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SPE 54719A Fundamentally New Model of Acid Wormholing in Carbonates
Rick Gdanski, SPE, Halliburton Energy Services, Inc.
Copyright 1999, Society of Petroleum Engineers, Inc.This paper was prepared for presentation at the 1999 European Formation Damage Confer-ence held in The Hague, The Netherlands, May 31-June 1.This paper was selected for presentation by an SPE Program Committee following review ofinformation contained in an abstract submitted by the author(s). Contents of the paper, aspresented, have not been reviewed by the Society of Petroleum Engineers and are subject tocorrection by the author(s). The material, as presented, does not necessarily reflect anyposition of the Society of Petroleum Engineers, its officers, or members. Papers presented atSPE meetings are subject to publication review by Editorial Committees of the Society ofPetroleum Engineers. Permission to copy is restricted to an abstract of not more than 300words. Illustrations may not be copied. The abstract should contain conspicuous acknowledg-ment of where and by whom the paper is presented. Write Librarian, SPE, P.O. Box 833836,Richardson, TX 75083-3836, U.S.A, fax 01-972-952-9435.
References at the end of the paper.
Abstract
A new theoretical model has been developed to describe thechemical reactions of acid in porous carbonate media. For thepast 20 years, the three major unanswered questions regardingwormholing when acid is pumped into carbonate formationshave been the following: (1) How many dominant wormholes arecreated? (2) What is the spatial distribution of those dominantwormholes along the wellbore? (3) What is the leakoff profilefrom the dominant wormholes under radial flow conditions?This paper presents a model that proposes answers to these threebasic questions. Once these questions are answered, the reactionof acid in the matrix and the interaction of wormhole develop-ment become straightforward. The new model requires a majorparadigm shift in the understanding of matrix carbonate acidizingand the variables that control wormhole growth. Parametricstudies that were conducted with the new theory are presented. Itwas found that wormhole length is predominantly controlled bymatrix porosity and volume of acid pumped—
not
by reactivity.It was found that wormhole diameter is predominantly con-trolled by reactivity and contact time. The new theory confirmsclassically held guidelines for matrix acidizing of carbonates,and gives insight on how to improve matrix carbonate acidizingtreatments. The new model also accommodates the effects of permeability anisotropy caused by natural fracturing or layeringeffects.
Introduction
In 1979, SPE published Monograph Volume 6 of the Henry L.Doherty Series
Acidizing Fundamentals
which was coauthoredby Bert Williams, John Gidley, and Robert Schechter.
1
Themonograph is an excellent reference about matrix acidizingsandstones, fracture acidizing carbonates, and matrix acidizingcarbonates. Subsequent improvements to fracture acidizingtheory include balancing mass transport and surface reaction forcalculation of acid spending,
2,3
incorporating heat of reaction,
4
measuring and using energy of activations for reactivity profilesas a function of temperature,
5
improving fracture geometrycalculations for fracture acidizing with multiple stages,
6
deter-mining reactivity for slow-reacting gelled acids,
7
and introduc-ing an improved method for using laboratory-measured conduc-tivity data in a fracture acidizing simulator.
8
Matrix acidizing of carbonates is also covered in the acidizingmonograph. A method is given for calculating the spending of acid down a dominant wormhole in either turbulent or laminarflow. Hand calculations of acid spending lengths can be per-formed either with or without fluid leakoff. Unfortunately, thefollowing three fundamental questions that have prevented useof the published concepts have remained unanswered for thepast 20 years:1.How many dominant wormholes are generated?2.What is the spatial distribution of these dominant worm-holes?3.What is the leakoff profile from the dominant wormholes?A new theory presented in this paper proposes to answerthese three fundamental questions.
