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Discrete-time stochastic models of management of groundwater resources have been extensively used for understanding a number of issues in groundwater management. Most models used suffer from two drawbacks: relatively simplistic treatment of the cost of water extraction, and a lack of important structural results (such as monotonicity of extraction in stock and concavity of the value function), even in simple models. Lack of structural properties impede both practical policy simulation and clarity of understanding of the resulting models and the underlying economics. This paper provides a unifying framework for these models in two directions; first, the usual cost function is extended to encompass cases where marginal cost of pumping depends on the stock and second, the analysis dispenses with assumptions of concavity of the objective function and compactness of the state space, using instead lattice-theoretic methods. With these modifications, a comprehensive investigation of which structural properties can be proved in each of the resulting cases is carried out. It is shown that for some of the richer models more structural properties may be proved than for the simpler model used in the literature. This paper also introduces to the resource economics literature an important method of proving convergence to a stationary distribution which does not require monotonicity in stock of resource. This method is of interest in a variety of renewable resource model settings.