  We construct two types of unital separable simple $C^*$-alebras $A_z^{C_1}$
and $A_z^{C_2},$ one is exact but not amenable, and the other is non-exact.
Both have the same Elliott invariant as the Jiang-Su algebra, namely,
$A_z^{C_i}$ has a unique tracial state, $$(K_0(A_z^{C_i}), K_0(A_z^{C_i})_+,
[1_{A_z^{C_i}} ])=(\mathbb Z, \mathbb Z_+,1)$$ and $K_{1}(A_z^{C_i})=\{0\}$
($i=1,2$). We show that $A_z^{C_i}$ ($i=1,2$) is essentially tracially in the
class of separable ${\cal Z}$-stable $C^*$-alebras of nuclear dimension 1.
$A_z^{C_i}$ has stable rank one, strict comparison for positive elements and no
2-quasitrace other than the unique tracial state. We also produce models of
unital separable simple non-exact $C^*$-alebras which are essentially tracially
in the class of simple separable nuclear ${\cal Z}$-stable $C^*$-alebras and
the models exhaust all possible weakly unperforated Elliott invariants. We also
discuss some basic properties of essential tracial approximation.
