Formation Testing: Pressure Transient and Contamination Analysis

By Wilson C Chin, Yanmin Zhou, Yongren Feng and 2 more

Traditional well logging methods, such as  resistivity, acoustic, nuclear and NMR, provide indirect information related to fluid and formation properties.  The “formation tester,” offered in wireline and MWD/LWD operations, is different.  It collects actual downhole fluid samples for surface analysis, and through pressure transient analysis, provides direct measurements for pore pressure, mobility, permeability and anisotropy.  These are vital to real-time drilling safety, geosteering, hydraulic fracturing and economic analysis.

Methods for formation testing analysis, while commercially important and accounting for a substantial part of service company profits, however, are shrouded in secrecy.  Unfortunately, many are poorly constructed, and because details are not available, industry researchers are not able to improve upon them.  This new book explains conventional models and develops new, more powerful algorithms for early-time analysis, and importantly, addresses a critical area in sampling related to “time required to pump clean samples” using rigorous multiphase flow techniques.  All of the methods are explained in complete detail.  Equations are offered for users to incorporate in their own models, but convenient, easy-to-use software is available for those needing immediate answers.

The leading author is a well known petrophysicist, with hands-on experience at Schlumberger, Halliburton, BP Exploration and other companies.  His work is used commercially at major oil service companies, and important extensions to his formation testing models have been supported by prestigious grants from the United States Department of Energy.  His new collaboration with China National Offshore Oil Corporation  marks an important turning point, where advanced simulation models and hardware are evolving side-by-side to define a new generation of formation testing logging instruments.  The present book provides more than formulations and solutions: it offers a close look at formation tester development “behind the scenes,” as the China National Offshore Oil Corporation opens up its research, engineering and manufacturing facilities through a collection of interesting photographs to show how formation testing tools are developed from start to finish.

• Language: English
• Publisher: Wiley
• Release date: Feb 14, 2014
• ISBN: 9781118831144

Wilson C Chin

Wilson C. Chin, PhD MIT, MSc Caltech, fluid mechanics, physics, applied math and numerical methods, has published twenty-five research books with John Wiley & Sons and Elsevier; more than 100 papers and 50 patents; and won 5 awards with the US Dept of Energy. He founded Stratamagnetic Software, LLC in 1997, an international company engaged in multiple scientific disciplines.

Book preview

Chapter 1

Basic Ideas, Challenges and Developments

The formation tester is a well logging instrument originally used to extract formation fluids from newly drilled wells for delivery to the surface, where detailed chemical analyses are performed. The pumping process is accompanied by flowline pressure transients that depend on, among other parameters, formation and fluid properties such as permeability, porosity, compressibility and viscosity. Pressure measurements are importantly used in two ways. Steady-state values are utilized in pressure gradient analysis, that is, determination of static pressure variations versus depth for hydrocarbon type identification. On the other hand, steady-state and transient values are used, separately and in concert, for reservoir characterization, in particular, in predicting pore pressure, horizontal and vertical permeabilities. This book deals with forward and inverse methods relevant to pressure analysis, with multiphase contamination remediation, and to special areas related to hardware design. Importantly, the methods apply to all manufacturers’ tools. For this reason, we do not specifically cite particular designs or trademarked or patented processes.

1.1 Background and Introduction

Two basic formulations arise, namely the forward or direct problem, in which pressures are sought when pumping schedules, formation, fluid and other properties are given, and the inverse or indirect problem, in which horizontal and vertical permeabilities (or their respective mobilities when fluid viscosities are uncertain) are sought when pressure drops or transients are specified. In some applications, pore pressures are to be extracted from early time pressure transients strongly affected by flowline storage. Different problem subsets dependent on time-scale and tool hardware constraints can be constructed which are rich in mathematical challenges. These methods form the subject matter of this book.

In the simplest limit, the fluid is a single-phase, compressible liquid, satisfying the standard Darcy flow model, a linear partial differential equation for pressure. Next in the hierarchy is gas flow, satisfying a nonlinear extension, one requiring additional thermodynamic information – gas pumping, rapidly becoming important, has never been addressed in a general and systematic way. At the upper end of the spectrum in mathematical complexity are multiphase flows, which can be miscible or immiscible in nature. These are modeled by equations for pressure, which couple to partial differential equations which describe space and time-varying contamination, namely, mixture concentration for miscible flows and oil saturation for immiscible applications.

