Computer Note/
Estimating Aquifer Transmissivity from Specific
Capacity Using MATLAB
by Stephen G. McLin1
Abstract
Historically, specific capacity information has been used to calculate aquifer transmissivity when pumping
test data are unavailable. This paper presents a simple computer program written in the MATLAB programming
language that estimates transmissivity from specific capacity data while correcting for aquifer partial penetration
and well efficiency. The program graphically plots transmissivity as a function of these factors so that the user
can visually estimate their relative importance in a particular application. The program is compatible with any
computer operating system running MATLAB, including Windows, Macintosh OS, Linux, and Unix. Two simple
examples illustrate program usage.
Introduction where B = formation loss coefficient (T/ L2), C =
A computer technique for estimating transmissivity well loss coefficient (T2/ L5 if n = 2), Q = well discharge
from specific capacity data is currently available (Bradbury (L3/ T), and n = an exponent related to wellbore turbu-
and Rothschild 1985). However, it is written in BASIC lence (typically, 1.5 n 3.5).
and does not graphically display results. This paper pres- When well efficiency (E) is defined as E = 100(sf /st)
ents a modified version of the Bradbury-Rothschild itera- and n = 2, then C is related to E by:
tive solution technique that is written in the MATLAB
st E
language and listed in the Appendix. A useful new feature C¼ 12 ð2Þ
includes a three-dimensional graphical display of results Q2 100
so that the user can quickly estimate the relative impor- When sf is given by the Jacob approximation for the
tance of aquifer penetration and well efficiency. Potential Theis solution, then B can be found from (Sternberg
users should be aware that MATLAB must be installed on 1973):
their computers before the program will function. Alter-
nately, users may convert either the original or revised Q 2:25Tt
sf ¼ BQ ¼ ln þ 2s p ð3Þ
code to any convenient programming language (e.g., 4pT rw2 S
C11, Fortran, Excel, or MathCad). However, MATLAB
where T = aquifer transmissivity (L2/T), S = aquifer stor-
is a powerful tool with numerous capabilities that are not
age coefficient (dimensionless), t = time since pumping
readily found in other languages.
began (T), rw = effective wellbore radius (L), and sp =
Recall that total drawdown (st) observed in a pro-
a partial penetration factor (dimensionless).
duction well can be written (Bouwer 1978) as the sum of
In Equation 3, the effect of partial penetration may
drawdown due to formation loss (sf) and drawdown due to
be represented by (Brons and Marting 1961):
well loss (sw), or:
D2L D L
st ¼ sf þ sw ¼ BQ þ CQn ð1Þ sp ¼ ln 2G ð4Þ
L rw D
where D = aquifer thickness (L), L = well screen
1Los Alamos National Laboratory, P.O. Box 1663 MS-K497,
length (L), and G = a function of the L /D ratio (dimen-
Los Alamos, NM 87544; sgm@lanl.gov sionless).
Received August 2003, accepted December 2003 Using available data, Bradbury and Rothschild
Copyright ª 2005 National Ground Water Association. (1985) expressed G as the polynomial G = a 1 b(L /D) 1
Vol. 43, No. 4—GROUND WATER—July–August 2005 (pages 611–614) 611
Table 1
Properties for Well 1 (metric) Were Used in the MATLAB Program to Generate
Figure 1. A Similar Figure Can Be Generated with Well 2 data
Parameter Well 1 (metric) Well 1 (U.S.) Well 2 (metric) Well 2 (U.S.)
Q (lpm or gpm) 37.853 10 37.853 10
st (m or feet) 4.572 15 2.743 9
t (min) 480 480 480 480
L (m or feet) 14.326 47 20.726 68
rw (cm or inch) 7.62 3 7.62 3
S (dimensionless) 0.0002 0.0002 0.0002 0.0002
D (m or feet) 62.484 205 35.052 115
C (min2/m5 or s2/ft5) 3.453 32.7 3.453 32.7
c(L /D)2 1 d(L /D)3, where the fitting coefficients were single best estimates for T and E are also obtained. Using
a = 2.948, b = 27.363, c = 11.447, and d = 24.675. well 1 metric units from Table 1, we find T = 46.6 m2/d at
Substituting Equation 1 into Equation 3 yields: E = 99.9% and L /D = 23% and for well 2, T = 36.2 m2/d at
E = 99.9% and L /D = 59%. Bradbury and Rothschild orig-
Q 2:25Tt inally reported T values of 47.6 and 36.7 m2/d for wells
T¼ ln þ 2s p ð5Þ 1 and 2, respectively. Well efficiencies were determined
4pðst 2sw Þ rw2 S
from Equation 2 using their C value.
