# A New Method for the Evaluation of Well Rehabilitation from the Early Portion of a Pumping Test

^{*}

## Abstract

**:**

## 1. Introduction

_{w}≠ 0), the “wellbore storage effect” (the volume of water inside a wellbore), and its influence on pumping test evaluations, was published by Papadopulos-Cooper [12,13], who presented a close form solution for the drawdown in a large-diameter well, while also accounting for the effect of wellbore storage.

_{F}

_{.}This factor is used to characterize all additional resistance in the pumping well and in the surrounding well: the larger the skin factor, the greater the drawdown loss. The permeability damage to aquifers caused by drilling operations with drilling mud ranges from a few millimeters to several centimeters and during ageing (exploitation) of the well, the permeability damage is caused by physical, chemical, and biological processes; moreover, the skin zone can reach several meters [11]. The additional resistance (the skin effect) and finite volume of a wellbore (wellbore storage) are the two main factors that influence the pumping test data measured at a well. The drawdown caused by additional resistance (the skin effect) was introduced, for the first time in petroleum engineering, by van Everdingen and Hurst [21,22,23]. Since then, many authors in the fields of petroleum and groundwater hydraulics have published articles focused on the problem of the influence of the skin effect and wellbore storage on the measured values of the real drawdown or pressure drops in a well [24,25,26,27,28,29,30,31]. In 1970, Agarwal [32] presented a fundamental study of the importance of wellbore storage and skin on the short-time transient flow and introduced the idea of log–log type curve matching to analyze pressure data for a well dominated by the skin effect and wellbore storage. Subsequently, many other type curve methods have been presented. A type curve method is a graphical representation of the theoretical solution to flow equations. A type curve analysis consists of finding the theoretical type curve that “matches” the actual response from a test well and reservoir when subjected to changes in production rates or pressures. These tests are usually presented in terms of dimensionless pressure, dimensionless time, dimensionless radius, and dimensionless wellbore storage rather than real variables. The reservoir parameters (such as permeability) and well parameters (for example, skin) can be evaluated from the dimensionless parameters defining that type of curve [33,34,35,36,37,38,39,40,41,42,43]. Among the many type curve methods, two are mentioned further: Gringarten type curves [36] and the pressure derivative method [3]. There are three dimensionless groups that Gringarten uses when developing a type curve: dimensionless pressure, the ratio of dimensionless time and dimensionless wellbore storage, and the dimensionless characterization group, C

_{D}e

^{2S}

_{F}. All type curve solutions are obtained for the drawdown solutions. The Gringarten type curve can also be used for a gas system, but the dimensionless pressure drop and time must be redefined. Bourdet defined the pressure derivative as the derivative of dimensionless pressure with respect to dimensionless time. From a mathematical point of view, the derivative of pressure with respect to time is even more fundamental than pressure itself, as the pressure’s partial derivative appears directly in the diffusivity equation describing the transient fluid flow in an aquifer system. Bourdet and his co-authors proposed that flow regimes can have clear characteristic shapes if the “pressure derivative”, rather than pressure, is plotted against time in the log-log coordinates. Since the introduction of the pressure derivative type curve, well-testing analysis has been greatly enhanced. The use of pressure derivative type curves offers the following advantages: flow regimes have clear characteristic shapes on the derivative plot, and the derivative plots are able to be displayed in a single graph with many separate characteristics that would otherwise require different plots. This derivative approach improves the definition of the analysis plots and, therefore, the quality of the interpretation and quantitative estimation of aquifer properties becomes more reliable. By using this method, interpretation is not only simple but also more accurate [10]. Interesting methods for evaluating the transmissivity, wellbore storage, and skin factor from the early-time portion of pumping test data was published in [31]. If the skin effect and wellbore storage dominate the drawdown data, and if testing has been conducted long enough, two semilogarithmic straight lines are normally obtained. For an infinite aquifer (where the boundary effect does not interfere with the pumping test data), the second straight line is the appropriate Cooper–Jacob straight line. The first straight line is the line of the maximum slope and is typical for early time points. The slope of this straight line is affected by wellbore storage and skin effect. The new method for evaluating the skin factor in a single well fully penetrating the confined aquifer from the early portion of a pumping test is derived under the following assumptions: non zero additional resistance at the pumping well, a finite well radius (wellbore storage effect), and the well being situated in an infinite aquifer (where no observation well is available). The solution of the general partial differential equation of a symmetrical radial water flow to a well in Laplace space was used [15]. The Laplace transform was numerically inverted by the Stehfest algorithm 368 [44], which is very often used in groundwater flow problems [32,43,45,46,47,48,49,50]. Here, we present a correlation of the dimensionless drawdown in dimensionless time for the intersection the first straight-line with the horizontal axis (log time) in a semilogarithmic graph as a function of the dimensionless wellbore storage constant and skin factor. Using this correlation, a new procedure for evaluating the skin factor is presented. This new method to evaluate the skin factor from the early portion of a pumping test is not intended to replace any of the approaches presented in the literature but rather to supplement them.

## 2. Materials and Methods

_{F}) from the early part of a pumping test, dimensionless parameters were used, as is commonly done in petroleum engineering [4,5,32,50]. Dimensionless parameters simplify the solution of hydrodynamic tests on wells by including parameters such as hydraulic conductivity, transmissivity, aquifer storage coefficient, time, well radius, etc. into the dimensionless parameters. By introducing dimensionless parameters, we reduce the number of unknowns, ensuring that the solution is independent of the unit systems being used. To solve the groundwater flow to the well, these dimensionless parameters were defined as follows [4,48,49]:

^{3}T

^{−1}]; T is the transmissivity [L

^{2}T

^{−1}]; H is the hydraulic head within the radius of well influence [L] and h(r, t) is the hydraulic head at time t and at a distance r from the well axis [L]; and r

_{D}is dimensionless radius and t

_{D}is dimensionless time.

