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Piezometric head distribution in sand-filled wells
Journal of Hydrology 1 (1963) 195-203; © North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
P I E Z O M E T R I C H E A D D I S T R I B U T I O N IN
SAND-FILLED WELLS DONALD J. BROWN Hartford Laboratories, General Electric Company, Richland, Washington, U.S.A. *)
Received 30 September, 1963
Introduction
The ground water monitoring program at Hanford**) provides continuous assessment of the status of radioactive contaminants in the ground water resulting from controlled waste disposal operations in the Chemical Processing Areas. More than 400 cased, eight-inch wells have been drilled on the Project to obtain ground-water samples for analysis and use in making the routine evaluations. These wells are drilled and perforated from 50 to 750 feet below the regional water table. In some cases, due to head differences, the water throughout a perforated well is not representative of the ground water in the respective formations outside the well; but because of the flow pattern established in and around the well, contaminants may not enter the casing and be detected. In other instances, contaminants may enter a well from a contaminated aquifer of higher head than other aquifers tapped by the well; flow throughout the well may lead to the conclusion that all aquifers are so contaminated. Such flow patterns can, and were found to exist where wells penetrate several aquifers having different hydraulic potentials. Many of the wells drilled to water on the Project tap aquifers of sufficient head difference to create flow patterns which may result in erroneous or incomplete information on the status of contamination in the ground water. It is desirable, therefore, to modify the deep wells to permit representative sampling of the various aquifers. A requirement following modification is that the hydraulic potential distribution everywhere *) Work performed under Contract No. AT(45-1)-1350 for the U.S. Atomic Energy Commission. **) Here after indicated as Project. 195
196
D.J. BROWN
within the casing is less than or equal to that naturally present in the formation outside the well. Such a potential distribution will insure that ground waters enter and leave the well at or near the same elevation regardless of the number of aquifers of different potential tapped by the well. The method originally used for equalizing the hydraulic head within a deep well was to seal plastic piezometer tubes inside the casing at selected intervals. This method involved the setting of a number (usually four) of smaU diameter tubes in the well with 50 to 60-foot intervals between the bottoms of the tubes. The casing was filled with sand in the zones where the tubes were perforated, and the casing sections between tubes were cement-filled. Thus, access to the sealed-off aquifers was provided, and reliable samples and head measurements of the ground water were obtained, since the movement of water up or down the casing was restricted. Eight wells were modified in this manner. The time required to install piezometer tubes in each well varied from 6 to 14 days. Because of the relatively long installation time, other methods for installing tubes were studied. One method investigated simply was to fill the well with sand, thereby eliminating the time-consuming cementing phase of the installation work. It was necessary, however, to evaluate the head dissipating characteristics of the sand as compared to the cement formerly used. For this evaluation, a series of three-dimensional cases was solved with a newly-developed computer program 1) simulating both a sandfilled well and one in which cemented sections were present. The boundary conditions selected were the same as some of those known to exist at Hanford. The results of the theoretical evaluation are summarized in this paper.
Summary and conclusions Two sets of boundary conditions were selected as input to a computer program to evaluate the potential distribution in a well filled with sand as compared to that in a well containing cemented sections. The first set of boundary conditions was chosen as that most representative of the hydrological conditions encountered in the greatest number of deep wells at Hanford. The results from this case showed that the piezometric head distribution in sand-filled wells is within 0.007 foot of the true potential distribution existing in the formation outside the well. The potential distribution within the well containing the cemented section varied from the true value in the formation by 0.01 foot. The second set of conditions was chosen to duplicate some of the more extreme hydrological conditions encountered by a few of the deep wells. The results again show potential distributions in wells filled with sand alone are as close to the true potential values in the
PIEZOMETRIC HEAD DISTRIBUTION IN SAND-FILLED WELLS
197
formation as are those in the wells with the cemented sections. In this case, however, the variance from the true values was on the order of 0.1 foot for both the sand and the cement at one point in the well. Another type of calculation, using Poiseuille's and Darcy's equations for the laminar flow of water in pipes, revealed that the ratio of head loss for an open pipe to that of a pipe filled with sand is about 8 x 1 0 - 9 ; whereas, the ratio for a pipe filled with cement is 8 x 10 -13. These ratios indicate that sand alone is extremely effective in dissipating most of the head differences in a well. Some flow of water naturally takes place between the aquifers within the formation. Sand, therefore, more nearly duplicates the actual conditions present in the formation because it does not restrict the flow as completely as does cement. Accordingly, sand in most cases gives more accurate results than does cement. This study demonstrates that a bank of piezometer tubes installed in an eight-inch well casing, using sand alone to dissipate the hydraulic head between aquifers encountered by the well, will provide a reliable structure for sampling purposes and head measurements.
