole (L) zero order modified Bessel function of first kind first order modified Bessel function of first kind hydraulic conductivity of isotropic soil (LT -l) zero order modified Bessel function of second kind first order modified Bessel function of second kind horizontal hydraulic conductivity (LT -1) vertical hydraulic conductivity (LT -1) (Kr/Kz) anisotropy ratio (dimensionless) summation indices 1,2,3,...oo 1,2,3,...c~ flow taking place through the bottom of the partially penetrating auger hole (Fig. 2) (L3T -1) total flow into the fully penetrating auger hole (Fig. 1) (L3T -l) total flow into the partially penetrating auger hole (Fig. 2) (L3T -l) radial coordinate (L) vertical coordinate (L)
Greek letters
¢ ¢1 ¢2 ¢1 ¢2
hydraulic head distribution for the fully penetrating auger hole problem (Fig. 1) (L) hydraulic head distribution for region 1 (Fig. 2) for the partially penetrating auger hole problem (L) hydraulic head distribution for region 2 (Fig. 2) for the partially penetrating auger hole problem (L) Stoke's stream function for the fully penetrating auger hole problem (Fig. 1) (L2T -1) Stoke's stream function for region 1 (Fig. 2) for the partially penetrating auger hole problem (LET-1) Stoke's stream function for region 2 (Fig. 2) for the partially penetrating auger hole problem (L2T -L)
of the flow medium is impervious and when the bottom boundary is a layer of infinite permeability. Healy and Laak (1973), as reported by Bouwer and Rice (1983), used broad and shallow auger holes (pits) for in-situ determination of the saturated hydraulic conductivity in stony and gravely soils. They called their technique a 'pitbailing' method and used conventional well-theory (Thiem's equation) to interpret their results. Bouwer and Rice (1983) argued that the equation used by Healy and Laak (1973) for calculating the saturated hydraulic conductivity of soil from experimental data on pits cannot be applied to pits that penetrate only partially to an impermeable layer. They extrapolated the piezometer theory of Youngs (1968) to obtain values of the shape factor (a shape factor is a factor which expresses the relationship between the observed rate of rise of water in a pumped auger hole and the hydraulic conductivity of the surrounding soil medium) to cases where the width to depth ratio is large. Boast and Langebartel (1984) extended the earlier work of
G. Barua, K.N. Tiwari / Journal of Hydrology 167 (1995) 1-22
Boast and Kirkham (1971) to find values of the shape factor for broad and shallow pits. Lomen, Warrick and Zhang (1987) provided a steady-state theory of flow into axially symmetric auger holes and pits of arbitrary geometries. However, none of the theories of seepage into auger holes (fully or partially penetrating) take into account, explicitly, the anisotropy of a soil medium. The usual procedure for solving a problem in an anisotropic flow medium when the solution in an isotropic medium is available, is, firstly, to convert the anisotropic medium into a fictitious isotropic medium by shrinking the dimensions in the direction of greatest permeability or by stretching the section in the direction of lesser permeability (Muskat, 1946; Childs, 1952; Maasland and Kirkham, 1955; Maasland, 1957); then the flow problem in this fictitious isotropic medium is solved using the isotropic solution of the problem. Finally, the results are transformed into anisotropic prototype space. Using the results of the theorems stated by Maasland and Kirkham (1955), Maasland (1956) (see also Maasland, 1957) developed formulae and graphs to calculate the horizontal and the vertical hydraulic conductivities of an anisotropic soil medium from the observed rate of rise of water in a pumped auger hole below a water table when the anisotropy quotient, i.e. the ratio of horizontal to vertical hydraulic conductivity of the soil under consideration, is known. There is, however, always an advantage in obtaining a solution which can address the problem of anisotropy explicitly since it can be applied directly, i.e. without the necessity of bringing about a transformation of coordinate axes, to estimate the directional conductivities of an anisotropic flow medium from auger hole experimental data below a water table. Reeve and Kirkham (1951) reported that long and thin auger holes tend to measure essentially the horizontal conductivity of the soil whereas broad piezometers of zero cavity length tend to measure essentially the vertical conductivity of the soil. Actually, however, both the auger hole and the piezometer measure the apparent hydraulic conductivity of the flow medium rather than its horizontal or vertical hydraulic conductivity. Childs et al. (1957) combined the two-well method of Childs (1952) and the tube method of Kirkham (1945) to calculate the directional conductivities of an anisotropic medium below a water table. Since this method depends on the tube method for sensing the apparent hydraulic conductivity of the soil, the conductivity values determined by this method may not be totally representative of the soil under consideration because the tube method yields an apparent hydraulic conductivity for only a small volume of the soil (Talsma, 1960). One way of avoiding this difficulty is by using a very large sampler but that would make the method cumbersome; also, in this way, a representative value for the apparent conductivity would still be obtained for only a small depth interval. Values of the horizontal and vertical hydraulic conductivities of an anisotropic medium can also be obtained from the results of measurements on two piezometer installations, one with a cavity of small height and the other with a cavity of large height. The theory for this procedure was given by Maasland (1957). Utilising this theory, Talsma (1960) calculated the directional conductivities of anisotropic media of infinite depth by installing piezometers of zero and known cavity lengths below the water table. This method also suffers from the same kinds of drawbacks as the Childs' (1957) method; like the Childs' method, a tube is used to determine the apparent hydraulic conductivity of the
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G. Barua, K.N. Tiwari / Journal of Hydrology 167 (1995) 1-22
soil. Bouwer and Rice (1983) described an iterative procedure to determine the directional conductivities of an anisotropic soil medium below a water table by considering two pits of different geometries. The main advantage of their method is that one need not know beforehand the anisotropy quotient of the soil under consideration to determine the directional conductivities of the soil. Boast and Langebartel (1984), however, reported that the shape factors obtained by Bouwer and Rice (1983) for estimating the saturated hydraulic conductivity of soil by the 'pitbailing' method are 20% greater than the shape factors obtained by Bouwer and Rice (1983) from sand tank model studies on broad and shallow pits. Because of the inability to measure accurately the shape factors for pits, errors may crop up in hydraulic conductivity estimations because an error in shape factor is reflected proportionately in the calculated hydraulic conductivity value. Instead of using Bouwer and Rice's shape factors for pits, one may also utilise the shape factors (greater by a factor of 2) given by Boast and Langebartel (1984) for broad and shallow auger holes, but even then an error in hydraulic conductivity estimation may result. Thus, there seems to be, at present, no suitable theory which can predict accurately, the shape factors for broad and shallow pits. The horizontal and vertical hydraulic conductivities of an anisotropic soil medium below a water table can also be determined in the laboratory with the help of suitable apparatus. The procedure is illustrated in detail by Kessler and Oosterbaan (1980). An obvious drawback of this method is that the samples used for experiments are small and therefore' a large number of samples need to be taken and experimented upon before any meaningful conclusions can be drawn from the experimental results. It is clear that there exists a need for suitable theories of flow into auger holes which take into account, explicitly, the anisotropy of the flow medium, and which should be easy to apply for calculating the directional conductivities of an anisotropic soil medium from auger hole experimental data below a water table. This paper addresses this need.
2. Mathematical formulation and solution 2.1. Case 1: fully penetrating auger hole problem
Fig. 1 shows an auger hole of radius a resting on an impervious base of a homogeneous and anisotropic aquifer of infinite radial extent. The depth from the water table to the impervious layer is h and the depth of water in the auger hole, as measured from the origin O, is HI. The saturated hydraulic conductivities of the aquifer in the horizontal and the vertical directions are Kr and Kz, respectively. Because of axial symmetry, the flow region to the right of the vertical axis only is considered. A righthand coordinate system is established with the origin as shown in Fig. 1. For convenience, the z axis is taken as positive vertically downward and the r axis is taken as positive towards the right. In the following analysis it is assumed that the flow is steady, the drawdown of the water table around the auger hole because of seepage into the hole is negligible,
G. Barua, K.N. Tiwari / Journal of Hydrology 167 (1995) 1-22
SOIL
5
SUR FACF..
