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Int. J. Soc. Mater. Eng. Resour. Vol.17, No.1, (Mar. 2010)

1.　Introduction

The permeability of rock is controlled by fissures [1], and is conventionally measured in situ [2] [3]. However, when rocks are porous, production flow rate behavior is controlled by the rock matrix, not by the fissures [4]. Therefore, evaluation of the permeability of the rock matrix is important for predicting the long-term behavior of groundwater production flow rate.

The constant head method and the falling head method are standard methods for measuring the permeability of soils and porous media, but are unsuitable for rocks. Consequently, many methods for measuring the permeability of rock matrix have been developed, relying on special techniques. The constant pressure method, based on the constant head method, is not so used for low-permeability rock because it needs a high-precision flow meter. The flow pump and transient pulse methods [5] [6], based on the falling head method, are not widely used, because the apparatus used in these methods are expensive and measurement techniques are complicated. Therefore, information on the permeability of rock matrices is limited, so it is difficult to predict groundwater flow. To solve this problem, we developed a new method that uses a cheaper apparatus and a simpler measuring technique.

Here we propose a method based on the falling head method for measuring rock matrix permeability, using a pressure accumulator as a pressure source instead of a standpipe.

2.　Measurement theory

2.1　Outline of measurement methodFigure 1 illustrates our measuring method. Hydraulic pressure

P (upstream pressure) in a pressure accumulator is applied to the surface of a rock specimen placed in a pressure vessel, and water flows from the surface to a hole drilled through the center of the specimen. The water drains out of the pressure vessel at

downstream pressure Pc, controlled by the backpressure-regulating valve installed in the drain line. The flow rate in the rock specimen is governed by the pressure difference between upstream and downstream. This method takes advantage of radial flow in cylindrical specimen and measuring time is relatively short.

The pressure P in the accumulator decreases when the water drains from the vessel. If the downstream pressure is assumed to be constant, the change in upstream pressure P under the condition of Boyle's law is equivalent to that in the head of a standpipe. Therefore, the permeability is calculated from the change in the upstream pressure over time.

2.2　Model of pressure change in accumulatorThis section describes the relation between pressure in the

accumulator and the rate of outflow from the accumulator.Figure 2 shows the change in pressure P and volume V of gas

in the accumulator with elapsed time t. The gas in the accumulator

New Permeability Test Based on Falling Head Method in Rock Core

Tadao IMAI1, Naoto KAMOSHIDA2, Shinji YAMAGUCHI1 and Fumio SUGIMOTO1

1Department of Earth Sciences and Technology, Akita University, 1-1 Tegata-gakuen, Akita 010-8502, Japan2Department of Civil and Environmental Engineering, Iwate University, 4-3-5 Ueda, Morioka 020-8551, Japan

The permeability of rock matrices is not routinely measured because the equipment is expensive and the measurements are time-consuming. We designed a new laboratory permeability test using low-cost apparatus and investigated its performance. The method is a variation of the falling head method, in which the standpipe used as the pressure source is replaced with a pressure accumulator. High pressure is applied to a rock specimen, making it possible to measure permeability within an hour. The apparatus produces radial flow from the surface to a central hole drilled in a cylindrical specimen. We determined that the apparatus measures hydraulic conductivity in the range of 3.5×10-10 to 2.6×10-4 cm/s. The error in hydraulic conductivity obtained from repeated measurements is ± ~10%, and the difference in conductivity between this method and the standpipe method is ~25%. For a rock with a hydraulic conductivity of 3.0×10-9 cm/s, measurement takes at least 800 s.

Key Words : Rock, Hydraulic Conductivity, Permeability Test, Falling Head Method

OriginalReceived October 7, 2009Accepted for Publication February 25, 2010©2010 Soc. Mater. Eng. Resour. Japan

Figure 1　Schema of new measurement method.

1

Imai, T., Kamoshida, N., Yamaguchi, S. and Sugimoto, F.

Figure 1 Schema of new measurement method Figure 2 Change in volume and pressure in accumulator.

Figure 3 Geometry of rock specimen Figure 4 Radial flow in rock specimen

79New Permeability Test Based on Falling Head Method in Rock Core

Int. J. Soc. Mater. Eng. Resour. Vol.17, No.1, (Mar. 2010)

is sealed by rubber film. Figure 2a shows the accumulator filled initially with nitrogen gas; the gas pressure is P0 and the gas volume is V0. When water is injected into the accumulator, the water and gas pressures increase to P1 and the gas volume decreases to V1. After the permeability test starts, pressure and volume change with elapsed time t. Figures 2b and c show the changes in pressure and volume of gas in the accumulator at times t1 and t2.

The volume V' of outflow from the accumulator from t1 to t2 is expressed as:

V'=V2－V1 (1)

If the change of state of the gas in the accumulator is isothermal, the relation between the pressure and volume of the gas is expressed as follows (based on Boyle's law):

P0V0 =P1V1=P2V2 (2)

Therefore the gas volumes V1 and V2 are expressed as:

P0V0V1 =―― (3) P1

P0V0V2=―― (4) P2

We assume that the differences in time (t2－t1) and pressure (P1

－P2) are infinitesimal.

dt=(t2－t1)dP=(P2－P1)

Since P1 and P2 are approximately P, the change in gas volume dV' for infinitesimal time dt is expressed as:

dV'=(V2－V1)

P1－P2= ――― P0・V0P1・P2

1=－P0・V0―dp (5)

P2

Equation (5) expresses the relation between the change in gas volume dV' and the change in pressure in the accumulator when the pressure decreases slowly. The change dV' for the infinitesimal time dt is equivalent to the rate at which the water flows out of the accumulator. We confirmed that the change of state of gas in the accumulator is approximately isothermal when the gas flow rate is <5×10-3×V0/s.

