Flow to a Well in a Two-Aquifer System

Transcription

1 Flow to a Well in a Two-Aquifer System Bruce Hunt 1 and David Scott Abstract: An approximate solution for flow to a well in an aquifer overlain by both an aquitard and a second aquifer containing a free surface is obtained from numerical inversions of exact analytical solutions for Laplace transforms. This study extends previous work on this topic in three ways: 1 it improves an eigenvalue decomposition solution method used earlier by Hemker and Maas in 1987 for unsteady flow in multiaquifer systems; it obtains an approximate solution that is simpler to evaluate than previously known solutions; and 3 it utilizes a much more general solution from a MODFLOW finite-difference model to explore both limitations of the approximate solution and the physical behavior of the two-aquifer system. In addition, criteria are obtained that show when the two-aquifer solution can be replaced with the simpler Boulton solution when analyzing pumping test field data. DOI: /ASCE :146 CE Database subject headings: Wells; Ground-water flow; Aquifers. Introduction The delayed-yield solution for unconfined flow to a well was originally obtained from an equation that was postulated empirically by Boulton 1954, A number of years later, Boulton 1973 and Cooley and Case 1973 showed that this solution can be interpreted as horizontal flow to a well in an aquifer underlain by an aquiclude and overlain by an aquitard containing a free surface. Flow in the aquitard was considered vertical and incompressible, and the delayed-yield effect resulted from vertical drainage through the overlying aquitard as the free surface drew down. Hunt and Scott 005 recently extended this idea to show that the Boulton solution also applies when the pumped aquifer is overlain by any number of aquitard layers provided that: 1 the top layer contains a free surface; the elastic storage coefficient of the aquifer is much less than the specific yield of the aquitard containing the free surface; and 3 none of the aquitard layers has a transmissivity that exceeds about 5% of the pumped aquifer transmissivity. However, during the process of testing this conclusion with comparisons between the Boulton solution and a more general finite-difference model, it was discovered that some interesting and significant differences occurred between these solutions when the transmissivity restriction was violated. These observations were the primary motivating factor for the following work, which considers flow to a well in an aquifer overlain by an aquitard and a second unpumped aquifer containing a free surface. The first solution for this problem was obtained by Hantush 1 Retired Reader, Dept. of Civil Engineering, Univ. of Canterbury, Private Bag 4800, Christchurch, New Zealand. Groundwater Hydrologist, Environment Canterbury, P.O. Box 345, Christchurch, New Zealand. Note. Discussion open until August 1, 007. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on April 0, 005; approved on June, 006. This paper is part of the Journal of Hydrologic Engineering, Vol. 1, No., March 1, 007. ASCE, ISSN /007/ /$ by assuming that flow in the bottom and top aquifers was horizontal and that flow in the aquitard between the two aquifers was both vertical and incompressible. The solution was obtained using the Laplace and Hankel transforms, and the end result contained integrals with infinite upper limits and integrands that behaved like J 0 x/x as x. Thus, the integrands decayed like 1/x 3/ as x, which is fast enough to ensure the existence of the integrals but too slow to approximate the integrals efficiently by replacing the upper limit with a finite number. Further, any transformation that compresses the semiinfinite integration interval into a finite interval will map all of the infinite number of oscillations on the semiinfinite interval onto the finite interval. Hantush resolved these formidable difficulties by obtaining asymptotic approximations for large and small times and by using graphical interpolation to obtain approximations for intermediate values of time. Neuman and Witherspoon 1969 extended the Hantush 1967 solution by including elastic storage within the aquitard. Like Hantush 1967, they also neglected vertical flow in each aquifer and horizontal flow in the aquitard. Their solution was obtained via the Laplace and Hankel transforms, so it is not surprising that their solution contained integrals with infinite upper limits and integrands that behaved like J 0 x/x as x. A few numerical values were calculated from these integrals by using the Zonneveld adaptation of the Adams-Moulton numeric method of integration. The Adams-Moulton integration method is a numerical method used to integrate ordinary differential equations, and it is not apparent to the writers how this method was adapted to evaluate a definite integral with an infinite upper limit and a slowly decaying, oscillating integrand. No reference was given for the Zonneveld adaptation of this method. Cheng and Morohunfola 1993 comment upon the difficulty of obtaining numerical values for integrals in the Neuman-Witherspoon solution. Hemker and Maas 1987 have also obtained a solution of the same problem solved by Neuman and Witherspoon 1969 in an example application of a method for obtaining unsteady solutions to general multiaquifer problems. Their solution method, which was originally devised by Hemker 1984 to describe steady flow through multiaquifer systems, required both a numerical solution of an ordinary eigenvalue problem and a numerical solution for the inverse of a matrix containing the eigenvectors in its columns. 