asymptotics of distance between points on geodesics moving with constant speed

Question

Let $(M,g)$ be a Riemannian manifold of unbounded diameter and let $\gamma_1, \gamma_2$ be two geodesics such that $\gamma_1(0)=\gamma_2(0)=x$. Suppose that $\gamma_1, \gamma_2$ are parametrized so that $||\dot{\gamma_1}||_g=||\dot{\gamma_2}||_g \equiv 1$. What can be said about the asymptotics of the function $d(\gamma_1(t), \gamma_2(t))$, where $d$ is the geodesic distance, in case $M$ has nonpositive sectional curvature? strictly negative sectional curvature?