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A new method to calculate effective porosity based on specific capacity data from drillers' logs.

Introduction

Several years ago, the author was part of a regional groundwater modelling team at the USGS and it was thought that the concept of a relationship between effective porosity and other known hydraulic parameters should exist. Several researchers have established various relationships between hydraulic properties of groundwater, but have yet to establish a quick and inexpensive method to determine effective porosity from recorded measurement data found in drillers' records, such as specific capacity (pumping rate and drawdown).

A variety of methods have been developed and used for estimating effective porosity. Using tracers is common when there are at least 1 or more wells available to sample [1-8]. This method is time consuming and expensive [3], as well as highly dependent upon the groundwater gradient and hydraulic conductivity [8]. Additionally it was concluded that laboratory tracer experiments did not accurately coincide with estimated results [4]. Furthermore most tracer tests are generally used for determining hydraulic conductivity or transmissivity to use for calculating effective porosity with one of the various existing equations. Remedial workers tend to favor this method over others, and it is widely used in remedial applications. However the hydraulic conductivity and transmissivity estimates contribute further uncertainty to the estimation of effective porosity [4].

Another common method for estimating or calculating effective porosity is through extensive laboratory testing of sediment properties such as particle size, shape, packing, sorting, pore space, etc. [9,10,11,12,13,14,15,16,17,18]. A similar type of analysis uses binary mixtures in laboratory experiments. However it has been pointed out that the weak point of this method is the inability to duplicate ideal packing of large and small sediments, not to mention the infinite combinations. Research based on glass beads to simulate sediments, reassures us that the results from the laboratory testing either overestimate or underestimate effective porosity with this method [18].

This leads us to published tables of effective porosity for rocks and sediments that can be found in many textbooks and references [1], [9], [19], [20]. Ranges of effective porosity have been established through extensive analyses of hundreds of field samples that have been compiled to establish these ranges of effective porosity for virtually all types of sediments and rocks. Although these ranges are useful for illustrative purposes, choosing a value from a range can introduce considerable error in models and simulations. It requires several iterations of trial and error to arrive at a value of effective porosity to calibrate the model. However, this method does not take into account the spatial variability of effective porosity that exists due to the inhomogeneous nature of the lithology inherent to the depositional environment.

Several other methods have been utilized to determine the effective porosity, total porosity, and other hydraulic parameters. The use of ground-penetrating radar, digital optical borehole images, core analyses, and thin sections are also used to determine effective porosity and hydraulic conductivity [21]. These methods tend to be expensive, time consuming, and require the proper equipment. Laser polarized xenon nuclear magnetic resonance (NMR) methods were used to simultaneously determine permeability and effective porosity of oil reservoir rocks with reasonable accuracy [22]. This method can be very useful but it does require equipment that makes it impractical for quick surveys. Other sources of interference with determining effective porosity are tidal and atmospheric pressure [23] and biological clogging from a form of bacteria referred to as slime [24].

A relationship between hydraulic conductivity and effective porosity was established [25] and coordinated with published effective porosity ranges [9], then an algorithm was applied to adjust the values [26,27]. However this method was limited to a maximum effective porosity of 35 percent.

The focus of this paper is to show how a reliable relationship between specific capacity and effective porosity was determined, calibrated, and tested, making calculation of effective porosities relatively simple.

Summary description of the data

Figure. 1 shows the location and distribution of the wells, along the Oregon Washington border in the United States, in the Portland area. The well database of 1586 located and confirmed wells includes the construction details of the wells, including location (latitude and longitude), altitude (feet), well depth (feet), open interval (feet), well diameter (inches), and well performance data that includes test method, yield (Gpm), drawdown (feet), test period (hours), and other miscellaneous information [28]. All units were converted to metric for use in this paper.

[FIGURE 1 OMITTED]

The location is a basin with nine hydrogeologic layers, some of which have been grouped together and some that have been divided into subunits [29]. The initial hydrogeology is defined and discussed in detail [30] and includes an appendix showing the altitude of each unit as intersected by the wells. The following is a brief summary description of each unit based on these sources.

Unit US (Unconsolidated Sediments aquifer) is a combination of flood deposits and glacial outwash. It lacks cementation and commonly has been disturbed by subsequent reworking from the local river and streams. This is a generally very productive source for groundwater; however since it is the uppermost unit it is highly susceptible to contaminants.

