J199236 DOI: 10.

2118/199236-PA Date: 9-June-20 Stage: Page: 1 Total Pages: 23

Decision Criterion for Acid-Stimulation

Method in Carbonate Reservoirs:
Matrix Acidizing or Acid Fracturing?
Mateus Palharini Schwalbert, Petrobras; Murtada Saleh Aljawad, King Fahd University of Petroleum and Minerals;
and Alfred Daniel Hill and Ding Zhu, Texas A&M University

Summary
Most wells in carbonate reservoirs are stimulated. Because of their low cost and simpler operations, acid-stimulation methods are usu-
ally preferred if they are sufficient. Matrix acidizing can effectively stimulate carbonate reservoirs, often resulting in skin factors on the
order of 3 to 4. In low confining stress and hard rocks, acid fracturing can yield better results than matrix acidizing. However, acid
fracturing is less effective in high permeability, high confining stress, or soft rocks. There is a combination of parameters, among them
permeability, confining stress, and rock geomechanical properties, that can be used as criteria to decide whether matrix acidizing or
acid fracturing is the best acid-stimulation technique for a given scenario.
This study compares the productivity of matrix-acidized and acid-fractured wells in carbonate reservoirs. The criterion used to
decide the preferred method is the largest productivity obtained using the same volume of acid for both operations. The productivity of
the acid-fractured wells is estimated using a fully coupled acid-fracturing simulator, which integrates the geomechanics (fracture propa-
gation), pad and acid transport, heat transfer, temperature effect on reaction rate, effect of wormhole propagation on acid leakoff, and
finally calculates the well productivity by simulating the flow in the reservoir toward the acid fracture. Using this simulator, the acid-
fracturing operation is optimized, resulting in the operational conditions (injection rate, type of fluid, amount of pad, and so forth) that
lead to the best possible acid fracture that can be created with a given amount of acid. The productivity of the matrix-acidized wells is
estimated using the most recent wormhole-propagation models scaled up to field conditions.
Results are presented for different types of rock and reservoir scenarios, such as shallow and deep reservoirs, soft and hard lime-
stones, chalks, and dolomites. Most of the presented results considered vertical wells. A theoretical extension to horizontal wells is also
presented using analytical considerations. For each type of reservoir rock and confining stress, there is a cutoff permeability less than
which acid fracturing results in a more productive well; at higher than this cutoff permeability, matrix acidizing should be preferred.
This result agrees with the general industry practice, and the estimated productivity agrees with the results obtained in the field. How-
ever, the value of the cutoff permeability changes for each case, and simple equations for calculating it are presented. For example, for
harder rocks or shallower reservoirs, acid fracturing is more efficient up to higher permeabilities than in softer rocks or at
deeper depths.
This method provides an engineered criterion to decide the best acid-stimulation method for a given carbonate reservoir. The deci-
sion criterion is presented for several different scenarios. A simplified concise analytical decision criterion is also presented: a single
dimensionless number that incorporates all pertinent reservoir properties and determines which stimulation method yields the most
productive well, without needing any simulations.

Introduction
A great part of the world’s conventional hydrocarbon reserves are found in carbonate reservoirs. The rocks that form these reservoirs
are composed of more than 50% of carbonate minerals (Economides and Nolte 2000), the most common being calcite and dolomite.
Most wells in these reservoirs are stimulated.
The most common stimulation methods applied in this scenario are matrix acidizing, acid fracturing, and propped hydraulic fractur-
ing (Economides and Nolte 2000). Matrix acidizing and acid fracturing take advantage of the fact that carbonate rocks are soluble in
most acids.
From an operational point of view, the execution of an acid-fracturing treatment is easier than the execution of a propped fracture
(Economides and Nolte 2000), mainly because of the risk of screenouts in propped hydraulic fracturing. It has been reported that the
propped hydraulic fracture is difficult to be concluded in hard offshore carbonates with high closure stresses because of screenouts
(Azevedo et al. 2010; Neumann et al. 2012). The stability of the rock layers above and below the reservoir when subjected to the high
pressure of the fracturing process is also an operational concern (Oliveira et al. 2014), as well as the integrity of wellbore equipment.
Because of their low cost and simpler operations, if acid-stimulation methods are sufficient to stimulate a given well, they are usu-
ally preferred instead of propped hydraulic fracturing. Especially in offshore wells, where operational problems lead to more costly con-
sequences, the methods that offer less risk are usually preferred. In the stimulation of wells in carbonate reservoirs, if matrix acidizing
or acid fracturing can give results similar to the propped hydraulic fracturing, the first two methods are usually preferred for
practical reasons.
There are studies regarding the selection of the hydraulic-fracturing method for a given scenario (selecting between acid and
propped fracture). Examples of such studies are Ben-Naceur and Economides (1989), Abass et al. (2006), Vos et al. (2007), Azevedo
et al. (2010), Neumann et al. (2012), Oliveira et al. (2014), Cash et al. (2016), Jeon et al. (2016), and Suleimenova et al. (2016).
Daneshy et al. (1998) mention that proppant is usually required in wells with closure stress greater than 5,000 psi. However,
Neumann et al. (2012) discuss the fact that the limit of 5,000 psi is just a general guideline based on the behavior of shallow soft carbon-
ates, while deeper carbonates might be mechanically more competent in some cases. Neumann et al. (2012) and Oliveira et al. (2014)
present some results that can expand the limit of 5,000 psi to higher values.

Copyright V
C 2020 Society of Petroleum Engineers

This paper (SPE 199236) was accepted for presentation at the SPE International Conference and Exhibition on Formation Damage Control, Lafayette, Louisiana, USA, 19–21 February 2020,
and revised for publication. Original manuscript received for review 13 January 2020. Revised manuscript received for review 24 February 2020. Paper peer approved 9 March 2020.