_{W}is the hydraulic head in the well at time t [L], and H is the initial hydraulic head in time equal zero [L].

_{w}is the well radius [L]; S is the storage coefficient (aquifer storativity) [–]; and t is time [T].

_{D}

_{w}is the well radius [L].

^{2}] defined in [20].

^{*}of the “first straight line” with the horizontal axis, log t

_{D}:

_{w}→ 0) and whose additional resistance in the well and its immediate vicinity is zero. In this article, we will not focus on ideal wells and the evaluation of hydrodynamic tests on these wells. The basic literature dealing with the evaluation of hydrodynamic tests on ideal wells includes [3,4,5,6,8,9,50].

#### 2.1. Real Wells

_{ws}from the well volume is equal to zero. In more detail, Papadopulos and Cooper [12] (for groundwater) and Ramey (for petroleum engineering) [20] described and discussed the impact of the well volume on the pumping test during the first part of a pumping (build-up) test. The effect of the well radius (wellbore storage) on dimensionless drawdown gradually diminishes (Figure 1) as the time of pumping increases. Figure 1 shows the relationship between the pumped amount from the well volume, Q

_{ws}

_{,}and the inflow rate from the aquifer, Q

_{aq}

_{,}as a function of time.

_{B}is time for the arbitrarily selected point B on the “unit” slope line (Figure 2) [T], and s

_{B}is the drawdown at time t

_{B}[L].

- (a)
- By the clogging of pores (s
_{1}) with, e.g., a fine material, which reduces the flow rate of the porous environment or disrupts the original internal structure of the porous environment in the vicinity of the wellbore during digging and equipping (it decreases the porous environment’s permeability) in rotary drilling, the result of which is so-called sludge bark; in the case of impact drilling, the porous environment in the vicinity of the well is compacted, thereby reducing throughput [52,53,54]. - (b)
- Through a reduction in the wellbore wall cross-section (s
_{2}) for the water inflow where the borehole wall is formed by a filter, perforated casing, etc., by trapping rock particles or backfill in filter openings, including chemical incrustation and the blockage of filter openings by microorganisms and bacteria [55,56,57]. - (c)
- Via the friction (s
_{3}) of water on the borehole walls and its internal friction (this group also includes the additional resistance arising from the turbulent flow regime of the water inside the borehole and the turbulent flow in the aquifer, especially in the vicinity of the pumping well. - (d)
- Where appropriate, other types of additional resistance occur.

_{skin}is the total additional drawdown in the well (“the skin drawdown”) caused by the additional resistance in the well and its immediate vicinity (in the so-called skin zone). The separation of the individual additional types of resistance involved in the skin effect is very problematic; therefore, the total dimensionless additional resistance coefficient, S

_{F}(in the petroleum literature, referred to as the skin factor), is commonly used to express the total additional resistance. The total drawdown of the water level measured in the borehole during the pumping test can be expressed (when neglecting the additional resistance resulting from the turbulent flow regime) by the relation:

_{w}is the total drawdown in the pumping well [L]; s

_{te}is the theoretical drawdown of the water level in an “ideal” well (zero additional resistances) [L]; and s

_{skin}is the additional drawdown of water in the wellbore due to additional resistance [L].

_{3}(its share in the total additional drawdown is negligible), the magnitude of the additional drawdown caused by additional resistance is dependent on the pumping rate, Q, with the linear relationship [21]

_{F}is the dimensionless coefficient of additional resistance (skin factor) [–]; and s

_{skin}is the drawdown caused by additional resistance [L].

_{w}is the wellbore radius [L]; and S

_{F}is the coefficient of additional resistance (skin factor) [–].

_{F}and adjusting, we obtain:

#### 2.2. Deriving the New Correlation to Evaluate the Skin Factor from the First Straight Line

- The gravitational forces are negligible;
- A constant density and viscosity of water;
- The aquifer has an infinite areal extent;
- The pumping well penetrates the full thickness of the aquifer;
- The flow to the pumping well is horizontal;
- The flow is unsteady;
- The diameter of a pumping well is very small (negligible), allowing the storage in the well to be neglected;
- The well is pumped with constant rate Q;
- The aquifer is horizontal and bounded on the bottom and top by impermeable layers (a confined aquifer);
- The aquifer flow to the pumped well is radial and laminar, so Darcy’s law is applied;
- The confined aquifer is homogeneous and isotropic;
- The height of an aquifer (where the flow to the well is constant and has a size b transmissivity, T, and storability (aquifer storage), S) is constant over time and space;
- The water supply from the aquifer to the well changes during the pumping test from Q
_{aq}= 0 to the final inflow, Q_{aq}= Q = const.; - Before pumping begins (i.e., for t = 0), the hydraulic head is constant in all points of the aquatic environment and equals H; this also applies to the water level at a well.

- The well possesses its final volume, and, at the beginning of pumping, the effect of the water volume in the wellbore influences the drawdown and must be considered;
- The wellbore storage coefficient is constant and does not change during pumping;
- The influence of additional resistance occurring in the wellbore itself and in its immediate vicinity is considered (the width of the perpendicular zone can reach up to about 2–7 m).

