Explicit fracture models often use analytical solutions for predicting flow in fractured media, usually assuming uniform fluid viscosity for simplicity. This assumption, however, can be inaccurate as fluid viscosity varies due to factors like composition, temperature, and dissolved substances. Our study, recognizing these discrepancies, abandons this uniform viscosity assumption for a more realistic model of variable viscosity flow, focusing on viscous displacement scenarios. This includes instances like injecting viscous surfactants for hydrocarbon recovery in fractured reservoirs or soil decontamination. This presents a significant challenge, enhancing our understanding of transport within fractures, mainly governed by advection. Our study centers on a low-permeability rock matrix intersected by two fractures with variable apertures. We employ two methods: an analytical approach with a new solution and numerical simulations with two distinct in-house codes, discretizing both the rock matrix and fractures with two-dimensional triangular elements. The first code uses a Discontinuous Galerkin finite element method, while the second utilizes a finite-volume method, allowing a comprehensive comparison of solutions. Additionally, we investigate parameter identifiability, like fracture apertures and viscosity ratios, using breakthrough curves from our analytical solution, applying the Markov Chain Monte Carlo technique.