2.3　Radial flow in rock specimenFigure 3 shows the shape of the rock specimens used. The

cylinder has radius rs0 and length L, and a hole of radius rsi and depth b is drilled at its center. Hydraulic pressure is applied to the outside surface of the specimen, and water flows from there to the central hole. Radial flow is most permeable in a cylindrical specimen (Figure 4). Given the r-axis in the direction of the radius, the regular pressure diffusion is expressed in cylindrical coordinates as:

d2Ps　 1　dPs―― +― ―― = 0 (6) dr2　 r　dr

where Ps is pressure.The general solution of differential equation (6) is expressed as:

dPsr――=C1 (7) dr

Ps =C1 ln r+C2 (8)

where C1 and C2 are integration constants.The boundary conditions in equation (6) are expressed as:

r= rs0 then Ps =Pr= rsi then Ps =Pc

From these boundary conditions, C1 is defined as:(P－Pc)C1 =―――― (9)ln(rs0 /rsi)

Figure 2　Change in volume and pressure in accumulator.

1

Imai, T., Kamoshida, N., Yamaguchi, S. and Sugimoto, F.

Figure 1 Schema of new measurement method Figure 2 Change in volume and pressure in accumulator.

Figure 3 Geometry of rock specimen Figure 4 Radial flow in rock specimen

Figure 3　Geometry of rock specimen.

1

Imai, T., Kamoshida, N., Yamaguchi, S. and Sugimoto, F.

Figure 1 Schema of new measurement method Figure 2 Change in volume and pressure in accumulator.

Figure 3 Geometry of rock specimen Figure 4 Radial flow in rock specimen

Figure 4　Radial flow in rock specimen.

1

Imai, T., Kamoshida, N., Yamaguchi, S. and Sugimoto, F.

Figure 1 Schema of new measurement method Figure 2 Change in volume and pressure in accumulator.

Figure 3 Geometry of rock specimen Figure 4 Radial flow in rock specimen

80 Tadao IMAI et al.

Int. J. Soc. Mater. Eng. Resour. Vol.17, No.1, (Mar. 2010)

From equations (9) and (7), the pressure gradient at an arbitrary point is expressed as:

dPs (P－Pc)――=――――― (10)dr r ln(rs0 /rsi)

The flow velocity v is expressed by Darcy's law as:

k dPsv=－――・―― (11)ρg dr

where k is hydraulic conductivity, ρ is water density, and g is gravitational acceleration.

The flow rate Q in the central hole is expressed as:

Q=2π rsi b|v|r=rsi (12)

where |v|r = rsi is the flow velocity on the surface area of the central hole with r= rsi.

Substituting equations (11) and (10) into equation (12) when r= rsi gives Q:

2πbkQ=――――― (P－Pc) (13)ρg ln(rs0 /rsi)

Because Q in equation (13) is the same as the change in volume of the gas dV' for infinitesimal time dt, Q is defined as:

dV'――=Q (14)dt

Equation (13) is rewritten as:

2πbkdV'=―――――― (P－Pc)dt (15)ρg ln(rs0 /rsi)

2. 4　Calculation of hydraulic conductivityEquation (5) is equivalent to equation (15), so the following

equation is derived:

1 2πbk－P0･V0――dp=―――――― (P－Pc)dt (16)P0

2 ρg ln(rs0 /rsi)

Equation (16) is rearranged for dt as:

ρg ln(rs0 /rsi) 1dt=－P0V0―――――― ―――――dP (17)

2πbk P2 (P－Pc)

If the pressure in the accumulator changes from P1 at time t1 to P2 at t2, both sides of equation (17) are integrated as follows:

ρg ln(rs0 /rsi) 　 1∫t1

t2

dt=－P0V0―――――― ∫P1

P2

―――――dP (18) 2πbk P2 (P－Pc)

Equation (18) is rearranged for hydraulic conductivity k (m/s) as:

P0V0 ρg ln(rs0 /rsi) (α2－α1)k=―――――――― ―――― (19)2πb (t2－t1)

where α1 and α2 are expressed as:

1　　1　　 P1α1 =―――－――ln ―――PcP1　Pc

2 P1－Pc 1　　1　　 P2α2 =―――－――ln ―――

PcP2　Pc2 P2－Pc

When Pc approximates atmospheric pressure (0.1013 MPa), and P1 and P2 are large enough compared with Pc, equation (19) is approximated as:

P0V0 ρg ln(rs0 /rsi) 1 1k=―――――――― ――－―― (20)

4πb (t2－t1) P22 P1

2

but where hydraulic conductivity k is expressed in cm/s (not m/s).Hydraulic conductivity k is converted into intrinsic permeability

K (m2) by:

μK=k ―― (21)ρg

where µ is the viscosity coefficient of water.Hydraulic conductivity k is calculated by using equation (19) or

(20) from the measurement of upstream pressure with elapsed time.

