Elastic Properties of Overpressured and 

Unconsolidated Sediments 

By Myung W. Lee 

U.S. Geological Survey Bulletin 2214 

U.S. Department of the Interior 

U.S. Geological Survey 
U.S. Department of the Interior 

Gale A. Norton, Secretary 

U.S. Geological Survey 

Charles G. Groat, Director 

Posted online September 2003, version 1.0 

For more information about the USGS and its products:


Telephone: 1-888-ASK-USGS


World Wide Web: http://www.usgs.gov/


This publication is only available online at: 

http://geology.cr.usgs.gov/pub/bulletins/b2214/ 

Any use of trade, product, or firm names in this publication 

is for descriptive purposes only and does not 

imply endorsement by the U.S. Government 
III


Contents 

Abstract .......................................................................................................................................................... 1


Introduction ................................................................................................................................................... 1


Theory ............................................................................................................................................................ 2


BGTL Parameters.......................................................................................................................................... 2


Biot Coefficient..................................................................................................................................... 2


Parameters G and n............................................................................................................................. 2


Constant m ............................................................................................................................................ 3


Analysis and Modeling ................................................................................................................................ 4


Velocity Ratio with Respect to Differential Pressure..................................................................... 4


Velocity Ratio with Respect to S-Wave Velocity ............................................................................ 5


Velocity with Respect to Porosity and Differential Pressure ....................................................... 5


Prediction of Pore Pressure............................................................................................................... 6


Discussion...................................................................................................................................................... 7


Velocity Ratio and Differential Pressure.......................................................................................... 7


Differential Pressure and S-Wave Velocity..................................................................................... 8


Clay Effect on Velocity Ratio .............................................................................................................. 9


BGTL Parameter m .............................................................................................................................. 9


Conclusions ................................................................................................................................................... 9


References..................................................................................................................................................... 9


Figures 

1. 	 Graphs showing porosity reduction with respect to differential pressure 


and variation of constant m ........................................................................................................ 3


2. 	 Graph showing computed and measured velocity ratio with respect to 


differential pressure..................................................................................................................... 5


3. Measured and calculated velocity ratios with respect to S-wave velocity....................... 5


4. 	 Calculated velocities by Biot-Gassmann theory by Lee, Duffy-Mindlin 


theory, and measured velocities by Prasad with respect to 


porosity and differential pressure ............................................................................................. 6


5. 	 Graph showing relationship among differential pressure, Vp/Vs

, 


and S-wave velocity..................................................................................................................... 7


6. Graph showing Vp/V with respect to differential pressure.................................................... 8


7. 	 Measured and computed velocities from BGTL with respect to 


differential pressure..................................................................................................................... 8


Table 

1. Elastic constants used in this paper ......................................................................................... 4

Elastic Properties of Overpressured and 


Unconsolidated Sediments 

By Myung W. Lee 

Abstract 

Differential pressure affects elastic velocities and Poisson’s ratio of sediments in such a way that velocities increase 

as differential pressure increases. Overpressured zones in sediments can be detected by observing an increase in Poisson’s 

ratio with a corresponding drop in elastic velocities. In highly 

overpressured sands, such as shallow water flow sands, the Pto S-wave velocity ratio (Vp/Vs

) is very high, on the order of 10 

or higher, due to the unconsolidated and uncemented nature of 

sediments. In order to predict elastic characteristics of highly 

overpressured sands, Biot-Gassmann theory by Lee (BGTL) 

is used with a variable exponent n that depends on differential 

pressure and the degree of consolidation/compaction. The 

exponent n decreases as differential pressure and the degree of 

consolidation increases, and, as n decreases, velocity increases 

and Vp/Vs decreases. The predicted velocity ratio by BGTL 

agrees well with the measured velocity ratio at low differential 

pressure for unconsolidated sediments. 

Introduction 

Knowledge of fluid pressure in the pore spaces of sedimentary rocks is important for the successful exploration and 

production of hydrocarbons. High pore pressure, commonly 

associated with shallow and unconsolidated sediments, presents a significant hazard during drilling and completion of 

offshore wells. Therefore, predicting overpressured intervals 

before drilling not only mitigates drilling hazard but also minimizes drilling cost. 

One exploration problem in deep water is encountering 

shallow-water-flow (SWF) sands (Sayers and others, 2001; 

Bruce and others, 2002; Mallick and others, 2002; Ostermeier 

and others, 2002). SWF sands are overpressured and unconsolidated (Huffman and Castagna, 2001; Bruce and others, 2002). 

