Page 119 - Well Logging and Formation Evaluation

Integration with Seismic 109

Definitions:
K o,g,w = bulk modulus of oil, gas, water measured, in Pascals (Pa)
K F,init,final = bulk modulus of combined fluid measured, in Pa
K matrix = bulk modulus of matrix, in Pa
U matrix = shear modulus of matrix, in Pa
K grain = bulk modulus of individual grains, in Pa
S oi,gi,wi = initial saturation of oil, gas, water (as fraction)
S of,gf,wf = final saturation of oil, gas, water (as fraction)
RHOF init,final = initial/final combined fluid density, in g/cc
RHO oil,gas,water = fluid density of oil, gas, water, in g/cc
Por = porosity (as fraction)
V Pinit,final = initial/final compressional velocity, in m/s
V Sinit,final = initial/final shear velocity, in m/s
V Pinit = 1e6/(3.281*DTP), in m/s
V Sinit = 1e6/(3.281*DTS), in m/s
-2
AI init = 1000*RHOB init *V Pinit , in kg·m ·s -1

For the following, a gas/water system is assumed, but the method works
equally well with oil/water.

K Finit = 1/[S gi /K g + S oi /K o + S wi /K w] [K in Pa]
RHOF init = RHO gas *S gi + RHO water *S wi + RHO oil *S oi [RHO in g/cc]
Por = (RHOM - RHOB init )/(RHOM - RHOF init )
VF init = sqrt(K Finit /RHOF init )/30.48
X 1 = RHOB init *((V Pinit *30.48)Ÿ2 - 1.3333*(V Sinit *

30.48)Ÿ2)/K grain
X 2 = 1 + Por*K grain /K Finit - Por
K matrix = K grain *(X 1 *X 2 - 1)/(X 1 + X 2 - 2)
U matrix = RHOB init *(V Sinit *30.48)Ÿ2 [matrix shear modulus]
X 3 = U matrix /(K matrix *1.5)
Matrix Poisson ratio (Mpoi) = (1 - X 3 )/(2 + X 3 )

Having determined K matrix and U matrix , the new V p and V s can be deter-
mined as follows:

RHOF final = RHO gas *S gf + RHO water *S wf + RHO oil *S of
RHOB final = RHOM*(1 - Por) + Por*RHOF final
K Ffinal = 1/[S gf /K g + S of /K o + S wf /K w]
V Ffinal = sqrt(K Ffinal /RHOF final )/30.48
Beta = K matrix /K grain