S. S. PAPADOPULOS & ASSOCIATES, INC. ENVIRONMENTAL & WATER-RESOURCE CONSULTANTS

INTERPRETATION OF CONSTANT-HEAD TESTS: RIGOROUS AND APPROXIMATE ANALYSES

Christopher J. Neville and Jeffrey M. Markle

Paper presented at the First Joint IAH-CNC/CGS Groundwater Specialty Conference Montreal, Quebec October 15-18, 2000

Affiliations:

Christopher J. Neville S.S. Papadopulos & Associates, Inc. 207 King St. S. Waterloo, Ontario N2J 1R1 [email protected]

Jeffrey M. Markle Department of Earth Sciences University of Western Ontario London, Ontario ON N6A 5B8 [email protected] INTERPRETATION OF CONSTANT-HEAD TESTS: RIGOROUS AND APPROXIMATE ANALYSES

Christopher J. Neville1 and Jeffrey M. Markle2

1: S.S. Papadopulos & Associates, Inc., 207 King St. S., Waterloo, ON N2J 1R1, [email protected] 2: Dept. of Earth Sciences, Univ. of Western Ontario, London, ON N6A 5B8, [email protected]

ABSTRACT The constant-head test is an essential technique of geotechnical and hydrogeological site characterization. The test is used frequently to estimate the of relatively low permeability formations (sparsely fractured rock and clayey soils). The interpretation of these tests has traditionally been based on approximate analyses. The approximate analyses neglect storage in the formation, assume that the penetrates the formation completely, and assume uniform material properties in the vicinity of the well. Novakowski (1993) developed a rigorous analysis for constant-head tests, considering explicitly storage in the formation, partial penetration, and the presence of a finite-thickness skin. The rigorous analysis involves a mixed boundary condition at the wellbore and cannot be solved by conventional integral transform approaches. Novakowski used a novel application of the Dirac-delta function to derive the Laplace-transform solution. In this study we check the predictions of the Dirac-delta solution against the results of high-resolution finite element analyses. We show that for cases of partial penetration, Novakowski’s solution yields erroneous results. We present a correct version of the rigorous solution for Novakowski’s problem. Finally, we use the correct, rigorous solution to examine the estimation of hydraulic conductivity from constant-head tests in partially penetrating . Our results show that significant errors can be made in interpreting tests when partial penetration is neglected.

