Poceedings of the 006 IASME/SEAS Int. Conf. on ate Resouces, Hydaulics & Hydology, Chalkida, Geece, May -3, 006 (pp7-) Analytical Solutions fo Confined Aquifes with non constant Pumping using Compute Algeba JUAN OSPINA, NICOLAS GUARIN and MARIO VELEZ. Logic and Computation Goup Depatment of Physical Engineeing School of Sciences and Humanities EAFIT Univesity Medellín COLOMBIA Abstact: - Using Compute Algeba Softwae (CAS) cetain wok about symbolic computational hydogeology is ealized. The explicit solutions fo the bounday value poblems that coespond to a confined aquife with vaious foms of non constant pumping ates, ae deived. Many diffeent cases ae consideed and the coesponding solutions ae given in tems of hittake functions which ae genealizations of the usual Theis well function. The method of solution is the Laplace Tansfom Technique. This method is implemented at fully using only CAS. The solutions ae obtained by means of cetain algoithm. As a esult we obtain the explicit foms of the dawdown pofiles fo the vaious pumping egimes that wee consideed. and also we do some execises with ou solutions applying the notion of image well. Fom ou esults is possible to deive new potocols fo estimating aquife popeties. Key-ods: - Symbolic Computational Hydology, Confined Aquifes, Pumping Tests, Laplace Tansfom. hittake functions, Compute Algeba.. Intoduction Mathematical Hydology is a souce of vey inteesting computational poblems at such way that is possible to speak about a new emegent scientific discipline named Computational Hydology (CH). The CH have two faces. The fist face is the Numeic Computational Hydology (NCH) which efes to hydologic poblems that ae solved using numeical analysis and softwae fo numeical computations (Matlab). The second face is named Symbolic Computational Hydology (SCH) and consists in to obtain analytical solutions fo hydologic poblems using mathematical analysis implemented by Compute Algeba Softwae (CAS)(Mathematica, Maple) [,]. The NCH is actually intensively studied but the SCH is a domain that emains pactically unexploed. The object of the pesent wok is to exploe the lands of the SCH. e intend to pesent, some examples of poblems on SCH. Such poblems ae concened with the computation of the analytical solutions fo the case of a confined aquife with a well which is woking accoding with a pumping ate vaiable on time. e obtain some genealizations of the well known Theis solution [3]. Such genealizations ae given in tems of hittake functions [4]. The mathematical method that is used in this wok is the Laplace Tansfom Technique fully implemented using CAS (Maple). It is veified that Maple is a vey poweful tool fo the development of the SCH. The Mathematical Poblem e conside a confined aquife with a well pumping at a ate that is changing with the time accoding to an abitay function. This configuation is a genealization of the configuation oiginally studied and solved by Theis fo the case of constant pumping ate [3]. e assume as valid the same assumptions oiginally intoduced by Theis, with the only diffeence that hee the pumping ate is an abitay function of time. The mathematical model that was used by Theis is a diffusion equation with a sink of a fom of Diac delta function, that epesents a well of infinitesimal adius. Such diffusion equation with Diac tem is complemented with adequate initial conditions without othe bounday conditions. But the Diac tem can be extacted fom the equation and can be conveted in a new bounday condition and such way that the mathematical model to conside hee is the following poblem [5,6] geneal bounday value
