New Dimensions in Modeling Resistivity

Over sixty years ago, resistivity modeling

e m e rged as a means to design electrical

logging tools and interpret their

responses in simple, layered form a t i o n s .

Computational advances now make it

possible to rapidly predict unexpected

responses from modern tools in complex

f o rmations, and to more accurately

extract true resistivities and stru c t u r a l

geometries from resistivity logs.

Spring 1997 41

B a r b a ra A n d e r s o nVladimir DruskinTarek HabashyPing LeeMartin LülingRidgefield, Connecticut, USA

Tom BarberGreg GroveJohn Love l lR i chard RosthalJacques Ta b a n o uSugar Land, Texas, USA

D avid Ke n n e dyMobil E&P Te chnical CenterDallas, Te x a s

Liang ShenU n iversity of HoustonHouston, Te x a s

In 1927, Conrad Schlumberger changed thecourse of petroleum exploration when hepurposefully tipped his surface electricalprospecting array vertically and sent it log-ging down a wellbore.1 Existing resistiv i t ymodeling methods were applied to the newgeometry to predict the vertical arrayr e s p o n s e .2 Thus modeling is not new; but itis finding new uses.

In the logging ve r n a c u l a r, the term model-ing refers to computing a logging instrumentresponse in the presence of the env i r o n m e n tsurrounding the logging instrument. Th emodel attempts to capture all the detailrequired to account for and duplicate ano b s e r ved instrument response, and it is usedas an aid to log interpretation or tool design.

A resistivity model consists of a resistiv i t ydistribution for the environment and adescription of the instrument sensors. Th er e s i s t ivity distribution includes the resistiv i t yof various borehole and formation regionsand locations of their boundaries. Depend-ing on the logging situation, the boundariesmight include the locations of geologicalbedding planes, borehole wall, inva s i o nfronts, and the relative angle between all ofthese and the instrument axis. The result of

modeling is a synthetic tool response, orlog, wh i ch can be compared to an observe dlog for interpretation or to the resistivity va l-ues in the model for tool design (see “Whatis a Model?” page 43) .

In the early days of resistivity modeling,scale models and analog computers—forexample resistor networks—helped to ch a r-acterize tool responses in cases too compli-cated for mathematical models, such aswhen radial and vertical resistivity va r i a t i o n swere to be considered simultaneously(a b ove). The resistor network became one ofthe earliest applications of resistivity modelsin the creation of log correction ch a r t sdescribing the effects of one-dimensional(1D) and two-dimensional (2D) env i r o n-mental heterogeneities (such as changes inthe vertical direction like shoulder beds andin the radial direction as with invasion) onlogs. Typical cases were computed for thecharts, but intermediate or extreme casesencountered in practice required interpola-tion within a chart or among seve ral ch a r t s .The draw b a ck to this kind of modeling wa sthe difficulty in varying the parameters ofthe problem; each new case required a newscale model to be built or hundreds to thou-sands of resistors to be exch a n g e d .

With the development of digital comput-ers, direct numerical solutions of the elec-tromagnetic equations can be used to com-pute resistivity response in models hav i n g

complex geometries. To ach i e ve this, theseequations are approximated by a large sys-tem of simultaneous equations; the next taskis to find numbers that simultaneously sat-isfy all the resulting equations. The geomet-rical restrictions are much less severe thananalytical and scale models. Realistic prob-lems can be formulated, and the solutionsappear as simple and unthreatening num-bers rather than as arcane and difficult-to-compute mathematical functions. Unfortu-n a t e l y, the computational burden isenormous; solutions for realistic three-dimensional (3D) problems by this methodrequire extensive machine computation (see“ The Vocabulary of Resistivity Modeling,”page 46). On the other hand, mach i n ecapability has steadily improved for morethan 50 years, and it is now possible toobtain useful numerical solutions to for-merly intractable problems.

Most log analysts appreciate that tool-response modeling plays an important rolein instrument design. Less well recognizedis that both the apparent resistivity recordedby the logging tool itself, and the va r i o u scorrections thereto, are derived from mathe-matical models. The models help relate toolresponses to true formation resistiv i t y, Rt.

■Resistor network. The resistor networks were analog computers for two-dimensional (2D)resistivity models. These devices led to predicting tool responses relative to true form a t i o nresistivity. The last resistor networks used at Schlumberger were two racks about 6 ft [1.8 m] tall and 12 ft [3.6 m] long with nearly half a million re s i s t o r s .

1. S chlumberger AG: The Schlumberger A dve n t u r e . N e wYork, New York, USA: ARCO Publishing, Inc., 1982.

2. Allaud LA and Martin MH: S ch l u m b e r g e r The Historyof a Te ch n i q u e. New York, New York, USA: Jo h nWiley & Sons, 1977.

For help in preparation of this article, thanks to Dav i dAllen, Robert Freedman and Koji Ito, Schlumberger Wi r e-line & Testing, Houston, Texas, USA; and T. S. Ramakrish-nan, Sch l u m b e r g e r-Doll Research, Ridgefield, Connecti-cut, USA.AIT (Array Induction Imager Tool), CDR (CompensatedDual Resistivity), DIL (Dual Induction Resistivity Log),DLL (Dual Laterolog Resistivity), FMI (Fullbore Fo r m a t i o nMicroImager), INFORM (Integrated Fo r ward Modeling),MicroSFL, Phasor, PL AT F O R M EX P R E S S, PrePlus, RAB (Resis-t ivity-at-the-Bit) and SFL (Spherically Focused Resistiv i t y )are marks of Sch l u m b e r g e r. RtBAN is a mark of Z&SConsultants, Inc.

42 Oilfield Review

In the last decade, modeling provided thebasis for automatic environmental correc-tion algorithms such as Phasor processing,and is the basis for the log products of theAIT A r ray Induction Imager Tool. For both ofthese tools, shoulder effect and thin-bedresponse are corrected to the resolution lim-its of the tool at low dip angles. Furthermodel-based processing allows responsecorrections for moderate dip angles (< 50°)for these tools.

Although modern resistivity tools performmost environmental corrections as the logis being recorded, oil company arch ives arebulging with logs run with older DIL DualInduction Resistivity Log and SFL Spheri-cally Focused Resistivity tools and equiva-lents from many service companies. Fo rthese tools, modeling is the only way tounderstand the complex effects that causedtheir logs to read completely wrong inm a ny situations.

Tool-response modeling, as practiced inmodern log interpretation, extends the inter-pretation of instrument responses beyo n dthe simple resistivity distributions env i s a g e din the apparent resistivity function and itsfirst-order corrections—as provided in ch a r t-books—to more realistic geometricala r rangement of the borehole and formatione nvironment. The idea is to discover theprobable distribution of resistivity thatwould give rise to an observed toolresponse. In many cases, only tool-responsemodeling can decode what the instrumentresponses reveal about the formation beinglogged—in many cases, if not most, intu-itions gained from study of, and experiencewith using, charts are misleading, becausethe charts are not applicable.

R e s i s t ivity modeling was reviewed in O i l-field Review f ive years ago.3 The two majora dvances chronicled at the time weregeosteering with logging-wh i l e - d r i l l i n g( LWD) resistivity tools and the combinationof analytical and numerical modeling meth-ods to speed calculations. At that time, twoneeds were identified. First, modelersrequired enhanced computational capabilityto develop models in the 3D domain formore realistic analysis in complex forma-tions. Such improve m e n t s — a l r e a dy on theh o r i z o n — would, in part, come from deve l-oping new advanced algorithms that speedup lengthy calculations. Second, furthert e chnological improvements were neededto help log analysts model logs quick l yenough to be practical. Some of this capa-bility was already under development insome oil company research groups.

This article discusses the progress made in the past five years. We first show how 3D modeling is used for strongly dippingformations and horizontal well eva l u a t i o n .Then, we discuss the impact of recent com-putational breakthroughs in modeling capa-bilities and practical interpretation usingr e s i s t ivity modeling.