Unit TG (Troutdale Gravel aquifer) is a sandy conglomerate with lenses of lava and soil horizons. This unit lacks cementation and is generally a very productive source of groundwater.

Unit UF (Undifferentiated Fine-Grained Sediments) is fine-grained and similar to the confining units. This unit is present where C1 and C2 are not separated by TS. It consists of clay, silt, and fine sand lenses. It is not considered to be a good source of groundwater except at the local level.

Unit C1 (Confining Unit 1) is composed of clay and silt, with local lenses of fine sand. It is not used for public water supplies, although some personal water supplies draw groundwater from this unit.

Unit TS (Troutdale Sandstone aquifer) is coarse-grained sandstone with lenses of finer-grained sands. This unit is poorly to well cemented and has primary and secondary effective porosity as the result of partial dissolution of the cementation.

Unit C2 (Confining Unit 2) is composed of clay and silt, with local lenses of fine sand. Similar to C1, it is not used for public water supplies, although some personal water supplies draw groundwater from this unit.

Unit SG (Sand and Gravel aquifer) is composed of sandy gravel with some finer-grained lenses. However, the SG unit is subdivided into an upper coarse grained unit designated as SC, and a lower fine grained unit designated as SF [27]. In this paper this unit is referred to as SG since the original data [30] didn't differentiate between the upper and lower units. This unit is generally a very productive source of old groundwater, meaning that it hasn't been subjected to anthropogenic influences.

Unit OR (Older Rocks) consists mostly of volcanic and marine sedimentary rocks. The volcanic rocks were deposited from several different episodes of volcanism, each with a different mineral profile. The marine sediments are very fine grained clay and silt. Generally this unit is not used as a source of groundwater except in outlying rural areas where it is used as a household source of water.

UF, C1, C2, SF, and OR are considered to be aquitards and generally poor sources for water supplies, while US, TG, TS, and SC are generally good aquifers and good sources for water supplies. This hydrogeologic environment represents a wide variety of conditions for effective porosity, which can vary extensively from one hydrogeologic unit to the next as well as spatially within the each hydrogeologic unit.

Methods and calculations

Laboratory Experiments

It was determined that it would be better to obtain very well sorted sediments for the laboratory experiments from a local supplier. Five sediments of specific sizes were used for the experiments. They are medium sand (MS), 0.25-0.5 mm, coarse sand (CS), 0.5-1.0 mm, fine gravel (FG), 4.0-8.0 mm, medium gravel (MG), 8.0-16.0 mm, and coarse gravel (CG) 16.0-32.0 mm. These sediments were used to determine the initial effective porosity ([[phi].sub.i]) using direct measurements. Pump tests were simulated with each sediment size. The data generated from these two steps were used to determine a preliminary relationship between effective porosity and specific capacity. It should be noted that packing and sorting affect the measurements; however, these effects were minimized during the experiments by using very well sorted sediments as well as packing the sediments.

Direct measurements

This method of measurement is based on the difference of the dry and saturated sediments divided by the total weight and expressed as a percentage. A 3 liter plastic container was weighed and calibrated, and then it was completely filled and packed with air dried sediment, and weighed. The weight of the dry sediments was determined by subtracting the weight of the container. Then the container of dry sediment was filled with water to the brim, completely saturating the sediment, and then weighed. Using the weights of the dry sediment and the water, effective porosities were calculated for each sediment size. The results are shown in Table 1.

Pump test simulations The equipment

To simulate pump tests with each sediment size, the main equipment consisted of a 200 liter tank, a 50 liter/minute pump, and a flow meter. Figure 2 shows the equipment and the layout.

[FIGURE 2 OMITTED]

Inside the tank, a 55 cm long, 4 cm wide slotted PVC tube, which represents the well, was installed in the center and connected to the external pump through the plumbing in the bottom of the tank. The pump was connected to the plumbing, a network of PVC tubing and valves to control and monitor the flow. One of the channels routed the water through the flow meter, which allowed direct observation and control of the flow up to 30 liters/min. The water was then routed back to the tank where the water was distributed around the perimeter of the tank, forming a recharge boundary. By bypassing the flow meter the pump could run at full capacity, 50 liters/min. Additionally 5 mm tubes were used at equal spacing from the center to the edge of tank to monitor water levels from the center of the tank to the perimeter of the tank. These tubes were connected to vinyl tubing attached to a board with calibration marks for the water levels.