3.　Measurement system and technique

3.1　Measurement systemFigure 5 illustrates the apparatus. The pressurizing part (A) is

composed of a pressure vessel with a drain line, accumulators (a, high-pressure; b, low-pressure; Table 1), a backpressure-regulating valve to control the downstream pressure, and a hydraulic hand pump to supply the pressure in the accumulators. The accumulator used depends on the permeability of the rock. The direction of flow

Figure 5　Schematic diagram of experimental apparatus.

2

Imai, T., Kamoshida, N., Yamaguchi, S. and Sugimoto, F.

Figure 5 Schematic diagram of experimental apparatus

81New Permeability Test Based on Falling Head Method in Rock Core

Int. J. Soc. Mater. Eng. Resour. Vol.17, No.1, (Mar. 2010)

in the pipe is controlled with three needle valves a, b, and c.The loading part (B) is composed of the loading hand pump and

the loading frame, and an axial load is applied to the rock specimen in the pressure vessel.

The measurement part (C) is composed of pressure transducers for upstream (a) and downstream pressure (b), a dynamic strain meter, an analog-to-digital converter, and a personal computer for data logging. The quantity of water from the specimen is measured in a graduated cylinder. When the backpressure-regulating valve is released, the downstream atmospheric pressure is measured with a barometer.

3.2　Structure of pressure vessel and water flow linePhoto 1 shows the pressure vessel, which has three joints. The

upper joint is connected with the backpressure-regulating valve, the middle joint is the ventilation duct, and the lower joint is connected with the accumulator.

Figure 6 illustrates the water flow in the pressure vessel. Pressurized water is injected via the lower joint of the vessel, and upstream pressure is applied to the side surface of the specimen. Water permeates from the surface to the central hole of the specimen, flows into the hole of the piston, and is drained from the upper joint via the piston. The specimen is held between the two end pieces with silicone resin, and the end pieces are fastened to a piston for axial loading.

3.3　Measurement procedureBefore the permeability test starts, the rock specimen in the

pressure vessel is saturated with water. Then the vessel is installed in the loading frame, and a prescribed axial load is applied to the specimen with hand pump (Figure 5). The upper and lower joints of the vessel are connected to the water flow line with high-pressure hoses, and a prescribed pressure is applied to the accumulator by the hydraulic pump while only the backpressure-regulating valve is shut. Next, valves a, b, and c in Figure 5 are shut, and the downstream pressure is set to a prescribed value with the backpressure-regulating valve.

The measurement starts when valve a is opened. The personal computer records elapsed time, upstream pressure, and downstream pressure. The upstream pressure is set within the range of 0.3-5 MPa depending on the permeability of rock specimen so that the pressure does not decrease rapidly. The measurement time is about 30 min until the upstream pressure decreases appreciably. In this study, we assume that the defamation of the apparatus by hydraulic pressure is neglected.

3.4　Rock specimens and dimensionsWe tested six rock specimens: two types of tuff (Towada and

Ogino), three types of sandstone (Kimachi, Tako, and Berea), and an andesite (Oga) (Photo 2). Ogino tuff had the lowest density and the largest porosity, and Oga andesite had the highest density and the smallest porosity (Table 2). The other specimens had nearly

Figure 6　Schematic diagram of vessel.

3

Imai, T., Kamoshida, N., Yamaguchi, S. and Sugimoto, F.

Photo. 1 Pressure vessel Figure 6 Schematic diagram of vessel

Photo. 2 Rock specimens used in the experiments

Photo 1　Pressure vessel.

3

Imai, T., Kamoshida, N., Yamaguchi, S. and Sugimoto, F.

Photo. 1 Pressure vessel Figure 6 Schematic diagram of vessel

Photo. 2 Rock specimens used in the experiments

Table 1　Specifications of accumulators.

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Imai, T., Kamoshida, N., Yamaguchi, S. and Sugimoto, F.

Figure 5 Schematic diagram of experimental apparatus

Pressure Type P0 (MPa) V0 (cm3)

High 1.200 1000

Low 0.103 1000

Table 1 Specifications of accumlators

Photo 2　Rock samples used in the experiments.

3

Imai, T., Kamoshida, N., Yamaguchi, S. and Sugimoto, F.

Photo. 1 Pressure vessel Figure 6 Schematic diagram of vessel

Photo. 2 Rock specimens used in the experiments

Table 2　Physical properties of rock specimens.

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Imai, T., Kamoshida, N., Yamaguchi, S. and Sugimoto, F.

Photo. 2 Rock specimens used in the experiments

Table 2 Physical properties of rock specimens

Table 3 Size of specimen

Rock Specimen Dry Density (g/cm3) Porosity (％)

Towada Tuff 2.04 19.4

Ogino Tuff 1.52 27.1

Kimachi Sandstone 2.03 19.6

Tako Sandstone 2.08 16.2

Berea Sandstone 2.08 16.6

Oga Andesite 2.23 11.1

Length Diameter (r s0) Diameter (r si) Depth (b)

50-80 39.0 3.0 40-60

Unit : mm

Configuration Center Hole

82 Tadao IMAI et al.

Int. J. Soc. Mater. Eng. Resour. Vol.17, No.1, (Mar. 2010)

equal density and porosity; all specimens had a porosity of more than 10% and relativity high permeability. Table 3 shows their dimensions. In the table, the length of the center hole is somewhat shorter than that of the specimen by drilling conditions of the center hole. But the length of the center hole hardly influences the hydraulic conductivity.