1. Introduction exists around the well, they may yield parameter estimates of formation properties that are not representative of the The single-well test involving injection at a constant head is formation. an essential technique of geotechnical and hydrogeological site characterization. The test was developed initially by Few systematic attempts have been made to quantify the Lugeon (1933) to assess grouting requirements for dams errors associated with conventional interpretations of founded on rock. It is still used for that purpose, but it has constant-head tests. This lack of attention may be due to also been adopted by geotechnical engineers and the fact that more realistic conceptual models have defied hydrogeologists to estimate the hydraulic conductivity of the development of analytical solutions. In 1993, K.S. fractured rock and relatively low permeability clayey soils Novakowski made an important attempt to rectify this (Shapiro and Hsieh, 1998); and Novakowski et al., 1999; situation by developing an analytical solution for a Tavenas et al., 1990). In the geotechnical literature, the physically-based boundary value problem that incorporates test is generally referred to as the Lugeon or packer test; in storage in the formation, partial penetration of the well, and the literature the test is generally designated the possibility of an altered zone around the well. the packer or constant-head test. In this study, we revisit Novakowski’s solution for the case Conventional interpretations of constant-head tests are of a partially penetrating well. We show that for cases of based on approximate analyses. Geotechnical applications partial penetration, Novakowski’s solution yields erroneous typically neglect storage in the formation a-priori from the results. This conclusion is supported by an examination of analyses. Hydrogeologic interpretations sometimes the properties of the infinite series appearing in consider transient effects, using the analysis of Jacob and Novakowski’s solution, and a re-consideration of the Lohman (1952), but more frequently the interpretations also boundary conditions implicit in Novakowski’s methodology. assume steady flow. Zero-storage analyses oblige the We present a correct form of the solution. The close match interpreter to assume an arbitrary radius of influence for with the results of high-resolution finite-element analyses each test. The radius of influence may have little physical demonstrates the correctness of our new solution. We use meaning in an extensive formation. Furthermore, in the the correct, rigorous solution to evaluate the cases of carefully instrumented tests, conventional appropriateness of the conventional methods of analysis. analyses require neglecting much of the transient data that In particular, we try to answer the following question: “How are collected. much of an error do we make in our estimation of hydraulic conductivity when we adopt a simplified model to interpret In many cases, the tested interval represents only a the data from a constant-head test?”. relatively small portion of the entire thickness of the formation. In these cases, conventional interpretations of constant-head tests idealize the formation as a confined 2. Conventional interpretations of constant-head tests of thickness equal to the length of the packed-off interval. The analysis of test data using solutions A schematic set-up for a constant-head test is shown on developed for fully penetrating wells can lead to erroneous Figure 1, along with an illustration of idealized data obtained results. Drilling and installation of wells frequently results in during a single-stage test. The principle of the test is the development of a zone of altered properties around the simple: the in a packed-off interval is raised wellbore, referred to as a “skin”. Conventional analyses suddenly and held constant. The injection rate required to assume that the formation has uniform properties. Since maintain the head rise is monitored through time. the conventional analyses neglect the possibility that a skin The interpretation of constant-head test data is described in It is important to note that because all of these shape factor the Earth Manual of the U.S. Bureau of Reclamation (1974) approaches share the same conceptual model of zero- and Ziegler (1976). The horizontal hydraulic conductivity KH storage in the formation, there is little to distinguish is estimated from the general formula for steady-state flow: between them. All models that ignore storage in the formation are based on an artificial conception of the Q hydraulics of a constant-head test, regardless of the K = (1) H F∆H “exactness” of the representation of flow around the wellbore. where Q and ∆H denote the flow rate and the applied Conventional approaches for interpreting constant-head change in hydraulic head, respectively. The term F is tests presume that steady-state conditions prevail. In generally referred to as the shape factor. The conventional formations where flow is dominated by a few fractures, this interpretation assumes purely radial flow within the interval may be a realistic approach since the contribution from of the packers. For this conceptual model, the flow rate is storage may be negligible. However, in compressible given by the Thiem solution: formations such as soft clays, or in densely fractured rock K L ∆H masses with relatively permeable matrix blocks, the Q = H (2) observed flow rate may continue to decline over the 2π  R  duration of a test. Under these circumstances, the ln  conventional approach will require ignoring much of the rw  data collected in a well-instrumented test, and choosing an arbitrary pseudo-steady flow rate. where L is the length of the packed-off interval, rw is the radius of the wellbore, and R is the radius of influence. Since the conventional approach for interpreting constant- Comparing (1) and (2), the shape factor is: head tests has a weak theoretical foundation, is not well- suited for interpreting data from partially penetrating tests, = L 1 and may require discarding data, there is motivation for F (3) developing a rigorous analysis of the constant-head test. 2π  R  ln  rw  3. Rigorous analysis of the constant-head test We refer to this method of interpretation as the The conceptual model for a rigorous analysis of a constant- “conventional approach”, as it appears frequently in the head test is shown on Figure 2. The boundary value project reports of research organizations that provide problem (BVP) considers a well of finite radius that partially important examples of practice, for example, Canada’s penetrates an aquifer of uniform thickness B. The aquifer is National Water Research Institute and the United States confined above and below by impermeable strata. A skin Geological Survey. zone of finite thickness, with hydraulic properties different The conventional approach requires specification of a from the formation surrounds the wellbore. radius of influence, R. In practice, a constant value for R is The governing equations for three-dimensional, transient assumed (Shapiro and Hsieh, 1998; Novakowski et al., flow in the skin and the formation are: 1999). Since R appears in the log term, the interpreted 2 2 hydraulic conductivity is not particularly sensitive to its  ∂ s 1 ∂s  ∂ s ∂s K  1 + 1  + K 1 = S 1 value. Nevertheless, the fact that a radius of influence must r1  ∂ 2 ∂  z1 ∂ 2 s1 ∂ be assumed, even in an extensive formation far from any  r r r  z t physical boundaries, demonstrates the weak theoretical foundation of the conventional approach. (4) ≤ In cases where the tested interval represents a relatively within the skin zone, rw rs and small portion of the entire thickness of the formation, 2 2 conventional interpretations of constant-head tests idealize  ∂ s 1 ∂s  ∂ s ∂s K  2 + 2  + K 2 = S 2 the formation as a confined aquifer of thickness equal to the r 2  ∂ 2 ∂  z2 ∂ 2 s2 ∂ length of each packed-off interval. This approach ignores  r r r  z t vertical gradients and and may lead to errors that are (5) difficult to quantify. In an attempt to generalize the ≤ ∞ conventional approach, several alternative formulae have within the formation, rs rw < been presented in the literature. These formulae differ in where si, Ki, and Ssi are the head change, hydraulic the shape factor used in Eq. (1). For example, the U.S. conductivity, and of the skin (i = 1) and Bureau of Reclamation (1974) presents a shape factor formation (i = 2). originally suggested by Hvorslev (1951). The particular shape factor drawn from the work of Hvorslev is based on the idealized geometry of an ellipsoid embedded in an infinite porous medium. This model is considered more representative of the geometry of a packed-off interval that is only a small portion of the full thickness of a permeable formation.

