Canadian Occidental Petroleum Ltd.modified theFORTRAN program. The first author also would like to thank R. James Brown for showing the theory explained ...
291
Physical parameter estimation for sandstone reservoirs Taiwen Chen, Donald C. Lawton and Fred Peterson
ABSTRACT
Through theoretical derivation and numerical modeling, this report shows that BiotGassman's theory can be used for predicting some physical parameters (e.g., velocity, density, Poisson's ratio, bulk modulus) as functions of porosity or water saturation. Also, it is shown that oil and gas have different Poisson's ratio curve as a function of water saturation. Several plots in this report suggest the importance of obtaining good initial velocity, porosity and dry rock Poisson's ratio estimates when modeling gas-water or oil-water saturated reservoirs. INTRODUCTION
The Biot-Gassman theory has drawn many authors' attention (Geertsma and Smit, 1961: Domenico, 1974: Gregory, 1977: Hampson and Russell, 1990). Gregory (1977) suggests that the Biot-Gassman theory is a technique through which some physical parameters (e.g., velocity, density, Poisson's ratio, bulk modulus) as functions of porosity or water saturation can be obtained. Based on this theory, some modified equations were developed in order to obtain Poisson's ratio for AVO (amplitude versus offset) studies. Comparing the Biot-Gassman's equations (Gregory, 1977) with modified ones, one may feel that the modified equations are a more direct approach to get some parameters such as Poisson's ratio. Several numerical models show the application of this theory. THEORY
The main issue of the modified Biot-Gassman theory is to obtain Poisson's ratio ((r) as a function of porosity ((_)or water saturation (Sw). Following are the list of symbols: V Vp Vs
- velocity of the rock; - P-wave velocity; - S-wave velocity;
Vp0 M Md M0
-
initial P-wave velocity (at _0 and Sw0); space modulus (or P-wave modulus); dry rock space modulus; initial space modulus;
- rigidity modulus (or S-wave modulus);
292
lad 6 ¢_d P
- dry rock rigidity modulus; - Poisson's ratio; - dry rock Poisson's ratio; - bulk density;
Po
- initial bulk density (at _o and Swo);
Ps Pf
- bulk density of solid matter;, - bulk density of fluid;
Pro
- initial bulk density of fluid (at ¢0 and Sw0);
Pw Ph
- bulk density of water;, - bulk density of hydrocarbon (oil or gas);
¢
- porosity;
¢_0 Sw Sw0 K Kd Kd0 Ks
-
initial porosity; water saturation; initial water saturation; bulk modulus; dry rock bulk modulus; initial dry rock bulk modulus; solid matter bulk modulus;
Kp Kf Kw Kh C Cd Cs Cf Cw Ch
-
pore bulk modulus; fluid bulk modulus; water bulk modulus; hydrocarbon (oil and gas) bulk modulus; compressibility (reciprocal of the bulk modulus); dry rock compressibility; solid matter compressibility; fluid compressibility; water compressibility; hydrocarbon (oil or gas) compressibility;
It is well known that a can be obtained by:
(___p)2_ 2 6=_(_--_Ps)2-1]
(e.g., Sheriff, 1991),
(1)
VP M - C
(e.g., Gregory, 1977),
(2)
Vs = "_/g-¥ P
(e.g., Sheriff, 1991),
(3)
and Vp, Vs can be obtained:
Therefore, o can be obtained if M, I.tand p are a/l known.
293
Calculation
of p
It is known that: p = (l-¢)ps + Cpf, and pf = Swpw+ (1-Sw)Ph
(e.g., Gregory, 1977),
then we have: p = (1-¢)ps + ¢SwPw + ¢(1-Sw)Ph,
(4)
Ps, Pw and Ph are known. Equation (4) tells us that bulk density p is the function of porosity ¢ and water saturation Sw. Calculation
of M
Geertsma and Smit (1961) gave the equation to obtain space modulus (M).
