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DIVISION S-1—SOIL PHYSICS

Vapor Flow to Horizontal Wells in Unsaturated Zones

Dep. of Geology & Geophysics, 3115 TAMU, Texas A&M Univ., College Station, TX 77843-3115

* Corresponding author (zhan{at}hydrog.tamu.edu)

	ABSTRACT

TOP ABSTRACT INTRODUCTION Mathematical Modeling Mathematical Model Vapor Flow to a... Vapor Flow to a... Results of Vapor Flow... Pressure Distribution for a... Pressure Distribution for an... Numerical Evaluation of Solution... Vapor Mass Flux at... DISCUSSION AND CONCLUSIONS APPENDIX A APPENDIX B APPENDIX C REFERENCES

 
We have solved the linearized vapor-flow equation in a transient, three dimensional form for a horizontal-well sink in an unsaturated zone. This is done by solving the vapor flow to a point sink first, then superposing the point-sink solution along a finite length of the horizontal-well axis to obtain the solution of flow to a horizontal-well sink. Vapor-pressure distributions near a horizontal well are provided for both covered and uncovered ground surface cases. A computer program VF3D (available from our website) is used to calculate the vapor pressure and to generate the type curves of vapor-pumping tests. The derived solution is used to calculate the specific vapor flux and the total vapor-mass flux across the uncovered ground surface. The type curves at early time for a covered surface case are influenced by the locations of the monitoring wells and vertical anisotropy of the unsaturated zone. These type curves converge to the vertical-well Theis curve at late time. The type curves for an uncovered surface case are significantly different from the Theis curve during the entire pumping time, and are similar to the type curves of leaky aquifers. The steady-state contour maps of mass fluxes across the uncovered surfaces are plotted. The transient total mass flux rate converges exponentially to the horizontal-well pumping rate, independent of the horizontal-well elevation. The derived analytical and numerical solutions are useful for assessing the performance of soil-vapor extraction, bioventing, enhanced bioremediation, air sparging, and for interpreting vapor-pumping tests.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION Mathematical Modeling Mathematical Model Vapor Flow to a... Vapor Flow to a... Results of Vapor Flow... Pressure Distribution for a... Pressure Distribution for an... Numerical Evaluation of Solution... Vapor Mass Flux at... DISCUSSION AND CONCLUSIONS APPENDIX A APPENDIX B APPENDIX C REFERENCES

 
VAPOR FLOW IS AN IMPORTANT subject in soil physics and soil-remediation technologies. An early study of vapor flow in a gas reservoir was carried out by Muskat and Botset (1931). Soil scientists have utilized injected air and pressure measurements to evaluate soil permeability (Kirkham, 1946; Evans and Kirkham, 1949; Grover, 1955; Tanner and Wengel, 1957). In recent years, numerous studies have been done regarding soil venting, bioventing, enhanced bioremediation, air sparging, and soil-vapor extraction that have become the common methods for treating subsurface contaminated soils (Baehr, et al., 1989; Johnson et al., 1990; USEPA, 1992; Falta, 1995; Hunt and Massmann, 2000). Previous studies of vapor flow are mostly limited to vertical wells (Massmann, 1989; McWhorter, 1990; Baehr and Hult, 1991; Cho and DiGiulio, 1992; Shan et al., 1992; Cho, 1993; Beckett and Huntley, 1994; Baehr and Joss, 1995). Illman and Neuman (2000) reported a type curve interpretation of single-hole pneumatic tests for vertical and slanted wells in unsaturated fractured rocks. They also presented type-curve interpretation of cross-hole pneumatic injection tests conducted at the same site (Illman and Neuman, 2001); however, recently work has been done on horizontal wells.

Horizontal wells have been found to be effective tools for collecting contaminated groundwater and vapor from the subsurface because of their large screen length and large contact area with groundwater and vapor (Cleveland, 1994; Falta, 1995; Sawyer and Lieuallen-Dulam, 1998; Zhan, 1999; Hunt and Massmann, 2000; Zhan and Cao, 2000). Horizontal wells have been extensively used for oil and gas production also because of the large contact area with oil or gas (Goode and Thambynayagam, 1987; Daviau et al., 1988; Ozkan et al., 1989; Rosa and Carvalho, 1989). In addition, horizontal wells are often used at sites where vertical wells cannot be used because of the difficulty in accessing the sites. The environmental applications of horizontal wells for vapor extraction are significant because of their advantages over vertical wells. However, vapor flow to a horizontal well is rarely studied and the knowledge base for understanding vapor extraction through horizontal wells is limited. Present available references include an analytical study by Falta (1995) and a recent investigation of vapor flow to a leaky trench by Hunt and Massmann (2000). Falta's (1995) solution is two-dimensional since the well is assumed to be infinitely long. Hunt and Massmann's (2000) solution is steady-state.