_{D}), where p is a complex variable; and F(p) is the image of the given object, f(t

_{D}). In Laplace space, the solution of Equation (17) has the following form:

_{D}and t

_{D}in Laplace space; p is a complex variable; K

_{0}is the modified Bessel function of the second kind with an order of zero (imaginary argument); K

_{1}is the modified Bessel function of the second kind with an order of one (imaginary argument); C

_{D}is the dimensionless coefficient of wellbore storage; and S

_{F}is the dimensionless coefficient of additional resistance (skin factor). When evaluating K

_{0}and K

_{1,}dimensionless drawdown values were taken from [59]. For the inverse Laplace transformation, the Stehfest algorithm 368 [44] was used, which is very often used in petroleum engineering and groundwater hydraulics [7,60]. From [7] we can obtain:

_{D}from Equation (23) by means of Equations (24)–(26), value n is 10, as used in [7]. Stehfest’s algorithm 368 was used to calculate the dimensionless drawdown, which depends on the skin factor and wellbore storage. Calculations were done for skin factor S

_{F}= 0, 2, 4, 6, 8, 10, 12.5, 15, 20, 30, 35, 40, 50, and for dimensionless wellbore storage, C

_{D}= 10, 50, 5 × 10

^{2}, 1 × 10

^{3}, 5 × 10

^{3}, 1 × 10

^{4}, 5 × 10

^{4}, 1 × 10

^{5}, 5 × 10

^{5}, 1 × 10

^{6}, 5 × 10

^{6}, and 1 × 10

^{7}. For clarity of the graphs, Figure 4 illustrates the dependencies of s

_{D}on log t

_{D}for C

_{D}= 1 × 10

^{2}, 1 × 10

^{4}, 1 × 10

^{7}, and S

_{F}= 0, 10, 20, 30, and 40.

_{ZD}. The “second straight line” in the semilogarithmic plot of s

_{D}vs. log t

_{D}appears only in the late periods of the graph and corresponds to the area that can be evaluated by the Cooper–Jacob semilogarithmic approximation [2]. The same two straight lines can be observed in field pumping tests on the semilogarithmic graph drawdown of s vs. log t [31].

_{ZD}section increases; also, if the dimensionless wellbore storage, C

_{D}, increases then the curves s

_{D}vs. log t

_{D}are moved in a positive direction (i.e., to the right).

_{D}

^{*}, for dimensionless time t

_{D}* was selected, which is the time of the intersection of the first straight line with the timeline axis (Figure 5). Dimensionless drawdown s

_{D}* depends on the dimensionless wellbore storage C

_{D}and the coefficient of additional resistance, S

_{F}. In this way, we can find a correlation between the dimensionless drawdown, s

_{D}*; the dimensionless wellbore storage, C

_{D}; and the skin factor, S

_{F}.

## 3. Results

_{D}* for the dimensionless wellbore storage, C

_{D}= 1 × 10

^{2}, 5 × 10

^{2}, 1 × 10

^{3}, 5 × 10

^{3}, 1 × 10

^{4}, 5 × 10

^{4}, 1 × 10

^{5}, 5 × 10

^{5}, 1 × 10

^{6}, 5 × 10

^{6}, and 1 × 10

^{7}and for the dimensionless skin factor S

_{F}= 1, 2, 4, 6, 8, 10, 15, 20, 25, 30, 35, 40, 45, 50, 60, 70, 80, 90, and 100 (see Table 1).

_{D}*, vs. the skin factor, S

_{F}(for dimensionless wellbore storage C

_{D}= 10

^{5}). For the sake of clarity, only the values S

_{F}= 0, 20, 30, and 50 are used in Figure 6.

_{D}

^{*}, vs. the skin factor, S

_{F}

_{,}for the selected values of C

_{D}

_{.}

_{D}* and S

_{F}are a set of parallel straight lines. The equations for the system of straight lines can be expressed by:

**and b**

_{si}**for all straight lines (as shown in Figure 7) were evaluated by means of the least squares method. Based on Figure 7 and Table 1, it follows that these coefficients are a function of the dimensionless skin factor, S**

_{si}_{F}. The dependence a

_{si}= f(S

_{F}) and also the dependence.

_{si}= f(S

_{F}) were evaluated by similar equations, as stated in [52]:

_{D}, the coefficient a

_{si}was constant and equal to 0.166.

_{si}for the given values of dimensionless wellbore storages, C

_{D,}were evaluated in the same way as those in [52]. The equation for the solved dependence has the form:

_{si}, using Equation (29) for dimensionless C

_{D}drift, are shown in Table 2.

_{S}i vs. the logarithm of the dimensionless wellbore storage coefficients represented by C

_{D,}from Table 2 was found to be linear (see Figure 8).

_{s}vs. log C

_{D}has the form:

_{1S}and b

_{2S}

_{,}we use a modified version of Equations (28) and (29):

_{1S}and b

_{2S}were evaluated via Equations (31) and (32):

_{1S}= 0.1908, b

_{2S}= 0.2681.

_{D}* in the form:

#### Field Test: Rehabilitation of the Dug S-V well—Veselí Nad Lužnicí

^{2}s

^{−1}and a storage coefficient S = 0.074.

_{7}was located outside the area affected by additional resistance with an internal diameter of 50 mm. The shape of the graph of the pumping test before rehabilitation showed that the yield of the well was very low (a large increase on the well’s shell subsequently emerged from the evaluation). The casing of the well had a large number of inlet openings, and the inspection showed that the inlet openings were, for the most part, clogged with sand, clay, and pebbles, or were overgrown with hard increments. The environment behind the borehole of the well (according to observation of the water level in the casing boreholes) was rounded (low permeability). Based on these findings, the following rehabilitation procedure was performed. First, the blasting of observation boreholes P

_{i}and R

_{i}was carried out using cleaning heads. Thereafter, pressure regeneration was carried out using the observation wells, which had an effect on the well surroundings. After the water level was lowered to approximately 2.5 m above the bottom, mechanical cleaning of the overgrown inlet holes was carried out in the well. At the same time, pressure regeneration of the surroundings of the well was carried out using pressure surges in the observation wells made especially for this purpose, and the impurities were drained from the inside of the well.

