Constraining the moment release of injection-induced earthquakes is of paramount importance to reduce the seismic hazard in the geo-energy industry. Recent studies suggest that a significant part of the moment release during fluid injections can be due to aseismic motion, namely, aseismic moment M0. Current models of injection-induced aseismic moment do not incorporate fault rupture mechanics. Here, we present a theoretical and numerical analysis that highlights a possible scaling relation between the aseismic moment and a key operational parameter, the injected volume of fluid V. The scaling relation emerges from the model of a stable frictional shear crack that propagates in mixed mode (II+III) on a planar fault interface. The interface is characterized by a constant hydraulic transmissivity and a shear strength that is equal to the product of a constant friction coefficient and the local effective normal stress. Fluid is injected right into the fault interface at a constant flow rate. The resulting relation between the aseismic moment and the injected volume is M0=A⋅ V^(3/2). The prefactor A accounts for the dependence of the aseismic moment on the pre-injection stress state, the parameters of the injection (notably, the injection flow rate), and the fault elasto-frictional and hydraulic properties. Unlike previous studies, our model accounts for the possibility that ruptures can propagate beyond the fluid-pressurized fault patch, a condition that is expected to occur in critically stressed and/or highly-pressurized fractures/faults. We test the scaling relation against estimates of moment release due to aseismic motion during fluid injections that vary in size from laboratory experiments to industrial applications. Our predictions are in good agreement with these observations. These results provide a simple means to quantify the size of aseismic ruptures in response to fluid injections related to both natural and human sources.