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However, there has not been much study regarding the selection of the stimulation method between matrix acidizing and acid frac-
turing. Oliveira et al. (2014) reported problems and unsatisfactory results in acid-fracturing operations when a matrix-acidizing opera-
tion had already been performed on the same well. They mention the importance of a criterion to select the best stimulation method
between acid fracturing and matrix acidizing, which is not obvious and does not yet exist in the industry.
The focus of this study is on matrix acidizing and acid fracturing in carbonate reservoirs. In both techniques, the enhancement in
well performance results from a dissolution structure created by acid, and the outcome is somewhat proportional to the volume of acid
injected. Therefore, it is expected that for a given well and volume of acid, one of these methods renders better results than the other.
A well-designed and executed matrix-acidizing treatment can result in a skin factor on the order of 4 (Burton et al. 2018). This
result depends on the reservoir mineralogy and existence of natural fractures, but it seems to be insensitive to reservoir permeability,
according to a large data set presented by Burton et al. (2018), as long as the reservoir is permeable enough to allow matrix injection.
The outcome of acid-fracturing operations, however, is very sensitive to reservoir permeability. In general, fracturing stimulation
results in better (more negative) skin factor when performed in lower-permeability reservoirs. This leads to the industry general rule of
thumb: to matrix acidize wells in high-permeability carbonate reservoirs and to acid fracture (or hydraulic fracture in general) wells in
lower-permeability carbonates. The value of the cutoff permeability, however, is not clearly defined in the literature, and different com-
panies use different cutoff values. Daneshy et al. (1998) mention that a general guideline is to use acid fracturing in carbonate reservoirs
with permeability less than 20 md. However, they do not present a source or a scientific reason for this value choice.
The objective of this study is to define a decision criterion to select the best method between matrix acidizing and acid fracturing for
a given scenario. In other words, for a given scenario and volume of acid, is it preferable to matrix acidize or to acid fracture a well?
In practice, this decision does not only depend on the achievable productivity index (PI). For example, in wells where zonal isolation
is important, if the geomechanics indicate that a hydraulic fracture can grow into undesired zones, it is common to avoid hydraulic frac-
turing. Because the pressures involved in fracturing are higher than in matrix acidizing, there can be also mechanical and logistical con-
straints to using hydraulic fracturing. In addition, matrix acidizing is a simpler stimulation method, with low risk of failure, low cost,
and longstanding results, as shown in Burton et al. (2018). In this sense, if the maximum PI is not a concern, matrix acidizing is often
the selected method.
However, mechanical or logistical constraints are not analyzed in this study. This work focuses on wells where both methods can be
applied, with the objective of determining which method has potential to result in greater PI using the same volume of acid. Other stim-
ulation methods, such as propped hydraulic fracturing, are not included in the analysis.
For the fractured wells, only the production from the biwing fracture is considered. No natural-fracture networks are considered. In
Ugursal et al. (2018), we studied the productivity of acid-fractured wells where the main acid fracture intersects natural fractures. As a
general rule, in all cases presented in Ugursal et al. (2018), the PI with or without the presence of natural fractures is on the same order
of magnitude. In most cases, the productivity is larger when the hydraulic fracture intersects natural fractures, but in some cases it is
lower because of the acid lost to the natural fractures. Because the study did not consider fracture propagation, those results should
ideally be revisited with a fracturing model that includes fracture propagation.
In this section, only the PI in the pseudosteady state is used for comparison. The same has been applied for most studies of produc-
tivity of conventional hydraulic fractured wells, such as Economides et al. (2002) and Meyer and Jacot (2005), and it is a consensus
that optimizing the PI for the pseudosteady state is enough for conventional reservoirs. In fact, Economides et al. (2002) mention that a
“common misunderstanding is related to the transient flow period. … In reality, the existence of a transient flow period does not change
the previous conclusions on optimal dimensions. Our calculations show that there is no reason to depart from the optimum compromise
derived for the pseudosteady state case, even if the well will produce in the transient regime for a considerable time (say months or
even years). Simply stated, what is good for maximizing pseudosteady state flow is also good for maximizing transient flow.”
In summary, the objective of this study is to create a decision criterion to estimate, for a given scenario and volume of acid, whether
matrix acidizing or acid fracturing is the stimulation technique to result in a higher PI in the pseudosteady state.

Methodology
The methodology used in this study consisted of using state-of-the-art models to simulate wormhole propagation and acid-fracturing
operations, and to calculate the dimensionless PIs of the stimulated wells. The entire modeling and analysis processes are detailed in
Palharini Schwalbert (2019).
The dimensionless PI used in this study is defined, in consistent units, as
q
Dpreservoir Bl
JD ¼ ¼ J; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð1Þ
2pkH h 2pkH h
Bl

where JD is the dimensionless PI, q is the production rate, Dpreservoir is the reservoir drawdown, B is the formation volume factor of the
produced fluid, l is the produced-fluid viscosity, kH is the effective horizontal permeability, h is the net pay (permeable-formation
thickness), and J is the PI (q=Dpreservoir ). The same equation applies for injector wells, in which case q is the injection rate and the fluid
properties are those of the injected fluid. The same definition of JD was used for both vertical and horizontal wells.
The model presented by Al Jawad (2018) was used to simulate the acid-fracturing operations, with the modifications presented in
Palharini Schwalbert (2019). The acid-fractured-well productivity model presented in Aljawad et al. (2020) was used to evaluate the
productivity of the fractured wells. The fully coupled model simulates the fracture propagation and dissolution of the fracture walls
caused by the acid reaction. The fracture conductivity is calculated using correlations such as Nierode and Kruk (1973) or Deng et al.
(2012). The temperature distribution is also solved at each timestep, and it is coupled with the acid transport and reaction, which is
especially significant in the simulation of dolomite formations, as shown in Aljawad et al. (2019). The fractured-well productivity is
calculated by simulating the flow in the reservoir-drainage region of the given well. Fig. 1 shows a flow chart illustrating the acid-
fracturing simulator.
The optimization of the acid-fracturing operations followed the procedure presented in Aljawad et al. (2020). The optimal acid-
fracturing design for each scenario is found by simulating different operational conditions (varying injection rate, type of acid system,
and amount of pad), until a maximum PI is achieved for a given amount of acid used. Fig. 2 shows a flow chart illustrating the proce-
dure to calculate the optimal acid-fracturing design, by running several simulations of the simulator illustrated in Fig. 1. Notice that the
flow chart shown in Fig. 1 is but one step of the flow chart presented in Fig. 2.

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Read input t= Geometry

Acid during
data t + Δt during injection Converge?
injection
Yes

Leakoff calculation, No
accounting for Leaked acid
wormholes creates
wormholes
No
Yes t = Final Yes
Acid during Geometry t= Temperature
treatment Converge?
closure during closure t + Δt time (te)? during injection

No

No
No
t= Fractured-well-productivity
Temperature Yes Fracture- Yes
Converge? Fracture calculation, simulating the
during closure closure time
conductivity reservoir including the
(tc)
fractured well

Fig. 1—Flow chart illustrating the acid-fracture simulator.