At the start of the test, the head change in the skin and the Novakowski’s Laplace-transform solution is given by: formation is zero, and the head change in the wellbore, s , 0 γξ 1/ 2 is applied instantaneously. Therefore, the initial conditions = − ( ) are written as: QDp ( p) 1/ 2 LD p s (r, z,0) = s (r, z,0) = 0 (6a) 1 2 [I (q )β + K (q )β ] × 1 3 3 1 3 4 (B − B ) + = ψ 2 1 sw (0 ) sw0 (6b) 2L ∞ q [I (q )β + K (q )β ] In the general case of an interval open between elevations − D ∑ 1 1 1 1 1 1 2 z1 and z2, the inner boundary conditions along the wellbore 2 π = pψ are written as: n 1 ∂ 1 2 s1 × [sin(ωB ) − sin(ωB )] = ; 0 ≤ z ≤ z (7a) 2 2 1 (rw , z,t) 0 1 n (B − B ) ∂r 2 1 (12) = ≤ ≤ s1 (rw , z,t) sw (t) ; z1 z z2 (7b) Readers are referred to Novakowski (1993) for definitions of the notation. Note that (12) is identical to Novakowski‘s ∂ Eq. [13], with the exception of the term (pψ) appearing in s1 the summation. The omission of this term from (r , z,t) = 0 ; z2 ≤ z ≤ B (7c) ∂r w Novakowski’s Eq. [13 is a typographical error, as the correct form of the summation appears in Novakowski’s Eq. [15]. where sw is the head change in the wellbore itself. Novakowski presented no verification of his solution, apart The formation is assumed to be areally extensive and the from showing that it collapsed to the solution of Jacob and outer boundary condition is written as: Lohman (1952) for a fully penetrating well with no skin ∞ = zone. s2 ( , z,t) 0 (8)