Current State-Of-Affairs
At five research centers throughout the world, significant stud-ies of acid wormholing in carbonates have been conducted. Thetwo most prolific places have been the University of Michiganin Ann Arbor under the direction of Scott Fogler and theUniversity of Texas in Austin under the direction of Dan Hill.The other locations are the Mining University Leoben withThomas Frick under the direction of Michael Economides, theInstitute Français du Pétrole with Brigitte Bazin, and theHalliburton Energy Services, Inc. (HES) European ResearchCenter (ERC) with Martin Buijse.Scott Fogler and graduate students at the University of Michigan have conducted many experiments and have visual-ized the wormhole patterns with metal castings
9
and later with
2SPE 54719A FUNDAMENTALLY NEW MODEL OF ACID WORMHOLING IN CARBONATES
neutron radiography.
10
They conducted linear-flow experimentsat various flow rates with carbonates of different reactivities,and modeled the general pattern of the wormholes with anetwork model.
11
Unfortunately, such a computer model hasgreat difficulty in scaling to the real world. Nonetheless, theydemonstrated that the wormholing patterns were a function of the Damköhler number, which is essentially the ratio of reactiv-ity to mass transport. They recently introduced a generalizedDamköhler number and an optimum Damköhler number formost efficient generation of wormholes in carbonates.
12
Theconcept is apparently useful for both slowly reacting organicacids as well as HCl on either limestones or dolomites. Unfor-tunately, the concept has been developed through the use of linear-flow experiments, and it is uncertain how it will scale toradial flow.Dan Hill and his graduate students at the University of Texas conducted a large number of linear-flow experiments andhave attempted to define an optimum flow rate for acidizing.
13,14
This approach is very similar to that taken at the University of Michigan. Recently, the University of Texas group has intro-duced some new mathematics and charts for calculating theoptimum injection rate for initiating efficient wormholing at thewellbore. However, their predictions for the optimum injectionrate for acidizing carbonate reservoirs still seem unrealisticallylow, and the method is cumbersome.Thomas Frick, at the Mining University Leoben under thedirection of Michael Economides, published work a few yearsago in which fractal mathematics was used to describe thebranching nature of the wormhole patterns observed in radialacidizing experiments of a quarried carbonate.
15,16
Unfortu-nately, such calculations were exceedingly time-intensive andrequired weeks of CPU time on a powerful workstation. Theresults of these calculations were still not general enough toallow use in other carbonate reservoirs of differing lithology andreactivity. Later, Behdokht Mostofizadeh, while at the MiningUniversity Leoben under the direction of Michael Economides,published an approach to predict the optimum injection ratefrom radial acidizing experiments.
17
Brigitte Bazin at Institute Français du Pétrole has investi-gated the implications of linear wormholing on the leakoff properties of acid during fracture acidizing treatments.
18,19
While the work is interesting, it is not clear that there is a readyapplication in either fracture acidizing or matrix acidizing of carbonates.Martin Buijse at the HES ERC has been performing bothlinear and radial flow experiments in an attempt to understandthe difference of wormholing behavior in linear flow vs. radialflow. In addition, he recently published an approach for calcu-lating acid penetration distances down a wormhole under lami-nar flow conditions.
20
Interestingly, the recent work indicatesthat the classic “pore volumes of acid to breakthrough” in linear-flow experiments may be an artifact of the core diameter.Therefore, the apparent artifact suggests that the linear experi-ments may not be capable of contributing much more to theunderstanding of wormhole growth in radial flow.Gérard Daccord with Dowell-Schlumberger in Saint Etienne,France has published several papers on the wormhole patternsformed by the dissolution of plaster with water.
21,22,23
Hisimportant work, though seemingly unrelated to acid wormholingin carbonates, provided fundamental knowledge for the devel-opment of the new theory presented in this report.In the nearly 20 years since the SPE acidizing monographwas published, a great deal of work has been performed, but thethree fundamental questions have remained unanswered.
The Breakthrough
In the mid-1980’s, the author attended a meeting of the Indus-trial Affiliates Program at the University of Michigan under thedirection of Scott Fogler. Results of carbonate wormholingstudies were presented, and pictures of some radial flow disso-lution experiments were displayed. The experiments by Daccordwere actually dissolution patterns created by water injection intoplaster casts. The pictures displayed an amazingly high degreeof symmetry that was clearly evident in the wormholing pat-terns. Daccord published a picture (
Fig. 1
, Page 9) in
SPEPE
in1989.