Change
Start: read Try another Try another Try another
reservoir
input data acid system injection rate pad volume
permeability
Loop over
Loop over
Loop over pad volumes
injection rates
No acid systems No No Acid-fracturing simulation
(fracture propagation acid
All pad transport, wormhole
Yes All acid All pad creation, heat transfer,
injection
Loop over permeabilities

systems volumes fracture conductivity,

rates
simulated? Yes Yes simulated? and finally
simulated?
fractured-well-productivity
calculation, as shown
in Fig.1)

Whole range
of reservoir Yes
End. Full map of optimal
permeabilities acid-fracturing
simulated? operational parameters and
maximum possible
No acid-fractured-well productivity
for each permeability.

Fig. 2—Flow chart illustrating the acid-fracture simulator.

In the case studies presented in this study, the acid-fracture-conductivity model by Nierode and Kruk (1973) was used. This is an
empirical correlation that relates the acid-fracture conductivity to the rock-embedment strength (property of the rock) and amount of
rock dissolved by acid. The results obtained, however, do not apply only to that model. They can be generalized to other conductivity
models using the generalized conductivity correlation presented in Palharini Schwalbert (2019). Any of the available correlations can
be written in the form of Eq. 3 (presented later in this text), and the criterion presented in this paper uses that generalized form of con-
ductivity correlation for its calculations. Palharini Schwalbert (2019) presents the equations to calculate the generalized parameters,
Parameters A and B, for a number of correlations, not only Nierode and Kruk (1973).
The matrix-acidizing operations were simulated using global models of wormhole propagation. The model used in the cases pre-
sented in this paper was the one presented in Palharini Schwalbert et al. (2019b). Palharini Schwalbert (2019) also presents comparison
with the models by Buijse and Glasbergen (2005) and Furui et al. (2012). The wormhole model has a significant effect on the results.
Because these models are basically empirical or semiempirical correlations, it is suggested that the reader use the model that better fits
their matrix-acidizing data. The skin factor of the acidized well was calculated using either Hawkins’ formula (for the vertical wells) or
the equations presented by Palharini Schwalbert et al. (2019a) (for the horizontal wells). Fig. 3 shows a flow chart illustrating the loop
of simulations of matrix-acidizing operations.
The properties of the reservoir, well, formation, and acid used as the base case are presented in Table 1. All the cases are built using
the properties listed in Table 1, varying one or more properties case to case. The reaction-kinetics parameters and heat of reaction were
obtained from Schechter (1992), and are presented in Table 2. Three different acid systems were considered: straight, gelled, and emul-
sified acid. These acid systems might have different properties depending on the chemical-additives types and concentration. The prop-
erties used for each acid system in this study are presented in Table 3 and are considered representative of a reactive-acid system
(straight acid), retarded system (gelled acid), and very retarded system (emulsified acid).

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Calculate
Change acid injection Calculate
Start acidizing
Start: read reservoir t = t + Δt velocity at wormhole
with a small
input data permeability the propagation
injection rate
wormhole in Δt
Loop over front
permeabilities
No
Whole range
of reservoir Yes Reduce injection
t = tfinal? Recalculate skin
permeabilities rate and recalculate factor because
No
simulated? timestep of the wormholes

pwi > pbd

Increase injection Calculate injection

rate to the Compare pressure (updated by
End. maximum possible injection and injection-rate
Matrix-acidized- that still keeps breakdown superposition and new
well productivity pwi < pbd pressures skin factor caused by
for each the wormholes)
permeability pwi < pbd

Fig. 3—Flow chart illustrating the matrix-acidizing simulator.

Input Data Field Unit

Reservoir/Formation Properties
Reservoir permeability, k 0 .1 m d
Reservoir porosity, 15%
Reservoir initial pressure gradient 0.4333 psi/ft
Bottomhole flowing pressure gradient (during production) 0.3 psi/ft
M in i m u m h o r i z o n t a l s t r e s s g r a d i e n t 0 .6 psi /f t
Breakdown-pressure gradient 0.7 psi/ft
Biot poroelastic coefficient 1
Poisson’s ratio 0.25
6
Young’s modulus 4×10 psi
0.5
Toughness, KIc 1,200 psi-in.
RES, SRES 50,000 psi
M i n e ra l o g y 100% calcite
3
Formation-fluid density, ρf 53 lbm/ft
Reservoir thickness (netpay), h 100 ft
Drainage-region length in x-direction, Lx 3,281 ft
Drainage-region length in y-direction, Ly 3,281 ft
Formation-fluid viscosity, μf 1 cp
Formation volume factor, B 1.3 RB/STB
–5 –1
Total compressibility, ct 1×10 psi
Reservoir temperature, TR 212°F
3
Formation-rock density, ρma 162.24 lbm/ft
Formation specific heat capacity, cma 0.2099 Btu(lbm-°F)
Formation thermal conductivity, kma 0.907 Btu/(hr-ft-°F)
Wellbore Properties
Vertical Well
Wellbore radius, rw 0.3281 ft
Inner-casing radius, r1 4.0085 in.
Outer-casing radius, r2 4.3125 in.
Overall heat-transfer coefficient, Ut 0.039 Btu/(hr-ft-°F)
Ambient temperature, Tb 77°F

Table 1—Input data used for the base case. RES ¼ rock-embedment strength.

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Input Data Field Unit

Mechanical Properties of Layers Above and Below Pay Zone

Poisson’s ratio 0.25
6
Young’s modulus 4×10 psi
0.5
Toughness, KIc 2,200 psi-in.
Horizontal stress 400 psi greater than reservoir stress
Acid Properties
3
Density, ρ 67 lbm/ft
Acid initial concentration, Ci 15%
Acid volume, Vacid 500 bbl
Filter-cake wall-building leakoff coefficient, Cw 0 (no filter cake is formed)
2
Spurt loss, Sp 0 gal/ft
Fluid-loss multiplier outside pay zone, fm 0.25
Opening time-distribution factor, K 1 .5
Acid heat capacity, cp 0.964 Btu(lbm-°F)
Acid thermal conductivity, K 0.347 Btu/(hr-ft-°F)
Acid temperature at injection, TI 80.6°F
Acid-Wormholing Parameters
Wormhole model Palharini Schwalbert et al. (2019b)
PVbt,opt,core 0 .5
vi,opt,core 2 cm/min
dcore 1 in.
ε1 0.53
ε2 0.63
drep1 3 ft
drep2 1 ft
Pad Properties
0.5
Filter-cake wall-building leakoff coefficient, Cw 0.003 ft/min
2
Spurt loss, Sp 0 gal/ft
Fluid-loss multiplier outside pay zone, fm 0. 2 5
n 2
Consistency index, K 0.0082 lbf-sec /ft
Rheology-behavior index, n 0.55
Acid-Fracture-Conductivity Correlation
Correlation used Nierode and Kruk (1973)

Table 1 (continued)—Input data used for the base case. RES ¼ rock-embedment strength.

kg-mol HCl
k r0 nr
kg-mol HCl ∆E KJ
m 2- s 3 (K ) ∆ Hr
Mineral nr m acid solution R mol HCl
7 3
Calcite 0.63 7.314×10 7.55×10 7.5

6.32 × 10−4 T 5 3
Dolomite 4.48×10 7.9×10 6.9
1 − 1.92 × 10 −3 T

Table 2—Reaction kinetics constants and heat of reaction for the reaction between HCl and calcite/dolomite (Schechter 1992).