The boundary conditions along the top and bottom of the aquifer are: 4. Problems with the Novakowski solution ∂s ∂s In order to check the Novakowski (1993) solution for the 1 (r,0,t) = 2 (r,0,t) = 0 (9a) case of partial penetration, we compare results from ∂z ∂z Novakowski’s solution with the results of high-resolution numerical simulations using a finite element model (FEM). ∂s ∂s The FEM code was designed specifically for the analysis of 1 (r, B,t) = 2 (r, B,t) = 0 (9b) complex aquifer tests (Sudicky, MacQuarrie and Neville, ∂z ∂z 1990). Novakowski presents selected results in the form of dimensionless type curves. Figure 3 shows the results of The BVP is completed with compatibility conditions at the Novakowski’s solution and the FEM simulations. The interface between the skin and the formation: results show that while the solutions match closely for the s (r , z,t) = s (r , z,0) (10) case of full penetration, they diverge as the degree of 1 s 2 s penetration decreases. The disagreement has been confirmed by repeating the FEM simulations for increasingly ∂s ∂s 1 = 2 finer spatial discretizations. K r1 (rs , z,t) K r 2 (rs , z,t) (11) ∂r ∂r Inspection of Novakowski’s equation [13] reveals that it does not converge. For a partially penetrating well with no As pointed out by Ruud and Kabala (1997), the inner skin (i=1=2) Eq. (12) reduces to: boundary condition for the rigorous problem mixes Type I (specified head) and Type II (no-flow) conditions along the = K1(q4 ) − wellbore, over the portions of the well that are screened and QD ( p) (B2 B1) cased, respectively. This mixing of conditions precludes an Ld q4 K0 (q4 ) exact solution by conventional integral transform methods. 2L ∞ q K (q ) 1 + D ∑ 5 1 5 Novakowski (1993) developed an analytical solution for the π 2 2 − BVP using a novel application of the Dirac-delta function in n=1 pK0 (q5 ) n (B2 B1) conjunction with the Laplace transform. Novakowski used × ω − ω 2 the delta function to derive a Green’s function for the [sin( B2 ) sin( B1)] problem, and then integrated the building block solution (13) over the length of the open interval. The final solution was obtained by numerical inversion of the Laplace-transform solution.

