On Certain Singular Integral Equations Arising in the Analysis of Wellbore Recharge in Anisotropic Formations

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1 On Certain Singular Integral Equations Arising in the Analysis of Wellbore Reharge in Anisotropi Formations C. Atkinson a, E. Sarris b, E. Gravanis b, P. Papanastasiou a Department of Mathematis, Imperial College London, South Kensington Campus, London, UK. b International Water Researh Center (IWRC) NIREAS, University of Cyprus, 678 Niosia, Cyprus. Department of Civil and Environmental Engineering, University of Cyprus, 678 Niosia, Cyprus. Abstrat The problem of determining the steady state pressure field for single and multi-well onfigurations with non-trivial wellbore boundary onditions is onsidered in this work as an integral equation problem. The aquifer, where the well onfiguration resides, is assumed to have different vertial and horizontal intrinsi permeabilities and it is bounded above and below by impermeable geologial settings. The solutions of the integral equation, known as density funtions, are studied from two points of view. First, the singular behavior of the density funtion is investigated by studying the singular part of the kernel of the integral equation; on this basis the density funtion is suitably expressed in terms of a non-singular ounterpart, the redued density funtion, for whih a polynomial approximation is formulated and onstruted numerially. The onvergene of the approximation is studied with respet to the order of the polynomial and shown to be adequately fast. Seond, the density funtions for large depth to radius ratio and/or large horizontal to vertial permeability ratio exhibit similarity. The density funtions depend on the parameters of the singlewell problem through a single similarity parameter. For large values of, orresponding to the physial limits just mentioned, the density funtions are redued to essentially a single funtion, modulo a fator (log ). This property simplifies onsiderably the analysis of all the large ases. Considering the ase of two wells, as an illustrative example, we also show that the properties of the single well ase are also exhibited by the multi-well density funtions. Keywords: Wellbore reharge, singular integral equations, anisotropi formations.. Introdution Groundwater use is of fundamental importane to meet the rapidly expanding urban, industrial and agriultural water requirements partiularly in semi-arid limates. Reently, the water resoures management has also attrated onsiderable onern of sientists and engineers as it ensures availability around the year of groundwater for the various usages in semi-arid regions []. Reharge is defined in general sense as the downward flow of water reahing the water table, forming an addition to the groundwater reservoirs (i.e. aquifers) []. The priniple reharge mehanisms were originally defined by Learner et al. [3] and are as follows: ) diffuse perolation, as either an unsaturated flux or a saturated front (piston type flow), ) maro-pore flow

2 through root hannels, desiation raks and fissures, 3) preferential flow aused by unstable wetting fronts and differentiated soil physial harateristis within the soil. Modelling the reharge proess is thus a prerequisite for effiient and sustainable groundwater resoure management in dry areas. Analytial models are often developed for better modelling of the physial proesses involved in both surfae and wellbore reharge []. In wellbore reharge, analytial models aount for the effet of a single wellbore effiieny and wellbore geometry (radius, length) on the groundwater flow. Furthermore, models are utilized to deal with some groundwater wellbore reharge issues whih inlude: a) the finite well radius, b) wellbore storage, ) well partial penetration and d) the presene of skin [4]. There is a vast literature on alulating pseudo-steady flow (produtivity/injetivity) from wells espeially for petroleum related appliations. Muh of this is onerned with the pressure diffusion equation whih is appliable to laminar flow in porous medium dedued from using the Dary law [5]. To study the flow harateristis various approximations have been made when applying differential equations to problems solving flow to/from wells. Often a well is approximated as a line sink and this is suffiient for many pratial purposes (but, see for example [6]). In these works the omplex physial boundaries indue boundary onditions whih do not admit a straightforward analytial treatment and the respetive problems are formulated as suitable integral equations, whih are solved numerially [7]. In general, integral equation methods possess ertain a priori advantages: the unknown funtion is defined only on the boundary of the domain, omplex physial boundaries are easy to inorporate, the ill-onditioning assoiated with disretizing the governing equations is avoided, high-order auray is easy to attain and far-field boundary onditions are handled naturally. Reduing the problem to an integral equation introdues the onept of a density funtion, the solution of the integral equation, whih physially is interpreted as a point, or a distributed soure that generates the assumed boundary onditions [6]. The density funtions are mathematial objets whih are interesting in their own right. Their determination most usually requires effetive approximate/numerial methods. These methods may be improved onsiderably by exploiting fundamental properties of the density funtions following from the struture of the kernel of the integral equation they obey. Additionally, as the density funtions an always be given in a dimensionless form, studying their dependene on the similarity parameters of the problem may lead to interesting and possibly useful observations, espeially regarding the behavior of the density funtions near the limiting or ritial values of the similarity parameters. In this work we study the density funtions from this point of view onsidering a setting whih is as simple as it an be without being entirely trivial (trivial would be a setting suh that the hydrauli potential along the wellbore does not depend on the vertial oordinates). We model the well as a ylinder of given length and radius oriented along the vertial prinipal axis of the permeability tensor. Thus we keep our geometry simple. We also work under steady state onditions. The boundary onditions imposed on the hydrauli potential along the wellbore boundary are hosen

