Estimation of axisymmetric spatial distributions of permeability and porosity from pressure-transient data acquired with in situ permanent sensors

Abstract

Oil-producing companies have shown increased interest in instrumenting their hydrocarbon fields with in situ pressure sensors. As opposed to standard bottom-hole permanent pressure gauges, in situ pressure sensors are deployed behind casing to remain in direct hydraulic communication with rock formations. Prototype deployments have been tested in field operations that included intelligent completions. In situ pressure sensors allow the possibility of monitoring real-time dynamic variations of reservoir conditions due to primary or enhanced recovery. In consequence, a feedback loop can be enforced to modify the production scheme in a way that optimizes the recovery of existing hydrocarbon assets. While a great deal of laboratory and field work has been undertaken to advance hardware prototypes, relatively little has been done to quantify the spatial resolution and reliability of in situ permanent pressure data to detecting hydrocarbon reservoir properties. In this paper, we consider the inverse problem of simultaneously estimating spatial distributions of absolute permeability and porosity from transient measurements of pressure acquired with in situ permanent sensors. We pose the inverse problem as the minimization of a quadratic cost function that quantifies the misfit between the measured and numerically simulated data. A modified Gauss–Newton nonlinear optimization technique is used to minimize the quadratic cost function subject to physical constraints. We also make use of a dual-grid approach that alternates the use of coarse and fine finite difference grids to accelerate the inversion. Several examples of inversion are performed with noise-free and noisy synthetic measurements 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 spatial distributions of permeability. Following these proof-of-concept exercises of applicability, the inversion algorithm is used for the simultaneous estimation of spatial distributions of permeability and porosity. Results indicate that in situ permanent pressure measurements exhibit significantly more sensitivity than bottom-hole pressure sensors to detecting spatial variations of permeability and porosity. However, because of their diffusive nature, the spatial resolution and distance of penetration of noisy in situ pressure measurements rapidly decreases with increasing distances from the sensor and flow rate pulse locations. Finally, the dual-grid inversion technique is utilized for the quantitative interpretation of time records of pressure acquired during an experimental field deployment of in situ permanent sensors. The field experiment was conducted to demonstrate the feasibility of electrical, pressure, and fiber optic measurement technologies to monitoring water movement between an injection and an observation well. Only transient wellbore pressure measurements acquired in the injection well were deemed consistent with the assumptions of the forward model used in this paper. The absence of in situ pressure data severely reduced the sensitivity of the measurements to spatial variations of permeability and porosity in the reservoir. In consequence, the estimated distributions of permeability and porosity reflected only large-scale effective medium properties of the formation of interest.