Poceedings of the 006 IASME/SEAS Int. Conf. on ate Resouces, Hydaulics & Hydology, Chalkida, Geece, May -3, 006 (pp7-) k + S ( t, ) S ( t, ) S ( t, ) t 0 0) 0 lim t ) 0 () () (3) lim π k 0 Q( t ) (4) whee k is the hydaulic diffusivity, S(, is the obseved dawdown at a distance fom the pumping well, k is the aquife tansmissivity and Q( gives the vaiation on time of the pumping ate. The equation () is the diffusion equation fo the dawdown, the equation () epesents the initial condition of a non petubed aquife, the equation (3) gives the natual condition at infinite and the equation (4) is the condition fo a extacting well of vey small adius with a pumping ate Q(. e conside hee a geneal fom fo Q(, namely n 0 Q n t n Q( t ) (5) whee Q n ae constants. Some concete examples that we want to exploe hee ae: Q (6) Qt (7) Qt (8) Qt n (9) Qe ( ) (0) Qt e ( ) () Qt m e ( ) () The case of (6) is the oiginal poblem consideed by Theis [3]. The case (0) with ε i w, was studied in [5]. Then the mathematical poblem that is poposed and solved in this wok consist in to obtain the analytical solution of the equations ()-(4) with the specifications (5)-(). The Theis solution is edeived and altenative fomulas espect to that ae given at [5,6], ae obtained fo the case of sinusoidal pumping ate. 3 Poblem Solution The bounday value poblem ()-(4) with the specifications (5)-() is a linea poblem with computable analytical solution [6]. But such solution can not be obtained by means of the method of sepaation of vaiables. It is necessay to apply the Laplace Tansfom method. Since the necessay manipulations to solve the equations ()-(4) with (5)- () ae too voluminous as fo to be ealized by hand with pencil and pape, it is necessay to apply some type of system of compute algeba that allows symbolic computation [,]. In this wok the method of Laplace tansfom was implemented at fully using only CAS with an appopiate algoithm. A vey useful mathematical tools fo this wok was the theoy of hittake functions [4]. In the following subsection ou method of solution is descibed moe detailed. 3.. Method of Solution A sketch of the algoithm that we have used to solve ()-(4) fo all specifications (5)-() is as follows. The inputs of the algoithm ae: Eq, that epesents the equation (); I.C. that epesents the initial conditions () and B.C. that epesents the bounday conditions (3)-(4). The output fo the algoithm is the explicit pofile fo the dawdown it is to say the analytical fom of the function S(,. Ou algoithm opeates at the following way: The inputs Eq., I.C., and B.C, by means of a Laplace Tansfome ae tuned into a tansfomed equation denoted T.Eq. and a tansfomed bounday condition denoted T.B.C. Then, T.Eq and T.B.C. ae pocessed by a cetain Dsolve that geneates the tansfomed solution denoted Tsol. Next, Tsol is pocessed by means of an invese, and we obtain the explicit fom of the solution, denoted sol. e implement such algoithm using Maple []. Thee ae thee ways to do that: using a woksheet, using a pocedue and using a package. In this wok we use a pocedue which is constucted with Maple. Such pocedue is the following: > estat: (B) >Hydo:poc(Q):(B) >with(inttans):(b3) > eq:diff(s(,,- k/*(diff(s(,,)+*dif f(s(,,`$`(,)))0: B4
Poceedings of the 006 IASME/SEAS Int. Conf. on ate Resouces, Hydaulics & Hydology, Chalkida, Geece, May -3, 006 (pp7-) >eqp:subs({laplace(s(,,t,s)v(),s(,0)0},laplac e(eq,t,s)): (B5) >assume(s>0,k>0): >assume(m::positive):(b6) >sol:dsolve(eqp,v()):b7 >aux:_csolve(limit(*pi *k[]**diff(hs(sol),), 0)subs({laplace(t^n,t,s)n!