By other properties, we refer to flow characteristics related to the mudcake and to hardware properties associated with the tool. In multiphase flow modeling, which importantly addresses contamination issues in addition to those for pressures, math models attempt to answer a number of questions related to the role of invasion through the mudcake. What is the time required to produce a clean sample? What parameters can be manipulated to reduce contamination? Are there any limitations associated with particular hardware designs? What is the true formation pressure? The mudcake adhering to the borehole wall controls the invasion rate into the formation for permeable rocks – but when mudcake and rock permeabilities are comparable, as in tight formations, cake growth and reservoir flow are dynamically coupled. In either case, useful models are only available when the parameters controlling cake growth are understood and well characterized, and then, only when the dynamic coupling noted is properly formulated in a mathematical model and solved.

Every student of mathematics understands that complications in obtaining practical solutions to partial differential equations arise principally from geometrical complexities associated with the problem domain. In formation testing, these might include borehole shape, the method of pad contact, the positioning of source and multiple observation probes in the logging string, the dip angle orientation of the tool, and finally, the presence of formation layers.

This book reports, for the first time, a series of original researches undertaken to model a large subset of pressure transient and contamination problems arising in formation testing field applications. In all cases, existing models are reviewed and critiqued, and newer, more robust and computationally stable methods are given which are implemented in user-friendly software – detailed calculated examples are also given to provide readers with an intuitive engineering feel for the physical problem. All mathematical details are provided in order to widely disseminate the new formulations. These models, both in executable and source code form, are available to industry practitioners for general use. Ideas for additional possible extensions, which number many, are easily developed from the topics highlighted in the Table of Contents.

Early formation testers were single-probe in design and were exclusively used in wireline applications. Modern testers are available for both wireline and formation-testing-while-drilling (or, so-called FTWD) applications. Hardware configurations include different combinations of pad-type nozzles, e.g., as shown in schematics (a) – (f) of Figure 1.1a. Rubber pad nozzles extending from these formation testers press against the borehole wall to form a firm seal with the mudcake. Then, fluid withdrawal begins until termination by the operator – this process is accompanied by pressure drawdown and buildup.

Figure 1.1a. Formation tester tool string configurations.

Circular holes about one inch in diameter are found at the centers of typical pads. Because of small hole sizes, pad-type nozzles are not ideal for unconsolidated and fractured formations since contacts with the formation are not repeatable and lower in quality. Contacts may not be truly representative of the geological volume. For such applications, the dual packer or straddle packer nozzles shown in schematics (h) – (l) of Figure 1.1a are more useful; once the packers inflate, the intervening annular volume serves as an elongated source or sink through which formation fluid traverses. Dual packer axial extents may vary from one foot to tens of feet. The elongated pad shown in schematic (g) may be regarded as an intermediary between single-probe pad nozzles and dual packers – the slot, several inches long, is designed to overlap greater numbers of heterogeneities and allows increased pumping volume for improved pressure transient analysis. Pad nozzles sense small formation volumes while dual packer depths of investigation are significantly greater; elongated pad tools, of course, sense intermediate volumes. The terms dual packer and straddle packer are used synonymously in this book.

The configurations displayed in Figure 1.1a represent a broad range of tools available from many oil service companies. Figure 1.1b displays more recent designs. A single-pad tool is shown at the far left for reference; next to it resides the elongated pad. A recent tool, sketched in (o), is equipped with elongated nozzles every 90° azimuthally. A significantly increased flow area allows rapid pumping and fluid collection without the large pressure drawdowns that would obtain otherwise. This is a significant development for low permeability formations. Finally, at the far right, we have focused probes. Traditional pads contain a single centered nozzle approximately one inch in diameter. Focused probes contain an additional concentric ring probe that surrounds the usual nozzle. These are controlled by independent hydraulic systems. Essentially, the ring probe creates a cylindrical curtain barrier that prevents mud entering through the mudcake from contaminating the central flow collection mechanism. This importantly shortens the time needed to obtain a clean in situ sample. The models in this book apply to all manufactures’ tools.

Figure 1.1b. Formation testers, recent developments.

