Page 88 - Well Logging and Formation Evaluation

78 Well Logging and Formation Evaluation

M h = M ◊ exp - ( t T )
0
2
This pulse scheme is referred to as a CPMG (Carr, Purcell, Meiboom,
and Gill) excitation. In practice, not all the fluid in the pores will relax
according to the same T 2. Those lying close to the pore wall will relax
more quickly than those in the center of the pore. This means that there
is a series of contributions to M h, each decaying at a different rate. In addi-
tion, because of diffusion of the nuclei within the pores, nuclei that may
not initially be close to the wall may move toward the wall during the
measurement and relax more quickly. This introduces an extra term into
the behavior of M h, given by:

2
2
(
◊
Exp -◊ g 2 ◊ D G t 3 )
◊
t
where D = molecular self-diffusion coefficient and G = gradient of the
static magnetic field. Hence the full expression for M h is given by:
2
M h = M ◊ exp - ( t T )◊ Exp ( t - ◊ g 2 ◊ D G ◊ t 2 ) 3 .
◊
0
2
Because different fluids (oil, gas, water) have different values of D, if
measurements are done at different values of t, there is the possibility of
differentiating fluid type. This is the basis for what is called time domain
analysis (TDA). The influence of the pore wall on T 2 is assumed to follow
a relation of the form:

1 T = 1 T bulk + ◊ S V
r
2,
2
where T 2 , bulk = relaxation time of bulk fluid, r= surface relaxivity, and
S/V = pore surface-to-volume ratio. For spherical pores of radius r, S/V
reduces to r·3/r.
Note that the expression for capillary pressure, P c , is given by:

( ))
P c =◊ ( s ◊ cos q r.
2
Hence it can be shown that if T 2 , bulk >> r·S/V, which is the case close
to the pore wall, then:

PT = (s cos ( ) 15q . ◊ ) r .
◊
◊
c
2
This is important because it shows how a cutoff between bound and
free fluid, made on the basis of P c , can be translated into a cutoff based