Evaluating the late-time limiting form of equation (13) yields: To determine the flow rate into the well, we use Darcy’s law at the well face in the following dimensionless form: Lim 2 = z pQ D ( p) D 2 ∂s p → 0 π ()B − B = − D1 2 1 Q ( p) ∫ = dz (16) D ∂ rD 1 D ∞ rD 1 zD1 ×∑ []sin()ωB − sin ()ωB 2 2 1 The solution in the Laplace domain for the flow from the = n n 1 well is: (14)  ρ  = 1 + QD ( p)  A1'Io (q1') A2 'K0 (q1') The series in the last expression is divergent, indicating that p ∆'  equation (12) is the solution to an ill-posed problem. 2 ∞ []A I (q ) + A K (q ) 1 Independent confirmations of problems with the + ∑ 1 o 1 2 0 1 ∆ β Novakowski (1993) solution are provided by Cassiani and LDbD n=1 n Kabala (1998) and Cassiani et al. (1999). Cassiani and co- × []()()β − β 2 workers use Novakowski’s approach to develop an sin n zD2 sin n zD1 alternate solution for a semi-infinite aquifer. They (17) demonstrate that solutions developed using Novakowski’s methodology are influenced by Gibbs effects, and produce where the variables are defined in the notation. Complete total fluxes that are infinite. details of the derivation of the solution, and a FORTRAN code implementing it are available upon request. The fundamental problem with Novakowski’s solution for a partially penetrating well lies with the use of the Dirac-delta Figure 4 shows the results of our revised solution and the function to implement the boundary condition along the previous FEM simulations. The results of our solution wellbore. With Novakowski’s approach, the Dirac-delta agree much more closely with the numerical results. There function is used to describe the change in hydraulic head in are still some differences for the smallest degree of the formation due to a point source at the well screen. The penetration considered; however, these differences tend to delta function is then integrated along the open portion of be exaggerated by the logarithmic axes. The discrepancy the well to obtain the change in head in the formation due to may be due to an approximation in the representation of the a line source. Use of the delta function implies that the boundary condition along the wellbore that only becomes change in the hydraulic head is equal to a specified value significant as the relative length of the open interval along the open portion of the well, and is zero elsewhere becomes relatively small. along the wellbore. The delta function approach does not implement the boundary conditions as specified in Equations (7a-c); instead, the forcing of zero head change 6. Interpretation of data from tests in partially along the closed section of the wellbore creates an infinite penetrating wells sink, which gives rise to the infinite total flux observed by In order to gauge the magnitude of the errors that may arise Cassiani et al. (1999). when applying the conventional method of interpretation, we consider an idealized example. For the example, we consider a confined formation that is 10 m thick and has a 5. Development of an improved analytical solution uniform horizontal hydraulic conductivity of 10-6 m/sec and a -6 -1 We have revisited the BVP for a rigorous model of the specific storage of 10 m . These values are typical for constant-head tests and developed a new solution following intact clays. The open interval along the wellbore extends the approach of Dougherty and Babu (1984). The solution from the middle of the formation. The applied head change for the dimensionless hydraulic head in the skin and the at the wellbore is 10.0 m. formation is obtained by using the Laplace transform with For the example we use the correct rigorous solution to respect to dimensionless time, tD, and the finite Fourier generate “perfect” discharge vs. time records. We consider transform with respect to the dimensionless elevation, zD. an open interval that ranges from 10% to 100% of the The solutions in the Laplace domain are derived by formation thickness, and a vertical hydraulic conductivity application of the inverse finite Fourier transform. Final ranging over three orders of magnitude, from Kz/Kr = 1.0 to results are obtained by numerical inversion of the Laplace- 0.001. A typical set of results is shown on Figure 5; results transform solutions using the algorithm of Talbot (1979). are shown for the case of isotropic hydraulic conductivity, In addition to the governing equations (4-5) and (10-11), the for a range of penetration ratios ρ (=L/B), from 1.0 (full BVP is completed with a statement of mass conservation penetration) down to 0.01. All of the curves shown on for the well: Figure 5 are generated with the same specific storage for the formation. The results illustrate a general result: the z2 ∂ s1 (rw , z,t) discharge curves become progressively flatter as the Q (t) = 2π r K ∫ dz (15) degree of penetration decreases. Rather than providing w w r1 ∂r z1 “proof” of the validity of the zero-storage conceptual model, this response illustrates the difficulties in interpreting data when multiple hydraulic processes interact.

