Abstract

The transient problem for formation tester pressure response in anisotropic media with flowline storage and skin at arbitrary dip, solved in exact, closed analytical form assuming ellipsoidal sources (in terms of the complex complementary error function) in an earlier paper, is used to derive exact solutions to several inverse problems where permeabilities are sought when dip angle and source and observation probe pressure drops are given. First, the zero-skin forward solution is evaluated in the steady-state limit for constant rate pumping. Explicit inverse formulas are derived for all horizontal and vertical permeabilities and dip angles. With pressure drop data computed at various dip angles from the forward simulation, our derived formulas are used to successfully predict both assumed permeabilities - demonstrating their utility in field interpretation. Neglect of dip angle can lead to significant errors in anisotropy prediction. Moreover, multi-valued inverse solutions exist: for a given set of pressure drops, three permeability pairs are found which require resolution from additional logging data. Second, the "with-skin" forward solution is evaluated at steady-state for constant rate pumping to develop formulas relating source and observation probe pressure drop, both permeabilities and skin factor. An algorithm giving possible solutions for horizontal and vertical permeability and skin at any dip angle when both pressure drops are known is derived. However, because only two pressure data points are assumed, additional logging information is needed to render a unique determination. Third, short duration pulse interactions at the observation probe are used to determine anisotropy. These are strongest and most advantageous at low permeabilities where diffusion predominates. Short pulses, high in frequency content, provide detailed information. Multi-pulse wavetrains with different flowrates, pulse durations and separations enable multiple fast test suites at the rigsite without requiring new hardware - they are economical and reduce tool sticking risks. Finally, a full three-dimensional horizontal well model for single-probe, dual-probe and packer and oval pad tools with real mandrels in layered media is described, with computations showing effects of azimuth and bed boundary on pressure response and their implications on permeability prediction. While source models require dual-probe measurements for inverse application, single-probe tools utilizing azimuthal pressure measurements together with mandrel-based interpretation models do not and both permeabilities are uniquely determined. The latter method is ideal in Formation-Testing-While-Drilling when only single probe data is available.

Introduction

In formation tester pressure transient analysis, two general types of practical field applications arise, namely, "forward modeling," in which source and observation probe responses are sought when fluid, formation and tool parameters are specified, and "inverse modeling," in which kh and kv permeabilities are required when all other parameters are given. A number of formulations are possible. These range from simple point source models, which "blow up" at the "r = 0" origin - to finite radius models, which apply flowline storage and skin boundary conditions at source surfaces (ellipsoidal in transversely isotropic flow) - and finally, to fully three-dimensional models which account for pad, mandrel and bedding plane effects. This paper introduces new physical concepts in pressure transient interpretation using innovative math models developed for their implementation. So that the ideas are clearly explained,

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