KdZ (1-.v-) M=Md +
Ks
(5)
¢ +I-¢ Kf
Ks
Ks
Gregory (1977) gave the equation to get Kf: Cf = SwCw + (1-Sw)Ch,
and
1 =Sw.¢ Kf Kw Kf =
1-Sw Kh '
1 Sw + 1-Sw" Kw Kh
(6)
In equation (5), Md=4_-_ + Kd
(e.g., Sheriff, 1991),
3(1-2Od)KA I_ = 2(]_a)_'_
(e.g., Sheriff, 1991),
and
then (5) becomes:
Kd 2 (I- _--_-s ) M =_d
+Kd t
¢+__1-¢ Kd Kf K_ K_
294
(1 M- 4x3 l'2_dKd + Ka I
Ks)2
-72 l+od
e 1-¢ Kd' Kf
(1 - K_?2
M=SKi•
Ks
K2
,
1-¢ Kd Kf
Ks
(7)
K_
here S = 3(1-t_d) l+_d " In equation (6), Kw, Kh are known. For example, 2.38 x 10a°dynes/cm 2 and 0.0208 x 1010
dynes/cm 2 are suggested values for bulk moduli of water(Kw), and gas, respectively
(Hilterman; Hampson and Russell, 1990). Hence, Kf is a function only of Sw. In equation (7), Ks, _d are known.
Values of 40.0 x 10l° dynes/cm 2 and 0.12 are suggested for sandstone
Ks and ad, respectively (Hilterman; Hampson and Russell, 1990). From equation (5) and (6), it is clear that space modulus M is a function of porosity _, water saturation Sw and bulk modulus of dry rock Kd. However,
Kd can be obtained by knowing Kd0. From Appendix A, we can obtain:
Cd = 0Cp + (l-O)Cs, then 1=¢ +1-¢ Ka Kp K's
Kd-
1 ¢ +--__¢ Kp
(8)
Ks
similarly, Kd0 =
I
¢o-I--1-Oo
(9)
Ks
Kp
From equation (9), we can obtain Kp.
_0 Kp=
.
Kao
Ks
(10)
295
Substituting (10) into (8): Kd=
1
1-¢"
_._..(1 ¢0 _
(11)
1-*0) ._ Ks Ks
Equation (7) can be used to solvc for Kdo. For the initial porosity _o, initial fluid hulk modulus Kf0, initial bulk modulus of the dry rock Kdo and initial space modulus Mo, equation (7) becomes:
(1 - ._--_°) 2 Mo = SKd0 t
Ks
,
(12)
¢o _1-¢o Kao Kro
Ks
Ks2
to separate the factor Kao in equation (12):
MO(K_ m + 1-¢o. __, ¢0 + 1-¢0 Ks Kd0) K_ =o°Kao___.fo Ks Kd0) K_ +(1 - Kd0)2, Ks further written as: S-_(Kdo)2
Mo S¢o S(1-0o)],_
+[MoOo+Mo(1-¢o) 1 =0,
then we have: A(I_o) 2+BI_o+C=0, where: A=S-1.
K_' B=
2 .Mo.S¢° S(1-¢o). Ks Ks2 Kf0 Ks
C = M0¢0 4 M0(1-_) -1; Kfo Ks 3 1-_d;
S = y-+b--rid Kf0 =
1
Swo + 1-Swo" Kw Kh
(13)
296
Mo = (Vpo)2po; and Po = (1-¢o)ps + ¢oSwoPw+ ¢o(1-Swo)Ph. By solving the equation (13), we can get Kd0: Kd0-
-B ___/-_- 4AC 2A
04)
Knowing Kd0 from equation (14), and by substituting (11) into (7), we can see clearly that M is only the function of porosity _ and water saturation Sw.
Calculation
of p.
Gregory (1977) gave the equation: 3(1-2(_d)v p.=gd=_X,.d.
(15)
¢_dis known, and from equation (11), Kd is the function of porosity _. Hence, p. is the function of porosity _ only. In equations (2) and (3), M, p and }.tcan be obtained by equations (7), (4)and (15), respectively.
From the above, it is clear that P-wave velocity (Vp) and S-wave velocity (Vs) are
the functions of porosity _ and water saturation Sw. By giving one value of ¢ or Sw, we can get the curves of Vp (Vs) versus Sw or _. Then from equation (1), Poisson's ratio (¢_)versus Sw or curves can be obtained. DISCUSSION
Figure 1 and Figure 2 are two examples for the application of the Biot-Gassman theory for water-gas and water-oil saturated sandstone. Parameters (most values were suggested by Hampson and Russell, 1990) are listed in Table 1. Table
Water-gas Water-oil
1:
Physical
K(x 1010 dynes/cm 2) Ks Kw Kh 40.0 2.38 0.0208 40.0 2.38 1.0
Parameters
Ps 2.65 2.65
used for Sandstone
P(g/cm3) Pw Ph 1.089 0.103 1.089 0.75
(_d
Vp0
Sw0
1_0
0.12 0.12
(m/s) 2500 3000
1.0 1.0
0.33 0.33
3000
"
7000 I_rmlty
P-wave
,
= :_1%
-
6000 _,
2000
Wat_
Saturation.
