Semi-analytical numerical calculation method for flow in confined and unconfined pumping wells considering the hysteresis effect

Abstract

A semi-analytical numerical calculation method for confined-unconfined pumping well flow considering the delayed water supply effect includes the following steps: (1) Based on the hydraulic parameters of the confined aquifer obtained from actual experiments, including initial head, aquifer thickness, hydraulic conductivity, specific yield, and pumping rate, the distance from the confined-unconfined zone transition interface to the pumping well is predicted; (2) By comparing the distance from the observation well to the pumping well with the predicted transition interface distance, it is determined whether the observation well is located in a confined or unconfined zone; Based on the result of the transition interface distance, the region is determined, and different equations are selected for different regions to realize the head prediction work of confined-unconfined pumping well flow based on aquifer hydraulic parameters. The calculation method of this invention considers the delayed water supply effect, which can determine the distance from the confined-unconfined flow transition interface to the pumping well at any time, thereby more accurately determining the region where the observation well is located and facilitating the use of the head prediction model.

Description

Technical Field

[0001] This invention belongs to the field of hydrogeological engineering, and specifically relates to a semi-analytical numerical calculation method for flow in confined and unconfined pumping wells that considers the delayed water supply effect. Background Technology

[0002] Unsteady confined-unconfined pumping well flow is a classic groundwater dynamics problem in the drainage process of coal mines. In recent decades, numerous scholars at home and abroad have conducted targeted research on the confined-unconfined well flow problem, and established a large number of valuable numerical and analytical models.

[0003] Confined-unconfined pumping well flow occurs when the pumping flow rate is too high or the pumping time is too long in a confined aquifer, causing the water head in some areas to drop below the upper impermeable plate. In this case, two different flow regimes exist simultaneously in the aquifer: unconfined flow near the pumping well, where the water head drops below the impermeable plate, and confined flow further away from the pumping well, where the water head remains above the impermeable plate. During pumping in the unconfined zone, due to the delayed release of gravity feedwater, an unsaturated zone can form above the free water surface, replenishing the aquifer below. This replenishment is called the delayed response. In the mid-20th century, scholars, through observation of unconfined aquifers in natural environments, proposed the first mathematical model considering the delayed response. This model can be used to predict the water level drop caused by constant-flow pumping from fully permeable wells in unconfined aquifers. Subsequently, many scholars conducted extensive research on the delayed water supply phenomenon of unpressurized pumping well flow based on mathematical models and field experiments, and achieved many valuable results.

[0004] Existing research on unsteady confined-unconfined well flows reveals that most confined-unconfined well flow models are derived based on the Dupuit assumption. The Dupuit assumption posits that unconfined pumping well flow is a one- or two-dimensional seepage field with only gravity-driven water release from the aquifer, neglecting the influence of aquifer anisotropy and elastic water release. This leads to poor prediction of drawdown in the initial stages of pumping. Therefore, further elucidating the delayed water supply mechanism in the unconfined zone of unsteady confined-unconfined well flows is of significant importance for the design of coal mine drainage projects.

[0005] Currently, scholars have proposed the first semi-analytical numerical model that considers the anisotropy of confined-unconfined aquifers and the influence of delayed vertical flow feed. This model uses the recharge boundary condition proposed by Moench to simulate delayed feed. However, Moench's recharge boundary condition approximates recharge to unsaturated zones by incorporating an empirical parameter α1, which lacks a clear physical meaning. This empirical parameter is highly controversial and difficult to determine in practical engineering applications, limiting the applicability of models based on this boundary condition in real-world engineering. Furthermore, existing research only considers the pumping operation of complete wells, neglecting the pumping of incomplete wells, which are more common when aquifer thickness is large. Currently, no published literature has successfully established a precise calculation model for incomplete well pumping that can account for delayed feed effects in confined-unconfined aquifers. Therefore, proposing corresponding research methods has significant engineering application value for coal mine drainage design. Summary of the Invention

[0006] The technical problem to be solved by this invention is to address the above-mentioned shortcomings of existing pressurized-unpressurized pumping well flows by proposing a semi-analytical numerical calculation method for pressurized-unpressurized pumping well flows that considers the lag effect of water supply. By predicting the distance between the pressurized-unpressurized flow transition interface and the pumping well at any time, the method can more accurately determine the area where the observation well is located and facilitate the use of the head prediction model.

