Constraint modeling is known to play an important role in solving problems efficiently. Remarkable improvements of SAT solvers have been made over the last decade. Such improvements encourage researchers to solve Constraint Satisfaction Problems (CSPs) by encoding them into SAT. In this paper, we study SAT-based constraint modeling of the Packing Array (PA) problem in Combinatorial Designs. PA is also called mutually orthogonal partial latin squares and has been applied to optimal disk allocation in Databases. We first present four constraint models from different viewpoints of the PA problem, and then present their SAT encodings based on the order encoding. Particularly, basic alldiff model is designed to reduce the number of clauses for the packing constraints of a given PA. To evaluate the efficiency of our proposed models, we carried out experiments on the PA problems in Handbook of Combinatorial Designs. We succeeded in proving the optimality of previously known upper bounds for two arrays and in obtaining improved lower bounds for five arrays.