100%
_._ 4000
1500 •
1000 2500
oo
1000
012
014 Water
016
018
Lo
"
'
0.0
0.2
0.4
0.6
0.8
i.o
Saturation Porosity
04"
0.4 •
I_rmlly.
_
0
33%
0.2'
•
0.2-
0.3"_
i
03-
Oi
o.o
o12
_,_ MD
oi,_
olo
o18
_.o
OI
o.o
o12
Water Saturation
Figure 1: Water-gas Saturated Sandstone
o'.4
oi_
Porosity
o'.8
,.o
6000,
Por_lty
3500
5000 "
= 33%
•
_
P-wave
_ 2500
3000 -
2000
1500 0.0
0:2
, 0.4 Water
016
0:8
1000 0.0
1.0
_
0.22 '
0.18
'
0.0
0:4
0:6
0]8
1.0
Porosity
0.3
0.24 •
0.20
0:2
saturation
0,28 •
"_
-
012
: ,
0.4
Water
0:6
_
ha
0.2
.
o'.8
,.o
O,
oo
0:2
,
o._
Saturation
Porosity
Figure 2: Water-oil Saturated Sandstone
0:6
0:8
i.o
299
Shown in the left side diagrams of Figure 1, it is clear that a small percentage of gas in a sand has a strong effect on the Vp and a, which is consistent with Ostrander's (1984) and Hilterman's results. Oil, however, does not have the same effect. In the left side diagrams of Figure 2, Vp and ff change smoothly with the change of Sw; P increases with the increasing of Sw, thus causes the Vs to decrease with increasing Sw, which can be seen in the plots on the upper left hand side both in Figure 1 and Figure 2. Generally Vp and Vs decrease with the increasing _, increases with the increasing of _. These can be seen in the fight-hand side diagrams of both Figure 1 and Figure 2. Also, it can be noticed that o changes nonlinearly with the change of _ for small values of 0.
Effect of Initial
Estimation
of Velocity
From the theoretical derivation, it is clear that initial velocity Vp0 is one of the important input parameters for the calculation of Poisson's ratio. Therefore, it is needed to know how would the initial estimation of velocity affect the final results. Figure 3 shows the plots of three models with Vpo changed (arbitrary values) and other parameters kept constant. were suggested by Hampson and Russell, 1990) are listed below:
Parameters (most values
p(g/cm3): Pw = 1.089, Ph = 0.75, Ps = 2.65; K(x 101° dynes/cm2): Kw = 2.38, Kh = 1.0, Ks = 40.0; (_d= 0.12, Sw0 = 0.30, _0 = 0.15;
Vp0(m/s): Vpm = 4000, Vp02 = 3600, Vp03= 3200. The plot on the upper left hand side of Figure 3 represents the Vp change with the change of Sw for different Vpo, and the plot on the upper right shows the Vs change. From these two, it is clear that for a given Sw, decreasing Vp0 will decrease both Vp and Vs- t_ increases with the decreasing Vpo for a given Sw, which can be seen on the lower left hand side plot. The lower right hand plot shows that with the decreasing of Vp0, Kd decreases. compressible, in other words, less compatible. Effect of Initial Estimation
This means that the rock is more
of Porosity
Initial estimation of porosity (¢0) is another important factor for the calculation.
Figure 4
shows the plots of three models with O0 changed (arbitrary values) and other parameters
5000 3200 -
----o---VpO =4000 VpO =3600 _, 4500
Pore_ty=15%
3000-
-_
_
------o---VpO = 4000 _ VpO=3600
Por_s_ty=15%
2800 -
VpO=3200
"--'_--
VpO= 3200
2400 _
__.c_.c._e...o.__e--.e-.-c_c.r" 3000
0.0
012
'
04
'
0.6
018
20001800
1.0
oo
WaterSaturation
,
012
04
0'.6
018
10
WaterSaturation
0.34.