,,/,6¢~
///~:'4e',i,P,~"
r
WATER
TABLE
Or0,0 ) Kr ~
HI
I4
-
~ Kr
Kt
"111'
q
IV ;'-; / - -
////
/ / ,
/= 212 2 ; - / - ' ~ " . " "I(IMPE~VIOU$/ /" / / / / LAYER
///
Fig. 1. Geometry of the flow system of a fully penetrating auger hole. the capillary affects above the water table are non-existent and the principal directions of anisotropy of the medium coincide with the horizontal and vertical directions, respectively. The following form of Fourier sine series (Barua, 1991) will be used to obtain solutions to both the problems considered in this paper. In theorem form, the series can be stated as: If f (x) is piecewise continuous in an interval (O, L) and has right- and left-hand derivatives at all points of that interval, then the following infinite trigonometric series E m=l
Am sin (1 - 2m) 2
converges to (a) f ( x ) ,
(b)
if x is a point o f continuity
f(x + O) +f(x 2
- O) if x is a point of discontinuity
6
G. Barua, K.N. Tiwari / Journal of Hydrology 167 (1995) 1-22
where
jf(x)sin L
Am= 2
[(1 2 2 m ) L - - ] dx
0
and f ( x + 0) and f ( x - 0) are right- and left-hand limits o f f ( x ) at x. The steady-state continuity equation to be solved in cylindrical coordinate system is
Kr O2(b
K~ Oq~ 020 _ + 7 + Kzb- z2 - 0
(11
subject to the following boundary conditions (BCs): = -z;
r = a;
0 < z < H1
(BCIa)
~b = -H1;
r = a;
Hi < z < h
(BCIb)
~b = 0;
z = 0;
a < r < oo
(BCII)
Or
O;
r
oo;
0< z< h
(BCIII)
0q~ Oz
0;
z
h;
a
(BCIV)
where ~b is the hydraulic head measured with respect to the reference level passing through the origin O and Kr, Kz (r and z have already been defined). Using the separation of variable technique (see Kirkham and Powers, 1972) a solution to Eq. (1) satisfying boundary conditions II, III and IV as mentioned above, for any values of the constants Bin, can be expressed as Oc
_Nmr
m=l
where
K~ = (Kr/Kz) U2
(4)
K0(.) = zero order modified Bessel function of section kind and m = 1,2, 3 , . . . To make Eq. (2) reducible to the form of Fourier sine series developed by Barua (1991) along r = a, take sm =
Am
(s)
G. Barua,K.N. Tiwari/ Journalof Hydrology 167 (1995) 1-22 Substituting the right-hand side (r.h.s.) of Eq. (5) for
= ~ Am
=,
Bm into
Eq. (2),
K°\ K~ )sin(Nmz) {-Nma\ :
(6)
where Am are arbitrary constants. Let
k [-Nmr\ --
/'-Nmr\ ) (7a)
and
/'-Nmr\ Kl~--~o ) k l ~ K~ / = K° (__Nmcx k K~ ~ ) (-Nmr ~
(7b)
From Eqs. (6) and (7a),
Am
-Nmr
sin (Nmz)
(8)
m=l
Now, finally, applying boundary conditions Ia and Ib to Eq. (8), at r = a Am sin (Nmz)
-z = ~
for
0 < z < Hl
for
Hi < z < h
m=l
oo
-H1 = Z Am sin (Nmz) m=l
Hence, the constants
Am =
~
Am are
given by
( - z ) sin (Nmz) dz +
( - H , ) sin (Nmz) d~
(9)
Hi
Eq. (9), upon simplification, gives
Am =
-2 - - - - - 2 sin (NmHl) tl(Atm)
(lo)
8
G. Barua,K.N. Tiwari/ Journalof Hydrology 167 (1995) 1-22
Now, from Eqs. (3) and (10), Am
_
-8h~ I(1 - 2m)TrH1] (1 - zm)ZTr2sin 2h J
(11)
Substituting the values of A m and N m into Eq. (6), the expression for total hydraulic head, ~b, finally becomes
- -= l m
( - ( 1 - 2__m)Trr.'~ -8h sin [ ( 1 - 2re)Trill. ] 2 h Ko\ 2K6h ] sin(N,nz) ( 1 - 2m)eTr2 J K0 ( . - ( 1 - 2m)Tro{'~
\
2K h
(12)
J
Eq. (12) gives the distribution of hydraulic head for all points throughout the porous medium of Fig. 1. Equations connecting the hydraulic head function, ~b, with the Stoke's stream function, ~b, in axis-symmetric flow for the anisotropic case are as given below (Bear, 1972): 0~b
1 0~b
(13a)
Kr T r - r T z
and
O~ Kz-'~z--
(13b)
1 0¢ r Or
where ~b is the Stoke's stream function. Application of equations (13a) and (13b) to Eq. (8), after simplification, gives ~b(r,z) = - ZAmr(KrKz)½kl
cos (Nmz) + C
(14)
m=l
where C is a constant to be determined and K 1(.) is first order modified Bessel function of second kind. As the quantity of interest is the flow between two stream lines, the constant C can be chosen arbitrarily; for the sake of simplicity, consider the stream line given by the following expression as the zero stream line, i.e. ¢(r--~ oc, z) = 0
(15)
Now, application of Eq. (15) to Eq. (14) gives (after using the L' Hospital's theorem and Dwight's (1961) Formula No. 816.2 for n = 1) C = 0. Therefore, ~b takes the form OO
~b(r,z) = - Z
m=l
Amr(KrK~)½kl ( ~
\ K6,/
cos
(Nmz)
(16)
The quantity of water, Qf, entering the auger hole can be calculated by either of the
G. Barua, K.N. Tiwari / Journal of Hydrology 167 (1995) 1-22
9
equations given below: h
(17)
Qf = Kr 0
Qf = -Kz
(o0)2=027rrdr
(18)
Ozz
Eq. (17), after simplification~ reduces to
(-Umr
Qf=27rc~--~I Z Amkl • K[) /I
(19)
m=l
Similarly, Eq. (18), after simplification yields (2~3
Of = 27reeKzKo Z Amkl (-Nmr'~ m=l
(20)
~ K; /
Substituting the r.h.s, of Eq. (4) for K~ into Eqs. (19) and (20), respectively, (3O
Qf = 27r°~(KrKz)½ Z Amkl (-Nmr~ m=l
(21)
~ K~ )
For an isotropic soil medium, Kr = Kz = K (say) and K~ = (KJK2) ½= 1; thus, ¢, for the isotropic case reduces to (X3
0 = Z Am ko(-N~r) sin (Nmz)
(22)
m=l
and the stream function, z), for the isotropic condition can be expressed as (X?
¢ = - K ~ rAmkl (-Nmr) cos (Nmz)
(23)
m=l
Also, the quantity of water seeping into the auger hole for the isotropic soil condition is given by
Qf = 27reeKy ~ Amk1(-Nmo0
(24)
m=l
2.2. Case 2."partially penetrating auger hole problem Fig. 2 illustrates the problem under consideration. An auger hole of radius penetrates an aquifer of infinite horizontal extent to a depth of H 3 below the water table. The symbols h, H 1, Kr and Kz carry the same meaning as in the fully penetrating auger hole problem. A coordinate system similar to that shown in Fig. 1 is also
10
G. Barua, K.N. Tiwari / Journal of Hydrology 167 (1995) 1-22
considered here and, as before, because of axial symmetry, only the flow region to the right of the vertical axis passing through the origin O is considered. Following K_irkham's (1958) mode of analysis, the flow domain is divided into two regions: (1) the region between the bottom of the hole and the impermeable layer and (2) the remaining region. The hydraulic head in region (1) is designated by (91 and in region (2) by (92. The boundary conditions for the problem, as illustrated in Fig. 2, can be expressed as
0(gt =
r = 0;
H3 < z < h
(BC I)
z=Hs; r = a;
0
(BCII) (BC Ilia)
r=a;
H3 < z < h
(BCIIIb)
o;
z=h;
0
(BC IV)
-H,;
r=a; r=a;
Or
0;
(9,=-H1; ¢1 = (92;
K r0(9, ~=K,
0(9, =
Oz (92 --Z;
~rz ;
(92 = o;
z=0;
O
0(92 Or =
0;
r=~;
0
(BE VII)
0(92 Oz =
0;
z = h;
a < r < cx~
(BC VIII)
=
(92 =
(BC Va) 3 (BC Vb)
(BC Vl)
The function, ¢, must be chosen such that
/~ 02(9, K~ 0(91 02(9, --ff~-- + --;- - ~ - + Xz --ff~-z2= 0
(25)
in region (1) and K 02(92 Kr 0(92 02(92 r --ff~--+ -~ --~r + Kz ---~Tz2= 0
(26)
in region (2) must be satisfied together with the boundary conditions I to VIII as listed above. Using the separation of variable technique (see Kirkham and Powers, 1972) solutions to Eqs. (25) and (26) can be expressed in the following forms: [-(1 - 2m)Trrq (9, =
sin [.(l - 2m)Tr(z- H3)]
G. Barua, K.N. Tiwari / Journal of Hydrology 167 (1995) 1-22
11
SOIL SURFACE
1
WATER
t
TABLE
VJ
I
II
Kp@ Kp
H
Kz
I 1
(~
Vb
, _ _
Vll "Z_-Z-
2 2"- Z - - - - _ - _ "
I
REGION
ma
®
llllb
I
,
"///////2///"/"/"
7//////
REGION
IV
Vlll IM PERVIOUS
/////////2
LAYER
Fig. 2. Geometry of the flow system of a partially penetrating auger hole.
and
u ~ [-(1 - 2n)~-,l - ~ ~b2 = ~ B, ~ -L - 2K~h J sin [ ( 1 - 2n)Trz] =
-o[
-j
2h
J
(28)
where M and N are any integers, i.e. M = 1,2,3,...; N = 1,2,3,...;
(29)
m and n are summation indices; I0(.)= zero order modified Bessel function of first kind and K~, h, H] , 1-13, a, K o (.), r and s have already been defined. Let , [(1 - 2m)Tr] N~n = [2(h - H3)J
(30)
iV. = [(1 - 2 n ) r ] 2h J
(31)
G. Barua, K.N. Tiwari / Journal of Hydrology 167 (1995) 1-22
12
{-N~r)
;or
j
I°(---~o ) -
)
_NSd~
(32a)
-N'r'~
il\ K; J=
(32b)
Io(- o ) - N'~'~
where I1(.) is the first order modified Bessel function of first kind and the other symbols carry the same meaning as defined before. Substitution of Eqs. (7a), (32a), (30) and (31) into Eqs. (27) and (28), respectively, gives
051= Z Ami°
sin[Nm(z - / / 3 ) ] - HI
(33)
m=l
and 052 = ~ ~ B , k0 ~(-N.r) - - ~ ,] sin(N,z)
(34)
m=l
In Eq. (33), 051 satisfies boundary conditions I, II, and IV for any values of the constants A,,. Similarl3,, 052 in Eq; (34), satisfies boundary conditions VI, VII and VIII for any values of the constants/in. Boundary conditions IIIa, IIIb, Va and Vb remain to be satisfied by proper choice of the constants Am and B,. To satisfy Va, Vb and IIIa, at r = N
Z / i n sin(N,z) = -z;
0 < z < HI
(35)
n=l N
Z
B, sin(Nnz) = - H I ;
H1 < z < / / 3
(36)
n=l N
M
Z B, sin(Nnz) = E n=l
Am sin[N~m(Z - H3)] - HI ;
H 3 <
z< h
(37)
m=l
and to satisfy boundary condition IIIb, after applying boundary condition IIIb to Eqs. (33) and (34) M
N
Amsin[Nm(z -
Y~
m=l
H3)] = ~ n=l
Bn
kl (~Ko~) ] sin(Nnz);H3 < •
- N
I c~
-
z < h -
(38)
If N ~ ~ (in view of Eq. (29)), then the functions as shown on the r.h.s, of
G. Barua, K.N. Tiwari / Journal of Hydrology 167 (1995) 1-22
13
Eqs. (35), (36) and (37), respectively, become suitable for expansion by the form of Fourier sine series developed by Barua (1991) along the boundary 0 < z < h. The constants Bn, can then be evaluated as l
Bn=-~
(-z)sin(N~z)dz + (-H1)sin(N~z)dz H~ h
h M
_]
-IHlsin(N,z)dz + I ~-~Amsin[N~(z-H3)]sin(Nnz)dzj //3
(39)
H3 m=l
After carrying out the above integrations using Dwight's (1961) Formulae Nos. 435 and 465, for Nn ~ N~,
Bn -.tt~n),~--~22.sin(N, Hl) + ~./-~--h--~-~Am l I (N, -1 Arm)sin((N~'H3) + (N, - N~)h) 1
- (Nn - N~m)sin((N~mH3) + (Nn 1
Nt + (Nn + N~) sin((mH3) •
-
-
N~n)H3)
(N. + N~m)h)
-(NnlN~)sin((N~H3)-(Nn+Ntm)H3)] (40) where n = 1 , 2 , 3 , . . . N - - . oc; m--- 1,2,3,...M. For N n = N~, the integrals on the r.h.s, of Eq. (39), after simplification, reduce to B , - -2sin(N, Hl)