A general transient, three-dimensional solution of vapor flow to a finite length horizontal well in an anisotropic unsaturated zone does not exist now. This solution is very useful for understanding dynamics of vapor flow in the unsaturated zone, developing vapor extraction remediation plans, and interpreting vapor pumping and injecting tests using horizontal wells. In this paper, we will provide such a general solution of transient vapor flow to a finite-length horizontal well in a three-dimensionally anisotropic unsaturated zone. We consider two practical ground surface boundaries: a covered surface and an uncovered surface. If the ground surface is covered with impermeable materials, it is treated as a no-flow boundary for vapor. If the ground surface is uncovered, it is treated as a boundary of constant pressure equal to atmospheric pressure. The top of the capillary fringe underneath the unsaturated zone serves as the lower no-flow boundary for vapor.

	Mathematical Modeling

TOP ABSTRACT INTRODUCTION Mathematical Modeling Mathematical Model Vapor Flow to a... Vapor Flow to a... Results of Vapor Flow... Pressure Distribution for a... Pressure Distribution for an... Numerical Evaluation of Solution... Vapor Mass Flux at... DISCUSSION AND CONCLUSIONS APPENDIX A APPENDIX B APPENDIX C REFERENCES

 
Problem Description
Figure 1 is a diagram of a finite-length horizontal well, underneath a covered surface. The origin of the coordinate system is at the lower boundary. The z-axis is through the center of the horizontal well and the x-axis is parallel to the well axis. The distance from the horizontal well to the lower boundary is z_w and the distance from surface to the lower boundary is d. The well screen length is L.

View larger version (15K): [in this window] [in a new window]   Fig. 1. Schematic diagram of a horizontal well in a zone with a covered surface.

Several of these assumptions deserve comment. The first assumption is necessary for the analytical solution. It can be relaxed to include a heterogeneous unsaturated zone if a numerical simulation is used. The second assumption assures that no heat driving flow is considered. This is reasonable if the temperature distribution is relatively uniform in the subsurface; otherwise, both pressure-gradient driven flow and heat-gradient driven flow should be included in the governing equation. Also, the vapor's viscosity is not a constant if the temperature is nonuniform. The third assumption assures that the vapor flow permeabilities can be treated as constants. The fifth assumption allows us to deal with a single-phase flow problem rather than a two-phase flow problem. The last assumption assures a linear relationship between the vapor flux and pressure gradient. This assumption is quite restrictive and may not always be satisfied for studying vapor flow problems, particularly when the vapor flow rate is large. A high vapor flow rate could exist near a vertical pumping well. However, the usually low pumping rates per unit screen length for most vapor-extraction horizontal wells will make this assumption less restrictive in this study.

	Mathematical Model

TOP ABSTRACT INTRODUCTION Mathematical Modeling Mathematical Model Vapor Flow to a... Vapor Flow to a... Results of Vapor Flow... Pressure Distribution for a... Pressure Distribution for an... Numerical Evaluation of Solution... Vapor Mass Flux at... DISCUSSION AND CONCLUSIONS APPENDIX A APPENDIX B APPENDIX C REFERENCES

 
On the basis of above assumptions, a mathematical model of vapor flow is established. We discuss the problem in two steps. At first, we discuss vapor flow to a point sink, then we superpose the point-sink solution along the horizontal well to obtain the solution of a horizontal line sink. The point-sink solution developed here can be utilized as a starting point to find solutions of any sink shapes.

The transient vapor flow for a point source or sink located at (x₀, y₀, z_w) in an anisotropic unsaturated zone is described by the following equation (Bear, 1972, p. 200; Baehr et al., 1989; Johnson et al., 1990):

[1]

where n_v equals the volumetric vapor content; µ is the dynamic vapor viscosity (M [LT]^-1); p represents the vapor pressure (Pa or M [LT²]^-1); t equals the time (T); k_x, k_y, k_z represents the effective vapor permeabilities in the x, y, and z directions, respectively (L²); R is the universal gas constant (8.3145 ML² [T²mole K]^-1); T equals the absolute temperature (K); is the vapor mass extraction or injecting rate ( > 0 for extraction and < 0 for injecting) (M T^-1); M_wt represents the molecular weight of the soil gas (M mol^-1); is the Dirac delta function (L^-1); x₀, y₀, z_w equals the coordinates of the sink point (L).

The vapor mass-extraction rate is related to the vapor pumping rate Q (L³ T^-1) at a given pressure p_a as

[2]

where p_a is the ambient atmospheric pressure, which is 1.01 3 x 10⁵ Pa at 25°C.

