Kinematic Wave Theory - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 69

This “analytical continuation” eliminates the subscript 0s, and allows us to
return Equations 2.90 and 2.91 to functional forms having complete k' s. The
generalized phase (or eikonal ) relation that results is
r (k) a X i k (k)/a + 1/2 k X i kk - 1/2 2 a XX r kk /a
- 1/2 2 a X k X r kkk /a - 1/8 2 k X 2 r kkkk - 1/6 2 k XX r kkk (2.92)
Since i is small, the terms on the first and second lines of Equation 2.92 other
than r (k) itself are O( 2 ). Those on the first line represent high-order effects
due to dissipation; those on the second are high-order effects due to dispersion.
The corresponding equation governing a 2 takes the form

a 2 / T + (

r k a 2 )/ X =

2

i (k)a 2 /

(2.93)

2 (aa XX i kk / + aa X k X i kkk /

-

+ 1/4 a 2 k X 2

i kkkk /

+ 1/3 a 2 k XX

i kkk / )

+

2 (1/3 aa XXX r kkk + 1/2 aa XX k X r kkkk

+ 1/3 aa X k XX

r kkkk

+ 1/12 a 2 k XXX

r kkkk

+ 1/6 a 2 k X k XX

r kkkkk + 1/4 aa X k X 2 r kkkkk

+ 1/24 a 2 k X 3

r kkkkkk )

The left side of Equation 2.93 appears in conservation form . The right “sink-
like” side indicates the expected “2 i (k)a 2 / ” damping, but higher order terms
related to wave dissipation and dispersion are both present.
Equations 2.92 and 2.93 appear in slow (X,T) variables. Returning to
physical coordinates (x,t), we have, to two higher orders beyond classical ray
theory (with O(

i (k)/ r (k)) << 1), the phase relationship

r (k)

a x

i k (k)/a + 1/2 k x i kk - 1/2 a xx r kk /a

(2.94)

r kkk /a - 1/8 k x 2 r kkkk - 1/6 k xx r kkk

- 1/2 a x k x

for the slowly varying wave, to be contrasted with the dispersion relationship
(k) r (k) i i (k) given by Equation 2.71, which only applies to the
uniform plane wave . We also obtain the generalized amplitude equation

a 2 / t +

r k a 2 )/ x = i (k)a 2

( 2 (2.95)

- (aa xx

i kk + aa x k x i kkk

+ 1/4 a 2 k x 2

i kkkk + 1/3 a 2 k xx i kkk )

+ (1/3 aa xxx

r kkk + 1/2 aa xx k x r kkkk

+ 1/3 aa x k xx

r kkkk + 1/12 a 2 k xxx r kkkk

+ 1/6 a 2 k x k xx

r kkkkk + 1/4 aa x k x 2 r kkkkk

+ 1/24 a 2 k x 3

r kkkkkk )