4　Results

4.1　Accuracy of the permeability testing apparatusWe investigated the lower and upper limits of hydraulic

conductivity that the apparatus could measure (Figure 7). The lower limit is restricted by water leakage from the apparatus, and the upper limit is imposed by flow resistance in the apparatus. The equivalent hydraulic conductivity of the apparatus was calculated based on the standard size of specimen in this test.

Figure 7a shows the relationship between upstream pressure and elapsed time when the specimen is not set up and all valves are shut, and the high-pressure accumulator was used. After 1800 s (30 min), the upstream pressure decreased by 0.006 MPa from the initial 4.507 MPa owing to water leakage. The apparent hydraulic conductivity by water leakage was almost constant. Therefore, the lower limit of hydraulic conductivity measured with the apparatus is estimated to be 3.5×10-10 cm/s (mean value).

Figure 7b shows the relationship between upstream pressure and elapsed time when the specimen is not set up and all valves are opened, and the low-pressure accumulator was used. For the first 70 s, the upstream pressure decreased gradually from the initial 0.308 to 0.167 MPa. Then it decreased rapidly to atmospheric pressure after 70 s. The rapid decrease in pressure is caused by the elastomeric force of the rubber film in the accumulator. Therefore we used the data from the first 70 s to calculate the apparent

hydraulic conductivity by flow resistance in the apparatus, which was almost constant. Therefore the upper limit of hydraulic conductivity measured with the apparatus is estimated to be 2.6×10-4 cm/s (mean value).

4.2　Reliability of the apparatus(1) Stability of measurements of hydraulic conductivity

Figure 8 shows the results of the permeability tests of Kimachi and Berea sandstones. For Kimachi sandstone (Figure 8a), the high-pressure accumulator was used, and the downstream pressure was set to atmospheric pressure. The test ran for 4800 s. The hydraulic conductivity calculated at intervals of 800 s was almost constant during the test, with a mean value of 3.0×10-9 cm/s. For the Berea sandstone (Figure 8b), the low-pressure accumulator was used, and the downstream pressure was set to atmospheric pressure. Only the data for the first 1000 s was suitable. The hydraulic conductivity calculated at intervals of 100 s was initially high but became almost constant, with a mean value of 2.23×10-5 cm/s. These results show that the hydraulic conductivity remained stable during measurement.(2) Reliability and reproducibility of measurements

The flow volume through the above specimen of Kimachi sandstone was measured. Table 4 shows the flow rate per hour, the change in upstream pressure (from P1 to P2) over an hour, and the hydraulic conductivity k calculated from P1 and P2.

The flow rate Q (m3/s) from the accumulator is calculated (based on Boyle's law) as:

P1－P2Q=V2－V1 =P0V0 ――― (22)P1P2

The average flow rate (m3/s) over the measurement time is estimated from the hydraulic conductivity by equation (13) using the mean of P1 and P2 as the upstream pressure and a hole length b of 60 mm. Figure 9 shows the measured flow rate, and the flow rates calculated from the change in upstream pressure and hydraulic conductivity. All three values are approximately 7 cm3/h, with an error of within 5%.

The above specimen of Kimachi sandstone was tested three times (Figure 10). The reproducibility of the method was high, and

Figure 7　Relationship between upstream pressure and elapsed time in test of apparatus leakage and resistance: (a) valve a, b and c shut; (b) valve a, b and c open.

! 6 !

Imai, T., Kamoshida, N., Yamaguchi, S. and Sugimoto, F.

Figure 7 Relationship between upstream pressure and elapsed time in test of apparatus leakage and

resistance: (a) valve a, b and c shut; (b) valve a, b and c open

Figure 8 Relationship between upstream pressure and elapsed in test

of rock specimens: (a) Kimachi sandstone; (b) Berea sandstone

(a) valve a, b and c shut (b) valve a, b and c open

(b) Berea sandstone(a) Kimachi sandstone

4.40

4.45

4.50

4.55

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

0 500 1000 1500 2000

Upst

ream

Pre

ssure

(M

Pa)

Hydra

uli

c C

onduct

ivit

y (c

m/s

)

Elapsed Time (sec.)

Hydraulic Conductivity (Lower limit)

Upstream Pressure

Without Specimen

0.10

0.15

0.20

0.25

0.30

0.35

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

0 500 1000 1500 2000 2500

Upst

ream

Pre

ssure

(M

Pa)

Hydra

uli

c C

onduct

ivit

y (c

m/s

)

Elapsed Time (sec.)

Berea Sanstone

Hydraulic Conductivity

Upstream Pressure

Calculating Range

Rubber Effect

Downstream Pressure : 0.101MPa

0.10

0.15

0.20

0.25

0.30

0.35

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

0 20 40 60 80 100 120

Upst

ream

Pre

ssure

(M

Pa)

Hydra

uli

c C

onduct

ivit

y (c

m/s

)

Elapsed Time (sec.)

Hydraulic Conductivity (Upper limit)

Upstream Pressure

Without Specimen

Calculating Range

Rubber Effect

4.0

4.5

5.0

5.5

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

0 1000 2000 3000 4000 5000

Upst

ream

Pre

ssure

(M

Pa)

Hydra

uli

c C

onduct

ivit

y (c

m/s

)

Elapsed Time (sec.)

