Thermo-viscous feedback in heat transport through fracture–matrix systems with finite thermal capacity

In geothermal systems, injected fluid flows along preferential fracture pathways and exchanges heat with matrix blocks. Reduced models often neglect temperature-dependent viscosity and assume unlimited matrix storage, although the thermal capacity accessible to the flow is finite. This work analyzes thermo-viscous feedback in a planar fracture–matrix system where viscosity controls discharge through the integrated hydraulic resistance, while finite capacity enters through an effective single-timescale memory fixed by initial uptake and total capacity. The reduced model gives parsimonious semi-analytical predictions with physically based parameters, verified against the nonlinear solution. A perturbation hierarchy separates matrix storage from the hydraulic response it induces, while an exponential velocity reconstruction preserves the exact terminal discharge and bounds front-moment errors within the front-moment limit model. At fixed pressure difference, heating accelerates and broadens the front, whereas cooling delays and sharpens it. The terminal discharge depends only on the endpoint viscosity ratio, while matrix uptake, storage and contrast determine its approach time. At fixed rate, viscosity alters only the pressure drop, not the thermal response. Finite thickness controls the transition to capacity-limited equilibration, and strong matrix uptake collapses the isoviscous response onto a two-variable similarity surface.