Electroseismics is a procedure that uses the conversion of electromagnetic to seismic waves in a fluid-saturated porous rock due to the electrokinetic phenomenon. This paper concerns the time-domain analysis of such an electroseismic conversion problem in an unbounded structure in three dimensions. Using an exact transparent boundary condition and suitable interface conditions, we study an initial- boundary value problem for the coupling of Maxwell's equations and the Biot's equations. The well-posedness and stability are established for the reduced problem. Our proof is based on the method of energy, the Lax-Milgram theorem, and the inversion theorem of the Laplace transform. Moreover, a priori estimates with explicit dependence on the time are achieved for the quantities of electric filed and solid-fluid fields by taking special test functions for the time-domain variational problem.