Blocking Moving Window algorithm: Conditioning multiple-point simulations to hydrogeological data

Connectivity constraints and measurements of state variables contain valuable information on aquifer architecture. Multiple-point (MP) geostatistics allow one to simulate aquifer architectures, presenting a predefined degree of global connectivity. In this context, connectivity data are often disregarded. The conditioning to state variables is usually carried out by minimizing a suitable objective function (i.e., solving an inverse problem). However, the discontinuous nature of lithofacies distributions and of the corresponding objective function discourages the use of traditional sensitivity-based inversion techniques. This work presents the Blocking Moving Window algorithm (BMW), aimed at overcoming these limitations by conditioning MP simulations to hydrogeological data such as connectivity and heads. The BMW evolves iteratively until convergence: (1) MP simulation of lithofacies from geological/geophysical data and connectivity constraints, where only a random portion of the domain is simulated at every iteration (i.e., the blocking moving window, whose size is user-defined); (2) population of hydraulic properties at the intrafacies; (3) simulation of state variables; and (4) acceptance or rejection of the MP simulation depending on the quality of the fit of measured state variables. The outcome is a stack of MP simulations that (1) resemble a prior geological model depicted by a training image, (2) honor lithological data and connectivity constraints, (3) correlate with geophysical data, and (4) fit available measurements of state variables well. We analyze the performance of the algorithm on a 2-D synthetic example. Results show that (1) the size of the blocking moving window controls the behavior of the BMW, (2) conditioning to state variable data enhances dramatically the initial simulation (which accounts for geological/geophysical data only), and (3) connectivity constraints speed up the convergence but do not enhance the stack if the number of iterations is large.