Computer program and boundary conditions for model A computer program was developed at Hanford to analyze one-, two-, and three-dimensional and axi-symmetrical problems with up to 8000 grid points1). With only minimum input effort, a wide range of boundary conditions needed for various problems is available. The system is completely flexible in that every grid point can be individually controlled for boundary type. Descriptive data on soil permeability can be easily designated at every spatial grid point. The above-mentioned computer program was used to evaluate the effectiveness of sand in dissipating hydraulic head in a well. A model of slightly over 7000 grid points was chosen. The permeability and the boundary conditions were assumed to be symmetrical about the well so that only half of the flow system needed to be studied. The physical size of the saturated block of soil represented by the model was seven feet (x direction) by four feet (y direction) by sixty feet (z direction). The well was located in the center of the x z plane at Yl and Y2. The potential distribution was calculated for two separate sets of boundary conditions. The first set was that which most nearly duplicated the conditions believed to be present in most of the deep wells at Hanford. In this instance a potential head difference of 3 feet in the z direction and 0.035 foot in the x direction was set at the two side boundaries, y z plane at Xmin and y z plane at xmax. A soil of uniform permeability was selected for input throughout the model.
198
D.J. BROWN
The more extreme hydrological conditions believed present in a few of the deep wells were represented by the second set of boundary conditions. In the second model the permeabilities were selected to represent a uniformly thick horizontal clay bed in the middle half of the model. The permeability in the upper and lower quarters was held the same as those in the first model. The head difference in the z and x directions were again set at the two side boundaries. The vertical head component across the clay bed was 15 feet while the horizontal component was the same as in the first model, 0.035 foot.
Experimental results Three cases were run on the IBM 7090 computer using the boundary conditions of the first model. The first case determined the potential distribution in the model with no well present. These data provided a standard for comparison of the results from the sanded well and the cemented well cases. The second case solved for the potential distribution in a well filled entirely with sand. In this instance the ratio of permeability of sand to the permeability used throughout the model was selected as 15. This ratio is typical of that which would exist in a deep well at Hanford if it were filled with sand. (Flow of ground water into a well would not occur if the ratio of permeabilities is less than 1). The third case was for a well in which the middle half of the casing was filled with cement and the upper and lower quarters filled with sand. In this case the ratio of the permeability of the cement in the well and the material in the model was 0.001. The cement acts as a dam restricting almost all flow up or down the casing. The results from the three cases were compared and the variance from the true potential pattern calculated. Variances are plotted in Fig. 1 to show
-T
Depth iO° i
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io° I
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zo'
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-
S a n d - F i l l e d Well With Cemented Section Sand-Filled
Well
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.
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Fig. 1. Variance of potential in well from true potential in formation (ff w e l l q~formation).