h(Nn)z
+~
1
~-~Am [ + ~ h (h - H3) cos(N~H3)
/
sin(NmH3)
, cos(ZNrmh) ] + 2 -1~ sin(N~,H3) l
(41)
Although Eqs. (40) and (41) relate Bn and A m to two different situations, viz. Nn ¢ N~m and Nn = N~,, in actual practice the occurrence of the situation Nn = N" can easily be avoided by a judicious selection of the parameters//3 and h. Eq. (40) yields a set of equations giving values of B1, B2, B3,... Bn in terms of A1, A2,... Am. To have the same number of Am as there are B,, take M = N
(42)
G. Barua, K.N. Tiwari / Journalof Hydrology 167 (1995) 1-22
14
There remains Eq. (38) from which to find another set of relations between Am and B, and hence their individual values. Let - N o~
Substitution of Eq. (43) into Eq. (38) gives M
Z
N
Am sin[NIm(z - n 3 ) ] = Z
m=l
nnPnm sin(NnZ)
(44)
n=l
Now, if M ~ cx~ (in view of Eq. (29)), then A m of Eq. (44) can be evaluated as h
2
Am -
N
J Z. = , BnPnm sin(Nnz) sin [N~(z - H3)] dz H3
(45)
By carrying out the above integration and simplifying, for Nn ¢ N~
N P~mBn [ 1 sin((N~H3) + (N~ - N~)h) Am = Zn=l ( h - H3) (Nn - N~m) 1
(N n _ Nim) sin((N~H3) + (Am - N~m)H3) 1
-~ (Nn + )Vim)sin((N~mH3) - (Nn + N~m)h) 1 sin((N~nH3)- (Nn + N~m)H3)] (N, + N~)
(46)
where m = 1 , 2 , 3 , . . . M ~ oe;n = 1 , 2 , 3 , . . . N ~ ec. For Nn = N~, Eq. (45) yields, after simplification, N
Am = Z
PnmBn [ 1 . (-h--H33) (h - H3) cos(N.H3) +~N~nS,n(N.H3)
n=l
+~
1
sin(N~H3) cos(2Nnh)
(47)
Eqs. (46) (for Nn ¢ N~) and (47) (for Nn = N~m) give a second set of relations between A m and B,. As before, the situation Nn = N~ is avoided by a proper selection of the parameters h and H3. It should be noted that for the hydraulic head functions, ¢1 and ¢2, developed for regions (1) and (2), respectively, to be exact, both N and M should tend to infinity; however, for most situations, expansion of five to six terms gives a reasonably accurate value of the hydraulic heads around a
G. Barua, K.N. Tiwari / Journal of Hydrology 167 (1995) 1-22
15
partially penetrating auger hole. Linear equations stemming from Eqs. (40) and (46) (for Nn # N~m)can be solved easily by following a Gauss elimination procedure or by any other suitable method (see Scarborough, 1966). To get the stream function, ~bl, for region (1), Eqs. (13a) and (13b) are applied to Eq. (33); the resultant expression is
M , (_Ntmr) , ~1 = ZAmr(KrKz) ~i] cos[Nm(z -//3)] + CI ,,,=,
\K6)
(48)
where ~31 is the stream function for region (1) and Ci is a constant to be determined. For simplicity, ~1 (0, z) = 0
(49)
is considered as the zero stream line. From Eqs. (48) and (49), CI = 0. Hence, ~bI is given by M
, (_N~mr,~
~21 = Z Amr(KrKz)~il
\ K~,]
m=l
c°s[NIm(Z - H3)]
(50)
Similarly, application of Eqs (13a) and (13b) to Eq. (34) and using the limit O2(r--~ oo, z) = 0 yields N
~b2 = - ~
enr(KrKz)llZkl
n=l
(_Nnr)
\ K~ .,/
cos(Unz)
(51)
The quantity of water, Qp, entering the partially penetrating augerhole is given by co
Qp = _ K z f (O~2~ J
27rrdr
(52)
Eq. (52), upon simplification, yields
Qp = 27ra(KrKz)½ Z B"kl n=l
(53) \
K~ J
The quantity of water, Qb, entering through the bottom of the auger hole is given by
Qb = Kz j \ Oz /z=n327rrdr
(54)
0
From Eqs. (33) and (54), after simplification M
Qb = -2~ra(KrKz) k ~ a,,i~ m--1
(_N~ma~
(55)
kKo)
The solution presented here for the isotropic soil condition is simpler than Kirkham's (1958) isotropic solution to the partially penetrating auger hole
16
G. Barua, K.N. Tiwari / Journal of Hydrology 167 (1995) 1-22
problem. Kirkham had to define an identity (see Kirkham, 1958, Eq. (26)) to obtain a second set of relations between Am and Bn; Eq. (15) of his paper (Kirkham, 1958), which is equivalent to Eq. (38) of this paper, alone was unsuitable for numerical work whereas Eq. (38) of this paper alone is sufficient to give the other set of relations between A m and Bn. Moreover, in Kirkham's (1958) analysis, the occurrence of the situation n = m for the case n / h ~ m / ( h - / - / 3 ) (in the notation of this paper) necessitates the utilisation of a different formula from that of the situation in which n ¢ m (see Eqs. (31a) and (31b), Kirkham, 1958). However, in the theory proposed here, the occurrence of the situation n = m for the case N, ~ N~mdoes not require any new formula from that of the situation in which n ~ m and Eqs. (40) and (46) alone have been found suitable to evaluate the constants A m and B,. In a later paper, Boast and Kirkham (1971) solved this problem (and also the problem arising from a situation when the impervious layer is replaced by an infinitely permeable layer) for the isotropic soil condition by using the normal equations and the Gram-Schmidt orthonormalisation methods. Boast has used the number of multiplications and divisions required by the two methods as a rough estimate of the relative calculation time required and found that, for large values of N, the normal equations method is about twice as fast as the Gram-Schmidt orthonormalisation method. Boast and Kirkham (1971) further reported that calculations using Kirkham's (1958) approach require an amount of time of the same order of magnitude as calculations using the normal equations method. Since the proposed theory has been shown to be simpler than that of Kirkham's (1958) solution and since the time required for calculation using Kirkham's (1958) method and Boast and Kirkham's (1971) normal equations method is of the same magnitude and furthermore since the normal equations method is faster (for large N) than that of the Gram-Schmidt orthonormalisation method, the proposed theory is expected (for large N) to be simpler than that of the normal equations and Gram-Schmidt orthonormalisation methods of Boast and Kirkham (1971). Finally, the theory proposed here for the partially penetrating auger hole problem is of a more general nature than presently available solutions to this problem; the theory presented here takes into account the anisotropy of the soil medium explicitly.
3. Field applications The theories proposed here can be utilised for calculating the saturated hydraulic conductivity of an isotropic medium and the directional conductivities of an anisotropic flow medium from auger hole experimental data below a water table. To determine the saturated hydraulic conductivity of an isotropic medium or the directional conductivities of an anisotropic medium when the ratio of horizontal to vertical hydraulic conductivity of the medium under consideration is known, the proposed theories can be applied directly to the data obtained from an auger hole experiment (Van Bavel and Kirkham, 1948) in the field (see Examples 1 to 4 below). However, if the anisotropy ratio of the medium is not known beforehand, then the directional
G. Barua, K.N. Tiwari / Journal o f Hydrology 167 (1995) 1-22
17
conductivities of the medium may be determined by an iterative procedure using two auger holes of different geometries. This procedure is illustrated in Example 5.
3.1. Example 1."fully penetrating auger hole in isotropic soil Let the data obtained from an auger hole experiment on a fictitious fully penetrating auger hole be as follows: soil isotropic, i.e. K6 = 1, h = 126.5 cm, H1 = 126.5 cm, = 5cm, and rate of rise of water in the auger h o l e = 0.112cms -l, therefore, Qf = 7r(5) 2 0.112 = 8.7964594cm 3 s -1. F r o m Eq. (24), K=
Qf
oo
AmklI-Nm ) m=l
where Nm and Am of Eq. (24) are given by Eqs. (3) and (10), respectively. Substituting the given values of the parameters in the r.h.s, of the above equation, 8.7964594
K=
oo
2 x 7r × 5 ~
A~kl ( - N m × 5)
m=l
where
2m) ] Nm =
x 126.51
and
-2 Am - 126.5(N,,) 2 sin(Nm × 126.5) Simplifying the above expressions, after summing up to only five terms of the infinite series K = 5.17810 × 10-4cms -1 = 0 . 4 5 m d a y -1 Now the hydraulic conductivity of the medium may be estimated by applying Van Bavel and Kirkham's (1948, p. 93, Fig. 4) nomograph to the data to find K = 0 . 4 3 m d a y -1. Alternately, applying the modified Ernst's curve of Van Beers (1958) (see Bouwer and Jackson, 1974, p. 619, fig. 23-5), to the data, the K value is 0 . 4 1 m d a y -1. It is to be noted that both the K values of 0 . 4 3 m d a y -1 and 0.41 m day -l, obtained by applying Van Bavel and Kirkham's nomograph and the modified Ernst's curve o f Van Beers, are quite close to the K value of 0.45 m day -1 obtained by applying Eq. (24) developed here.
3.2. Example 2." partially penetrating auger hole in &otropic soil Let the data obtained from an auger hole experiment on a partially penetrating
G. Barua, K.N. Tiwari / Journal of Hydrology 167 (1995) 1-22
18
auger hole be as follows: soil isotropic, i.e. K~ = 1, h = 200cm, H1 = 75cm; H 3 = 100cm, a = 50cm, and rate of rise of water in the pumped auger hole = 0.112 cm s - 1, therefore, Qp = zr(50) 20.112 = 879.64594 cm s- 1. F r o m Eq. (53), for the isotropic condition K=
N Qp
(56)
2~r&Z B.kl(-N.a) n=l
Substituting the values of the given parameters into Eq. (56), 879.64594
K =
N
2 ×
× 50
B.kI(-N.
× 50)
n=l
where
x 200 J and Nm =
2 x-(260 Z ~ 0 ) J
and Bn of Eq. (56) can be evaluated by solving Eqs. (40) and (46) simultaneously. Summing up to only five terms (M = N = 5) of the series occurring in Eq. (56), i.e. solving only five simultaneous linear equations stemming from Eqs. (40) and (46), K = 1.847 × 10-2cms -1 = 15.95mday -1 Summing up to ten terms (M = N = 10) of the series, K = 1.797 × 10-2cms -1 --- 15.53mday -l Using Boast and Kirkham's (1971, p. 371, table 1; also reproduced by Bouwer and Jackson, 1974, p. 615, table 23-1) table to calculate the hydraulic conductivity of the medium from the data, the K value is 15.45 m d a y - 1, which is very close to the K value of 15.53mday I (for M = N = 10) calculated above by using Eq. (56).
