Numerical Simulation and Inversion of Pressure Data Acquired With Permanent Sensors

Abstract

Abstract We have developed a new axisymmetric solution for the numerical simulation of single-phase fluid flow in permeable media. In this new solution, the governing parabolic partial differential equation (PDE) is transformed into a linear operator problem that is subsequently solved with an Extended Krylov Subspace Method (EKSM). The algorithm yields solutions for pressure at a multitude of times with the same efficiency of that of a single-time solution. Extensive comparisons with analytical solutions show that our simulator yields pressure results with accuracies better than 1%, and favorably competes with commercial simulators in CPU execution times and memory efficiency. Examples of numerical simulation are considered for water injection and variable flow-rate problems. In both cases, we have focused our attention to the description of pressure data acquired with permanent sensors deployed in direct hydraulic contact with the producing formation. The inverse problem considered in this paper consists of mapping time and space variations of pressure and flow rate into a local distribution of permeabilities. We solve this inverse problem by minimizing a cost function written as the sum of the square differences between the measured pressure data and the corresponding pressure data yielded by our Krylov subspace simulator. A nonlinear Gauss-Newton fixed-point iteration search is used to minimize the quadratic cost function. We also test a dual coarse- and fine-grid approach to accelerate the solution of the inverse problem. Examples of inversion are shown with noise-free and noisy data aimed at understanding the role played by the flow-rate function and the location, spacing, and number of permanent sensors into the accuracy and stability of the inverted permeability values. Introduction The availability of permanently installed downhole pressure and temperature sensors has opened a new window of opportunities to probe hydrocarbon reservoirs. A continuous stream of time-domain data is now available to perform real-time monitoring of the variation of fluid-flow parameters resulting from production. In turn, time-domain monitoring provides the basic hardware for a much-needed active feedback loop to control hydrocarbon production in real time. Very recently, prototypes of permanent sensors have been constructed to be deployed behind casing and hence to be in direct contact with the producing formations. When commercially available, these in-situ sensors will allow the possibility of delivering real-time images of the spatial distribution of fluid-flow parameters in the vicinity of a well and also between existing wells. Interpretation work is therefore in order to quantify how a variation in the measurements acquired by in-situ permanent sensors will translate into a variation in the spatial distribution of fluid-flow properties. It is also imperative to optimally design the spacing and number of permanent sensors in light of existing reservoir conditions and potentially deleterious noisy measurements. The work described in this paper is an attempt to quantify the spatial resolution properties of in-situ permanent pressure measurements. To do so, we consider a hypothetical water injection experiment and make use of the extensive body of estimation techniques available from the field of geophysical inverse theory. Our goal is to construct an efficient algorithm to quantify the sensitivity of permanent pressure data to lateral and vertical variations in the distribution of permeability around the injection well.