[0007] The technical solution adopted by the present invention to solve the above-mentioned technical problems is as follows:

[0008] The semi-analytical numerical calculation method for flow in confined and unconfined pumping wells, considering the delayed water supply effect, includes the following two steps:

[0009] (1) Based on the actual test, the hydraulic parameters of the confined aquifer are obtained, including the initial head, aquifer thickness, hydraulic conductivity, specific yield and pumping rate. First, the distance from the confined-unconfined zone transition interface to the pumping well is predicted.

[0010] (2) By comparing the distance (r) between the observation well and the pumping well with the predicted transition interface distance (R), it can be determined whether the location of the observation well is in a confined area (r≥R) or a non-confined area (r≤R). Based on the regional determination results, different equations are selected for different regions to realize the prediction of the flow head of confined and non-confined pumping wells based on the hydraulic parameters of the aquifer.

[0011] According to the above scheme, step (1) is as follows:

[0012] 11) Based on the initial head, aquifer thickness, hydraulic conductivity, specific yield, and pumping rate parameters, an implicit function equation considering the characteristics of the interface boundary conditions is proposed. The implicit function equation is as follows:

[0013]

[0014]

[0015] In the formula, p is the Laplace operator; m is the Fourier operator; I z (v) and K z (v) respectively represent the first and second z-order modified Bessel functions of v; b is the aquifer thickness [L]; K r is the horizontal hydraulic conductivity [L / T]; K z is the vertical hydraulic conductivity [L / T]; h0 is the initial water head of the confined area [L]; Q is the pumping flow rate [L 3 / T]; l and d are the vertical distances from the highest and lowest points of the filter pipe at the well wall to the bottom of the aquifer [L], and (l - d) represents the length of the filter pipe at the well wall; R is the distance from the confined-unconfined conversion interface of the aquifer to the pumping well [L]; ε n is an additional parameter, and the value of ε n is calculated through the following implicit function formula:

[0016]

[0017] In the formula, S r represents the elastic storage coefficient; S y represents the gravity storage coefficient;

[0018] 12) Approximately obtain the parameter value of the distance from the confined-unconfined conversion interface to the pumping well at the required time through the parameter inversion module (Equations and System Solver) of MATLAB for the described implicit function equation.

[0019] According to the above scheme, the prediction of the flowing water head of the confined-unconfined pumping well based on the hydraulic parameters of the aquifer in step (2) is as follows:

[0020] 21) When the distance from the observation well to the pumping well is less than the distance of the conversion interface (r < R), the water flow in the observation well is in the unconfined area, and the prediction of the water head of the observation well is completed relying on the unconfined area model, specifically as follows:

[0021] As the pumping time increases, the range of the unconfined area gradually expands from near the pumping well, that is, the value of R slowly increases with time; when the range of the unconfined area reaches the position of the observation well, r < R appears, and the water flow at the observation well changes to the unconfined state. According to the determined aquifer characteristic parameters, let

[0022]

[0023]

[0024]

[0025]

[0026] At this point, the head of the observation well in the Laplace and Fourier domains is expressed as:

[0027]

[0028] In the formula, denoted by [L], representing the head in the unconfined zone under the Laplace and Fourier domains; r is the horizontal distance [L] between the observation well and the pumping well.

