Furthermore, we can calculate the amounts of the exotic matter threading the wormhole by using the “volume integral quantifier ” (VIQ) method. It is noteworthy that the amounts of exotic matter can be described effectively by the definite integral $\int T_{\mu\nu}k^\mu k^\nu dV$ and that here we just consider the finite range of the traversable wormhole, $r\in[r_0, r_m]$. Subsequently, using the corresponding quantity $I_V=\int[p_r(r)+\rho]dV$, we can obtain the following relation: $$I_V=\int^{r_m}_{r_0}(r-b)\left[\ln\left(\frac{e^{2\Phi}}{1-b/r}\right)\right]'dr. \label{16}$$ It follows that $$I_V=\frac{[1+\omega_{eff}(z)](r_0-r_m)}{\omega_{eff}(z)}. \label{17}$$ Obviously, this physical quantity is sensitive to the redshift $z$. Like the case of the constant redshift function, we also take the best-fitting values (0.2684, 0.1912) of the model parameter pair ($\Omega_{m0}$, $\Omega_{r_c}$) from the SNe Ia+OHD+Planck+GW constraints and obtain $$I_V=0.0689013(r_0-r_m). \label{18}$$ It is not difficult to verify that the physical quantity $I_V\rightarrow0$ when $r_m\rightarrow r_0$, which means that, in theory, we can construct a traversable wormhole with infinitesimal amounts of averaged-NEC-violating dark energy fluid in the DGP braneworld scenario.

![Relations between different energy conditions and the wormhole radius $r$. The green (long-dashed) line represents the expression $(p+3\rho)/\rho=3\omega_{eff}+1$, the blue (short-dashed) line represents the expression $(p+\rho)/\rho=\omega_{eff}+1$, the pink (dot-dashed) line represents the expression $(\rho-p)/\rho=1-\omega_{eff}$, and the red (solid) line represents the expression $\rho/\rho=1$.[]{data-label="f6"}](6.jpg)