146 / JOURNAL OF HYDROLOGIC ENGINEERING ASCE / MARCH/APRIL 007

2 The paper by Hemker and Maas 1987 extended this method to unsteady flow by first taking the Laplace transform of the governing equations and then obtaining a numerical inversion of their solutions for the Laplace transform of drawdown in each aquifer. The methods of Hemker 1984 and Hemker and Maas 1987 require a numerical solution for an eigenvalue problem with a nonsymmetrical matrix. Consequently, there is no a priori assurance that the eigenvalues will be either real or positive, and in general the eigenvectors will not be orthogonal. Eigenvalues that are complex or negative lead either to unwanted oscillations or to solutions that increase exponentially with time. It might be assumed beforehand that either of these two types of behavior is unlikely for a physically based problem, but there is no mathematical assurance that this behavior will be avoided for a nonsymmetrical matrix. Furthermore, since the eigenvectors are not orthogonal, boundary conditions are imposed by using numerical methods to calculate the inverse of an eigenvector matrix. Hunt 1985, 1986 developed a modification of the Hemker 1984 method for steady flow that avoided the two disadvantages just discussed. This modification used the solution of a generalized eigenvalue problem in which both matrices were symmetric and positive definite. Consequently, as shown by Hildebrand 1965, real positive eigenvalues always exist, and the eigenvectors are orthogonal relative to one of the matrices. The orthogonality property is particularly important as it obviates the need to use numerical methods to calculate the inverse of an eigenvector matrix, which is required for the methods of Hemker 1984 and Hemker and Maas The following work breaks new ground for this problem in three different ways. First, the modification originally devised by Hunt 1985, 1986 for steady flow in multiaquifer systems will be extended to show its application to an unsteady problem in a two-aquifer system. Second, a closed-form analytical solution for the problem first considered by Hantush 1967 will be obtained for the Laplace transforms of drawdowns in both aquifers. Then, an application of the Stehfest 1970 inversion algorithm will lead to a solution that is efficient and simple enough to be included in a collection of freely available user-defined Excel spreadsheet programs for groundwater resource analysis. Third, a very general MODFLOW finite-difference model, which includes aquitard compressibility and vertical and horizontal flow components in the aquitard and in both aquifers, will be compared with the simpler solution to check the solution accuracy and to shed light both on the physical behavior of the system and on the effect of simplifying assumptions made in the problem first formulated by Hantush Problem Statement Fig. 1 shows flow to a well in a two-layer aquifer. The well is screened in the bottom aquifer, which has an elastic storage coefficient S, a transmissivity T, and drawdowns denoted by s. The top aquifer is unconfined with a specific yield S y, a transmissivity T 0, and free surface drawdowns denoted by. The aquitard between the two aquifers has a thickness B and a conductivity K. The solution obtained herein also applies if the top boundary of the upper aquifer is impermeable provided that the specific yield, S y, is replaced with an elastic storage coefficient. It is assumed that a large conductivity contrast exists between the pumped aquifer and aquitard, which implies that flows in the pumped aquifer and aquitard are horizontal and vertical, respectively. If vertical flow through the aquitard is considered incompressible, then the Dupuit approximation allows this flow to be described by the following set of equations: T r rr r s = S s T 0 r rr Fig. 1. Flow to a well in two-aquifer system = S r y t t + K/Bs 0 r, 0 t 1 + K/B s 0 r, 0 t lim r 0r r s = Q T 0 t 3 lim r 0r r =0 0 t 4 s,t =,t =0 0 t 5 sr,0 = r,0 =0 0 r 6 where Q=flow abstracted from the well; r=radial coordinate; and t=time. Eqs. 1 and combine Darcy s law with volume conservation statements for horizontal flow in the bottom and top aquifers, respectively. Eq. 3 requires