_{w}= 4.5 m. After rehabilitation, a pumping test of 240 min at a pumping rate of 3.7 L/s was performed, and the drawdown at the end of pumping was s

_{w}= 1.03 m. One year after the rehabilitation was performed, a pumping test of 75 min at a constant pumped rate of 3.52 L/s was carried out to determine the development of the well’s ageing. The drawdown at the end of the pumping test was s

_{w}= 1.29 m (see Table 3). During the pumping tests, water samples were taken for complete chemical and bacteriological analyses (alkalinity, acidity, pH, and fixation of iron (II)), which were performed on site.

_{z}) segment with the timeline axis (see Figure 5).

## 4. Discussion and Conclusions

_{D}

^{*}of the first straight-line (see Figure 3) of the pumping test in a semilogarithmic graph as a function of the dimensionless wellbore storage, C

_{D}

_{,}and the skin factor, S

_{F}(Equation (12)). The derivation was undertaken by applying an approximate solution of the partial differential equation using dimensionless parameters for the symmetrical radial flow problem in an infinite confined aquifer with a constant thickness to the fully penetrating well, pumped with a steady flow rate. The solution was obtained by using the Laplace transform in conjunction with the Stehfest algorithm 368, the approximate method for the numerical inversion of the solution in Laplace space. The skin factor, S

_{F}, was evaluated from the early portion of pumping test data (the part of the pumping test before the Cooper–Jacob semilogarithmic straight-line was achieved) by means of Equation (34).

_{w}vs. log t) of the early portion of the pumping test can be evaluated without using type curve matching. This new procedure is particularly useful when the early-time slope of the first straight-line has been observed. The usefulness of the derived correlation s

^{*}(vs. S

_{F}and C

_{D}) has been demonstrated in the field example through the evaluation effect of mechanical rehabilitation at a S-V well (Veselí nad Lužnicí). The validity of this new method was checked by comparing the skin factor calculated by the new method with the Cooper–Jacob semilogarithmic method (Equation (16)) (see Table 1). We evaluated the differences between the skin factor determined from Equation (34) and that determined from Equation (16). Comparing these skin factors demonstrates that the estimate of S

_{F}is in fair agreement with the other values. For the field example, the agreement is surprisingly good. The differences ranged between 7% and 12% (Table 4).

_{w}vs. log t can fully develop into “the first straight line” on the semilogarithmic plot. This method may be used when other techniques do not provide satisfactory answers or if the late-time portion of the pumping test is missing and the Cooper–Jacob method or type curve cannot be used. For the application-derived procedure, the storage coefficient and transmissivity of the aquifer must be known, but to evaluate the effects of rehabilitation on the well and accurately control age, this procedure is sufficient. We applied the stated method to several other evaluations of rehabilitation wells. In all instances, the results of the Cooper–Jacob method were as good as those in the previous field example comparisons. The differences were a maximum of 25% (in groundwater, this is very good agreement). However, the derived procedure presented in this paper is not proposed to replace the current methods used in groundwater hydraulics and petroleum engineering. To evaluate the skin factor from the initial pumping test section, it is necessary to record a decrease in the water level in the well at sufficiently short intervals immediately from the start of pumping. It is advisable to select a monitoring interval of 1 s until the beginning of the first straight line in the semilogarithmic chart, and then it is possible to observe the reduction in time steps of 5–10 s. This is very important for the application of the method. By correctly applying the derived method and correctly determining the first straight line (the first straight line can be determined in almost all instances), the intersection time of the first slope with the log t axis can be determined with a high degree of confidence. The application of this method eliminates the main problem of the type curve method because lines of different types are often so similar that it is very difficult, and often impossible, to find conformity with the actual pump test curve. For this reason, quantitative information cannot be obtained with the same degree of accuracy as information obtained via calculations using conventional Cooper–Jacob approximations. Due to the cost of extended pumping tests, the modern trend has been towards the development of hydrodynamic tests for the pertinent analysis for early-period data for information obtained prior to the usual straight-line of the well test [5]. These data can be analyzed using the “derived method” for economic or practical reasons. The method derived here contributes to the possible evaluation of the skin factor. The advantage of this method is that it can be used in cases where the application of other methods is difficult or even impossible.

_{t}is the total compressibility (psi

^{−1}), r

_{w}is the wellbore radius (ft), and C is the wellbore storage coefficient (STP/psi).