We do point out interesting subtleties. Schematic (a) is a single-probe tool – its only existing probe not only pumps but serves as the observation probe as well. Such tools are typically used in formation-testing-while-drilling to minimize hardware complexity. When pressure measurements are interpreted using ideal source models of the problem, which assume spherical symmetry, one at most recovers a single permeability known as the spherical permeability ks. Note that spherical symmetry applies to isotropic media, whereas ellipsoidal symmetries apply to transversely isotropic problems (when we speak of spherical sources, we loosely refer to both situations). Source models with symmetries are but idealizations. In reality, the pad is applied to one point on the borehole wall. Schematic (a) shows a single probe, but this is not to say that a single permeability is at most possible. Now, a real pad extends at one point from a physical drill collar or wireline mandrel. If the single-pad tool can be rotated and operated at different azimuthal angles in an anisotropic formation and the corresponding pressures are recorded, then it is in principle possible to obtain both kh and kv inferred from a suitable three-dimensional model. Refer to Figure 6.16 and the accompanying discussion for more details.

On the other hand, schematic (b) shows two probes, a source probe that pumps (sometimes referred to as a sink probe) and a passive observation probe. The probes on such dual-probe tools are aligned axially at the same angular location. When pressure drops are available at both probes, both permeabilities can in principle be estimated, although with varying degrees of accuracy depending on the methodology used. These issues are raised and resolved in this book for different types of tools operating in different kinds of fluids. A logging string can be equipped with multiple source and observation probes, not unlike resistivity array tools with multiple transmitters and receivers.

Schematic (d) displays still another multi-probe configuration in which a bottom source probe appears diametrically opposite to a bottom observation probe. A second observation probe appears higher in the tool. This three-probe tool is useful in obtaining both kh and kv from pressure measurements, although to the author’s knowledge, the interpretation method for permeability prediction applies to vertical wells only. Schematics (h) - (l) show obvious counterparts involving dual packers – flow produced from cylindrical borehole surfaces contained by inflatable packers enters orifices in the tool body. The packers may be spaced, say one to tens of feet apart, so that for permeability prediction purposes, vertical resolution is limited. Large amounts of fluid may also be pumped from the packer into the formation with pressure signatures to be measured at distant pad-type probes. Schematic (g) displays an elongated pad, which may be used alone or in concert with one or more pad-type observation p probes. Figure 1.1c displays tool designs offered by CNOOC/COSL. A single-probe tool appears at top left while a dual-probe tool is shown at top center. One focused probe design is give at top right and in greater detail at the bottom.

Figure 1.1c. Example CNOOC/COSL hardware designs.

We also note that several point source liquid and gas models introduced in this book, for both forward and inverse problems, apply to tool strings with dual packer tools as well as to multi-probe pad tools. For this to be correct, the separation between packers should be small compared to the length joining the observation probe to the center point of the dual packer. If the two distances are comparable, more complicated three-dimensional analysis is required.

A number of excellent publications are available which provide historical surveys, develop best practices, and describe basic logging problems and advances more comprehensively than is possible in this monograph. Among these are the recent book Fundamentals of Formation Testing (Schlumberger, 2006) as well as the company’s earlier review articles, e.g., Ireland et al (1992), Crombie et al (1998), Andrews et al (2001), Ayan et al (2001) and Akkurt et al (2006). Also of interest are proceedings from two industry symposiums, namely, Wireline Formation Testing Technology Workshop, 38th SPWLA Annual Symposium, Society of Professional Well Log Analysts, Houston, Texas, June 15-18, 1997, and Formation Testing: Applications and Practices, Spring Topical Conference, Society of Petrophysicists and Well Log Analysts, Taos, New Mexico, Mar. 28 – April 1, 2004. However, none of these publications provides detailed math expositions such as those given in this book.

1.2 Existing Models, Implicit Assumptions and Limitations

The literature on well testing is highly developed, comprising of thousands of papers and dozens of books written over several decades. Authors have successfully modeled fully and partially-penetrating wells which may be vertical, deviated or horizontal, and have accounted for effects related to reservoir boundary shape, well-to-well interaction, and so on. A balanced exposition covering modeling and interpretation is offered in the work of Raghavan (1993). There, concepts related to wellbore storage and skin are developed – ideas that also arise in formation testing forward and inverse models. The book illustrates several modeling techniques and gives formulas useful in cylindrical-radial borehole pressure transient interpretation.