3 to be Dirihlet and dependent on the vertial position along the boundary. The dependene of the potential, or essentially equivalently of the injetion rate, on the vertial position along the wellbore is not usual in hydrologi reharge. We have borrowed it from petroleum geomehanis where, while fraturing a formation from deep wells, a tehnique alled multistage fraturing is used. With this tehnique, parts of the wellbore are isolated with pakers in order to frature the formation at the desired length of the deep well [8]. Apart from its physial interpretation this boundary ondition serves mostly as a mathematially attrative hoie as, in partiular, we shall assume that the hydrauli potential along the wellbore boundary is a polynomial funtion of the vertial oordinate. The problem is redued to a singlevariable integral equation with the polynomial boundary ondition as the soure term. The kernel of the integral equation possesses a logarithmi singularity. This is also true for a muh more general lass of related problems (see e.g. [7]). For the general theory of singular integral equations see e.g. [9]. In general, the presene of singularities makes the numerial solution of an integral equation very timeonsuming and it is also rather diffiult to ahieve aurate results. There exist effiient methods for dealing with the problem of singular kernels, suh as the hybrid Gauss-trapezoidal quadrature rules []. Nonetheless, the type of the singularity arising in wellbore reharge problems provides the resolution of the diffiulty indued by the very presene of the singularity. The logarithmi kernel leads to soluble integral equations. Thus the singular behavior of the density funtion an be analyzed. We shall show that indeed, after fatoring out a suitable singular funtion assoiated with the logarithmi kernel, what remains of the density funtion is a well behaved non-singular funtion. (Compare with [7]). The method used in order to determine the laimed-to-be non-singular part of the density funtion, whih we all the redued density funtion, is to approximate it with polynomials of inreasing order. One of our aims is to investigate how fast the polynomial approximations of the redued density funtions onverge to a single well-behaved funtion. Clearly, the dual purpose of suh an investigation is both to establish the approximate method as suh and also to answer the theoretial question whether the redued density funtion is indeed non-singular as laimed. We shall show that the sought for onvergene is adequately fast. From the pratial point of view, suh results suggest that, most possibly, if the singular part of the kernel leads to a soluble integral equation then the polynomial approximation an be used as an effetive tool in produing aurate solutions in that partiular problem. In the simple setting used in this work the density funtions depend on the parameters of the problem essentially only through a single dimensionless parameter, defined in equation (4). The similarity parameter is interpreted as the apparent halfdepth of the well in oordinates whih are suitably defined so that the medium will appear isotropi, i.e. defined through suitable resaling of the physial oordinates by the intrinsi permeability. When the well is deep and/or the horizontal permeability of the medium is muh larger than the vertial then is large. The behavior of the density for large is an attrative mathematial question whih is also of physial interest. In fairly usual situations, where also there is an assumed exess of horizontal 3

4 permeability relatively to the vertial, the number may easily be in the viinity of, whih as we shall see qualifies as a large in the following sense. For values of the similarity parameter of that order the density funtions exhibit similarity i.e. they are essentially redued to a single one-variable funtion, modulo a fator (log ). From a pratial point of view the similarity simplifies the analysis of the all the large ases. Thus, one needs only to determine the density funtion for a ertain large. The struture of this paper is as follows: in setion, we onstrut the single well integral equation for the reharge problem and dedue the singular behavior of the density funtion. Setion 3 desribes the formulation of the numerial solution of the integral equation via the polynomial approximation of the density funtion, while in setion 4 we extend the methodology to aount for the well interation in multiwell arrangements. Setion 5, presents the results laimed above for both the single well and the well interation ases.. Formulation of the problem. The single well integral equation We shall treat first the single well problem. Figure shows the graphial representation of the problem under onsideration in Cartesian oordinates. We onsider a vertial ylindrial wellbore of radius r in a homogeneous medium whih is bounded above and below by impermeable geologial settings. We assume anisotropy between the horizontal and vertial diretion. The horizontal diretions X, Y are assumed as isotropi. Let X, Y, Z be the physial oordinates, denoted olletively by X i. The formulation is based on steady state onditions. The equation for the hydrauli potential follows from the Dary veloity q i given by (see e.g. []): kh qi kij, p Z, kij kh X j k v () where k ij is the matrix of intrinsi permeability of the medium, is the dynami visosity of the water, p is the pore pressure and the speifi weight of the water. Everywhere outside the wellbore holds that: qi i.e. kij X X X i i j () We define the isotropi oordinates x, yz, by: 4

5 X Y Z kh x, y, z (3) r r r k v The isotropi oordinates are also dimensionless. The radial variable r x y expresses the distane from the enter of the wellbore in units of the wellbore radius. (Note that often in what follows we will denote the position vetor on the x,y plane by r i.e. r ( x, y )). Therefore in the isotropi oordinates (Eq. 3), the wellbore has unit radius. Also its length is parameterized as z where: d kh r k (4) v where d is the wellbore depth. The parameter is the similarity parameter in this problem. It an also be viewed as the dimensionless half-depth of the well. Note that, the more permeable is the medium in the horizontal diretion the more the well appears longer in the isotropi oordinates. Therefore, the apparent length of the well in the isotropi oordinates expresses the vertial-horizontal anisotropy of the medium. Hydrauli potential = b(z) Confining Layer Z + d/ + d/ X,Y Wellbore Formation reharge zone -d/ -d/ k v k h Aquifer Wellbore axis (Axisymmetri) r Confining Layer Fig. Representation of the single wellbore reharge problem In the isotropi oordinates the hydrauli potential satisfies the boundary value problem with the following set of equations (5a-d): r z (5a),, lim ( rz, ) bz ( ) (5b) r lim ( rz, ) (5) rr (,) r (, r ) (5d) z z 5