3.3. Example 3: partially penetrating auger hole in isotropic soil Consider the same example of Kessler and Oosterbaan (1980, p. 276, example 5) for estimating saturated hydraulic conductivity from auger hole experimental data below a water table using the Ernst's chart. The example in the present notation can be expressed as: soil isotropic, i.e. K~=(Kr/Kz)½=I, a = 4 c m , H l = 2 8 . 4 c m , //3 = 126cm, average rate of rise of water in the auger h o l e = 0.112cms -l, therefore, Qp = rr(4)2(0.112) = 5.629734 cm 3 s -1, the depth to the impervious layer is not given in the example directly but it is given that h - / / 3 > / / 3 / 2 , assume h = 303 cm. Proceeding in exactly the same way as in Example (2), after summing up to four
G. Barua, K.N. Tiwari / Journal of Hydrology 167 (1995) 1-22
19
terms (M = N = 4) of the series occuring in Eq. (56) K = 8.0118 x 10-4cms -l = 0.69mday -1 Expanding up to ten terms of the series, K = 7.54 × 10-4cms -1 = 0.65mday -1 These K values agree fairly well with the K value of 0.66mday -l calculated by Kessler and Oosterbaan using the Ernst's chart.
3.4. Example 4." partially penetrating auger hole in anisotropic soil, anisotropy ratio known Consider the same situation as in Example 3 but with one minor difference. Here, instead of K r = 1, K ~ = x / ~ . Thus, for this example a = 4 c m , h=303cm, H1 -----28.4cm, H3 = 126cm, Qp = 5.629734cm 3 s -l and K~ = x/~. From Eqs. (4) and (53), QpK~ Kr=
N
27reeZ Bnkl ( - N " e ~ n:l
\
K~ ,]
Substituting the given value of the parameters into the above equation, 5.629734 × v ~ Kr =
(57)
N
2×Tr×4ZBnkl(.-N~
×4)
n=l
where N.
[ ( ~ - 2n)Tr] , [. ( 1 S 2m)Tr ] × 303 ] and Nm = [2(303 - 28.4)J
and as before (see Example 2 above), the constants Am and Bn can be determined by making use of Eqs. (40) and (46). Summing up to five terms (M = N = 5) of the series occurring in the r.h.s, of Equation (57),. Kr = 9.689 x 10-4cms -1 = 0.84mday -l Hence,
Kr Kz -
(K~) 2 -
0.837 _ 0.28 m d a y _ 1 3
3.5. Example 5." partially penetrating auger hole in anisotropic soil, anisotropy ratio unknown The directional conductivities of an anisotropic medium, when the anisotropy ratio
20
G. Barua, K.N. Tiwari / Journal of Hydrology 167 (1995) 1-22
of the medium is unknown, can be estimated by performing the auger hole experiment on two auger holes of different geometries. Since the vertical hydraulic conductivity tends to vary more than the horizontal hydraulic conductivity, the auger holes dug in the field should be of nearly the same depth but their radii should vary. Let the auger hole experimental data obtained from two fictitious auger holes be as follows: Auger hole (1)
Auger hole (2)
O~(1) = 50 cm h0) = 152.3cm /-/3(1) = 105cm HI(l) = 75cm Qp(1) -- 72.22cm3 s -1
oL(2) = 15cm h(2) = 152.3cm H3(2) = 105 cm Hi(2) = 75 cm Qp(2) -- 36.11 cm 3 s -1
where the subscripts (1) and (2) refer to auger holes (1) and (2), respectively. F r o m Eq. (53),
N
(_Nn(l)a(1)~ ~00 ]
QpO) 72.22
C~(1)~n=lBn(1)kl ~-
Qp(2)
N a(2) ~ Bn(2)kl ( 7Nn(2)°l(2)'~
36.11
.=1
\
(58)
/
where
[(1 - 2n)Tr] N"(1) = N'(z) = [27-15--2~.3J and ,
,
r
(1-2m)rr
]
N;"/I/= N;"/2/= /2(152.3 - 105)] In Eq. (58), except K~, all the other parameters are known; hence K~ can be calculated. Expanding up to ten terms (M = N = 10) of the series occurring in the r.h.s, of Eq. (58), by trial and error K~ = 1.275 (approximately) Hence, (K~) 2 ---Kr/Kz = 1.625. Kr and Kz can now be calculated by considering the discharge expression (Eq. 53) for either the first or the second auger hole. Considering the first auger hole, after simplification
Kr = 1.794 x 10-3cms -1 = 1.55mday -1 Hence, Kz = KJ(K~) 2 = 0.95m day -1 . Examples 1 to 3 show that the hydraulic conductivity values estimated by using the theories developed for the isotropic condition agree fairly well with the hydraulic
G. Barua, K.N. Tiwari / Journal of Hydrology 167 (1995) 1-22
21
conductivity values calculated by using other known methods. This proves the correctness of the proposed analytical models. Parallel numerical modes (not given in this paper) for both the boundary value problems as shown in Figs. 1 and 2, respectively, have been developed using the finite difference technique (Rushton and Redshaw, 1979), and comparisons have been made among predictions from these numerical models with respective predictions from the analytical models. The numerical results match their analytical counterparts perfectly thereby proving the validity of the proposed analytical models.
4. Summary and conclusion The hydraulic head function and from it the stream function and the flow rate have been derived for groundwater seeping into fully penetrating and partially penetrating auger holes in a homogeneous and anisotropic soil medium. The theories proposed here can be utilized directly (i.e. without transforming the coordinate axes) to translate the rate of rise of water in a pumped auger hole into directional conductivities of the soil when the anisotropy ratio of the medium under consideration is known. If the anisotropy ratio of the medium is not known beforehand, an iterative method using two auger holes of different geometries may be employed to determine the directional conductivities of the medium. The seepage theory proposed here for the partially penetrating auger hole for the isotropic soil is expected to be simpler than that o f Kirkham's (1958) and Boast and Kirkham's (1971) solutions to the same problem. The theories proposed here apply not only to auger holes but also to any cylindrical cavity in soil of the type as shown in Figs. 1 and 2, respectively. These theories are also important in connection with well problems because a well can be considered as a special case of an auger hole.
References Barua, G., 1991.Analyticalmodelfor prediction of flowto ditches from pondedfields. Thesis (unpublished) Indian Institute of Technology,Kharagpur. Bear, J., 1972. Dynamicsof Fluids in Porous Media. Elsevier, New York, 235 pp. Boast, C.W. and Kirkham, D., 1971. Auger hole seepage theory. Soil Sci. Soc. Am. Proc., 35: 365-373. Boast, C.W. and Langebartel, R.G., 1984. Shape factors for seepage into pits. Soil Sci. Soc. Am. J., 48: 10-15. Bouwer, H. and Jackson, R.D., 1974. Determining Soil Properties. In: Jan Van Schilfgaarde (Editor), Drainage for Agriculture. Am. Soc. Agron. No. 17, Madison, WI, pp. 611-672. Bouwer, H. and Rice, R.C., 1983. The pit bailing method for hydraulic conductivity measurement of isotropic or anisotropic soil. Trans. Am. Soc. Agric. Eng., 26: 1435-1439. Childs, E.C., 1952.The measurement of the hydraulicconductivityof saturated soil in situ. I. Principles of a proposed method. Proc. R. Soc., Lond., A 215: 525-535. Childs, E.C., Collis-George, N. and Holmes, J.W., 1957. Permeability measurements in the field as an assessment of anisotropy and structure development. J. Soil Sci., 8: 27-41. Dwight, H.B., 1961. Tables of Integrals and other Mathematical Data. Fourth edn., The Macmillan Company, New York.
22
G. Barua, K.N. Tiwari / Journal of Hydrology 167 (1995) 1-22
Johnson, H.P., Frevert, R.K. and Evans, D.D., 1952. Simplified procedure for the measurement and computation of soil permeability below the water table. Agr. Eng., 33(5): 283-286. Kessler, J. and Oosterbaan, R.J., 1980. Determining hydraulic conductivity of soils. In: Drainage Principles and Applications, 2nd edn., Publ. 16, Vol. III. Edited from lecture notes of the international course on land drainage, Wageningen, Netherlands, pp. 254-296. Kirkham, D., 1945. Proposed method for field measurements of permeability of soil below the water table. Soil Sci. Soc. Am. Proc., 10: 58-68. Kirkham, D., 1958. Theory of seepage into an auger hole above an impermeable layer. Soil Sci. Soc. Am. Proc., 22: 204-208. Kirkham, D. and Van Bavel, C.H.M., 1948. Theory of seepage into auger holes. Soil Sci. Soc. Am. Proc., 13: 75-82. Kirkham, D. and Powers, W.L., 1972. Advanced Soil Physics. Wiley-Interscience, New York, pp. 71-74, 120-130. Lomen, D.O., Warrick, A.W. and Zhang, R., 1987. Determination of hydraulic conductivity from auger holes pits - an approximation. J. Hydrol., 90: 219-229. Luthin, J.N., 1957. Drainage of Agricultural Lands. Am. Soc. Agron., Vol. 7, Madison, WI, pp. 420-432. Maasland, M., 1956. Measurement of hydraulic conductivity by the auger hole method in anisotropic soil. Soil Sci., 81: 379-388. Maasland, M., 1957. Soil anisotropy and land drainage. In: J.N. Luthin (Editor), Drainage of Agricultural Lands. Am. Soc. Agron., Vol. 7, Madison, WI, pp. 216-285. Maasland, M. and Kirkham, D., 1955. Theory and measurement of anisotropic air permeability in soil. Soil Sci. Soc. Am. Proc., 20: 395-400. Muskat, M., 1946. The Flow of Homogeneous Fluids through Porous Media. J.W. Edwards, Ann Arbor, MI, pp. 225-227. Reeve, R.C. and Kirkham, D., 1951. Soil anisotropy and some field methods for measuring permeability. Amer. Geophys. Union Trans., 32: 582-590. Rushton, K.R. and Redshaw, S.C., 1979. Seepage and Groundwater Flow. Numerical Analysis by Analog and Digital Methods. Wiley, New York, pp. 91-92. Scarborough, J.B., 1966. Numerical Mathematical Analysis, 6th edn. Oxford & IBH Publishing Co., New Delhi, pp. 269-296. Talsma, T., 1960. Measurement of soil anisotropy with piezometers. J. Soil. Sci., 11: 159-171. Van Bavel, C.H.M. and Kirkham, D., 1948. Field measurement of permeability using auger holes. Soil Sci. Soc. Am. Proc., 13: 90-96. Youngs, E.G., 1968. Shape factors for Kirkham's piezometer method for determining the hydraulic conductivity of soil overlying an impermeable floor or infinitely permeable stratum. Soil Sci., 106: 235-237.