The pump tests

The 200 liter tank was filled and packed with the selected sediment size, then filled with water to saturate the sediments. Using the flow meter to control and monitor the flow rate, several pump tests were conducted at different pumping rates for each sediment size. Each pump test continued until the water levels in the tank had stabilized, which at that scale of pump testing didn't take very long. A high-speed digital camera photographically recorded the water levels for each pump test. These pump tests generated values for Q, s, and b, as summarized in Table 2. Note that b is constant, but it is not a constant in the equations. It represents the open interval corresponding to the saturated zone of the sediments. In this experiment the equipment did not change, only the sediments and pumping rates, therefore the value of b remained the same for all tests..

Calculations

Ahuja et al. [25] developed this relationship between saturated hydraulic conductivity (cm/hour) and effective porosity. It is based on a variety of soil types, which limits the measurements and data to a relatively shallow and limited environment.

[K.sub.s] = [[phi].sub.e.sup.3.29] (1)

Razack and Huntley [31] developed this relationship between transmissivity and specific capacity.

T = 15.3[(Q/s).sup.0.67] (2)

T is transmissivity in [m.sup.2]/day,

[K.sub.s] is saturated hydraulic conductivity in cm/hour,

[[phi].sub.e] is the dimensionless effective porosity,

Q is the yield in [m.sup.3]/day,

s is the drawdown in meters, and

Q/S is specific capacity in [m.sup.2]/day.

Solving equation 1 for T in the corrected units gives us

T = 764.5b[[phi].sub.e.sup.3.29] (3)

By substituting equation 2 into equation 3, and solving for [[phi].sub.e] , we get

[[phi].sub.e][3.29th root of [(Q/s).sup.0.67]/11.99b] (4)

Equation 4 was applied to the test data and the effective porosity results were too high. These results were plotted against the original effective porosity and the resulting fit to the plot was a polynomial solution. Through several iterations of applying the fit to the original effective porosity, an equation was arrived at which fit the test data with an [R.sup.2] of 1.0.

[[phi].sub.e] = 0.1301 + [0.154 x [[3.29th root of [(Q/s).sup.0.67]/11.992b]] - (0.0165 x [[3.29th root of [(Q/s).sup.0.67]/11.992b]].sup.2] (5)

Calibration and revision of the relationship

The selection of wells was based on completeness of the data in the published database from both sources [28,30]. The criteria for valid data were location, altitude, depth, open interval, yield, and drawdown. The resultant selection of wells was cross-referenced to the hydrogeologic descriptions [30] to assign the hydrogeologic unit to the open interval of the wells. Not all of the initially selected wells matched with hydrogeology data and the final result was 610 wells that met all of the criteria.

Equation (5) was applied to specific capacity data calculated from the 610 selected wells, and the fit for this plot has an [R.sup.2] of 0.71 and generated equation (6)

[[phi].sub.e] = 0.15108 x [(Q/s).sup.0.0826] (6)

Equation (6) was applied to calculate effective porosity using calculated specific capacity and is shown in figure 3. The fit that was generated for figure 3 has an [R.sup.2] of 1.0.

[FIGURE 3 OMITTED]

Application of the relationship to calculate effective porosity

Equation 6 was used to calculate the effective porosity using the calculated specific capacity from the 610 selected wells, and the results are summarized in Table 3.

As an example, the calculated effective porosity was plotted for the TG unit and is shown in figure 4. Figure 5 shows the distribution of hydraulic conductivity in the TG unit [26,27]. The areas outlined in red denote zones that had hydraulic conductivities above the cut off point of 4.6 m/day (15 ft/day) resulting in the application of the method used by Hinkle and Snyder [26] to cap effective porosity at 31%.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

Discussion

The initial equation was based on a laboratory investigation that used 5 different sizes of sediments. The initial equation satisfied the conditions of the sediments in the laboratory and established a basis for testing with a well database. However, the hydrogeology in the study area is complex and does not reflect the very basic sediments and conditions used in the laboratory experiments. This was anticipated and the subsequent revisions were expected.