The initial condition is

[3]

The lower boundary condition is

[4]

The upper boundary condition for a covered surface is

[5]

and for an uncovered surface is

[6]

The lateral boundary condition is

[7]

Equation [1] is a nonlinear partial-differential equation. It is usually approximated by a linear equation to achieve a closed-form solution. The linearization is done by replacing p in the denominator of the left-hand side of Eq. [1] by the ambient pressure, p_a. Such an approximation is commonly used in previous study of vapor flow (Falta, 1995; Hunt and Massmann, 2000). Illman and Neuman (2000)( 2001) provided a detailed discussion on the linearization of the vapor flow equation. Previous studies indicated that when the maximum pressure drop in the unsaturated zone is less than one-half of the ambient pressure, the error resulting from this linearization is small (Kidder, 1957; Falta, 1995). Massmann (1989) and Hunt and Massmann (2000) also pointed out that most vapor transport problems have small enough pressure changes to allow use of this linearized equation.

With the following transformations

[8]

the governing equation and the boundary and initial conditions become

[9]

[10]

[11]

[12]

[13]

[14]

Notice that Eq. [9] has the same form as the groundwater-flow equation although the coefficients there have very different meaning from those in the groundwater-flow equation (Bear, 1972).

The above equation and initial and boundary conditions are usually converted to dimensionless forms using the following notation:

[15]

where the subscript D denotes the dimensionless term. Transformation of h_a is the same as that of h, and transformations of x₀, y₀, and z_w are the same as that of x, y, and z, respectively.

The dimensionless equation is solved in Appendix A. The results for covered and uncovered surface cases are discussed separately in the following.

	Vapor Flow to a Point Sink with a Covered Surface

TOP ABSTRACT INTRODUCTION Mathematical Modeling Mathematical Model Vapor Flow to a... Vapor Flow to a... Results of Vapor Flow... Pressure Distribution for a... Pressure Distribution for an... Numerical Evaluation of Solution... Vapor Mass Flux at... DISCUSSION AND CONCLUSIONS APPENDIX A APPENDIX B APPENDIX C REFERENCES

 
The vapor flow to a point sink with a covered surface has the following representation in Laplace domain:

[16]

where H_D is the Laplace transform of h_aD-h_D, h_aD, and h_D are the dimensionless h_a and h defined in Eq. [15], s is the Laplace transform parameter referred to the dimensionless time, K₀ is the second kind, zero-order modified Bessel function, r_D is the dimensionless horizontal distance between the monitoring point and the sink point, _n = n, n = 1, 2,..., z_wD and z_D are the dimensionless z_w and z, respectively. The inverse Laplace transformation of Eq. [16] yields the solution in real time

[17]

where W(u) and W(u, ß) are the confined well function and leaky well function, respectively and they are defined as (Hantush, 1964):

[18]

The following inverse Laplace transformations are used when deriving Eq. [17] from Eq. [16] (Hantush, 1964, p. 303):

[19]

The tables and graphical curves of confined well functions and leaky well functions can be found in standard ground water hydrology textbooks; thus, the calculation of Eq. [17] is straightforward.

	Vapor Flow to a Point Sink with an Uncovered Surface

TOP ABSTRACT INTRODUCTION Mathematical Modeling Mathematical Model Vapor Flow to a... Vapor Flow to a... Results of Vapor Flow... Pressure Distribution for a... Pressure Distribution for an... Numerical Evaluation of Solution... Vapor Mass Flux at... DISCUSSION AND CONCLUSIONS APPENDIX A APPENDIX B APPENDIX C REFERENCES

 
The vapor flow to a point sink with an uncovered surface has the following representation in Laplace domain:

[20]

where _n = n + /2 n = 0, 1, 2,... The inverse Laplace transformation of Eq. [20] yields the real time solution:

[21]

Equations [16] through [21] are used as the bases to find the vapor flow to a horizontal well.

	Results of Vapor Flow to a Horizontal Well

TOP ABSTRACT INTRODUCTION Mathematical Modeling Mathematical Model Vapor Flow to a... Vapor Flow to a... Results of Vapor Flow... Pressure Distribution for a... Pressure Distribution for an... Numerical Evaluation of Solution... Vapor Mass Flux at... DISCUSSION AND CONCLUSIONS APPENDIX A APPENDIX B APPENDIX C REFERENCES

 
A horizontal well can be treated as a summation of infinite point sinks along the well axis, the superposition of solutions from these point sinks will yield the solution of a horizontal well. Before we proceed, we must know how the total pumping rate is distributed along the well axis.