Fluid flow formulations for formation testing applications are not usually cylindrical-radial, but what we term spherical. By spherical flow, we refer to the spatially symmetric field about a point source in a homogeneous, infinite, isotropic medium, e.g., see Carslaw and Jaeger (1946) for discussions and early techniques for flowline storage modeling. When the formation is transversely isotropic, ellipsoidal symmetries apply – we often refer to these flows, for convenience only, as spherical also (of course, in the presence of boreholes, layers, or both, symmetries are generally no longer found).

In contrast to cylindrical flow, the literature on spherical flow is somewhat lacking. Early work on source point modeling for liquids is reviewed in detail in Chapter 5 and is quickly summarized here. Originally, source representations of the pressure field were singular at the origin r = 0, meaning that pressure formulas, which gave infinities at the source location, were unusable there and at early times – also, flowline storage and skin effects were not considered. The latter were later incorporated into improved source models that applied pumping boundary conditions at nonzero effective radii characterizing the nozzle – these models more realistically simulated pressure transient behavior near the source probe. Laplace transform methods were typically employed to solve these models, with inversion to the time domain often performed by inaccurate numerical routines. Problems involving boreholes and layers were (and are) typically solved numerically by finite element or finite difference analyses.

Chapters 5 and 6 obtain exact, analytical, closed form solutions for non-zero source radius models for isotropic and transversely isotropic media, for both forward and inverse problems, with flowline storage, with and without skin effects, and at any dip angle, assuming liquids. The work in these chapters represents state-of-the-art capabilities only recently available to the industry. We also address the important subject of formation testing while drilling, or FTWD, in which real-time pore pressures and spherical mobilities are determined from short time pressure transient data. Having said this, it is important to clarify the roles of these solutions to several recent methods that are popular but often misused and misquoted – and also to explain and resolve the limitations of all source-based models in general.

1.2.1 Exponential tight zone approximation.

The first of several exact source solutions for single-phase, liquid Darcy flows appeared in Proett and Chin (1996a,b). This initial model, which included the effects of flowline storage, assumed homogeneous isotropic media without skin damage. It importantly showed that a popular exponential time formula used to describe source probe pressure drops in tight zones, derived earlier from empirical considerations, followed as a special limit of the complete analytical solution (e.g., refer to Equation 5.128). We emphasize that the exponential model applies to the source probe only, and again, to tight, isotropic media without skin, and then, to zero dip applications.

Use of this model is very restrictive, however, when all requirements are met, it does allow permeability estimation at early times. Bear in mind that, if a dataset is marked by anisotropy, skin effect or both, interpreted results using this model will lead to incorrect or inconsistent predictions. Data processing procedures can sometimes add uncertainty to predictions. In some applications, pressure transient data is smoothed by eye before actual formulas are applied. Visual curve-fitting, performed differently by differently individuals and algorithms, provides a source of diffusion in addition to that in the reservoir. Thus, results can be questionable; it is preferable to apply interpretation formulas to limited sets of unsmoothed data and then average the results.

We might note, under the same assumptions, that Equation 5.130 provides a more accurate approximation in terms of the real complementary error function – rapidly computed solutions are easily implemented using algorithms in readily accessible publications, e.g., Abramowitz and Stegun, 1970. When possible, of course, complete exact solutions developed in this book should be used, which avoid the pitfalls associated with approximations.

The last paper in a series tackling problems of increasing physical complexity, published four years later by Proett, Chin and Mandal (2000), again allowing flowline storage, included extensions to transversely isotropic media and non-zero skin effects, but applied only at zero dip angle. This single-phase fluid model does not simulate the borehole and does not consider invasion. An analogous exponential formula was not derived and does not generally apply. We emphasize that the exponential model is not an exact model, but to the contrary, limited in scope as indicated. However, this is not important since the liquid flow formulas in Chapters 5 and 6 apply with few restrictions; Chapters 7 and 8 deal with general gas flows and applications.

1.2.2 Permeability and anisotropy from steady-state dual-probe data.

An important formation testing objective is permeability and anisotropy prediction from pressure drop data recorded at source and observation probes. We focus our discussion for now on homogeneous media and on methods using steady-state pressure drop data. These methods are not ideal. For example, they may require long wait times in low permeability formations, not to mention the use of inaccurate data with low dynamic resolution. In high permeability reservoirs, small pressure drops may be difficult to measure precisely.

