The symmetry of wormholing

A new carbonate matrix wormholing model proposed for acidizing takes advantage of the experimentally observed symmetry of wormholing under radial flow conditions.

Acid wormholing through carbonate formations can provide significant stimulation, with resulting skins in the range of -3 to -3.5 common for matrix carbonate acidizing treatments. Considerable laboratory research has been conducted to understand wormhole initiation and penetration distances. Most of the studies have been conducted on linear cores, and resultant predictions were that field treatment conditions had to be controlled carefully to obtain good stimulation results. However, field experience indicates good stimulation results are easy to achieve across a range of conditions.
Matrix acidizing of carbonates is covered in the SPE monograph Acidizing Fundamentals.1 A method is given for calculating the spending of acid down a dominant wormhole in either turbulent or laminar flow. Hand calculations of acid spending lengths can be performed with or without fluid leakoff. Unfortunately, three fundamental questions have remained unanswered for the past 20 years, preventing use of the published concepts:
• How many dominant wormholes are generated?
• What is the spatial distribution of these dominant wormholes?
• What is the leakoff profile from the dominant wormholes?
While the wormholing theories developed for laboratory flow tests on small cores seem sound, they have not scaled successfully to field-size treatments. In fact, the well-established laboratory theories predict optimum field injection rates should be about one-tenth as fast as those used with great success. The reason for the failure of theory seems to be the fundamental difference between linear and radial flow conditions and errors due to end effects that can dominate short core laboratory experiments. A new theory proposes to answer these three fundamental questions.2,3
The breakthrough
G. Daccord has published work on the wormhole patterns formed by the dissolution of plaster with water.4-6 His work set the stage for the new theory. Daccord's pictures of radial flow dissolution patterns created by water injection into plaster casts showed a high degree of symmetry that was clearly evident in the wormholing patterns (Figure 1).
The presence of the high degree of symmetry in the radial flow dissolution experiments was of profound significance. Symmetry occurs in highly coupled systems, and its presence can dramatically affect the ease of understanding and modeling of a particular phenomenon. For example, organic and inorganic chemists use symmetry to predict certain reactivity characteristics. In a similar way, the presence of symmetry in the dominant wormholing patterns for radial flow dissolution allows for significant insight and powerful simplification of the phenomenon.
In Daccord's radial dissolution experiments, water was flowed from a wellbore installed in a cylinder of hardened plaster. The observed symmetry existed radially from the center and linearly along the length of the wellbore (Figure 2). Dominant wormholes occurred as sets of either five or six in a single horizontal plane. The wormholes in sets containing six had a radial separation of 60° from each other. Such a distribution essentially formed a series of isosceles triangles. As a result, the distance between the tips of the separate dominant wormholes in each set was the same as the length of the wormholes.
These horizontal sets occurred in a periodic fashion along the vertical length of the wellbore but had different radial penetrations. Several sets of wormholes about 1 in. in radial length were separated from each other by about 1 in. Inspection showed that every other set had ceased to grow, and the remainder continued to grow until their length was equal to the separation distance between them (about 2 in.). Once again, about half of those sets quit growing while the remaining sets continued to grow until one wormhole broke through the edge of the plaster cylinder.
The presence of the symmetry means all dominant wormholes must "know" about each other somehow. That is, the wormholing process is a highly coupled phenomenon, and dominant wormholes do not form randomly from each other in radial flow. The mechanism for coupling is through the pressure field induced by matrix flow of fluid through the porous media.
For example, 1,000 gal of 15% HCl can dissolve 10.9 cu ft (0.3 cu m) of limestone, but that same volume of acid occupies 134 cu ft (3.8 cu m) of space. Only 8% of the injected acid can occupy space created by limestone dissolution. Therefore, 92% of the injected acid must flow through the original matrix porosity and be influenced by permeability contrasts within the matrix. In other words, the patterns must observe the influences of Darcy's law for matrix radial flow.