There are primarily two different possible treatments of a pumping well: a uniform-flux or a uniform-pressure sink. The uniform-flux treatment implies that the total pumping rate is evenly distributed along the well. Such a treatment is easy to handle analytically, but it does not consider the well end effect, thus may yield inaccurate solutions for points very close to the well end. The uniform-pressure treatment implies that the well has an infinitely large permeability, thus the pressure change along the wellbore is always the same. This treatment is close to physical reality but difficult to handle analytically. The flux distribution along a uniform-pressure horizontal well is nearly uniform for most parts of the well except at the two well ends where the fluxes are higher than that in the central part of the well.

It is important to test how different is the pressure change because of these two different views of a horizontal well. We have done this numerically using a standard groundwater flow modelling software Visual Modflow (Waterloo Hydrogeologic, 2000). The rationale for using a groundwater flow model to simulate vapor flow is that the linearized vapor flow Eq. [9] is identical to the groundwater flow equation. The numerical result is briefly described below.

First, we simulate a uniform-flux horizontal well. Because a horizontal well cannot be directly simulated using Visual Modflow, each cell representing a part of the horizontal well is simulated using a partially penetrating vertical well. The total pumping rate of the horizontal well is evenly distributed among the series of partially penetrating vertical wells. Pressure distribution is found by using the model. Secondly, we simulate a uniform-pressure horizontal well. The cells representing the horizontal well are assigned an extremely large, for instance, 10000 times larger, hydraulic conductivity than the hydraulic conductivity of the aquifer. The remaining parameters are the same as those used in the uniform-flux case.

The numerical simulation results show that when the distance between a monitoring point and a well end is ten times the horizontal-well diameter, the difference between the uniform-flux and uniform-pressure results is <5%. When that distance increases, the difference of the uniform-flux and uniform-pressure solutions decreases quickly to an undetectable level. If assuming that the horizontal well has a well diameter of 15.24 cm (6 inches) (a standard well diameter used in environmental application), this implies that when the monitoring well is 152.4 cm (60 inches) away from the well end, there is <5% difference between the uniform-flux and uniform-pressure solutions. Therefore, uniform-flux treatment can be used in practical cases.

	Pressure Distribution for a Covered Surface

TOP ABSTRACT INTRODUCTION Mathematical Modeling Mathematical Model Vapor Flow to a... Vapor Flow to a... Results of Vapor Flow... Pressure Distribution for a... Pressure Distribution for an... Numerical Evaluation of Solution... Vapor Mass Flux at... DISCUSSION AND CONCLUSIONS APPENDIX A APPENDIX B APPENDIX C REFERENCES

 
Using the coordinate system shown in Fig. 1 and considering the horizontal well as a uniform-flux sink, the dimensionless solution to the horizontal well is obtained through the following integration of the point sinks.

[22]

where r_D = and the dimensionless well length L_D is defined as L_D = . Equation [22] can be calculated using the existing well function tables. Equation [22] can also be written in an alternative form using the dimensionless time as the argument in the integration. This is done using the well function definitions (Eq. [18]), changing the order of integration, and changing the variable of integration from v to . The result is:

	Pressure Distribution for an Uncovered Surface

TOP ABSTRACT INTRODUCTION Mathematical Modeling Mathematical Model Vapor Flow to a... Vapor Flow to a... Results of Vapor Flow... Pressure Distribution for a... Pressure Distribution for an... Numerical Evaluation of Solution... Vapor Mass Flux at... DISCUSSION AND CONCLUSIONS APPENDIX A APPENDIX B APPENDIX C REFERENCES

 
Using the point-sink solution Eq. [21] as the starting point and following the same procedure as that in the covered surface case, we obtained the following solution of vapor flow to a horizontal well for an uncovered surface:

[24]

Equation [24] has the following alternative form using dimensionless time as the argument in the integration:

Equation [25] is evaluated using the numerical program VF3D. Converting all the parameters in Eq. [23] or [25] to the dimensional vapor flow parameters using Eq. [8] and [15], we obtain the vapor pressure near a horizontal well.

	Numerical Evaluation of Solution of Vapor Flow to a Horizontal Well

TOP ABSTRACT INTRODUCTION Mathematical Modeling Mathematical Model Vapor Flow to a... Vapor Flow to a... Results of Vapor Flow... Pressure Distribution for a... Pressure Distribution for an... Numerical Evaluation of Solution... Vapor Mass Flux at... DISCUSSION AND CONCLUSIONS APPENDIX A APPENDIX B APPENDIX C REFERENCES

 
VF3D is written in a MATLAB (Mathworks, 2000) script file that uses the Gaussian Quadrature method (Press et al., 1989) to perform the integrations over time. This program calculates the transient vapor pressure for both covered and uncovered surface cases, and the total vapor mass flux across the uncovered surface, and generates dimensionless vapor flow type curves. Type curve is a concept broadly used in groundwater hydrology in which it plots the dimensionless drawdown versus dimensionless time (or dimensionless time over the square of the dimensionless radial distance) in a log-log plot (Hantush, 1964; Bear, 1972). It has been used for understanding aquifer response to pumping and for finding the hydraulic conductivity (or transmissivity) and storativity. This concept is also used for understanding air pumping or injecting in an unsaturated zone and for finding air flow permeability (Illman and Neuman, 2000, 2001).

