The Determination of Biot's Parameters for Sandstones Part 1 �9 Static Tests

Experimental procedure for determining Blot's parameters for four rock specimens is presented and theoretical values of wave speeds propagating through them are compared with experimental values

by C.H. Yew and P.N. Jog

ABSTRACT--One of the acceptable theories in describing the wave motions in fluid saturated porous rocks is Biot's theory. 2 In this paper, we present the procedures in the determination of the parameters in Blot's theory. One artificial rock 'Alundum' and three types of natural sandstones were tested. The calculated wave speeds in the rock specimens, based on the experimentally determined Blot's parameters, were found in good agreement with the measured dilatationat wave speeds l l e i n n t h ~ , d t r a e n n l n m~ l lo~ m ~ f h n r q

Introduction

The mechanical and physical properties of a fluid- saturated porous rock such as porosity, permeability, viscosity of the contained fluid and the elastic properties of the medium are important information for the petroleum engineers. The commonly used technique in obtaining this information is to test the extracted rock samples. In order to simulate the actual environment of the sample, the test is performed with the rock sample subjected to a high confining pressure. An experiment of such kind is usually very complicated and expensive. Although the test yields valuable information, it has two major disadvantages: (1) the properties of the rock are changed by the process of extraction; and (2) since the size of the extracted rock sample is usually small, the test results represent the properties of only a small portion of the rock mass. An alternative method for obtaining the physical and mechanical properties of the rock mass is to study the wave motions in the rock generated by a pressure pulse. This method, when fully developed, would provide not only useful information of the rock mass but also a means of in-situ measurement.

It has been shown analytically ~ that the desirable information about a fluid-saturated rock mass could be obtained by analyzing the measured wave motions generated by a wide-band pulse using Biot's theory. Biot's equations for wave motions in a fluid saturated porous medium have the following form :2

C.H. Yew is Associate Professor, Department o f Aerospace Engineering and Engineering Mechanics and P.N. Jogi is- Graduate Student, Department of Petroleum Engineering, The University of Texas' at Austin, TX 78712. Original manuscript submitted : May 11, 1977. Final version received : September 24, 1977.

(I) For dilatational waves,

0 5 A~ ( P e + Qe) = ~ y - ( ~ , , e

0 5 A S ( Q e + Re ) = ~ ( o , s e

0 + ~o,se) + b ~ ( e - e)

0 + ~ - O - b ~ - ( e - e)

(1)

(2) For rotational waves,

02 o 0T2(0,l~ + 0,sfl) + b (~ - ~2) = NAko

0 5 (~,5~o + 0 . ~ 2 ) - b O (~0 ~2) = 0

Ot 5 OI (2)

In eqs (1) and (2), e, ~0 and e, f~ are the dilatational and rotational components of motion of the solid portion and the fluid portions, respectively; the parameters P , Q, R, known as the Biot-Willis constants, represent the mechanical properties of fluid-saturated porous medium; parameters ~,,, O,z and ~25 are the dynamic coefficients that take into account the inertia and interactive effects of the solid and fluid portions; finally, the coefficient b which relates the permeability K, porosity/3 and viscosity #, is defined as

b - #/32 K (3)

In this paper, we present the values of Blot's parameters in eqs (1) and (2) of an artificial rock 'Alundum' and three kinds of natural s~ndstones (Berea, Pecos and Ohio) using the 'standard' laboratory test. The standard test involves (1) determination of the porosity of the specimen; (2) determination of the bulk moduli of the material under the jacketed and unjacketed conditions; (3) determination of the permeability of the specimen with a permeameter and (4) determination of the wave speeds in the material using an ultrasonic pulser. The material constants so determined will be used, in turn, as a standard in the comparison of material constants determined by analyzing the measured wave motions in conjunction with eq (1). This portion of the investigation will be presented in another report. 3

Exper imental Mechanics �9 167

Descript ion of Specimen

The artificial rock Alundum and three types of natural sandstones, Berea, Ohio and Pecos sandstones, were considered in this investigation. All the specimens were machined to a cylinder of approximate dimensions of 2 in. (5.08 cm) in diameter and 1 inch (2.54 cm) in length with a diamond-tipped core cutter:

Alundum is a sintered material with a metallic skeleton. Individual grains contain 99.28% A1203, 0.5% SiO2, 0.15% Fe203, 0.02% TiO2 and 0.05% Na20. By controlling the sintering process, the pore distribution can be made uniform within a certain porosity range.