We consider two typical dual-probe hardware configurations. In the first, shown in Figure 1.2, the bottom probe represents the source probe, which performs the pumping; the upper probe may also pump fluid, but very often, it is a passive observation probe. Probe separation is typically several inches. The second is shown in Figure 1.3, with a dual packer source at the bottom and an observation probe at the top. It is assumed, in order for source models to apply, that the distance from observation probe to dual packer center greatly exceeds the distance between packers – if not, numerical models are necessary. Liquid with a constant viscosity μ is pumped at a constant volume flowrate Q.

Figure 1.2. Dual-probe tester with idealizations.

Figure 1.3. Dual-probe tester with dual packer.

Let Rw denote the effective probe radius and ΔPs the steady-state source probe pressure drop relative to ambient conditions. For single-probe pad-nozzle tools such as those used in formation-testing-while-drilling, source flow modeling shows that at most one can obtain information about the spherical permeability ks = kh²/³ kv¹/³ only. In fact,

(1.1) equation

The foregoing formula holds at the source and is independent of dip angle. For dual probe systems, a popular method for kh and kv prediction requires the additional use of

(1.2) equation

where ΔPd is the observation probe pressure drop and L is the probe separation. Since we have two equations in the two unknown permeabilities, their values can be solved by simple algebra. It is important to emphasize that the second formula applies to vertical well applications only since it contains but kh – it does not apply at arbitrary dip angles. Pressures must fully equilibrate before the method can be used. While pressures attain steady-state relatively quickly at the source probe, they do not at observation probes, particularly in low permeability media. Furthermore, the method does not apply to gases, which satisfy equations nonlinear in pressure. Later in this book, we model gases generally. All source based models, however, fail in layered media when the source is located close to a boundary.

1.2.3 Three-probe, vertical well interpretation method.

The setup shown in Figure 1.4 is designed to interrogate the formation for kh and kv. The pumping probe, called the sink probe, appears at the bottom right, with a vertical observation probe positioned above it as in Figure 1.2. Diametrically opposed to the sink probe is the so-called horizontal observation probe. Formulas for kh and kv analysis are given in Goode and Thambynayagam (1992) and other related publications. These assume constant rate pumping, neglect the effects of flowline storage and skin, and like those given previously, are restricted to zero dip, vertical well applications. Like source probe formulations, the three-probe model applies to homogeneous media only.

Figure 1.4. Triple-probe formation tester.

1.2.4 Gas pumping.

In many reservoirs, formation testers will pump primarily gas. We emphasize that almost all formation testing pressure simulators in the literature have been developed for slightly compressible liquids – use of these computer programs for gas pumping (say, by entering a very small gas viscosity, a common industry practice) is not correct since the equations available until now do not model the pressure-dependent compressibility of the gas. Also, essential thermodynamic information, e.g., adiabatic or isothermal production, is not accounted for. Formulations for gas pumping are given in Chapters 7 and 8 for source models and for more complicated three-dimensional flows.

1.2.5 Material balance method.

An approach we will term the material balance method, originally introduced by Kasap et al (1999), in this author’s opinion, has led to some confusion in modeling and interpretation. However, its rationale and limitations can be clarified by returning to fundamental principles. We will attempt to understand the method by comparing with existing exact simulation models and then identifying where important formulation differences arise and have their greatest effect.

To understand the method, we first consider the simplest problem in Chapter 5 for formation testing, that of isotropic flow with flowline storage but without skin damage. Without flowline effects, the boundary condition is just (4πRw²) {k/μ ∂P(Rw,t)/∂} = Q(t), nothing more than spherical area x Darcy velocity = volume flow rate (here, Rw is the effective probe radius, k is permeability, μ is viscosity and Q(t) is the prescribed time-dependent volume flow rate at the probe). When flowline compressibility and storage are important, the volume flux (4πRw²) {k/μ ∂P(Rw,t)/∂r} is reduced by VC ∂P/∂t, where V is flowline volume and C is flowline fluid compressibility – the explanation is analogous to the one for wellbore storage in well testing (Raghavan, 1993). This flowline mass balance, used in both Chapter 5 and all prior cited references there, is the standard simulation boundary condition. In Chapter 5, (4πRw²k/μ) ∂P(Rw,t)/∂r – VC ∂P/∂t = Q(t) is given as Equation 5.4.