6 for some funtion bz ( ) whih we shall take to be a polynomial. This funtion bz ( ) has dimensions of pressure. The medium has been assumed bounded above and below by impermeable layers (Fig. ). This fat is introdued in the problem through the Neumann boundary onditions (5d). We also introdue a boundary ondition at infinity i.e. there is a potentiometri surfae at some adequately large distane R whih is hosen as the referene datum where the elevation is set to zero. The Neumann boundary onditions in the z-diretion impose that there is a solution whih is independent of z. In an integral representation of the solution Φ, this part of the solution needs to be isolated. Expliitly, this part is the vertial mean () r of the field Φ that satisfies the radial boundary onditions. Indeed, by the ondition (5) one finds that () r satisfies the Laplae equation: r () r r r r (6) whih implies () r C Dlogr. The integration onstants C and D are determined through the boundary onditions. By the ondition (5d) one finds: log r () r C, C b() z dz log R (7) The integral representation of the non-trivial (z-dependent) part of the solution is built as follows. We introdue first the Neumann Green s funtion g (, z z ) whih satisfies the boundary onditions and g (, z) g (, z) and the differential equation: z l z l l z l g l( z, z) l gl( z, z) l ( z z) (8) where is the Dira delta funtion. The additive onstant l in the equation subtrats the zero-mode (z-independent solution of (Eq.)) from the definition of the (Neumann) Green s funtion. As a result the Green s funtion satisfies the ondition: g ( z l, z ) dz (9) In the Appendix A we show that the following funtion is a solution of the Laplae equation for r r : Gz (, r; z, r) dlg( zz, ) J( l r r ) () l 6

7 where r and r are the position vetors of two points in the x,y plane i.e. r = (x,y) and r = (x,y ) and J (x) is the zeroth order Bessel funtion. The integral representation of the solution amounts to forming a superposition of suh solutions by introduing a funtion f( z, ) over the boundary of the wellbore, the density funtion, and write the solution as: (, rz) () r G(, rr;, zz) f ( z, ) d dz () where the vetor r (os,sin ) parameterizes the irumferene of the wellbore. The funtion f has the dimensions of pressure and the funtion G is dimensionless. The expression an, though, immediately be simplified beause the problem of a single well possesses ylindrial symmetry and the density funtion does not depend on. The angular integration is merely redued to the alulation of the integral J ( l rr ) d () By the Bessel funtion addition theorem []: m im ( ) ( ) m( ) m( ) m J l r r e J lr J l (3) (where i is the imaginary unit) we have that this integral an be alulated expliitly J( l rr) d J( lr) J( l ) (4) The representation () takes then the form (, rz ) () r dlg (, zz ) J ( lrj ) () l f ( z ) dz (5) l The funtion Φ as defined satisfies the Laplae equation for all r. The boundary ondition for r, (Eq. 5b), leads to an integral equation for the unknown density funtion. This integral equation reads Gzz (, ) f( z) dz bz ( ) C, Gzz (, ) dlj( l) g( zz, ) (6) l where, we introdue a new funtion, Gzz (, ), the kernel of the fundamental integral equation. Solving Eq. (8) under the assoiated boundary onditions (Neumann) we find that gl (, z z) is given by: 7

8 osh[ l ( z)]osh[ l ( z)], z z sinh( l) l gl (, z z) osh[ l ( z)]osh[ l ( z)], z z sinh( l) l (7) This ompletes the expliit formulation of the problem at hand. We shall frequently refer to (6) as the fundamental integral equation.. Parity properties of the kernel The Green s funtion equation implies that: g ( z, z ) g ( z, z ) (8a) l l Thus g ( z, z ) g ( z, z ) (8b) l l Subsequently the kernel of the integral equation satisfies G( z, z ) G( z, z ) (9) This is important beause the parity properties of the funtion f ( z ) are related to the those of bz ( ) C : The even and odd part of the density funtion f ( z ) may generate only the even and odd part of bz ( ) C respetively. The linearity of the integral equation allows one to exploit this property and break the problem up into its even and odd parts as it will be desribed next. Let b ( z) and b ( z) be the even and the odd part of the funtion bz ( ). Then the fundamental integral equation an be rewritten as the pair of integral equations: Gzz (, ) f( z) dz b( z) (a) Gzz (, ) f( z) dz b( z) C (b) where f ( z) and f ( z) are the even and odd parts of the density funtion, respetively. This greatly failitates the numerial solution of the problem and we shall use it expliitly when alulating the unknown density funtions f ( z ) in setion 5. 8

9 .3 The singularity of the kernel and the density funtion The kernel Gzz (, ) defined in equation (6) is logarithmially singular when z and z are lose. Clearly this behavior omes from the large l ontribution to the defining integral of the kernel. Indeed, for large l the Green s funtion gl (, z z ) an be written as g z z 4exp( l)osh[ l( z)]osh[ l( z )], z z l (, )~ 4exp( l ) osh[ l ( z )]osh[ l ( z )], z z () This gives the following form to the kernel 4 4 Gzz (, ) K z z z z K K zz ( zz) zz ( zz) 4 4 K 4 zz 4 z z () where K( x ) is the omplete ellipti integral of the first kind. Its definition and ertain properties of this funtion are given in the Appendix B. Note that this expression respets the Neumann boundary onditions at z. Consider first small separations, zz, in the interior of the interval z, i.e. z and z are not lose to the boundary points of the interval. Then, in the small separations limit, the behavior of the kernel is governed only by the first term in (). Using the formula (B.4) we find zz Gzz (, )~ log (3) 8 The logarithmi kernel gives rise to a standard problem in the theory of integral equations. Differentiating the integral equation we obtain: f( z ) dz b( z) z z (4) where the prime denotes derivative. The solution to (4) depends on the assumed behavior of the solution near the boundaries z. The most singular solution to this equation an be written as [4]: 9

