Somers' D for model validation Somers' D as defined for example in "The predictive accuracy of credit ratings:
Measurement and statistical inference" by Walter Orth is defined for the case when Y is predicted by X as 
$$
D_{XY} = \frac{1}{n_u} \sum_{i=1}^n\sum_{j>i} c_{ij},
$$
where $c_{ij}$ counts 1 for concordant pairs of $(X,Y)$ and -1 for discordant pairs and 0 else.
So far it all rather simple.
The denominator $n_u$ is defined as
$$
n_u = \sum_{i=1}^n\sum_{j>i} 1_{ \{Y_i \neq Y_j \}}.
$$
Thus it counts all pairs but the ones where the Y-values are equal. Thus ties in $X$ are counted but ties in $Y$ reduce the denominator and therefore increase $D_{XY}$.
How does this carry over to the case of validation of a model? 
Can we just set $X = \hat{Y}$ by which I mean the model prediction and $Y$ are the realized values?
Then I look at concordant and discordant pars of the prediction $\hat{Y}$ and the realization. For the denominator I don't count realization pairs $i,j$ with $Y_i = Y_j$. The author says that in such case one can not judge the performance of the prediction.
If $Y$ is a default indicator of a credit scoring model, then I would interpret $X$ as the probability of default or a risk class from my model and $Y$ the default observation.
In the paper linked it is mentioned that in this case the above gives the accuracy ratio (= Gini coeffient).
My question: do you agree that a useful interpretation of Somers' D is as above: $X$ is the prediction and $Y$ the observed realization with the stated consequences? Especially, on this case ties of predictions are counted in the denominator but ties in the relization not.
 A: Somers' $D_{xy}$ is a general measure of predictive discrimination which can handle binary, ordinal, and censored time-to-event outcome variables.  In the binary $Y$ case it is just $2 (c - 0.5)$ where $c$ is the concordance probability, AKA area under the ROC curve.  It is a very interpretable measure of predictive discrimination but like AUROC is not sensitive enough for choosing or comparing models.
To your question, yes we compute $D_{xy}$  by playing $\hat{Y}$ against $Y$ and using the ordinary Somers' rank correlation formula you have listed, where ties on $Y$ are ignored (not penalized against).  The only difference is that the estimate of $D$ will be biased when you have overfit the data patterns.  For that reason we don't concentrate on the "apparent $D_{xy}$" but rather on the overfitting-corrected version of it.  For most cases this is most efficiently estimated using the bootstrap.  Alternatively, 100 repeats of 10-fold cross-validation may be used.  
Details are in Regression Modeling Strategies book and course notes.  Start with the chapter on binary logistic regression where I cover $D_{xy}$.  The R rms package validate.* functions compute apparent and validated (overfitting-corrected) $D_{xy}$.
The "not sensitive enough" comment relates to rank measures not giving a sufficient reward for extreme predictions that are "right".  Sensitive measures are discussed here.