Kimachi Sanstone

Hydraulic Conductivity

Upstream Pressure

Downstream Pressure : 0.101MPa

Table 3　Sizes of specimens.

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Imai, T., Kamoshida, N., Yamaguchi, S. and Sugimoto, F.

Photo. 2 Rock specimens used in the experiments

Table 2 Physical properties of rock specimens

Table 3 Size of specimen

Rock Specimen Dry Density (g/cm3) Porosity (％)

Towada Tuff 2.04 19.4

Ogino Tuff 1.52 27.1

Kimachi Sandstone 2.03 19.6

Tako Sandstone 2.08 16.2

Berea Sandstone 2.08 16.6

Oga Andesite 2.23 11.1

Length Diameter (r s0) Diameter (r si) Depth (b)

50-80 39.0 3.0 40-60

Unit : mm

Configuration Center Hole

83New Permeability Test Based on Falling Head Method in Rock Core

Int. J. Soc. Mater. Eng. Resour. Vol.17, No.1, (Mar. 2010)

the measurement error was ±~10% of the mean.(3) Comparison of measurement methods

Figure 11a shows the hydraulic conductivity of Kimachi sandstone as measured by our method and by the transient pulse method by Sugimoto et al. [7] and Xue et al. [8]. Our result agrees

well with that of Sugimoto et al., but is approximately twice that of Xue et al. We know that the reliability of our method is high, because the measurement values of our study and the other studies are the same order of magnitude though the different rock samples and measurements.

Figure 11b shows the hydraulic conductivity of Berea sandstone was measured by our new method and the standpipe method, in which the accumulator was replaced with a standpipe. The hydraulic conductivity in the standpipe method is approximately 25% less. Therefore the new method has the same degree of reliability as the standpipe method.

Table 4　Pressure change and flow rate in Kimachi sandstone.

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Imai, T., Kamoshida, N., Yamaguchi, S. and Sugimoto, F.

(b) Berea sandstone

Figure 9 Comparison of measured and calculated

flow rates.

Figure 10 Effect of repeated measurement on

hydraulic conductivity.

(a) Kimachi sandstone

Figure 11 Comparison of hydraulic conductivity of rocks among

methods: (a) Kimachi sandstone; (b) Berea sandstone

P 1 (MPa) P 2 (MPa) Q (cm3/h) k (cm/s)

4.94 4.80 7.0 2.7E-09

Table 4 Pressure change and flow rate in Kimachi

sandstonet.

6.5

7.0

7.5

Flo

w R

ate

(cm

3/h

)

Mesurement Conversionfrom Pressure

Conversionfrom Hydraulic Conductivity

Kimachi Sandstone

10-5

10-4

OurMethodH

ydra

uli

c C

onduct

ivit

y (c

m/s

)

StandpipeMethod

Berea Sanstone

10-9

10-8

Hydra

uli

c C

onduct

ivit

y (c

m/s

)

Repeat Times

1st 2nd 3rd

Kimachi Sandstone

10-9

10-8

Hydra

uli

c C

onduct

ivit

y (c

m/s

)

This Study Sugimoto et. al.[7]

Xue et. al.[8]

Kimachi Sandstone

Figure11　Comparison of hydraulic conductivity of rocks among methods: (a) Kimachi sandstone; (b) Berea sandstone.

! 7 !

Imai, T., Kamoshida, N., Yamaguchi, S. and Sugimoto, F.

(b) Berea sandstone

Figure 9 Comparison of measured and calculated

flow rates.

Figure 10 Effect of repeated measurement on

hydraulic conductivity.

(a) Kimachi sandstone

Figure 11 Comparison of hydraulic conductivity of rocks among

methods: (a) Kimachi sandstone; (b) Berea sandstone

P 1 (MPa) P 2 (MPa) Q (cm3/h) k (cm/s)

4.94 4.80 7.0 2.7E-09

Table 4 Pressure change and flow rate in Kimachi

sandstonet.

6.5

7.0

7.5

Flo

w R

ate

(cm

3/h

)

Mesurement Conversionfrom Pressure

Conversionfrom Hydraulic Conductivity

Kimachi Sandstone

10-5

10-4

OurMethodH

ydra

uli

c C

onduct

ivit

y (c

m/s

)

StandpipeMethod

Berea Sanstone

10-9

10-8

Hydra

uli

c C

onduct

ivit

y (c

m/s

)

Repeat Times

1st 2nd 3rd

Kimachi Sandstone

10-9

10-8

Hydra

uli

c C

onduct

ivit

y (c

m/s

)

This Study Sugimoto et. al.[7]

Xue et. al.[8]

Kimachi Sandstone

Figure10　Effect of repeated measurement on hydraulic conductivity.

! 7 !

Imai, T., Kamoshida, N., Yamaguchi, S. and Sugimoto, F.

(b) Berea sandstone

Figure 9 Comparison of measured and calculated

flow rates.

Figure 10 Effect of repeated measurement on

hydraulic conductivity.

(a) Kimachi sandstone

Figure 11 Comparison of hydraulic conductivity of rocks among

methods: (a) Kimachi sandstone; (b) Berea sandstone

P 1 (MPa) P 2 (MPa) Q (cm3/h) k (cm/s)

4.94 4.80 7.0 2.7E-09

Table 4 Pressure change and flow rate in Kimachi

sandstonet.