Well efficiency is embedded in Equation 5 since sw = One may question the choice of having partial pene-
CQ2, and C is defined by Equation 2. Hence, a step- tration as a variable in Figure 1 since a single value for
drawdown test is not required if E can be estimated. In this parameter should be known from the driller’s log.
addition, the effect of partial penetration is represented by However, we often have difficulty actually deciding
Equation 4 using the Bradbury-Rothschild polynomial for where aquifer boundaries are located. This is especially
G. In Equation 5, T appears on both sides of the equation; true in horizontally stratified aquifers where vertical
hence, an iterative solution is required (Bradbury and changes in hydraulic conductivity may not be obvious. In
Rothschild 1985). Initially, a guess is made for T (Tguess
in the program) on the right-hand side of Equation 5, and
an updated solution for T (Tcalc in the program) is ob-
tained from the left-hand side. This updated solution
is again used on the right-hand side of Equation 5, and
a new T is again computed. This iterative process
continues until some suitable tolerance criterion for error
(Err in the program) is reached. For the MATLAB pro-
gram shown in the Appendix, either metric or customary
U.S. units may be employed.
Transmissivity (m2/day)
Program Usage
The program is executed from the MATLAB com-
mand line by typing in the m-file program name (i.e., [A,
T] = TQs). The user is prompted to select a system of
units and then enter input values for Q, st, t, L, rw, S, D
(optional), and C (optional). Walton (1970) showed that T
is relatively insensitive to variations in S; hence, this
value may be estimated. Tabulated and graphed output
consists of a range of T values that correspond to a range
of expected well efficiencies and aquifer penetration
)
values. The two original examples shown in Bradbury and %
W
ell on(
Rothschild (1985) are used as illustrations. Input data Ef ati
fic etr
ien Pen
for these tests are summarized in Table 1. The MATLAB cy fer
(% ui
program is executed once for each test, and the user is ) Aq
prompted to enter appropriate data from Table 1. Figure 1
is a graphical representation of the tabulated output for
Figure 1. Transmissivity as a function of aquifer penetra-
well 1. Output for well 2 was omitted because it is similar tion and well efficiency for well 1.
to Figure 1. If known values for D and C are entered, then
612 S.G. McLin GROUND WATER 43, no. 4: 611–614
addition, step-drawdown tests that determine C are the Acknowledgments
exception rather than the rule, especially in monitoring This work was supported by the Groundwater Protec-
well applications. This program simply provides a range tion Program at Los Alamos National Laboratory (LANL).
of estimated T values that can assist us in overcoming LANL is operated by the University of California for
these difficulties. As aforementioned, we can narrow the the U.S. Department of Energy under contract W-7405-
range of possible T values to a single best estimate if we ENG-36. Special thanks are extended to David Schafer
know partial penetration and well efficiency. Alternately, for fruitful discussions during this effort. Review com-
we may determine partial penetration from Figure 1 if we ments by Ken Bradbury and Mary Anderson are especially
have independent estimates for T and E. The real value of appreciated. The LANL document number for this com-
this exercise, however, may be the recognition of uncer- puter note is LA-UR-03-6116.
tainty in the estimation process.
Editor’s Note: The use of brand names in peer-reviewed
papers is for identification purposes only and does not
Conclusions constitute endorsement by the authors, their employers, or
Specific capacity data are often used in hydrogeo- the National Ground Water Association.
logical studies to estimate T. The major criticism of this
method is that it assumes a quasi–steady state condition
has been established. This is in contrast to a conventional References
aquifer test where transient s and t values are matched Bouwer, H. 1978. Groundwater Hydrology. New York:
to an appropriate theoretical type-curve. However, the McGraw-Hill.
Bradbury, K.R., and E.R. Rothschild. 1985. A computerized tech-
MATLAB program presented here is really a parameter nique for estimating the hydraulic conductivity of aquifers
sensitivity analysis because it translates specific capacity from specific capacity data. Ground Water 23, no. 2: 240–246.
into a range of T values that reflect the combined influ- Brons, F., and V.E. Marting. 1961. The effect of restricted fluid
ence of the formation, aquifer penetration, and well effi- entry on well productivity. Journal of Petroleum Technol-
ciency. This type of analysis simply gives us another ogy 13, no. 2: 172–174.
Sternberg, Y.M. 1973. Efficiency of partially penetrating wells.
way to determine T. These T estimates can be valuable in Ground Water 11, no. 3: 5–7.
those situations where conventional aquifer tests are Walton, W.C. 1970. Groundwater Resource Evaluation. New
unavailable. York: McGraw-Hill.
Appendix
function [A, T]=TQs
%TQs computes Transmissivity (T) from Specific Capacity (Q/s) data.