During drilling, SWF sand may flow into the well if their 

pore pressures are not balanced by mud weight, causing large 

and long-lasting uncontrolled flows, well damage, foundation 

failure, damaged casing, reentry, and control problems (Ostermeier and others, 2002). Seismic methods have been used to 

predict these important overpressured intervals (Huffman and 

Castagna, 2001; Dutta, 2002; Mallick and others, 2002; Sayers 

and others, 2002). 

Physical properties of SWF sands differ from most reservoir and seal rocks for petroleum. These materials exist near 

transition zones between rock and sediment (Huffman and 

Castagna, 2001). SWF sands usually have high porosities and 

very low differential pressure or high pore pressure; they are 

unconsolidated and uncemented. Therefore, interval velocities 

(P-wave and S-wave velocity) of SWF sands are low compared to adjacent normally pressured sands, and the decrease 

of velocity is related to the amount of overpressure (Huffman 

and Castagna, 2001; Prasad, 2002). With increasing pore 

pressure, SWF sands lose cohesion, causing Vs

 to drop faster 

than Vp. This causes Vp/Vs to increase rapidly. A typical Vp/Vs 

ratio of SWF sand is on the order of 10 or higher (Mallick and 

others, 2002). 

Sayers and others (2002) predicted pore pressure using 

reflection tomography and four-component (4-C) seismic 

data. Reflection tomography gives higher spatial resolution 

for interval velocities, and 4-C data may help to reduce the 

ambiguity between variations in pore pressure and variations 

in lithology and fluid content. Dutta (2002) used prestack 

inversion of large-offset P-wave data and rock models to predict deep water geohazards or overpressure. Both approaches 

utilize the reduction of P- and S-wave velocities and the 

high Vp/Vs

 ratio for overpressured intervals. Because large 

changes in Poisson’s ratio of SWF sands also result in significant changes in the amplitude versus offset (AVO) response, 

high-resolution AVO and inversion methods should be able to 

resolve changes in pressure (Huffman and Castagna, 2001). 

One key element in predicting overpressure is the relationship between differential pressure and Vp/Vs

. Empirical 

relationships between differential pressure, Vp/Vs, and velocities with respect to differential pressure are presented by Huffman and Castagna (2002) and Prasad (2002). Zimmer and 

others (2002) investigated pressure and porosity influence on 

Vp/Vs in unconsolidated sands using glass bead samples. In 

this paper, elastic velocities of overpressured unconsolidated 

sediments are predicted by the Biot-Gassmann theory by Lee 

or BGTL (Lee, 2002, 2003). Velocities or velocity ratios with 

respect to differential pressure are incorporated by using a 

variable exponent n as a function of differential pressure and 

consolidation/compaction. Predicted velocities or velocity 

1

2 Elastic Properties of Overpressured and Unconsolidated Sediments 

ratios by BGTL agree well with measured data complied by 

Prasad (2002) and Huffman and Castagna (2001). 

Theory 

Based on an assumption that the velocity ratio (Vp/Vs) 

depends on porosity, Lee ( 2002, 2003) derived the following 

elastic moduli for water-saturated sediments: 

k = km a(1− β) + β 2M (1) 

μ = μmakma (1− β)G2

(1−φ )

2n + μmaβ 2

MG2(1−φ )

2n

(2) kma + 4μma[1− G2

(1−φ )

2n

]/ 3

with 

1 (β −φ)

+ φ = M k k m a fl

where 

k and μ are bulk and shear moduli of water-saturated 

sediments, respectively, 

kma , μma, and kfl are bulk and shear moduli of matrix 

material and bulk modulus of fluid, respectively, 

β and φ are the Biot coefficient and porosity, respectively, 

and 

n and G are parameters that depend on the degree of 

consolidation/compaction, differential pressure, and 

clay content. 

Elastic velocities, Vp for P-wave and Vs

 for S-wave velocities, of water-saturated sediments can be computed from the 

elastic moduli by the following formulas: 

Vp ρ

k + 4 μ/ 3

and Vs = ρ

μ (3) 

where 

ρ is density of the formation. 

The formation density is given by 

ρ = (1− φ) ρm a + φρfl (4) 

where 

φ, ρma and ρfl are porosity, matrix density , and pore fluid 

density, respectively. 