Note ∂P(Rw,t)/∂r is the derivative in Darcy’s law velocity = k/μ ∂P/∂r, and so, can be approximated by ∂P(Rw,t)/∂r ≈ (Pformation – Pprobe)/Δr, in which Pprobe(t) is the pressure at the probe, but Pformation is the formation pressure at a very small distance Δr from the probe – it is not the farfield formation pressure (we have merely invoked the definition of the spatial derivative). We substitute this into Equation 5.4 and re-organize the terms as follows:

(1.3)

equation

Next we turn to our evaluation of the material balance method. The flowline material balance equation

(1.4)

equation

is offered in Kasap et al (1999) where, according to the authors, p(t) is the pressure at the probe, p* is the formation pressure, qdd is the pumping volume flowrate, and csys and Vsys correspond to our flowline constants. The key idea in the Kasap et al (1999) analysis is to interpret Equation 1.4 as the straight line y = mx + b where m is the slope and b is the y-intercept. With p on the vertical axis and (csysVsys dp/dt + qdd) on the horizontal, the intercept would be the formation pressure p* while the slope would be directly related to μ/(kG0ri) – thus, knowing G0ri yields estimates for the mobility k/μ.

Equations 1.3 and 1.4 are almost term-by-term identical (note that G0ri replaces our 4πRw²/Δr, where G0 is an empirical dimensionless geometrical factor). However the derivation of our Equation 1.3 indicates that p* cannot possibly be the formation pressure, but is instead, the pressure in the formation at a small distance Δr away. In order for p* to represent true formation pressure, Δr would have to be infinite, in which case the ∂P/∂r appearing in Darcy’s law would not represent the derivative. By our argument, the meaning of p* is uncertain, although there is some latitude in its interpretation since the geometrical factor G0 is not rigorously defined: p* may well be correlated to the formation pressure, but what the exact correlation is, is not apparent. This author does, however, concur with Kasap et al (1999) in noting that Equation 1.4 is not restricted to buildup, drawdown or any particular range of permeabilities – like Q(t) in Equation 1.3, qdd may be a general function of time.

If we extend Equation 1.4 the benefit of the doubt, then the method could be a useful diagnostic tool. For instance, it implicitly assumes laminar flow and perfect pad contact – if field data do not satisfy it, the integrity of the data is questioned. Subsequent papers using computed field results have attempted to justify Equation 1.4. However, there is really no need to do this since the equation, a material balance statement like Equation 1.3, is independently invoked to constrain solutions to partial differential equations. Justification using simulation data, which must have been obtained using Equation 1.3 or an equivalent alternative, only provides a programming consistency check.

Our discussion does point to difficulties in the extension of the method to formulations bearing greater physical complexity. For example, Chapter 5 shows that isotropic flows with both storage and skin effects modeled, where S is a dimensionless skin coefficient, must solve Equation 5.76,

(1.5)

equation

that is, a boundary condition containing second-order mixed partial derivatives. This is also the case in well testing (Raghavan, 1993).

On the other hand, for transversely isotropic problems with storage considered but no skin, the volume flow rate 4πRw² k/μ ∂P/∂r noted above, per detailed analyses undertaken in Chapter 5, would be replaced by one with a term proportional to (4πr*w² Pref kv¹/² kh/μ) (∂p*/∂r*)w which includes the spherical permeability. What forms the extensions to Equation 1.4 would take in these limits, and in cases where anisotropy, storage and skin are all important, are not clear. Where and how would ∂²P/∂r∂t appear in the straight line plot, if at all? How would the measured slope relate to horizontal and vertical permeability? Are additional constraints needed? If the extensions involve an over-simplifying constant geometric factor G0, the method might conceal additional possible dependencies on kh and kv. We close with two notes. First, our explanation of p* as a near-probe formation pressure directly conflicts with the interpretation given in Kasap et al (1999) as farfield formation pressure. And second, because the method is not solved with a partial differential equation, it is not predictive in the sense of traditional methods, e.g., it cannot be used to calculate pressures a given distance away. In any event, it is always best to use both simulation results (from partial differential equation solutions) for history matching and boundary conditions separately for diagnostic analysis.

