Solution with equilibrium

Solution for xe and ze → ∞

Sensitivity tests of morphologic breach model

Kraus2_CEJ_03
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which, for a small initial perturbation, yields xy ∼ 2at, a special case of Eq. (8).

Equations (9) and (10) make apparent the significant role the dimensions of the

initial perturbation take on the course of breach development. Although Eqs. (6)

(10) govern incipient breaching, they have limited value in being restricted to very

short elapsed time. More informative solutions are obtained by solving Eqs. (4) and

(5), which incorporate modification and control of breach growth by the inclusion

of equilibrium.

3.2. Solution with equilibrium

Closed-form solution of Eqs. (4) and (5) is not found feasible for arbitrary values of

the factors a and b, but, for the situation of a = b (signifying QS = QB = Q), the

following solution emerges:

x = xe(1 - f (x)e-t/τ )

(11)

and

z = ze(1 - g(z)e-t/τ )

(12)

in which,

τ=

(13)

where Ve = xezeL is the volume of the breach at equilibrium, and

1/α

- αx/xe

f (x) =

(14)

1 - αx0/xe

x0/xe - z0/ze

α=

(15)

(1 - z0/ze)x0/xe

1/β

- βz/ze

g(z) =

(16)

1 - βz0/ze

z0/ze - x0/xe

(17)

β=

(1 - x0/xe)z0/ze

Iteration is required to evaluate Eqs. (11) and (12) because of the appearance of the

dependent values in Eqs. (14) and (16). Equations (4) and (5) can also be solved

numerically, as through a Runge-Kutta procedure, and this was done to confirm the

validity of the closed-form solutions above, as well as to perform calculations for

a = b.

In addition to making basic dependencies of breach growth explicit, Eqs. (11) and

(12) provide closed-form solutions with which to check numerical solutions of the

simultaneous, non-linear governing Eqs. (4) and (5). This simple morphologic model

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