The VF3D program and user's manual are in the website: http://geoweb.tamu.edu/Faculty/Zhan/Research.html and can be freely down loaded.

Figures 2 through 5 are type curves of horizontal-well pumping for covered and uncovered surfaces. The monitoring point is at (1, 1, and 2.5 m) for a near field case and at (10, 10, and 2.5 m) for a far field case. These type curves plot h_aD - h_D = versus in log-log scale, where r_D = is the dimensionless horizontal radial distance from the monitoring point to the center of the horizontal well.The k_h in the legends of Fig. 2 through 5 is the horizontal permeability. The parameters used in these figures are as follows: the horizontal well is 40 m long in the center of the unsaturated zone. The thickness of the unsaturated zone is 5 m. The Theis curve of a fully penetrating vertical well is also included in the figures for reference. Theis curve is a simple type curve that is commonly used in groundwater hydrology as a reference to compare with other type curves (Hantush, 1964; Bear, 1972). It refers to a fully penetrating vertical line sink (a pumping well) in a homogeneous, isotropic confined aquifer whose lateral boundaries are infinitely far from the sink. The dimensionless drawdown in the Theis curve only depends on a single variable: , where r_D is the dimensionless horizontal radial distance from the monitoring point to the line sink. The Theis curve only depends on horizontal flow and is independent of vertical anisotropy of the aquifer.

View larger version (21K): [in this window] [in a new window]   Fig. 2. Type curves at a near field point (1, 1, and 2.5 m) in an isotropic (Kh/Kv = 1) and an anisotropic (Kh/Kv = 10) zone with covered surfaces.

View larger version (21K): [in this window] [in a new window]   Fig. 5. Type curves at a far field point (10, 10, and 2.5 m) in an isotropic (Kh/Kv = 1) and an anisotropic (Kh/Kv = 10) zone with uncovered surfaces.

View larger version (18K): [in this window] [in a new window]   Fig. 3. Type curves at a far field point (10, 10, and 2.5 m) in an isotropic (Kh/Kv = 1) and an anisotropic (Kh/Kv = 10) zones with covered surfaces.

View larger version (21K): [in this window] [in a new window]   Fig. 4. Type curves at a near field point (1, 1, and 2.5 m) in an isotropic (Kh/Kv = 1) and an anisotropic (Kh/Kv = 10) zone with uncovered surfaces.

The way we plot the type curves in this paper is similar to that used in the Theis curve, which honors the horizontal flow. Therefore, the type curves for a covered surface may have similar behavior as the Theis curve at a late time because of the primarily horizontal flow to a horizontal well at that time; but the type curves for an uncovered surface should be sharply different from the Theis curve at late time because of the strong vertical as well as horizontal flows to a horizontal well in this case at late time. Type curves for both covered and uncovered surfaces are different from the Theis curve at early time because of the contrast of vertical flow to a horizontal sink and horizontal flow to a vertical sink at early time. Bearing this in mind, we will not be surprised to see the difference of the horizontal-well type curves with the Theis curve at early time in the Fig. 2 through 5, and the difference of the type curves with the Theis curve at late time in the Fig. 4 through 5.

Figure 2 presents the type curves at a near field point (1, 1, and 2.5 m) in an isotropic and an anisotropic unsaturated zone with covered surfaces. The difference between the type curve and the Theis curve at early time confirms the different nature of flow to a horizontal well and to a vertical well at that time. The type curves show a trend of gradual convergence to the Theis curve at late time in Fig. 2, indicating the similar nature of flow to a horizontal well and to a vertical well at late time. The main difference between type curves in the isotropic and anisotropic unsaturated zones is at the early time when the vertical permeability greatly influences the vertical component of flow. Such a difference diminishes when pumping time gets longer and the flow becomes primarily horizontal.

Figure 3 presents the type curves at a far field point (10, 10, and 2.5 m) for isotropic and anisotropic unsaturated zones with covered surfaces. Because vapor flow is primarily horizontal in the far field, the vertical anisotropy has very mild influence upon the type curves.

We should point out that the x-axis in Fig. 2 and 3 is , not t_D. Given the same t_D, is larger for a near field (r_D is smaller) than that for a far field (r_D is larger). This explains why it appears to take a much larger "" in Fig. 2 for the type curve to converge to the Theis curve than that in Fig. 3.

