The Welch-Satterthwaite d.f. can be shown to be a scaled weighted harmonic mean of the two degrees of freedom, with weights in proportion to the corresponding standard deviations.

The original expression reads:

$$\nu_{_W} = \frac{\left(\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}\right)^2}{\frac{s_1^4}{n_1^2\nu_1}+\frac{s_2^4}{n_2^2\nu_2}}$$

Note that $r_i=s_i^2/n_i$ is the estimated variance of the $i^\text{th}$ sample mean or the square of the $i$-th [standard error of the mean][1]. Let $r=r_1/r_2$ (the ratio of the estimated variances of the sample means), so

\begin{align}
\nu_{_W} &= \frac{\left(r_1+r_2\right)^2}{\frac{r_1^2}{\nu_1}+\frac{r_2^2}{\nu_2}}  \newline
\newline
&=\frac{\left(r_1+r_2\right)^2}{r_1^2+r_2^2}\frac{r_1^2+r_2^2}{\frac{r_1^2}{\nu_1}+\frac{r_2^2}{\nu_2}}  \newline
\newline
&=\frac{\left(r+1\right)^2}{r^2+1}\frac{r_1^2+r_2^2}{\frac{r_1^2}{\nu_1}+\frac{r_2^2}{\nu_2}}
\end{align}

The first factor is $1+\text{sech}(\log(r))$, which increases from $1$ at $r=0$ to $2$ at $r=1$ and then decreases to $1$ at $r=\infty$; it's symmetric in $\log r$.

The second factor is a [weighted harmonic mean](http://en.wikipedia.org/wiki/Harmonic_mean#Weighted_harmonic_mean):

$$H(\underline{x})=\frac{\sum_{i=1}^n w_i }{ \sum_{i=1}^n \frac{w_i}{x_i}}\,.$$

of the d.f., where $w_i=r_i^2$ are the relative weights to the two d.f.

Which is to say, when $r_1/r_2$ is very large, it converges to $\nu_1$. When $r_1/r_2$ is very close to $0$ it converges to $\nu_2$. When $r_1=r_2$ you get twice the harmonic mean of the d.f., and when $s_1^2=s_2^2$ you get the usual equal-variance t-test d.f., which is also the maximum possible value for $\nu_{_W}$.

--

With an equal-variance t-test, if the assumptions hold, the square of the denominator is a constant times a chi-square random variate.

The square of the denominator of the Welch t-test isn't (a constant times) a chi-square; however, it's often not too bad an approximation. A relevant discussion can be found [here](http://stats.stackexchange.com/questions/16921/how-to-understand-degrees-of-freedom/17148#17148).

A more textbook-style derivation can be found [here](http://sections.maa.org/okar/papers/2014/Lloyd.pdf).

  [1]: https://en.wikipedia.org/wiki/Standard_error_of_the_mean