PIEZOMETRIC HEAD DISTRIBUTION IN SAND-FILLED WELLS
199
the contrast throughout the well. The potential in both the sanded well and the partially-cemented well is less than the true potential between the 53and 60-foot depths. This would indicate flow into the well throughout this section in both cases. Between the depths of 46 feet and 53 feet, the potential in the sanded well approaches the true potential; however, the potentials in the well still remain slightly on the negative side. In the well with the cemented zone, the potential inside the casing becomes greater than the true potential which indicates outflow of water from the well. (This is expected as the water, not being able to flow through the cemented zone inside the casing, would flow out into the formation and upward around the outside of the casing.) From a depth of 18 feet down to the 46-foot depth, little or no variance is noted from the true potential. The potential in the sanded well is about 0.001 foot less than the true potential, and the potentialin the well with the cemented section is approximately 0.004 foot greater than the true values. In the latter case water, if present in that section, would tend to flow out of the cemented section and into the formation. From the 18-foot depth to the top of the model the potential in the sanded well gradually increases until it is greater than the potential in the formation. From a depth of about 13 feet, where the potential in the sanded well equals the true potential, to the top of the model, flow takes place out of the well into the formation. The maximum variance from the true potential in the sanded well thus occurs at the very top and bottom of the model; there the difference is about 0.03 foot. This is also true for the well with the cemented section. In the case of the cemented well the potential in the upper 18 feet of the model first decreases and then increases sharply. From 18 feet up to about 9 feet the water flowing up around the casing from below re-enters the well. From 9 feet to the top of the model the water again leaves the well. A study of this model with the first set of boundary conditions shows that the potential distributions measured in wells filled with sand alone are as close, and often closer, to the actual potential distributions in the formation as are those measured in wells having cemented sections. The volume of ground water moving up the well through the sanded section would be less than one gallon per day for the boundary conditions used in this model. Three cases were solved on the 7090 computer using the boundary conditions set up in the second model: the thick clay bed and higher potential head difference. In the first case the potential distribution of the model was determined with no well present. The second and third cases were solved for the sand-filled well and the cemented section, respectively. The permeability ratios used in the second and third cases in the upper and lower
200
D.J.
BROWN
sections of the model were the same as those used in the first model. In the middle clay section, the ratio of the permeability between the sand and the clay was 150. The ratio of the permeability of the cement to that of the clay was 0.01. The results of these three evaluations were compared and analyzed in a manner similar to that used in the first model. A graph comparing the variances was prepared and is exhibited as Fig. 2. Only small differences existed between the measured potentials in the formation in the topmost, middle, and the lowermost sections. The largest differences in potentials Depth 0*
;:!::
.z;"
Sand-Filled Well W~th
Cemented Section
S a n d - F i l l e d Well
Q
!o
_m
,o'
I .Oe
.OO
(-)
.04
;:i
.g2 .o'~ .do ~,
.02 (Feet)
(+l
Fig. 2. Variance of potential in well from true potential in formation (ff well-formation).
appeared at the intermediate zones where the upper and lower clay interfaces occurred. In the case of the cemented well the potential distribution was quite similar to that seen in the first model. Water enters the lower part of the well, flows up the well until it nears the cemented section, and then leaves the well. The water which flows out of the well moves through the clay section outside the well and then re-enters the well near the top of the clay interface. There it flows up the well for a short distance and then leaves the well again. The maximum variance from the true potential in the formation in this instance was 0.08-0.09 foot. The potential distribution in the sanded well was also quite similar to that observed in the first model; however, at the clay interfaces some differences exist. The potential in the well just below the lower interface is much lower than that in the formation; there the greatest inflow occurs. At the lower clay interface the potential difference decreased sharply similar to that observed in the well with the cemented section. Throughout the lower
PIEZOMETRIC HEAD DISTRIBUTION IN SAND-FILLED WELLS
201
clay section the potential difference decreased gradually until in the middle section the variance was less than 0.001 foot. Starting several feet below the upper clay interface, there was a rapid increase in potential differences up to the clay boundary and then these differences became smaller until the variance was insignificant. One explanation for this is evident from examination of the relationship between the permeability of the clay and the permeability of the formation above the clay. Because there is no flow out of the top of the model, and because flow does occur up the well, the potential in the upper half of the well gradually increases upward. The permeability of the clay, however, is so low that water can not leave the well at a rate fast enough to dissipate the gradually rising potential. The potential, therefore, appears to increase exponentially. When the clay interface is passed the water can then flow out of the well at a much faster rate because of the higher permeability in this zone. This outflow thus relieves most of the pressure to bring it closer to the true potentials in the formation. The maximum potential difference in the sanded well was approximately 0.1 foot; however, this potential difference occurred in the clay zone near the upper interface and would have little effect on the potential measured by a piezometer tube in the upper sanded section. The average variance is probably closer to 0.01 foot. For the second model it was calculated that less than one gallon of water per day flows from the lower aquifer through the sanded well to the upper aquifer. That flow rate probably would have little effect on the monitoring results from those wells.