23
It was only a side view, and the full symmetry was notas obvious. Scott Fogler also had copies of top views that werenot published.The high degree of symmetry in the radial flow dissolutionexperiments was profoundly significant. Symmetry occurs inhighly coupled systems, and its presence can dramatically affectthe ease of understanding and modeling of a particular phenom-enon. For example, organic and inorganic chemists use symme-try to predict certain reactivity characteristics. The existence of symmetry in electron wave functions of certain reactive mol-ecules allows one to predict whether certain types of reactionswill proceed. When such symmetry is present, it is not necessaryto spend the months of CPU time that might be required tocalculate the reaction potential. Rather, one can look at thestructure drawn on paper and say “this will proceed” or “this willnot proceed.” Likewise, the existence of symmetry and/or thebreaking of symmetry in the study of high-energy nuclearphysics phenomena can have an enormous impact on the theo-ries that physicists may accept or dismiss. In fact, in grandunified theories on the nature of the universe, the breaking of symmetry gives us gravity vs. electromagnetic force vs. strongnuclear force, etc. Similarly (but certainly much less signifi-cantly), the symmetry in the dominant wormholing patterns forradial flow dissolution allows for significant insight and simpli-fication of the phenomenon.The radial dissolution experiments of Daccord consisted of water being flowed from a “wellbore” installed in a hardenedplaster cylinder. The observed symmetry existed both radiallyfrom the center and linearly along the length of the wellbore.Dominant wormholes occurred as “sets” of either five or six ina single horizontal plane. The wormholes in sets of six had a
SPE 547193R. GDANSKI
radial separation of 60° from each other. Such a distributionessentially formed a series of equilateral triangles. Thus, thedistance between the tips of the separate wormholes in each setwas the same as the length of the wormholes. These horizontalsets occurred in a periodic fashion along the vertical length of the wellbore but had different radial penetrations. A number of sets of wormholes approximately 1 in. in radial length wereseparated from each other by approximately 1 in. Inspectionshowed that every other set had ceased to grow, and that theremainder continued to grow until their length was equal to theseparation distance between them (approximately 2 in.). Again,about half of those sets stopped growing while the remainingsets continued to grow until one wormhole broke through theedge of the plaster cylinder.The presence of the symmetry indicates that all
the domi-nant wormholes must know about each other
—thewormholing process is a highly coupled phenomenon, anddominant wormholes do not form randomly from each other inradial flow. The mechanism for coupling is through the pressurefield induced by matrix flow of fluid through the porous media.This includes interferences between developing dominant worm-holes through the matrix flow of both displaced pore fluids andinvading spent fluids. Minimizing these interferences requiresthat the dominant wormholes stay as far apart as possible. As aconsequence, the development of multiple wormholes in radialflow must be symmetrical both radially and vertically. In addi-tion, the wormholes must also observe the influences of Darcy’slaw for matrix radial flow. Most investigators assume that thedirection of fluid flow through the matrix is governed by thedeveloping wormhole pattern. As a result, they focus on thephysics of the wormhole growth and ignore the matrix itself.The breakthrough in thinking is that
the developing wormholepattern is governed by fluid flow through the matrix
. Oncethis concept is accepted, the diameter and length of the develop-ing dominant wormholes are readily calculated from acid reac-tivity and spending procedures currently used in advancedfracture-acidizing theory.
The New Theory
The new theory introduced in this paper is that wormholing of acid in carbonates is highly symmetrical when it is performed onthe scale of field acid treatments. In addition, the secondaryshape of this symmetry is governed by the native formationpermeability. This symmetry was experimentally observablewith radial flow of water through plaster casts because of the lowsolubility of plaster in water and the fairly homogeneous natureof hardened plaster, thereby minimizing “end effects.” In thefew radial carbonate flow tests conducted at the ERC, the highsolubility of carbonate in HCl, the generation of a CO
2
gasphase, and the fairly nonhomogeneous nature of carbonate coresprevented the high degree of symmetry from forming beforewormhole breakthrough occurred. Therefore, end effects werea dominant part of the wormholing pattern. However, a closeinspection of casts from these radial HCl flow experimentsindicated that the symmetrical acidizing was initially trying toform.