The Berea sandstone is of Mississippian age, and is light tan in color. It is classified as a fine-grained gray- wacke with a clastic texture composed of subgranular grains--quartz grains form 99 percent of the rock with traces of carbonate minerals. It is quarried in Berea, Ohio.

The Ohio sandstone is a very hard and brittle clastic rock. It is a very fine-grained rock. Most of the matrix material is quartz, clay and shale.

The Pecos sandstone used in this investigation came from western Texas. The formation lies between thick beds of limestone. It is a reddish brown, calcareous sandstone with thin shaley streaks along the bedding planes.

Figure 1 shows the photographs of the surface of the four samples made with an electronic scanning micro- scope. With the exception of perhaps Berea sandstone which shows an irregular matrix structure, the pore distribution of the Alundum, Pecos and Ohio sandstones may certainly be regarded as macroscopically homogeneous.

Determinat ion of Parameters o,, , o,, and 022

According to Biot, 2 parameters 0,, , 0,2 and 022 are related to the total mass density of the fluid saturated medium, Q, by equation

0 = 0 - + 20,2 + 022. (4)

When there is no relative motion between the" fluid and solid skeleton, the coupling parameter 0~2 = O, eq (4) reduces to

0 = 0~, + Q22 (5)

Using the definition of porosity as the ratio of the volume of the voids to the total volume of the specimen, parameters 0 - and 022 in eq (5) may be written as

0 . = (l - ~ ) o ,

Q22 = /30s, (6)

where 0, and 0s are the mass densities of the solid skeleton material and the fluid, respectively.

Parameters 0,, and 022 were thus determined by the following procedures : (1) Measure the dimensions of the specimen, and calculate the volume of the specimen I1",. (2) Obtain the weight of 'dry specimen'. The dry specimen was prepared by placing the specimen in an oven at 200~ for four hours. (3) Obtain the weight of wet specimen (fluid-saturated

PECOS SANDSTONE X80 OHIO SANDSTONE X80

Fig. 1 ~Sur face structure of samples made with an electronic scanning microscope

ALUNDUM X80 BEREA SANDSTONE X80

168 �9 May 1978

specimen). The wet specimen was prepared by placing the dry specimen in the fluid. In order to obtain a full saturation, the specimen with its fluid container was placed in a vacuum chamber (p = 10 mm Hg) for removing the air bubbles that might be trapped in the specimen. The density of the wet specimen is therefore

weight of wet specimen (7) 0 = volume of the specimen

(4) The volume of the voids can be determined from the expression

weight of wet specimen - weight of dry specimen ~vold - - weight of density of the fluid

(8)

(5) The porosity of the specimen /3 was calculated from the expression

Vvoid = (9) v,

(6) The mass density of the fluid Q~ was taken to be 1 gm/cc (water). The mass density of the solid skeleton was calculated from eq (10),

o, - 1 - ~ do)

Parameters 0,, and 0= were calculated using eq (6). The coupling parameter O,z can be determined from the

sound-wave-speed measurement. The determination of this parameter is postponed to a later section.

Determination of Parameters P, Q and R

The parameters P, Q and R are known as the Biot- Willis constants. The physical meanings of these constants and the method of obtaining them are discussed in detail by Biot and Willis ~ and Fatt. ~ These constants were determined by following a similar procedure described in Ref. 5, but with a slight variation due to equipment availability.

Based on eq (21) of Ref. 5, the Biot's constants P , Q and R in eq (1) can be expressed as

P = X + 2 #

6 / 3 (1 - # - ~- )

O =

R -

~2 v + f - - -

k

~)2 v + 5 - - -

k

+ /32 + (1 - 2 /3 ) (1 - ~i k T ) 2

X = /~ 62 3

v + ~ j - - - k

v = /~ (c - 6) (11)

Where/3 is the porosity, k is the modulus of the jacketed compressibility, 6 is the modulus of the unjacketed compressibility, /~ is the shear modulus of the bulk material, and c is the compressibility of the fluid.

Water was used as the saturating fluid and its com-

pressibility was taken as c = 3.378 x 10 6 (psi) -~ at 200 psi ' . The moduli of the jacketed and the unjacketed compressibility of the fluid ~5 and k were determined using a specially designed apparatus. Figure 2 shows a schematic diagram of the loading chamber of this apparatus. The fluid-saturated rock specimen (approxi- mately 1 in. in length and 2 in. in diameter) was placed in the central portion of a thick-walled cylindrical chamber, and the inside of the chamber was filled with lubricating oil. The fluid pressure (or the confining pressure) in the chamber was controlled by an external hydraulic pumping system. The axial and the radial displacements of the specimen were monitored by a pair of specially designed displacement sensors. The resolution of the displacement sensor was + 10 #in.

