  Making use of the system of pulling a spherical intruder in static
three-dimensional granular media, we numerically study the scaling law for the
drag force $F_{\rm drag}$ acting on the moving intruder under the influence of
the gravity. Suppose if the intruder of diameter $D$ immersed in a granular
medium consisting of grains of average diameter $d$ is located at a depth $h$
and moves with a speed $V$, we find that $F_{\rm drag}$ can be scaled as
$(D+d)^{\phi_\mu} h^{\alpha_\mu}$ with two exponents $\phi_\mu$ and
$\alpha_\mu$, which depend on the friction coefficient $\mu$ and satisfy an
approximate sum rule $\phi_\mu+\alpha_\mu\approx 3$. This scaling law is valid
for the arbitrary Froude number (defined by $\mathrm{Fr}={2
V}\sqrt{{2D}/{g}}\big/(D+d)$), if $h$ is sufficiently deep. We also identify
the existence of three regimes (quasistatic, linear, and quadratic) at least
for frictional grains in the velocity dependence of drag force. The crossovers
take place at $\mathrm{Fr}\approx 1$ between the quasistatic to the linear
regimes and at $\mathrm{Fr}\approx 5$ between the linear to the quadratic
regimes. We also observe that Froude numbers at which these crossovers between
the regimes happen are independent of the depth $h$ and the diameter of the
intruder $D$. We also report the numerical results on the average coordination
number of the intruder and average contact angle as functions of intruder
velocity.