10 b( t) ( ) z z t f z C t dt (5) C is an arbitrary onstant. We have mentioned that b(z) shall be assumed to be a polynomial. One may expliitly show eah monomial z k of b(z) produes polynomials of order k. This is shown in Appendix C. Therefore, then the quantity in the brakets is a polynomial (of the same order). Thus the asymptoti form of the kernel Gzz (, ) for small separations in the interior of the interval zsuggests the following general form for the density funtion: f( z) F( z) z (6) where F( z) is a non-singular funtion (with dimensions of pressure) whih we shall all as the redued density funtion. A non-singular funtion an be approximated by polynomials. A priori, at the level of the exat integral equations (), the relation (6) is a mere hange of variable from f to F. Following the lead oming from the previous analysis we shall look for polynomial approximations of the (exat) redued density funtion F( z ). We turn now to the study the behavior of the kernel for small separations near to the boundary points of the interval z. We must make sure that the singular funtion ( z ) whih we have fatored out by (6) is at least as singular as the atual behavior of the density funtion f ( z ) near the boundary points. Otherwise the redued density funtion F( z ) would be itself singular. Indeed, we argue below that f ( z ) diverges only logarithmially near the boundary points i.e. it exhibits a milder singular behavior. Consider small separations in the viinity of the boundary point z. That is, we take zz and also zz. The behavior of the kernel is governed now by the first term as well as the third term of (). Using formula (B.4) we have zz zz 8 8 Gzz (, )~ log log (7) whih leads to the integral equation f ( z ) ( ) zz zz dz b z (8)

11 in plae of equation (4). Define variables ttby, z t, z t (9) The boundary point z is reahed for vanishingt. Equation (8) beomes ( ) ( ) f z t dt b z t t t (3) where the dependene of z and z on t and t is left understood. This equation an be solved by a formula analogous to equation (5) for the interval [,]. The derivation is given in the Appendix D. The solution is b( s ) f ( z) C s ds t t s (3) Near the boundary point z (i.e. t ) the quantity of importane is the integral s ds t s (3) as the b-dependent part of the integrant beomes onstant near the boundary points for every monomial bz ( ) z k k. This integral, whih is expliitly defined by its Cauhy prinipal value, an be written as t ts t ts t lim ~ log log log ds ds t s s (33) for t near. Therefore the density funtion f is logarithmially singular for any polynomial bz ( ), as promised. An analogous analysis an be given for the boundary point z. Hene the redued density funtion introdued by the relation (6) should be a regular funtion whih tends to vanish as we approah the boundary points z. This behavior seems to be indeed borne out by the numerial results shown below. Finally, the hange of variable z os, failitates alulations in both the exat and asymptoti integral equations. Then, presumably, the denominator in the definition (6) beomes part of the natural integration measure dz z d (34)

12 Introduing the even F ( z) and odd F ( z) part respetively of the redued density funtion F( z ), the fundamental pair of integral equations (a) and (b) now take the form: ( os, os ) ( os ) ( os ) (35a) G F d b ( os, os ) ( os ) ( os ) (35b) G F d b C (setting z os ) also and these two equations, (35a) and (35b), are to be solved for the unknown even and odd parts of the redued density funtion F( z ), i.e. the density funtions F ( z). 3. Formulation of the numerial solution - The polynomial approximation of the density funtion 3. The polynomial approximation It is adequate, as we shall expliitly show, to determine polynomial approximations of the redued density funtion and therefore of its parts F ( z) as: 4 n z z z F ( z) F F F 3 4 Fj j j (36a) 3 5 n z z z z F ( z) F F F Fj j j (36b) That is, one must determine the oeffiients F j, j,,3,, n. This an be done with the following methodology. In order to determine a polynomial expression of F ( z) with n terms, we should hoose n suitable values of the angle : i, i,,3,, n. We shall speifially hoose: i i n (37) Then we have to solve the two deoupled linear systems: n j n j M F b ij j i M F b ij j i (38)

13 where j ij l i i i M dlj ( l) g ( os, os ) os d, b b ( os ) (39a) ( ) j ij ( os, os )os, ( os ) l i i i M dlj l g d b b C (39b) That is, the problem has been redued to the alulation of the elements M ij of the n n matries M. The matries M amount to a disrete version of the kernel of the integral equation in any given polynomial approximation of the redued density funtion F( z ). Note that the matries M depend solely on the similarity parameter of the problem. 3. Charateristi density funtions As disussed earlier in this work, we may write the polynomial bz ( ) as: z z bz ( ) b b b (4) where b, b, b,... are quantities of the same dimension, and they all have the dimensions of pressure. The linearity of the integral equation (6), or more speifially (35), allows us to determine the redued density funtion F( z ) assoiated with bz ( ) by determining the density funtions assoiated with eah monomial z k k. Indeed, if by ( k F ) ( z ) we denote the density funtion assoiated with the monomial z k k then the density () () ( k funtion reads F( z) bf ( z) bf ( z). The funtions F ) ( z ) assoiated with the monomial of order k will be alled harateristi density funtions. The funtions ( k F ) ( z ) are dimensionless. That means that ( k F ) ( z ) depend only on the similarity parameter. The harateristi density funtions are then the main objets one needs to study and in partiular their dependene on the similarity parameter. ( k We shall study the polynomial approximations of the funtions F ) ( z ) through method disussed in setion 3.. We know from the study of the kernel singularity that these polynomial approximations must be of order k or higher. Moreover, the parity of the monomial z k k ( k ditates the parity of F ) ( z ). If k is even or odd then ( k ) F ( z ) is approximated by an even or odd polynomial, F ( z ) or F ( z ) respetively, of degree k or higher. Expliitly, for every monomial z k k with even k, one has to determine n oeffiients F with n greater or equal to k, and every monomial i k with odd k, one has to determine n oeffiients k z equal to k. F with n greater or i 3

