The idea of radius of investigation has been used in conventional pressure transient analysis for various applications such as calculation of distance to a flow barrier. In multistage fractured horizontal wells reservoirs, linear flow predominates for most of the production period, prompting the development of an analogous concept. In this work, we present the derivation of the depth of investigation for linear flow, and analyze the difference between time of detection and time of arrival for various production conditions held at the well.
We derive the unit impulse function for linear flow rigorously using the method of Green's functions, and then obtain the expression for depth of investigation using the concept of maximum pressure disturbance. Similar concepts that have historically been used to develop depth of investigation type expressions, such as the successive steady states method, probe radius method, and methods based on asymptotic solutions analogous to wave propagation are explored and compared. Using the concept of source functions, we analyze the discrepancy between time of arrival of the pressure pulse at the boundary and the time of detection of the boundary at the well.
We show that the unit impulse function method is the most reliable method for determining depth of investigation (and the distance of investigation, in general). We also compare the depth of investigation calculated from the impulse function method with the time to the end of linear flow. The discrepancy between the time of detection of the boundary and time of arrival of the pressure pulse at the boundary is purely a function of the resolution of the pressure measurements at the well, and the analysis technique used to account for changing bottomhole conditions.
Distance of investigation can be used to estimate drainage areas and map depletion levels in the reservoir. It can therefore be used to calculate optimal well spacing, and, in the case of multistage fractured horizontal wells, optimal fracture spacing. In this work, we demonstrate the determination of permeability from the end of linear flow once the depth of investigation has been estimated, and how this method can lead to errors if end of linear flow is not properly identified.