Lee (2003) referred to the above method of computing 

shear modulus as BGTL , Biot-Gassmann theory by Lee, to 

differentiate from the classical Biot-Gassmann theory (BGT) 

(Biot, 1956; Gassmann, 1951), which computes shear modulus 

by μ = μma(1 – β). Note that, as indicated in Lee (2003), there 

is no difference between BGT and BGTL for a dry or gassaturated rock. 

BGTL Parameters 

Biot Coefficient 

Within the poroelastic framework, skeleton or frame 

moduli, k and μ, are undetermined and must be specified a 

priori. The moduli of a dry frame are related to the moduli 

of matrix material through the Biot coefficient, β. Usually, β

depends on differential pressure as well as porosity, but the 

following Biot coefficient as a function of porosity for unconsolidated sediments (Lee, 2002) is used in this paper: 

−184.05 β = 1+ e(φ+0.56468)/ 0.09425+ 0.99494 (5) 

For consolidated sediment, the equation by Raymer and others 

(1980), which is written in the following form by Krief and 

others (1990), can be used. 

β =1− (1− φ)

3.8 (6) 

Because the Biot coefficient shown in equations 5 and 6 is 

independent of differential pressure, it is not suitable to model 

velocities with respect to differential pressure. However, the 

dependence of velocity on differential pressure can be incorporated by using a variable n as a function of differential pressure. 

Parameters G and n 

In Lee (2003), it is demonstrated that the scale G depends 

on clay content in sediments and is given by the following 

formula: 

G = 0.9552 +0.0448e−Cv / 0.06714

(7) 

where 

Cv is the clay-volume content. 

Equation 7 indicates that G is equal to 1 for clean sands. Without explicit use of Cv

, clean sandstones are assumed in this 

investigation. 

The exponent n is the most important parameter in BGTL 

and is proposed by the following function: 

n = [10(0.426−0.235Log10 p)]/ m (8) 
where

p is differential pressure (difference between pore pressure 

and overburden pressure) in MPa, and

m is a constant to be determined.

Equation 8 is derived from Prasad’s data (2002) measured at 

differential pressure between 0.001 and 50 MPa with the Biot 

coefficient given in equation 5.

Constant m

To a first-order approximation, the constant m appears 

to depend on the degree of consolidation of sediments, and 

m becomes larger as the degree of consolidation/compaction 

increases. As can be seen from equation 8, as m increases, n

decreases. As indicated in Lee (2003), elastic velocities mainly 

depend on the exponent n and increase as n decreases or the 

degree of consolidation increases. Also, as n increases or m

decreases, Vp/Vs increases.

Figure 1 shows the porosity variation with respect to differential pressure and constant m. Porosity decreases as depth 

or differential pressure increases by packing, compaction and 

cementation (Schön, 1996). In this paper, in order to quantify 

the constant m, the degree of compaction is defined as the rate 

of porosity change due to elastic rebound with respect to differential pressure (∂φ/∂p). Generally, the magnitude of ∂φ/∂p

is related to the degree of consolidation as well as the degree 

of compaction, making it difficult to differentiate one from the 

other. Therefore, in this paper, the degree of consolidation is 

used interchangeably with the degree of compaction. 

Porosity reduction by compaction is shown in fi gure 

1A as solid lines; sandstone and shale lines are from Sclater 

and Christie (1985). The shale compaction curve is given by 

φ = 0.803e–5.1d/10000, and the sandstone compaction curve 

is given by φ = 0.49e–2.7d/10000, where d is depth in meters. 

Porosity change with respect to depth or differential pressure due to compaction is large. However, porosity changes 

with respect to differential pressure for samples examined by 

Prasad (2002), Domenico (1977), and Han and others (1986) 

are much smaller than that shown on the compaction curves 

because compaction is predominantly an inelastic process 

and only a small amount of elastic rebound occurs when difA B

∂φ/∂p

e ∂φ/∂p

Figure 1. Graph showing porosity reduction with respect to differential pressure and variation of constant m. A, Porosity versus differential 

pressure for measured data by Domenico (1977), Han and others (1986), and Prasad (2002). Sand and shale compaction curves are calculated 

from Sclater and Christie (1980). B, Constant m with the rate of porosity reduction with respect to differential pressure. Average rate of porosity 

change with respect to differential pressure for Domenico, Han and others (only five clean sandstones), and Prasad data, shown as large solid 

circles, and a least-squares fitting curve are shown.

