General Solution for Tidal Behavior in Confined and Semiconfined Aquifers Considering Skin and Wellbore Storage Effects

14 Tidal analysis provides a cost-effective way of estimating aquifer properties. Tidal response 15 models that link aquifer properties with tidal signal characteristics, such as phase and ampli16 tude, have been established in previous studies, but none of the previous models incorporate 17 the skin effect. It is found in this study that the skin effect and the wellbore storage ef18 fect can have significant influence on the results of tidal analysis and should be included 19 in tidal response models. New models are proposed with skin and wellbore storage effects 20 fully incorporated, so that aquifer information can be assessed more accurately based on 21 tidal analysis. The models can be applied to confined aquifers with only horizontal flow or 22 semiconfined aquifers with both horizontal flow and vertical flow. For confined aquifers, the 23 new model indicates that positive skin leads to larger phase lag between the tidal response 24 the the theoretical tide, and negative skin can reduce the phase lag or even cause a phase 25 advance. For semiconfined aquifers, both the skin effect and the vertical flow affect the phase 26 difference between the tidal response and the theoretical tide, and with the proposed model, 27 contribution from these two sources can be separated and analyzed independently, making 28 it feasible to evaluate semiconfined aquifer properties considering both factors. Increasing 29 wellbore storage causes larger phase lag or smaller phase advance for both types of aquifers. 30 Real-world examples for confined and semiconfined aquifers are analyzed respectively to 31 demonstrate practical applications of the proposed models. 32

1. Incorporation of wellbore storage and skin effect into tidal analysis and its application, 89 considering both confined and semiconfined aquifer. 2. If the aquifer transmissivity is known, wellbore storage or skin can be quantified 91 using the Earth tide analysis without the need for well flow test or pumping test. 92 Determination of wellbore storage and skin is important for pressure transient analysis 93 and wellbore and aquifer property estimation. The conventional way of evaluating 94 wellbore storage and skin involves interpretation of well test data or pumping test data 95 (Gringarten et al., 1979;Ramey Jr, 1970;Cinco-Ley & Samaniego, 1977). However, 96 when well test data or pumping test data are not available (e.g. when the well is 97 closed and static), conventional methods cannot be applied. As proposed in this 98 study, an alternative method to determine wellbore storage or skin takes advantage 99 of the Earth tide analysis. The Earth tide effect influences pressures in closed wells or 100 water levels in open wells regardless of whether the well is active or not. By modelling  where the subscript i corresponds to each harmonic tidal constituent. H is the amplitude, 128 and ω is the tidal frequency. χ is an astronomical argument (Schwiderski, 1980). δ is the  Table 1 (Melchior, 1966; D. E. Cartwright & Tayler, 1971; D. Cartwright & Eden, 1973). 133 The observed tidal responses (e.g. water level or pressure variations) follow a form When the aquifer is confined, vertical flow is prevented by impermeable layers both 146 above and below the aquifer. A schematic of a confined aquifer system is shown in Figure   147 1. The well in Figure 1 is closed, and the target aquifer is penetrated by the well entirely.

148
Tidal forces cause a cyclic pressure fluctuation at the wellbore. r c and r w are casing radius 149 and wellbore radius respectively. h is the thickness of the target aquifer. The damaged zone 150 that causes the skin effect has a radius of r s , and the skin effect results in a pressure drop 151 ∆p s at the wellbore. The skin factor can be defined as: where p is the excess pressure in the aquifer above the initial baseline pressure.

154
It is assumed that outside the damaged zone the aquifer is isotropic, homogeneous and 155 laterally extensive. The flow transient in a confined aquifer system ( Figure 1) under the 156 cubic tidal stress σ t is governed by the following equation: where µ is the fluid viscosity. c t is the total compressibility, and B is Skempton's coefficient.

159
The outer boundary condition for the laterally extensive aquifer (r = ∞), as shown in Figure   160 1, can be expressed as: The inner boundary conditions involving the skin effect and the wellbore storage effect are: also follow a cyclic pattern.
where i is the imaginary unit. ω is the tidal frequency, and σ 0 (ω), p 0 (r, ω) and p w0 (ω) are 184 complex amplitudes of the cubic tidal stress, fluid pressure and wellbore pressure fluctua-185 tions, respectively. By inserting Equations 13-15 into Equations 9-12, we can reduce the 186 governing equation to an ordinary differential equation: with the following boundary conditions: The general solution to Equation 16 is: where I 0 and K 0 are the zero-order modified Bessel function of the first and second kind re-197 spectively. The outer boundary condition gives that A 1 = 0. From the boundary condition,

198
Equation 19, the constant A 2 can be obtained as: where C D is the dimensionless wellbore storage coefficient and τ = 2π/ω is the period of 206 fluctuation. In this case, T D and S D can be comprehended as dimensionless transmissivity 207 and storativity, and they are related with conventional aquifer transmissivity and storativity 208 through the following equations: where T is the conventional aquifer transmissivity (T = kh µ ρg), which is defined as the prod-212 uct of hydraulic conductivity and aquifer thickness. S is the conventional aquifer storativity 213 (S = φc t hρg) and is the product of aquifer specific storage and thickness. ρ is water density, 214 and g is the gravitational acceleration.