2 n 2
Acid T (°F) N K (lb/ft -sec ) DA (cm /s) Reference
–4
Straight 84 1 0.00002 1.00×10 Roberts and Guin (1975)
–6
Gelled 84 0.55 0.0082 8.00×10 De Rozieres et al. (1994)
–8
Emulsified 83 0.675 0.0066 2.66×10 De Rozieres et al. (1994)

Table 3—Properties of the acid systems.

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Fig. 4 illustrates the acid-fracturing optimization for the base case with properties presented in Table 1 and a permeability of
0.1 md. Different acid systems were simulated (straight, gelled, and emulsified acid) for injection rates varying from 5 to 100 bbl/min,
using different amounts of pad. In Fig. 1, Npad is the ratio between the volume of pad and volume of acid. Fig. 1 shows only one
amount of pad for each acid system, but other amounts were tested, from Npad ¼ 0 (no pad) to Npad ¼ 1 (volume of pad equal to the
volume of acid). For this particular case, the maximum possible dimensionless PI for the acid-fractured well is 0.6, obtained by inject-
ing gelled acid at 60 bbl/min.

0.6

0.5

JD
0.4

Straight acid, Npad = 1

0.3 Gelled acid, Npad = 0

Emulsified acid, Npad = 0

0.2
0 20 40 60 80 100
q (bbl/min)

Fig. 4—Example of optimization of acid-fracturing treatment in a reservoir of 0.1 md.

A property of great importance in the acid-fracturing modeling presented in this paper is the rock-embedment strength (RES), SRES .
It is a standard measurement for acid fracturing, such as that presented by Nierode and Kruk (1973) and reviewed by Neumann (2011).
The standard procedure to measure it involves inserting a sphere-tipped needle into the rock sample and measuring the strength required
for the sphere to penetrate a distance equal to the sphere’s radius. The RES (SRES ) is the force divided by the sphere-projected area.
Unfortunately, it is a quantity measured only for hydraulic fracturing, so it is not broadly known in other areas. It is, however, very
well-related to Brinell hardness, a more commonly known material property, and it can also be correlated to Young’s modulus, as pre-
sented by Neumann (2011). If the SRES cannot be measured experimentally because of unavailability of downhole samples, it can be
estimated using a correlation with Young’s modulus (which can be estimated through sonic logs).
The maximum possible PI of the matrix-acidized well is obtained when the acid is injected at an injection rate to maintain the intersti-
tial velocity at the wormhole front as close as possible to the optimal interstitial velocity of wormhole propagation. Operationally, it is usu-
ally desirable to inject at more than the optimal injection rate, because it is much more efficient to inject at greater than the optimal
injection rate than to inject at less than the rate. To estimate the maximum achievable productivity, however, the optimal injection rate is
used. If the permeability of the rock is high enough to allow injection at the optimal injection rate while keeping the injection pressure less
than the fracturing pressure, the maximum-possible productivity can be achieved. If the permeability is too low, matrix injection might
not be possible at the optimal injection rate without reaching the fracturing pressure. In these low-permeability cases, the outcome of
matrix-acidizing treatments tends to be unsatisfactory because the suboptimal injection rate does not propagate wormholes efficiently.

Results
The usual industry practice is to matrix acidize wells in reservoirs of high permeability, and acid fracture when the reservoir per-
meability is low. To investigate this rule of thumb, in this study the outcome of both stimulation methods was calculated for wells in a
reservoir with a given set of properties, but varying permeability. In all cases, the acid-fractured-well dimensionless PI decreases with
increasing reservoir permeability. For matrix-acidizing treatments, some minimum permeability is required to be able to inject at the
optimal injection rate to propagate efficient wormholes. If the permeability is high enough to allow injection at the optimal injection
rate, the outcome of the matrix-acidizing job is not very sensitive to permeability. In this sense, there might be a cutoff permeability
above which the PI achievable by a matrix-acidized well is greater than that achievable by an acid-fractured well. This cutoff per-
meability is denoted in this study by kcutoff .

Case Studies of Vertical Wells. Scenario 1: Base Case. The base case is that with properties presented in Table 1. Fig. 5 shows the
maximum dimensionless PI achievable with each stimulation method as a function of permeability. The initial, unstimulated well is
assumed to be damaged with a skin factor of þ9, with an unstimulated dimensionless PI of JD0 ¼ 0:059. As expected, the acid-
fractured-well dimensionless PI decreases as the permeability increases. Notice that the PIs shown in Fig. 5 are the maximum possible
achievable with each optimized stimulation technique (the maximum value of JD ¼ 0:6 from Fig. 1, for a reservoir of 0.1 md, is just
one point in Fig. 2).
If the permeability is too low, matrix acidizing is not efficient because the injection velocity is too small. In this scenario, for perme-
abilities less than 0.1 md, matrix acidizing cannot even remove all the formation damage, and it cannot achieve the optimal injection
rate for permeability greater than 6 md. At greater than 6 md, it is possible to inject at an optimal injection rate, and the maximum
dimensionless PI with matrix acidizing can be achieved, being approximately 0.3. The method used to estimate the maximum injection
rate allowable without fracturing the formation calculated the wellbore pressure using the superposition of injection rates with a vari-
able skin factor, as detailed in Palharini Schwalbert (2019).
The cutoff permeability for this scenario is 22 md (intersection of the two curves in Fig. 2). This means that if a reservoir has the
properties listed in Table 1 and a permeability of less than 22 md, it can be better stimulated with acid fracturing. If it has a permeability
of more than 22 md, it can be better stimulated with matrix acidizing.
Scenario 2: Shallow Reservoir. Scenario 2 has all the same properties as the base case (Table 1), except that it is shallower, at a
depth of only 3,000 ft (compared with 10,000 ft for the base case). The pressure and stress gradients are the same as presented in
Table 1, but because the depth is shallower, the pressures and stresses are lower. For example, the minimum horizontal stress gradient

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is 0.6 psi/ft. In Scenario 1, this resulted in a minimum horizontal stress of 6,000 psi. In the shallow reservoir of Scenario 2, the mini-
mum horizontal stress is only 1,800 psi. Because the fracture conductivity decreases exponentially with the confining stress, the
acid-fractured-well productivity is greater in the shallow reservoir. This causes the acid-fracturing technique to be preferable over a
larger range of permeability, so a higher cutoff permeability is expected. Fig. 6 shows the maximum dimensionless PI achievable with
each stimulation method as a function of permeability.