BGTL Parameters 3
4 Elastic Properties of Overpressured and Unconsolidated Sediments 

ferential pressure acting on a formation is reduced (unloading) 

(Bowers, 2002). An interesting point shown in figure 1A is 

that the rate of porosity decrease with respect to differential 

pressure is different among samples by Prasad, Domenico, 

and Han and others. The average ∂φ/∂p values for data by 

Prasad, Domenico, and Han and others are –3.2×10–3/MPa, 

–4.4×10–4/MPa, and –4.1×10–5/MPa, respectively. (The rate 

for data by Han and others data is the average of five clean 

sand samples for a differential pressure range between 10 and 

40 MPa). As shown later, velocities or velocity ratios computed by BGTL using equation 8 with m = 1, m = 1.5, and 

m = 5 agree well with those of Prasad, Domenico, and Han 

and others. Based on the above ∂φ/∂p and m, the following 

equation for m is derived by a least-squares method and is 

shown in figure 1B. The equation is: 

m =1.0 +4.95289e5212 ∂φ /∂p (9) 

Equation 9 indicates that, as ∂φ/∂p approaches zero, m 

approaches about 6, and, as ∂φ/∂p approaches a large negative 

number, m approaches 1. In deriving equation 9, only three 

input points by Domenico, Han and others, and Prasad are 

used and the least-squares fitting to the exponential function is 

exact. Therefore, it is not known whether equation 9 is appropriate for other values of ∂φ/∂p. Note that ∂φ/∂p in equation 

9 applies only to porosity changes owing to elastic rebound 

(unloading process). Therefore, in general, equation 9 may not 

be used for the sand or shale compaction curve. In practice, 

∂φ/∂p is rarely known; thus, a direct application of equation 

9 is limited. Measured data suggest that m ≈ 4–6 is appropriate for consolidated sediments at differential pressures above 

about 30 MPa and m ≈ 1–2 is suitable for unconsolidated sediments at differential pressures of about 10 MPa or less. 

Analysis and Modeling 

The elastic constants used in this study are given in 

table 1. 

Table 1. Elastic constants used in this paper. 

Velocity Ratio with Respect to 

Differential Pressure 

Velocities in sediments depend primarily on differential 

pressure and porosity. For a given sediment, as differential 

pressure increases, porosity decreases as shown in figure 1. 

In order to accurately predict elastic velocities with respect to 

differential pressure, porosity changes owing to differential 

pressure should be incorporated, particularly for unconsolidated sediments. The general behavior of porosity variation 

with respect to differential pressure is not accurately known. 

Therefore, in this paper, data in table 4 by Prasad (2002) are 

used to derive a relationship between porosity and differential 

pressure; this relationship is given as follows: 

φ =0.38452− 0.00319p (10) 

The porosity range for Prasad’s data is between 0. 382 at p = 

0.89 MPa and 0.321 at p =19.67 MPa (actually Prasad, 2002, 

provided density instead of porosity—porosity is calculated 

assuming a matrix density of 2.65 g/cm3 and a fluid density of 

1.0 g/cm3). Therefore, all the calculated results are accurate 

for these porosity and differential pressure ranges. 

Figure 2 shows measured and computed velocity ratios 

with respect to differential pressure. Circles are measured 

ratios complied by Prasad (2002). The measured velocity ratio 

appears to be a linear function of pressure in log-log scale, and 

Prasad derived a least-squares fitting curve, which is given by 

Vp/Vs

 = 5.6014p–0.2742. As indicated in figure 2, the predicted 

ratio by BGTL using n given by equation 9 with m = 1 is close 

to the measured ratio at a differential pressure greater than 

about 0.5 MPa, but the computed ratio is much larger than that 

predicted by the linear function for a differential pressure of 

less than about 0.5 MPa. The dashed line is the least-squares 

fitting curve for the data analyzed by Huffman and Castagna 

(2001) and agrees well with the prediction of BGTL with m = 

1.3 in equation 9 for a differential pressure greater than about 

0.2 MPa. As differential pressure decreases, the difference 

between the two curves increases. 