Why Model Resistivity? R e s e r ves estimates are based on log-derive dmeasurements of formation resistiv i t y. A l lother factors being constant, as the forma-tion hydrocarbon content increases, so doesthe formation resistiv i t y. In fact, using Rt,A rch i e ’s law allows computation of the fra c-tional volume of the formation containinghydrocarbons. How e ve r, a few opera t o r suse raw resistivity logs as a hy d r o c a r b o ni n d i c a t o r — where resistivity is high, theyp e r f o rate the well. This is not the best use ofr e s i s t ivity data.

The log analyst looking at resistivity logsfrom older tools—uncorrected for env i r o n-mental effects—often has little to serve as aguide for resistivity interpretation. Manytimes the uncorrected resistiv i t y, for exam-ple from the deep induction tool, is consid-ered an accurate enough estimator of Rt.This may sometimes be true, but all resistiv-ity instrument responses, whether frominduction or laterolog tools, are influencedby the resistivity distribution in a large vo l-ume surrounding the logging instrument.Correct interpretation would require manycorrections for bed boundaries, borehole,i nvasion and other environmental or geo-metrical effects.

This means that what looks like the best,or highest, resistivity reading on a rawinduction log may simply be an artifact ofsome nearby boundary layer of contra s t i n gr e s i s t iv i t y. Many high-resistivity anomaliesdo not correspond to resistive beds at all,but rather to the interface between twobeds, each with lower Rt than the apparentr e s i s t iv i t y, Ra, indicated on the log. Undersimple enough conditions, such as in a ve r-tical well through thick beds with simplei nvasion, chartbook corrections can suffice.4For these applications, many software sup-pliers offer chartbook-based corrections forborehole and invasion for the DIL tool.

Wells are not always ideal for logging.L ow-cost drilling efforts frequently lead toboreholes that are not uniform. Deep inva-sion also affects tool responses. In the lastdecade, thousands of deviated and horizon-tal wells have been drilled to optimize pro-d u c t iv i t y. These wells have logging tool-to-formation orientations not contemplated inthe design of the apparent resistiv i t y

response and the corresponding ch a r t b o o kcorrections. Resistivity responses in suchformations appear counterintuitive, eve nweird, and are not correctable by ch a r t-book algorithms.

H ow e ve r, the physical principles—embod-ied in Maxwell’s equations—of such instru-ment responses are well understood, andwith modeling capable of honoring enoughdetail of the tool and environment, suchresponses are predictable. Artifacts such aspolarization horns, wh i ch appear as larget ransient overshoots in the tool resistiv i t yresponse at bed boundaries, can be under-stood on the basis of such models.

Another Dimension in Resistivity ModelingThe growing interest in complex formationgeometries, such as dipping beds, inva s i o nand anisotropy, has led to progress in deve l-opment of sophisticated models andenhanced computational efficiency. A sinterpretation problems have become moredetailed, model requirements have grow nfrom 1D, to 2D, and finally to 3D codes, tohandle the geometry of the problems athand. Significant recent developments havebeen the use of new numerical tech n i q u e s ,wh i ch make solutions in 3D models muchf a s t e r. Now, they are beginning to be usedfor difficult log interpretation problems.5

Compared to current methods, early mod-eling codes were slow in execution andtherefore cumbersome to use for forwa r dmodeling. Models had to be modified ate a ch iteration by editing detailed text files,and comparison of observed and syntheticresponses required hard-copy plotting of thesynthetic log, and visually inspecting for dif-ferences at each itera t i o n .6 N e vertheless, thevalue added was significant.

3. Anderson B, Minerbo G, Oristaglio M, Barber T,Freedman B and Shray F: “Modeling ElectromagneticTool Response,” Oilfield Review 4, no. 3 (July 1992):2 2 - 3 2 .Gianzero S: “The Mathematics of Resistivity andInduction Logging,” originally published in The Te ch-nical Review 29, no. 1 (March 1981): 4-32;reprinted in R e s i s t ivity Logging. SPWLA Reprint Vo l-ume Series 33, Houston, Texas, USA: University ofHouston (June 1992): 1-19.

4. The logging corrections are for parasitic effectsincluding adjacent beds, simple invasion of drillingmud, borehole, and skin effect (wh i ch causes a nonlinear decrease in signal strength in conductivef o r m a t i o n s ) .

5. Koelman JMVA, van der Horst M, Lomas AT, Ko e l e m i jAT and Bonnie JHM: “Interpretation of Resistiv i t yLogs in Horizontal We l l s ,” Transactions of the S P W L A37th Annual Logging Symposium, New Orleans,Louisiana, USA, June 16-19, 1996, paper G.

6. Anderson B, Barber T, Singer J and Broussard T:“ELMOD—Putting Electromagnetic Modeling toWork to Improve Resistivity Log Interpretation,”Transactions of the SPWLA 30th Annual LoggingS y m p o s i u m, Denve r, Colorado, USA, June 11-14,1989, paper M.

P redicting the response of a logging tool is a com-

plicated, detailed process. The boundaries of

regions of differing resistivity (or other material

e l e c t r omagnetic pr o p e rties) and the pr o p e rt i e s

within each region must by specified by the model.

The tool must also be introduced into the model

with its transmitters and receivers located at spec-

ified points, with the transmitters exciting the

model medium in some specified manner. The

interaction of the disturbance produced by the

transmitters proceeds through the medium and is

detected by the tool’s receiver coils or electrodes.

The response of the each receiver is thus pre-

dicted. Resistivity modeling belongs to the class of

so-called boundary-value problems. In r e l a t i v e l y

simple cases, these problems may yield analytic

f o rmulas, but, in general, a system of equations

must be solved explicitly (though appr o x i m a t e l y )

for each new case considered—such as when the

tool is moved or the model changed—and the

result is a set of numbers rather than a for m u l a .

This type of mathematical modeling is called

numerical modeling. Two examples of numerical

modeling in use today are the finite-element

method (FEM) and finite-dif f e rence method (FDM).

Given a system of equations governing a tool

response, it is straightfor w a rd to compute the tool

response. One simply plugs in model parameters

and after computation, the desired tool r e s p o n s e

quantities are written down. The result is an ar r a y

of tool responses that would be observed if a physi-

cal experiment were per f o rmed. This is called the

f o rw a rd p roblem. For cases where the gover n i n g

equations are linear in the resistivities, this pro-

cess is partially reversible—given the tool

response, the model parameters can be estimated

by multiplying the vector of observed tool

responses by the generalized inverse of the same

matrix used in the corresponding for w a rd pro b l e m .

This is called the i n v e r s e p roblem.

T h e re are variations upon this theme. For some

systems, the output of the system—tool r e s p o n s e —

can be re p resented as the mathematical operation

of c o n v o l u t i o n of the tool input with the impulse

response of the tool. Given the output (a log), the

input (a formation) can be correctly determined if

the transfer function (the Fourier transform of the

impulse response) does not pass through zero for

any value of its argument. Under this restriction, the

input is determined from the output by the pr o c e s s

known as d e c o n v o l u t i o n.

U n f o rt u n a t e l y, this is not the case for many

tools. For example, the 6FF40 is known to have a

blind frequency corresponding to about 5-ft inter-

v a l s . 1 As a result, there is a nonunique inver-

sion—for practical purposes, the thin-bed

response at blind frequencies turns out to be indis-

tinguishable from the whole-space r e s p o n s e .

M o re o v e r, the response of resistivity tools is

usually nonlinear with respect to formation r e s i s t i v -

ity because of skin effects. Thus, in the strict sense

mentioned above, neither deconvolution nor inver-

sion is possible—at least not in a single, deter m i n-

istic step. Fur t h e rm o re, the for w a rd problem itself

is relatively difficult to solve. It typically is r e p re-

sented by a large system of equations. The solution

of the equations may be relatively straightfor w a rd ,

but since the system is large, extensive computa-

tional r e s o u rces and time are r e q u i re d .