This mechanism includes pressure interferences between developing dominant wormholes through the matrix flow of displaced pore fluids and invading spent fluids. Minimizing these interferences requires that the dominant wormholes stay as far apart as possible. As a consequence, the development of multiple wormholes in radial flow must be symmetrical in the radial and vertical directions.
Most investigators assume the direction of fluid flow through the matrix is governed by the developing wormhole pattern. As a result, they focus on the physics of the wormhole growth and ignore the matrix. The breakthrough in thinking is that the developing wormhole pattern is actually governed by fluid flow through the matrix. Once this concept is accepted, the diameter and length of the developing dominant wormholes are readily calculated from acid reactivity and spending procedures used in advanced acidizing theory.
The new theory
The new theory is that wormholing of acid in carbonates is highly symmetrical when performed on the scale of field acid treatments. In addition, the secondary shape of this symmetry is governed by the native formation permeability.
In the few radial carbonate flow tests reported, the high solubility of carbonate in HCl, the generation of a CO2 gas phase and the fairly nonhomogenous nature of carbonate cores prevented the high degree of symmetry from forming before wormhole breakthrough occurred. Therefore, end effects were still a dominant part of the wormholing pattern. However, a close inspection of casts from these radial HCl flow experiments indicated the symmetrical acidizing was initially trying to form. Such symmetry could never be observed in linear cores.
Verification of the new theory with laboratory experiments presents serious challenges. Large pieces of cylindrical quarried limestone and special large-scale core holders would be required. Instead, a different approach was taken to develop the new theory. The thought process was: "If symmetry is correct...and if Darcy's Law is true...then what assumptions must be imposed to generate symmetry?" The assumptions for the new theory are technical and have been presented elsewhere.2,3 Instead of laboratory experiments, the assumptions were used to create a computer model.
The new model
Extensive parametric studies using the new model provided unexpected results. The model was written with the expectation that a spent acid front would precede the wormhole development front, as suggested by the multiple pore volumes of acid required in laboratory linear flow experiments. In addition, the spending routine was written assuming laminar flow in the wormholes. However, as the model was studied for its behavior, it was found that the flow in the wormholes was usually turbulent, and live acid stayed very close to the fluid invasion front. This behavior required incorporation of turbulent flow spending and a closer examination of fluid leakoff. In addition, it was initially puzzling that live acid could penetrate more than 30 ft (9 m) into a high-reactivity matrix. This observation temporarily put the spending routine in question.
As a check, hand calculations performed by the author several years ago using mathematics and charts from the Acidizing Fundamentals monograph were reviewed. HCl is capable of traveling down a wormhole for more than 100 ft (31 m) in a high-reactivity carbonate, depending on conditions such as wormhole diameter, fluid velocity and fluid leakoff. So it seems the monograph and the new wormholing theory are in agreement.
Findings from the model
The role of reactivity for most practical matrix treatments is to generate wormhole diameter. Specifically, live acid penetration to 20 ft (6 m) from the wellbore is governed by pumping enough acid to fill the accessible matrix porosity to a radial distance of 20 ft (6 m), regardless of reactivity. If the carbonate is of low reactivity, the wormholes will be small in diameter, and production increase could be disappointing. If the carbonate is of moderate to high reactivity, the wormholes will be of moderate to large diameter, and production increase will be as expected. In both cases, the live acid will have penetrated the 20 ft (6 m). This is also true even though the same rock-dissolving power is used. The difference is that low-reactivity carbonates are dissolved uniformly throughout the matrix, while high-reactivity carbonates are dissolved more in the dominant wormholes than in the matrix.
Since wormhole length is governed simply by fluid invasion distance, higher acid concentrations only affect wormhole diameters - not wormhole length. This consideration might be important for a low-reactivity carbonate where wormhole diameter development is slow. However, for a high-reactivity carbonate, there would be no advantage of 28% HCl over 15% HCl because the wormhole diameter would already be large enough if 15% HCl were used.