Journal of Hydrology 167 (1995) 1-22
Hydrology
[3]
Theories of seepage into auger holes in homogeneous anisotropic soil G a u t a m B a r u a * , K . N . Tiwari Department of Agricultural and Food Engineering, LL T. Kharagpur, West Bengal, India
Received 6 December 1993;revision accepted 26 September 1994
Abstract Steady-state theories have been developed for flow into an auger hole in a homogeneous and anisotropic soil medium when (i) there is an impermeable layer at the bottom of the auger hole and (ii) there is an impermeable barrier at a finite distance below the bottom of the auger hole. The proposed theories can be modified easily for an isotropic soil condition and the partially penetrating auger hole theory presented here is simpler than the existing theories (Kirkham, 1958; Boast and Kirkham, 1971) for the same soil condition. Furthermore, the theories can be applied directly, i.e. without transformation of coordinate axes, to determine the directional conductivities of an anisotropic medium (and hence also the hydraulic conductivity of an isotropic medium since an anisotropic medium can be transformed into an isotropic one simply by treating the horizontal hydraulic conductivity of the medium as equal to its vertical hydraulic conductivity) from auger hole experimental data below a water table.
1. Introduction The auger hole method for determining the saturated hydraulic conductivity of soil below a water table has been described by Van Bavel and Kirkham (1948), Luthin (1957), Bouwer and Jackson (1974) and Kessler and Oosterbaan (1974). A steadystate theory of flow into an auger hole fully penetrating a homogeneous and isotropic porous medium of infinite horizontal extent was given by Kirkham and Van Bavel (1948). Kirkham (1958) gave an exact theory of seepage into an auger hole partly penetrating a porous medium bounded above by a water table and below by an impermeable layer. Boast and Kirkham (1971) developed exact theories of flow into a partially penetrating auger hole both for situations when the bottom boundary
* Corresponding author. 0022-1694/95/$09.50 © 1995 - Elsevier Science B.V. All rights reserved SSDI 0022-1694(94)02629-7
2
G. Barua, K.N. Tiwari / Journal of Hydrology 167 (1995) 1-22
Notation
A~,,Bm,B, h HI tt3
It(.) l~(.) K
Ko(.) K~(.) K, K~ (K~)2 rn~n
M N
Qb Qf Qp r z
constants with m = 1,2, 3,... to and n = 1,2, 3,... ec radius of auger hole (L) depth to the impermeable layer from the water table (L) water level depth in auger hole (L) depth of penetration of the partially penetrating auger hole (L) zero order modified Bessel function of first kind first order modified Bessel function of first kind hydraulic conductivity of isotropic soil (LT -l) zero order modified Bessel function of second kind first order modified Bessel function of second kind horizontal hydraulic conductivity (LT -1) vertical hydraulic conductivity (LT -1) (Kr/Kz) anisotropy ratio (dimensionless) summation indices 1,2,3,...oo 1,2,3,...c~ flow taking place through the bottom of the partially penetrating auger hole (Fig. 2) (L3T -1) total flow into the fully penetrating auger hole (Fig. 1) (L3T -l) total flow into the partially penetrating auger hole (Fig. 2) (L3T -l) radial coordinate (L) vertical coordinate (L)
Greek letters
¢ ¢1 ¢2 ¢1 ¢2
hydraulic head distribution for the fully penetrating auger hole problem (Fig. 1) (L) hydraulic head distribution for region 1 (Fig. 2) for the partially penetrating auger hole problem (L) hydraulic head distribution for region 2 (Fig. 2) for the partially penetrating auger hole problem (L) Stoke's stream function for the fully penetrating auger hole problem (Fig. 1) (L2T -1) Stoke's stream function for region 1 (Fig. 2) for the partially penetrating auger hole problem (LET-1) Stoke's stream function for region 2 (Fig. 2) for the partially penetrating auger hole problem (L2T -L)
of the flow medium is impervious and when the bottom boundary is a layer of infinite permeability. Healy and Laak (1973), as reported by Bouwer and Rice (1983), used broad and shallow auger holes (pits) for in-situ determination of the saturated hydraulic conductivity in stony and gravely soils. They called their technique a 'pitbailing' method and used conventional well-theory (Thiem's equation) to interpret their results. Bouwer and Rice (1983) argued that the equation used by Healy and Laak (1973) for calculating the saturated hydraulic conductivity of soil from experimental data on pits cannot be applied to pits that penetrate only partially to an impermeable layer. They extrapolated the piezometer theory of Youngs (1968) to obtain values of the shape factor (a shape factor is a factor which expresses the relationship between the observed rate of rise of water in a pumped auger hole and the hydraulic conductivity of the surrounding soil medium) to cases where the width to depth ratio is large. Boast and Langebartel (1984) extended the earlier work of
G. Barua, K.N. Tiwari / Journal of Hydrology 167 (1995) 1-22
Boast and Kirkham (1971) to find values of the shape factor for broad and shallow pits. Lomen, Warrick and Zhang (1987) provided a steady-state theory of flow into axially symmetric auger holes and pits of arbitrary geometries. However, none of the theories of seepage into auger holes (fully or partially penetrating) take into account, explicitly, the anisotropy of a soil medium. The usual procedure for solving a problem in an anisotropic flow medium when the solution in an isotropic medium is available, is, firstly, to convert the anisotropic medium into a fictitious isotropic medium by shrinking the dimensions in the direction of greatest permeability or by stretching the section in the direction of lesser permeability (Muskat, 1946; Childs, 1952; Maasland and Kirkham, 1955; Maasland, 1957); then the flow problem in this fictitious isotropic medium is solved using the isotropic solution of the problem. Finally, the results are transformed into anisotropic prototype space. Using the results of the theorems stated by Maasland and Kirkham (1955), Maasland (1956) (see also Maasland, 1957) developed formulae and graphs to calculate the horizontal and the vertical hydraulic conductivities of an anisotropic soil medium from the observed rate of rise of water in a pumped auger hole below a water table when the anisotropy quotient, i.e. the ratio of horizontal to vertical hydraulic conductivity of the soil under consideration, is known. There is, however, always an advantage in obtaining a solution which can address the problem of anisotropy explicitly since it can be applied directly, i.e. without the necessity of bringing about a transformation of coordinate axes, to estimate the directional conductivities of an anisotropic flow medium from auger hole experimental data below a water table. Reeve and Kirkham (1951) reported that long and thin auger holes tend to measure essentially the horizontal conductivity of the soil whereas broad piezometers of zero cavity length tend to measure essentially the vertical conductivity of the soil. Actually, however, both the auger hole and the piezometer measure the apparent hydraulic conductivity of the flow medium rather than its horizontal or vertical hydraulic conductivity. Childs et al. (1957) combined the two-well method of Childs (1952) and the tube method of Kirkham (1945) to calculate the directional conductivities of an anisotropic medium below a water table. Since this method depends on the tube method for sensing the apparent hydraulic conductivity of the soil, the conductivity values determined by this method may not be totally representative of the soil under consideration because the tube method yields an apparent hydraulic conductivity for only a small volume of the soil (Talsma, 1960). One way of avoiding this difficulty is by using a very large sampler but that would make the method cumbersome; also, in this way, a representative value for the apparent conductivity would still be obtained for only a small depth interval. Values of the horizontal and vertical hydraulic conductivities of an anisotropic medium can also be obtained from the results of measurements on two piezometer installations, one with a cavity of small height and the other with a cavity of large height. The theory for this procedure was given by Maasland (1957). Utilising this theory, Talsma (1960) calculated the directional conductivities of anisotropic media of infinite depth by installing piezometers of zero and known cavity lengths below the water table. This method also suffers from the same kinds of drawbacks as the Childs' (1957) method; like the Childs' method, a tube is used to determine the apparent hydraulic conductivity of the
4
G. Barua, K.N. Tiwari / Journal of Hydrology 167 (1995) 1-22
soil. Bouwer and Rice (1983) described an iterative procedure to determine the directional conductivities of an anisotropic soil medium below a water table by considering two pits of different geometries. The main advantage of their method is that one need not know beforehand the anisotropy quotient of the soil under consideration to determine the directional conductivities of the soil. Boast and Langebartel (1984), however, reported that the shape factors obtained by Bouwer and Rice (1983) for estimating the saturated hydraulic conductivity of soil by the 'pitbailing' method are 20% greater than the shape factors obtained by Bouwer and Rice (1983) from sand tank model studies on broad and shallow pits. Because of the inability to measure accurately the shape factors for pits, errors may crop up in hydraulic conductivity estimations because an error in shape factor is reflected proportionately in the calculated hydraulic conductivity value. Instead of using Bouwer and Rice's shape factors for pits, one may also utilise the shape factors (greater by a factor of 2) given by Boast and Langebartel (1984) for broad and shallow auger holes, but even then an error in hydraulic conductivity estimation may result. Thus, there seems to be, at present, no suitable theory which can predict accurately, the shape factors for broad and shallow pits. The horizontal and vertical hydraulic conductivities of an anisotropic soil medium below a water table can also be determined in the laboratory with the help of suitable apparatus. The procedure is illustrated in detail by Kessler and Oosterbaan (1980). An obvious drawback of this method is that the samples used for experiments are small and therefore' a large number of samples need to be taken and experimented upon before any meaningful conclusions can be drawn from the experimental results. It is clear that there exists a need for suitable theories of flow into auger holes which take into account, explicitly, the anisotropy of the flow medium, and which should be easy to apply for calculating the directional conductivities of an anisotropic soil medium from auger hole experimental data below a water table. This paper addresses this need.
2. Mathematical formulation and solution 2.1. Case 1: fully penetrating auger hole problem
Fig. 1 shows an auger hole of radius a resting on an impervious base of a homogeneous and anisotropic aquifer of infinite radial extent. The depth from the water table to the impervious layer is h and the depth of water in the auger hole, as measured from the origin O, is HI. The saturated hydraulic conductivities of the aquifer in the horizontal and the vertical directions are Kr and Kz, respectively. Because of axial symmetry, the flow region to the right of the vertical axis only is considered. A righthand coordinate system is established with the origin as shown in Fig. 1. For convenience, the z axis is taken as positive vertically downward and the r axis is taken as positive towards the right. In the following analysis it is assumed that the flow is steady, the drawdown of the water table around the auger hole because of seepage into the hole is negligible,
G. Barua, K.N. Tiwari / Journal of Hydrology 167 (1995) 1-22
SOIL
5
SUR FACF..
