  In this paper, we demonstrate the use of physical models to evaluate the
production of $^{39}$Ar and $^{40}$Ar underground. Considering both cosmogenic
$^{39}$Ar production and radiogenic $^{40}$Ar production in situ and from
external sources, we can derive the ratio of $^{39}$Ar to $^{40}$Ar in
underground sources. We show for the first time that the $^{39}$Ar production
underground is dominated by stopping negative muon capture on $^{39}$K and
($\alpha,n)$ induced subsequent $^{39}$K(n,p)$^{39}$Ar reactions. The
production of $^{39}$Ar is shown as a function of depth. We demonstrate that
argon depleted in $^{39}$Ar can be obtained only if the depth of the
underground resources is greater than 500 m.w.e. below the surface. Stopping
negative muon capture on $^{39}$K dominates over radiogenic production at
depths of less than 2000 m.w.e., and that production by muon-induced neutrons
is subdominant at any depth. The depletion factor depends strongly on both
radioactivity level and potassium content in the rock. We measure the
radioactivity concentration and potassium concentration in the rock for a
potential site of an underground argon source in South Dakota. Depending on the
probability of $^{39}$Ar and $^{40}$Ar produced underground being dissolved in
the water, the upper limit of the concentration of $^{39}$Ar in the underground
water at this site is estimated to be in a range of a factor of 1.6 to 155 less
than the $^{39}$Ar concentration in the atmosphere.
  The calculation tools presented in this paper are also critical to the dating
method with $^{39}$Ar.