Similar conclusions such as the difference of flow to a horizontal well and to a vertical line sink at early time and similarity of flow to a horizontal well and to a vertical line sink at late time were reported in previous studies of oil and gas flow in petroleum reservoir (Goode and Thambynayagam, 1987; Daviau et al., 1988; Ozkan et al., 1989; Rosa and Carvalho, 1989) and in studies of vapor and groundwater flow in unsaturated zone and aquifers (Falta, 1995; Zhan, 1999; Zhan and Cao, 2000).

Figures 4 and 5 presents the type curves for isotropic and anisotropic unsaturated zones with uncovered surfaces at a near field point (1, 1, and 2.5 m) and a far field point (10, 10, and 2.5 m), respectively. These two figures show that the type curves of horizontal well pumping in an uncovered surface unsaturated zone are significantly different from the Theis curve during all time. This reflects the fact that vertical flow in an uncovered surface unsaturated zone becomes important and vapor can replenish the unsaturated zone from the atmosphere. The type curves in Fig. 4 through 5 are similar to those of a leaky aquifer (Hantush, 1964). The flat type curves indicate that the pumped vapor is supplied entirely from the atmosphere through crossing surface flow at late time. Thus, no additional pressure changes will be observed in the unsaturated zone at late time. Because of the strong vertical component of flow, the vertical anisotropy affects the type curves significantly.

	Vapor Mass Flux at the Ground Surface

TOP ABSTRACT INTRODUCTION Mathematical Modeling Mathematical Model Vapor Flow to a... Vapor Flow to a... Results of Vapor Flow... Pressure Distribution for a... Pressure Distribution for an... Numerical Evaluation of Solution... Vapor Mass Flux at... DISCUSSION AND CONCLUSIONS APPENDIX A APPENDIX B APPENDIX C REFERENCES

 
It is interesting to see how much vapor mass flux occurs at the ground surface for the uncovered surface case. Defining the following dimensionless pressures:

[26]

and neglecting gravity (which is much smaller than the pressure gradient term), the specific vapor flux at the ground surface is

[27]

Using Eq. [25] and the fact that p|_z=d = p_a and sin (_n) = (-1)ⁿ, one obtains:

[28]

where the negative sign means a downward flow because of the pumping.

The maximal specific discharge occurs at x = y = 0. It is interesting to see the vertical vapor flux distribution at the ground surface. Figures 6a and 6b show the contour maps of steady-state normalized vertical mass fluxes q_z/q_0z across the uncovered surface (z_wD = 1) with two different well locations at z_wD = 0.2 and 0.6, respectively, where q_0z = . The dashed line from x_D = 0 to 2 represents half of the horizontal well screen (another half is from x_D = -2 to 0). Because these contour maps are symmetric with respect to both x and y axes in a planar view, we only show the upper-right quarters of the maps. The following attributes are observed from these figures:

View larger version (21K): [in this window] [in a new window]   Fig. 6. The upper-right quarter of the contour map of steady-state normalized vertical mass flux qz/q0z across the uncovered surface, where q0z = -ß. The dashed line from xD = 0 to 2 represents half of the horizontal well screen (another half is from xD = -2 to 0). (a) The horizontal well is at zwD = 0.2, and (b) the horizontal well is at zwD = 0.6.

2. A shallower well (Fig. 6b, z_wD = 0.6) withdraws more vapor from the central region than the deeper well (Fig. 6a, z_wD = 0.2), seen from the higher q_z/q_0z value near the well in Fig. 6b. However, q_z/q_0z drops more rapidly in Fig. 6b than in Fig. 6a when moving away from the central region, reflected from the steeper slope of q_z/q_0z over y_D in Fig. 6b than that in Fig. 6b near the well.

Using Eq. [2], we can find the total mass flux across the ground surface owing to the pumping of the horizontal well:

[29]

where _s denotes the mass flux across the ground surface. In Appendix B, we obtain

[30]

where is the mass flux to the pumping well and the negative sign implies a downward flux. It is interesting to see from Appendix B that the steady-state (t_D ) mass flux across the ground surface is

[31]

This result means that the pumping rate of the horizontal well equals the total mass flux across the ground surface at steady-state. Thus, the atmosphere is the only source for supplying vapor to the horizontal well after a sufficiently long time of pumping. This finding agrees with what has been observed in the type curves of Fig. 4 and 5.

Figure 7 shows the total vapor mass fluxes across the uncovered surface with well locations at z_w = 1, 2.5, and 4 m, respectively. An isotropic unsaturated zone is assumed and the thickness of the vapor zone is d = 5 m. It is clear that the total vapor mass fluxes across the surface approach the horizontal well pumping rate at around dimensionless time t_D = 1 for all three elevations, showing that the converging time is very insensitive to the well elevations. Before converging to the pumping rate, the shallowest horizontal well (z_w/d = 0.8) has the highest cross-surface flow at any time t_D < 1. This is simply because the shallowest well is closest to the source of vapor supply, which is the atmosphere.