,,/,6¢~
///~:'4e',i,P,~"
r
WATER
TABLE
Or0,0 ) Kr ~
HI
I4
-
~ Kr
Kt
"111'
q
IV ;'-; / - -
////
/ / ,
/= 212 2 ; - / - ' ~ " . " "I(IMPE~VIOU$/ /" / / / / LAYER
///
Fig. 1. Geometry of the flow system of a fully penetrating auger hole. the capillary affects above the water table are non-existent and the principal directions of anisotropy of the medium coincide with the horizontal and vertical directions, respectively. The following form of Fourier sine series (Barua, 1991) will be used to obtain solutions to both the problems considered in this paper. In theorem form, the series can be stated as: If f (x) is piecewise continuous in an interval (O, L) and has right- and left-hand derivatives at all points of that interval, then the following infinite trigonometric series E m=l
Am sin (1 - 2m) 2
converges to (a) f ( x ) ,
(b)
if x is a point o f continuity
f(x + O) +f(x 2
- O) if x is a point of discontinuity
6
G. Barua, K.N. Tiwari / Journal of Hydrology 167 (1995) 1-22
where
jf(x)sin L
Am= 2
[(1 2 2 m ) L - - ] dx
0
and f ( x + 0) and f ( x - 0) are right- and left-hand limits o f f ( x ) at x. The steady-state continuity equation to be solved in cylindrical coordinate system is
Kr O2(b
K~ Oq~ 020 _ + 7 + Kzb- z2 - 0
(11
subject to the following boundary conditions (BCs): = -z;
r = a;
0 < z < H1
(BCIa)
~b = -H1;
r = a;
Hi < z < h
(BCIb)
~b = 0;
z = 0;
a < r < oo
(BCII)
Or
O;
r
oo;
0< z< h
(BCIII)
0q~ Oz
0;
z
h;
a
(BCIV)
where ~b is the hydraulic head measured with respect to the reference level passing through the origin O and Kr, Kz (r and z have already been defined). Using the separation of variable technique (see Kirkham and Powers, 1972) a solution to Eq. (1) satisfying boundary conditions II, III and IV as mentioned above, for any values of the constants Bin, can be expressed as Oc
_Nmr
m=l
where
K~ = (Kr/Kz) U2
(4)
K0(.) = zero order modified Bessel function of section kind and m = 1,2, 3 , . . . To make Eq. (2) reducible to the form of Fourier sine series developed by Barua (1991) along r = a, take sm =
Am
(s)
G. Barua,K.N. Tiwari/ Journalof Hydrology 167 (1995) 1-22 Substituting the right-hand side (r.h.s.) of Eq. (5) for
= ~ Am
=,
Bm into
Eq. (2),
K°\ K~ )sin(Nmz) {-Nma\ :
(6)
where Am are arbitrary constants. Let
k [-Nmr\ --
/'-Nmr\ ) (7a)
and
/'-Nmr\ Kl~--~o ) k l ~ K~ / = K° (__Nmcx k K~ ~ ) (-Nmr ~
(7b)
From Eqs. (6) and (7a),
Am
-Nmr
sin (Nmz)
(8)
m=l
Now, finally, applying boundary conditions Ia and Ib to Eq. (8), at r = a Am sin (Nmz)
-z = ~
for
0 < z < Hl
for
Hi < z < h
m=l
oo
-H1 = Z Am sin (Nmz) m=l
Hence, the constants
Am =
~
Am are
given by
( - z ) sin (Nmz) dz +
( - H , ) sin (Nmz) d~
(9)
Hi
Eq. (9), upon simplification, gives
Am =
-2 - - - - - 2 sin (NmHl) tl(Atm)
(lo)
8
G. Barua,K.N. Tiwari/ Journalof Hydrology 167 (1995) 1-22
Now, from Eqs. (3) and (10), Am
_
-8h~ I(1 - 2m)TrH1] (1 - zm)ZTr2sin 2h J
(11)
Substituting the values of A m and N m into Eq. (6), the expression for total hydraulic head, ~b, finally becomes
- -= l m
( - ( 1 - 2__m)Trr.'~ -8h sin [ ( 1 - 2re)Trill. ] 2 h Ko\ 2K6h ] sin(N,nz) ( 1 - 2m)eTr2 J K0 ( . - ( 1 - 2m)Tro{'~
\
2K h
(12)
J
Eq. (12) gives the distribution of hydraulic head for all points throughout the porous medium of Fig. 1. Equations connecting the hydraulic head function, ~b, with the Stoke's stream function, ~b, in axis-symmetric flow for the anisotropic case are as given below (Bear, 1972): 0~b
1 0~b
(13a)
Kr T r - r T z
and
O~ Kz-'~z--
(13b)
1 0¢ r Or
where ~b is the Stoke's stream function. Application of equations (13a) and (13b) to Eq. (8), after simplification, gives ~b(r,z) = - ZAmr(KrKz)½kl
cos (Nmz) + C
(14)
m=l
where C is a constant to be determined and K 1(.) is first order modified Bessel function of second kind. As the quantity of interest is the flow between two stream lines, the constant C can be chosen arbitrarily; for the sake of simplicity, consider the stream line given by the following expression as the zero stream line, i.e. ¢(r--~ oc, z) = 0
(15)
Now, application of Eq. (15) to Eq. (14) gives (after using the L' Hospital's theorem and Dwight's (1961) Formula No. 816.2 for n = 1) C = 0. Therefore, ~b takes the form OO
~b(r,z) = - Z
m=l
Amr(KrK~)½kl ( ~
\ K6,/
cos
(Nmz)
(16)
The quantity of water, Qf, entering the auger hole can be calculated by either of the
G. Barua, K.N. Tiwari / Journal of Hydrology 167 (1995) 1-22
9
equations given below: h
(17)
Qf = Kr 0
Qf = -Kz
(o0)2=027rrdr
(18)
Ozz
Eq. (17), after simplification~ reduces to
(-Umr
Qf=27rc~--~I Z Amkl • K[) /I
(19)
m=l
Similarly, Eq. (18), after simplification yields (2~3
Of = 27reeKzKo Z Amkl (-Nmr'~ m=l
(20)
~ K; /
Substituting the r.h.s, of Eq. (4) for K~ into Eqs. (19) and (20), respectively, (3O
Qf = 27r°~(KrKz)½ Z Amkl (-Nmr~ m=l
(21)
~ K~ )
For an isotropic soil medium, Kr = Kz = K (say) and K~ = (KJK2) ½= 1; thus, ¢, for the isotropic case reduces to (X3
0 = Z Am ko(-N~r) sin (Nmz)
(22)
m=l
and the stream function, z), for the isotropic condition can be expressed as (X?
¢ = - K ~ rAmkl (-Nmr) cos (Nmz)
(23)
m=l
Also, the quantity of water seeping into the auger hole for the isotropic soil condition is given by
Qf = 27reeKy ~ Amk1(-Nmo0
(24)
m=l
2.2. Case 2."partially penetrating auger hole problem Fig. 2 illustrates the problem under consideration. An auger hole of radius penetrates an aquifer of infinite horizontal extent to a depth of H 3 below the water table. The symbols h, H 1, Kr and Kz carry the same meaning as in the fully penetrating auger hole problem. A coordinate system similar to that shown in Fig. 1 is also
10
G. Barua, K.N. Tiwari / Journal of Hydrology 167 (1995) 1-22
considered here and, as before, because of axial symmetry, only the flow region to the right of the vertical axis passing through the origin O is considered. Following K_irkham's (1958) mode of analysis, the flow domain is divided into two regions: (1) the region between the bottom of the hole and the impermeable layer and (2) the remaining region. The hydraulic head in region (1) is designated by (91 and in region (2) by (92. The boundary conditions for the problem, as illustrated in Fig. 2, can be expressed as
0(gt =
r = 0;
H3 < z < h
(BC I)
z=Hs; r = a;
0
(BCII) (BC Ilia)
r=a;
H3 < z < h
(BCIIIb)
o;
z=h;
0
(BC IV)
-H,;
r=a; r=a;
Or
0;
(9,=-H1; ¢1 = (92;
K r0(9, ~=K,
0(9, =
Oz (92 --Z;
~rz ;
(92 = o;
z=0;
O
0(92 Or =
0;
r=~;
0
(BE VII)
0(92 Oz =
0;
z = h;
a < r < cx~
(BC VIII)
=
(92 =
(BC Va) 3 (BC Vb)
(BC Vl)
The function, ¢, must be chosen such that
/~ 02(9, K~ 0(91 02(9, --ff~-- + --;- - ~ - + Xz --ff~-z2= 0
(25)
in region (1) and K 02(92 Kr 0(92 02(92 r --ff~--+ -~ --~r + Kz ---~Tz2= 0
(26)
in region (2) must be satisfied together with the boundary conditions I to VIII as listed above. Using the separation of variable technique (see Kirkham and Powers, 1972) solutions to Eqs. (25) and (26) can be expressed in the following forms: [-(1 - 2m)Trrq (9, =
sin [.(l - 2m)Tr(z- H3)]
G. Barua, K.N. Tiwari / Journal of Hydrology 167 (1995) 1-22
11
SOIL SURFACE
1
WATER
t
TABLE
VJ
I
II
Kp@ Kp
H
Kz
I 1
(~
Vb
, _ _
Vll "Z_-Z-
2 2"- Z - - - - _ - _ "
I
REGION
ma
®
llllb
I
,
"///////2///"/"/"
7//////
REGION
IV
Vlll IM PERVIOUS
/////////2
LAYER
Fig. 2. Geometry of the flow system of a partially penetrating auger hole.
