An enriched finite element for simulation of groundwater flow to a well or drain — Comment

381

Journal of Hydrology, 60 (1983) 381--382 Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands

Discussion [21 AN ENRICHED FINITE ELEMENT F O R SIMULATION OF G R O U N D W A T E R FLOW TO A WELL OR DRAIN -- COMMENT

MARIAN KEMBLOWSKI

Kansas Geological Survey, Lawrence, KS 66044 (U.S.A.) (Received June 1,1982; accepted for publication July 6, 1982)

In his recent paper, Tharp (1982) gave a solution for simulating flow to a well or drain using finite-element method (FEM). There are two reasons for inaccurate determination of the head distribution in the vicinity of wells b y FEM. First, c o m m o n l y used linear basis functions do n o t properly approximate radial flow. Secondly, even if the higher-order basis functions are used, it is convenient, and therefore often neglected, to fit the mesh to finite diameter of the wells. Tharp has proposed to modify the basis function at the elements surrounding a well in order to include the specific properties of radial flow. The m e t h o d has proven accurate and may be successfully applied. There is another method of solving this problem which seems to be simpler and equally efficient. Instead of using special-basis functions, the value of the transmissivity at the elements surrounding a well is modified, so the numerical simulation gives the same results as the analytical solution. After imposing the special finite-element mesh around the well node the value of the transmissivity in the eight elements surrounding this node is calculated as: T* = c t g a - 7rT ~-

In a -

= 2.414 ~T

In

where a = 2 2 ° 3 0 ' ; a = the distance between the well node and surrounding nodes; T = aquifer transmissivity; and rw = well radius. Then, assuming the head difference between each surrounding node and the well node equals Ah, the flow throughout one element towards the well is given b y (Fig. 1):

bT.

-Qe = Ah l

=

Ah sin~ --ctga cosc~

~rT ( a a ) -~ 7rT ( ~ _ w a ) -1 In = Ah-In -44

0022-1694/83/0000--0000/$03.00 © 1983 Elsevier Scientific Publishing Company

382

C\ / /

WELL

I

Fig. 1. Division of well elements. The discharge Q0, calculated for all surrounding elements, is given by: Qo = ~

Qe = 2Ah~cT

i=l

ln-w

and is equal to the discharge calculated from the analytical solution. The m e t h o d proposed above has been compared with the analytical solutions for confined and unconfined aquifers and the results have shown a very good agreement. Obviously, this m e t h o d is applicable only during pumping periods.

REFERENCE Tharp, T.M., 1982. An enriched finite element for simulation of groundwater flow to a well or drain. J. Hydrol., 55: 237--245.