that flow entering the well screen in the bottom aquifer equal flow abstracted by the well. Eq. 4 requires that the well not be screened in the top aquifer. Eq. 5 requires that drawdowns in both aquifers vanish at an infinite distance from the well, and Eq. 6 requires zero drawdown for all points in both aquifers when pumping starts. The governing equations can be simplified by introducing the following set of dimensionless variables: s *, *,r *,t *,K *,T * 0, = st Q, T Q, r L, tt K/BL SL, T, T 0 T, S S y 7 where L=arbitrarily chosen length that cancels out when Eq. 7 is introduced into Eqs. 1 6 to obtain 1 r rr r s = s + Ks 0 r, 0 t 8 t T 0 r rr r = 1 t + K s 0 r, 0 t 9 JOURNAL OF HYDROLOGIC ENGINEERING ASCE / MARCH/APRIL 007 / 147

3 lim r 0r r s = 1 0 t 10 lim r 0r r =0 0 t 11 s,t =,t =0 0 t 1 sr,0 = r,0 =0 0 r 13 The asterisk superscript has been omitted in Eqs for notational convenience. Problem Solution The Laplace transforms of s and will be denoted by and, respectively r,p =0 sr,te pt dt 14 r,p r,te =0 pt dt 15 where p = dimensionless transform parameter. Taking the Laplace transform of Eqs. 8 1 and writing the result in matrix notation gives the following problem: A 1 r rr r ȳ = B ȳ 16 ȳ,p =0 where the vector ȳ is defined as lim r 0r r = ȳ 1/p 17 0 ȳ = A = T 0 0 p + K B = K 1 K p/ + K Underbars and overbars are used in Eqs to denote matrices and vectors, respectively. The matrices A and B are both symmetric. Furthermore, if ē is a vector defined by ē =v u then it is easy to show that ē T A ē = u + T 0 v 3 ē T B ē = pu + p/v + Ku v 4 As the right-hand sides of Eqs. 3 and 4 are positive for all real nonzero values of u and v, the matrices A and B are also positive definite. Therefore, the generalized characteristic value problem B ē = A ē 5 always has two real positive eigenvalues, 1 and, and two corresponding eigenvectors, e 1 and e, that are orthogonal relative to A e it A e j = 0 i j l i 6 i = j where l i =real number. Proofs of these properties can be found in Hildebrand The orthogonality property given by Eq. 6 suggests that a solution for ȳ might be sought in the form ȳr,p = e i F i r,p 7 where e i =eigenvector i; and F i r, p=scalar function yet to be determined. Substituting Eq. 7 into Eq. 16 and using Eq. 5 gives the following equation for determining F i : A e i 1 r rr F i r i F i =0 8 Eqs. 6 and 8 show that F i is a solution of the following form of Bessel s equation: 1 r rr F i i F i =0 9 r Eq. 18 shows that F i, p=0. Therefore, the solution of Eq. 9 that vanishes at infinity can be substituted into Eq. 7 to obtain an expression for ȳr, p: ȳ = e i C i K 0 r i 30 where K 0 x=zero-order, modified Bessel function of the second kind; and C i =constant. An equation to determine C i can be obtained by substituting Eq. 30 into Eq. 17 e i C i = 1/p 31 0 Eqs. 6 and 31 show that C i is given by C i = u i pl i where u i =first component of the eigenvector e i e i = u i v i and l i is seen from Eqs. 3, 6, and 33 to be given by l i = u i + T 0 v i Therefore, the solution for ȳr, p is ȳ u = i K 0 r i e i 35 pl i Finally, equating components on both sides of Eq. 35 gives scalar expressions for the Laplace transforms of s and 148 / JOURNAL OF HYDROLOGIC ENGINEERING ASCE / MARCH/APRIL 007

4 r,ptransform sr,t = r,ptransform r,t = K 0 r i p K 0 r i p u i l i u i v i l i Eq. 5 must be solved before numerical values can be computed from Eqs. 36 and 37. The exact solution of this generalized characteristic value problem follows: 1 = b + b c = b b c u i =1 i =1, 40 v i = p + K i /K i =1, 41 where b and c are given by b = p + K + p/ + K/T 0 / 4 c = p + K1+p/T 0 43 Eqs give the exact closed-form solutions of Eqs , and numerical approximations in the solutions for s and are made only when Eqs. 36 and 37 are inverted with a numerical method. Inversion of the Transforms Eqs. 36 and 37 were inverted numerically with the Stehfest 1970 algorithm. This algorithm, which is used extensively by people working in oil recovery and groundwater hydrology, works extremely well for some transforms and poorly for others. Further, the accuracy of the final result depends upon the number of terms retained in an infinite sum. Too few terms give an inaccurate result, but increasing the number of terms beyond a certain point also creates inaccuracies that often become evident when a plotted curve that should be smooth begins to oscillate. In fact, Stehfest 1970 himself was very aware of accuracy limitations for his algorithm, stating It was found that with increasing N the number of correct figures first increases nearly linearly and then, owing to the rounding errors, decreases linearly. The optimum N is approximately proportional to the number of digits the machine is working with. The accuracy of the numerical inversion of Eqs. 36 and 37 will be assessed in the next section by comparisons with both a MODFLOW finite-difference model and with known exact solutions for the limiting case T 0 =0. Examples of the use of the Stehfest inversion algorithm are given by Stehfest 1970 and Moench It is also important to obtain some understanding of the behavior of the solutions for large and small values of time. Expansion of the solution for r, p as p gives p + K r,p K0r p 44 p which can be inverted with a formula given by Hantush 1964 to obtain the behavior of sr,t as t 0 sr,t 1 r W 4 4t,r K t 0 45 where Wx,y=Hantush-Jacob leaky aquifer function. The importance of the result given by Eq. 45 becomes more evident when it is rewritten in dimensional coordinates to obtain sr,t Q 4T W Sr 4Tt,rK/B T t 0 46 Thus, the pumped aquifer at small values of time behaves as a leaky aquifer with an elastic storage coefficient S and a transmissivity T, both of which characterize the bottom pumped aquifer. In particular, parameters for the top aquifer do not appear in the asymptotic solution for small time. Expansion of the solutions for r, p and r, p as p 0 gives 1 r,p r,p 1+T 0 0r p K 1+1/p 1+T 0 p 0 47 Inverting Eq. 47 gives a result for the behavior of s and as t 1 sr,t r,t 41+T 0 W 1+1/r 41+T 0 t t 48 where Wx=Theis well function. Again, the importance of Eq. 48 becomes more evident when it is rewritten in dimensional variables to obtain W Q sr,t r,t S y + Sr 4T + T 0 4T + T 0 t 49 Since S y S, Eq. 49 shows that drawdown in both aquifers behaves at large values of time as drawdown in an unconfined aquifer with a specific yield S y and a transmissivity equal to the sum of transmissivities of both aquifers. The next term in the asymptotic expansion on the right-hand side of Eq. 49 is a constant, and numerical calculations will show later that this constant has different values for sr,t and r,t. Eqs. 46 and 49 will form an important part of a later discussion concerning the solution behavior. Solution Accuracy Check Eqs. 36 and 37 give exact closed-form solutions for the Laplace transforms of s and. However, the inversion of these transforms is performed numerically with the Stehfest 1970 algorithm and is approximate. Therefore, it is important to obtain a check on the accuracy of the inversion process. Hunt and Scott 005 show that the Boulton 1954, 1963 solution for delayed-yield flow to a well can be applied to the problem shown in Fig. 1 when the transmissivities of the top aquifer and the aquitard do not exceed about 5% of T and when S y is much larger than the elastic storage coefficient of any of the aquifer and aquitard layers. They also show that the Boulton solution can be obtained by solving Eqs. 1 6 after setting T 0 =0 if K/B is interpreted as an equivalent value given by JOURNAL OF HYDROLOGIC ENGINEERING ASCE / MARCH/APRIL 007 / 149

5 Fig. 3. An elevation view of the MODFLOW finite-difference model Fig.. The Boulton solution compared with solutions obtained from Eqs. 36 and 37 for T 0 /T= K/B eq = 50 B 1 /K 1 + B/K where B 1 and K 1 =initial thickness and conductivity of the top aquifer, respectively. Under most circumstances the first term in the denominator of Eq. 50 is small enough relative to the second term to be neglected. Thus, solutions obtained from Eqs. 36 and 37 for sr,t and r,t, respectively, should approach the Boulton solution for these variables as T 0 0. It is not possible to set T 0 equal to zero in Eqs. 36 and 37 since b and c in Eqs. 4 and 43 become infinite in this limit. However, drawdowns plotted in Fig. for a very small value of T 0 /T agree closely with corresponding values calculated from the Boulton solution. This result suggests that the numerical inversion of Eqs. 36 and 37 has been carried out accurately for at least smaller values of T 0 /T. A reviewer has suggested that it would be desirable to explain briefly how the Boulton solution was calculated in this study. Presumably, this is because the Boulton solution is a relatively difficult function to calculate. Excel spreadsheet user-defined functions developed by the senior writer were used for this calculation. Programs for these user-defined functions are based upon mathematical results obtained by Hunt 003, 005a and can be obtained without charge from a website given at the end of this paper. Accuracy checks for larger values of T 0 /T were carried out using the USGS MODFLOW model McDonald and Harbaugh Because this model will also be used to explore limitations of the solution as well as the physical behavior of a two-aquifer system, this model was constructed to allow compressible flow in the aquitard together with vertical and horizontal flow in the aquitard and in both aquifers. Therefore, the numerical model was more general than any of the models considered in previous studies, all of which have assumed horizontal flow in both aquifers and vertical flow in the aquitard. The model uses an eight-layer finite-difference grid to represent a three-layer geological system, as shown in Fig. 3. The upper two model layers represent a water table aquifer, the middle four model layers