_{F}can be evaluated.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- 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
**1935**, 16, 519–524. [Google Scholar] [CrossRef] - Cooper, H.H.; Jacob, C.E. A generalized graphical method for evaluating formation constants and summarizing well-field history. Trans. Am. Geophys. Union
**1946**, 27, 526–534. [Google Scholar] [CrossRef] - Bourdet, D. Well test analysis: The use of advanced interpretation models. Handb. Pet. Explor. Prod.
**2002**, 3, 1–426. [Google Scholar] [CrossRef] - Lee, J. Well Testing. In SPE Textbook Series Vol. 1; Society of Petroleum Engineers: Richardson, TX, USA, 1982; p. 159. ISBN 978-089520-317-5. [Google Scholar]
- Horne, R.N. Modern Well Test Analysis: A Computer Aided Approach, 4th ed.; Alto, P., Ed.; Petroway, Inc.: Palo Alto, CA, USA, 1995; p. 185. ISBN -0-9626992-09. [Google Scholar]
- Kresic, N. Hydrogeology and Groundwater Modeling, 2nd ed.; CRC Press, Taylor and Francis: Boca Raton, FL, USA, 2007; ISBN 13 978-0849333484. [Google Scholar]
- Watlton, W.C. Aquifer Test Modeling, 1st ed.; CRC Press: Boca Ralton, FL, USA, 2007; p. 240, ISBN-13: 978-1-4200-4292-4. [Google Scholar]
- Batu, V. Aquifer Hydraulics: A Comprehensive Guide to Hydrogeologic Data Analysis; John Wiley & Sons: New York, NY, USA, 1998. [Google Scholar]
- Kruseman, G.P.; de Ridder, N.A. Analysis and Evaluation of Pumping Test Data, 2nd ed.; IILRI: Wageningen, The Netherlands, 2008; pp. 1–372. [Google Scholar]
- Tongpenyai, Y.; Raghavan, R. The effect of wellbore storage and skin on interference test data. J. Pet. Technol.
**1981**, 33, 151–160. [Google Scholar] [CrossRef] - Novakowski, K.S. A Composite analytical model for analysis of pumping tests affected by well bore storage and finite thickness skin. Water Resour. Res.
**1989**, 25, 1937–1946. [Google Scholar] [CrossRef] - Pasandi, M.; Samani, N.; Barry, D.A. Effect of wellbore and finite thickness skin on flow to a partially penetrating well in a phreatic aquifer. Adv. Water Resour.
**2008**, 31, 383–398. [Google Scholar] [CrossRef] - Papadopulos, I.S.; Cooper, H.H. Drawdown in a well of large diameter. Water Resour. Res.
**1967**, 3, 241–244. [Google Scholar] [CrossRef] - Moench, A.F. Transient flow to a large-diameter well in an aquifer with storative semiconfining layers. Water Resour. Res.
**1985**, 21, 1121–1131. [Google Scholar] [CrossRef] - Chen, C.; Lan, C. A simple data analysis method for a pumping test with skin and wellbore storage effect. Terr. Atmos. Ocean. Sci.
**2009**, 20, 557–562. [Google Scholar] [CrossRef][Green Version] - Ramey, H.H., Jr. Short-time well test data interpretation in the presence of skin effect and wellbore storage. J. Pet. Technol.
**1970**, 22, 97–104. [Google Scholar] [CrossRef] - Yeh, H.D.; Yang, S.Y.; Peng, H.Y. A new closed-form solution for a radial two-layer drawdown equation for groundwater under constant-flux pumping in a finite-radius well. Adv. Water Resour.
**2003**, 26, 747–757. [Google Scholar] [CrossRef] - McKinley, R.M. Wellbore transmissibility from afterflow-dominated pressure buildup data. J. Pet. Technol.
**1971**, 23, 863. [Google Scholar] [CrossRef] - Kabala, Z.J. Sensitivity analysis of a pumping test on a well with wellbore storage and skin. Adv. Water Resour.
**2001**, 24, 483–504. [Google Scholar] [CrossRef] - Ramey, H.J.; Agarwal, R.G. Annulus unloading rates as influenced by wellbore storage and skin effect. Soc. Pet. Eng. J.
**1972**, 12, 453–462. [Google Scholar] [CrossRef] - Van Everdingen, A.F. The skin effect and its influence on the productive capacity of a well. J. Pet. Technol.
**1953**, 5, 171–176. [Google Scholar] [CrossRef] - Hurst, W. Establishment of skin effect and its impediment to fluid flow into a well bore. Pet. Eng.
**1953**, 25, B6–B16. [Google Scholar] - Hawkins, M.F., Jr. A note on the skin effect. Trans. Am. Inst. Min. Met. Eng.
**1956**, 8, 356–357. [Google Scholar] [CrossRef] - Park, E.; Zhan, H.B. Hydraulics of a finite-diameter horizontal well with wellbore storage and skin effect. Adv. Water Resour.
**2002**, 25, 389–400. [Google Scholar] [CrossRef] - Barua, G.; Bora, S.N. Hydraulics of a partially penetrating well with skin zone in a confined aquifer. Adv. Water Resour.
**2010**, 33, 1575–1587. [Google Scholar] [CrossRef] - Jargon, J.R. Effect of wellbore storage and wellbore damage at the active well on interference test analysis. J. Pet. Technol.
**1976**, 28, 851–858. [Google Scholar] [CrossRef] - Yang, S.Y.; Yeh, H.D. Solution for flow rates across the wellbore in a two-zone confined aquifer. J. Hydraul. Eng. ASCE
**2002**, 128, 175–183. [Google Scholar] [CrossRef][Green Version] - Chen, C.S.; Chang, C.C. Theoretical evaluation of non-uniform skin effect on aquifer response under constant rate pumping. J. Hydrol.
**2006**, 317, 190–201. [Google Scholar] [CrossRef] - Basak, P. Steady non-Darcian seepage through embankments closure. J. Irrig. Drain. Div. ASCE
**1979**, 102, 435–443. Available online: http://cedb.asce.org/CEDBsearch/record.jsp?dockey=0007101 (accessed on 4 September 2019). - Sen, Z. Nonlinear flow toward wells. J. Hydraul. Eng. ASCE
**1989**, 115, 193–209. [Google Scholar] [CrossRef] - Chu, W.C.; Garcia-Rivera, J.; Raghavan, R. Analysis of interference test data influenced by wellbore storage and skin at the flowing well. J. Pet. Technol.
**1980**, 32, 623–630. [Google Scholar] [CrossRef] - Agarwal, R.G.; Al-Hussainy, R.; Ramey, H.J. An investigation of wellbore storage and skin effect in unsteady liquid flow: I. Analytical treatment. Soc. Pet. Eng. J.
**1970**, 10, 279–291. [Google Scholar] [CrossRef] - Earlougher, R.C., Jr.; Kersch, K.H. Analysis of short-time transient test data by type-curve matching. J. Pet. Technol.