The most important factor in determining acid penetration is the matrix invasion distance of the acid. The formation porosity, permeability contrast and the volume of acid pumped into the formation control the invasion distance. Therefore, it becomes important to know the formation porosity and the accessible porosity to flow. Calibration of the model should be possible if the accessible porosity and permeability contrasts are adjusted to match post-stimulation skins. The impact of porosity can finally help explain why high-porosity chalks can be so difficult to stimulate to the same level as classical carbonates with porosities of 10% to 15%. In addition, the impact of natural fracturing, or permeability contrast, to cause oriented acid invasion can finally help explain why some carbonates generate such impressive production improvements with moderate volumes of acid.
Improved wormhole penetration distance will likely depend on methods that prevent acid from filling the entire matrix porosity, thereby reducing the effective or accessible porosity for the treatment. This effect could be accomplished with fluid-loss control measures such as oil-soluble resin, gel diverters, emulsified acid or foam diversion.
Designs made easy
Two rules of thumb govern matrix acidizing of carbonates. First, expect to pump the acid at about 0.1 bbl/min/ft of zone. Higher rates are better if the wellhead and friction pressure limitations permit. Second, expect to pump about 100 gal/ft of acid. Doubling the volume of acid doesn't double the production over the results from 100 gal/ft. These two guidelines have been useful, but do not allow for optimizing a matrix treatment based on formation properties. The new theory finally allows optimization of acid treatments.
The real purpose of the new wormholing theory is not to simply provide a new description of how wormhole patterns might be described. Rather, it is to exploit the simplicity of the theory to provide easy design criteria for acidizing carbonate formations. As such, consider the two cases, circular acidizing and elliptical acidizing, and how they affect stimulation results.
Circular acidizing. Carbonate acidizing creates wormhole patterns that dramatically improve the permeability of the formation. A great deal of the research presented in the literature was directed at describing the type of wormhole formation or the nature of the branching under different flow rate conditions. However, once the permeability of the formation has been increased by a factor of 100, the nature of the wormhole pattern becomes secondary to knowing how far from the wellbore the pattern reaches. This concept can be demonstrated if the apparent stimulation skin is calculated for various permeability improvements.
Figure 3 shows stimulated (negative) skin for permeability improvement to various depths assuming circular (uniform) radial flow. For example, if an acid treatment improves the permeability of the formation by a factor of three to a depth of 20 ft (6 m), then the apparent skin would be -3. If the permeability were improved by a factor of 10, then the skin would be -4. However, once the permeability is improved by a factor of 100, additional improvement does not further decrease the stimulated skin beyond -4.3.
The hundreds of linear flow tests of acid through carbonate cores clearly show that permeability improvement resulting from wormhole formation almost always exceeds 100-fold. The only time this improvement is slow to happen is with highly retarded acid systems such as emulsified acid7 or very slow-reacting dolomites. Even in these cases, the permeability improvement exceeds 100-fold; it is just a little slower developing. It becomes apparent, therefore, that once a wormhole pattern is created, it is only necessary to know how far from the wellbore the pattern extends.
Figure 4 takes the new concept that the wormhole distance is predominantly controlled by volumetric fill and combines it with stimulated skin, assuming in excess of 100-fold permeability improvement by the wormholing. The figure is a design chart for typical cases where the invaded porosities are fairly large and the acid volumes are relatively small to moderate. The porosities listed in the figure are the invaded or accessible porosities. One recent case history suggests only 30% to 50% of the formation porosity is invaded by acidizing in carbonates.