24
Such symmetry was never, and could never, be observedin linear cores because of the fundamental difference betweenlinear and radial flow.The assumptions for the new theory follow:1.The spatial distribution of the dominant wormhole patternis governed by the native permeability of the carbonateformation.2.The fluid velocity through the wormhole pattern convergesat the tip to the fluid velocity calculated by uniform matrixradial flow as if no wormhole pattern existed. (This condi-tion is essentially the leading edge boundary condition, andit is a fundamental assumption for calculating the fluidleakoff profile from the developing dominant wormholes.)3.Dominant wormholes form as sets of six in an x-y planeperpendicular to the wellbore and independent of perfora-tion density.4.The radial separation of the wormholes in each set is 60°.5.Two opposing dominant wormholes in each set align them-selves with the direction of highest permeability, which isassumed to be in the x direction.6.The tip length undergoing leakoff for each dominant worm-hole is the same as one-half the length of the 60° leadingedge arc assigned to that wormhole.7.The remaining wormhole length (closest to the wellbore) issimply a conduit for transporting acid from the wellbore; itdoes not undergo leakoff but does undergo reaction andgrowth by dissolution.8.The secondary shape of the dominant wormhole pattern ineach set is governed by the permeability contrast betweenthe x and y directions.9.The frequency of the sets of wormholes along the z direc-tion parallel to the wellbore is governed by the permeabilitycontrast between the x and z directions.10.The separation of growing wormhole sets cannot be closerthan the dominant wormhole length when the x and zpermeabilities are the same. (This is essentially how onedetermines when to reduce the number of developing worm-hole sets.)These assumptions are rather technical but can help providethe answers to the three unanswered questions. The assumptionsalso allow a straightforward programming of the model. Inaddition, the reaction of the acid down the wormholes is calcu-lated according to the balanced acid reactivity/mass transportapproach of Roberts and Guin.
2
The mass transport coefficientsfor turbulent and transitional flow regimes in tubes were ob-tained from subroutines in an HES fracture acidizing program.These coefficients had been obtained from literature sourcescited by Roberts and Guin. The fracture acidizing program usesmass transport coefficients for tubes and converts them for use
4SPE 54719A FUNDAMENTALLY NEW MODEL OF ACID WORMHOLING IN CARBONATES
in parallel plates (fractures). The mass transport for laminar flowthrough reactive tubes was obtained from Levich.
25
General Algorithm
A computer model was written based on the assumptions listedabove. The first step was to determine the initial number of wormhole sets. For a homogeneous formation, the number of sets was chosen to be a value of (
2
i
– 1
), where
i
is an integer, thatwould give an even distribution of sets along the zone heightwith a separation of 18 in. or less. Each wormhole set consistedof six dominant wormholes at a radial separation of 60° fromeach other. The wormholes were initially defined as being 6 in.long with a radius at the wellbore of 0.03 in. and a radius at thetip of 0.01 in. The matrix pore throats ahead of the growingdominant wormholes were also assumed to have an initial radiusof 0.01 in.An acid volume was then injected into the formation. Theacid volume was calculated to be the incremental volumerequired to fill the matrix porosity for the next radial length stepahead of the growing wormhole sets. The radial length step wasset at 0.20 in. Fluid distribution per wormhole was calculated tobe the volume step divided by the total number of dominantwormholes.Fluid leakoff was calculated as the injected fluid was moveddown the length of the dominant wormholes. In a homogeneousformation, the length of each dominant wormhole experiencingleakoff was calculated as one-half the arc length associated witheach wormhole. For example, a wormhole set having dominantwormholes extending 5 ft from a 6-in. diameter wellbore wouldhave a circumference of 33 ft. Each wormhole has an associatedarc length of 5.5 ft. Leakoff from each dominant wormholewould be calculated along the leading length of 2.75 ft, or 55%of the leading length of each dominant wormhole. No leakoff would be allowed for the first 45% of the wormhole length.Enough acid leakoff would be calculated to leave just enoughfluid at the tip to fill the associated radial increment ahead of thewormhole tip. This leakoff procedure ensured compliance withAssumption 2 and is fundamental to the new theory. It is themethod by which a homogeneous pressure field is establishedand through which “coupling” of the wormhole network be-comes symmetrical.Acid spending down the length of the wormholes was thencalculated. The procedure was a finite element spending routinethat used the balanced surface reactivity/mass transport ap-proach of Roberts and Guin.