represent an aquitard, and the bottom two model layers represent a pumped aquifer with centrally placed well cells. Each of the layers in the three-layer system is homogeneous and isotropic and has properties that are specified in Fig. 3, where B i =thickness of aquifer layer i; K i =hydraulic conductivity of aquifer layer i; S si =specific storage of layer aquifer i; and S y =specific yield of the water table aquifer. The aquitard specific storage and hydraulic conductivity have not been assigned numerical values in Fig. 3 as they have been treated as variables in the assessment. A horizontal view of the finite-difference grid is shown in Fig. 4. The model consists of a 185-row by 185-column grid with constant head boundaries specified in Columns 1 and 185 of the lowest model layer. The central 81 rows by 81 columns of the model employ a constant grid spacing of 10 m. Beyond that central model zone, grid spacing increases geometrically to produce a maximum spacing of approximately.5 km. Transient simulations have been started from a uniform initial head of 0.0 m and involved 77 time steps to simulate a 10,000 -day pumping period time step multiplier of 1.. The model grid is sufficiently extensive 50 km50 km to ensure that extended pumping periods up to 10,000 days can be simulated without being significantly affected by the constant head boundary conditions, as explained by Hunt and Scott 005. All drawdowns have been calculated at a point 00 m from the pumped well. Fig. 5 shows a comparison between solutions calculated from Eqs. 36 and 37 and numerical solutions for a range of aquitard specific storage values. Numerical solutions for an incompressible aquitard, which are shown with filled squares for Run 1, differ very little from the solutions calculated from Eqs. 36 and 37 Fig. 4. Finite-difference grid for the MODFLOW model with coordinates in meters from the central pumped well 150 / JOURNAL OF HYDROLOGIC ENGINEERING ASCE / MARCH/APRIL 007

6 Fig. 5. Solutions calculated from Eqs. 36 and 37 and numerical solutions for a range of aquitard specific storage values Fig. 8. Solutions calculated from Eqs. 36 and 37 and numerical solutions for Run 7 K /K 3 =10 for drawdown in the pumped aquifer and even less for drawdown in the top unpumped aquifer, which is shown by the second curve from the bottom. Numerical solutions are compared with solutions calculated from Eqs. 36 and 37 for a range of aquitard hydraulic conductivity values in Figs The aquitard specific storage has been set equal to zero in these calculations. The comparisons are all very close until K /K 3 increases to 0.1, at which point drawdowns become noticeably different at larger values of time but are probably still close enough to be acceptable for most practical purposes. The solutions differ considerably in Fig. 10 for K /K 3 =1, which is not surprising as horizontal flow carried by the aquitard at this point is about 33% of the total horizontal flow carried by both aquifers. It is not possible to say from this study whether the small differences observed between numerical solutions and solutions calculated from Eqs. 36 and 37 in Figs. 5 8 are the result of numerical errors in either the Stehfest inversion or the finitedifference solution or are simply the result of neglecting horizontal flow in the aquitard and vertical flow in both aquifers in the two-aquifer solution. However, the comparisons are undoubtedly close enough to suggest that numerical inversions of Eqs. 36 and 37 are sufficiently accurate for almost all practical applications. Behavior of a Two-Aquifer System Previous studies by Hantush 1967, Neuman and Witherspoon 1969, and Hemker and Maas 1987 have concentrated almost entirely upon obtaining solutions or else presenting and illustrating solution techniques. Therefore, solutions calculated from Eqs. 36 and 37 and the more general MODFLOW solution will be Fig. 6. Solutions calculated from Eqs. 36 and 37 and numerical solutions for Run 1 K /K 3 =10 4 Fig. 7. Solutions calculated from Eqs. 36 and 37 and numerical solutions for Run 6 K /K 3 =10 3 Fig. 9. Solutions calculated from Eqs. 36 and 37 and numerical solutions for Run 8 K /K 3 =10 1. Pumped aquifer and water table drawdowns are indistinguishable for all times in both solutions. JOURNAL OF HYDROLOGIC ENGINEERING ASCE / MARCH/APRIL 007 / 151

