Chapter 2

Single-Phase Flow Forward and Inverse Algorithms

Because the authors' prior book Formation Testing: Pressure Transient and Contamination from John Wiley & Sons is relatively new, appearing only in 2014, it is fitting to provide a concise summary of the methods and algorithms introduced there so that the contributions of the present book can be viewed and quickly understood in context. The work provided in this chapter sets the stage for the advanced drawdown-buildup and phase delay models discussed in Chapters 3 and 4.

2.1 Overview

We describe a comprehensive set of integrated formation testing forward and inverse analysis tools developed for wireline and "formation testing while drilling" (FTWD) applications in hardware design and pressure transient interpretation. The methods, based on rigorous Darcy flow formulations, are solved analytically in closed form whenever possible and cross-checked in different limits to ensure physical consistency and accuracy.

The transient problem for formation tester liquid pressure response in anisotropic media with flowline storage and skin at arbitrary dip, earlier solved in exact, closed analytical form assuming ellipsoidal sources (using complex complementary error functions), 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 drops computed at various dip angles from the forward simulation, derived formulas are used to 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, vertical and horizontal permeabilities and skin factor. An algorithm giving possible solutions for both permeabilities and skin at any dip angle when both pressure drops are known is derived. 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 (analogous to the pulses emanating from acoustic tools). These are strongest and most advantageous at low permeabilities where diffusion predominates. Short pulses, high in frequency content, provide detailed information. Multi-pulse wave-trains with different flow rates, pulse durations and separations enable multiple fast test suites at the rigsite without requiring new hardware. They are accurate, economical and reduce tool sticking risks in tight zones.

Fourth, "phase delay" approaches for permeability prediction analogous to electromagnetic logging methods are described. Sinusoidal pressure transients are created at the pumping probe. Their amplitudes and phases are measured at one or more observation probes. These are interpreted using Darcy analysis models. As with pulse interaction methods, phase delay approaches allow short-duration tests that are economical, safe and characterized by high signal-to-noise ratios.

Fifth, a full 3D horizontal well model for single-probe, dual-probe, dual-packer and elongated pad tools with real mandrels in layered media is given, with computations showing effects of azimuth and bed boundary on pressure response and their implications on permeability prediction. While source models require dual-probe data for inverse application, full 3D models can be used with single-probe FTWD tools (measuring azimuthal pressures) to provide clues related to permeability, anisotropy and bed thickness.

Finally, real-time FTWD pore pressure and mobility prediction is discussed. Such problems, key to drilling safety and rapid economic evaluation, involve transient data distorted by flowline storage effects. Accurate predictions are possible using a minimum of pressure data. We develop rational polynomial expansion methods that do not require exponential, real or complex complementary error functions, and moreover, do not use regression or least-squares smoothing filters that introduce diffusive assumptions beyond those implicit in Darcy's laws. Rapid analysis frees microprocessor resources for other important control and interpretation functions needed during drilling.

2.2 Basic Model Summaries

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 (and possibly pore pressure) are required when all other parameters are given.

Forward models have been developed to a high degree of sophistication. Almost twenty years ago, Proett and Chin (1996) published the first full three-dimensional finite element analysis assuming realistic borehole environments with pad, probe, mandrel, flowline storage and bedding plane effects. These are reviewed in Formation Testing Pressure Transient and Contamination Analysis (Chin et al, 2014), which also cites extensions to include coupled dynamic mudcake growth and supercharge corrections.

Chin and Proett (2005) provided finite difference models that included multiphase miscible and immiscible effects. In downhole sampling, the time required to pump until clean in-situ fluids are obtained is important. This time scale differs from that used for pressure transient interpretation. Figures 2.1a,b illustrate capabilities that have been used to design new tools and interpret transient data obtained in complicated environments. Figure 2.1c shows Catscan experiments in which dynamic cake growth is measured in cores with different permeabilities. This is important to supercharge and contamination corrections.

Figure 2.1a. Finite element model, flow vectors near probe.

Figure 2.1b. Pressures for drawdown with two probes (left) and one (right).

Figure 2.1c. Catscan results (flow from top to bottom, darkening lines at center indicate cake growth in time).

While fully three-dimensional models are important in their own right, they do require complicated inputs, numerical simulation expertise, not to mention sophisticated and expensive computing environments that host high-overhead software and long calculations. Thus, fast methods that retain the basic elements of the physics are desirable, particularly for field office use and real-time downhole analysis. A number of simpler formulations are possible. These range from elementary point source models, which unfortunately "blow up" at the "r = 0" origin, to finite radius models, which apply flowline storage and skin boundary conditions at spherical source surfaces (ellipsoidal in transversely isotropic flow).

This chapter also introduces new physical concepts in pressure transient interpretation using innovative math models first described in Chin et al (2014). So that the ideas are clearly explained, math details are omitted in favor of examples, although shorter summaries are offered. For readers interested in analytical and numerical details, reference to the book is necessary. Several new capabilities and modules are available and, for convenience, are referred to by a "FT-" designation with "FT" referring to formation testing.

2.2.1 Module FT-00

This deals with exact transient liquid response in homogeneous anisotropic media. It solves for the unsteady Darcy pressure field about an ellipsoidal source surface immersed in a transversely isotropic infinite homogeneous medium allowing full skin effect and flowline storage boundary conditions. This "nonzero radius source" model is more powerful than limited point source approximations because it handles nearfield boundary conditions without becoming singular at the origin as do point source models. We emphasize that cylindrical borehole effects and drilling fluid invasion are not directly incorporated.

The earlier exact analytical solution, posed in terms of the complex complementary error function erfc(z), applied to a single drawdown or buildup and was limited to zero dip angles. To be effectively used for job planning and inverse applications, such as those considered here, improvements were needed.