215
From the inner boundary condition, Equation 18, p w0 is obtained as: We can see from Equation 27 that at the wellbore the pressure response to tidal forces is a 219 function of not only aquifer storativity and transmissivity but wellbore storage coefficient 220 and skin factor as well. As a result, the solution provides a way to estimate wellbore storage 221 coefficient and skin factor from tidal analysis given aquifer storativity and transmissivity. On  The key difference between a semiconfined aquifer and a confined aquifer is the existence  It is assumed that both the overlaying permeable aquitard and the target aquifer are 240 laterally extensive, and that the permeable aquitard has negligible storage and is incom- 245 The inner boundary conditions with skin effect and wellbore storage are given by Equa-247 tions 6 and 7. The outer boundary condition at r = ∞, however, is not Bσ t any more due 248 to the influence of the vertical flow. Instead, the outer boundary follows a cyclic function 249 p ∞ (t) with the same frequency as σ t but a different amplitude p ∞0 (ω): The relationship between p ∞0 and σ 0 can be found by inserting Equations 13 and 29 into 252 the governing Equation 28, which yields: and p ∞0 is given as: By defining: with the homogeneous outer boundary condition: and p w is given by Equation 15. Then Equation 33 becomes an ordinary differential equation 266 in terms of p 0 (r, ω): with the following boundary conditions: The solution to Equation 36 considering the outer boundary condition, Equation 37, is: From the boundary condition, Equation 39, we have: where β D is defined as: From the inner boundary condition, Equation 38, p w0 is obtained as: Using Equation 31, we can express p w0 as: where |z| and arg(z) are the modulus and the argument of the complex number z, respec-309 tively (Sato, 2015). A is the ratio of amplitude of wellbore pressure fluctuation p w0 to the 310 theoretical tidal fluctuation Bσ 0 , which is also the outer boundary condition. The phase 311 shift η is the difference in phase angles of p w0 and σ 0 . When the wellbore pressure response 312 lags behind the tidal stress disturbance, η becomes negative. In contrast, positive phase 313 shift (phase advance) indicates wellbore pressure response leads the tidal stress.

314
For fixed values of skin factor and S D , the amplitude ratio A and the phase shift η can 315 be plotted as a function of T D , as shown in Figure 3 when the skin factor is set to zero.

316
Note that when the skin effect does not exist (i.e. skin factor is zero), the profiles of A and 317 η shown in Figure 3 are exactly the same as those in the paper by Hsieh et al. (1987).

318
When the skin factor is nonzero, however, the profiles of A and η deviate from those 319 when s is zero. Figure 4 a Table 2 and Table 3. : manuscript submitted to Water Resources Research A decreases and η becomes more positive (larger phase advance) when s decreases from zero to a negative value positive skin A decreases and η becomes more negative (larger phase lag) when s increases from zero to a positive value    Under the second condition, T D varies with ω and k is kept constant. When K /b 395 is 10 −5 , which is relatively small in this case, the profile of A is almost the same as that 396 shown in Figure 4a. When K /b increases to 10 −3 , however, the curves become bell-shaped.

397
In Figure 9, the solid curves and dashed curves represent results for confined aquifer and 398 semiconfined aquifer respectively, and each color represents a skin factor value. It can be 399 seen from Figure 9 that the difference between A for confined aquifer and A for semiconfined had existed in this case. In this study, it is found that with the same input parameters, 438 nonzero skin factor can result in a significantly different estimation of aquifer transmissivity.

439
In Figure 11, the solid blue curve and the solid red curve represent the relation between 440 the aquifer transmissivity T and the phase difference η when skin is assumed to be zero, and  shown by the dash-dot lines and the dashed lines respectively. It can be seen from Figure   500 12 that when s = −5, the estimated range of K /b reduces to 9.2 × 10 −11 − 9.1 × 10 −10 501 m 2 /s, and when s = 5, the range increases to 1.5 × 10 −10 − 1.4 × 10 −9 m 2 /s.

502
Positive skin factor increases the estimation, because with a larger skin factor, it requires  is not always negative -a phase advance can occur when the skin factor is negative 517 with enhanced permeability around the wellbore, and the phase advance with more 518 negative skin is larger.
The dimensionless forms are: where p i is the initial aquifer pressure, and the rest of the variable are the same as those only. The boundary conditions, Equations A11 and A12, give that: The solution in the Laplace space is: is the general solution when the flow rate is nonzero. When the flow rate is 570 zero, however, we cannot transform the variables into the dimensionless form and need to 571 solve the system of equations in its dimensional form using the Laplace transform, which 572 process is the same as the steps illustrated in this appendix. The solution for closed well in 573 the dimensional form is: can be inverted numerically to attain the profiles of A and η. The numerical 577 inversion is based on the method introduced by Talbot (1979). In Figure A1, the dots 578 represent results from numerical inversion of the Laplace transform, and the solid lines 579 represent results from Section 2. It can be seen that the Laplace transform gives results 580 consistent with those from Section 2, thus verifying our solution.