  Let C be a hyperelliptic curve of good reduction defined over a discrete
valuation field K with algebraically closed residue field k. Assume moreover
that char k \ne 2. Given d \in K^*\K^*2, we introduce an explicit description
of the minimal regular model of the quadratic twist of C by d. As an
application, we show that if C/Q is a nonsingular hyperelliptic curve given by
y^2 = f(x) with f an irreducible polynomial, there exists a positive density
family of prime quadratic twists of C which are not everywhere locally soluble.