2OO £.o
0.30"
o---- VpO=4oO0 * VpO= 3600 , ,[7---VpO= 3200
Poro_ty = 15%
._.$
cr"
_" _
_ 180
_
160
o
o 0
VpO=4oo0 VpO= 3600 VpO 320O
0
i
0.26" 120
o.18.
"g
0,22' ,.._.,._,__
_,_
0.14
oo
012
014
016
Water Saturation
&
1.0
_
ioo 140 80
0.0
o12
o14
o16
Water Saturation
Figure 3: Effect of Initial Estimation of Velocity
o18
,0
2320 ----o---
3800 •
--
{
_
I.l_-asity=2O%
•
/
• LPormlty=15%
_-
_
jCj
_ 3600 3700 y
0'.6
018
_
2180 2240'! 2160" "-°" _+_d"_'_ 2140
014
.
LPereslty=15% L pe.re_ty = 20% LPorostty=10%
2200
_
012
*
2220 - _ 2260-
_
].l:'oresity:lO%
_,
_00 0.0
-_
2300 2280
_.0
0.0
_'_"a"'>"t_'t_'_
0'.2
014
0'.6
0'.8
,.0
Water Saturation
Water Saturation 0.28 0.26'
_ ----o---
L porosl_ pere*l_ == 20% _% L
,_.f,_.,,1
_
0"22I
_
I 17
_
L Porosity = 20%
--o--e --
1. por_ty = 10% L poro61ty= 15%
o
I 15
0.20' 0A8 2 0.24- _ 0.16 0,0
"
_
0',2
" 0.4
I
0.6
I
1.0
0"0
0.8
0_2
014
Water Saturation
Figure 4: Effect of Initial Estimation
0"6
Water Saturation
of Porosity
. 0"8
I'O
302
unchanged. below:
Parameters (most values were suggested by Hampson and Russell, 1990) are listed
p(g/cm3): Pw = 1.089, Ph = 0.75,Ps= 2.65; K(× 10I°dynes/cm2): Kw = 2.38,Kh = 1.0, Ks = 40.0; crd = 0.12,Swo = 0.30,Vpo = 3600m/s;
(_ot = 0.20, 002 = 0.15, (_03= 0.10. From the two upper plots, we can see that there is a crossing of the Vp curve with Sw, and for a given Sw, Vs decreases with the decreasing value of t_o. The change in Vs is greater than that of Vp, therefore for a given Sw, _ increases with the decreasing value of _0. Effect of initial estimation
of dry rock Poisson's
ratio
Figure 5 are plots of three models with (_dvarying (arbitrary values) while all other parameters remain constant. Parameters (most values were suggested by Hampson and Russell, 1990) are listed below: p(g/cm3): Pw = 1.089, Ph = 0.75, Ps = 2.65; K(x 101° dynes/cm2): Kw = 2.38, Kh = 1.0, Ks = 40.0; Swo = 0.30, Vpo = 3600 m/s, _0 = 0.15; t_d: Odl
=
0.12, t_d2 = 0.14, t_d3 = 0.16.
From the two upper plots in Figure 5, it is obvious that Vp remains almost constant with the change of ad, while Vs decreases with the increasing _d. Hence, for a given Sw, _ increases with the increasing value of od. The lower right hand side plot shows the bulk modulus increasing with the increasing t_a. This infers that the rock is more compacted (or less compressible). Assumptions
for Biot-Gassman
Theory
The Biot-Gassman theory is only used for water-gas and water-oil saturated reservoirs, but it is not valid for water-oil-gas saturatedreservoirs. Also, as Hilterman cited in his notes, the Biot-
2280.
3750
i_"
----o---•
P.R. (dry rock)= 0.12 P.R.(dryro_k)=0.14
_
_
2200.
-----o----
P.R. (dry rock) = 0.12
2240.
_
•
P.R. (dry rock) = 0.14 P.R. (dry rock) = 0.16
os 2200. _
3650
- -
2100.