14 4. Interation of multiple (distant) wells In this setion we will extend the methodology desribed in previous setions to aount for the wellbore interation. Let us onsider an arrangement of N w vertial wells, of the same radius r for simpliity, whose enters are loated at ( xi, y I), I,..., Nw. We shall work under the assumption that the distane between the wells is muh larger than the radius of the wells. This allows us to introdue an approximation whih is mathematially enoded into two onditions. First, the singlewell boundary ondition (5b) is replaed by the ondition: d b( z ) (4) I for all I,..., Nw. (We assume that all wellbores have the same boundary ondition bz ( ). That an easily be generalized to different boundary onditions but we shall not need it.) The variable I is an angle that parameterizes the irumferene of the wellbore. This ondition expresses the fat that the hydrauli potential hanges very little around the irumferene of the wellbore for distant wells []. The seond ondition is that the N w density funtions f I assoiated with the multi-well arrangement depend only on the vertial position z along the boundary of the wellbore and not on the angular position I. Speifially, as the linearity of the Laplae equation allows us to write the hydrauli potential as a sum of one-well type of expressions, the potential an be written as: N w I r r ( x, y, z) ( x, y) dl g (, ) ( ) l z z J l ( ) I d I f I z dz (4) i.e. there is one term for eah well that involves a density funtion that depends only on z. One should note that the two onditions are strongly related: Under an exat treatment, the density funtions would be two-variable funtions fi( z, I) and the boundary ondition (4) would not suffie to determine the field Φ in spae. The vetor ri ri (os I,sin I ) parameterizes the points of the I-th wellbore irumferene and r I ( x I, y I ) is the position of the enter of the well on the x,y plane. Also we have: ( x, y) (,, ) x y z dz (43) Integrating out the angle variables I by using the identity (3) as before we obtain: 4

15 N w r r l I I I ( x, yz, ) ( xy, ) dlg( zz, ) J( l ) J( l) f( z) dz (44) ( x, y ) satisfies the Laplae equation: x y (45) and the relevant solution is given from: N w ( xy, ) Cmulti-well D J log rrj (46) J At the I-th wellbore, the position vetor r in the argument of the logarithm is set equal to ri ri (os I,sin I ). Therefore, r rj ri r J (os I,sin I ). Implementing the boundary ondition (4) the following identity is useful: d log r(os,sin ) log r (47) unless r in whih ase the result is learly. Then the boundary ondition (4) implies N w C D log R b( z) dz C multi-well J IJ J JI (48) where RIJ ri rj ( xi xj ) ( yi y J ) is the distane between the enters of the wells labeled by I and J. Note that, if the arrangement of the wells is symmetri then all the DJ oeffiients are the same. The boundary ondition at infinity i.e. the ondition (5), whih involves some large radius R, gives N Cmulti-well w DJ log R (49) J These onditions determine ompletely the onstants C multi-well and D I. The solution is now redued to the linear system of N w unknowns as: 5

16 RIJ DI log R DJ log C R N w J JI (5) We may now deal with the non-trivial part of the solution of equation (4) i.e. the part that involves the density funtions. If the wells are infinitely distant then f ( z) f ( z ), whih obeys equation (6). The interation of distant wells I single-well lies in the deviation of fi ( z ) from f single-well ( z ). This means that we an write fi ( z) as: f ( z) f ( z) f ( z ) (5) I single-well I where f I ( z ) is by definition a small quantity. We shall use this deomposition of the density funtions to form perturbation series. The funtions f I ( z) essentially quantify the interation of the distant wells. We insert the previous deomposition of the density funtions into the general expression (44) and impose the boundary ondition (4). In simplifying the result, one needs the single well integral equation (6), the equation (48) deriving from the appliation of the boundary ondition (4) on the part of the hydrauli potential, and the identity: J ( l ri r J (os I,sin I )) di J( lrij ) J( l) (5) whih is merely one more appliation of the identity (4). One eventually finds: Nw J JI Nw J JI dlg ( z, z ) J ( l) f ( z ) dz l l I dlg ( z, z ) J ( l) J ( lr ) f ( z ) dz IJ single-well dlg ( z, z ) J ( l) J ( lr ) f ( z ) dz l IJ J (53) This expression defines a perturbation expansion for f I, with J ( ) lr IJ understood as the perturbation order parameter. The first order result for f I is obtained by dropping the last term in the expression. The higher order orretions are derived by iterations. For the first order result, whih is the result of interest, we obtain: 6

17 Gzz (, ) f( z) dz b() z I I Nw I() (, l ) () ( IJ) single-well( ) J JI b z dlg zz J lj lr f z dz (54) where Gzz (, ) is the kernel (9) of the fundamental integral equation (6). One the single well density funtion f ( ) single-well z is alulated, the funtion b ( z) an also be omputed as well. (Note that only the fator J ( lr IJ ) is atually summed in the expression (54) as no other quantity depends on the well number index J.). Therefore the problem again is redued to the solution of the fundamental integral equation (6) and one may proeed as explained in setion 3. I 5. Numerial solution of the harateristi density funtions F (), F (), F (3), F (4) As disussed in setions. and. the density funtion f ( z) assoiated with the boundary ondition bz ( ) an be derived by determining its even and odd parts, f ( z ), whih are assoiated with the even and odd parts of bz ( ), respetively. The same applies in the redued density funtion F( z ), defined by equation (6). Thus, as disussed in setion 3, given that we are interested in a polynomial bz ( ), we may determine and study separately the density funtions assoiated with eah monomial term of bz ( ). We have alled these funtions as the harateristi density ( k funtions ) ( k F ( z ). The harateristi density funtions F ) ( z ), are assoiated with the k k k-th power term of bz ( ), speifially with the monomial z. We shall restrit out attention to the first four harateristi densities funtions, i.e. those assoiated with the linear, the quadrati, the ubi and the quarti term of bz ( ). One should bear in mind that the possible onstant (zeroth order) term in bz ( ) is assoiated with the z- independent part of the solution (Eqn. 7) and there is no harateristi density funtions assoiated with it. We shall be onerned with two matters. As disussed in setion 3, the ( k funtions F ) ( z ) will be determined by onstruting polynomial approximations ( k given by equations (36a-b). The analysis of setion.3 suggests that F ) ( z ) requires a priori an at least order k polynomial approximation, as suh a polynomial is produed by the singular part of the kernel alone. The first issue of interested is then how fast the approximating polynomials onverge to the exat funtion as the order of the polynomial inreases. It turns out that the odd harateristi funtions onverge rather faster than the even ones. Also, as mentioned in setion 3. the funtions ( k F ) ( z ) depend only on the dimensionless grouping, the similarity fator of the problem defined in equation (4). The seond issue of interest is then the behavior of ( k F ) ( z) as hanges and espeially as inreases to large values. Physially, the large 7