0.9
0.8 Matrix acidizing

0.7 Acid fracturing

0.6

JD (max)
0.5
0.4
0.3
0.2
0.1
0
0.01 0.10 1.00 10.00 100.00 1,000.00
k (md)

Fig. 5—Comparison of matrix-acidized and acid-fractured well productivity: Scenario 1, base case.

1.2
Acid fracturing
1.0
Matrix acidizing

0.8
JD (max)

0.6

0.4

0.2

0
0.01 0.10 1.00 10.00 100.00 1,000.00
k (md)

Fig. 6—Comparison of matrix acidizing and acid fracturing: Scenario 2, shallow reservoir.

In this shallow reservoir, the cutoff permeability is 120 md. If the permeability is less than this value, acid fracturing results in
higher productivity than matrix acidizing in this scenario.
Scenario 3: Deep Reservoir. Scenario 3 has all the same properties as the base case (Table 1), except that it is deeper, at a depth of
20,000 ft (compared with 10,000 ft for the base case), which results in greater pressures and stresses. For comparison, the minimum hor-
izontal stress is 12,000 psi, compared with 6,000 psi for the base case. This causes the acid-fracture conductivity to be smaller, and con-
sequently the acid-fracturing technique to be preferable over a shorter range of permeability. Fig. 7 shows the maximum dimensionless
PI achievable with each stimulation method as a function of permeability.

0.7
0.6 Acid fracturing
Matrix acidizing
0.5
JD (max)

0.4

0.3

0.2

0.1

0
0.01 0.10 1.00 10.00 100.00 1,000.00
k (md)

Fig. 7—Comparison of matrix acidizing and acid fracturing: Scenario 3, deep reservoir.

For this deep reservoir, the cutoff permeability is between 2 and 3 md, meaning that in this scenario, for a reservoir of 3 md or
greater, matrix acidizing results in better productivity than acid fracturing.
Scenarios 1S, 2S, and 3S: Soft Limestones. Scenarios 1S, 2S, and 3S are derived from Scenarios 1, 2, and 3 (medium, shallow, and
deep limestones, respectively), except that the RES is lower, SRES ¼ 20,000 psi (compared with 50,000 psi of the base cases). These cal-
culations use the fracture-conductivity model by Nierode and Kruk (1973), in which the conductivity depends on SRES. The smaller the

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RES, the softer the rock, and the smaller the fracture conductivity. The results for Scenario 1S (soft limestone at medium depth) are pre-
sented in Fig. 8. The results for Scenario 2S (shallow soft limestone) are presented in Fig. 9. The results for Scenario 3S (deep soft
limestone) are presented in Fig. 10.

0.9
0.8 Matrix acidizing

0.7 Acid fracturing

0.6

JD (max)
0.5
0.4
0.3
0.2
0.1
0
0.01 0.10 1.00 10.00 100.00 1,000.00
k (md)

Fig. 8—Comparison of matrix acidizing and acid fracturing: Scenario 1S, soft limestone at medium depth.

1.2
Matrix acidizing
1.0
Acid fracturing

0.8
JD (max)

0.6

0.4

0.2

0
0.01 0.10 1.00 10.00 100.00 1,000.00
k (md)

Fig. 9—Comparison of matrix acidizing and acid fracturing: Scenario 2S, shallow soft limestone.

0.6
Acid fracturing
0.5
Matrix acidizing
0.4
JD (max)

0.3

0.2

0.1

0
0.01 0.10 1.00 10.00 100.00 1,000.00
k (md)

Fig. 10—Comparison of matrix acidizing and acid fracturing: Scenario 3S, deep soft limestone.

For Scenario 1S (Fig. 8), the cutoff permeability is 11 md. For Scenario 2S (Fig. 9), the cutoff permeability is 100 md. For Scenario
3S (Fig. 10), the cutoff permeability is 1.3 md.
As expected, in all three scenarios, the soft rock results in a less-productive acid fracture compared with the base cases. This results
in a lower cutoff permeability, compared with the scenarios of harder rock. Comparing Scenario 1S with Scenario 1, for example, the
cutoff permeability decreased from 22 to 11 md. Comparing Scenario 2S with Scenario 2 (shallow reservoir), it decreased from 120 to
100 md, and comparing Scenario 3S with Scenario 3 (deep reservoir), it decreased from approximately 2.5 to 1.3 md.
Scenarios 1H, 2H, and 3H: Hard Limestones. Scenarios 1H, 2H, and 3H are again derived from Scenarios 1, 2, and 3 (medium,
shallow, and deep limestones, respectively), except that the rocks are harder, with a higher RES, SRES ¼ 200,000 psi (compared with
50,000 psi of the base cases). According to the fracture-conductivity model by Nierode and Kruk (1973), the higher RES (harder rock)
leads to higher fracture conductivity. The results for Scenarios 1H, 2H, and 3H (hard limestone at medium, shallow, and deep depth,
respectively) are presented in Figs. 11, 12, and 13, respectively.
For Scenario 1H (Fig. 11), the cutoff permeability is 55 md. For Scenario 2H (Fig. 12), it is 160 md. For Scenario 3H (Fig. 13),
15 md.

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0.9
0.8 Matrix acidizing

0.7 Acid fracturing

0.6

JD (max)
0.5
0.4
0.3
0.2
0.1
0
0.01 0.10 1.00 10.00 100.00 1,000.00
k (md)

Fig. 11—Comparison of matrix acidizing and acid fracturing: Scenario 1H, hard limestone at medium depth.

1.2
Matrix acidizing
1.0
Acid fracturing
0.8
JD (max)

0.6

0.4

0.2

0
0.01 0.10 1.00 10.00 100.00 1,000.00
k (md)

Fig. 12—Comparison of matrix acidizing and acid fracturing: Scenario 2H, shallow hard limestone.

0.8
Matrix acidizing
0.7
Acid fracturing
0.6

0.5
JD (max)

0.4

0.3

0.2

0.1

0
0.01 0.10 1.00 10.00 100.00 1,000.00
k (md)

Fig. 13—Comparison of matrix acidizing and acid fracturing: Scenario 3H, deep hard limestone.