The laboratory generated initial equation was applied to the selected well data and initially produced values of effective porosity ranging from near 0% to over 400%. The solution to the fit turned out to be a polynomial; hence, the subsequent iterations were based on polynomial variations. Three iterations were needed to achieve an [R.sup.2] of 1.0.

The wells in the area yield from 27 [m.sup.3]/day up to 55,000 [m.sup.3]/day [30], indicating very a large distribution of effective porosity values in the groundwater system and within each of the hydrogeologic units.

To illustrate that point, the TG unit has been isolated and the well locations plotted in figure 4, showing effective porosity values. This unit was chosen because it is aerially extensive and has values of effective porosity ranging from 0.14 up to 0.32. These data are based on over 200 well records. It can be seen that effective porosity values vary considerably within the TG unit. The effective porosity varies spatially for each of the units in the basin and is not limited to 0.31. This type of analysis shows that there are variations in the composition of the TG aquifer, and possible stratification or facies changes within this aquifer. A similar analyses revealed that the hydraulic conductivity in 99% of the wells in the US aquifer, 41% of the wells in the TG aquifer, and 55% of the wells in the TS aquifer are above the cut-off level of 4.6 m/day, meaning that the effective porosity was assigned a value of 0.31 for those model cells.

Results and conclusions 1 unit

Even though effective porosity is highly variable, figure 3 shows that a relationship between effective porosity and specific capacity does exist and is represented in the metric system by equation 6. However, it should be pointed out that this relationship is only as good as the data used for the calculations. The well database contained wells completed in all 9 units, ranging from unconsolidated sediments to volcanic rocks. The relationship was used to calculate effective porosity values in each of these hydrogeologic units. This shows that the relationship is valid in all of these environments, and therefore valid for nearly any environment. Additionally, the equations in this paper are in SI units, but can be easily modified to use in other units of measure [31]

The implications of this relationship should be considered for future modelling, whether it is for general modelling, transport modelling, contaminant modelling, or particle tracking, etc. Effective porosity affects the flow of groundwater and therefore will affect the results of models. This relationship should have a positive impact on parameter estimation and accuracy of modelling results.

Acknowledgements

Although the well data is in the public domain, the authors would like to acknowledge the US Geological Survey for the use and interpretation of their data. Funding for this research and publication were provided by Keio University. Additionally numerous unnamed people helped with suggestions and reviews.

References

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* James M. Wilkinson and Naotatsu Shikazono

Keio University, School of Science for Open and Environmental Systems, 3-14-1 Hiyoshi, Kokuku-ku, Yokohama, Kanagawa, 223-8522, Japan

E-mail: jwilkinsonrg@gmail.com; sikazono@applc.keio.ac.jp

* Corresponding Author E-mail address: jwilkinsonrg@gmail.com.
Table 1: Measured effective porosity for each
sediment size

Lithology MS CS FG MG CG

Porosity 28.9% 22.4% 29.8% 30.7% 34.6%

Table 2: Pump test results
for each sediment size

Lithology Q b s
 l/min cm cm

CG 30.0 55.0 2.0
CG 50.0 55.0 3.0
MG 30.0 55.0 2.1
MG 50.0 55.0 3.6
FG 30.0 55.0 3.0
FG 50.0 55.0 5.0
CS 1.0 55.0 4.1
CS 2.0 55.0 7.9
CS 3.0 55.0 11.4
CS 4.0 55.0 15.3
CS 8.0 55.0 25.1
CS 10.0 55.0 30.1
MS 0.5 55.0 8.8
MS 1.0 55.0 12.2
MS 1.5 55.0 18.7
MS 2.0 55.0 28.0
MS 2.5 55.0 36.8

Table 3: Average values of effective porosity
for each hydrologic unit

Hydrogeologic Average Maximum Minimum
unit Effective Effective Effective
 Porosity Porosity Porosity

US 0.29 0.39 0.17
TG 0.22 0.32 0.14
C1 0.20 0.21 0.19
TS 0.22 0.30 0.18
C2 0.21 0.24 0.17
UF 0.21 0.23 0.17
SG 0.23 0.27 0.19
OR 0.19 0.27 0.11
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Author:Wilkinson, James M.; Shikazono, Naotatsu
Publication:International Journal of Applied Environmental Sciences
Date:Jul 1, 2012
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