[0029] Performing an inverse Fourier transform on equation (3e) yields the head in the unconfined zone under the Laplace domain. for

[0030]

[0031] After performing an inverse Laplace transform on formula (3f), the head h1 in the unconfined zone is obtained as follows:

[0032]

[0033] The required head change model in the non-confined zone can be obtained from formula (3g);

[0034] 22) When the distance between the observation well and the pumping well is greater than the transition interface distance (r≥R), the water flow at the observation well is in a confined zone. A confined zone head prediction model is used for this study. Based on the above known parameters, let...

[0035]

[0036]

[0037] At this point, the head of the observation well in the Laplace and Fourier domains is expressed as:

[0038]

[0039] In the formula, Let [L] represent the confined zone head in the Laplace and Fourier domains; then, perform an inverse Fourier transform on equation (4c) to obtain the confined zone head in the Laplace domain. for

[0040]

[0041] After performing an inverse Laplace transform on the obtained formula (4d), the head h2 of the confined zone can be obtained as follows:

[0042]

[0043] The required head change model in the confined zone can be obtained from formula (4e).

[0044] Compared with the prior art, the beneficial effects of the present invention are:

[0045] 1. Based on the research of Neuman (1974) and Xiao et al. (2022), and based on the Theis flow hypothesis and the hysteresis effect, a three-dimensional unsteady confined-unconfined incomplete well flow semi-numerical analytical model was first proposed. In this model, Neuman (1974) transient hysteresis theory was introduced to ensure that all hydraulic parameters in the scheme are reasonable and measurable. Furthermore, the universality of the model was enhanced by considering the pumping well as an incomplete well. The constructed model was simplified and derived based on Laplace transform and Fourier transform, and the required Laplace-Fourier domain constant flow pumping head prediction model for confined and unconfined zones was obtained.

[0046] 2. Based on measurable aquifer hydraulic parameters and the proposed semi-analytical model, a predictive calculation method for the dynamic distribution of the confined-unconfined flow transition interface in a single-well constant-rate pumping test is derived. The proposed scheme relies on Laplace and Fourier transform derivations, and the resulting implicit function equation can be used to determine the distance between the confined-unconfined flow transition interface and the pumping well at any given time, thereby determining the location of the observation well and facilitating the use of the head prediction model. Attached Figure Description

[0047] Figure 1 This is a schematic diagram of constant flow pumping in an incompletely permeable well considering the transition from pressurized to non-pressurized conditions, according to the present invention.

[0048] Figure 2 This is a plan view showing the locations of the pumping and observation wells for testing the Cape Cod aquifer;

[0049] Figure 3 This is a schematic diagram comparing the Cape Cod aquifer test with the model proposed in this invention. Detailed Implementation

[0050] The specific embodiments of the present invention will now be described in detail with reference to the accompanying drawings and examples.

[0051] The semi-analytical numerical calculation method for flow in pressurized-unpressurized pumping wells, which considers the delayed water supply effect, as described in this invention, includes two steps: predicting the distance from the pressurized-unpressurized zone transition interface to the pumping well and predicting the flow head of the pressurized-unpressurized pumping well.

[0052] Reference Figure 1The diagram illustrates constant-flow pumping from an incompletely permeable well, considering the confined-unconfined transition. Before pumping begins, the aquifer is fully confined with an initial head of h0. As pumping continues, the head gradually decreases. If pumping time is long enough, the head near the well can drop below the upper impermeable layer, forming an unconfined zone within a radial radius R of the well. The transient confined-unconfined flow characteristics resulting from this incompletely permeable well pumping are as follows: Figure 1 As shown.

[0053] Under normal circumstances, relevant parameters of confined aquifers, including initial head, aquifer thickness, hydraulic conductivity, specific yield, and pumping rate, can be directly measured and used as known coefficients in actual experiments. Based on these known coefficients, the distance from the confined-unconfined zone transition interface to the pumping well can be predicted first. By comparing the distance (r) between the observation well and the pumping well with the predicted transition interface distance (R), it can be determined whether the observation well is located in a confined zone (r≥R) or an unconfined zone (r≤R). Based on the region determination results, different equations are selected for different regions to achieve the prediction of the flow head of the confined-unconfined pumping well.