Head loss in an open pipe A calculation was made to determine the ratio of head loss in an unobstructed pipe to the head loss in a pipe filled with sand. Using Poiseuille's equation for head loss in open pipe (assuming laminar flow) and Darcy's equation for head loss in a pipe filled with sand, the ratio for both head loss and velocity can be obtained by equating the two expressions:
3 2 p l I/1 hLl y d 2
#1 V2 hL2 k y
(1)
V1 and hLl designate the velocity and head loss in the open pipe, and Vz, hL2 the velocity and head loss in the sand-filled pipe. The intrinsic permeability in Darcy's equation, represented by k, has units offt 2, and should not be confused with the permeability or hydrauhc conductivity which is usually represented by K and has units of volume per time flowing through a unit cross section at unit gradient. The diameter of the pipe is
202
D.J. BROWN
d, the viscosity/z, the specific weight y, and the length of the pipe 1. From equation (1), if we let V1 = V2, then: hLl hL2
32 k dr-, or if
(2)
hL~ = hL2 then V2
32k
(3)
The head loss ratio (or the velocity ratio) in the case of the open pipe and the pipe filled with sand is 8.4 x 10 -9. A similar ratio for an open pipe and a pipe filled with cement is 8.4 x 10 -13. The intrinsic permeability of the sand used in making this calculation was 1.2 x 10-l°ft2, and was determined from permeameter data. The intrinsic permeability (k) of the cement was calculated to be 1.2 x 10-1aft 2. A comparison of these two ratios emphasizes the effectiveness of the sand alone in dissipating head differences. Unconsolidated sediments in a formation are much more permeable than cement; hence, we may reasonably assume that the potential distribution in a sand-filled well will more closely approximate the true potential distribution in a formation.
Field experience with the sand-pack method In the southeastern part of the Hartford Project near the Columbia River, ground water temperature profiles measured in four wells showed no change in temperature with depth. The data suggested that the wells penetrated two or more aquifers of different piezometric head, and that vertical flow was occurring in the casing. Piezometer tubes were installed in two of these wells using the cement method to seal off the sections of casing between aquifers. When equilibrium was re-established, water level recordings showed head differences of 25 to 26 feet between the lowermost aquifer and the free ground water in both wells. The highest head was found to occur in the lowermost aquifer, and thus upflow did take place in the casing prior to the installation of the piezometer tubes. After completing the theoretical evaluation of potential distributions in sand-filled wells, the sand-pack method was used to install piezometer tubes in one of the remaining two wells. The location of this well was about midway between the two wells containing the cemented-in piezometers. The head difference measured between the lowermost aquifer and the free ground water in this well was
PIEZOMETRIC HEAD DISTRIBUTION IN SAND-FILLED WELLS
203
almost 30 feet. Although this was not a direct comparison of the piezometric-head measurements obtained by the cement and sand-pack methods, such as would result from successively installing tubes by the two methods in a single well, it does point out the effectiveness of the sand-pack method for dissipating large head differences between aquifers penetrated by a well.
Other advantages of using sand to install piezometer tubes It has been demonstrated that accurate head measurements and representative water samples can be obtained from wells where the sand-pack method is used to install piezometer tubes. It should also be pointed out that two other significant advantages are also realized by using this method. The first advantage is the savings of time and material. Normally, about six man-days were required to install four piezometer tubes in a well when cement was used to seal off the zones between tubes. Using sand alone, the same amount of installation work can be accomplished in two mandays. The second important advantage is the ability to restore the well to its original condition by simply jetting the sand back out of the well. The restored well can be used again for other essential test purposes thereby possibly eliminating the need for drilling a new well at the site.
Acknowledgments The author is greatly indebted to R. W. Nelson, who originally proposed the sand-pack method as a possible means of dissipating head differences throughout the deep wells at Hanford, and to A. E. Reisenauer for his invaluable assistance with the programming work. The assistance of F. B. Steele in planning and carrying out the field work is also greatly appreciated.
