Tony,

I'm not sure which variety, but it's definitely a geranium. I've got
something similar growing in my garden although it doesn't sound as invasive
as yours. I looked in the RHS Encyclopaedia and there is only one with a tap
root ( G. farreri ) but it has pink flowers. A better match if they are just
long roots is maybe G.pylzowianum which it describes as ( no picture ) :

G. pylzowianum
Spreading perennial with underground runners and tiny tubers. H 12-25cm,
S25cm or more. Fully hardy. bears semi-circular, deeply cut, dark green
leaves and in summer, trumpet-shaped, green centred, deep rose-pink flowers.
May be invasive.

Martin.