View larger version (14K): [in this window] [in a new window]   Fig. 7. Normalized total vapor mass fluxes across an uncovered surface with dimensionless time.

	DISCUSSION AND CONCLUSIONS

TOP ABSTRACT INTRODUCTION Mathematical Modeling Mathematical Model Vapor Flow to a... Vapor Flow to a... Results of Vapor Flow... Pressure Distribution for a... Pressure Distribution for an... Numerical Evaluation of Solution... Vapor Mass Flux at... DISCUSSION AND CONCLUSIONS APPENDIX A APPENDIX B APPENDIX C REFERENCES

Combining the analytical and numerical results of this paper, we can calculate pressure distributions of horizontal well pumping in an unsaturated zone for a given point at given time.

Theories of vapor flow to vertical wells have been developed before and been verified by field experiments (Illman and Neuman, 2000, 2001). Unfortunately, although horizontal wells have been broadly used for vapor extraction, to our knowledge, very little attention is given to record the time-space dependent pressure data in field experiments. One possible reason for the lack of field experiments is that there is no existing theory to compare with. The analytical and numerical results of this paper provide the capability to compare the theory with future experiments.

The following conclusions can be drawn from this study:

1. In a covered surface unsaturated zone, the early time-type curves are different from the vertical well Theis curve. This reflects the fact that the pressure change at early time is caused by strong vertical radial flow to the horizontal well.

2. The late time-type curves converge to the Theis curve in both near and far fields in a covered surface unsaturated zone. This indicates that the late time pressure changes are caused by horizontal flow and lateral withdrawal of vapor from progressively farther distances.

3. The main difference between type curves in the isotropic and anisotropic covered unsaturated zones is at the early time when the vertical permeability greatly influences the vertical component of flow. Such a difference diminishes when pumping time gets longer and the flow becomes primarily horizontal.

4. The type curves at an uncovered surface unsaturated zone are significantly different from the Theis curve during all the pumping time. This is caused by the strong vertical flow across surface. These type curves approach asymptotic limits at late time and are similar to those observed in leaky aquifers.

5. The steady-state vapor mass flux rate across the uncovered surface is the highest at the point right above the center of the horizontal well and gradually decreases when moving away from the well. The withdrawal of vapor by a shallower horizontal well from across the uncovered surface near the horizontal well is more than that by a deeper well, but the withdrawal rate decreases more rapidly than that by a deeper well when moving away from the well.

6. The transient total vapor mass fluxes across the uncovered surface approach the horizontal well pumping rates exponentially, independent of well elevation.

	APPENDIX A

TOP ABSTRACT INTRODUCTION Mathematical Modeling Mathematical Model Vapor Flow to a... Vapor Flow to a... Results of Vapor Flow... Pressure Distribution for a... Pressure Distribution for an... Numerical Evaluation of Solution... Vapor Mass Flux at... DISCUSSION AND CONCLUSIONS APPENDIX A APPENDIX B APPENDIX C REFERENCES

 
Solving the Dimensionless Vapor Flow Equation
After changing h = h - h_a, converting the governing Eq. [9] into the dimensionless format, and conducting the Laplace transform, we have

[A1]

[A2]

[A3]

[A4]

[A5]

where s is the Laplace parameter referred to the dimensionless time, and H_D is the Laplace transformation of dimensionless h.

The solution satisfying boundary condition [A2] is written as (Dougherty and Babu, 1984; Moench, 1997):

[A6]

Substituting [A6] into [A3] results in:

[A7]

for a covered surface case.

Substituting Eq. [A6] into Eq. [A4] results in:

[A8]

for an uncovered surface case.

Substituting Eq. [A6] into Eq. [A1], multiplying by cos(_n z_D), and integrating from z_D = 0 to z_D = 1 will result in a function of F_n. The equations and solutions for F_n are different for the cases of covered surface and uncovered surface, thus they are discussed separately.

Case 1: Covered Ground Surface
In this case, ₀ = 0, and _n 0 if n > 0, thus the equation of F₀ is separated from the equation of F_n with n > 0.

[A9]

[A10]

where _n in Eq. [A10] is defined in Eq. [A7].

Equations [A9] and [A10] are the modified Helmholz equations with delta-function sinks. The solutions satisfying Eq. [A9], [A10], and boundary condition [A5] are obtained through the Green's function (Arfken and Weber, 1995, Table 8.5):

[A11]

[A12]

where r_D = [(x_D - x_0D)² + (y_D - y_0D)²]^1/2 is the dimensionless distance between the measured point and the sink point.