7 Fig. 10. Solutions calculated from Eqs. 36 and 37 and numerical solutions for Run 9 K /K 3 =1. Pumped aquifer and water table drawdowns are indistinguishable for all times in both solutions. used to explore some limitations of solutions calculated from Eqs. 36 and 37 and to make a few observations about the physical behavior of flow to a well in a two-aquifer system. Solutions calculated from Eqs. 36 and 37 neglect compressibility within the aquitard. Therefore, it becomes important to know when aquitard compressibility effects become important. Fig. 5 shows the result of increasing the aquitard specific storage in the finite-difference model from a value of zero to a value that is 1,000 times greater than the specific storage of the pumped aquifer. Drawdown solutions in the pumped aquifer are virtually identical when the aquitard specific storage has the same order of magnitude as the pumped aquifer specific storage, and additional calculations not shown here have indicated that the small differences that do occur can be removed by slightly increasing the elastic storage coefficient of the pumped aquifer in the twoaquifer solution. However, significant differences start to appear in pumped aquifer drawdowns at early and intermediate times when S s /S s3 O10. It is also interesting to note that drawdowns in the top unpumped aquifer remain unchanged as S s /S s3 increases until Run 5, when S s /S s3 =O10 3. Apparently this is because the ordinate of the pseudosteady-flow portion of the pumped aquifer drawdown curve does not change until S s /S s3 =O10 3. One of the more surprising results of these calculations is that horizontal flow within the aquitard does not start to become significant in the two-aquifer solution until K /K 3 O0.1, as shown by Figs This is a conclusion that could only have been reached by using the MODFLOW finite-difference model, since all previous models neglect horizontal flow in the aquitard. One of the most important questions in applications is how can a practitioner determine from experimental drawdown measurements alone whether a pumping test analysis should be carried out with a solution for a multiaquifer system or with the simpler Boulton solution, which has fewer aquifer parameters to determine. The answer to this question is illustrated in Fig. 11, where drawdowns for the top and bottom aquifers have been plotted as a function of log time for both the Boulton solution and the twoaquifer solution. In both cases, drawdown curves for the pumped aquifer consists of three segments: a an initial segment that is virtually identical to the Theis 1935 solution for fully confined flow; b a second segment of pseudosteady flow which, when combined with the first segment, is closely approximated by the Fig. 11. The Boulton solution compared with solutions obtained from Eqs. 36 and 37 for T 0 /T=0.5 Hantush-Jacob 1955 solution for flow to a well in a leaky aquifer; and c a third segment which can be approximated with the Theis 1935 solution after replacing the elastic storage coefficient with the specific yield of the top aquifer. However, in a semilog plot, the Boulton solution has an asymptotic slope for the third segment that either equals or exceeds the asymptotic slope of the first segment, while the two-aquifer solution has an asymptotic slope for the third segment that is smaller than the asymptotic slope of the first segment. The proof of this conclusion is given by Eqs. 46 and 49, which show that the asymptotic slopes of semilog plots of the first and third segments of the two-aquifer solution are inversely proportional to T and T+T 0, respectively. Thus, since the Boulton solution applies when T 0 /T1, the asymptotic slopes of both segments are identical for the Boulton solution but the third segment has a smaller asymptotic slope than the first segment when T 0 /TO1, in which case the two-aquifer solution is needed. The fact that the slope of the first segment for the Boulton solution in Fig. 11 has not quite reached its true asymptotic value for the particular combination of parameters used in the figure does not alter this conclusion as this leads to an apparent asymptotic slope of the first segment that is smaller than the asymptotic slope of the third segment. If drawdown measurements are made in both aquifers during a pumping test, then Fig. 11 shows an even easier way to distinguish between data that can be analyzed with the Boulton solution and data that must be analyzed with a two-aquifer solution. This is to observe whether asymptotes of the third segment are identical or parallel. A two-aquifer system, which by definition has T 0 /TO1, carries a significant portion of horizontal flow toward the well. Since this flow can only enter the well in the pumped aquifer if it seeps downward through the aquitard, a drawdown difference must be maintained across the aquitard even at large values of time, in which case asymptotes of the third segment are parallel but not identical. On the other hand, the Boulton 1954, 1963 solution applies when T 0 /T1. In this case a negligible amount of horizontal flow is carried by the overlying layers, and asymptotic drawdowns become identical at large times in both the pumped aquifer and in the overlying layers. This result is illustrated again in Fig. 1, which shows the result of decreasing the aquitard conductivity by an order of magnitude for the two-aquifer system considered in Fig. 11. Asymptotic slopes of the third segment remain unchanged, but the parallel asymptotes are separated by a larger distance in Fig. 1 to increase the 15 / JOURNAL OF HYDROLOGIC ENGINEERING ASCE / MARCH/APRIL 007

