Simple lumped parameter DC motor model

How about a series R-L-C, with another resistor across the C.

R1 and L1 represent the real resistance and inductance that you'd measure with a stalled rotor, namely C1 shorted. C1 represents the rotational inertia of the motor and any load, and R2 (across C1) is the load, including motor losses.

Current in R1 becomes torque, and the resulting voltage across C1 represents both speed and back EMF.

Modeling what? Do you want the model to simulate the load on the motor, or to simulate the motor as an electromechanical device?

The model that I generally use is an ideal motor with armature resistance and (if really necessary) friction and windage. You can almost always ignore windage, and friction is nonlinear so you generally have to deal with it on a per-case basis.

The ideal motor model is easy-peasy:

T_a = k_t * i_a, e_a = k_t * {\omega}_a

where T_a is the torque exerted on the armature by the coils, k_t is the motor's torque constant, i_a is the motor armature current, e_a is the induced armature voltage (the back-emf, in other words), and {\omega}_a is the armature speed.

If you keep everything in mks units, and armature speed in radians/ second, then the k_t in both equations is the same: 1 Newton-meter/Amp is exactly equal to 1 volt-second/radian.

Around this you wrap the circuit model on the electrical side, to take into account the voltage drop in the armature resistance and (if necessary) the armature inductance. You also (if necessary) add whatever mechanical behavior you need to add; often you'll find that the mechanical dynamics are dominated by the inertia of the motor and whatever its attached to, so you can just model this as

d/dt {\omega}_a = (T_a - T_d)/J_m,

where T_d is your disturbance torque and J_m is the moment of inertia of the motor and stuff.

If you're just interested in the electrical side, then model k_t * {\omega}_a as an unknown voltage and be happy.

I don't think I've done a control loop where the mechanical dynamics weren't dominated by the motor inertia once things got going; there are times (particularly if you've got a gear train in the mix) where the start-stop dynamics are significantly influenced by friction and/or backlash: that's an advanced subject and I've already splattered enough words into this response.