and
u ~ [-(1 - 2n)~-,l - ~ ~b2 = ~ B, ~ -L - 2K~h J sin [ ( 1 - 2n)Trz] =
-o[
-j
2h
J
(28)
where M and N are any integers, i.e. M = 1,2,3,...; N = 1,2,3,...;
(29)
m and n are summation indices; I0(.)= zero order modified Bessel function of first kind and K~, h, H] , 1-13, a, K o (.), r and s have already been defined. Let , [(1 - 2m)Tr] N~n = [2(h - H3)J
(30)
iV. = [(1 - 2 n ) r ] 2h J
(31)
G. Barua, K.N. Tiwari / Journal of Hydrology 167 (1995) 1-22
12
{-N~r)
;or
j
I°(---~o ) -
)
_NSd~
(32a)
-N'r'~
il\ K; J=
(32b)
Io(- o ) - N'~'~
where I1(.) is the first order modified Bessel function of first kind and the other symbols carry the same meaning as defined before. Substitution of Eqs. (7a), (32a), (30) and (31) into Eqs. (27) and (28), respectively, gives
051= Z Ami°
sin[Nm(z - / / 3 ) ] - HI
(33)
m=l
and 052 = ~ ~ B , k0 ~(-N.r) - - ~ ,] sin(N,z)
(34)
m=l
In Eq. (33), 051 satisfies boundary conditions I, II, and IV for any values of the constants A,,. Similarl3,, 052 in Eq; (34), satisfies boundary conditions VI, VII and VIII for any values of the constants/in. Boundary conditions IIIa, IIIb, Va and Vb remain to be satisfied by proper choice of the constants Am and B,. To satisfy Va, Vb and IIIa, at r = N
Z / i n sin(N,z) = -z;
0 < z < HI
(35)
n=l N
Z
B, sin(Nnz) = - H I ;
H1 < z < / / 3
(36)
n=l N
M
Z B, sin(Nnz) = E n=l
Am sin[N~m(Z - H3)] - HI ;
H 3 <
z< h
(37)
m=l
and to satisfy boundary condition IIIb, after applying boundary condition IIIb to Eqs. (33) and (34) M
N
Amsin[Nm(z -
Y~
m=l
H3)] = ~ n=l
Bn
kl (~Ko~) ] sin(Nnz);H3 < •
- N
I c~
-
z < h -
(38)
If N ~ ~ (in view of Eq. (29)), then the functions as shown on the r.h.s, of
G. Barua, K.N. Tiwari / Journal of Hydrology 167 (1995) 1-22
13
Eqs. (35), (36) and (37), respectively, become suitable for expansion by the form of Fourier sine series developed by Barua (1991) along the boundary 0 < z < h. The constants Bn, can then be evaluated as l
Bn=-~
(-z)sin(N~z)dz + (-H1)sin(N~z)dz H~ h
h M
_]
-IHlsin(N,z)dz + I ~-~Amsin[N~(z-H3)]sin(Nnz)dzj //3
(39)
H3 m=l
After carrying out the above integrations using Dwight's (1961) Formulae Nos. 435 and 465, for Nn ~ N~,
Bn -.tt~n),~--~22.sin(N, Hl) + ~./-~--h--~-~Am l I (N, -1 Arm)sin((N~'H3) + (N, - N~)h) 1
- (Nn - N~m)sin((N~mH3) + (Nn 1
Nt + (Nn + N~) sin((mH3) •
-
-
N~n)H3)
(N. + N~m)h)
-(NnlN~)sin((N~H3)-(Nn+Ntm)H3)] (40) where n = 1 , 2 , 3 , . . . N - - . oc; m--- 1,2,3,...M. For N n = N~, the integrals on the r.h.s, of Eq. (39), after simplification, reduce to B , - -2sin(N, Hl)
h(Nn)z
+~
1
~-~Am [ + ~ h (h - H3) cos(N~H3)
/
sin(NmH3)
, cos(ZNrmh) ] + 2 -1~ sin(N~,H3) l
(41)
Although Eqs. (40) and (41) relate Bn and A m to two different situations, viz. Nn ¢ N~m and Nn = N~,, in actual practice the occurrence of the situation Nn = N" can easily be avoided by a judicious selection of the parameters//3 and h. Eq. (40) yields a set of equations giving values of B1, B2, B3,... Bn in terms of A1, A2,... Am. To have the same number of Am as there are B,, take M = N
(42)
G. Barua, K.N. Tiwari / Journalof Hydrology 167 (1995) 1-22
14
There remains Eq. (38) from which to find another set of relations between Am and B, and hence their individual values. Let - N o~
Substitution of Eq. (43) into Eq. (38) gives M
Z
N
Am sin[NIm(z - n 3 ) ] = Z
m=l
nnPnm sin(NnZ)
(44)
n=l
Now, if M ~ cx~ (in view of Eq. (29)), then A m of Eq. (44) can be evaluated as h
2
Am -
N
J Z. = , BnPnm sin(Nnz) sin [N~(z - H3)] dz H3
(45)
By carrying out the above integration and simplifying, for Nn ¢ N~
N P~mBn [ 1 sin((N~H3) + (N~ - N~)h) Am = Zn=l ( h - H3) (Nn - N~m) 1
(N n _ Nim) sin((N~H3) + (Am - N~m)H3) 1
-~ (Nn + )Vim)sin((N~mH3) - (Nn + N~m)h) 1 sin((N~nH3)- (Nn + N~m)H3)] (N, + N~)
(46)
where m = 1 , 2 , 3 , . . . M ~ oe;n = 1 , 2 , 3 , . . . N ~ ec. For Nn = N~, Eq. (45) yields, after simplification, N
Am = Z
PnmBn [ 1 . (-h--H33) (h - H3) cos(N.H3) +~N~nS,n(N.H3)
n=l
+~
1
sin(N~H3) cos(2Nnh)
(47)
Eqs. (46) (for Nn ¢ N~) and (47) (for Nn = N~m) give a second set of relations between A m and B,. As before, the situation Nn = N~ is avoided by a proper selection of the parameters h and H3. It should be noted that for the hydraulic head functions, ¢1 and ¢2, developed for regions (1) and (2), respectively, to be exact, both N and M should tend to infinity; however, for most situations, expansion of five to six terms gives a reasonably accurate value of the hydraulic heads around a
G. Barua, K.N. Tiwari / Journal of Hydrology 167 (1995) 1-22
15
partially penetrating auger hole. Linear equations stemming from Eqs. (40) and (46) (for Nn # N~m)can be solved easily by following a Gauss elimination procedure or by any other suitable method (see Scarborough, 1966). To get the stream function, ~bl, for region (1), Eqs. (13a) and (13b) are applied to Eq. (33); the resultant expression is
M , (_Ntmr) , ~1 = ZAmr(KrKz) ~i] cos[Nm(z -//3)] + CI ,,,=,
\K6)
(48)
where ~31 is the stream function for region (1) and Ci is a constant to be determined. For simplicity, ~1 (0, z) = 0
(49)
is considered as the zero stream line. From Eqs. (48) and (49), CI = 0. Hence, ~bI is given by M
, (_N~mr,~
~21 = Z Amr(KrKz)~il
\ K~,]
m=l
c°s[NIm(Z - H3)]
(50)
Similarly, application of Eqs (13a) and (13b) to Eq. (34) and using the limit O2(r--~ oo, z) = 0 yields N
~b2 = - ~
enr(KrKz)llZkl
n=l
(_Nnr)
\ K~ .,/
cos(Unz)
(51)
The quantity of water, Qp, entering the partially penetrating augerhole is given by co
Qp = _ K z f (O~2~ J
27rrdr
(52)
Eq. (52), upon simplification, yields
Qp = 27ra(KrKz)½ Z B"kl n=l
(53) \
K~ J
The quantity of water, Qb, entering through the bottom of the auger hole is given by
Qb = Kz j \ Oz /z=n327rrdr
(54)
0
From Eqs. (33) and (54), after simplification M
Qb = -2~ra(KrKz) k ~ a,,i~ m--1
(_N~ma~
(55)
kKo)
The solution presented here for the isotropic soil condition is simpler than Kirkham's (1958) isotropic solution to the partially penetrating auger hole
16
G. Barua, K.N. Tiwari / Journal of Hydrology 167 (1995) 1-22
problem. Kirkham had to define an identity (see Kirkham, 1958, Eq. (26)) to obtain a second set of relations between Am and Bn; Eq. (15) of his paper (Kirkham, 1958), which is equivalent to Eq. (38) of this paper, alone was unsuitable for numerical work whereas Eq. (38) of this paper alone is sufficient to give the other set of relations between A m and Bn. Moreover, in Kirkham's (1958) analysis, the occurrence of the situation n = m for the case n / h ~ m / ( h - / - / 3 ) (in the notation of this paper) necessitates the utilisation of a different formula from that of the situation in which n ¢ m (see Eqs. (31a) and (31b), Kirkham, 1958). However, in the theory proposed here, the occurrence of the situation n = m for the case N, ~ N~mdoes not require any new formula from that of the situation in which n ~ m and Eqs. (40) and (46) alone have been found suitable to evaluate the constants A m and B,. In a later paper, Boast and Kirkham (1971) solved this problem (and also the problem arising from a situation when the impervious layer is replaced by an infinitely permeable layer) for the isotropic soil condition by using the normal equations and the Gram-Schmidt orthonormalisation methods. Boast has used the number of multiplications and divisions required by the two methods as a rough estimate of the relative calculation time required and found that, for large values of N, the normal equations method is about twice as fast as the Gram-Schmidt orthonormalisation method. Boast and Kirkham (1971) further reported that calculations using Kirkham's (1958) approach require an amount of time of the same order of magnitude as calculations using the normal equations method. Since the proposed theory has been shown to be simpler than that of Kirkham's (1958) solution and since the time required for calculation using Kirkham's (1958) method and Boast and Kirkham's (1971) normal equations method is of the same magnitude and furthermore since the normal equations method is faster (for large N) than that of the Gram-Schmidt orthonormalisation method, the proposed theory is expected (for large N) to be simpler than that of the normal equations and Gram-Schmidt orthonormalisation methods of Boast and Kirkham (1971). Finally, the theory proposed here for the partially penetrating auger hole problem is of a more general nature than presently available solutions to this problem; the theory presented here takes into account the anisotropy of the soil medium explicitly.
3. Field applications The theories proposed here can be utilised for calculating the saturated hydraulic conductivity of an isotropic medium and the directional conductivities of an anisotropic flow medium from auger hole experimental data below a water table. To determine the saturated hydraulic conductivity of an isotropic medium or the directional conductivities of an anisotropic medium when the ratio of horizontal to vertical hydraulic conductivity of the medium under consideration is known, the proposed theories can be applied directly to the data obtained from an auger hole experiment (Van Bavel and Kirkham, 1948) in the field (see Examples 1 to 4 below). However, if the anisotropy ratio of the medium is not known beforehand, then the directional
G. Barua, K.N. Tiwari / Journal o f Hydrology 167 (1995) 1-22
17
conductivities of the medium may be determined by an iterative procedure using two auger holes of different geometries. This procedure is illustrated in Example 5.