8 Fig. 1. The result of decreasing the ratio K/B in Fig. 11 drawdown gradient across the aquitard and maintain the same rate of downward leakage through the aquitard. Fig. 13 shows the result of decreasing the transmissivity ratio T 0 /T from ten to zero for a two-aquifer system. The solution for T 0 /T=10 is almost identical to the Hantush-Jacob 1955 solution for a leaky aquifer, and the solution for T 0 /T=0, as observed previously, is identical to the Boulton 1954, 1963 solution. The important point about this plot is that it confirms the conclusion reached earlier by Hunt and Scott 005, by using order of magnitude arguments and a more limited number of calculations, that the Boulton solution is applicable when T 0 /T0.05. A curve calculated for T 0 /T=0.01 was virtually identical to the curve for T 0 /T=0 and is not shown. The case of T 0 /T=10 is unlikely to appear in practice since a driller will not usually deepen a well after encountering a layer with a relatively high transmissivity value. Hantush 1967 first pointed out that Eqs. 1 6 can be used to describe flow to a well when the top aquifer is confined provided that the elastic storage coefficient of the top aquifer is substituted for S y. An example of this is shown in Fig. 14, where drawdown curves for a fully confined system are plotted for S/S y =1. Fig. 14 also shows drawdown curves plotted for S/S y =0.001, which is a typical value for this parameter when a free surface exists in the top aquifer. The interesting feature of these plots is that a horizontal segment of pseudosteady flow occurs for drawdown in the pumped aquifer only when a free surface exists in the top unpumped aquifer. If the upper boundary of the top aquifer is an aquiclude, then a pseudosteady-flow segment does not occur in the drawdown curve for the pumped aquifer. Fig. 13. The result of changing the transmissivity ratio T 0 /T Fig. 14. Drawdowns compared when the top aquifer is confined S/S y =1 and unconfined S/S y =0.001 The physical reason for this behavior is that the volume of recharge available for release from elastic storage in an upper confined aquifer is much smaller than the volume of water available for release from an unconfined aquifer as a free surface drops. The mathematical reason is that a time derivative appears twice in the governing equations, and when the coefficients of these time derivatives differ by an order of magnitude or more, then a fast time scale and a slow time scale emerge in the solution. Conclusions Flow to a well in a two-aquifer system has been a subject for investigation by a number of workers since the original paper by Hantush This work adds to results from previous investigations in three ways: 1. A solution method devised by Hunt 1985, 1986 to improve upon a method originally proposed by Hemker 1984 for steady flow in multiaquifer systems has been extended for use in unsteady flow. This method is an improvement of a similar eigenvalue decomposition technique used by Hemker and Maas 1987 for unsteady flow since it can be stated a that real positive eigenvalues always exist for this formulation; and b that the resulting eigenvectors satisfy an orthogonality property that can be used to avoid using numerical methods to calculate the inverse of an eigenvalue matrix.. A closed-form analytical solution has been obtained for the Laplace transform of drawdowns in both aquifers, and this allows drawdowns to be calculated from a numerical inversion of these transforms. The inversion process has been tested and found to be sufficiently accurate for applied work, and the end result is a solution that is evaluated more easily and efficiently than previously known solutions of this problem. 3. This solution and much more general solutions obtained from a MODFLOW finite-difference model have been used to explore both limitations of the solution and the physical behavior of a two-aquifer system, neither of which have been considered in comparable detail in previous studies. This work has led to the following conclusions: Compressibility effects for flow within the aquitard can be neglected in a solution when the ratio of aquitard storage coefficient to pumped aquifer storage coefficient has an order of magnitude of one or less. JOURNAL OF HYDROLOGIC ENGINEERING ASCE / MARCH/APRIL 007 / 153