[0054] The specific steps are as follows:

[0055] (1) Prediction of the distance from the pumping well to the confined-unconfined zone transition interface based on aquifer hydraulic parameters

[0056] Considering that the distance (R) between the pressure-non-pressure zone transition interface and the pumping well is difficult to observe in practice, the model proposes an implicit function equation based on the characteristics of the interface boundary conditions, namely formula (1), so that the parameter value of this distance at the required time can be approximately obtained by using the parameter inversion module (Equations and System Solver) of MATLAB.

[0057] The implicit function equation is:

[0058]

[0059] Equation (1) is obtained through Laplace and Fourier transformation. The specific derivation process is as follows: When considering vertical flow, the three-dimensional flow control equations for the non-pressure region and the pressure region are expressed as Equation (5a) and (5b), respectively.

[0060]

[0061]

[0062] In the formula, q r1 and qr2 q represents the specific flow rate [L / T] in the horizontal direction for the non-pressure zone and the pressure zone, respectively; z1 and q z2 , respectively, are the specific flow rates [L / T] in the vertical direction of the unconfined zone and the confined zone; h1 and h2 are the water heads [L] of the observation wells in the unconfined zone and the confined zone, respectively; S r is the elastic storage coefficient; b is the aquifer thickness [L]; t is the pumping time [t]; r is the horizontal distance from the observation well to the pumping well [L]; z is the vertical distance from the bottom of the aquifer [L]. The head and specific flow rate are both functions of r, z, and t.

[0063] Assuming the pumping flow obeys Darcy's law, its flow rate is...

[0064]

[0065]

[0066]

[0067]

[0068] In the formula, K r and K z Let [L / T] be the permeability coefficient in the horizontal and vertical directions.

[0069] The initial conditions of the model are

[0070] lim t→0 h1 = b(7a)

[0071] lim t→0 h2 = h0(7b)

[0072] In the formula, h0 is the initial head [L] in the confined zone.

[0073] Under the condition of a constant pumping rate, the boundary conditions are as follows:

[0074]

[0075]

[0076] lim r→R q r1 =lim r→R q r2 (10a)

[0077] lim r→R h1 = lim r→R h2(10b)

[0078]

[0079]

[0080] In the formula, l and d are the vertical distances from the highest and lowest points of the filter tube to the bottom of the aquifer, respectively [L]; Q is the pumping flow rate, which is a known positive number during the pumping process [L]. 3 / T];S y denoted by , where represents the gravity storage coefficient; R is the distance [L] from the transition interface between the confined and unconfined zones of the aquifer to the pumping well, which is a function of the pumping time t.

[0081] Substituting formulas (6a)-(6d) into formulas (5a) and (5b), the governing equations for the non-pressure zone and the pressure zone can be written as formulas (12a) and (12b).

[0082]

[0083]

[0084] To simplify the calculation process, we can assume in formula (12a) that... The error caused by this assumption has been shown to be very small and can be ignored in modeling (Moench & Prickett, 1972). Therefore, equation (12a) can be rewritten as

[0085]

[0086] From Table 1, we can see that the equations in formulas (7)-(12) after dimensionless transformation are:

[0087]

[0088]

[0089] and

[0090]

[0091]

[0092]

[0093]

[0094]

[0095]

[0096]

[0097]

[0098] Formula (14g) is a Laplace equation, and its general solution is:

[0099]

[0100] In the formula, The dimensionless descent of an infinite region in the Laplace field; p is the Laplace operator; and f n (r D p) is a group containing r D It is a function of p. n is an arbitrary constant that can be determined by the boundary conditions at the free surface.

[0101] Table 1. Dimensionless Conversion Table

[0102]

[0103] Combining formula (15), the dimensionless governing equations of the non-pressure region, and the boundary conditions of the upper free surface, namely formulas (13a), (14c), (14g), and (14f), we can further refine the equations for time t. D After performing the Laplace transform, they are rewritten as (16), (17a), (17b) and (17c).

