  In this paper a new technique for modeling non-linear force-free fields
directly from line of sight magnetogram observations is presented. The
technique uses sequences of magnetograms directly as lower boundary conditions
to drive the evolution of coronal magnetic fields between successive force-free
equilibria over long periods of time. It is illustrated by applying it to MDI
observations of a decaying active region, NOAA AR 8005. The active region is
modeled during a 4 day period around its central meridian passage. Over this
time, the dispersal of the active region is dominated by random motions due to
small scale convective cells. Through studying the build up of magnetic energy
in the model, it is found that such small scale motions may inject anywhere
from $2.5-3 \times 10^{25}$ erg s$^{-1}$ of free magnetic energy into the
coronal field. Most of this energy is stored within the center of the active
region in the low corona, below 30 Mm. After 4 days the build-up of free energy
is 10% that of the corresponding potential field. This energy buildup, is
sufficient to explain the radiative losses at coronal temperatures within the
active region. Small scale convective motions therefore play an integral part
in the energy balance of the corona. This new technique has wide ranging
applications with the new high resolution, high cadence observations from the
SDO:HMI and SDO:AIA instruments.