Reference 1) R. W. Nelson and A. E. Reisenauer, Hanford Studies on Flow in Porous Media, Proc. Second Ground Disposal of Radioactive Wastes Conference, Atomic Energy of Canada Ltd. TID-7628, (September 26-29, 1961) 130-144
P I E Z O M E T R I C H E A D D I S T R I B U T I O N IN
SAND-FILLED WELLS DONALD J. BROWN Hartford Laboratories, General Electric Company, Richland, Washington, U.S.A. *)
Received 30 September, 1963
Introduction
The ground water monitoring program at Hanford**) provides continuous assessment of the status of radioactive contaminants in the ground water resulting from controlled waste disposal operations in the Chemical Processing Areas. More than 400 cased, eight-inch wells have been drilled on the Project to obtain ground-water samples for analysis and use in making the routine evaluations. These wells are drilled and perforated from 50 to 750 feet below the regional water table. In some cases, due to head differences, the water throughout a perforated well is not representative of the ground water in the respective formations outside the well; but because of the flow pattern established in and around the well, contaminants may not enter the casing and be detected. In other instances, contaminants may enter a well from a contaminated aquifer of higher head than other aquifers tapped by the well; flow throughout the well may lead to the conclusion that all aquifers are so contaminated. Such flow patterns can, and were found to exist where wells penetrate several aquifers having different hydraulic potentials. Many of the wells drilled to water on the Project tap aquifers of sufficient head difference to create flow patterns which may result in erroneous or incomplete information on the status of contamination in the ground water. It is desirable, therefore, to modify the deep wells to permit representative sampling of the various aquifers. A requirement following modification is that the hydraulic potential distribution everywhere *) Work performed under Contract No. AT(45-1)-1350 for the U.S. Atomic Energy Commission. **) Here after indicated as Project. 195
196
D.J. BROWN
within the casing is less than or equal to that naturally present in the formation outside the well. Such a potential distribution will insure that ground waters enter and leave the well at or near the same elevation regardless of the number of aquifers of different potential tapped by the well. The method originally used for equalizing the hydraulic head within a deep well was to seal plastic piezometer tubes inside the casing at selected intervals. This method involved the setting of a number (usually four) of smaU diameter tubes in the well with 50 to 60-foot intervals between the bottoms of the tubes. The casing was filled with sand in the zones where the tubes were perforated, and the casing sections between tubes were cement-filled. Thus, access to the sealed-off aquifers was provided, and reliable samples and head measurements of the ground water were obtained, since the movement of water up or down the casing was restricted. Eight wells were modified in this manner. The time required to install piezometer tubes in each well varied from 6 to 14 days. Because of the relatively long installation time, other methods for installing tubes were studied. One method investigated simply was to fill the well with sand, thereby eliminating the time-consuming cementing phase of the installation work. It was necessary, however, to evaluate the head dissipating characteristics of the sand as compared to the cement formerly used. For this evaluation, a series of three-dimensional cases was solved with a newly-developed computer program 1) simulating both a sandfilled well and one in which cemented sections were present. The boundary conditions selected were the same as some of those known to exist at Hanford. The results of the theoretical evaluation are summarized in this paper.
Summary and conclusions Two sets of boundary conditions were selected as input to a computer program to evaluate the potential distribution in a well filled with sand as compared to that in a well containing cemented sections. The first set of boundary conditions was chosen as that most representative of the hydrological conditions encountered in the greatest number of deep wells at Hanford. The results from this case showed that the piezometric head distribution in sand-filled wells is within 0.007 foot of the true potential distribution existing in the formation outside the well. The potential distribution within the well containing the cemented section varied from the true value in the formation by 0.01 foot. The second set of conditions was chosen to duplicate some of the more extreme hydrological conditions encountered by a few of the deep wells. The results again show potential distributions in wells filled with sand alone are as close to the true potential values in the
PIEZOMETRIC HEAD DISTRIBUTION IN SAND-FILLED WELLS
197
formation as are those in the wells with the cemented sections. In this case, however, the variance from the true values was on the order of 0.1 foot for both the sand and the cement at one point in the well. Another type of calculation, using Poiseuille's and Darcy's equations for the laminar flow of water in pipes, revealed that the ratio of head loss for an open pipe to that of a pipe filled with sand is about 8 x 1 0 - 9 ; whereas, the ratio for a pipe filled with cement is 8 x 10 -13. These ratios indicate that sand alone is extremely effective in dissipating most of the head differences in a well. Some flow of water naturally takes place between the aquifers within the formation. Sand, therefore, more nearly duplicates the actual conditions present in the formation because it does not restrict the flow as completely as does cement. Accordingly, sand in most cases gives more accurate results than does cement. This study demonstrates that a bank of piezometer tubes installed in an eight-inch well casing, using sand alone to dissipate the hydraulic head between aquifers encountered by the well, will provide a reliable structure for sampling purposes and head measurements.