The inversion or deconvolution of a nonlinear

system re q u i r es that the for w a rd problem be “lin-

earized” by some means. Phasor processing of an

induction log response provides an example of lin-

earizing an instrument response. The induction

t o o l ’s receiver in-phase or resistive voltage, the R-

signal, is deconvolved using a filter designed to

c o rrect the zero resistivity response. The tool’s

q u a d r a t u re or reactive voltage, the X-signal, is

p rocessed with filters and fitting functions to

a p p roximate the dif f e rence between the actual R-

signal and a linear estimate of the R-signal. 2 T h e

two are added, and the result is a linear output

with the shoulder effect corrected. In enhanced

resolution Phasor processing, the log resolution is

c o rrected for beds as thin as 3 ft [1 m].

A more versatile technique finds a solution by

iteratively solving the for w a rd problem. If the itera-

tions are per f o rmed manually, the process is called

f o rw a rd modeling. If the iterations are carried out

by an algorithm seeking to minimize the dif f e r-

ences between the observed log and a synthetic log

calculated in a model by adjusting the model to

reduce the dif f e rences at each step, the process is

called inverse modeling. Note that for either tech-

nique, an important part of the process is an ef f i-

cient means of solving the for w a rd problem so that

the dif f e rences can be obtained rapidly and the

iteration carried for w a rd as fast as possible.

Equally important, especially for inverse model-

ing, is the construction of the initial model. If the

initial model is good enough, the dif f e re n c e s

between observed and synthetic logs vanish on the

first comparison. Failing this, the closer the initial

model is to the actual formation, the fewer the

iterations r e q u i r ed, and the final solution is

reached rapidly. Thus, the best possible initial

model is essential to inversion. An initial model

would be based upon the highest resolution data

available plus any available estimates—such as

What is a Model?

1. Kennedy WD: “Induction Tool Forw a rd Modeling: A Rigor-ous and Systematic Approach to Model Constru c t i o n , ”Transactions of the SPWLA 36th Annual Logging Sympo-s i u m, Paris, France, June 26-29, 1995, paper G.

2. Anderson BI and Barber TD: Induction Logging. SugarLand, Texas, USA: Schlumberger Wi reline & Testing, 1997.

Spring 1997 43

using FMI Fullbore Formation MicroImager r e a d-

ings to get bed boundaries—of radial r e s i s t i v i t y

variation within the bed boundaries.

The type of inverse modeling described above is

known as m o d e l - b a s e d inversion. This means that

the form that the solution is most likely to take is

decided in advance, and only solutions of that

f o rm will be found by the technique.

D i ff e rences Between FEM and FDM

In the finite-element method, Maxwell’s equations

a re written as integral equations of the r e s i s t i v i t y

and electromagnetic fields. The next step is a pro-

cess of discretization throughout the medium sur-

rounding the tool and formation—the step of con-

v e rting continuous equations to a finite

dimensional system of equations that can be

solved with a digital computer. For FEM, dis-

c retization is done based on a variational princi-

ple. For example, this could mean the total ener g y

of the system is minimized at points on a gridded

network throughout the media environment ( a b o v e

r i g h t ). This is a reasonable way to solve the pr o b-

lem, since the static electromagnetic potentials

always adjust themselves to seek a minimum

e n e rg y. The topology of the discretization is usu-

ally selected to conform with the problem geome-

t ry. The discretization step leads to a large set of

simultaneous equations for the electr o m a g n e t i c

field in terms of the media resistivities, which are

re p resented by a set of large matrices.

The flexibility of FEM translates to complexity in

numerical implementation and the codes can be

m o re difficult to optimize. The form of the matrices

is unstru c t u red, having its nonzero elements

s p read throughout the matrix, because these

matrices reflect the topology of the discr e t i z a t i o n . 3

The final step is to find a solution to the matrix

equations r e p resenting the electromagnetic fields.

For a problem of a given size, the FEM method is

capable of the highest conformity to geometrical

complexity of any numerical method, but is com-

putationally slow.

The finite-dif f e rence method uses discr e t i z a t i o n

based on a direct dif f e rence approximation of the

d i ff e rential operator form of Maxwell’s equations.

This leads to a grid r e q u i r ement restricted to a

C a rtesian topology ( r i g h t ). Although the grid topol-

ogy does not conform to the geometry of the r e s i s -

tivity discontinuities found in the formation envi-

ronment, it can be made to approximate this

g e o m e t ry. The real benefit of this approach lies in

the next step, in which the matrix equations for the

d i s c retization of the dif f e rential equations are usu-

ally str u c t u red because of the Cartesian topology,

and always sparse because the derivatives are

local operators. With these advantages, matrices

lend themselves to fast, specialized computational

methods that allow rapid solutions to r e s i s t i v i t y

models with extremely complex geometry. 4

■Grid and coordinate system used for 3D finite-dif-ference method calculations. The staircase showshow a dipping boundary is implemented in the Carte-sian grid system. The material properties (resistivity)of the cells intersected by the staircase boundary areaveraged in the calculation.

■Typical 2D finite-element grid for resistivity prob-lems. The grids used for defining the nodes used infinite-element models usually conform to the naturalgeometry of the environment, in which edges andintersections between different geometrical regionswith different properties (resistivities) lie on the grid.

3. Lovell, re f e rence 12, main text.

4. Druskin V, Knizhnerman L and Lee P: “Solutions ofM a x w e l l ’s Equations Using Krylov Subspaces from InversePowers of the Stiffness Matrix,” presented at the Thirt e e n t hAnnual Review of Pro g ress in Applied Computational Elec-t romagnetics, Montere y, California, USA, March 17-21,1 9 9 7 .

44 Oilfield Review

Spring 1997 45

Consumers of well log data concludedthat to gain maximum benefit from resistiv-ity logs, the numerical modeling of toolresponses would be the standard for the1990s. Seve ral oil companies, among themShell, BP, Chevron and A R C O, began tod e velop or acquire in-house modeling capa-bilities. And one major oil company, Mobil,formed a task force of researchers, pra c t i c-ing petrophysicists, computer scientists andacademic consultants to make logging toolresponse modeling convenient, pra c t i c a land routinely applicable. This commitment,combined with seve ral technical break-throughs, brought rapid success to this effort(see “An Efficient User Interface for Resistiv-ity Modeling,” page 49).

As early as the late 1980s, computationale f f i c i e n cy was benefiting from the newlyd e veloped fast Hankel transform (FHT)applied to induction log modeling, wh e nR i chard Hardman invented a method ofoptimizing the use of the FHT.7 The numberof computations was further reduced byexploiting various symmetries of the induc-tion antenna array and reciprocity. Ava i l a b l eby 1991, the resulting 1D computer pro-g rams were from 100 to 600 times faster,depending on dip, than the standard codesused by University of Houston Well LoggingL a b o ra t o r y.8 A recent revision to Hardman’s1D induction modeling will be available inGeolog6 software from MINCOM, Pty. Ituses shallow measurements, such as SFLlogs, as input to assign bed boundaries.Model viewing and editing are performedrapidly through a graphical interface.

During the last decade, at the Centra lG e o p hysical Expedition in Moscow,Vladimir Druskin and Leonid Knizhner-man—using 3D modeling techniques origi-nally developed for surface electromagneticprospecting problems—developed a break-through in a new, efficient 3D finite-differ-ence method (FDM) for resistiv i t ym o d e l i n g .9 Their approach, along with otherrecent developments in their mathematicalt e chniques, has led to remarkable improve-ments in the speed and accura cy of full 3Dr e s i s t ivity modeling capability for bothinduction and 2-MHz propagation tools.1 0

Horizontal and Highly Deviated We l l s —3D Resistivity ModelingInterpretation of resistivity tools in horizon-tal or highly deviated wells is far more com-plicated than in vertical wells, particularlywhen drilling-induced resistivity ch a n g e sare important to the tool response. Resistiv-ity tools have been designed for focusing invertical wells, where horizontal layers andaxial symmetry are assumed. When theseconditions hold, interpretation proceduresare stra i g h t f o r ward—the recommended pro-cedure is to apply borehole corrections,process to improve or match vertical resolu-tion, and correct for invasion, in sequence.Although dip-induced shoulder effects canbe corrected for angles up to 60°, in pra c-tice invasion complicates the inter-p r e t a t i o n .1 1 At higher angles, shouldereffects can be different on all sides of thesonde, and invasion may be highly nonsym-metrical. As a result, apparent resistiv i t yresponse interpretation using the standardmethods is not possible in horizontal andhighly deviated wells.