6.5

7.0

7.5

Flo

w R

ate

(cm

3/h

)

Mesurement Conversionfrom Pressure

Conversionfrom Hydraulic Conductivity

Kimachi Sandstone

10-5

10-4

OurMethodH

ydra

uli

c C

onduct

ivit

y (c

m/s

)

StandpipeMethod

Berea Sanstone

10-9

10-8

Hydra

uli

c C

onduct

ivit

y (c

m/s

)

Repeat Times

1st 2nd 3rd

Kimachi Sandstone

10-9

10-8

Hydra

uli

c C

onduct

ivit

y (c

m/s

)

This Study Sugimoto et. al.[7]

Xue et. al.[8]

Kimachi Sandstone

Figure 9　Comparison of measured and calculated flow rates.

Figure 8　Relationship between upstream pressure and elapsed in test of rock specimens: (a) Kimachi sandstone; (b) Berea sandstone.

! 6 !

Imai, T., Kamoshida, N., Yamaguchi, S. and Sugimoto, F.

Figure 7 Relationship between upstream pressure and elapsed time in test of apparatus leakage and

resistance: (a) valve a, b and c shut; (b) valve a, b and c open

Figure 8 Relationship between upstream pressure and elapsed in test

of rock specimens: (a) Kimachi sandstone; (b) Berea sandstone

(a) valve a, b and c shut (b) valve a, b and c open

(b) Berea sandstone(a) Kimachi sandstone

4.40

4.45

4.50

4.55

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

0 500 1000 1500 2000U

pst

ream

Pre

ssure

(M

Pa)

Hydra

uli

c C

onduct

ivit

y (c

m/s

)

Elapsed Time (sec.)

Hydraulic Conductivity (Lower limit)

Upstream Pressure

Without Specimen

0.10

0.15

0.20

0.25

0.30

0.35

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

0 500 1000 1500 2000 2500

Upst

ream

Pre

ssure

(M

Pa)

Hydra

uli

c C

onduct

ivit

y (c

m/s

)

Elapsed Time (sec.)

Berea Sanstone

Hydraulic Conductivity

Upstream Pressure

Calculating Range

Rubber Effect

Downstream Pressure : 0.101MPa

0.10

0.15

0.20

0.25

0.30

0.35

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

0 20 40 60 80 100 120

Upst

ream

Pre

ssure

(M

Pa)

Hydra

uli

c C

onduct

ivit

y (c

m/s

)

Elapsed Time (sec.)

Hydraulic Conductivity (Upper limit)

Upstream Pressure

Without Specimen

Calculating Range

Rubber Effect

4.0

4.5

5.0

5.5

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

0 1000 2000 3000 4000 5000

Upst

ream

Pre

ssure

(M

Pa)

Hydra

uli

c C

onduct

ivit

y (c

m/s

)

Elapsed Time (sec.)

Kimachi Sanstone

Hydraulic Conductivity

Upstream Pressure

Downstream Pressure : 0.101MPa

! 7 !

Imai, T., Kamoshida, N., Yamaguchi, S. and Sugimoto, F.

(b) Berea sandstone

Figure 9 Comparison of measured and calculated

flow rates.

Figure 10 Effect of repeated measurement on

hydraulic conductivity.

(a) Kimachi sandstone

Figure 11 Comparison of hydraulic conductivity of rocks among

methods: (a) Kimachi sandstone; (b) Berea sandstone

P 1 (MPa) P 2 (MPa) Q (cm3/h) k (cm/s)

4.94 4.80 7.0 2.7E-09

Table 4 Pressure change and flow rate in Kimachi

sandstonet.

6.5

7.0

7.5

Flo

w R

ate

(cm

3/h

)

Mesurement Conversionfrom Pressure

Conversionfrom Hydraulic Conductivity

Kimachi Sandstone

10-5

10-4

OurMethodH

ydra

uli

c C

onduct

ivit

y (c

m/s

)

StandpipeMethod

Berea Sanstone

10-9

10-8

Hydra

uli

c C

onduct

ivit

y (c

m/s

)

Repeat Times

1st 2nd 3rd

Kimachi Sandstone

10-9

10-8

Hydra

uli

c C

onduct

ivit

y (c

m/s

)

This Study Sugimoto et. al.[7]

Xue et. al.[8]

Kimachi Sandstone

84 Tadao IMAI et al.

Int. J. Soc. Mater. Eng. Resour. Vol.17, No.1, (Mar. 2010)

4.3　Effect of upstream pressure, downstream pressure, and axial stress

We investigated the effect of initial upstream and downstream pressure on the hydraulic conductivities of Ogino tuff, Kimachi sandstone, and Tako sandstone, using the high-pressure accumulator (Figure 12). When the initial upstream pressure ranged from 2 to 9 MPa and the downstream pressure was set to atmospheric pressure, the hydraulic conductivities in the three rocks were almost constant (Figure 12). When the downstream pressure was set to 2.19 MPa, the hydraulic conductivities were again almost constant in the range of upstream pressure from 2 to 9 MPa. But, in Figure 12, we cannot clarify the effect of the difference between upstream and downstream pressures on hydraulic conductivity.