1.2.6 Conventional three-dimensional numerical models.

The wide availability of commercial three-dimensional finite element flow simulators might suggest that highly accurate simulations of formation testing are within the reach of all users if computational resources were not problematic. However, this is far from the truth. Formation testing involves non-standard boundary conditions which require custom development, e.g., flowline storage for dual-probe and dual packer tools, dynamic mudcake growth coupled to Darcy flow in the formation, mixed second-derivative skin damage models such as that in Equation 1.5, and so on.

All of these specialized models are derived in this book and implemented numerically – validation examples are offered to highlight the physics. Recent authors indicate that important parameters of physical interest have been difficult to model. For example, the axisymmetric immiscible flow simulator described in Angeles et al (2007), with mudcake coupling based on the invasion work of Chin (1995), ignores flowline storage and skin effects – even though storage is especially significant in the annular space within dual packers. The authors only note that, although not investigated here, skin damage can be readily implemented in the inversion method by using a similar approach to the multilayer formation example presented in this work. In addition, even though we ignore tool storage effects, the latter can be studied with time-variable flow rates of fluid production.

The paper by Liu et al (2004) states, for instance, that to model the tool storage effect is key to eliminating or minimizing the uncertainty in the calculated pressure response during the pressure transient period, prior to the onset of fully-developed radial flow. Rigorous wellbore modeling, however, is one of the most challenging aspects of reservoir simulation and often requires consideration of transient fluid flow. Thus, the authors instead use an approximation in which the drawdown or production rate is known or can be calculated explicitly, and the fluid flow is static or steady-state inside the wellbore or FT tool. Invasion is treated as a simple injection process rather than one dynamically coupled between formation and mudcake. Quoting from the paper, in our current model, we can impose a dynamic flow rate in this period through adjusting the thickness and permeability of the mud cakes. More accurate simulation of the mud cake, however, would require a dynamic cake model, a consideration for our future work. In this book, modeling of storage, skin and dynamic mudcake coupling are addressed from first principles, and various issues raised by recent authors are discussed and solved in detail.

1.2.7 Uniform flux dual packer models.

Dual packer sources are useful in the formation testing of fractured or unconsolidated formations – field applications in which pad-type nozzle samples may not be repeatable or effective in rock contact. Recent investigators have introduced uniform flux models to simulate flows from dual packers. When the total volume flow rate Q(t) is given for dual packer length L and a borehole radius R, the boundary condition used assumes a uniform radial velocity Q(t)/(2πRL) about the entire flow surface for mathematical simplicity. This completely neglects the effects of annular fluid compressibility. Also, specification of uniform velocity requires that pressure vary axially, which is untrue when the minimal effects of gravity are neglected.

In standard pad-type tools, understanding the role of internal flowline volume is significant in pressure transient interpretation. When the formation tester starts withdrawing fluid, two contributions to production enter the local mass balance: the compression of the fluid already in the flowline and the Darcy flow entering through the nozzle. The former is proportional to VC ∂P/∂t, where V is the volume of the flowline, C is the compressibility of the fluid within it, and ∂P/∂t is the time rate of pressure increase within the tool (the latter depends on kh and kv, the formation fluid compressibility c and the viscosity μ, plus spatial pressure gradient effects). When the tool stops pumping, the reverse, namely, the expansion of fluid in the flowline, is seen first. Thus pressure transducers in the tool feel flowline storage effects before they do formation effects. Flowline storage hides formation properties from pressure transient interpretation. One can generally argue that the greater the storage volume, the longer the waiting time must be before kh and kv effects can be detected in pressure traces – this lengthened wait time is associated with high well logging costs and increased risks in tool sticking. Pressure distortions associated with storage increase with increases in V and with decreases in permeability.

For dual packer tools, the flowline volume additionally includes the annular volume – this volume dwarfs that of the hardware and its effect on pressure must be understood and modeled accurately. In this book, we introduce a dynamically correct model: the total volume flow rate Q(t) is specified, in such a way that pressure is constant everywhere in the annular volume – this constant will generally vary with time, with its time-dependent values determined as part of the complete numerical solution. The model is derived in Chapter 4, where sample calculations are offered; a detailed analysis of one miscible flow simulation with respect to contamination and clean-up is given in Chapter 2.

