Temperature measurements have been used by a variety of researchers to gain insight into groundwater discharge patterns. However, much of this research has reduced the problem to heat and fluid flow in one dimension for ease of analysis. This approach is seemingly at odds with the goal of determining spatial variability in specific discharge, which implies that the temperature field will vary in more than one dimension. However, it is unclear how important the resulting discrepancies are in the context of determining groundwater discharge to surface water bodies. In this study, the importance of these variations is examined by testing two popular one-dimensional analytical solutions with stochastic models of heat and fluid flow in a two-dimensional porous medium. For cases with low degrees of heterogeneity in hydraulic conductivity, acceptable results are possible for specific discharges between 10-7 and 10-5 m/s. However, conduction into areas with specific discharges less than 10-7 m/s from adjacent areas can lead to significant errors. In some of these cases, the one-dimensional solutions produced estimates of specific discharge of nearly 10-6 m/s. This phenomenon is more likely in situations with greater degrees of heterogeneity.