/s^(n+)},laplace(Q,t,s)), _C): (B8) >sol:subs({_c0,aux},so l): (B9) >S(,invlaplace(hs(sol ),s,; (B0) > end poc: B() This Maple pocedue, is named Hydo as we can see in the block (B). The block (B3) loads the Maple package fo integal tansfoms. In the block (B4) the equation () is intoduced inside the Maple envionment. In the block (B5) the laplace tansfomation of the equation () with () is made. The block (B6) intoduces the necessay assumptions fo computations and the block (B7) is the Dsolve of the tansfomed equation. The block (B8) applies the bounday condition (4) and the block (B9) intoduces the bounday condition (3). Finally the block (B0), makes the invese laplace tansfomation and then the analytical fom of S(, is obtained. The block (B) is the contol command fo the finalization of the pocedue. Ou pocedue Hydo, can be visualized as a black box that tansfom cetain pumping ate vaiable on time Q(, in to the analytical fom of S(,. Fo the applications, being peviously loaded the pocedue Hydo, the activation command is: > Hydo(Q(); 3.. Results of Computations Hee we pesent the esults of computations fo pumping ates coesponding to the equations (5)- (). Explicitly, fo Q( given by (6), we use the command > Hydo(Q); and the esult is Q Ei 4 kt 4 π k (3) whee Ei(x) is the Exponential integal function [7]. This is justly the oiginal Theis solution fo a constant pumping ate of a well within a confined aquife [3]. Now fo Q( given by (7), we apply the command > Hydo(Q*; and the answe is S ( t, ) Q kt ( / ) 3 8 kt e -3,, 0 4 kt π k (4) whee (u,v,x) is the hittake function with paametes u and v and with agument x [4]. hen the pumping ate ae changing on time as (8), we use the command > Hydo(Q*t^); and the esult is Q kt ( / ) 5 8 kt e -5,, 0 4 kt π k (5) whee the hittake functions appea again. Moe geneally fo the pumping ate given by (9) the command is > Hydo(Q*t^n); and the esult is S ( t, ) Q n! kt ( n + / ) 8 kt e,, n 0 4 kt π k (6) Now fo the geneal Q( with the fom (5) we use the command >Hydo(Sum(Q[n]*t^n,n0..infinity); and the esult is n + / 8kt S ( t, ) n 0 e Q n n! kt ( ),, n 0 4kt πk (7) Similaly fo the pumping ate of the fom (0) we use the command
Poceedings of the 006 IASME/SEAS Int. Conf. on ate Resouces, Hydaulics & Hydology, Chalkida, Geece, May -3, 006 (pp7-) >Hydo(Sum(Q*(epsilon*^n/n!,n0..infinity)); and the esult is S ( t, ) Qε n kt ( n + / ) n 0 8kt e πk,, n 0 4kt (8) Fo the pumping ate of the fom () we use the expansion Qte ( ε Q ( ε t ) ( ) ( n )! n n t (9) and then we apply the command >simplify(hydo(sum(q*(epsilon*^(n -)*t/(n-)!,n..infinity))); fo hence the esult is 8kt e Qε ( n ) n kt ( n + / ),, n 0 4kt n S, ( πk (0) Finally fo the moe geneal pumping ate that is consideed hee, given by (); we use the expansion Qt m e ( ε Q ( ε t ) ( n m) t m ( n m)! n m () and we can use the following command > Hydo(Sum(Q*(epsilon)^(nm)*t^n/(n-m)!,nm..infinity)); which poduces the esult S, ( Qε ( n m )! n m n kt ( n + / ) 8kt e ( n m)! πk,, n 0 4kt () 3.3. Analysis of Results It is clea that () contains as paticula cases all the esults (3)-(0). In paticula, the Theis solution can be obtained fom () with m0 and ε0. Fom the othe side the case m0 and εiw, was consideed in [5] and the esult was established in tems of Bessel functions [7]. Hee we have an altenative fomula in tems of hittake functions [4]. Now we want to show the effective expansions fo the solutions (3)-() when t. The esults ae displayed at Table. In this table the equations (3)- (30) ae espectively the appoximations of the exact equations (3)-(8), (0) and (). In the expansions of the Table, γ is the Eule constant [7], namely, γ 0.577 and Ψ(x) is the Digamma function [8]. As the eade can note, the leading tem of (3) is obtained fom (6) with n0 and with Ψ()-γ. Also, the leading tem of (3) is obtained fom (30) with m0, ε 0 and again with Ψ()-γ. As we can obseve fo all expansions (3)-(30), the leading tem is logaithmical espect to the distance and the time t. 3.3. Applications of Results All the esults that wee deived, the equations (3)- () ae genealizations of the oiginal Theis equation and hence these equations ae valid only fo infinite and confined aquifes. In the case of semi-infinite