[0104]

[0105]

[0106]

[0107]

[0108] Substituting formula (15) into formula (17b), we obtain an implicit function expression, as follows:

[0109]

[0110] In the formula, Based on formula (18), ε n The solution can be obtained using a built-in module in MATLAB called "Equation and System Solver".

[0111] Based on the general solution of the Bessel function, we can derive the following from formula (16):

[0112]

[0113] In the formula, C1 and C2 are arbitrary constants; I0(v) and K0(v) represent the first and second kind of modified Bessel functions with respect to v, respectively.

[0114] Substitute formula (19a) into formulas (17a) and (17c), and adjust z D Perform a Fourier transform, and the constants C1 and C2 are expressed as follows.

[0115]

[0116]

[0117] After Fourier transform, formulas (15), (19a), (19b), and (19c) are combined to obtain the dimensionless drawdown of the non-pressure region in the Laplace-Fourier domain, which can be expressed as:

[0118]

[0119] in

[0120]

[0121]

[0122] In the formula, is the dimensionless depth of the non-pressured region in the Laplace-Fourier domain; m is the Fourier operator.

[0123] Similarly, the Laplace and Fourier transforms of the governing equations and boundary conditions in the pressure zone are as follows.

[0124]

[0125] and

[0126]

[0127]

[0128] In the formula, It is the dimensionless drawdown of the pressure zone in the Laplace-Fourier domain.

[0129] Since formula (22) satisfies the general solution of the Bessel function, it can be rewritten as follows:

[0130]

[0131] Where C3 and C4 represent arbitrary constants.

[0132] Substituting formula (24a) into formula (23a), we obtain the arbitrary constant C3 as follows:

[0133] C3 = 0 (24b)

[0134] Combining formulas (24a) and (23b), we can obtain

[0135]

[0136] Therefore, in the Laplace-Fourier domain, the dimensionless drawdown of the pressure zone can be expressed as:

[0137]

[0138] in

[0139]

[0140]

[0141] To assess the dynamic development of the transient non-pressured region represented by R, a method based on the boundary condition equation (14e) at the interface is proposed. Similarly, for t D and z D Performing Laplace and Fourier transforms, equation (14e) can be rewritten as follows:

[0142]

[0143] Combining formulas (20), (25) and (27), we can obtain the implicit form as shown in formula (28).

[0144]

[0145] Assume R D The value of is a root of formula (28) at any time point, and can be accurately obtained using the Equations and System Solver module in MATLAB. The drawdown within the observation well can be calculated using l. D to d D The average drop within the filter tube interval is described by the following formula:

[0146]

[0147] In the formula, For relative to z D Average dimensionless descent depth; s D For dimensionless reduction of s D1 or s D2 .

[0148] Combining the dimensionless transformation in Table 1, substituting the corresponding parameters into formulas (20), (25), and (28) yields the following results:

[0149]

[0150]

[0151]

[0152] Substituting (21a), (21b), (26a), and (26b) into equation (30) gives the following equation

[0153]

[0154]

[0155]

[0156]

[0157] (2) Prediction of the flowing water head of a confined-unconfined pumping well based on the hydraulic parameters of the aquifer

[0158] In the prediction of the aquifer water head, this model will discuss the water heads in the confined and unconfined areas using different methods respectively. Which prediction formula to select for an observation well at a specific location at a specific time needs to be adjusted according to the distance of the conversion interface (R). Specifically, when the distance between the observation well and the pumping well is greater than the distance of the conversion interface (r ≥ R), it is considered that the observation well is in the confined area, and the confined area water head prediction model is used for research. Correspondingly, when the distance between the observation well and the pumping well is less than the distance of the conversion interface (r < R), the unconfined area model needs to be relied on to complete the prediction.