Computer program and boundary conditions for model A computer program was developed at Hanford to analyze one-, two-, and three-dimensional and axi-symmetrical problems with up to 8000 grid points1). With only minimum input effort, a wide range of boundary conditions needed for various problems is available. The system is completely flexible in that every grid point can be individually controlled for boundary type. Descriptive data on soil permeability can be easily designated at every spatial grid point. The above-mentioned computer program was used to evaluate the effectiveness of sand in dissipating hydraulic head in a well. A model of slightly over 7000 grid points was chosen. The permeability and the boundary conditions were assumed to be symmetrical about the well so that only half of the flow system needed to be studied. The physical size of the saturated block of soil represented by the model was seven feet (x direction) by four feet (y direction) by sixty feet (z direction). The well was located in the center of the x z plane at Yl and Y2. The potential distribution was calculated for two separate sets of boundary conditions. The first set was that which most nearly duplicated the conditions believed to be present in most of the deep wells at Hanford. In this instance a potential head difference of 3 feet in the z direction and 0.035 foot in the x direction was set at the two side boundaries, y z plane at Xmin and y z plane at xmax. A soil of uniform permeability was selected for input throughout the model.
198
D.J. BROWN
The more extreme hydrological conditions believed present in a few of the deep wells were represented by the second set of boundary conditions. In the second model the permeabilities were selected to represent a uniformly thick horizontal clay bed in the middle half of the model. The permeability in the upper and lower quarters was held the same as those in the first model. The head difference in the z and x directions were again set at the two side boundaries. The vertical head component across the clay bed was 15 feet while the horizontal component was the same as in the first model, 0.035 foot.
Experimental results Three cases were run on the IBM 7090 computer using the boundary conditions of the first model. The first case determined the potential distribution in the model with no well present. These data provided a standard for comparison of the results from the sanded well and the cemented well cases. The second case solved for the potential distribution in a well filled entirely with sand. In this instance the ratio of permeability of sand to the permeability used throughout the model was selected as 15. This ratio is typical of that which would exist in a deep well at Hanford if it were filled with sand. (Flow of ground water into a well would not occur if the ratio of permeabilities is less than 1). The third case was for a well in which the middle half of the casing was filled with cement and the upper and lower quarters filled with sand. In this case the ratio of the permeability of the cement in the well and the material in the model was 0.001. The cement acts as a dam restricting almost all flow up or down the casing. The results from the three cases were compared and the variance from the true potential pattern calculated. Variances are plotted in Fig. 1 to show
-T
Depth iO° i
J
io° I
<::".
zo'
w u. a
:~i: o
3O'
:
50'
-
S a n d - F i l l e d Well With Cemented Section Sand-Filled
Well
'i'v in
:iJ.
o N
.De
.ol6
.04
(-I
I
.
•0 ~
(Fo e t)
}
.04
I
I
.06
.Oe
60'
(+)
Fig. 1. Variance of potential in well from true potential in formation (ff w e l l q~formation).