As expected, in all three scenarios, the hard rock results in a more-productive acid fracture compared with the base cases. This
results in a higher cutoff permeability compared with the softer-rock scenarios. Comparing Scenario 1H with Scenario 1, for example,
the cutoff permeability increased from 22 to 55 md. Comparing Scenario 2H with Scenario 2 (shallow reservoir), it increased from 120
to 160 md, and comparing Scenario 3H with Scenario 3 (deep reservoir), it increased from approximately 2.5 to 15 md.
The deeper the reservoir, the higher the effect of the RES. In the shallow reservoir, when SRES increases from 50,000 to 200,000 psi,
the cutoff permeability increases 33%, from 120 to 160 md. In the deep reservoir, however, the cutoff permeability goes from 2.5 to
15 md, an increase of 600%. The reason for this behavior is the confining stress, which is much higher in the deeper reservoir, requiring
a hard rock to withstand the open fracture. In the shallow reservoir, the confining stress is small enough for the rock hardness to be
less significant.
Fig. 14 shows the comparison of different RESs at the same depths. The same plot presents the results for Scenarios 1S, 1, and 1H,
with RESs of 20,000, 50,000, and 200,000 psi, respectively. This plot shows how the acid-fracturing outcome varies with the RES.
Scenarios 4 and 5: Different Acid Volumes. Scenarios 4 and 5 consist of the same reservoir as the base case (Scenario 1) but use
different volumes of acid. While the base case used 500 bbl of 15% hydrochloric acid (HCl), in Scenario 4, that volume is reduced to
100 bbl (1 bbl/ft), and in Scenario 5, it is increased to 2,000 bbl (20 bbl/ft).
The outcome of both stimulation methods for all three volumes of acid is presented in Fig. 15. As expected, the PI increases with
the volume of acid for both methods. Comparing Fig. 15a with Fig. 14 shows that for these scenarios the volume of acid has a more sig-
nificant effect on the acid-fracturing outcome than the RES. The comparison between the methods for each volume of acid is presented
in Fig. 16 (Scenario 4) and in Fig. 17 (Scenario 5).

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0.9 SRES = 200,00 psi

0.8 SRES = 50,00 psi
SRES = 20,00 psi
0.7
Matrix acidizing
0.6 Unstimulated
0.5

JD (max)
55 md
0.4

0.3
0.2
22 md
0.1 11 md
0
0.01 0.10 1.00 10.00 100.00 1,000.00
k (md)

Fig. 14—Comparison of different RESs.

1.8 0.40
1.6 Acid fracturing, 20 bbl/ft
0.35
1.4 Acid fracturing, 5 bbl/ft
0.30
1.2 Acid fracturing, 1 bbl/ft
0.25

JD (max)
JD (max)

1.0 Unstimulated
0.20 Matrix acidizing, 20 bbl/ft
0.8
0.6 0.15 Matrix acidizing, 5 bbl/ft
Matrix acidizing, 1 bbl/ft
0.4 0.10
Unstimulated
0.2 0.05
0 0
0.01 0.10 1.00 10.00 100.00 1,000.00 0.01 0.10 1.00 10.00 100.00 1,000.00
k (md) k (md)
(a) Acid fracturing (b) Matrix acidizing

Fig. 15—Impact of acid volume on (a) acid fracturing and (b) matrix acidizing.

0.5
Matrix acidizing
0.4 Acid fracturing

0.3
JD (max)

0.2

0.1

0
0.01 0.10 1.00 10.00 100.00 1,000.00
k (md)

Fig. 16—Comparison of matrix acidizing and acid fracturing: Scenario 4, 100 bbl of acid.

1.8
1.6 Matrix acidizing
1.4 Acid fracturing
1.2
JD (max)

1.0
0.8
0.6
0.4
0.2
0
0.01 0.10 1.00 10.00 100.00 1,000.00
k (md)

Fig. 17—Comparison of matrix acidizing and acid fracturing: Scenario 5, 2,000 bbl of acid.

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Although the larger acid volume results in higher productivity for both matrix acidizing and acid fracturing, the results show that the
effect on acid fracturing is more pronounced. The larger volumes of acid result in higher cutoff permeabilities. This means that when
more acid is used, acid fracturing is the preferable technique over a wider range of permeability. Comparing Figs. 16, 5, and 17, it can
be seen that the cutoff permeability goes from 9 md (with 100 bbl of acid), to 22 md (with 500 bbl of acid), to 49 md (with 2,000 bbl of
acid). This shows that the preferable stimulation method is not only a function of the reservoir properties, but also of the volume of acid
that can be used in the operation.
Scenario 6: Injector Well. All scenarios presented so far considered a producer well, with a bottomhole flowing pressure gradient
of 0.3 psi/ft. In an injector well, this bottomhole flowing pressure gradient would be higher, resulting in a smaller effective confining
stress. Because of this, the acid-fracture conductivity is expected to be higher in an injector well than in a producer well. In Scenario 6,
the initial reservoir-pore-pressure gradient of 0.4333 psi/ft was used for calculating the effective confining stress. All other properties
are the same as the base case, Scenario 1. The results are presented in Fig. 18.

0.9
Matrix acidizing
0.8
Acid fracturing, injector (Scenario 6)
0.7
Acid fracturing, producer (Scenario 1)
JD (max) 0.6
0.5
0.4
0.3
0.2
0.1
0
0.01 0.10 1.00 10.00 100.00 1,000.00
k (md)

Fig. 18—Comparison of matrix acidizing and acid fracturing: Scenario 6, injector well.