18 values of are related to deep wells and large ratio of the horizontal to the vertial permeability. Both limits are of interest. Regarding this matter we shall argue, based on our numerial results, that the density funtions exhibit similarity. Speifially, the density funtions appear to onverge to a single funtion for large values of one they have been saled with (log ). () () (3) (4) The polynomial approximations of the funtions F, F, F, F are shown in Figures -5 for four different values of the similarity parameter (i.e.,5,,5 ). The approximating polynomials are onveniently plotted as funtions of z. In the odd ases, i.e. in the ase of the funtions F () and F (3), the number n labeling the urves in the plots is the number of terms in the approximating polynomial whose order is n (see Eqn. 36a). In the even ases, F () and F (4), the approximating polynomial are of order is n (see Eqn. 36b), where n is again the number labeling the urves in the plot. Therefore in Figures and 4 the approximating polynomials of () (3) the funtions F and F up to order shown, while in Figures 3 and 5 are shown () (4) the approximating polynomials of the funtions F and F up to order 6. Figures and 4 show that the approximating polynomials of the odd density funtions onverge fairly fast and in a monotoni manner to the exat funtion. On the other hand, Figures 3 and 5 show that the approximating polynomials of the even density funtions onverge less fast to the exat funtion through diminishing up and downs, reating the relatively fuzzy piture shown in Figures 3 and 5. Atually, the urves orresponding to the ases n 7,8,9 are hard to distinguish visually. Additionally, the onvergene in the even ases is somewhat subtle. The basi property of the Green s funtion gl (, z z) expressed by equation (9) is inherited by the kernel Gzz (, ). Through the fundamental equation (6) this property implies that the density funtions are defined up to an arbitrary additive onstant. Replaing f ( z ) with f( z ) onstant leaves the integral equation (6) invariant. This fat is another illustration of the assoiation of the density funtions solely with the z-dependent part of the hydrauli potential Φ. Clearly, this observation is important only for the even part of the density funtion. Realling equation (6), this arbitrariness translates at the level of the redued density funtion F( z ) as the replaement F( z) F( z) onstant z (55) whih leaves the integral equation (6) invariant and produes a ompletely equivalent redued density funtion. The onvergene shown in Figures 3 and 5 is ahieved only after subtrating from eah even polynomial F ( z ) the quantity F () z, as a normalizing proedure that produes an equivalent density funtion. Finally, one may observe that the urves in the 5 graph in every one of the Figures -5 are muh loser to one another than the urves for lower 8

19 , suggesting a faster onvergene of the polynomial approximation when the similarity parameter is large. =.5 = 5.3 F () n= n= n= n= n=3 n=3 n=4 n=4 n=5 n=5 n=6 n= z/ z/. = = 5 F ().5 F () F () n= n= n=3 n=4 n=5 n=6 -. z/ n= n= n=3 n=4 n=5 n=6 -.5 z/ Fig. Polynomial approximations of the harateristi density funtions for the linear term of b(z). =.8 = 5.35 F () n= n=3 F () n= n=3 n=4 n=4 n=5 n=5 n=6 n=6 n=7 n=7 n=8 n=8 n=9 n=9 z/ = = 5. z/.5 F () n= n=3 F () n= n=3 n=4 n=4 n=5 n=5 n=6 n=6 n=7 n=7 n=8 n=8 n=9 n=9 z/ Fig. 3 Polynomial approximations of the harateristi density funtion for the quadrati term of b(z) z/ 9

20 =. = 5.4 F (3) F (3) n= n= n= n= n=3 n=3 n=4 n=4 n=5 n=5 -. n=6 -.4 n=6 z/ z/ = = 5.5. F (3) n= n= n= n= n=3 n=3 n=4 n=4 n=5 n=5 n=6 n= z/ z/ Fig. 4 Polynomial approximations of the harateristi density funtions for the ubi term of b(z) F (3) = = 5.3 n= n=.7 n=3 n=3 n=4 n=4 n=5 n=5 n=6 n=6 F (4) n=7 n=8 n=9 F (4) n=7 n=8 n= z/ z/ = = 5.35 n= n=.6 n=3 n=3 n=4 n=4 n=5 n=5 n=6 n=6 n=7 n=7 F (4) n=8 n=9 F (4) n=8 n=9 -.5 z/ -. Fig. 5 Polynomial approximations of the harateristi density funtions for the quarti term of b(z) z/ Next we turn to the seond issue of interest i.e. the behavior of the density funtions under the variation of the similarity parameter and espeially their behavior for large values of. For this study we use the highest order polynomial approximation derived previously in eah ase. That is, we use the n 6 polynomials

21 () (3) in the ase of the odd density funtions F, F and the n 9 polynomials in the () (4) ase of the even density funtions F, F. The results are depited in Figures 6 and 7. The similarity parameter is hosen to vary from 5 to. The Figures 6A, 6C, 7A and 7C show the indiated density funtion multiplied by log. In Figures 6B, 6D, 7B and 7D the peak of the saled density funtions is used as a onvenient indiator of onvergene. The onvergene of the saled density funtions to a single funtion, as shown in Figures 6A, 6C, 7A and 7C, an be desribed as exellent arising in a lean, monotonous manner with inreasing. Perhaps more desriptively, the suggested onvergene is also exhibited through the variation of the peak of the saled density funtions with, shown in the Figures 6B, 6D, 7B and 7D. For the ase of the funtion F () the onvergene is essentially self-evident. For the higher k density funtions one observes that the rate of onvergene slows down nonetheless the onvergene is still rather evident. The suggested, or onjetured, onvergene, whih is observed numerially, is perhaps the most interesting result of this work. Speifially, if F (k) (z) is a ( k ) harateristi density funtion then log F ( u ) approahes a well-defined funtion of the variable u for infinitely large. (A) (B).5.45 F () =5 = = =3 =4 =5 =6 =7 =8 =9 = F () peak val ues.3.5 = -.5 = (C).5 z/ (D) F (3) =5 = = =3 =4 =5 =6 =7 =8 =9 = F (3) peak val ues = = -.5 z/ Fig. 6 Large onvergene of saled harateristi density funtion log F (odd ases)