PIEZOMETRIC HEAD DISTRIBUTION IN SAND-FILLED WELLS
199
the contrast throughout the well. The potential in both the sanded well and the partially-cemented well is less than the true potential between the 53and 60-foot depths. This would indicate flow into the well throughout this section in both cases. Between the depths of 46 feet and 53 feet, the potential in the sanded well approaches the true potential; however, the potentials in the well still remain slightly on the negative side. In the well with the cemented zone, the potential inside the casing becomes greater than the true potential which indicates outflow of water from the well. (This is expected as the water, not being able to flow through the cemented zone inside the casing, would flow out into the formation and upward around the outside of the casing.) From a depth of 18 feet down to the 46-foot depth, little or no variance is noted from the true potential. The potential in the sanded well is about 0.001 foot less than the true potential, and the potentialin the well with the cemented section is approximately 0.004 foot greater than the true values. In the latter case water, if present in that section, would tend to flow out of the cemented section and into the formation. From the 18-foot depth to the top of the model the potential in the sanded well gradually increases until it is greater than the potential in the formation. From a depth of about 13 feet, where the potential in the sanded well equals the true potential, to the top of the model, flow takes place out of the well into the formation. The maximum variance from the true potential in the sanded well thus occurs at the very top and bottom of the model; there the difference is about 0.03 foot. This is also true for the well with the cemented section. In the case of the cemented well the potential in the upper 18 feet of the model first decreases and then increases sharply. From 18 feet up to about 9 feet the water flowing up around the casing from below re-enters the well. From 9 feet to the top of the model the water again leaves the well. A study of this model with the first set of boundary conditions shows that the potential distributions measured in wells filled with sand alone are as close, and often closer, to the actual potential distributions in the formation as are those measured in wells having cemented sections. The volume of ground water moving up the well through the sanded section would be less than one gallon per day for the boundary conditions used in this model. Three cases were solved on the 7090 computer using the boundary conditions set up in the second model: the thick clay bed and higher potential head difference. In the first case the potential distribution of the model was determined with no well present. The second and third cases were solved for the sand-filled well and the cemented section, respectively. The permeability ratios used in the second and third cases in the upper and lower
200
D.J.
BROWN
sections of the model were the same as those used in the first model. In the middle clay section, the ratio of the permeability between the sand and the clay was 150. The ratio of the permeability of the cement to that of the clay was 0.01. The results of these three evaluations were compared and analyzed in a manner similar to that used in the first model. A graph comparing the variances was prepared and is exhibited as Fig. 2. Only small differences existed between the measured potentials in the formation in the topmost, middle, and the lowermost sections. The largest differences in potentials Depth 0*
;:!::
.z;"
Sand-Filled Well W~th
Cemented Section
S a n d - F i l l e d Well
Q
!o
_m
,o'
I .Oe
.OO
(-)
.04
;:i
.g2 .o'~ .do ~,
.02 (Feet)
(+l
Fig. 2. Variance of potential in well from true potential in formation (ff well-formation).
appeared at the intermediate zones where the upper and lower clay interfaces occurred. In the case of the cemented well the potential distribution was quite similar to that seen in the first model. Water enters the lower part of the well, flows up the well until it nears the cemented section, and then leaves the well. The water which flows out of the well moves through the clay section outside the well and then re-enters the well near the top of the clay interface. There it flows up the well for a short distance and then leaves the well again. The maximum variance from the true potential in the formation in this instance was 0.08-0.09 foot. The potential distribution in the sanded well was also quite similar to that observed in the first model; however, at the clay interfaces some differences exist. The potential in the well just below the lower interface is much lower than that in the formation; there the greatest inflow occurs. At the lower clay interface the potential difference decreased sharply similar to that observed in the well with the cemented section. Throughout the lower
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clay section the potential difference decreased gradually until in the middle section the variance was less than 0.001 foot. Starting several feet below the upper clay interface, there was a rapid increase in potential differences up to the clay boundary and then these differences became smaller until the variance was insignificant. One explanation for this is evident from examination of the relationship between the permeability of the clay and the permeability of the formation above the clay. Because there is no flow out of the top of the model, and because flow does occur up the well, the potential in the upper half of the well gradually increases upward. The permeability of the clay, however, is so low that water can not leave the well at a rate fast enough to dissipate the gradually rising potential. The potential, therefore, appears to increase exponentially. When the clay interface is passed the water can then flow out of the well at a much faster rate because of the higher permeability in this zone. This outflow thus relieves most of the pressure to bring it closer to the true potentials in the formation. The maximum potential difference in the sanded well was approximately 0.1 foot; however, this potential difference occurred in the clay zone near the upper interface and would have little effect on the potential measured by a piezometer tube in the upper sanded section. The average variance is probably closer to 0.01 foot. For the second model it was calculated that less than one gallon of water per day flows from the lower aquifer through the sanded well to the upper aquifer. That flow rate probably would have little effect on the monitoring results from those wells.