Elastic constant Value used Source 

Shear modulus of quartz 44 Gpa Carmichael (1989) 

Bulk modulus of quartz 38 Gpa Carmichael (1989) 

Shear modulus of clay 6.85 Gpa Helgerud and others (1999) 

Bulk modulus of clay 20.9 Gpa Helgerud and others (1999) 

Bulk modulus of water 2.29 Gpa 

Density of quartz 2.65 g/cm3 Helgerud and others (1999) 

Density of clay 2.58 g/cm3 Helgerud and others (1999) 

Density of water 1.0 g/cm3 
Velocity Ratio with Respect to S-Wave Velocity

Differential pressure, clay content, porosity, degree of 

consolidation, and other factors affect Vp/Vs and velocities in 

the sediment. Figure 3 shows Vp/Vs with respect to S-wave 

velocity. BGTL with a variable n computed from equation 

8 fits most of the measured data complied by Prasad (2002) 

and the least-squares fitting curve by Huffman and Castagna 

(2002). The trend of Vp/Vs with respect to differential pressure 

for data compiled by Prasad (2002) shows a different trend 

compared to that by Huffman and Castagna (2001), whereas 

the trend with respect to S-wave velocity is the same. This 

implies that the best predictor of Vp/Vs for sediments including SWF sand or overpressured sand is using S-wave velocity 

as an independent variable. In this way, variability of Vp/Vs

associated with differential pressure, porosity, and degree of 

consolidation is reduced.

Velocity with Respect to 

Porosity and Differential Pressure

The predicted velocity ratio (Vp/Vs) in the previous section indicates that BGTL is accurate for differential pressures 

greater than about 0.1–0.2 MPa. However, the accuracy of the 

velocity ratio predicted from the BGTL does not imply that 

the BGTL is accurate for the velocities themselves. In order 

to examine the accuracy of BGTL, elastic velocities are computed with respect to porosity and differential pressure, and 

are shown in fi gure 4. 

Figure 4A shows velocities with respect to porosity 

computed from equation 10. Open and solid circles represent 

measured velocities for differential pressures between 0.89 

and 19.67 MPa (table 4 from Prasad, 2002). Overall, velocities predicted by BGTL are slightly lower than measured 

velocities, but BGTL accurately predicts the trend between 

Vp /Vs

Figure 2. Graph showing computed and measured velocity ratio 

(Vp/Vs

) with respect to differential pressure. Open circles are measured Vp/Vs

 from data complied by Prasad (2002, table 3 ); the dotted 

line is a least-squares fitting curve to the measured data by Prasad; 

and the dashed line is a least-squares fitting curve to the measured 

data by Huffman and Castagna (2001).

Vp /Vs S

Figure 3. Measured and calculated velocity ratios with respect to 

S-wave velocity. Open circles represent data in table 3 from Prasad 

(2002); the solid line is the least-squares fit to the measured data by 

Huffman and Castagna (2001); and the dashed line is the calculated 

velocity ratio from Biot-Gassmann theory by Lee (BGTL).

Analysis and Modeling 5
6 Elastic Properties of Overpressured and Unconsolidated Sediments

velocity and porosity. Figure 4B shows computed velocities 

with all measured velocities (table 3 from Prasad, 2002). As 

in fi gure 4A, predicted velocities are slightly lower than measured velocities and BGTL predicts S-wave velocities more 

accurately than P-wave velocities. Note that predicted S-wave 

velocities near p = 0.001 MPa are close to measured S-wave 

velocities.

Prediction of Pore Pressure

It is generally recognized that pore pressure can be 

detected prior to drilling using velocity analysis because 

overpressured intervals show significantly reduced interval 

velocities and large Vp/Vs. From AVO analysis (Huffman and 

Castagna, 2001), full waveform inversion (Dutta, 2002), or 

multicomponent seismic data (Sayers and others, 2002), velocities and Vp/Vs

 can be estimated from seismic data. If velocity 

ratios or S-wave velocities are available, differential pressures 

can be estimated from fi gure 5. 

Figure 5 shows calculated Vp/Vs

, S-wave velocity, and 

differential pressure from BGTL with m = 1 and m = 1.3 using 

porosity given by equation 10. Figure 5 is a combination of fi gures 3 and 4 using S-wave velocity as an independent variable. 