2 Each discharge-time record is analyzed using the q1 √(αγp+φ1βm ) conventional approach. We assume a radius of influence q1’ √(αγp) 2 of 10 m (the same value as used in Novakowski et al., q2 √(p+φ2βm ) 1999), and use as a proxy for the steady flow the discharge q2’ √p rate after 1000 seconds. We note that the selection of Qw volumetric flow rate into or out of the well bore 1000 seconds is arbitrary, and that the discharge rate has [L3T-1] not necessarily stabilized at this time. The results of the QD dimensionless volumetric flow rate into or out of conventional analyses are summarized on Figure 6. The the wellbore (Q /2π(z -z )K H ) results for this example demonstrate that significant errors w 2 1 r1 0 r radial distance [L] can be made in the estimation of the horizontal hydraulic r dimensionless radius (r/r ) conductivity when the conventional approach is applied. D w r well radius [L] The errors are most significant as the degree of penetration w r radial co-ordinate of the outer boundary of the skin decreases, and as the vertical hydraulic conductivity s zone [L] approaches the magnitude of the horizontal conductivity. r dimensionless radial co-ordinate of the outer For highly anisotropic formations (K /K < 0.01, say), the Ds z r boundary of the skin zone (r /r) error arising from the use of the conventional method of s s hydraulic head change in region i [L] interpretation is relatively insensitive to the degree of i s dimensionless hydraulic head change in region i (s penetration. Di i /H0) sw hydraulic head change in the wellbore [L] 7. Conclusions sDw dimensionless hydraulic head change in the wellbore (sw /H0) Conventional approaches for interpreting the results of -1 Ssi specific storage in region i [L ] constant-head tests are based on a conceptual model of t time [T] aquifer response that has a weak theoretical foundation. 2 tD dimensionless time (Kr2t/Ss2rw ) The inability of these approaches to consider storage in the z vertical co-ordinate [L] formation and partial penetration of the test interval yield zD dimensionless vertical co-ordinate errors that are difficult to quantify. We have reviewed the z lower z co-ordinate of well screen [L] rigorous solution of Novakowski, and shown that it provides 1 zD1 dimensionless lower z co-ordinate of well screen incorrect results for cases of partial penetration. Our results (z /z) confirm the recent suggestion of Cassiani et al. (1999) that 1 z upper z co-ordinate of well screen [L] Novakowski’s solution is not well founded, and that use of 2 z dimensionless upper z co-ordinate of well screen the Dirac-delta function solution methodology is incorrect. D2 (z /z) We have derived a correct version of the rigorous problem 2 α radial hydraulic conductivity ratio (K /K ) considered by Novakowski. The good agreement between r2 r1 β our solution and the results of high-resolution FEM analyses n eigenvalue for finite Fourier transform π demonstrates that our solution is correct for the case of (n /BD) φ partial penetration. We use the correct rigorous solution to i vertical to radial hydraulic conductivity ratio (Kzi /Kri) examine the errors that can arise from the application of a γ specific storage ratio (Ss1 /Ss2) conventional interpretation approach, in the context of a λ q1/(αq2) synthetic example. The results from the example λ’ q1’/(αq2’) or √(γ /α) demonstrate that significant errors can be made in the ρ partial penetration ratio (b/B) estimation of the horizontal hydraulic conductivity when the ∆ q1I1(q1)[ K0(q1 rD1)K1(q2 rD1) conventional approach is applied to data from partially - λK0(q2 rD1)K1(q1 rD1) ] penetrating wells. + q1K1(q1)[ I0(q1 rD1)K1(q2 rD1) + λK0(q2 rD1)I1(q1 rD1) ] ∆ 8. Notation ’ q1’I1(q1’)[ K0(q1’ rD1)K1(q2’ rD1) - λ’K0(q2’ rD1) K1(q1’ rD1) ] λ A1 K0(q2 rD1)K1(q1 rD1) - K0(q1 rD1)K1(q2 rD1) + q1’K1(q1’)[ I0(q1’ rD1)K1(q2’ rD1) λ A1’ ’K0(q2‘rD1)K1(q1‘ rD1)-K0(q1‘ rD1)K1(q2‘ rD1) + λK0(q2’ rD1)I1(q1’ rD1) ] A λI (q r )K (q r ) + I (q r )K (q r ) 2 1 1 D1 0 2 D1 0 1 D1 1 2 D1 A2’ λ’I1(q1‘ rD1)K0(q2‘ rD1)+ I0(q1‘ rD1)K1(q2‘ rD1) b screen length (z2-z1) [L] bD dimensionless screen length (b/z) B aquifer thickness [L] BD dimensionless aquifer thickness (B/rw) H0 head change at the wellbore [L] Kri radial hydraulic conductivity in region i [LT-1] Kzi vertical hydraulic conductivity in region i [LT-1] n finite Fourier transform variable (n=0,1,2…∞) p Laplace transform variable

