Numerical Simulation of Bentonite Extrusion Through a Narrow Planar Space
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a,(e) is the coefficient of compressibility [m2/N], which is defined as a, = -de/da'. The effective stress a' [N/m 2] is considered to be carried by the solid skeleton of porous bentonite. For water-saturated bentonite, (Y'is equal to the swelling pressure, which results from the effects of electrical repulsion and attraction between bentonite particles. A plot of e versus log a' is approximately a straight line [3]. a,(e) and k(e) can be approximated in a form of an exponential function of e [2] (See Table I for the assumed functional forms of k(e) and a,(e)). From the mass conservation of water in the region R(0) < r < R(t), shown in Figure 2, the equation for the location of the tip in the gap is written as
dR(t) dt
100). O-C
Figure 1
-
[D(e) l , for t>O. L& r=R(t)
(2)
I
Geometrical definition of the problem and boundary conditions. R = R(t)
R = R(O)
z
1 AV. = 21rR(0)Lq. n r=R(O)At
.4 -
2rR(t)Lq. n4,=-(, At
SAV, of bentonite with e, Gap L
Water = AV,, +e,, Solid = AV,
I I+e
Bulk bentonite
Figure 2
Assumed mass flow at the inner boundary at r = R(O) where the gap intersects with the bulk bentonite, and at the outer boundary, r = R(t). q is the Darcy flux, and n is a unit normal to the boundaries at r = R(O) and r = R(t). AVw(t) is the volume of water crossing the boundary at r = R(O) from the gap to the bulk bentonite during the time interval At. 186
NUMERICAL SOLUTION SCHEME The numerical algorithm to calculate e(rt) is based on a fully implicit Galerkin-Finite Element technique combined with a predictor-corrector scheme. The temporal accuracy is controlled by automatic time step modification. The void ratio e(rt) is approximated by j(r,t), a sum of N shape functions Ok(r) weighted by N nodal values ek(t), k = 1, 2 ... , N. ek(t) satisfies the following system of non-linear, first-order ordinary differential equations in time, (3)
M(t)_(t) + K(t)e(t) = F(t),
where e(t) is the nodal value vector composed of ei(t), k = 1, 2 ... , N. The dot symbol denotes the derivative with respect to time. The double underline denotes a matrix. Elements of M, K, and F are written as R(t) I RMt) d4i Okd ar .' - --R(4) (4) k andf 1+- Okrdr, Kk(t)= r f Mi(=
J
dr dr
R(O)
R(O)
drkJ=R(O)
With linear shape functions, only the terms for k = i-1, i and i+1 in the matrices Mik and are non-zero. Only the two extreme terms for i = 1, and N in Fi are non-zero. The motion of the tip reads :
R(t) = D(e
d4N)
dr
"* "* "•
"* "* "*
+ r=R(e)
eN]
N
dr
for t.O.
Kik
in (3)
(5)
r=R(t)
We solve (3) and (5). The scheme is described as follows: From the knowledge of the tip position R(t,) and the void ratio e(tj) at time t,, R(t,, I) and e(t.+1 ) at time t,,+I are predicted by the Adams-Bashforth formula [4]. The internal nodes are relocated by the predicted tip position obtained in the previous step. The nodal vector e(t,,+,) of the void ratio on the relocated finite-element grid is computed by the implicit trapezoid rule. The nonlinear algebraic system is solved by the Newton-Raphson iterative technique with
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