As shown in figure 5, the predicted Vp/Vs

 by BGTL is insensitive to the degree of compaction (compare solid and dotted 

lines of Vp/Vs), but differential pressure strongly depends on 

the degree of compaction. Open circles in figure 5 represent the 

measured Vp/Vs, and solid circles represent measured differential pressures compiled by Prasad (2002). Because Vp/Vs

 with 

respect to S-wave velocity is insensitive to different data sets, 

as opposed to Vp/Vs with respect to differential pressure, it has 

the advantage of using S-wave velocity as a common variable to 

relate Vp/Vs to differential pressure. Let us assume that the measured Vp/Vs

 is 8. Figure 5 indicates that Vp/Vs = 8 corresponds 

to S-wave velocity of about 0.2 km/s and differential pressure 

is about 0.3 MPa if m = 1 is applicable, or p = 0.1 MPa if m = 

1.3 is applicable. Either Vp/Vs or S-wave velocity can be used to 

estimate differential pressure.

The P-wave velocity also can be used to predict overpresA B

Vp

Vs

Vp

Vs

P

S

P

S

Figure 4. Calculated velocities by Biot-Gassmann theory by Lee (BGTL), Duffy-Mindlin theory (1957), and measured velocities by Prasad (2002) 

with respect to porosity and differential pressure. Measured velocities are from tables 3 and 4 in Prasad. A, Velocity versus porosity for measured data in table 4 of Prasad (2002). B, Velocity versus differential pressure for measured data in table 3 of Prasad (2002). The dependence of 

differential pressure on S-wave velocity from Duffy-Mindlin theory is a power of 1/6 (p1/6), whereas it is p0.337 based on a least-squares fitting to 

the measured data.
sure. However, as indicated in fi gure 4B, the P-wave velocity 

is almost constant for differential pressures of less than about 

1 MPa. Therefore, the P-wave velocity is not an accurate indicator at low differential pressures. 

Discussion

Velocity Ratio and Differential Pressure

Results shown on figure 2 indicate that BGTL accurately 

predicts the Vp/Vs ratio with respect to differential pressure 

for data compiled by Prasad (2002) except for four anomalous 

points. Two anomalous points near p = 0.001 MPa are measured ratios for near-surface marine sediments of the continental slope of the Bearing Sea by Ayers and Theilen (1999). 

Prasad’s linear equation fits one of the anomalous points, 

whereas the predicted ratio by BGTL is much larger than the 

measured ratio. Because there are no other measurements 

between p = 0.5 MPa and p = 0.001 MPa, it is diffi cult to 

assess the accuracy of BGTL or the linear function of Prasad 

for differential pressures less than about 0.5 MPa. 

Higher Vp/Vs ratios—predicted from BGTL for p less 

than about 0.2 MPa—come from the fact that Vp is almost 

constant for differential pressures between 1 and 0.001 MPa, 

while Vs decreases monotonically to zero (fi g. 4B). This kind 

of behavior of P- and S-wave velocity is expected for watersaturated sediments. Therefore, the prediction of BGTL for 

differential pressures less than 2 MPa appears to be reasonable. However, the accuracy of BGTL for this low differential 

pressure cannot be established until more data are available in 

the low-pressure range.

Both Prasad (2002) and Huffman and Castagna (2001) 

demonstrate a linear relationship between log (Vp/Vs) and 

log (p), although Prasad’s equation predicts a higher Vp/Vs at 

a given differential pressure. Huffman and Castagna (2001) 

suspected that this difference is due to higher porosities in 

Prasad’s samples. Usually, higher porosity samples yield 

higher Vp/Vs

 ratios, so this conjecture is reasonable. However,

figure 2 indicates that BGTL with m = 1.3 instead of m = 1, 

which is good for the Prasad data, predicts accurate Vp/Vs

values for the Huffman and Castagna data, although the same 

porosities (equation 9) are used. As mentioned in Lee (2003), 

the constant m is introduced in order to incorporate the degree 

of compaction. Therefore, another interpretation of Huffman 

and Castagna data is that the degree of compaction of samples 

used by Prasad is less than that used by Huffman and Castagna.

The interpretation that lower Vp/Vs values at a given differential pressure implies a more consolidated nature of sediment may be explained using figure 6, where the Domenico 

(1977) and Prasad (2002) data are shown. The data set used to 

derive a relationship between Vp/Vs and differential pressure 

by Huffman and Castagna (2001) included the data measured 

by Domenico (1977). Although porosities at low differential 

pressure are similar, the rate of porosity decrease with respect 

to differential pressure in the Prasad data is much higher than 

that of the Domenico data, as previously mentioned. In other 

words, the Prasad sample is softer or easier to compact than 

the Domenico sample. 