Head loss in an open pipe A calculation was made to determine the ratio of head loss in an unobstructed pipe to the head loss in a pipe filled with sand. Using Poiseuille's equation for head loss in open pipe (assuming laminar flow) and Darcy's equation for head loss in a pipe filled with sand, the ratio for both head loss and velocity can be obtained by equating the two expressions:
3 2 p l I/1 hLl y d 2
#1 V2 hL2 k y
(1)
V1 and hLl designate the velocity and head loss in the open pipe, and Vz, hL2 the velocity and head loss in the sand-filled pipe. The intrinsic permeability in Darcy's equation, represented by k, has units offt 2, and should not be confused with the permeability or hydrauhc conductivity which is usually represented by K and has units of volume per time flowing through a unit cross section at unit gradient. The diameter of the pipe is
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d, the viscosity/z, the specific weight y, and the length of the pipe 1. From equation (1), if we let V1 = V2, then: hLl hL2
32 k dr-, or if
(2)
hL~ = hL2 then V2
32k
(3)
The head loss ratio (or the velocity ratio) in the case of the open pipe and the pipe filled with sand is 8.4 x 10 -9. A similar ratio for an open pipe and a pipe filled with cement is 8.4 x 10 -13. The intrinsic permeability of the sand used in making this calculation was 1.2 x 10-l°ft2, and was determined from permeameter data. The intrinsic permeability (k) of the cement was calculated to be 1.2 x 10-1aft 2. A comparison of these two ratios emphasizes the effectiveness of the sand alone in dissipating head differences. Unconsolidated sediments in a formation are much more permeable than cement; hence, we may reasonably assume that the potential distribution in a sand-filled well will more closely approximate the true potential distribution in a formation.
Field experience with the sand-pack method In the southeastern part of the Hartford Project near the Columbia River, ground water temperature profiles measured in four wells showed no change in temperature with depth. The data suggested that the wells penetrated two or more aquifers of different piezometric head, and that vertical flow was occurring in the casing. Piezometer tubes were installed in two of these wells using the cement method to seal off the sections of casing between aquifers. When equilibrium was re-established, water level recordings showed head differences of 25 to 26 feet between the lowermost aquifer and the free ground water in both wells. The highest head was found to occur in the lowermost aquifer, and thus upflow did take place in the casing prior to the installation of the piezometer tubes. After completing the theoretical evaluation of potential distributions in sand-filled wells, the sand-pack method was used to install piezometer tubes in one of the remaining two wells. The location of this well was about midway between the two wells containing the cemented-in piezometers. The head difference measured between the lowermost aquifer and the free ground water in this well was
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almost 30 feet. Although this was not a direct comparison of the piezometric-head measurements obtained by the cement and sand-pack methods, such as would result from successively installing tubes by the two methods in a single well, it does point out the effectiveness of the sand-pack method for dissipating large head differences between aquifers penetrated by a well.
Other advantages of using sand to install piezometer tubes It has been demonstrated that accurate head measurements and representative water samples can be obtained from wells where the sand-pack method is used to install piezometer tubes. It should also be pointed out that two other significant advantages are also realized by using this method. The first advantage is the savings of time and material. Normally, about six man-days were required to install four piezometer tubes in a well when cement was used to seal off the zones between tubes. Using sand alone, the same amount of installation work can be accomplished in two mandays. The second important advantage is the ability to restore the well to its original condition by simply jetting the sand back out of the well. The restored well can be used again for other essential test purposes thereby possibly eliminating the need for drilling a new well at the site.
Acknowledgments The author is greatly indebted to R. W. Nelson, who originally proposed the sand-pack method as a possible means of dissipating head differences throughout the deep wells at Hanford, and to A. E. Reisenauer for his invaluable assistance with the programming work. The assistance of F. B. Steele in planning and carrying out the field work is also greatly appreciated.
Reference 1) R. W. Nelson and A. E. Reisenauer, Hanford Studies on Flow in Porous Media, Proc. Second Ground Disposal of Radioactive Wastes Conference, Atomic Energy of Canada Ltd. TID-7628, (September 26-29, 1961) 130-144