To determine the modulus of the unjacketed com- pressibility of the specimen, the cylindrical specimen was placed in the test position inside the chamber, as shown.

PORE PRESSURE / / / /A

PORUS / SPECIMEN . .JACKET

AXIAL ~

Ir / , . :@ oRE

Fig. 2 - - S c h e m a t i c d iagram of loading chamber (radial- d isp lacement sensor is not shown)

- 8 . O ~ TEST)

-'~ A/ Fy / /3 l //// / / / I

-"ol l ll/ / / / I t.e CL

-,.ollll / / 1-.oo 0 . 2 . 0

0 . . .t . . . . . . . . . 5 AXIAL STRAIN (X I0 "~)

Fig. 3 - -P ressure -ax ia l stra in curves of samples tested in the jacketed and un jacketed cond i t i ons (A = A lundum, B = Berea sandstone, O = Ohio sandstone, P = Pecos sandstone)

Exper imental Mechanics �9 169

Fig. 4--Arrangement of gas permeameter

L .. FLOWMETER,

lira

BLEED VALVE

HASSLER CORE HOLDER

WATER MANOMETER

r).'• )

MERCURY MANOMETER

NITROGEN CYLINDER

The pore pressure was maintained at confining pressure and the deformation of the specimen was measured at each increment of the confining pressure.

For determining the modulus of the jacketed com- pressibility, the specimen was placed inside a rubber tubing of 1/8-in. wall thickness, approximately. The rubber tube with a proper opening is connected to the pore-pressure line. This arrangement allows pressure inside the porous specimen to be controlled. Since the pore pressure is always less than the confining pressure in this type of test, the use of a flexible rubber tube as the specimen jacket seemed to be satisfactory. The pore pressure in this investigation was kept at atmospheric pressure (Pa = 14.7 psi).

Figure 3 shows the representative p - e curves of the Alundum, Berea, Ohio and Pecos sandstones tested in the jacketed and unjacketed conditions. The pressure-strain curves of all the samples tested in the unjacketed condition are linear. In the jacketed test condition, the pressure strain curves of Berea and Ohio sandstone showed a considerable nonlinearity in the low-pressure region. Based on the measured dilatational wave speed which will be described in detail in a later section, we found, however, the modulus of compressibility of the jacketed specimen can be best represented by the slope of the linear pressure-strain port ion in the high-pressure region (shown as the dotted lines in Fig. 3). The volumetric strain of the specimen was calculated from the definition

A = e. + e~ + e0 = 3co, (12)

where co is the measured axial strain. The moduli of the jacketed and the unjacketed compressibility of the speci- men were calculated from the following equations :

6 = '~ (unjacketed) (13) P

and

k = 'SOacke ted ) (14) P

respectively.

The shear modulus # of the fluid saturated rock, in principle, should be obtained from a torsion test. The torsion tests were not performed in this investigation due to lack of equipment. Instead, the shear modulus of the specimen was calculated from equation

3 k (1 - 2u) (15) # = 2 (1 + u)

where u is the Poisson's ratio of a jacketed specimen and k'is the bulk modulus.

Determination of Permeability K The ability of porous medium to allow fluid through

its interstices is known as the permeability which is defined as

q K Pi - P~ - ( 1 6 )

A # L

where q = volume flow rate (cc/s) A = cross-sectional area of the specimen (cm 2)

p , - p~ pressure drop per unit length (a tm/cm)

L # = viscosity (centipoises)

K = permeabili ty (Darcy)

The permeability of the specimen was determined with a gas permeater. 7 Figure 4 shows a schematic diagram of its arrangement. The cylindrical specimen (2 in. diameter) was placed in the Hassler core holder. The pressure drop through the specimen length, P l - P2 was measured with a manometer . The volume flow rate was measured with a soap bubble or variable area flowmeter. Accounting

170 �9 May 1978

X EMITTER ' \ \ l ,,,~/'~ /TRANSDUCER

" ' ~ ~ x ~ i f (P-WAVE)

/ b } '

I I I I l l / I l l /

BET ECTO R xb~ TRANSDUCER

(P- WAVE) '~ I ~ , ~ 0 ~ '

I /

DETECTOR TRANSDUCER (S- WAVE}

Fig. 5--Schematic diagram showing geometry and orientation of dual-mode transducer and sample assembly 8

for the compressibility of the gas, the flow rate q in eq (16) is related to the measured flow rate Q by equation

P, + P2 Q = q - (17) 2p,

where p , is the atmospheric pressure. Substituting eq (17) into eq (16), the equation for calculating the gas permeability K~ has the form

2 L t ~ P . Q K~ - (18)

(p2 _ p~) A

Since the fluid-saturated rock is of the main interest here, it is necessary to convert the gas permeability K~ to the fluid permeability K e.