22 (A).6 (B).6 F () =5 = = =3 =4 =5 =6 =7 =8 =9 = = F () peak val ues = (C) z/.6 (D) F (4) =5 = = =3 =4 =5 =6 =7 =8 =9 = = F (4) peak val ues = -. z/ Fig. 7 Large onvergene of saled harateristi density funtion log F (even ases) 6. Two well interation Consider the ase of multiple wells on eah of whih we apply the non-trivial injetion onditions bz ( ). For simpliity we assume that the wells are distant i.e. the distane between them is muh larger than their radius. This is realistially quite meaningful. We will try to answer the following question. Do the density funtions of the multi-well onfigurations share the properties found in the single well density funtions? We are interested partiularly in the property of similarity. The formulas required to study this question were derived in setion 4. The density funtion of the well labeled by I an be written as f ( z) f ( z) f ( z ). I single-well f ( ) single-well z is the isolated (single) well density funtion assoiated with the injetion onditions bz ( ) i.e., the funtion we onstruted and studied in the previous setion 5; f I ( z ) is the perturbation of the single well density funtion due to the presene of the other wells. The question posed above onerns essentially the perturbation f I ( z ), whih is determined through equation (54). For definiteness we shall use the example of a onfiguration onsisting of two wells of the same radius and length. Equation (54) suggests that, due to the symmetry of the onfiguration, the perturbations of the density funtions assoiated with the two wells are idential, f( z) f( z ). We shall denote both these perturbations by a single symbol, f ( z ). Let us re-write equation (54) expliitly for this onfiguration: I Gzz (, ) f( z) dz bz () bz () dlg (, zz) J() lj( lr) f ( z ) dz l single-well (49)

23 where R denotes the distane, or spaing, between the two wells. The perturbation f ( z ) depends solely on the similarity parameter and the spaing R. Realling equation (3), one should bear in mind that R is dimensionless, expressing the distane between the wells relatively to the well radius r. We study the large behavior of the perturbation f ( z ) for four different values of spaing (i.e. R 3,4,5,6) for ranging from 5 to as in the single well examples. We onsider the perturbation to the harateristi density () () funtions F and F i.e. those assoiated with the linear and the quadrati term in the boundary ondition bz ( ). Figures 8A-D show the saled perturbation, i.e. the () quantity log F ( z ), as a funtion of z for the four different values of the spaing R respetively. Figures A-D present the saled perturbation () log ( ( )) F z as a funtion of z for the same values of the spaing R. As in the single well ase, for large one observes a lean and monotoni onvergene of the saled density funtions to a single funtion. (Presumably, one should observe that the perturbation has always a different sign than the single well density funtion i.e. it ontributes always negatively in the full density funtion. Indeed, in the ase of () F the perturbation is positive for the negative z while the single well funtion is () positive for the positive z, while in the ase of F the perturbation has simply the opposite sign. This means that the two-well onfiguration produes a lower hydrauli potential than the potential produed by algebraially adding up the ontributions of two isolated wells, in the same positions). Also, following our pratie in the single well ase, the peak of the saled density funtions is plotted as a funtion of as a more quantitative measure of the onjetured similarity. In Figures 9A-D the peak of () log F ( z ) for the four different values of spaing we onsider, R 3,4,5,6, is () plotted against. In Figures A-D the peak of log F ( z ) is plotted against for the same values of well spaing. Here, a slower onvergene is observed for large ompared to the single well density funtions, nonetheless the onvergene an be () regarded as evident. In fat, the saled perturbation log F ( z ) appears to () onverge faster than the saled perturbation log F ( z ), as opposed to the way the assoiated single well density funtions behave, as seen by omparing Figures 9 and with Figures 6B and 6D. Finally, an obvious similarity is observed in the Figures 9 and for different values of spaing R. Up to an R-dependent fator, analogous to (log ), the density funtions appear entirely similar. The dependene, though, of this fator on R does not seem to as simple and elegant as it is in the ase of the similarity parameter. 3

24 (A) (B) =5 =5 = = = = F () =3 =4 =5 =6 F () =3 =4 =5 =6 =7 =7 =8 =8 =9 =9 = = = = - -4 z/ = -4-5 z/ = (C) 7-6 (D).5-6 F () -7-6 z/ =5 = = =3 =4 =5 =6 =7 =8 =9 = = = F () z/ =5 = = =3 =4 =5 =6 =7 =8 =9 = = = Fig. 8 The saled perturbations log δf () for well spaing R equal to (A) 3, (B) 4, (C) 5, (D) 6 (A) (B) F () peak val ues F () peak val ues (C) (D) F () peak val ues F () peak val ues Fig. 9 The peak of the saled perturbations log δf () plotted against the similarity parameter for well spaing R equal to (A) 3, (B) 4, (C) 5, (D) 6 4