9. References

Cassiani, G., and Z.J. Kabala, 1998: Hydraulics of a Novakowski, K., P. Lapcevic, G. Bickerton, J. Voralek, L. partially penetrating well: Solution to a mixed-type Zanini, and C. Talbot, 1999: The Development of a boundary value problem via dual integral equations, Jl. Conceptual Model for Contaminant Transport in the of Hydrology, 211, pp. 100-111. Dolostone Underlying Smithville, Ontario, National Water Research Institute, Burlington, ON. Cassiani, G., Z.J. Kabala, and M.A. Medina, Jr., 1999: Flowing partially penetrating well: Solution to a mixed- Rudd, N.C., and Z.J. Kabala, 1997: Response of a partially type boundary value problem, Advances in Water Res., penetrating well in a heterogeneous aquifer: Integrated 23, pp. 59-68. well-face vs. uniform well-face flux boundary conditions, Jl. of Hydrology, 194, pp. 76-94. Dougherty, D.E., and D.K. Babu, 1984: Flow to a partially penetrating well in a double- reservoir, Water Shapiro, A.M., and P.A. Hsieh, 1998: How good are Resources Research, 20, pp. 1116-1122. estimates of transmissivity from slug tests in fractured rock? Ground Water, 36(1), pp. 37-48. Hvorslev, M.J., 1951: Time lag and soil permeability in groundwater observations, U.S. Army Waterways Sudicky, E.A., K. MacQuarrie and C.J. Neville, 1990: Experiment Station, Bulletin 36, Vicksburg, MS. AQUITEST, v. 3.2, Dept. of Earth Sciences, University of Waterloo, Waterloo, ON, Jacob, C.E., and S.W. Lohman, 1952: Nonsteady flow to a well of constant in an extensive aquifer, Talbot, A., The accurate numerical integration of Laplace Trans. American Geophysical Union, 33(4), pp. 559- transforms, Jl. Institute of Mathematical Applications, 569. 23, pp. 97-120. Lugeon, M., 1933: Barrages et Geologie, Dunod, Paris. Tavenas, F., M. Diene, and S. Leroueil, 1990: Analysis of the in situ constant-head permeability test in clays, Novakowski, K.S., 1993: Interpretation of the transient flow Canadian Geotechnical Jl., 27, pp. 305-314. rate obtained from constant-head tests conducted in situ in clays, Canadian Geotechnical Jl., 30, pp. U.S. Bureau of Reclamation, 1974: Earth Manual, U.S. 600.606. Dept. of the Interior, Washington, DC. Ziegler, T.W., 1976: Determination of Rock Permeability, U.S. Army Waterways Experiment Station, Technical

Report S-76-2, Vicksburg, Mississippi.

Figure 1. Schematic of constant-head test

Figure 2. Definition sketch for rigorous analysis

1 1

Novakowski (1993) Correct analytical solution

Finite Element Solution Finite Element Solution

Dw Dw

0.1 0.1

0.4 0.4

Dimensionless flow rate, Q rate, flow Dimensionless

Dimensionless flow rate, Q Dimensionlessrate, flow

0.8 0.8 fully penetrating, ρ = 1 fully penetrating, ρ = 1

0.1 0.1 1 2 4 6 8 1 2 4 6 8 10 10 10 10 10 10 10 10 Dimensionless time, t Dimensionless time, tD D

Figure 3. Comparison between Novakowski (1993) Figure 4. Comparison between correct and and FEM solutions for ρ = 1, 0.8, 0.4, and 0.1 FEM solution for ρ = 1, 0.8, 0.4, 0.1

-3 10 10

Isotropic Aquifer Kz = Kr

ρ = 1, fully penetrating Kz /Kr =1 0.5 -4 Kz /Kr =0.1 10 Kz /Kr =0.01

/s) 0.2 3 Kz /Kr =0.001 0.1 true

/ K 1

0.05 est K 0.02 discharge (m -5 0.01 10

-6 10 1 2 4 6 0.1 10 10 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time (s) L/B Figure 5. "Perfect" data for problem of a partially Figure 6. Conventional interpretation of data penetrating well with ρ = 1,0.5, 0.2, 0.1, 0.05, from a partially penetrating well

0.02, 0.01, and Kz=Kr