Substituting F_n into Eq. [A6] results in the solution of H_D.

Case 2: Uncovered Ground Surface
In this case, _n 0, when n = 0, 1, 2,... thus the equation of F_n is identical to Eq. [A10] except that _n is defined in Eq. [A8]. The solution is identical to Eq. [A12] except that n = 0, 1, 2,...

	APPENDIX B

TOP ABSTRACT INTRODUCTION Mathematical Modeling Mathematical Model Vapor Flow to a... Vapor Flow to a... Results of Vapor Flow... Pressure Distribution for a... Pressure Distribution for an... Numerical Evaluation of Solution... Vapor Mass Flux at... DISCUSSION AND CONCLUSIONS APPENDIX A APPENDIX B APPENDIX C REFERENCES

 
Derivation of Ground Surface Mass Flux
Using the following identities

[B1]

then Eq. [28] becomes

[B2]

Substituting Eq. [B2] into Eq. [29], changing dxdy to dx_Ddy_D, applying integrations to dx_D and dy_D first, and using the following identity

[B3]

one arrives the following solution

[B4]

Noticing that _n = n + /2, and using the following identity

[B5]

one immediately obtains the steady-state (t_D ) result:

[B6]

	APPENDIX C

TOP ABSTRACT INTRODUCTION Mathematical Modeling Mathematical Model Vapor Flow to a... Vapor Flow to a... Results of Vapor Flow... Pressure Distribution for a... Pressure Distribution for an... Numerical Evaluation of Solution... Vapor Mass Flux at... DISCUSSION AND CONCLUSIONS APPENDIX A APPENDIX B APPENDIX C REFERENCES

 
Nomenclature
d = distance from the ground surface to the top of capillary fringe zone aquifer (m)

h = p² (kg² m^-2 s^-4)

h_a = p²_a(kg² m^-2 s^-4)

h_D, h_aD = dimensionless h and h_a

k_x, k_y, k_z = permeabilities in the x, y, and z directions, respectively (m²)

k_h = permeabilities in the horizontal direction (m²)

K₀ = the second-kind zero-order modified Bessel function

L = horizontal well screen length (m)

L_D = (L d^-1) , dimensionless horizontal well screen length

= vapor mass extraction or injecting rate ( > 0 for extraction and < 0 for injecting) (kg s^-1)

_s = vapor mass flux across the ground surface (_s > 0 for upward flow and _s < 0 for downward flow) (kg s^-1)

M_wt = molecular weight of soil vapor (kg mol^-1)

n_v = volumetric vapor content

p = vapor pressure (Pa or kg m^-1 s^-2)

p_a = ambient atmospheric pressure (Pa or kg m^-1 s^-2)

p_D = , p_aD = , dimensionless pressure and atmosphere pressure, respectively

q_z = vertical vapor mass flux (m s^-1)

q_0z = -ß/ (m s^-1)

Q = volume pumping rate (Q > 0 for pumping; Q < 0 for injecting) (m³ s^-1)

r = [(x - x₀)² + (y - y₀)²]^1/2, horizontal distance between the measured and the sink point (m)

r_D = dimensionless r

R = 8.3145 (kg m² s^-2 mol^-1 K^-1), universal gas constant

s = parameter of Laplace transform referred to the dimensionless time

t = time (s)

t_D = dimensionless time

T = absolute temperature (K)

W(u) = confined well function

W(u, ß) = leaky well function

x, y = horizontal coordinates (m)

x₀, y₀ = horizontal coordinates of the point sink (m)

x_D, y_D = dimensionless horizontal coordinates

x_0D, y_0D = dimensionless horizontal coordinates of the point sink

z = vertical coordinate (m)

z_D = dimensionless vertical coordinate

z_W = vertical distance from the horizontal well to the top of the capillary fringe (m)

= n_vµ/p_a (s)

ß = 2µRT/M_wt (kg² m s^-4)

= Dirac delta function (m^-1)

µ = dynamic vapor viscosity (kg m^-1 s^-1)

_n = n, n = 0, 1, 2,... for the covered surface;

n + /2, n = 0, 1, 2,... for the uncovered surface

	ACKNOWLEDGMENTS

 
Support for this work was provided by the U.S. National Science Foundation Grant No. BES-9909964 and by the Office of Vice President for Research and Associate Provost for Graduate Studies, Texas A&M Univ. through the Interdisciplinary Research Initiatives. We thank Dr. Tony Gangi for his prereview of the manuscript. We thank Dr. John Nimmo and two anonymous reviewers for their detailed and insightful comments for revising the manuscript.

Received for publication April 18, 2001.

	REFERENCES

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