[0159] 21) Prediction model of the water head of an observation well in the unconfined area

[0160] When r < R, the water flow at the observation well is in an unconfined state, and the unconfined area model is relied on to complete the prediction of the water head of the observation well. According to the determined aquifer characteristic parameters, the water head of the observation well can be represented by the unconfined area water head h1 at this time:

[0161]

[0162] 22) Prediction model of the water head of an observation well in the confined area

[0163] When r ≥ R, the water flow at the observation well is in the confined area, and the confined area water head prediction model is used for research. Based on the above known parameters, the water head of the observation well can be represented by the confined area water head h2:

[0164]

[0165] To verify the feasibility of the model in practical engineering. Take a constant-rate pumping test carried out in a completely unconfined aquifer in Cape Cod, Massachusetts, USA as an example. To facilitate comparison with the measured data, the change in the drawdown in the unconfined area can be used to reflect the change in the water head in the unconfined area.

[0166] According to research by Moench et al. (2001, 2004), Cape Cod is located in the northeastern United States, southeast of Massachusetts, and close to the Atlantic Ocean (e.g., Figure 2 (As shown). Its geological structure is mainly composed of sand and gravel, and the region contains a large amount of sediment formed by glacial movement. The constant-flow pumping test studied was conducted in a non-confined aquifer near the surface (Moench et al., 2001). The pumping test determined the aquifer thickness to be 51.82 m and the pumping rate to be 0.0202 m. 3 / s. The pumping wells are incompletely penetrating wells, and the filter section extends from 4.02 to 18.35 m below the initial head (Mishra and Neuman, 2010). There are a total of twenty observation wells in the test area (e.g., Figure 2 (As shown). Among them, the drawdown time data of observation wells F505-032, F504-032, F478-061, F450-061, and F434-060 were used for aquifer parameter inversion, including K. r K z S r and S y Then, using the results of the parameter inversion, the proposed model was used to simulate the drawdown time curves of five other observation wells, namely F505-059, F505-080, F504-060, F504-080, and F377-037, verifying the reliability of the established model. The radial distances of F505-059, F505-080, F504-060, F504-080, and F377-037 from the pumping well are 5.944m, 6.584m, 15.179m, 16.185m, and 25.939m, respectively.

[0167] Table 2 Estimated values ​​of observation well parameters

[0168]

[0169] As shown in Table 2, the calculated K r K z S r S y The resulting ranges are 0.0013–0.002 m / s, 0.0006–0.001 m / s, 0.002–0.005 m / s, and 0.25–0.36 m / s, respectively. Since the simulation results do not contain any unreasonable abrupt changes, the average value of each data sequence can be used to represent the values ​​of each parameter of the aquifer, i.e., K. r =0.0016m / s, K z =0.0008m / s, S r =0.003 and S y=0.3 (Table 1), after dimensionless transformation, this series of parameters can be expressed as β=0.5, σ=0.01, M=0, h D0 =1, l D =0.92 and d D =0.65. This estimation result is similar to previous studies (Moench et al., 2001; artakovsky and Neuman, 2007), and the proposed model should be considered reliable in the inversion of the aquifer parameters.

[0170] Based on the above hydraulic parameter calculations, the drawdown time curves of observation wells F505-059, F505-080, F504-060, F504-080, and F377-037 were simulated and compared with measured data. For example... Figure 3 As shown. The results show that the drawdown time curve obtained by the proposed model is basically consistent with the field measurement results, with only a small deviation in the initial stage of pumping. This difference is caused by the anisotropy of the pumped aquifer and the measurement error in the initial stage of pumping (Moench, 2004; Mishra & Neuman, 2010). Therefore, the deviation in the comparison is acceptable, and the proposed solution is reliable. Compared with the model in Mishra and Neuman's (2010) paper, the proposed solution is simpler and easier to use. In addition, unlike the study of Moench (2010), the proposed model does not rely on empirical parameters, thus having a stronger theoretical foundation.