For laterolog-type tools, such as the DLLDual Laterolog Resistivity tool and the RABR e s i s t ivity-at-the-Bit tool, effects of the bore-hole and possible invasion cannot beignored. In the presence of dip, such toolsmust be modeled with 3D codes. Fo r t u-n a t e l y, their low - f r e q u e n cy operation meansthat they can be modeled with Po i s s o n ’s orL a p l a c e ’s equation—a significant simplifica-tion of Maxwell’s equations—and finite-ele-ment method (FEM) codes for these equa-tions are relatively fast to compute. Fo rexample, a full 3D FEM solution for the DLLresponse in the presence of dip, pinch o u t s ,a n i s o t r o py, invasion and tool eccentricitytakes 10 to 15 seconds per tool position.1 2

For induction-type tools, such as the A I Tresponse and CDR Compensated DualR e s i s t ivity response, the effects of resistivemud drilling-induced alterations and bore-hole signals can be negligible, but the fre-q u e n cy effects on the tools are not. Thus, thefirst attempts to model their response inhighly deviated wells were based on solvingfor the electromagnetic fields in layered 1Dgeometries, wh i ch omitted both boreholeand invasion. These early attempts have metwith reasonable success.1 3 This class ofmodels is widely applicable, fast to com-pute, and commercially available. Subse-quent efforts based on 2D models haveincluded a borehole and axisymmetric inva-sion; these are also commercially ava i l a b l eand fast enough to be practical for con-

structing model-based interpretations.U n f o r t u n a t e l y, nonaxisymmetric inva s i o nalso exists, even at early stages of inva s i o nencountered by LWD tools.1 4 E ven wh e ni nvasion is axisymmetric, in the presence ofdip this geometry requires a 3D solution.

Fast FDM modeling techniques are nowavailable for solving the full 3D electromag-netic problem necessary for interpretinginduction logs in deviated wells with inva-sion, and for understanding tool responsesin other difficult environments includingcomplex fracture systems, faults and pin-chouts, and effects of resistivity anisotropy.Fi n a l l y, realistic logging problems, withcombined effects of invasion, dip and shoul-der beds, can be analyzed within reason-able time using the new 3D FDM inductionf o r ward-modeling progra m s .1 5

7. Richard Hardman, personal communication, 1997.8. Anderson B and Gianzero S: “Induction Sonde

Response in Stratified Media,” The Log A n a l y s t 2 4no. 1 (Ja n u a r y - February 1983): 25-31. Anderson B, Safinya KA and Habashy T: “Effects ofDipping Beds on the Response of Induction To o l s ,”paper SPE 15488, presented at the 61st SPE A n n u a lTe chnical Conference and Exhibition, New Orleans,Louisiana, USA, October 5-8, 1986.

9. Druskin V and Knizhnerman L: “A Spectral Differen-tial-Difference Method for the Numeric Solution of3D Nonstationary Problems of Electrical Prospect-i n g ,” Izvestia A c a d e my of Sciences, U.S.S.R. P hy s i c sof Solid Earth 24, no. 8 (1988): 641-648. (Englishedition published by American Geophysical Union.)

10. Druskin V and Knizhnerman L: “Krylov SubspaceApproximation of Eigenpairs and Matrix Functions inExact and Computer A r i t h m e t i c ,” Numerical LinearA l g e b ra with A p p l i c a t i o n s 2, no. 3 (1995): 205-217.See also Druskin V and Knizhnerman L: “Spectra lA p p r o a ch to Solving Three-Dimensional Maxwell’sDiffusion Equations in the Time and Fr e q u e n cyD o m a i n s ,” Radio Science 29, no. 4 (Ju l y - Au g u s t ,1994): 937-953.

11. Hardman RH and Shen LC: “Charts for CorrectingEffects of Formation Dip and Hole Deviation onInduction Logs,” The Log A n a l y s t 28, no. 4 (Ju l y -August,1987): 349-356.

12. L ovell JR: Finite Element Methods in Resistivity Log-g i n g. Ridgefield, Connecticut, USA. Sch l u m b e r g e rTe chnology Corporation, 1993.

13. Hardman RH and Shen LC: “Theory of InductionSonde in Dipping Beds,” G e o p hy s i c s 51 (March1986): 800-809.Anderson et al, 1986. reference 8.

14. H oward AQ and Chew WC: “Electromagnetic Bore-hole Fields in a Layered, Dipping-Bed Env i r o n m e n twith Inva s i o n ,” G e o p hy s i c s 57 (March 1992): 451-4 6 5 .

15. Anderson B, Barber T, Druskin V, Lee P, Dussan V,Knizhnerman L and Dav y dy ch e va S: “The Responseof Multiarray Induction Tools in Highly Dipping Fo r-mations with Invasion and in A r b i t rary 3D Geome-t r i e s ,” Transactions of the SPWLA 37th Annual Log-ging Symposium, New Orleans, Louisiana, USA,June 16-19, 1996, paper A .

(continued on page 49)

46 Oilfield Review

Geometrical Dimensions in Modeling

The physics of resistivity modeling is described by

M a x w e l l ’ s equations. Depending on the assump-

tions made about the spatial distribution of r e s i s -

t i v i t y, these solutions can take the form of simple

f o rmulas, giving the electromagnetic field for any

d e s i red point in space. In complicated cases, the

field cannot be expressed in terms of simple for-

mulas, but must be laboriously computed at many

points in space simultaneously. The degree of

complexity is customarily summarized by r e f e rr i n g

to geometries of differing dimensionality—zer o -

dimensional (0D), one-dimensional (1D), two-

dimensional (2D) and three-dimensional (3D).

One way to decode this jargon is to consider how

resistivity varies in a coordinate system. 1 C o n s i d e r,

for example, the cylindrical coordinate system

specified by the coordinates ρ, φ and z. If the re s i s-

t i v i t y, R, is constant ever y w h e re, then it is r e f e rre d

to as 0D. If the r e s i s t i v i t y , R, in a geometry varies

with only a single coordinate direction, then that

g e o m e t ry is re f e r red to as 1D. The familiar layer-

cake geometry of sedimentary geological str u c t u re

is an example of how the resistivity varies only in

the z- c o o rdinate direction or, R = R(z) (below left).

In essence, each layer has a constant r e s i s t i v i t y ,

but the resistivity varies between the layers.

Another commonly used geometrical ar r a n g e-

ment in well logging features a cylindrical wellbore

s u rrounded by coaxial cylindrical shells bounding

cylindrical regions of differing resistivity ( l e f t ). In

this case, R = R(ρ). Thus, 1D geometries are either

v e rtically or radially layered geometries. 2

S i m i l a r l y , if resistivity varies in the vertical and

radial directions and is axisymmetric, then the

g e o m e t ry is two-dimensional (below right). In this

case, R = R(ρ, z) in the notation introduced above.

When R = R(ρ, φ, z), the geometry is said to be

t h ree-dimensional (next page, top). This system of

n o m e n c l a t u r e can be extended without modifica-

tion to cases in which resistivity at a point varies

with dire c t i o n — resistivity is a tensor.

To illustrate this, Maxwell’s equations are for-

mulated for the electric field of a magnetic dipole

in an infinite and homogeneous conductive

medium, and the result is a simple formula. The

medium is zero-dimensional, and the formula is

the basis for calibrating the induction tool’s volt-

age measurement to apparent r e s i s t i v i t y . If the

equations are formulated for the electric field of a

v e rtical magnetic dipole on the axis of a radially

l a y e r ed medium or within a horizontally layer e d

medium, the results are still formulas for a speci-

The Vocabulary of Resistivity Modeling

■1D layered formation model geometry. The familiar layer-cake geometry of as e d i m e n t a ry geological stru c t u re is a 1D resistivity geometry if the re s i s t i v i t yvaries only in the z- c o o rdinate dire c t i o n .

■2D layered formation with invasion model geometry. If resistivity varies inboth vertical and radial directions (and is axisymmetric), then the geometry isreferred to as two-dimensional. This 2D geometry is used in many commercialforward modeling programs today.