**1974**, 26, 793–800. [Google Scholar] [CrossRef][Green Version] - Earlougher, R.C., Jr. Advances in well test analysis. Monogr. Ser. Soc. Pet. Eng. AIME
**1977**, 5, 264. Available online: https://trove.nla.gov.au/version/13397770 (accessed on 24 August 2019). - Bourdet, D.; Whittle, T.M.; Douglas, A.A.; Pirard, Y.M. A new set of type curves simplifies well test analysis. World Oil
**1983**, 196, 95–106. [Google Scholar] [CrossRef][Green Version] - Gringarten, A.C.; Bourdet, D.P.; Landel, P.A.; Kniazeff, V.J. A comparison between different skin and wellbore storage type-curves for early–time transient analysis. Soc. Pet. Eng. SPE Ann. Technol. C Exh.
**1979**, 1–16. [Google Scholar] [CrossRef] - Yeh, H.D.; Chang, Y.C. Recent advances in modeling of well hydraulics. Adv. Water Resour.
**2013**, 51, 27–51. [Google Scholar] [CrossRef] - Yeh, H.D.; Chen, Y.H. Determination of skin and aquifer parameters for a slug test with wellbore-skin effect. J. Hydrol.
**2007**, 342, 283–294. [Google Scholar] [CrossRef] - Mashayekhizadeh, M.D.; Ghazanfari, M.H. The application of numerical Laplace inversion methods for type curve development in well testing: A comparative study. Pet. Sci. Technol.
**2011**, 29, 695–707. [Google Scholar] [CrossRef] - Kuchuk, F.J.; Kirwan, P.A. New skin and wellbore storage type curves for partially penetrated wells. SPE Eval.
**1987**, 2, 546–554. [Google Scholar] [CrossRef] - Wattenbarger, R.A.; Ramey, H.J. An investigation of wellbore storage and skin effect in unsteady liquid flow: II. Finite difference treatment. Soc. Pet. Eng. J.
**1970**, 10, 291–297. [Google Scholar] [CrossRef] - Moench, A.F. Specific yield as determined by type-curve analysis of aquifer-test data. Ground Water
**1994**, 32, 949–957. [Google Scholar] [CrossRef] - Wen, Z.; Huang, H.G.; Zhan, H.B. Non-Darcian flow in a single confined vertical fracture toward a well. J. Hydrol.
**2006**, 330, 698–708. [Google Scholar] [CrossRef] - Stehfest, H. Algorithm 368: Numerical inversion of Laplace transforms. Commun. ACM
**1970**, 13, 47–49. [Google Scholar] [CrossRef] - Van Everdingen, A.F.; Hurst, W. The application of the Laplace transformation to flow problems in reservoirs. J. Pet. Technol.
**1949**, 1, 305–324. [Google Scholar] [CrossRef] - Pech, P.; Novotny, R. Analytical solutions for well drawdown with well losses: 1. Multiple well system near a boundary. J. Environ. Hydrol.
**2005**, 13, 1–10. Available online: http://www.hydroweb.com/journal-hydrology-2005-paper-27.html (accessed on 5 November 2019). - Novotny, R.; Pech, P. Analytical solutions for well drawdown with well losses: 2. Real well near boundary—solution by image well. J. Environ. Hydrol.
**2005**, 13, 1–7. Available online: http://www.hydroweb.com/jeh/jeh2005/pech2.pdf (accessed on 5 November 2019). - Pech, P. Estimation of a well damage or stimulation. Sci. Agric. Bohem.
**2004**, 35, 131–134. Available online: https://sab.czu.cz/en/r-8044-archive/r-8064-older-issues/r-11689-04-2004#i-28f1287e11d0452b811e7a4937989a20 (accessed on 22 September 2019). - Holub, J.; Pech, P.; Kuraz, M.; Maca, P.; Kahuda, D. Evaluation of a pumping test with skin effect and wellbore storage on a confined aquifer in the Bela Crkva, Serbia. Int. J. Water
**2019**, 13, 1–11. [Google Scholar] [CrossRef] - Raghavan, R. Well Test Analysis, 1st ed.; PTR Prentice Hall: Englewood Cliffs, NJ, USA, 1993; p. 554. [Google Scholar]
- Houben, G.; Treskatis, C. Water Well Rehabilitation and Reconstruction, 3rd ed.; McGraw Hill Professional: Two Penn Plaza, NY, USA, 2007; ISBN 0-07-148651-8. [Google Scholar]
- Adebayo, A.R.; Bageri, B.S. A simple NMR methodology for evaluating filter cake properties and drilling fluid-induced formation damage. J. Pet. Explor. Prod. Technol.
**2019**, 9. [Google Scholar] [CrossRef][Green Version] - Elkatatny, S.; Jafarov, T.; Al-Majed, A.; Mahmoud, M. Formation Damage Avoidance by Reducing Invasion with Sodium Silicate-ModifiedWater-Based Drilling Fluid. Energies
**2019**, 12, 485. [Google Scholar] [CrossRef][Green Version] - Iscan, A.G.; Kok, M.V.; Bagci, A.S. Permeability Reduction Due to Formation Damage by Drilling Fluids. Energy Sources
**2007**, 29, 851–859. [Google Scholar] [CrossRef] - Ralph, D.E.; Stevenson, J.M. The role of bacteria in well clogging. Water Res.
**1995**, 29, 365–369. [Google Scholar] [CrossRef] - Houben, G.J. Iron oxide incrustations in Wells—Part 1: Genesis, mineralogy and geochemistry. Appl. Geochem.
**2003**, 18, 927–939. [Google Scholar] [CrossRef] - Patel, M.C.; Singh, A. Near Wellbore Damage and Types of Skin Depending on Mechanism of Damage. In Proceedings of the Society of Petroleum Engineers SPE International Conference and Exhibition on Formation Damage Control, Lafayette, LA, USA, 24–26 February 2016. [Google Scholar] [CrossRef]
- Pech, P. Determination of the skin factor in the early portion of an aquifer test. J. Environ. Hydrol.
**2003**, 11, 1–9. Available online: http://www.hydroweb.com/jeh/jeh2002/pech.pdf (accessed on 5 November 2019). - Abramowitz, M.; Stegun, I.A. Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, 10th ed.; Government Printing: Wahington, DC, USA, 1972; p. 470.
- Yang, S.Y.; Yeh, H.D. Laplace-domain solutions for radial two-zone flow equations under the conditions of constant-head and partially penetrating well. J. Hydraul. Eng. ASCE
**2005**, 131, 209–216. [Google Scholar] [CrossRef] - Hošek, J.; Pokorný, P.; Prach, J.; Šída, P.; Křížek, M. Fossil thermokarst in South Bohemia (Czech Republic). Geosci. Res. Rep.
**2018**, 51, 131–139. Available online: http://www.geology.cz/img/zpravyvyzkum/fulltext/27_Hosek_180903.pdf (accessed on 5 September 2019).