As expected, the injectivity index that can be obtained by acid fracturing the injector well is higher than the corresponding PI of the
producer well (for this case, approximately 9% higher on average). The productivity of the matrix-acidized well is the same for injector
and producer wells. This causes the acid-fracturing technique to be preferable up to a higher permeability for the injector well. For this
scenario, the cutoff permeability is 50 md for the injector well compared with 22 md for the producer well.
Scenario 7: Dolomite Formation. All scenarios presented so far consisted of limestones composed of pure calcite. Scenario 7 con-
sists of a dolomite formation. The main difference between limestones and dolomites concerning acid treatments is the reactivity of
these minerals with the acid. The reaction rate of dolomite with HCl is significantly lower than with limestone in usual wellbore condi-
tions. Table 2 shows the reaction kinetics constants for both minerals, and kr0 is two orders of magnitude smaller for dolomite. While
the reaction rate of calcite with HCl can be regarded as infinite (mass-transfer limited) for the temperatures of most reservoirs, for dolo-
mites the reaction kinetics are the limiting step unless the temperature is high (approximately 300 F).
HCl also has a lower dissolving power for dolomite, so the same volume of acid can dissolve a smaller volume of mineral. The
gravimetric dissolving power is b100 ¼ 1.27 mass of dolomite dissolved per unit mass of HCl (compared with 1.37 for calcite). The vol-
umetric dissolving power is v15 ¼ 0.071 volume of dolomite dissolved per unit volume of 15% HCl solution, compared with 0.082
for calcite.
Another implication of the lower reactivity and dissolving power of HCl with dolomite is a smaller wormholing efficiency. The
volume of acid required to create wormholes in dolomite is larger than the corresponding volume for calcite, as reported by Hoefner
and Fogler (1988) and Wang et al. (1993). The data used in this study are those from Ali and Nasr-El-Din (2018) for 15% HCl:
PVbt,opt,core ¼ 3.3 and vi,opt,core ¼ 3.3 cm/min, measured with 1.5  6-in. cores, with porosity of approximately 15%.
Scenario 7 consists of a formation composed of pure dolomite, so that the corresponding reaction kinetics, heat of reaction, dissolv-
ing power, and wormholing parameters were used. All other parameters are the same as Scenario 1 (Table 1).
The major difference between the two types of rocks is in the matrix-acidizing efficiency. Fig. 19 shows the comparison of the best-
possible post-stimulation dimensionless PI for limestone (Scenario 1) and dolomite (Scenario 7). Fig. 19a shows that the optimal
outcome of acid fracturing does not change significantly for the different rocks. However, as expected, Fig. 19b shows that matrix acid-
izing is more efficient in limestones, which have smaller PVbt values. Fig. 20 shows the comparison between matrix acidizing and acid
fracturing for the dolomite scenario.

0.90 0.35
0.80 Dolomite 0.30
0.70 Limestone 0.25
JD (max)

0.60
JD (max)

Unstimulated 0.20
0.50
Dolomite
0.40 0.15 Limestone
0.30
0.10
0.20 Unstimulated
0.10 0.05
0 0
0.01 0.10 1.00 10.00 100.00 1,000.00 0.01 0.10 1.00 10.00 100.00 1,000.00
k (md) k (md)
(a) Acid fracturing (b) Matrix acidizing

Fig. 19—Comparison of acid-stimulation outcomes in limestone and dolomite.

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0.9
0.8 Matrix acidizing
0.7 Acid fracturing
0.6

JD (max)
0.5
0.4
0.3
0.2
0.1
0
0.01 0.10 1.00 10.00 100.00 1,000.00
k (md)

Fig. 20—Comparison of matrix acidizing and acid fracturing: Scenario 7, dolomite.

Because of the smaller wormholing efficiency in dolomites, acid fracturing is the preferable stimulation technique over a larger
range of permeability. Fig. 20 shows a cutoff permeability of 350 md for the dolomite rock, compared with 22 md for the limestone of
Scenario 1 with the same geomechanical properties.
It is important to notice that although the acid-fracturing maximum productivity in the dolomite case is not significantly different
from the limestone (Fig. 19a), the operational conditions to obtain the maximum productivity are significantly different. In general, the
optimal injection rate is smaller for dolomites, especially for the higher permeabilities (1 md and greater). To illustrate this, Fig. 21
shows the outcome of acid fracturing for limestone (Scenario 1) and dolomite (Scenario 7).

0.6
Limestone, straight acid, no pad
0.5 Dolomite, straight acid, no pad

0.4
JD

0.3

0.2
0 20 40 60 80 100
q (bbl/min)

Fig. 21—Optimal acid-fracturing conditions for dolomite and limestone (1 md).

The maximum productivity achievable is similar for both rocks, but the optimal injection rate is significantly different: 10 bbl/min
for dolomite, and 100 bbl/min (maximum value considered) for limestone. If 100 bbl/min is used to stimulate the dolomite formation,
the resulting productivity is 40% smaller.
This behavior is thoroughly explored by Aljawad et al. (2019). The reason for this behavior is the low reactivity of dolomite at low
temperatures, and the fact that the acid is heated as it is transported inside the fracture. This causes the maximum reaction rate to
happen at a position inside the fracture away from the wellbore. Consequently, the etched width and conductivity profiles have maxi-
mum values away from the wellbore, and the near-wellbore conductivity might be small. At the higher injection rates, the acid is
pushed far away from the wellbore before it has time to react, and the resulting near-wellbore etched width and conductivity are too
small. Fig. 22 presents the etched-width and conductivity profiles for a 1-md reservoir, for dolomite at the injection rates of 10 bbl/min
(optimal for this case, as shown in Fig. 21) and 100 bbl/min, and for a limestone reservoir with injection rate of 100 bbl/min (optimal
injection rate for limestone). This is a good example of the importance of a fully coupled acid-fracturing simulator that integrates the
modeling of fracture propagation, acid reaction, heat transfer, and productivity.

0.10 2,000
Dolomite, 10 bbl/min Dolomite, 10 bbl/min
Dolomite, 100 bbl/min Dolomite, 100 bbl/min
0.08 Limestone, 100 bbl/min
1,500 Limestone, 100 bbl/min
kf w (md-ft)
we (in.)

0.06
1,000
0.04

500
0.02

0 0
0 100 200 300 400 500 0 100 200 300 400 500
x (ft) x (ft)
(a) Etched Width (b) Fracture Conductivity

Fig. 22—(a) Etched-width and (b) conductivity profiles for dolomite and limestone.

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Summary of the Case Studies for Vertical Wells. Table 4 presents a summary of the case studies’ results. Acid fracturing is the
preferable method if the reservoir permeability is less than kcutoff, while matrix acidizing is preferable at greater than kcutoff.

Producer or
Scenario Depth (ft) SRES (psi) Vacid (bbl/ft) Injector Mineral kcutoff (md)
1 10,000 22
2 3,000 50,000 120
3 20,000 2.5
1S 10,000 12
2S 3,000 20,000 5 100
3S 20,000 Producer 1
Limestone
1H 10,000 55
2H 3,000 200,000 160
3H 20,000 15
4 1 9
50,000
5 20 48
10,000
6 Injector 50
50,000 5
7 Producer Dolomite 350

Table 4—Summary of case studies.