■1D invasion geometry. A common geometricala rrangement in well logging is a cylindrical wellbores u rrounded by coaxial cylindrical shells boundingcylindrical regions of differing re s i s t i v i t y.

Spring 1997 47

fied ρ and z. The field can be computed without

re f e rence to the field at other points, but the for-

mulas are complicated and contain semi-infinite

integrals to be per f o rmed, after substituting

parameter values, to get a numerical answer. The

geometries are one-dimensional; these computa-

tions form the basis for invasion corrections (radi-

ally layered 1D geometry) and bed-thickness cor-

rections (vertically layered 1D geometry), and

most commercially available 1D codes.

Models with geometries of higher dimensionality

usually cannot be solved analytically—the solutions

cannot be expressed as simple formulas. In these

higher dimensional cases, numerical methods must

be used to obtain the field values. Although there

a re many dif f e rent numerical methods, they all

s h a re the same pr o p e rty: to compute the field at a

single point, the field must be known at many adja-

cent points. For some methods, the field must be

computed at every point, and these computations

a re time-consuming. Consequently, computations

in 2D are about three times slower than 1D compu-

tations, and 3D geometries r e q u i re two orders of

magnitude more time than 1D computations.

A possible source of confusion arises when

m o re realistic modeling is attempted. To illustrate,

consider a vertically oriented, point magnetic

dipole on the axis of an axisymmetric, two-dimen-

sional medium (top right). At each point in space,

the electromagnetic field has six components:

t h ree components of electric field and three com-

ponents of magnetic field. However, in the plane

of the source, as long as the source is on the axis

of symmetry, two of the electric field components

will be identically zer o .

The magnetic field components can be obtained

f rom the single component electric field using

F a r a d a y ’ s law of induction. Thus, it is sufficient to

f o rmulate this problem in terms of the azimuthal

component of electric field alone. However, if the

dipole is moved off the axis of symmetry to r e p re-

sent an eccentered tool in a borehole, the cylindri-

cal symmetry of the electric field is broken; now

two components of the electric field are

re q u i r ed—radial and azimuthal—to completely

describe the electric field ( b o t t o m ).

Numerical modelers sometimes refer to the

dimensionality of their models according to the

number of non-zero field components necessary to

completely specify the field. However, in this

example, the medium remains 2D irrespective of

the source location or orientation, even though the

mathematical formulation may have to account for

a greater or fewer number of field components

a c c o rding to the source orientation and position.

■3D dipping formation with asymmetric invasion model geometry. 3D geometries such as this and those forfractures, nonuniform invasion fronts and dipping beds with laminated anisotropy are used for most realisticformation models.

1. This geophysics convention is adopted in this art i c l e .

2. The third possibility for a 1D geometry, R = R(φ ), does notc o rrespond to a case of much utility for resistivity modelingof induction tools, which have no sensitivity to φ v a r i a t i o n s ,and is not used.

■Electric field arounda point magneticdipole on the boreholeaxis. The magneticfield components canbe obtained from thesingle component ofelectric field usingFaraday’s law ofinduction. The electricfield is concentric andsymmetrical with theaxis of the boreholeand is therefore com-pletely independent ofthe azimuthal angle φ.

■Electric field aro u n da point magneticdipole off axis from theb o rehole, re p re s e n t i n gan eccentered tool in ab o rehole. In thise c c e n t e red tool geom-e t ry, the cylindricals y m m e t ry of theinduced electric field isd e s t royed. Now, twocomponents of theelectric field arere q u i red (a radial Eρ,and an azimuthal Eφcomponent) to com-pletely describe thenonconcentric electricfield in the form a t i o n .

■T h ree-dimensional model for an invaded, dipping re s e r v o i r. The overlying and underly-ing formations are shales, while the re s e r v o i r, containing 90% oil saturation, is invaded to15 in. [38 cm] by saline water.

■AIT Array Induction Tool logs in dippingbed with invasion. The results of the 3Dmodeled induction responses (blue curves)show the 10-in. response in agre e m e n twith the invaded zone resistivity, Rx o,(dashed green line). However, in the cen-ter of the thick hydrocarbon zone, thedeeper tool responses are reading onlyabout 40% of the true formation re s i s t i v i t y ,Rt, (solid green line). This corresponds toan oil saturation of only 60%, a significante r ro r. The deepest reading 90-in. re s p o n s ealso shows strong polarization horns at thebed boundaries.

■The effect of invasion without dip. In theabsence of dip, the case of a bore h o l et h rough an invaded layer can be com-pletely modeled in 2D ( red curves). Tre a t-ing the problem in 3D yields the sameresults (blue curves). Note that the deep90-in. tool response is greater than the Rt i nthe center of the bed, while the shallower60-in. tool response is nearly equal to Rt.Also note the absence of horns at the bedboundaries. These effects are diff e re n tf rom those seen in dipping form a t i o n s .

■The effect of dip without invasion. In theabsence of invasion, the case of a bore h o l et h rough a dipping layer can be com-pletely modeled in 1D (red curves). Tre a t-ing the problem in 3D should, and does,yield the same results (blue curves). In thiscomparison, the bed boundary horn sappear on the shallow 10-in. reading aswell as on the deep 90-in. tool re s p o n s e .The deep 90- and 60-in. tool responses arereading much less than Rt, while the shal-low 10- and 20-in. responses are close to Rtin the center of the thick bed.

48 Oilfield Review

B e f o re computational codes were improved, the

time the log analyst spent editing the model geom-

e t ry file between computations was comparable to

computational time. By 1991, 1D induction

responses could be computed at the rates of hun-

d reds of feet per second. As a result, editing time

subsequently came to be viewed as a major bottle-

neck. The Mobil Resistivity Modeling Task For c e

was formed to specify and develop a graphical

user interface capable of displaying observed log

c u rves in a user-selectable format, and also capa-

ble of rapidly creating, deleting and other w i s e

modifying resistivity models in the same windows

as the data using the point, click and drag featur e s

available in the then new X-windows system.

The interface is not itself a numerical modeling

code; rather, it is a graphical editor useful for

quickly constructing and modifying models. It

incorporates facilities to rapidly and automatically

build initial models using user-specified log

response curves, and is useful for the specification

of 2D axisymmetric models and 1D models with

constant relative dip. Commercially available

modeling codes currently adapted for use include

f o rw a rd and inverse 1D for 6FF40 and ILD deep

induction logging, 2D for conventional electric sur-

v e y, 2D for laterolog and spherically focused logs,

and 2D for induction. Other suitable codes are

under development by various vendors. The Okla-

homa benchmark formation can be computed in

well under one second; log responses in 2D

geometries take significantly longer (though not

p rohibitive) times to compute over similar inter-

vals. However, the incentive (and r e w a rd) for 2D

modeling is the same as for 1D modeling—hydro-

carbon pore volume increases on the order of 5 to

15% from a more accurate Rt a n a l y s i s .

The resulting program was being routinely used

by Mobil log petrophysicists by 1992. The mar-

riage of fast, 1D induction for w a rd modeling to the

graphical user interface made for w a rd modeling a

practical tool for routine log interpretation. Mobil

licensed its graphical user interface to Z&S Con-

sultants, Inc. where it has been available under

the trade name R tBAN since about 1993; to date it

has been licensed to a number of major and inde-

pendent oil companies. Petrophysicists experi-

enced with the program and familiar with the

6FF40 response regularly model 3000 to 5000 ft

[900 to 1500 m] per day.

An Efficient User Interface for Resistivity Modeling

Spring 1997 49

Modeling the Effects of Dip and Inva s i o n.An example comes from a case in wh i ch theAIT tool logged through a 20-ft [6.1-m]r e s i s t ive bed at an angle of 60° from ve r t i c a l(previous page, top).1 6 The shallow 10-in.log reads near flushed zone resistivities, Rx o,as expected. How e ve r, the deep 90-in.c u r ve falls way below Rt, and the 20-, 30-and 60-in. curves cross each other in a dis-orderly fashion, indicating a mixture ofshoulder bed and invasion effects ( p r e v i o u spage, bottom left).