%
% This m-file was written in the MATLAB language by:
% Stephen G. McLin, 8 May 2003, e-mail: sgm@lanl.gov
%
% A = a matrix of T values as a function of R and E.
% Note that R is the last row of A and E is the last column of A
% T = transmissivity (sq m/day or sq ft/day).
% Q = well pump rate (lps or gpm).
% s = wellbore drawdown (m or ft).
% t = time (minutes).
% D = aquifer thickness (m or ft).
% L = well screen length (m or ft).
% R = L/D (dimensionless penetration).
% r = wellbore radius (cm or in).
% S = aquifer storage coefficient (or specific yield).
% E = well efficiency (%).
% C = well loss coefficient (min2/m5 or sec2/ft5).
%
format short;
Units=input(‘Enter 1 for metric units and 2 for US units.......');
if Units = =1
Q=input(‘Enter Q (lpm) now.......'); conv=1000;
s=input(‘Enter drawdown (m) now.......');
t=input(‘Enter time (minutes) now.......');
S.G. McLin GROUND WATER 43, no. 4: 611–614 613
Appendix (continued)
L=input(‘Enter well screen length (m) now.......');
r=input(‘Enter wellbore radius (cm) now.......'); r=r/100;
S=input(‘Enter storage coefficient S now.......');
Do=input(‘Enter observed aquifer thickness (m) now (enter 1 if unknown).......');
Co=input(‘Enter step-test C (min2/m5) now (enter 1 if unknown).......');
if Co~=1; Co=Co*3600; end; str='Transmissivity (sq m/day)';
elseif Units = =2
Q=input(‘Enter Q (gpm) now.......'); conv=7.48;
s=input(‘Enter drawdown (ft) now.......');
t=input(‘Enter time (minutes) now.......');
L=input(‘Enter well screen length (ft) now.......');
r=input(‘Enter wellbore radius (in) now.......'); r=r/12;
S=input(‘Enter storage coefficient S now.......');
Do=input(‘Enter observed aquifer thickness (ft) now (enter 1 if unknown).......');
Co=input(‘Enter step-test C (sec2/ft5) now (enter 1 if unknown).......');
str='Transmissivity (sq ft/day)';
else
error('You have entered an incorrect response. Please start again.');
end
E=[50:2:100]'; [n1,m1]=size(E);
R=[0.1:0.05:1.0]'; [n2,m2]=size(R); D=L./R;
A=zeros(n111,n211); err=0.000001; Tguess=1.0;
a=2.948; b=27.363; c=11.447; d=24.675;
C=(12E./100).*(s/Q^2); sw=C.*Q^2;
G=(a1b*(L./D)1c*(L./D).^21d*(L./D).^3);
sp=((D2L)./L.*(log(D./r)2G));
for j=1:n2; for i=1:n1;
Tcalc(i,j)=1440*Q*(log(2.25*Tguess*t/(1440*r^2*S))12*sp(j))/(4*conv*pi*(s2sw(i)));
diff=abs(Tcalc(i,j)2Tguess); test=diff;
while test>err
Tcalc(i,j)=1440*Q*(log(2.25*Tguess*t/(1440*r^2*S))12*sp(j))/(4*conv*pi*(s2sw(i)));
diff=abs(Tcalc(i,j)2Tguess); Tguess=Tcalc(i,j); test=diff;
end; A(i,j)=Tcalc(i,j);
end; end
A(1:n1,(n211))=E; A((n111),1:n2)=100.*R';
z=A(1:n1,1:n2); x=100.*R; y=E; h=figure;
set(h,'PaperPosition',[0.25,0.25,8.00,10.50]);
meshz(x,y,z); zlabel(str);
ylabel('Well Efficiency (%)'); xlabel('Aquifer Penetration (%)');
if Do = =1; T=1; return;
elseif Co = =1; T=1; return;
else
fac=60*60*conv*conv;
Eo=100*(12Co*Q^2/(s*fac)); swo=Co*Q^2/fac;
Go=a1b*(L/Do)1c*(L/Do)^21d*(L/Do)^3;
spo=(Do2L)/L*(log(Do/r)2Go);
Tcalco=1440*Q*(log(2.25*Tguess*t/(1440*r^2*S))12*spo)/(4*conv*pi*(s2swo));
diff=abs(Tcalco2Tguess); test=diff;
while test>err
Tcalco=1440*Q*(log(2.25*Tguess*t/(1440*r^2*S))12*spo)/(4*conv*pi*(s2swo));
diff=abs(Tcalc2Tguess); Tguess=Tcalco; test=diff;
end; T=[Tcalco Eo L*100/Do]; end;
% Tcalco=best single estimate for transmissivity;
% Eo=well efficiency; 100L/Do=aquifer penetration;
614 S.G. McLin GROUND WATER 43, no. 4: 611–614