**Figure 1.**Pumping rate from wellbore storage, Q

_{ws}

_{,}and groundwater inflow rate from aquifer, Q

_{aq}

_{,}as a function of time.

**Figure 2.**Graph log s vs. log t at the beginning of the pumping test when the water is pumped only from the wellbore’s own volume.

**Figure 4.**Demonstration of dimensionless drawdown s

_{D}vs. log t

_{D}(for the selected values, S

_{F}and C

_{D}).

**Figure 6.**Example of drawdown s

_{D}* from graph s

_{D}vs. log t

_{D}(for S

_{F}= 0, 20, 30, 50, and for C

_{D}= 1 × 10

^{5}).

**Figure 7.**Dependence s

_{D}* vs. skin factor, S

_{F}

_{,}for values of C

_{D}(1 × 10

^{2}, 5 × 10

^{2}, 1 × 10

^{3}, 5 × 10

^{3}, and 1 × 10

^{4}) and S

_{F}(0, 2, 4, 6, 8, and 10).

**Figure 10.**Cross-section of the S-V well with geology [61].

**Figure 12.**Semilogarithmic graphs of the pumping tests on the S-V well (

**a**) before rehabilitation, (

**b**) after rehabilitation, and (

**c**) one year after rehabilitation.

**Table 1.**Dimensionless drawdown, s

_{D}*, for the selected skin factors, S

_{F,}and the dimensionless wellbore storage, C

_{D}.

C_{D} | 1 × 10^{2} | 5 × 10^{2} | 1 × 10^{3} | 5 × 10^{3} | 1 × 10^{4} | 5 × 10^{4} | 1 × 10^{5} | 5 × 10^{5} | 1 × 10^{6} | 5 × 10^{6} | |
---|---|---|---|---|---|---|---|---|---|---|---|

S_{F} | |||||||||||

0 | 0.6 | 0.73 | 0.79 | 0.93 | 0.99 | 1.13 | 1.19 | 1.33 | 1.39 | 1.52 | |

2 | 0.94 | 1.08 | 1.14 | 1.28 | 1.33 | 1.47 | 1.53 | 1.67 | 1.73 | 1.87 | |

4 | 1.28 | 1.42 | 1.48 | 1.62 | 1.67 | 1.82 | 1.86 | 2.01 | 2.07 | 2.20 | |

6 | 1.62 | 1.76 | 1.81 | 1.96 | 2.02 | 2.15 | 2.21 | 2.35 | 2.41 | 2.54 | |

8 | 1.96 | 2.10 | 2.16 | 2.30 | 2.36 | 2.49 | 2.55 | 2.69 | 2.74 | 2.88 | |

10 | 2.31 | 2.44 | 2.50 | 2.64 | 2.69 | 2.83 | 2.89 | 3.02 | 3.08 | 3.21 | |

15 | 3.14 | 3.26 | 3.34 | 3.47 | 3.53 | 3.66 | 3.73 | 3.84 | 3.92 | 4.05 | |

20 | 3.98 | 4.12 | 4.18 | 4.31 | 4.37 | 4.50 | 4.56 | 4.70 | 4.76 | 4.89 | |

25 | 4.82 | 4.95 | 5.01 | 5.14 | 5.20 | 5.34 | 5.40 | 5.53 | 5.58 | 5.71 | |

30 | 5.64 | 5.78 | 5.84 | 5.98 | 6.04 | 6.17 | 6.22 | 6.35 | 6.40 | 6.54 | |

35 | 6.47 | 6.61 | 6.67 | 6.81 | 6.87 | 7.00 | 7.05 | 7.18 | 7.23 | 7.37 | |

40 | 7.30 | 7.44 | 7.50 | 7.64 | 7.70 | 7.83 | 7.88 | 8.01 | 8.06 | 8.20 | |

45 | 8.12 | 8.27 | 8.33 | 8.46 | 8.52 | 8.66 | 8.71 | 8.84 | 8.90 | 9.03 | |

50 | 8.96 | 9.09 | 9.14 | 9.28 | 9.34 | 9.48 | 9.54 | 9.68 | 9.73 | 9.86 | |

60 | 10.62 | 10.76 | 10.82 | 10.95 | 11.00 | 11.13 | 11.19 | 11.32 | 11.37 | 11.49 | |