3.1. Example 1."fully penetrating auger hole in isotropic soil Let the data obtained from an auger hole experiment on a fictitious fully penetrating auger hole be as follows: soil isotropic, i.e. K6 = 1, h = 126.5 cm, H1 = 126.5 cm, = 5cm, and rate of rise of water in the auger h o l e = 0.112cms -l, therefore, Qf = 7r(5) 2 0.112 = 8.7964594cm 3 s -1. F r o m Eq. (24), K=
Qf
oo
AmklI-Nm ) m=l
where Nm and Am of Eq. (24) are given by Eqs. (3) and (10), respectively. Substituting the given values of the parameters in the r.h.s, of the above equation, 8.7964594
K=
oo
2 x 7r × 5 ~
A~kl ( - N m × 5)
m=l
where
2m) ] Nm =
x 126.51
and
-2 Am - 126.5(N,,) 2 sin(Nm × 126.5) Simplifying the above expressions, after summing up to only five terms of the infinite series K = 5.17810 × 10-4cms -1 = 0 . 4 5 m d a y -1 Now the hydraulic conductivity of the medium may be estimated by applying Van Bavel and Kirkham's (1948, p. 93, Fig. 4) nomograph to the data to find K = 0 . 4 3 m d a y -1. Alternately, applying the modified Ernst's curve of Van Beers (1958) (see Bouwer and Jackson, 1974, p. 619, fig. 23-5), to the data, the K value is 0 . 4 1 m d a y -1. It is to be noted that both the K values of 0 . 4 3 m d a y -1 and 0.41 m day -l, obtained by applying Van Bavel and Kirkham's nomograph and the modified Ernst's curve o f Van Beers, are quite close to the K value of 0.45 m day -1 obtained by applying Eq. (24) developed here.
3.2. Example 2." partially penetrating auger hole in &otropic soil Let the data obtained from an auger hole experiment on a partially penetrating
G. Barua, K.N. Tiwari / Journal of Hydrology 167 (1995) 1-22
18
auger hole be as follows: soil isotropic, i.e. K~ = 1, h = 200cm, H1 = 75cm; H 3 = 100cm, a = 50cm, and rate of rise of water in the pumped auger hole = 0.112 cm s - 1, therefore, Qp = zr(50) 20.112 = 879.64594 cm s- 1. F r o m Eq. (53), for the isotropic condition K=
N Qp
(56)
2~r&Z B.kl(-N.a) n=l
Substituting the values of the given parameters into Eq. (56), 879.64594
K =
N
2 ×
× 50
B.kI(-N.
× 50)
n=l
where
x 200 J and Nm =
2 x-(260 Z ~ 0 ) J
and Bn of Eq. (56) can be evaluated by solving Eqs. (40) and (46) simultaneously. Summing up to only five terms (M = N = 5) of the series occurring in Eq. (56), i.e. solving only five simultaneous linear equations stemming from Eqs. (40) and (46), K = 1.847 × 10-2cms -1 = 15.95mday -1 Summing up to ten terms (M = N = 10) of the series, K = 1.797 × 10-2cms -1 --- 15.53mday -l Using Boast and Kirkham's (1971, p. 371, table 1; also reproduced by Bouwer and Jackson, 1974, p. 615, table 23-1) table to calculate the hydraulic conductivity of the medium from the data, the K value is 15.45 m d a y - 1, which is very close to the K value of 15.53mday I (for M = N = 10) calculated above by using Eq. (56).
3.3. Example 3: partially penetrating auger hole in isotropic soil Consider the same example of Kessler and Oosterbaan (1980, p. 276, example 5) for estimating saturated hydraulic conductivity from auger hole experimental data below a water table using the Ernst's chart. The example in the present notation can be expressed as: soil isotropic, i.e. K~=(Kr/Kz)½=I, a = 4 c m , H l = 2 8 . 4 c m , //3 = 126cm, average rate of rise of water in the auger h o l e = 0.112cms -l, therefore, Qp = rr(4)2(0.112) = 5.629734 cm 3 s -1, the depth to the impervious layer is not given in the example directly but it is given that h - / / 3 > / / 3 / 2 , assume h = 303 cm. Proceeding in exactly the same way as in Example (2), after summing up to four
G. Barua, K.N. Tiwari / Journal of Hydrology 167 (1995) 1-22
19
terms (M = N = 4) of the series occuring in Eq. (56) K = 8.0118 x 10-4cms -l = 0.69mday -1 Expanding up to ten terms of the series, K = 7.54 × 10-4cms -1 = 0.65mday -1 These K values agree fairly well with the K value of 0.66mday -l calculated by Kessler and Oosterbaan using the Ernst's chart.
3.4. Example 4." partially penetrating auger hole in anisotropic soil, anisotropy ratio known Consider the same situation as in Example 3 but with one minor difference. Here, instead of K r = 1, K ~ = x / ~ . Thus, for this example a = 4 c m , h=303cm, H1 -----28.4cm, H3 = 126cm, Qp = 5.629734cm 3 s -l and K~ = x/~. From Eqs. (4) and (53), QpK~ Kr=
N
27reeZ Bnkl ( - N " e ~ n:l
\
K~ ,]
Substituting the given value of the parameters into the above equation, 5.629734 × v ~ Kr =
(57)
N
2×Tr×4ZBnkl(.-N~
×4)
n=l
where N.
[ ( ~ - 2n)Tr] , [. ( 1 S 2m)Tr ] × 303 ] and Nm = [2(303 - 28.4)J
and as before (see Example 2 above), the constants Am and Bn can be determined by making use of Eqs. (40) and (46). Summing up to five terms (M = N = 5) of the series occurring in the r.h.s, of Equation (57),. Kr = 9.689 x 10-4cms -1 = 0.84mday -l Hence,
Kr Kz -
(K~) 2 -
0.837 _ 0.28 m d a y _ 1 3
3.5. Example 5." partially penetrating auger hole in anisotropic soil, anisotropy ratio unknown The directional conductivities of an anisotropic medium, when the anisotropy ratio
20
G. Barua, K.N. Tiwari / Journal of Hydrology 167 (1995) 1-22
of the medium is unknown, can be estimated by performing the auger hole experiment on two auger holes of different geometries. Since the vertical hydraulic conductivity tends to vary more than the horizontal hydraulic conductivity, the auger holes dug in the field should be of nearly the same depth but their radii should vary. Let the auger hole experimental data obtained from two fictitious auger holes be as follows: Auger hole (1)
Auger hole (2)
O~(1) = 50 cm h0) = 152.3cm /-/3(1) = 105cm HI(l) = 75cm Qp(1) -- 72.22cm3 s -1
oL(2) = 15cm h(2) = 152.3cm H3(2) = 105 cm Hi(2) = 75 cm Qp(2) -- 36.11 cm 3 s -1
where the subscripts (1) and (2) refer to auger holes (1) and (2), respectively. F r o m Eq. (53),
N
(_Nn(l)a(1)~ ~00 ]
QpO) 72.22
C~(1)~n=lBn(1)kl ~-
Qp(2)
N a(2) ~ Bn(2)kl ( 7Nn(2)°l(2)'~
36.11
.=1
\
(58)
/
where
[(1 - 2n)Tr] N"(1) = N'(z) = [27-15--2~.3J and ,
,
r
(1-2m)rr
]
N;"/I/= N;"/2/= /2(152.3 - 105)] In Eq. (58), except K~, all the other parameters are known; hence K~ can be calculated. Expanding up to ten terms (M = N = 10) of the series occurring in the r.h.s, of Eq. (58), by trial and error K~ = 1.275 (approximately) Hence, (K~) 2 ---Kr/Kz = 1.625. Kr and Kz can now be calculated by considering the discharge expression (Eq. 53) for either the first or the second auger hole. Considering the first auger hole, after simplification
Kr = 1.794 x 10-3cms -1 = 1.55mday -1 Hence, Kz = KJ(K~) 2 = 0.95m day -1 . Examples 1 to 3 show that the hydraulic conductivity values estimated by using the theories developed for the isotropic condition agree fairly well with the hydraulic
G. Barua, K.N. Tiwari / Journal of Hydrology 167 (1995) 1-22
21
conductivity values calculated by using other known methods. This proves the correctness of the proposed analytical models. Parallel numerical modes (not given in this paper) for both the boundary value problems as shown in Figs. 1 and 2, respectively, have been developed using the finite difference technique (Rushton and Redshaw, 1979), and comparisons have been made among predictions from these numerical models with respective predictions from the analytical models. The numerical results match their analytical counterparts perfectly thereby proving the validity of the proposed analytical models.
4. Summary and conclusion The hydraulic head function and from it the stream function and the flow rate have been derived for groundwater seeping into fully penetrating and partially penetrating auger holes in a homogeneous and anisotropic soil medium. The theories proposed here can be utilized directly (i.e. without transforming the coordinate axes) to translate the rate of rise of water in a pumped auger hole into directional conductivities of the soil when the anisotropy ratio of the medium under consideration is known. If the anisotropy ratio of the medium is not known beforehand, an iterative method using two auger holes of different geometries may be employed to determine the directional conductivities of the medium. The seepage theory proposed here for the partially penetrating auger hole for the isotropic soil is expected to be simpler than that o f Kirkham's (1958) and Boast and Kirkham's (1971) solutions to the same problem. The theories proposed here apply not only to auger holes but also to any cylindrical cavity in soil of the type as shown in Figs. 1 and 2, respectively. These theories are also important in connection with well problems because a well can be considered as a special case of an auger hole.
References Barua, G., 1991.Analyticalmodelfor prediction of flowto ditches from pondedfields. Thesis (unpublished) Indian Institute of Technology,Kharagpur. Bear, J., 1972. Dynamicsof Fluids in Porous Media. Elsevier, New York, 235 pp. Boast, C.W. and Kirkham, D., 1971. Auger hole seepage theory. Soil Sci. Soc. Am. Proc., 35: 365-373. Boast, C.W. and Langebartel, R.G., 1984. Shape factors for seepage into pits. Soil Sci. Soc. Am. J., 48: 10-15. Bouwer, H. and Jackson, R.D., 1974. Determining Soil Properties. In: Jan Van Schilfgaarde (Editor), Drainage for Agriculture. Am. Soc. Agron. No. 17, Madison, WI, pp. 611-672. Bouwer, H. and Rice, R.C., 1983. The pit bailing method for hydraulic conductivity measurement of isotropic or anisotropic soil. Trans. Am. Soc. Agric. Eng., 26: 1435-1439. Childs, E.C., 1952.The measurement of the hydraulicconductivityof saturated soil in situ. I. Principles of a proposed method. Proc. R. Soc., Lond., A 215: 525-535. Childs, E.C., Collis-George, N. and Holmes, J.W., 1957. Permeability measurements in the field as an assessment of anisotropy and structure development. J. Soil Sci., 8: 27-41. Dwight, H.B., 1961. Tables of Integrals and other Mathematical Data. Fourth edn., The Macmillan Company, New York.
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