aquifes with boundaies it is possible to use the method of images. Fo example in the case of an aquife bounded by a ive and with a well with pumping ate given by (), the total dawdown is the sum of the dawdown that is poduced by the pumping well and the dawdown coesponding to the image well. The total dawdown fo this configuation is showed at Figue. The esult is pesented inside a Maple envionment using the pocedue Hydo(Q) and the esults () and (30). In the Figue, is the distance fom the obsevation point to the pumping well and is the distance fom the obsevation point to the image well. e obseve in Figue that the total dawdown S(,, is quasi-stationay when t. Anothe possible application of ou esults (4)-() is concened with the elaboation of a potocol fo the expeimental detemination of the geo-hydaulic popeties of aquifes using pumping tests [5]. This issue can be the object of a possible futue wok. Anothe line of futue eseach is the extension of the esults that wee obtained hee fo the case of semi-confined o leaky aquifes [9]. 4 Conclusions This wok was an intend to bing the eade some of the taste of Symbolic Computational Hydogeology. The example that was chosen coesponds to the case of a confined aquife with a non constant pumping ate. The authos believe that some of the esults that wee pesented hee ae new in Mathematical Hydo-geology. Such esults can be consideed as genealizations of the Theis fomula. Ou esults can have pactical applications in
Poceedings of the 006 IASME/SEAS Int. Conf. on ate Resouces, Hydaulics & Hydology, Chalkida, Geece, May -3, 006 (pp7-) hydogeology, paticulaly in the field of new potocols fo chaacteization of aquifes. e confim the high impotance of Compute Algeba Softwae, paticulaly Maple, inside the domain of Mathematical Hydology. Using Maple all the esults that wee pesented, ae easily obtained. Fom the wok that was done we can note the geat elevance of the theoy of hittake functions fo the Mathematical Hydogeology. Finally we conclude that the new emegent discipline, named, Symbolic Computational Hydo-geology has a vey pominent futue. Refeences: [] en.wikipedia.og/wiki/compute_algeba_system [] www.maplesoft.com [3] Theis CV, The elation between the loweing of the piezometic suface and the ate and duation of dischage of a well using gound-wate stoage. Am Geophys Un Tans (935) 6: 59-54. [4] mathwold.wolfam.com/hittakefunction.html [5] Todd C. Rasmussen, Kevin G. Haboak and Michael H. Young, Estimating aquife hydaulic popeties using sinusoidal pumping at the Savannah Rive site, South Caolina, USA. Hydogeology Jounal (003) :466-48. Spinge- Velag 003. [6] Buggeman GA, Analytical solutions of geohydological poblems. Elsevie, New Yok, 999. [7] Bowman F., Intoduction to Bessel functions, Dove Publications Inc., 958 [8] mathwold.wolfam.com/digammafunction.html [9] Hantush MS, Jacob CE, Non-steady adial flow in an infinite leaky aquife. Am Geophys Un Tans (955) 36 (): 95-00.. S ( t, ) Table. Expansions fo the solutions (3)-() Q γ + ln 4 kt Q Q 4 π k 4 π k 4 kt Q 6 π k 4 kt 3 + + 7 π k 4 kt + O 4 kt 4 Qt t ) ln 4 kt γ + O 4 π k Qt t ) ln 4 kt 3 γ 4 + O π k 4 kt 6 ( 3 / ) 4 kt 6 Qt n ln 4 kt Ψ ( + n) γ + O 4 π k ( 3 / ) 4 kt 6 ( 3 / ) (3) (4) (5) (6)
Poceedings of the 006 IASME/SEAS Int. Conf. on ate Resouces, Hydaulics & Hydology, Chalkida, Geece, May -3, 006 (pp7-) n 0 n 0 n n m Q n t n ln 4 k t Ψ ( + n) γ 4 π k (7) ε n t n Q ln 4 k t Ψ ( + n) γ 4 n! π k (8) 4 4 n t n Q ln 4 k t Ψ ( + n) γ ( n )! ε ( ) n m t n ln 4 k t Ψ ( + n) γ ( n m)! Q ε ( ) (9) (30) Pumping ell: Using () with Q Image ell: Using () with -Q Using (30) Simplification Quasi-stationay flow Figue. Total dawdown fo a semi-infinite confined aquife bounded by a ive and with a well with pumping ate of the fom (). The method of image well is applied.