Klinkenberg's empirical equation ~ was used to perform this conversion. Klinkenberg's equation has the form

0.77Kg -~ K~ = K r (1 + ) (19)

P~

where Pm is the mean pressure on gas flowing through the rock. The numerical value of K r was computed from this equation.

Wave.speeds Measurement

The experimental arrangement for the measurement of P and S wave speeds is sketched in Fig. 5. This arrange- ment is essentially the same as that developed by Gregory and Podio. 8 The P wave generated by the emitter trans- ducers is refracted to the P and S waves at interface as shown in Fig. 5. The generated P or S wave, after traversing the specimen, was refracted again to P waves and it was monitored by the detector transducers. The wave speed was computed by dividing the specimen length with the transit time for the wave passing through the specimen. During the test, the whole assembly was placed in a confining chamber as shown in Fig. ,2; and the water saturated specimen was jacketed. The measurement of S wave in a low confining pressure was difficult. This was probably due to the poor coupling between the interface of the specimen and the aluminum wedges. Consequently,

TABLE 1--BLOT'S PARAMETER

Permeability Q,,, Porosity (K) Ib/in? x 10 -2

Specimen (~) Milli-Darcy (kg/M J x 103)

022, P, Q, Ib/in? x 10 -~ Ib/in? x 106 Ib/in} x 10 ~

(kgfM ~ x 103) (N/M 2 x 10 '~ (N/M 2 x 10 TM)

Alu ndum 0.316 2706 8.321 1.139 9.5216 0.0827 (2.303) (0.315) (6.5649) (0.0570)

Berea 0.187 125.5 7.681 0.674 3.3183 0.1 51 9 Sandstone (2.126) (0.186) (2.2879) (0.1047) Ohio 0.191 5.6 7.829 0.691 2.9489 0.1383 Sandstone (2.167) (0.191 ) (2.0332) (0.0953) Pecos 0.195 0.8 7.510 0.705 2.6774 0.1791 Sandstone (2.078) (0.195) (1.8460) (0.1235)

Specimen

R, /x, b, Ib/in. 2 x 10 ~ Ib/in? x 106 psi-sec/in 2

(N/M 2 x 10 '0 ) (N/M s x 10 'o ) N-sec/M 4 x 108

Alundum 0.0914 3.6737 3.48 (0.0630) (2.5330) (0.372)

Be rea 0.0480 1.1034 26.33 Sandstone (0.0331) (0.7608) (2.813) Ohio 0.0480 0.9920 61 7.10 Sandstone (0.0331) (0.6840) (65.94) Pecos 0.0550 0.8480 4436.7 Sandstone (0.0379) (0.5846) (474.1 3)

Experimental Mechanics �9 171

the measurement of wave speeds was made at a confining pressure of 3000 psi.

Results and Discussion

The Biot's coefficients, 0,1, 052, P , Q, R and b of the Alundum and the sandstones determined by the methods described in the preceding sections are tabulated in Table 1. To ensure that the results are repeatable, the results presented in Table 1 are the representatives from the results of two to three tests.

The most convenient means in checking the validity of these so-determined coefficients would be to compare the measured wave speed with that predicted by the theory. Equations (1) and (2) predict the existence of two dilatational waves and one shear wave. ''2 The wave speed of the fast and slow dilatational waves are

2 ( P R - Q2)

(Ro,,+Poz~ 2QQ,2)~:[(R~,,+PQ22 2Q~hz) 2 4(PR QZ)(0.022-02,2)]'/2

(2o) and the shear wave speed is

V,~ = / z / ( O l , - 0?5 ) . (21) Q22

In Ref. 1 it has been shown that the slow wave (II.) was highly attenuative; only the fast wave could be monitored in the ultrasonic wave-test arrangement as described in the section on "Wave-speed Measurement". Further, it was proved that this fast wave was lightly dispersed at the test frequency (1 MHz). The fast-wave speed cannot, however, be readily calculated from eq (20), since the parameter 0.5 which describes the interaction between the solid and the fluid portion of the medium, is still an un- known. Using the measured wave speed, the magnitude of 012 can be computed from eq (20). The computed values of 012 are tabulated along with the measured wave speeds and the wave speeds calculated from eq (20) and eq (21) by assuming 012 = 0 in Table 2. The effect of 012 on the dilatational wave speed was found to be very small (1- percent maximum). Since the ultrasonic wave is a wave of low amplitude, the interaction between the fluid and solid portion of the medium is not expected to be strong in this type of test.