To disentangle the different effects andcompare the relative effects of invasion andshoulder beds, the limiting cases of inva s i o nwith no dip (previous page, bottom center),and dip with no invasion were modeled(previous page, bottom right).

First, in the case of invasion with no dip,the expected curve separation occurs, withthe 10-in. curve reading close to Rx o and thedeeper reading curves increasing insequence towards Rt, with the 90-in. curvereading above Rt. Next, dip with no inva-sion results show that there is a significantlylarge shoulder-bed effect on the deep curve sin this 20-ft bed at 60° dip. At center- b e d ,the readings decrease in sequence from thes h a l l ow to deep curves, with the 90-in. cen-t e r-bed curve reading only 25 ohm-m in the100-ohm-m bed.

Comparing the combined dip and inva s i o nresults with the results of the nondippingcalculations shows the 10-in. curves readthe same in the nondipping bed. The 20-and 30-in. dipping curves read a bit low e rthan those in the nondipping bed, indicatinga slight shoulder effect. Comparing the com-bined dip and invasion results with those ofthe noninvaded dipping bed model show sthat the 60- and 90-in. curves behave thesame, in both cases reading below Rt. Th i sindicates that the deep-reading logs are notinfluenced by the shallow invasion, but thatthere is a considerable shoulder-bed effect,wh i ch in this case of conductive inva s i o n(Rx o less than Rt) could be mistaken fordeeper invasion. Modeling of dip and inva-sion together and separately helps to inter-pret this difficult case. It is important toa rch ive all AIT raw tool logging responses inorder to take full advantage of the new, faster3D modeling capability.

16. As a test, to verify that the curve crossover was notsimply an artifact of 3D code inaccura cy, the FDMcode was compared with an analytical solution inboth cases, since each limiting case can be solved ina 2D analytical model. Agreement within 3% wa sa ch i e ved in all cases.

50 Oilfield Review

■Shallow (24-in. diameter) invasion in ahorizontal well approaching an overlyingshale. Here the deep-reading 3D modeled60- and 90-in. curves ( b l u e ) both track theno-invasion 1D results ( re d ) and appro a c hRt as the tool moves farther away from theboundary, while the shallower 10-, 20-,and 30-in. curves are all shifted toward sRx o. In all cases, far from the bed bound-ary, the FDM results approach those of theno-bed-boundary, invasion-only 2D model(black points).

■Deep (48-in. diameter) invasion in a hori-zontal well approaching an overlyingshale. In this deep-invasion geometry, onlythe deepest 3D modeled 90-in. curve ( b l u e )tracks the no-invasion 1D result ( re d ). The10-in. curve reads Rx o, and the 20-, 30-,and 60-in. intermediate curves are shiftedt o w a rd Rx o. The 3D FDM calculation is car-ried up to, but not through, the bed bound-ary because of uncertainty about theshape of the invasion just below theboundary. In all cases, far from the bedboundary, the FDM results approach thoseof the no-bed-boundary, invasion-only 2Dmodel (black points).

■A n i s o t ropic invasion profile. The effect ofinvasion at 1-week intervals is shown forsix borehole deviations from vertical tohorizontal in a formation with a perm e-ability anisotropy kh o r i z o n t a l/ kv e r t i c a l of tento one. The invasion profile is character-ized by its aspect (long axis to short axis)ratio, which depends on perm e a b i l i t ya n i s o t ropy, time and dip.

Spring 1997 51

I nvaded Horizontal Well Near a Cap Shale.Horizontal well logs in any environment aredifficult to analyze without 3D modeling. Atypical and interesting problem is the caseof invasion in a permeable sand below acap shale interface (previous page, top leftand center).1 7 In these examples, the log-ging sonde remains parallel to the bedb o u n d a r y, while the distance between thed rainhole and the boundary is varied.

The 3D FDM model is computed for shal-l ow (24-in. diameter) and deep (48-in.diameter) invasion depths and comparedwith the results of a 1D analytical modelwithout invasion. The 1D model shows thetypical curve order reversal and polariza-tion-induced horns on the inductionresponses at the bed boundary. Also, forcomparison, the no-bed-boundary, inva s i o n -only limiting responses were computed witha numerical 2D model.

These results show that for shallow tom o d e rate invasion, the deepest curve canbe used to infer Rt and proximity to theshale cap, while the shallowest curve indi-cates Rx o. The relative separation betweenthe intermediate curves can be used withcaution to estimate depth of invasion. It isclearly still possible to get good results fromthe AIT induction in horizontal wells, eve nthough the tool was designed for ve r t i c a lwells with axially symmetric geometries.

Asymmetric Inva s i o n . As a refinement onthe above model geometry, wh i ch assumesa cylindrical invasion front, it is interestingto examine the causes of nonuniform inva-sion, and then forward model the toolresponse in such environments. Nonuniformeffects can be caused by gravity segregationor permeability anisotropy. As a well devi-ates from vertical to horizontal, gravity cancause invading fluid to behave differently inthe top and bottom halves of the borehole.Permeability anisotropy causes an ellipticali nvasion shape in deviated wells. The inva-sion shape becomes more elongated relativeto the borehole as permeability anisotropy,well deviation and quantity of filtrate inva-sion increase (previous page, right).

The 3D FDM computes the effects of theshape of the invasion profile, after threeweeks of elliptical invasion, on A I Tresponses ( l e f t ). The results show that vo l u-metric contribution of Rt is increasing as theshape of the equal area invasion frontbecomes more elongated. The sensitivity toRt becomes greater because a larger portionof the induction current circulates outsidethe invaded zone as the front elongates.

The modeled AIT responses directly reflectthe relatively shallow invasion depth in thevertical and horizontal portions of the well.The filtrate invades preferentially in the hori-zontal direction along the higher kh o r i z o n t a l,and less filtrate invades in the well’s ve r t i c a ldirection. In the horizontal borehole, theformation fluid remains closer to the loggingtool above and below the drainhole, wh i chincreases the influence of Rt on the shallowr e s i s t ivity curve s .

17. A cap shale is frequently called a s h a l e - s e a l or sim-ply a s e a l, because it represents an impermeablebarrier on top of a reservo i r.

■AIT log sensi-tivity to theinvasion pro f i l easpect ratio forresistive andc o n d u c t i v ea n i s o t ropic inva-sion. With re s i s-tive invasion,the four deepest(20-, 30-, 60- and90-in.) curvesa re fairly flat,while the shal-low 10-in. curvealone decre a s e st o w a rds Rt a sthe invasion axisratio incre a s e s .For conductiveinvasion, theshallow 10-in.curves read sys-tematically highand incre a s et o w a rd Rt as theinvasion shapebecomes moreeccentric. Them e d i u m - r a n g e20- and 30-in.curves alsoi n c re a s et o w a rds Rt in asimilar manner.

52 Oilfield Review

B u oya n cy, caused by grav i t y, is anotherphenomenon that can greatly distort theshape of the invasion front ( t o p ). The wa t e rf i l t rate invading an oil zone in a horizontalwell behaves much differently on the bot-tom side than on the top side of the dra i n-hole. As a result, the resistivities of thei nvaded zones at the top and bottom of thed rainhole are quite different.

On the top, Rx o is confined to a ra t h e rs h a l l ow invaded zone, but below the resis-t ivity is a gradual transition—from Rx o t oRt— over a relatively thick region.1 8 M o d e l-ing the gravity-induced asymmetric inva s i o nprofile in a horizontal well shows that theeffect on AIT responses is noticeable onlyon the most shallow curve—the 10-in.response ( a b ove ).

Field Logs and 3D Modeling. Computermodeling predicts many 3D effects, butwhat do real logs look like? A field log fromthe Middle East in a horizontal well withm o d e rately salty mud invasion illustrates theneed for 3D modeling in interpreting induc-tion logs in deviated wells (next page, topr i g h t ). Below 150 ft, the logs reveal signs ofthe horizontal wellbore approaching a shalebed from below—the deep 90-in. curvewith signs of horns and the 20- and 30-in.c u r ves in reverse order read lower than thes h a l l ow 10-in. AIT and the MicroSFL logs.H ow e ve r, the low-reading 20-in. curvecould not be modeled by a single interfacewith the known shale resistivity of 4 ohm-m.