70 | 12.28 | 12.41 | 12.46 | 12.59 | 12.65 | 12.77 | 12.82 | 12.95 | 13.01 | 13.15 | |

80 | 13.93 | 14.06 | 14.11 | 14.24 | 14.29 | 14.42 | 14.47 | 14.60 | 14.66 | 14.80 | |

90 | 15.58 | 15.71 | 15.76 | 15.89 | 15.94 | 16.07 | 16.12 | 16.25 | 16.30 | 16.43 | |

100 | 17.13 | 17.36 | 17.33 | 17.55 | 17.54 | 17.73 | 17.74 | 17.91 | 17.93 | 18.09 |

N. | C_{D} | b_{s} |
---|---|---|

1 | 1.0 × 10^{2} | 0.6448 |

2 | 5.0 × 10^{2} | 0.7690 |

3 | 1.0 × 10^{3} | 0.8420 |

4 | 5.0 × 10^{3} | 0.9754 |

5 | 1.0 × 10^{4} | 1.0375 |

6 | 5.0 × 10^{4} | 1.1689 |

7 | 1.0 × 10^{5} | 1.2325 |

8 | 5.0 × 10^{5} | 1.3644 |

9 | 1.0 × 10^{6} | 1.4272 |

10 | 5.0 × 10^{6} | 1.5581 |

11 | 1.0 × 10^{7} | 1.6195 |

**Table 3.**Pumping tests at S-V well in Veselí nad Lužnicí before rehabilitation (a), after rehabilitation (b), and one year after rehabilitation (c).

(a) | (b) | (c) | |||||
---|---|---|---|---|---|---|---|

N. | Time t(s) | Drawdown s (m) | Time t(s) | Drawdown s(m) | N. | Time t(s) | Drawdown s(m) |

1. | 130 | 0.02 | 30 | 0.01 | 1. | 60 | 0.08 |

2. | 215 | 0.04 | 45 | 0.025 | 2. | 120 | 0.13 |

3. | 240 | 0.05 | 60 | 0.03 | 3. | 180 | 0.16 |

4. | 300 | 0.08 | 90 | 0.045 | 4. | 300 | 0.21 |

5. | 420 | 0.14 | 150 | 0.08 | 5. | 600 | 0.305 |

6. | 560 | 0.20 | 180 | 0.09 | 6. | 900 | 0.4 |

7. | 800 | 0.28 | 240 | 0.115 | 7. | 1200 | 0.47 |

8. | 900 | 0.33 | 300 | 0.135 | 8. | 1800 | 0.62 |

9. | 1180 | 0.46 | 600 | 0.25 | 9. | 2700 | 0.75 |

10. | 1800 | 0.67 | 900 | 0.34 | 10. | 3600 | 0.84 |

11 | 2700 | 1.02 | 1200 | 0.415 | 11 | 4500 | 0.9 |

12. | 3600 | 1.25 | 1500 | 0.475 | 12. | 9000 | 1.09 |

13. | 5600 | 1.76 | 1800 | 0.54 | 13. | 15,000 | 1.18 |

14. | 7200 | 2.20 | 2100 | 0.585 | 14. | 27,000 | 1.25 |

15. | 10,800 | 2.78 | 2700 | 0.65 | 15. | 50,000 | 1.27 |

16. | 14,200 | 3.14 | 3600 | 0.72 | 16. | 80,000 | 1.285 |

17. | 20,000 | 3.51 | 5400 | 0.82 | |||

18. | 30,000 | 3.91 | 7200 | 0.875 | |||

19. | 44,000 | 4.15 | 10,800 | 0.955 | |||

20. | 70,000 | 4.32 | 14,400 | 0.98 | |||

21. | 100,000 | 4.41 | 25,000 | 1.01 | |||

22. | 200,000 | 4.47 | 40,000 | 1.02 | |||

23. | 400,000 | 4.50 | 72,000 | 1.03 |

Before Rehabilitation | After Rehabilitation | 1 Year after Rehabilitation | |
---|---|---|---|

Pumping rate, Q (m^{3}·s^{−1}) | 0.00335 | 0.0037 | 0.00352 |

Pumping test duration (s) | 400,000 | 72,000 | 80,000 |

s* (m) | 0.62 | 0.135 | 0.18 |

New method (Equation (34)) | 47 | 6.8 | 10.8 |

Cooper–Jacob method (Equation (16)) | 51 | 7.9 | 12 |

Difference (%) | 8.4 | 15 | 11.1 |

Additional drawdown s_{skin} (Equation (12)) | 3.51 | 0.56 | 0.85 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kahuda, D.; Pech, P. A New Method for the Evaluation of Well Rehabilitation from the Early Portion of a Pumping Test. *Water* **2020**, *12*, 744.
https://doi.org/10.3390/w12030744

**AMA Style**

Kahuda D, Pech P. A New Method for the Evaluation of Well Rehabilitation from the Early Portion of a Pumping Test. *Water*. 2020; 12(3):744.
https://doi.org/10.3390/w12030744

**Chicago/Turabian Style**

Kahuda, Daniel, and Pavel Pech. 2020. "A New Method for the Evaluation of Well Rehabilitation from the Early Portion of a Pumping Test" *Water* 12, no. 3: 744.
https://doi.org/10.3390/w12030744