Next, we investigated the effect of axial stress on the hydraulic conductivities of Ogino tuff, Kimachi sandstone, and Tako sandstone, using the high-pressure accumulator, initial upstream pressures of 2 MPa for Tako sandstone and 5 MPa for Kimachi sandstone and Ogino tuff, axial stress of 2-9 MPa (half of compressive strength of rock), and downstream pressure set to atmospheric pressure. The hydraulic conductivities of all three

rocks were almost constant irrespective of axial stress (Figure 13). Thus, the hydraulic conductivity of porous rock is not substantially influenced by axial stress if the axial stress is less than half the compressive strength of the rock.

5　Discussion

5.1　Validity of measurement resultsThe hydraulic conductivities of all six rocks samples ranged

between the order of 10-4 and 10-9 cm/s, within the upper and lower

Figure12　Effect of initial upstream and downstream pressure on hydraulic conductivity.

! 8 !

Imai, T., Kamoshida, N., Yamaguchi, S. and Sugimoto, F.

Figure 13 Effect of axial stress on hydraulic conductivity

Figure 12 Effect of initial upstream pressure and downstream pressure on hydraulic

conductivity.

10-10

10-9

10-8

10-7

10-6

0 2 4 6 8 10

Hydra

uli

c C

onduct

ivit

y (

cm/s

)

Initial Upstream Pressure (MPa)

Tako Sandstone

Kimachi Sandstone

Ogino Tuff

Downstream Pressure0.101MPa: 2.19 MPa:

10-10

10-9

10-8

10-7

10-6

0 2 4 6 8 10

Hydra

uli

c C

onduct

ivit

y (

cm/s

)

Axial Stress (MPa)

Tako Sandstone

Kimachi Sandstone

Ogino Tuff

Upstream Pressure : 2MPaDownstream Pressure : 0.101MPa

Upstream Pressure : 5MPaDownstream Pressure : 0.101MPa

Figure 13　Effect of axial stress on hydraulic conductivity.

! 8 !

Imai, T., Kamoshida, N., Yamaguchi, S. and Sugimoto, F.

Figure 13 Effect of axial stress on hydraulic conductivity

Figure 12 Effect of initial upstream pressure and downstream pressure on hydraulic

conductivity.

10-10

10-9

10-8

10-7

10-6

0 2 4 6 8 10

Hydra

uli

c C

onduct

ivit

y (

cm/s

)

Initial Upstream Pressure (MPa)

Tako Sandstone

Kimachi Sandstone

Ogino Tuff

Downstream Pressure0.101MPa: 2.19 MPa:

10-10

10-9

10-8

10-7

10-6

0 2 4 6 8 10

Hydra

uli

c C

onduct

ivit

y (

cm/s

)

Axial Stress (MPa)

Tako Sandstone

Kimachi Sandstone

Ogino Tuff

Upstream Pressure : 2MPaDownstream Pressure : 0.101MPa

Upstream Pressure : 5MPaDownstream Pressure : 0.101MPa

Table 5　Hydraulic conductivities of rocks (mean value).

! 9 !

Imai, T., Kamoshida, N., Yamaguchi, S. and Sugimoto, F.

Figure 14 Relationship between hydraulic conductivity and pore properties of rocks:

(a) porosity; (b) mean pore radius [9]

(a) porosity (b) mean radius of pore [9]

Table 5 Hydraulic conductivity of rocks (mean value)

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10 15 20 25 30

Towada Tuff

Ogino Tuff

Kimachi Sandstone

Tako Sandstone

Berea Sandstone

Oga Andesite

Hydra

uli

c C

onduct

ivit

y (

cm/s

)

Porosity (%)

Lower Limit

Upper Limit

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

0.01 0.1 1 10 100

Ogino Tuff

Kimachi Sandstone

Tako Sandstone

Berea Sandstone

Hydra

uli

c C

onduct

ivit

y (

cm/s

)

Mean pore radius ( m)

Lower Limit

Upper Limit

Rock Specimen Hydraulic Conductivity (cm/s)

Towada Tuff 1.52.E-08

Ogino Tuff 1.62.E-09

Kimachi Sandstone 3.00.E-09

Tako Sandstone 1.40.E-07

Berea Sandstone 2.23.E-05

Oga Andesite 1.41.E-04

Figure 14　Relationship between hydraulic conductivity and pore properties of rocks: (a) porosity; (b) mean pore radius [9].

! 9 !

Imai, T., Kamoshida, N., Yamaguchi, S. and Sugimoto, F.

Figure 14 Relationship between hydraulic conductivity and pore properties of rocks:

(a) porosity; (b) mean pore radius [9]

(a) porosity (b) mean radius of pore [9]

Table 5 Hydraulic conductivity of rocks (mean value)

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10 15 20 25 30

Towada Tuff

Ogino Tuff

Kimachi Sandstone

Tako Sandstone

Berea Sandstone

Oga Andesite

Hydra

uli

c C

onduct

ivit

y (

cm/s

)

Porosity (%)

Lower Limit

Upper Limit

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

0.01 0.1 1 10 100

Ogino Tuff

Kimachi Sandstone

Tako Sandstone

Berea Sandstone

Hydra

uli

c C

onduct

ivit

y (

cm/s

)

Mean pore radius ( m)

Lower Limit

Upper Limit

Rock Specimen Hydraulic Conductivity (cm/s)