[0171] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A semi-analytical numerical calculation method for flow in confined and unconfined pumping wells considering the delayed water supply effect, characterized in that, Specifically, it includes the following two steps: (1) Based on the hydraulic parameters of the confined aquifer obtained from actual tests, including initial head, aquifer thickness, hydraulic conductivity, specific yield, and pumping rate, the distance from the confined-unconfined zone transition interface to the pumping well is predicted first, as follows: 11) Based on the initial head, aquifer thickness, hydraulic conductivity, specific yield, and pumping rate parameters, an implicit function equation considering the characteristics of the interface boundary conditions is proposed. The implicit function equation is as follows: （1） In the formula, For the Laplace operator; For Fourier operators; and They represent about Class I and Class II Order-corrected Bessel function; The aquifer thickness is [L]. The horizontal hydraulic permeability coefficient [L / T]; The vertical hydraulic permeability coefficient [L / T]; The initial head [L] of the confined zone; Pumping flow rate [L] 3 / T]; and Let L be the vertical distance from the highest and lowest points of the filter pipe at the well wall to the bottom of the aquifer. Indicates the length of the filter pipe at the well wall; The distance [L] from the transition interface between the confined and unconfined zones of the aquifer to the pumping well; As an additional parameter, The value is calculated using the following implicit function formula: （2） In the formula, Indicates the elastic water storage coefficient; Indicates the gravity water storage coefficient; 12) The implicit function equation is used to approximate the parameter value of the distance from the pumping well to the pressure-non-pressure zone transition interface at the required time by using the parameter inversion module of MATLAB; (2) By comparing the distance between the observation well and the pumping well with the predicted distance to the transition interface, it can be determined whether the location of the observation well is in a confined area or a non-confined area. Based on the regional determination results, different equations are selected for different regions to realize the prediction of the flow head of the confined-non-confined pumping well based on the hydraulic parameters of the aquifer.

2. The semi-analytical numerical calculation method for flow in confined and unconfined pumping wells considering the delayed water supply effect as described in claim 1, characterized in that, The specific steps (2) for predicting the flow head of confined and unconfined pumping wells based on aquifer hydraulic parameters are as follows: 21) When the distance between the observation well and the pumping well is less than the transition interface distance, the water flow in the observation well is in the unconfined zone. The water head prediction of the observation well is completed based on the unconfined zone model, as follows: As pumping time increases, the area of ​​the non-pressurized zone gradually expands from the vicinity of the pumping well, that is... R The value increases slowly over time; when the non-pressure zone reaches the location of the observation well, a [problem occurs]. r < R The water flow at the observation well transitions to a non-pressurized state. Based on the determined aquifer characteristic parameters, let... （3a） （3b） （3c） （3d） At this point, the head of the observation well in the Laplace and Fourier domains is expressed as: （3e） In the formula, [L] represents the head in the unconfined region under the Laplace and Fourier domains; r The horizontal distance [L] between the observation well and the pumping well; Performing an inverse Fourier transform on formula (3e) yields the head in the unconfined zone under the Laplace domain. for （3f） After performing an inverse Laplace transform on formula (3f), the head in the unconfined zone is obtained. for （3g） Let t be the pumping time. The required head change model in the unconfined zone can be obtained from the formula (3g). 22) When the distance between the observation well and the pumping well is greater than the transition interface distance, the water flow at the observation well is in a confined zone. A confined zone head prediction model is used for this study. Based on the known parameters mentioned above, let... （4a） （4b） At this point, the head of the observation well in the Laplace and Fourier domains is expressed as: （4c） In the formula, Let [L] represent the confined zone head in the Laplace and Fourier domains; then, perform an inverse Fourier transform on formula (4c) to obtain the confined zone head in the Laplace domain. for （4d） The inverse Laplace transform of the obtained formula (4d) yields the head in the confined zone. for （4e） The required head change model for the confined zone can be obtained from formula (4e).

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