The low-reading 20-in. curve along withthe reversed curve order suggests an annulusi nvasion profile.1 9 A short zone from 100 to250 ft was chosen for modeling. With poros-ity log data, a known formation water resis-t iv i t y, Rw, and the tool response, a formationmodel with an annulus profile was gener-ated. From the earlier discussion on capshales and invasion, it is reasonable toassume that the shallow AIT logs (10-, 20-and 30-in. curves) respond to the inva s i o nprofile, while the deeper logs (60- and 90-in.c u r ves) respond to nearby bed boundaries.

In the formation model, invasion occurs inthe sands with only slight variations in thedepth of invasion to fit with variations inporosity and deep induction response ( n e x tpage, left). The effect of the shale was to cutoff the upper part of the invasion. Thus asthe invasion and annulus profilea p p r o a ched the cap shale, they are trun-cated at the bed boundary (next page, bot-tom right). As modeling progressed, itbecame clear that the behavior of the 90-in.log is influenced as much by invasion as it isby the bed boundaries. Only when the inva-sion radius was adjusted appropriately werethe excursions to 2000 ohm-m reproduced.

The qualitative agreement between thefield logs and modeled logs suggests that theannulus profile is real. The behavior of allthe logs is a complicated mixture of inva-sion effects and high-angle, bed-boundaryeffects. This model ignored any gravity orpermeability anisotropy effects, becausethese are negligible compared to the first-order effects of bed boundaries and inva-sion, and because the sensitivity of the A I Tlogs to these effects is not large.

All the unknowns surrounding invasion inhorizontal wells lead to the non-uniquenessof any solution that matches the field logs.Additional information, such as Rx o or shaler e s i s t iv i t y, Rs h a l e, becomes essential to helplimit the uncertainty. This example show sthat, just as in vertical wells, it is inadequateto assume a simple invasion model in hori-zontal wells. Only modern multiarray toolsand full 3D modeling can successfully pro-vide quantitative interpretation of resistiv i t ylogs in horizontal wells.

18. The invasion depths are strongly influenced by theformation fluid mobility (permeability/viscosity): highmobility results in shallow top and deep bottom inva-sion depths. See Anderson et al, 1996, reference 1 5 .

19. Terry R, Barber T, Jacobson S and Henry K: “The Useof Modern Logging Measurements and New Process-ing Algorithms to Provide Improved Evaluation inDeeply Invaded Gas Sands,” Transactions of theSPWLA 35th Annual Logging Symposium, Tu l s a ,Oklahoma, USA, June 19-22, 1994, paper FFF.

■AIT log response to gravity segregation in a horizontal well. Becausethe invasion is pulled downward away from the drainhole, the sensi-tivity to Rx o is decreased compared to cases in which buoyancy is nota factor. This means that in isotropic formations with high mobility,gravity prevents the filtrate from penetrating the formation over a sub-stantial region around the horizontal drainhole (top and sides), thusresulting in a greater influence of Rt on all the AIT curves.

■Gravity segre g a-tion of invasion in ahorizontal well.Gravity sweeps thedenser water down-w a rd ( l e f t ), re d u c-ing the invasion onthe top side whilep roducing a hydro-dynamically unsta-ble mixing of theinvading filtratewith movable oil onthe bottom side ofthe drainhole.Geometry for grav-ity segregation in ahorizontal well ismodeled as a box( r i g h t ).

Spring 1997 53

■Field logs from a horizontal well underlying a cap shale. The GRresponse increases from 300 ft to 150 ft as the borehole appro a c h e sthe cap shale.

■3D formation model with annulus profile geometry in ahorizontal well. Rt was 200 ohm-m. The borehole was mod-eled at the shale-formation interface ( t o p ) and at 6 in. ( m i d-dle) and 12 in. (bottom) below the interface. No invasionwas permitted into the impermeable shale.

■3D simulation results of AIT logs (top) in a horizontal well with capshale. The borehole trajectory and invasion profile relative to the capshale are shown (bottom). The model simulates the observed fieldlogs (top), and suggests that the borehole entered a steeply dippingc ross-bedded formation below the cap shale, known to exist in there s e r v o i r.

54 Oilfield Review

A n i s o t r o py. In highly deviated wells, induc-tion and propagation tools can detect resis-t ivity anisotropy—largely invisible to thesetools in vertical wells—because they havebeen designed to measure currents or fieldsin planes normal to the tool axis. The originsof electrical anisotropy are linked with thesame phenomena that cause permeabilitya n i s o t r o py—formation bedding geometry org rain size in homogeneous sand beds.2 0

Often thin, anisotropic, conductive shalebeds or laminations are mixed with high-r e s i s t ivity pay zones.2 1

Although some new properties can beobtained from the use of both horizontaland vertical resistiv i t y, it is horizontal resis-t iv i t y, Rh, that is most desired in log interpre-t a t i o n .2 2 In vertical wells, inductive toolsread the geometric mean of the horizontalbedding resistivities. In deviated or horizon-tal wells, it is difficult to define a simplemixing law to represent the way the induc-tion tools read an ave rage of the horizontaland vertical resistiv i t y, Rv, wh i ch will bevery different from the ave rage horizontalr e s i s t ivity the tool reads in a vertical well.

A version of the 3D FDM model has beend e veloped that can account for any direc-tional variation in resistiv i t y, wh i ch willaccount for any differences in vertical andhorizontal resistivities. A simple dipping bedexample, with no invasion, shows an induc-tion tool’s responses, logging at a dip of 45°as it moves from an isotropic bed into aanisotropic bed ( l e f t ). The induction curves all are in good agreement with Rt deep in theisotropic bed, but as the tool moves into theanisotropic bed, all the curves tend towards aweighted ave rage of Rh and Rv. In a ve r t i c a lwell, the log curves would all read Rh.

To investigate how anisotropy further com-plicates the already complex interpretationof invasion in a horizontal well, anisotropywas added to the previous example of ahorizontal well near a cap shale with inva-sion. The invaded zone was assumedisotropic; the ove rall responses are similar tothe previous case—polarization hornsappear at the bed boundary, and in thei nvaded zone shallow tool response curve sread close to Rx o and the induction logs sep-a rate—but there is a difference in the deeptool responses. In the anisotropic bed, thedeep tool responses tend towards an ave r-age of Rv and Rh, instead of reading close toRh. Without the model, it would be impossi-ble to determine the presence of anisotropyor accurate formation resitivities based onthe behavior of the resistivity curves alone.

K n owledge of vertical and horizontal resis-t ivities is important for analyzing thinly lam-inated dipping formations, where both resis-t ivity values are crucial for estimating thesand lamina resistivity and the net-to-grossratio (the sand fraction) simultaneously.2 3

Armed with the 3D modeling capability andk n owledge of the relative dip, the formationr e s i s t ivity anisotropy can be determined. Ofcourse, the inverse is also true, if the forma-tion resistivity anisotropy is known, then ther e l a t ive dip or deviation angle can bed e r ived from the induction tool responses. ■Characteristics of typical modeled inva-

sion profile geometries. The simple stepinvasion profile and the ramp pro f i l ere q u i re three parameters; the slope pro f i l eis a four-parameter model; the annulusp rofile is a five-parameter model.

■Comparison of AIT logs using 3D and 1Dmodels in an anisotropic dipping bed. Inthe lower, isotropic layer, the inductioncurves read Rt, but in the anisotro p i c ,upper zone the tool reads neither Rh nor Rv,but a weighted average of the two.