Towada Tuff 1.52.E-08

Ogino Tuff 1.62.E-09

Kimachi Sandstone 3.00.E-09

Tako Sandstone 1.40.E-07

Berea Sandstone 2.23.E-05

Oga Andesite 1.41.E-04

85New Permeability Test Based on Falling Head Method in Rock Core

Int. J. Soc. Mater. Eng. Resour. Vol.17, No.1, (Mar. 2010)

limits of our method (Table 5). Hydraulic conductivity appears to have a negative correlation with porosity (Figure 14a). It had a good positive correlation with mean pore radius measured with a mercury intrusion porosimeter [9] (Towada tuff and Oga andesite are excluded), being almost proportional to the second power of the radius (Figure 14b). The hydraulic conductivity of sandstone has a positive correlation with porosity when the mean grain size remains constant, and with mean grain size when the porosity remains constant [10]. Hydraulic conductivity depends on porosity and, in particular, means pore radius [11]. Therefore, the equivalent channel theory, which holds that hydraulic conductivity is directly proportional to the second power of the pore (pipe) radius [12], is applicable to our results (Figure 14b).

5.2 Characteristics of our methodTable 6 shows the advantages and disadvantages of five types of

laboratory permeability tests, including our method. The constant head method is an easy measurement technique and uses cheap apparatus, but it can measure only extremely high hydraulic conductivity. The standpipe method also uses cheap apparatus and can measure low hydraulic conductivity, but it needs a long measurement time. The transient pulse method and the flow pump method can measure the hydraulic conductivity of most types of rock, but the apparatus are extremely expensive and the measurement techniques are complicated.

6　Conclusions

We developed and tested a new laboratory permeability testing method based on the falling head method but using a pressure accumulator as the pressure source instead of a standpipe. We measured the hydraulic conductivities of various types of rock, and examined the performance of the apparatus and the reliability of the method.(1) The apparatus could measure hydraulic conductivity from 3.5×

10-10 to 2.6×10-4 cm/s.(2) The error in repeated measurement was ± ~10% and the

difference in hydraulic conductivity between this method and the standpipe method was about 25%.

(3) For rock with hydraulic conductivity of 3.0×10-9 cm/s, measurement took >800 s.

(4) The hydraulic conductivity measured by our method obeys the equivalent channel theory [12], that hydraulic conductivity is proportional to the second power of the pore (pipe) radius.

(5) The accumulator method can measure a wide range of

hydraulic conductivities, is easy to use, and requires cheap apparatus.

Acknowledgments

We are grateful to M. Nishimura and A. Ohbayashi who assisted in our experiments.

References

[1] Long, J. C. s., Remer, C. R. and Witherspoon, P. A., Porous Media Equivalents for Networks of Discontinuous Fractures, Water Resources Research, 18, 645-658, (1982)

[2] Ludes, M. L. and Jesus, C., A Methodology to interpret cross-hole tests in a granite block, Journal of Hydrology, 325, 222-240, (2006)

[3] Nappi, N., Esposito, L. Piscopo, V. and Rega. G., Hydraulic characterisation of some arenaceous rocks of Molise (Southern Italy) through outcropping measurements and Lugeon tests, Engineering Geology, 81, 54-64, (2005)

[4] Narasimhan, T. N., Multidimensional Numerical Simulation of Fluid Flow in Fractured Porous Media, Water Resources Research, 18, 1235-1247, (1982)

[5] Roger, H. M. and Harold W. O., Theoretical Analysis of the Transient Pressure Response From a Constant Flow Rate Hydraulic Conductivity Test, Water Resources Research, 23, 1461-1470 (1980)

[6] Brace, W. F. Walsh. J. B. and Frangos, W. T., Permeability of Granite under High Pressure, Journal of Geophysical Research, 73, 2225-2236, (1968)

[7] Sugimoto, F. and Furuzumi, M., Permeability Tests of Rock by Transient Pulse Method, Journal of MMIJ, 105, 727-731, (1989)

[8] Xue, Z., Isijima, Y. and Takahashi, M., Permeability and Microgeometry of Sandstones under Hydrostatic Pressure, Journal of MMIJ, 108, 769-775, (1992)

[9] Lin, W. and Takahashi, M., Mercury Intrusion Porosimetry and Its Application for Determination of Pore Volume Distribution of Rock, Chishitsu News, 549, 61-68, (2000)

[10] Schon, J. h. Physical Properties of Rocks : Fundamentals and Principles of Petrophysics, ELSEVIER, 42-49, (2004)

[11] Brace, W. F., Permeability from Resistivity and Pore Shape, Journal of Geophysical Research, 82, 3343-3349, (1977)

[12] Juang, C. H. Asce, A. M. Holtz R. D. and Asce, M., Fabric, Pore Size Distribution, and Permeability of Sanday Soils, Journal of Geotechnical Engineering, 112, 855-868, (1986)

Table 6　Comparison of the performance of other methods with the new method.

! 10 !

Imai, T., Kamoshida, N., Yamaguchi, S. and Sugimoto, F.

Table 6 Comparison of the performance of other

methods with the new method

Permeability Test TypeHigh

Permeability

Low

PermeabilityEasiness Cost

　Constant Head △ ○ ○　Stand Pipe ○ △ △ ○　Transient Pules ○ ○　Flow Pump ○ ○ △　Accumulator (This study) △ ○ ○ ○

○：Superior

△：Ordinary

： Inferior