3550
0.0
012
014
0;6
018
2140
_.0
0.0
_-_.__
012
Water Saturation
,
0._
01_
,
0.6
_.0
Water Saturation
Iz;0 0.26.
o---•
P.R. (dry rock) = 0.12 • ----c_---
135,
P.R. (dry rock) =0.14 P.l_ (dry rock) = 0.16
0.24"
_
_ R
J
to o t.o
P.R. (dry rock) = 0.12 P.R. (dry rock) = 0.14
----_--"P.R. (dry rock) = 0.16 130'
.¢_125, i_
0.22, --_
:::: :
: : : : : : : : : : : : : : ::
]15'
_
0.20. _ 0.16
0.0
"_ _ =
0:2
, 0.4
0;6
Water Saturation
0:6
_.o
_
IlO
o.o
0:2
, 0.4
0;6
Water Saturation
Figure 5: Effect of Initial Estimation of Dry Rock Poisson's Ratio
o;8
,.o
304
Gassman's theory assumes a homogeneous rock in which the pore fluid is uniformly distributed in the pores, the shear modulus is not affected by the pore fluid, and the pore shapes are spheroidal. CONCLUSION
Four conclusions can be obtained from this report: (1): The Biot-Gassman theory can be used for predicting physical parameters (e.g., veloci!y, density, Poisson's ratio, bulk modulus, etc.) as a function of porosity or water saturation. (2): Oil and gas have different Poisson's ratio change curve as a function of water saturation for a given porosity. (3): When modeling gas-water and oil-water saturated reservoirs, it is important to obtain good estimation of initial velocity, initial porosity and dry rock Poisson's ratio. (4): Generally, decreasing the initial P-wave velocity or initial porosity, or increasing dry rock Poisson's ratio, and keeping the rest of the parameters constant, will cause Poisson's ratio to increase for a given water saturation. ACKNOWLEDGEMENT
This study was initially suggested by Brian Russell of Hampson-Russell Software Services Ltd. We wish to give our special thanks to him for his suggestions and kind help. Marco Leal of Canadian Occidental Petroleum Ltd.modified the FORTRAN program. The first author also would like to thank R. James Brown for showing the theory explained in Appendix A. REFERENCES
Domenico,S.N.,1974,Effectof watersaturationonseismicreflectivityof sandreservoirsencasedin shale: Geophysics, 39, 759-769. Gardner,G.H.F.,Gardner,LW., and Gregory,A.R., 1974,Formationvelocityand density- the diagnosticbasics for stratigraphic traps: Geophysics, 39, 770 - 780. Geerstma, J., and Smit, D.C., 1961, Some aspects of elastic wave propagation in fluid - saturated porous solids:
Geophysics, 26, 169-181. Gregory,A.R., 1977,Aspectsof rockphysicsfromlaboratoryand logdatathat are importantto seismic interpretations: AAPG Memoir, 26, 15-46. Hampson, D., and Russell, B., 1990, AVO: Amplitude versus offset analysis: handbook for AVO version 4.1. Hilterman,F., Gassman-Biot-Geertma Equation: unpublishedcoursenotes. Russell, B., 1990, The theory of the amplitude versus offset analysis program: unpublished notes. Sheriff,R.E., 1991,Encyclopedicdictionaryof explorationgeophysics,third edition:SEG.
305
APPENDIX
A
According to the concept of the compressibility: C = AT?Z = AT., AP ZAP
(A-l)
i.e., the compressibility (C) is the relative change in volume (ATJZ) with pressure (AP). AZ is the absolute change in volume, and Z is the total volume of the rock. The total change in volume AZ is equal to the change in volume of the pore space (AZp) plus the change in volume of the solid matter (AZs), i.e.: AZ = AZp + AZs,
therefore:
Am_Amp AZs ZZ 4 Z •
(A-2)
According to the concept of the porosity: Zp = 0Z, Zs= (I-,)Z, Z= Zp, Z=
¢
Zs
1-¢'
where Zp is the total volume of pore space and Zs is the total volume for solid matter. equation (A-2) becomes:
Then
AZ = CAZp _ (1-t_)AZs Z Zp Z_ ' divided
by Ap:
AZ _ OAZp + (1-_b)AZs ZAP ZpAP ZsAP comparing the above equation with equation (A-I), we can get: C d = _bCp +
(1-¢)Cs).
(A-3)
Equation (A-3) was used to obtain Kd in equation (8). Equation (A-3) can also be written as:
Ca= ¢Cp+ Cs- ¢cs. Cs is quite small compared to Cp, and t)Cs is very small compared with both @Cpand Cs. Therefore the above equation can be simplified as: Cd = ¢Cp + Cs. Equation (A-4) is the same equation given by Domenico (1974).
(A-4)