Figure 4 is intended for use as a design aid and for understanding the difference in treatment performance. For example, suppose a chalk having 3-md permeability and 40% porosity was acidized and one-half the porosity (20%) was invaded by the acid. The figure shows that a treatment of 100 gal/ft of 15% HCl would be expected to penetrate about 4.5 ft (1.4 m) and provide a stimulation skin of about -2.8. This improvement would be significant. However, suppose another carbonate having 3-md permeability and 15% porosity was acidized as well. In this case, it is suspected that the acid only invaded one-third of the porosity (5%). The same 100 gal/ft of 15% HCl would now be expected to penetrate about 9 ft (2.7 m) for a stimulation skin of -3.6. Production from this latter formation would be significantly better than for the chalk. Without knowing about the concept of volumetric fill, one might mistakenly suspect reactivity, wormhole structure or emulsion damage for the difference in performance.
In fact, Figure 4 shows the chalk would require four times as much acid, or 400 gal/ft, to achieve similar stimulation results. Deeper penetration of wormholes with 100 gal/ft cannot be achieved by simple acid retardation; volumetric fill requirements do not allow reactivity to come into play at such short penetration distances. Slowing the reactivity predominantly reduces the diameter of the wormholes and hence the permeability improvement.
How does emulsified acid provide better stimulation in chalks? The answer is not through retardation, but by dramatically reducing the invaded porosity. Emulsions have significant viscosity that effectively reduces leakoff and enhances fingering through the matrix. The emulsion makes much of the porosity less accessible. Enhanced performance from emulsified acid in chalks comes from reducing the invaded porosity, thereby providing deeper penetration simply based on volumetric fill requirements. If the emulsified acid reduces the invaded porosity from one-half (20%) to one-fourth (10%), the stimulation skin improves from about -2.8 to about -3.2.
Only when wormholes grow to considerable length does reactivity become an important issue for treatment design. Unfortunately, it is difficult to achieve long wormhole lengths in uniform permeability formations. Calculations indicate that deep matrix invasion in uniform formations is only achievable with low invaded porosities and large acid volumes. For example, suppose a treatment of 1,000 gal/ft of 15% HCl is pumped into a formation in which the invaded porosity is 5%. The wormholes would extend less than 30 ft (9 m), but could still generate a stimulation skin of about -4.7. This treatment would be excellent and capable of long-term sustained production, assuming the reservoir pressure and permeability were high enough.
Even in this case, the new wormholing model indicates reactivity retardation would not be required. Unfortunately, this volume of acid is about 10 to 50 times greater than is economically viable in most cases.
Elliptical acidizing. Figure 4 would seem to indicate matrix acidizing of carbonates can only effectively generate stimulation skins limited to about -4. However, a few matrix stimulation treatments seem to indicate stimulation skins better than -5. Such performance does not seem possible unless preferential acidizing occurs in one direction. In fact, evidence is growing that perhaps many carbonates do treat preferentially in one direction. The new wormholing theory readily accommodates this type of performance. Permeability contrast in carbonates can occur from natural fracturing and depositional layering. The relative direction of the contrast will be dependent on which type of permeability effects are present and the orientation of the wellbore in the reservoir - horizontal, deviated or vertical.
The dominant effect of permeability contrast in the x and y directions, perpendicular to the wellbore, is to change the fluid invasion shape from circular to elliptical. Invasion profiles for 100 gal/ft of acid into a 20% porosity rock were calculated at four different permeability ratios. The calculations assume:
• 100% fluid invasion into the native porosity; and
• the pressure drop through the acidized matrix is insignificant.
The first assumption is obviously not correct. Instead, 20% must be viewed as the accessible porosity under matrix invasion conditions. The second assumption has no effect under uniform permeability conditions, but could have a small error in naturally fractured formations if acidized permeability development is slow. The error would only occur in low-reactivity carbonates or at very high injection rates.
Figure 5 reports calculations for the volumetric requirements to achieve elliptical wormhole pattern penetration in one direction to a certain distance. The figure can be considered a design chart for deep matrix acidizing of naturally fractured reservoirs or selective acidizing in some horizontal wells. The chart is considered valid for determining the volume requirements to achieve a certain elliptical penetration at the listed invaded porosities for any permeability contrast greater than 1. However, the stimulation skins marked on the figure are considered valid only for permeability contrasts, kx/ky, less than 10. At larger contrasts, such as a kx/ky of 100, one begins to expect the production increase to behave more like a fracturing treatment than a matrix treatment.
