Understanding the mechanisms underlying tumor metastasis is critical for designing effective anti-tumor therapies. This article focuses on the modeling and numerical analysis of cell migration by chemical signals and interstitial flow, two crucial factors in tumor metastasis. We consider a nonlinear Keller-Segel model that includes an elliptic equation based on Darcy's law for fluid flow. We propose a fully discrete method that combines an implicit-explicit method in time with a finite difference method in space. We establish the method's second-order superconvergence in space in a discrete \( H^1 \)-norm, optimal first-order convergence in time in a discrete \( L^2 \)-norm, and local nonlinear stability. Numerical simulations confirm the sharpness of the error analysis. We also look into the model's ability to reproduce laboratory experiments on the effects of flow and chemotaxis on tumor cell migration.