As stated in the section on "Determination of Parameters P , Q and R" , because of the nonlinearity in the p-e curves,

the moduli of compressibility of the jacketed Berea and Ohio sandstones were determined from the linear portion of the p-e curves in the high pressure region. The close agreement between the calculated and the measured sound-wave speeds seems to justify this undertaking. The nonlinearity of the p-E curves in the low-pressure region is probably due to the local deformation of the contact surfaces between the porous specimen and its supporting plates.

Equation (20) shows that the wave-speed measurement allows only the estimation of the mechanical properties (i.e., 011, 022, P , Q and R) of the fluid-saturated porous medium. Information concerning the permeability and viscosity of the medium does not reveal itself in the wave speed. Consequently, this information cannot be obtained by wave-speed measurement alone. It has been shown in Ref. 1 that information on permeability and viscosity can be obtained by studying the amplitude decrement and phase shift of the measured waves. This portion of the investigation will be presented in another report.

Acknowledgment The authors wish to express their appreciation to

K.E. Gray for his interest in this subject and for spending many hours in discussion during the course of investigation, and to N. Morita for his generous assistance in experi- mentation. This investigation was supported by the National Science Foundation (ENG76-09095).

References 1. Yew, C.H. and Jogi, P.N., "Study o f Wave Motions in Fluid

Saturated Rocks, "" J. Acous. Soc. Am., 60 (1), 2-8 (Jul. 1976). 2. Biot, M.A., "Theory o f Propagation o f Elastic Waves in a Fluid

Saturated Porous Solid, "" J. Acous. Soc. Am., 28 (2), 163-178 (Mar. 1956). 3. Yew, C.H. and Jogi, P.N., "Determination o f Biot's Constants

Based on the Measured Wave Motion, "" Report EMRL 1122, Department o f Engineering Mechanics, University o f Texas at Austin (1978).

4. Carmichael, C., "'Kent's Mechanical Engineers" Handbook, "' 12th ed., Wiley Handbook Series (1967).

5. Biot, M.A. and Willis, D.C., "The Elastic Coefficients o f the Theory o f Consolidation, " J. AppL Mech., 594-601 (Dec. 1957).

6. Fatt, L, "'The Biot-Willis Elastic Coefficients for a Sandstone, "" J. Appl. Mech., 296-297 (Jun. 1959).

7. Katz, D.L, Kobayashi, R., Poettmann, E.H., Vary, J.A., Elenbaas, J.R. and Neinaug, C.S., "Handbook o f Natural Gas Engineering," McGraw-Hill Book Co., 40-46 (1959)~

8. Gregory, A.R. and Podio, A . L , "Dual-Mode Ultrasonic Apparatus for Measuring Compressional and Shear Wave Velocities o f Rock Samples, "" Sonic and Ultrasonic Measurement, IEEE Trans., SU-17, 77-85 (1965).

TABLE 2--COMPARISON OF WAVE SPEEDS

Specimen

Bulk Measured Modulus, (k), Poisson's P Wave, Ib/in? x 106 Ratio, in./#s

(N/M z x 10 '~ ~, (M/s)

Measured Calculated Calculated S Wave* P Wave (e,2 = 0) S Wave e,2,

in./#s in./#s in./#s Ib/in? x 10 -~ (M/s) (M/s) (M/s) (Kg/M 3 x 103)

Alundum 4.548 .15 .213 (3.1358) (assumed) (5416)

Berea 1.366 15 .132 Sandstone (0.9418) (3346) Ohio 1.228 15 .128 Sandstone (0.8467) (3257) Pecos 0.8480 10 .130 Sandstone (0.5846) (3299)

.128 .210 .131 0.243 (3240) (5326) (3317) (0.067) .092 .131 .075 0.338

(2343) (3316) (1908) (0.094) .086 .122 .070 0.298

(2173) (3099) (1787) (0.083) .078 .120 .068 0.552

(1977) (3056) (1727) (0.153)

* Shear-wave measurement was made at 3,000 psi confining pressure

172 �9 May 1978