Figure 5 is not as easy to use as the previous design chart, but it is still not difficult to use. Notice that the x-axis is maximum wormhole penetration and the y-axis is acid volume times the permeability contrast. Suppose a matrix acidizing treatment is to be designed in a vertical well. Furthermore, it is known that the formation is naturally fractured and it is thought that the apparent permeability aligned with the natural fractures is 10 times the matrix permeability perpendicular to the natural fractures. In addition, the matrix porosity is 10%, and so it is assumed the invaded porosity will be about one half, or 5%. The treatment is to be designed for a skin of about -5.
The design chart shows that the wormholes will be required to penetrate about 37 ft (11.3 m) to achieve a skin of -5. Therefore, the volume requirement of acid will be about 1,700 (gal/ft)( kx/ky). If kx/ky were 1, then the treatment would require about 1,700 gal/ft, far too large to be economical. However, since kx/ky is considered to be about 10, the treatment should require about 170 gal/ft.
It is under conditions of deep, elliptical invasion that reactivity can become an issue. Calculations performed by the author, using classical reactivity design charts presented in the Acidizing Fundamentals monograph for acid spending down reactive tubes, indicate the potential for live acid penetration is greater than 100 ft (31 m). This potential is apparently true whether the flow is laminar or turbulent, or whether leakoff occurs or not. The ultimate penetration distance is clearly a function of these variables, but it is difficult to demonstrate potential live acid penetration to be only 30 ft (9 m) based on reactivity alone. However, it is reasonable to expect that under some conditions, it might be difficult to obtain live acid penetration beyond about 50 ft (15 m). A hot limestone with natural fracturing and a desired stimulation skin better than -5.5 is an example where reactivity might become an issue governing ultimate live acid penetration.
Use the charts
Operators and acidizing service providers may use these simple charts as aids to acid-stimulation job design. The charts shown in Figures 3 to 5 enable quick determination of:
• skin resulting from uniform radial acidizing;
• designs for shallow, uniform radial acidizing; and
• designs for deep, elliptical acidizing.
These charts are based upon a fundamentally new theory of acid wormholing in carbonates (i.e., that there is symmetry in acid wormholing patterns). The design charts also can be used for treatments in naturally fractured carbonates where permeability contrasts can be significant.
References
1. Williams, B.B., Gidley, J.L. and Schechter, R.S.: Acidizing Fundamentals, Monograph Volume 6, Henry L. Doherty Series, SPE-AIME, Dallas, Texas (1979).
2. Gdanski, R.D.: "A Fundamentally New Model of Acid Wormholing in Carbonates," paper SPE 54719 presented at the 1999 European Formation Damage Conference, The Hague, The Netherlands, May 31-June 1.
3. Gdanski, R.D.: "The Symmetry of Acid Wormholing in Carbonates," paper presented at the 2000 NIF Oil Field Chemicals Symposium, Fagernes, Norway, March 20-22.
4. Daccord, G.: "Chemical Dissolution of a Porous Medium by a Reactive Fluid," Phys. Rev. Lett. (1987) 58, 479-482.
5. Daccord, G. and Lenormand, R.: "Fractal Patterns from Chemical Dissolution," Nature (1987) 325, 41-43.
6. Daccord, G. Touboul, E. and Lenormand, R.: "Carbonate Acidizing: Toward a Quantitative Model of the Wormholing Phenomenon," SPEPE (February 1989) 63-68.
7. Buijse, M.A. and van Domelen, M.S.: "Novel Application of Emulsified Acids to Matrix Stimulation of Heterogeneous Formations," paper SPE 39583 presented at the 1998 International Symposium on Formation Damage Control, Lafayette, La., Feb. 18-19.