9 Horizontal flow within the aquitard can be neglected and the two-aquifer solution can be used when the ratio of aquitard transmissivity to pumped aquifer transmissivity has an order of magnitude less than 1/10. The applicability of the simpler Boulton solution to field data from a pumping test can be assessed from a semilog plot of drawdown measurements in the pumped aquifer by comparing asymptotic slopes of the first and third segments of the drawdown curve. The Boulton solution can be used if the asymptotic slope of the third segment is greater than or equal to the asymptotic slope of the first segment. Otherwise, the more complicated two-aquifer solution must be used. If drawdown measurements from a pumping test have been made in both aquifers, then the Boulton solution can be used if both sets of measurements for the third segment approach the same asymptote. If these measurements approach parallel asymptotes, then the two-aquifer solution must be used. A previous conclusion by Hunt and Scott 005 that the Boulton solution is applicable when the top aquifer has a transmissivity that is 5% of the pumped aquifer transmissivity has been confirmed by calculations made with the two-aquifer solution. The two-aquifer solution can also be used when the top aquifer is bounded above by an aquiclude rather than a free surface if the specific yield of the top aquifer is replaced with its elastic storage coefficient. Calculations for this case show that pseudo-steady flow occurs at intermediate values of time in the drawdown curve for the pumped aquifer only when a free surface exists in the top aquifer and that this pseudo-steady-flow portion of the drawdown curve is entirely missing when the top aquifer is bounded above by an aquiclude. Program Availability Programs used for drawdown calculations with Eqs. 36 and 37 have been incorporated in the Excel spreadsheet software Function.xls. This software, which has been described by Hunt 005b, and a manual explaining its use can be obtained, without charge, from the website bhunt.asp. Notation The following symbols are used in this paper: A, B matrices; B, B i aquitard or aquifer thicknesses L; C i constant; ē eigenvector; F i function; J 0 x Bessel function; K dimensionless variable defined in Eq. 7; K, K i hydraulic conductivities L/T; K 0 Bessel function; L arbitrary length L; l i length of eigenvector ē i relative to A ; p Laplace transform parameter; Q well abstraction rate L 3 /T; r radial coordinate L; S elastic storage coefficient; S y specific yield; s pumped aquifer drawdown L; T, T 0 transmissivities L /T; t time T; u, v components of ē, W well function; ȳ vector; dimensionless variable defined in Eq. 7; top aquifer drawdown L;, i eigenvalues; Laplace transform of s; and Laplace transform of Superscript * dimensionless variable defined in Eq. 7 References Boulton, N. S Unsteady radial flow to a pumped well allowing for delayed yield from storage. Int. Ass. Sci. Hydrol., Publication 37, Boulton, N. S Analysis of data from non-equilibrium pumping tests allowing for delayed yield from storage. Proc.-Inst. Civ. Eng., 66693, Boulton, N. S The influence of delayed drainage on data from pumping tests in unconfined aquifers. J. Hydrol., 19, Cheng, A. H.-D., and Morohunfola, O. K Multilayered leaky aquifer systems. 1: Pumping well solutions. Water Resour. Res., 98, Cooley, R. L., and Case, C. M Effect of a water table aquitard on drawdown in an underlying pumped aquifer. Water Resour. Res., 9, Hantush, M. S Hydraulics of wells. Advances in hydroscience, V. T. Chow, ed., Academic, New York. Hantush, M. S Flow to wells in aquifers separated by a semipervious layer. J. Geophys. Res., 76, Hantush, M. S., and Jacob, C. E Non-steady radial flow in an infinite leaky aquifer. Trans., Am. Geophys. Union, 361, Hemker, C. J Steady groundwater flow in leaky multiple-aquifer systems. J. Hydrol., 73/4, Hemker, C. J., and Maas, C Unsteady flow to wells in layered and fissured aquifer systems. J. Hydrol., 903/4, Hildebrand, F. B Methods of applied mathematics, Prentice-Hall, Englewood Cliffs, N.J. Hunt, B Flow to a well in a multiaquifer system. Water Resour. Res., 111, Hunt, B Solutions for steady groundwater flow in multi-layer aquifer systems. Transp. Porous Media, 14, Hunt, B Unsteady stream depletion when pumping from semiconfined aquifer. J. Hydrol. Eng., 81, Hunt, B. 005a. Calculation of groundwater integral. J. Hydrol. Eng., 106, Hunt, B. 005b. Visual Basic programs for spreadsheet analysis. Computer note. Ground Water, 431, Hunt, B., and Scott, D Extension of the Hantush and Boulton solutions. J. Hydrol. Eng., 103, / JOURNAL OF HYDROLOGIC ENGINEERING ASCE / MARCH/APRIL 007

10 McDonald, M. G., and Harbaugh, A. W A modular threedimensional finite-difference ground-water flow model. USGS techniques of water-resources investigations, Book 6, Chap. A1, United States Geological Survey, Reston, Va. Moench, A. F Computation of type curves for flow to partially penetrating wells in water-table aquifers. Ground Water, 316, Neuman, S. P., and Witherspoon, P. A Theory of flow in a confined two aquifer system. Water Resour. Res., 54, Stehfest, H Numerical inversion of Laplace transforms. Commun. ACM, 131, Theis, C. V The relation between the lowering of the piezometric surface and the rate and duration of discharge of a well using ground-water storage. Trans., Am. Geophys. Union, 16, JOURNAL OF HYDROLOGIC ENGINEERING ASCE / MARCH/APRIL 007 / 155