3d Stochastic Formulation for Travel Time Moments

Field data available to petroleum engineers can be sparse because they are generally obtained through discrete well locations. Therefore there is uncertainty in the knowledge of the reservoir’s physical properties such as the porosity or the permeability. In order to quantify the uncertainty in the predictions we are making for production quantities, what has traditionally been done is Monte Carlo Simulation or MCS. To capture the uncertainty we have in the reservoir knowledge, the MCS approach relies on multiple, equiprobable realizations of the reservoir. Each realization of the field is processed in a simulator which solves deterministic equations. Then the output from the flow simulations obtained from all the realizations is collected and post processed. By doing so, we are able to obtain the mean and the variance of the production quantities. This process can be computationally expensive and it is difficult to judge when statistical convergence has been reached; many simulations, possibly a hundred or a thousand have to be run to capture the prediction uncertainty due to the uncertainty in the reservoir description. An elegant alternative to this MCS approach is the Statistical Moment Equations (SME) approach. Predictions made by the SME approach should in theory be very close to the predictions obtained by the MCS method. Both methods should in the end yield the same statistics of the quantities of interest. The biggest advantage of SME over MCS is that SME only needs one simulation run, though being an expensive one. The stochastic framework provides a formal way for generating detailed models of reservoir description that honor available hard and soft data. In SME, moment equations, which derive from a stochastic mathematical statement of immiscible nonlinear two-phase flow in heterogeneous reservoirs are solved. The permeability is treated as a random space function and in turn, the velocity and the travel time are also random quantities. The streamline-based approach is used to deal with the transport problem. iv The SME simulator we extended in this work was tested both in 2D and 3D reservoirs of small size, due to memory and computation limitations. The moments of permeability, pressure, velocity, travel time and saturation were compared with MCS results. We also show results obtained with the well model integrated into the SME simulator and we investigated the influence of the input variance of log permeability on the prediction of the moments of the quantities of interest.