Figure 6 shows measured and calculated Vp/Vs

 with 

respect to differential pressure for sediments with differing 

porosities. Clearly, assuming the same value of m, calculated 

Vp/Vs values of higher porosity sediments is larger than those 

of lower porosity sediments (compare the solid and dashed 

lines). This implies that, for sediments having a similar degree 

of consolidation/compaction, the higher the porosity, the 

larger Vp/Vs will be. Also note that Vp/Vs of sediment having a 

porosity of 30 percent with m = 1 (dot-triangle-dot) is almost 

the same as that of sediments having a porosity of 40 percent 

with m = 1.5 (dashed line). This suggests that, as far as Vp/Vs

is concerned, it is difficult to determine whether the increase 

of Vp/Vs is owing to an increase of porosity or to a decrease of 

consolidation/compaction. 

Figure 6 indicates that Vp/Vs

 for Domenico’s data (open 

circles) is less than that for Prasad’s data (solid circles), even 

S-WAVE VELOCITY, IN KILOMETERS PER SECOND

(Vp /Vs)

Vp /Vs

Vp /Vs

m

m

S

Figure 5. Graph showing relationship among differential pressure, 

Vp/Vs

, and the S-wave velocity. Open and solid circles are measured 

Vp/Vs and differential pressure, respectively, for data complied by 

Prasad (2002). 

Discussiom 7
8 Elastic Properties of Overpressured and Unconsolidated Sediments

though porosity values in Domenico’s data are higher than 

those in Prasad’s data, as shown in fi gure 1A. The measured 

Vp/Vs

 shown in figure 6 (large solid circles are data from table 

4 and small solid circles are data from table 3 in Prasad, 2002) 

indicates that the discrepancy between Prasad’s and Domenico’s data increases as differential pressure decreases or the 

porosity of Prasad’s samples approach those of Domenico’s 

(1977). Therefore, figures 1 and 7 suggest that the difference 

in Vp/Vs between the Prasad and the Huffman and Castagna 

data may come from the difference in the degree of compaction, not from the difference in porosity as suggested by Huffman and Castagna (2001). 

Differential Pressure and S-Wave Velocity

In order to incorporate the effect of differential pressure on elastic velocities, a contact model based on the Hertz 

theory has been used. Duffy and Mindlin (1957) applied the 

Hertz theory to face-centered sphere space—most analogous 

to a clean, well-sorted, deep-marine, unconsolidated sand—

and showed that the velocity is proportional to 1/6 power of 

differential pressure. The prediction of the Duffy-Mindlin 

theory is shown in fi gure 4B as dash-dot-dot. The least-squares 

fi tting of S-wave velocities with respect to differential pressure 

is given by Vs = p0.337/0.332. The theoretical prediction of the 

effect of differential pressure on velocity based on the contact 

theory is about one-half of the observed effect. This observation is different from that by Merkel and others (2001), who 

demonstrated that the reservoir rocks, massive turbidites in the 

Gulf of Mexico, behave in accordance with the Duffy-Mindlin 

theory. 

In the BGTL formulation, the effect of differential pressure on velocity is incorporated by using the pressure-dependent exponent n. As indicated in equation 8, n is a function of 

differential pressure and m. As shown in previous sections, the 

data compiled by Prasad (2002) fit the prediction of the BGTL 

with m = 1.0 and with m = 1.3 for the Domenico data (1977). 

Vp /Vs

m

m

m

Figure 6. Graph showing Vp/V with respect to differential pressure. Measured data are shown as open circles (Domenico, 1977) 

and as solid circles (Prasad, 2002). Light small solid circles are for 

all samples by Prasad (his table 3) and large dark solid circles are 

for a subset of Prasad data (his table 4). Modeled velocity ratios are 

derived from BGTL with m = 1 and m = 1.5 with various porosities.

e

e

∂φ/∂p

P

S

Figure 7. Measured and computed velocities from BGTL with respect 

to differential pressure are shown. Measured velocities are for a 

clean sandstone having a porosity of 0.0670 at 5 MPa by Han and 

others (1986). Dotted lines represent predicted velocities using the 

Biot coefficient shown in equation 6 with a constant m = 5 and solid 

lines are with a variable m calculated from equation 9 with φ = 0.06329 

+ 0.00463e–p/21.439.
References 9 

The exponent of 1/6, predicted by Duffy-Mindlin theory, is 

approximately achieved by using BGTL with m = 4. Therefore, the degree of consolidation of reservoir rocks analyzed 

by Merkel and others (2001) may be much higher than that 

analyzed in this paper. 