Finally, we observe that while general commercial finite element or finite difference simulators may ultimately be constrained to account for all effects properly, they do not qualify as true interpretation tools because they lack portability and typically require expensive run-time licenses. The analytical and numerical models in this book, in contrast, are designed as portable standalone applications that efficiently address focused mathematical formulations with well-defined inputs and user objectives. Our methods clearly reveal key dimensionless physical parameters and how they control pressure. Recent summaries of our inverse modeling work are available in Chin (2013) and Zhou et al (2014a,b).

1.3 Tool Development, Testing and Deployment – Role of Modeling and Behind the Scenes at CNOOC/COSL

Very often, to those new to formation tester development, the scope and breadth of myriad design issues are not obvious. The formation tester is more than a pump: it is a sophisticated reservoir characterization instrument whose operating principles must be tuned to the laws governing fluid flow. At the same time, it must perform flawlessly in a hostile and unforgiving environment. In any final design, all of these factors must have been addressed in order to design the product successfully. While experienced engineers are familiar with these considerations, students and newcomers may not be. This final section of Chapter 1 provides a glimpse of the challenges, in both mathematics, hardware and field work that confront design teams.

1.3.1 Engineering analysis, design challenges, solutions.

Designing a formation tester from the ground up poses significant engineering questions. What properties characterize the formation, for instance, permeability, porosity, anisotropy, degree of matrix consoliation, and so on, and what properties does the underlying fluid possess, e.g., oil viscosity, in situ fluid contaminated by mud filtrate, potential gas liberation, and so on? What pressure drops are needed at the pump so that their effects can be monitored at an observation probe? What kinds of pump are actually available for hardware use? How is anisotropy determination affected by dip angle? Are steady-states realizable? When do steady and transient interpretation models apply? If used in downhole Measurement-While-Drilling (MWD) or Logging-While-Drilling (LWD) applications, the algorithm itself becomes a central issue. Does it require significant data processing? Memory storage? How do integrated circuits deteriorate with temperature, pressure and corrosion? What alternatives are available? What about mechanical complexity? How do shock and vibration considerations affect component layout? Then, of course, there’s the flowline … the culprit behind many a failed engineering project.

1.3.2 From physics to math to engineering – inverse problem formulation.

Despite the importance of inverse problems in formation evaluation, the majority of petroleum engineering students is not likely to encounter the mathematical formulation that scientists ultimately address in formation tester algorithm research. Many are likely to cite Darcy’s law "q = – k/μ ∇p and assume that this statement alone contains the answer. An appreciation for the analytical subtleties and nonunique solutions that are part of the problem can be developed by focusing on all-important formation testing while drilling" objectives requiring real-time pore pressure and spherical mobility. The problem statement, as we present it next, will challenge any mathematics professor. The constraints are more than the familiar initial-boundary conditions introduced in graduate courses. In fact, the formulation looks something like –

1.3.2.1 Simplified Theoretical Model Assumptions

Let x and y represent coordinates parallel to the bedding plane, with z being perpendicular; kh and kv denote horizontal and vertical permeabilities; ϕ, μ and C denote porosity, liquid viscosity and compressibility; p0 represent pore pressure; V denote flowline volume; and p(x,y,z,t) be the measured transient Darcy pressure. Then, the governing equations are

where the local velocity is q = kh ∂p/∂x i + kh ∂p/∂y j + kv ∂p/∂z k with i, j and k being the usual unit vectors in the x, y and z directions, dS is an incremental area on any control surface Σ surrounding formation tester, n is the unit normal to Σ, and Rw is the nozzle radius.

Despite the apparent generality, this formulation is nonetheless limited: it does not apply to single-phase gases, miscible or immiscible multiphase flows, problems with somewhat permeable mudcakes, actual borehole and pad geometries, and so on. Many of these issues are discussed in this book – however, we wish to emphasize here that even the most intimidating models will not address all applications. The inverse FTWD formulation is simply stated.

Problem

Given Q(t), V, Rw, and three-point (t1, p1), (t2, p2) and (t3, p3) data …

Find p0, kh/μ, kv/μ and C

Constraints

Very fast computing speed (seconds)

Compact code (suitable for downhole microprocessor)

Accurate predictions (confirm by exact forward analysis)

Does not infringe existing patents (legal)

All