25 (A) 3-6 (B).3-7 =5 =5 = = = = - F () =3 =4 =5 =6 () - F =3 =4 =5 =6 =7 =7 =8 =8 =9 =9 = = = = = = (C) 5-9 z/ (D) z/. - () - F =5 = = =3 =4 =5 =6 =7 =8 =9 = = = () - F =5 = = =3 =4 =5 =6 =7 =8 =9 = = = z/ z/ Fig. The saled perturbations log (-δf () ) for well spaing R equal to (A) 3, (B) 4, (C) 5, (D) 6 (A) (B) F () peak val ues F () peak val ues (C) (D) F () peak val ues F () peak val ues Fig. The peak of the saled perturbations log (-δf () ) plotted against the similarity parameter for well spaing R equal to (A) 3, (B) 4, (C) 5, (D) 6 7. Comparison of the proposed approah with a Fourier analytial solution We present here an example of omparison between the results of the present work and the related exat solution of the given ase. An exat solution is possible due to the simpliity of the problem we have hosen, whih failitates both the presentation of the method as well as its straightforward validation in hekable examples. We 5

26 shall onsider the ase of the linear term z/ of the funtion b(z) as explained in setion 3.. The assoiated exat solution is given in the Appendix E as a Fourier series. The solution through the present method is assoiated with the harateristi funtion F () (z), determined as explained in setion 3 and the results presented in setion 5. The harateristi density funtion is turned to a density funtion through equation (6) and to a hydrauli by equation (5). We onsider the ase = and use the highest polynomial approximation of F () (z) we onstruted and presented and setion 5 (orresponding to n =6, i.e. a polynomial of order 6 = ), in order to ahieve the highest preision results aording to the method. We alulate the hydrauli potential through our method at the following points i r j, z os 3 (5) where j =,,,5, that is, for the five different ase of distane r from the wellbore enter as multiple of the wellbore half-length, and i =,,,3, that is, for thirty points in the vertial diretion in the upper half of the wellbore (<z<), as the lower in the lower half is obtained by symmetry (b(z) = z/ and therefore the solution is an odd funtion of z). The results are given in Figure. We plot the funtion (, rz) () r, that is hydrauli potential minus the z-independent funtion () r given by in eq.(7). The ontinuous lines are the graphs of Fourier series exat solution while the irles orrespond to the results of the present numerial method. The agreement is indeed exellent. hydrauli potential r = r = r = 3 r = 4 r = z/ Fig. Plot of the funtion (, rz) () r versus z/ for five different distanes from the wellbore enter 8. Disussion This paper studies the solutions of integral equations, known as the density funtions, arising in single and multi-well onfigurations in steady state onditions. The hydrauli potential is assumed to be a polynomial funtion of the vertial oordinate along the boundary of the wellbore. The aquifer is assumed to have different vertial and horizontal intrinsi permeabilities and it is bounded above and below by impermeable planes. 6

27 The kernel of the integral equation for the single-well problem, equation (6), is singular. This property largely shapes the density funtions whih beome singular themselves at top and the bottom of the wellbore. The singular behavior an be identified by studying the singularity of the kernel and the integral equation assoiated with it. At this point it is not atually lear whether there are also subdominant singularities in the density funtion whih annot be unraveled by the presented analysis. On the other hand the analysis suggests that the density funtion an be fatorized into a part whih is adequately singular so that the other part, whih we have alled as the redued density funtion, is, most possibly, non-singular. This is investigated by studying polynomial approximations of the redued density funtions, whih are onstruted numerially. The approximating polynomials appear to onverge rather fast to a single non-singular funtion as the order of the polynomial inreases. Also, in full onsisteny with the singularity analysis, the approximating polynomials exhibit a tendeny to vanish near the top and the bottom of the wellbore, espeially as the order of the polynomial inreases. At the same time this investigation provides a test for the effiieny of the polynomial approximation as a pratial method for determining the density funtion, and therefore the pressure field, with adequate auray. The dependene of the density funtions on the parameters of the problem is realized through the similarity parameter, defined in equation (4). It an be interpreted as the apparent half-length of the well in isotropi oordinates, defined in equation (3). Large means large medium thikness to well-radius ratio and/or large horizontal to vertial permeability ratio. Large values of the number may arise quite easily in atual situations; hene this limit is of physial interest, apart from the obvious mathematial one. It is found that the density funtions exhibit similarity for large : in that limit the density funtions appear to onverge to a single funtion of z/, where z is the isotropi vertial oordinate, modulo a fator (log ). From a pratial point of view this property simplifies the analysis of the large ases. Based on this observation, one may onlude that similar simplifying properties may arise, in the assoiated limits, in muh more ompliated well reharge problems whih one may enounter in pratie. Regarding multi-well onfigurations, we use the symmetri two-well onfiguration of distant wells as an illustrative example. We show that the mentioned similarity, whih originated in the single-well problem, survives also in the problem of multiple distant wells. 7

28 Appendix A The zeroth order Bessel funtion J ( ) x satisfies the equation ( ) x x x x J x (A.) The (Neumann) Green s funtion satisfies equation (8). Multiplied by a fator this equation reads l e z l l l l e g (, l z z) l e gl(, z z) le ( zz) e l (A.) for some small number positive number ε. The variable r and z are the radial an regarded as the vertial oordinates, respetively, of a ylindrial system r, φ, z. It is straightforward to show that the funtion g (, z z) J( lr ) satisfies the equation l l l l l r { (, e gl z z) J( lr)} le J( lr) ( zz) e J( lr) (A.3) r r r z everywhere outside r =. The operator in the square brakets on the left hand side is the Laplaian operator in ylindrial oordinates. It also easy to verify that lim l ( ) lim dle l J lr. (A.4) 3 ( r ) Integrating (A.3) with respet to l from zero to infinity we see that the right hand side vanishes identially, therefore the funtion lim l dle g ( z, z ) J ( lr ) (A.5) l satisfies identially the Laplae equation outside r =. Shifting the origin of the x,y plane by an arbitrary vetor r hanges nothing but the form of the distane from the origin of the plane to an arbitrary point: r rr. As a result, the funtion l l Gz (,; r z, r ) lim dle g (, zz ) J ( l r r ) (A.6) 8