Clay Effect on Velocity Ratio 

Contribution of clay on Vp/Vs at low differential pressures 

is not analyzed in this paper because of the lack of available 

data. However, a brief and qualitative discussion is included 

in order to show a strange Vp/Vs of shaly sand. Huffman and 

Castagna (2002) indicate that Vp/Vs of sand containing 8 percent clay by weight is lower than that of clean sand—they 

interpreted that small amounts of clay act to bind loose sand 

grains, thereby decreasing Vp/Vs at a given pressure. Usually, 

clay in consolidated sand decreases S-wave velocity more than 

P-wave velocity, so clay increases the Vp/Vs

 of the sediment 

(Blangy and others, 1993; Lee, 2003). This is opposite to the 

observation by Huffman and Castagna (2002). Clay in sediments affects the rate of porosity reduction with respect to differential pressure as well as porosity itself; thus, both effects 

contribute to the variation of Vp/Vs. Therefore, without detailed 

knowledge about the porosity and porosity reduction of shaly 

sand, it is not conclusive whether clay increases or decreases 

Vp/Vs of unconsolidated sediments with respect to differential 

pressure. 

BGTL Parameter m 

As shown in figure 5, determination of an accurate m is 

important to reduce the uncertainty of the estimated differential 

pressure. As mentioned previously, m appears to depend on 

∂φ/∂p, and Prasad (2002) and Domenico (1977) data suggest 

that the proposed equation 9 for m is adequate. Figure 7 shows 

velocity versus differential pressure for a clean sandstone 

sample having a porosity of 0.067 at 5 MPa from Han and others 

(1986). Predicted velocities using BGTL with m = 5 agree well 

with the measured velocities for a differential pressure above 

20 MPa, but velocities are higher than measured velocity at 

low differential pressures. This is caused by using a constant 

m irrespective of ∂φ/∂p. By fitting the porosity with respect to 

differential pressure as φ = 0.06329 + 0.00463e –p/21.43901, a 

variable m using equation 9 is attempted to examine whether 

the proposed equation for m is really a function of ∂φ/∂p. 

Figure 7 shows the predicted velocities using a variable m, 

e.g., m = 5.11 at p = 40 MPa and m = 3.44 at p = 10 MPa, as 

solid lines. Predicted velocities with variable m are still higher 

than measured velocities at low differential pressure, but the 

predicted rate of velocity change with respect to differential 

pressure agrees reasonably well with the measured rate. 

The above analysis supports the idea that the constant 

m is a function of ∂φ/∂p. However, because ∂φ/∂p is rarely 

known in practice and because equation 10 for m is based on 

only three samples, it is difficult to use equation 10 directly. A 

proper choice of n and m is important to use BGTL effectively, 

and equations 8 and 9 provide general guidelines for the estimation of n and m. 

Conclusions 

Biot-Gassmann theory by Lee (BGTL) accurately predicts the velocity ratio (Vp/Vs

 ) with respect to differential 

pressure by using a variable exponent in the form of n = 

[10a+blog(p)]/m. Pore pressure or differential pressure can be 

estimated from Vp/Vs or the S-wave velocity by utilizing the 

fact that the Vp/Vs ratio with respect to the S-wave velocity 

depends strongly on differential pressure and is less sensitive 

to porosity and clay content. A proper choice of the constant 

m is important to reduce the uncertainty of the estimated differential pressure. As m increases, Vp/Vs

 decreases at a given 

differential pressure and appears to depend strongly on the rate 

of porosity reduction with respect to differential pressure. 

For differential pressures greater than about 0.5 MPa , 

BGTL with m = 1 accurately predicts the Vp/Vs

 ratio for measured data complied by Prasad (2002), and BGTL with m = 

1.3 is satisfactory for data analyzed by Huffman and Castagna 

(2001). Whether or not the prediction of Vp/Vs by BGTL is 

more accurate than that predicted by empirical relationships at 

low differential pressures is not clear because of the paucity of 

measured data at low differential pressure (less than 0.5 MPa). 

To use BGTL effectively, a proper choice of the exponent 

n is important, which can be regarded as a free parameter to fit 

observations. However, this study shows that general guidelines based on differential pressure and the degree of compaction work well for a variety of data sets. 

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Manuscript approved for publication August 28, 2003 

Published in the Central Region, Denver, Colorado 

Editing, layout, photocomposition—Richard W. Scott, Jr. 

Graphics by the author