Spring 1997 55

The response of propagation tools toa n i s o t r o py is even more pronounced thanthe response of induction tools. At high dip,the vertical component of resistivity for bothtools is multiplied by a term proportional tof r e q u e n cy. This term is an order of magni-tude greater for 2-MHz propagation toolsthan for induction tools. The anisotropicresponse also varies greatly with the tra n s-m i t t e r- r e c e iver spacing. The highly nonlineart ransforms from phase shift to resistivity andattenuation to resistivity behave differently,amplifying the anisotropic effects when thet wo logs are compared.2 4

Resistivity Modeling as a Log Analyst’s To o lModeling results in the dipping bed andhorizontal well examples indicate that resis-t ivity logs in complex formations containgeometrical information, but extracting it isa challenge. At the very least, the use of for-ward modeling has now become a key toolfor log analysts in understanding the forma-tion properties that combine to produce thelogging tool responses.

A new workstation program is currentlybeing developed for GeoQuest, calledI N VASION, to provide log analysts with thetools for invasion-based, resistiv i t y - m o d e l i n gformation analysis. The programs are basedon forward modeling for layered formations,and the interpretation allows for eva l u a t i n gdynamic reservoir properties—early-timep e r m e a b i l i t y, water cut and fractional fluidf l ow. The system helps take the drudgery outof the most time-consuming activities associ-ated with modeling. This is done with ag raphical interface and intera c t ive para m e t e rselection, wh i ch promotes a more accura t eRt e valuation from multiple resistivity mea-surements ( a b ove right).

The AIT induction logs are corrected fore nvironmental effects either at the wellsiteo r by a p r e p r o c e s s i n gp r o g ra mc a l l e dP r e P l u s ,wh i ch corrects for apparent dip and effectsof shoulders. The resulting logs are ve r t i c a l l ym a t ched in resolution.2 5 The logs are thenprocessed by a resistivity itera t ive inve r s i o np r o g ram, using 1D radial tool response func-tions as a forward model.2 6 The combinationof separate vertical processing and ra d i a lprocessing is called 1D+1D processing. Th i sstep, along with log squaring, gives the ana-lyst a first approximation of the formationbeds and resistivities. At this point, using ag raphical log display, the log analyst canreview the initial formation model andmake refinements to the model with ani n t e ra c t ive interface.

This intera c t ive task permits the analyst tomanually define invasion resistivity profilesand the formation bedding and resistiv i t ydescription through a tabular interface (p r e-vious page, right). With this intera c t ive task,the analyst can explore the sensitivity of logdata to changes in the formation model.

20. For a detailed introduction to anisotropy: A n d e r s o nB, Bryant I, Lüling M, Spies B and Helbig K: “OilfieldA n i s o t r o py: Its Origins and Electrical Chara c t e r i s-t i c s ,” Oilfield Review 6, no. 4 (October 1994): 48-56.

21. Klein JD: “Induction Log A n i s o t r o py Correction,”Transactions of the SPWLA 32nd Annual LoggingS y m p o s i u m, Midland, Texas, USA, June 16-19,1991, paper T.

22. H a g i wa ra T: “A New Method to Determine Horizon-t a l - R e s i s t ivity in Anisotropic Formations Without PriorK n owledge of Relative Dip,” Transactions of theSPWLA 37th Annual Logging Symposium, NewOrleans, Louisiana, USA, June 16-19, 1996, paper Q.

23. H a g i wa ra T: “Macroscopic A n i s o t r o py A p p r o a ch toE-Log Evaluation in Laminated Sand-ShaleS e q u e n c e s ,” presented at the 3rd SPWLA Reservo i rC h a racterization A rchie Conference, Galve s t o n ,Texas, USA, November 8-11, 1992.

24. Lüling MG, Rosthal RA and Shray F: “Processing andModeling 2-MHz resistivity Tools in Dipping, Lami-nated, Anisotropic Fo r m a t i o n s ,” Transactions of theSPWLA 35th Annual Logging Symposium, Tu l s a ,Oklahoma, USA, June 19-22, 1994, paper QQ.

25. Barber TA and Rosthal R: “Using Multiarray Induc-tion Tool to A ch i e ve High-Resolution Logs with Min-imum Environmental Effects,” paper SPE 22725, pre-sented at the 66th SPE Annual Te chnical Conferenceand Exhibition, Dallas, Texas, USA, October 6-9,1 9 9 1 .

26. G e n e ralized (finite conductivity) 1D tool geometricalresponse functions are derived using a solution simi-lar to the single scattering Born approximation for-malism traditionally used in quantum mech a n i c s .See Gianzero S and Anderson B: “A New Look atSkin Effect,” The Log A n a l y s t 23, no. 1 (Ja n u a r y -Fe b r u a r y, 1982): 20-34.Also see Thadani SG and Hall HE Jr: “PropagatedGeometrical Factors in Induction Logging,” Tra n s a c-tions of the SPWLA 22nd Annual Logging Symposium,Mexico City, Mexico, June 23-26, 1981, paper W W.

■First look. Deconvolution produces beds and resistivities for the initial formation model.The display shows the borehole and radii of invasion on the left track. AIT and Micro S F Lfield logs along with simulated logs are shown in the middle track. Rx o, and Rt a re alsoshown in this track. The right track contains log quality curves.

56 Oilfield Review

S e n s i t ivity analysis is especially importantwhen Rx o and Rt are dissimilar. With thesame task, the user can change Rt, and inva-sion profiles (invasion and annulus radii andr e s i s t ivity Rx o, and Ra n n) and quickly recom-pute, using forward modeling, the syntheticlog for verification with original logs. For lat-erolog responses, the forward model usesthe 3D FEM for dipping beds and a 2D FEMotherwise. For induction tools, the forwa r dmodel is based on 2D hybrid FEM, and a1D analytical code is used for dipping beds.Formation models can include inva s i o ngeometry as either a step function, ra m p e d ,annulus profile or no inva s i o n .

During the next phase of interpretation,after the analyst is satisfied that the final for-mation model accurately represents the log-ging environment, there are two new pre-liminary functions to compute invaded zonefluid properties.2 7 First, with inva s i o np a rameters, LWD and wireline resistiv i t i e sand porosity logs, the volume of mud filtra t ethat invaded the formation around the bore-hole can be computed. This allows a time-lapse permeability analysis. Also, if the wellis vertical in a clean or shaly sand and hasbeds thicker than 6 ft [1.8 m], the analystcan compute a formation wa t e r-cut log—useful for reservoir engineering, and a fra c-t i o n a l - f l ow log, both based on the fluiddynamics reflected in the formation inva s i o nprofile (a b ove ) .

This graphically intera c t ive, resistiv i t ymodeling-based formation evaluation p r o g ram facilitates interpretations in com-plex formations.

Outlook for Resistivity ModelingThe recent developments in code efficiencywill lead to full use of 3D models for inter-pretation applications. Induction, propaga-tion, laterolog and eventually other loggingtools, such as nuclear and acoustic, will bemodeled in more realistic formations. Pe t r o-p hysical relationships will be incorpora t e din the models, wh i ch means the formationwill be described in log analysis terms:l i t h o l o g y, porosity and satura t i o n .

Trends started with AIT and PL AT F O R M

EX P R E S S equipment will continue—sophisti-cated tool environmental corrections willbe built into the logs using results from

detailed physical and numerical modelingduring the tool design and deve l o p m e n tphases. The power of analysts’ tools basedon resistivity modeling, such as in theRtBAN, INFORM and INVASION progra m s ,will be brought to bear on petrophy s i c a linterpretation in complicated env i r o n m e n t s .M a ny consumers of well log data believethat to gain maximum value from resistiv i t ylogs in general, and induction logs in par-t i c u l a r, the numerical modeling of toolresponses will be an indispensable facet ofinterpretation for the 21st century.

— R C H

■Interpreting invasion. Model-based fractional flow inversion logs enable reservoirengineers to predict water cut throughout the reservoir. With porosity and the fractionalflow computed from the invasion profile, a water-cut log is computed that helps deter-mine which part of the reservoir to perforate. In many cases, it may be necessary tominimize water cut, and in other cases a higher water cut may be acceptable for opti-mum oil production.

27. Ramakrishnan TS, Al-Khalifa J and A l - Waheed HH:“Producibility Estimation from A r ray-Induction Logsand Comparison with Measurements—A CaseS t u dy,” to be presented at the SPWLA 38th A n n u a lLogging Symposium, Houston, Texas, USA, June 15-18, 1997.