2,3
New wormhole dimensions werecalculated, and the process was repeated.The number of sets of wormholes was reduced by aboutone-half when the length of the dominant wormholes becamethe same as the vertical separation of each wormhole set.However, the number of wormhole sets was always kept at somemultiple of (
2
i
– 1
).
Results and Discussion
General Isotropic Example.
Visualization of the new theory iscritical for understanding its simplicity as well as its impact andutility. An example will be given where the dominant wormholepattern has been calculated and visualized. The example is for a10-ft zone acidized with 15% HCl at a surface pumping rate of 0.1 bbl/min/ft of zone or 1 bbl/min. The treatment volume wasalso at the conventional value of 100 gal/ft of zone or 1,000 gal.The reactivity was assumed to be high at 150°F in limestone with10% porosity and uniform permeability.
Table 1
(Page 8) liststhe specific details.The results of the wormhole calculations are shown in
Figs.2
through
7
(Pages 9 and 10). Figs. 2 and 3 show the wormholingpattern from both a side view and a top view at the end of thetreatment. The vertical and radial symmetry is very similar to thesymmetry that was observed by the author several years ago atthe University of Michigan. This symmetry has now beenincorporated into the modeling of the new wormholing theory.Fig. 2 shows that at the end of the treatment there was only oneset of growing wormholes. Initially, there were seven sets, butfour of them died out within 1 minute and two more died outwithin 4 minutes of starting the treatment. This left one domi-nant wormhole set for a total of six wormholes taking theremainder of the acid for the final 20 minutes of pumping.It is also appropriate to use Fig. 3 (Page 9) to define primaryand secondary dominant wormholes. Figs. 4 through 7 (Page 10)show various calculated values for the dominant wormholes. InFig. 3, the primary dominant wormholes are the two wormholesaligned with the x-axis. The secondary dominant wormholes arethe four wormholes aligned at a 60° angle from the x-axis. Whenthe permeability in the x and y directions are the same, thecalculated values for the primary and secondary dominantwormholes are also the same. However, if a permeability con-trast exists between the x and y directions, the definition of primary and secondary wormholes becomes important, sincethe calculated values will be different.Fig. 4 (Page 10) shows the radius of the wormholes as afunction of length in the final growing wormhole set. Fig. 5(Page 10) shows the acid spending profile along the length of thedominant wormholes. The diameter of the wormholes was atleast 0.25 in. (>0.125-in. radius) for at least 60 in. (5 ft) from thewellbore and was over 1 in. in diameter for almost 4 ft from thewellbore. In addition, the bulk of the acid spending was in theleading 20% of the wormhole tip because of the longer residencetime in the developing wormholes as fluid leakoff occurred. Thesmaller wormhole diameters near the tip also allowed moreefficient acid spending since they had a higher surface-area-to-acid-volume ratio. Even though there was only one dominantwormhole set at the end of the treatment, their large size wouldstill have allowed very effective production improvement.Linear core testing has established that an optimumDamköhler number exists for creating wormholes most effi-ciently. This number is normally used in graphical form as itsinverse, 1/Da. The optimum 1/Da for high-reactivity limestone

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