Concise Decision Criterion for Vertical Wells. Palharini Schwalbert (2019) presented an analytical equation for the optimal acid-
fracture length,
(   )
1
A vfAe VA B B þ 1
xf ;opt ¼ ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð2Þ
CfD;opt k 2ð1  /Þh

where xf ;opt is the optimal acid-fracture conductive half-length, CfD;opt is the optimal dimensionless fracture conductivity, VA is the total
volume of acid solution injected, fAe is the fraction of the acid spent etching the fracture walls (subtracting the acid lost to leakoff), v is
the acid-solution volumetric dissolving power, / is the formation porosity, h is the reservoir thickness, and A and B are parameters of
the acid-fracture-conductivity correlation used. The acid-fracture-conductivity correlation is written in the form of Eq. 3. Most conduc-
tivity correlations can be written in the form of Eq. 3. For the Nierode and Kruk (1973) correlation, B ¼ 2:466 and A is given in field
units (md-ft-in.2.466) by Eq. 4. Other forms are presented by Palharini Schwalbert (2019) for other conductivity models. Eq. 2 and all
subsequent equations derived from it (Eqs. 5, 6, 7, 10, and 11) are valid in consistent units (e.g., SI units). To use it in field units, some
units conversion must be made. If using A in md-ft-inB, for example, then the permeability should be used in md, and the volume of
acid should be used in ft2-in (or use the volume in ft3, and include a number 12 multiplying the volume inside the brackets).

kf w ¼ Awe B ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð3Þ
(  
1:476  107 exp 0:001½13:9  1:3lnðSRES Þr0c ; if SRES < 20; 000 psi
A¼  
1:476  107 exp 0:001½3:8  0:28lnðSRES Þr0c ; if SRES  20; 000 psi;                           ð4Þ

where kf w is the fracture conductivity, we is the fracture etched width, SRES is the RES, and r0c is the effective confining stress.
Appendix A presents the derivation of the concise decision criterion to select the best acid-stimulation method for a given vertical
well in a carbonate reservoir. The full derivation was presented first in Palharini Schwalbert (2019).
The ratio of the PIs of the acid-fractured to the matrix-acidized vertical well is denoted here by RPI;V (PI ratio, for the vertical well),
and is given by
0 1
re
B s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi C 3
ln@ 2 VA A
rw þ 4
JD;af ;max p/hPV bt;field
RPI;V ¼ ¼ 0   1 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð5Þ
JD;ma;max p
B re þ f1 C
B CfD;opt gðkÞ C 3
B C
lnB(   ) 1 C
B A vfAe VA B B þ 1C 4
@ A
CfD;opt k 2ð1  /Þh

The decision criterion can be given by the value of RPI;V , or it can be summarized in a dimensionless number that groups all contri-
butions from the reservoir, acid type and volume, wormholing efficiency, and acid-fracture conductivity. This dimensionless number is
named here as the carbonate-acid-stimulation (CAS) number, or NCAS . For the simplified case in a square drainage area, with a fracture
that is short compared with the reservoir (Ix .0:2), which is usually the case when there is doubt between using acid fracturing or matrix
acidizing, the simplified equation for this dimensionless number is

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(   )
1
A vfAe VA B B þ 1
CfD;opt k 2ð1  /Þh
NCAS ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 ffi : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð6Þ
V A
4 rw2 þ
/phPVbt;field

The decision criterion is summarized as: If NCAS > 1, acid fracturing can result in a higher PI than matrix acidizing; if NCAS < 1,
matrix acidizing can result in a higher PI.
The cutoff permeability that divides the region of applicability of each technique, kcutoff , can be calculated by Eq. 6 when NCAS ¼ 1.
It is given by
 
A vfAe VA B
CfD;opt 2ð1  /Þh
kcutoff ¼ : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð7Þ
  B þ 1
Bþ1 2 VA 2
4 rw þ
/phPVbt;field

In Eqs. 2 and 5 through 7, all the inefficiencies of the matrix-acidizing and acid-fracturing operations are grouped in the terms fAe
and PVbt;field . For the case studies presented in this paper, in general PVbt;field was high for low permeabilities at which it was not possi-
ble to inject at optimal interstitial velocity, but once the permeability was high enough (plateau of matrix-acidizing curves in Figs. 5
through 20), it is very close to the optimal value of PVbt in the field scale [more on this subject is provided in Palharini Schwalbert
(2019)]. Most of the intersection between matrix acidizing and acid fracturing in Figs. 5 through 20 occurred on the plateau of the
matrix-acidizing curve, when PVbt;field  PVbt;opt;field .
The fraction of acid spent etching, fAe , depends on how much acid is lost because of leakoff and to nonpay zones. It is a function of
the leakoff coefficient and reservoir properties, especially reservoir permeability. It depends strongly on the pressure overbalance
between the fracturing pressure and the reservoir pressure ðpf  pi Þ. For the simulations performed in this study, it seems that fAe is usu-
ally less than 0.85, and at greater than 1 md, fAe decreases linearly proportional to the logarithm of the permeability and to the logarithm
of the overbalance squared. On the basis of the simulations that originated Figs. 5 through 20, we obtain the relation
n o
fAe  min 0:85; 1:8615  0:069  ln½kðrh;min  pi Þ2  ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð8Þ

where k is the permeability (in md), rh;min is the minimum horizontal stress (in psi), and pi is the initial reservoir pressure (in psi). The
function min (a,b) yields the minimum value between a and b, and it ensures that fAe  0:85. It was introduced because in the cases
tested, fAe was always less than 0.85.
Eq. 8 is not general, and fAe is expected to be different for different reservoirs and different fluids. In general, everything that causes
a large acid leakoff tends to cause a smaller fAe . The presence of natural fractures tends to cause a small fAe . On the other hand, when
using loss-control additives to reduce the acid loss, fAe tends to be increased.
For the scenarios analyzed in this paper, however, Eq. 8 is a good estimate of fAe , and Eqs. 5 and 6 can be used with this estimate of
Eq. 8 to have an excellent estimate of the cutoff permeability without running any simulation. Fig. 23 shows a comparison of the per-
meability cutoff obtained through the fully coupled simulations (shown in Figs. 5 through 22) and the quick estimate of Eqs. 6 and 7.
The black line is the diagonal. The fact that most points are aligned with the diagonal shows the good quality of the estimate of Eqs. 6
and 7.

1000.0
Analytical kcutoff (md)

100.0

10.0

1.0

0.1
0.1 1.0 10.0 100.0 1,000.0
kcutoff (md) from Simulations

Fig. 23—Comparison of the permeability cutoff estimated with analytical Eq. 7 and resulting from full simulations.

Uncertainty and Limitations. A great limitation of the presented criterion is the quality of data: Many of the input parameters
needed to use this criterion are subject to significant uncertainty. Parameters of special concern are the RES (not often measured), the
pore volumes (PVs) to breakthrough, horizontal stress, and permeability itself. The case studies, summarized in Table 4, can help assess
the importance of the uncertainty in each parameter. Fig. 14, for example, shows the effect of the uncertainty in RES. The cases of dif-
ferent depths show the effects of different horizontal stresses.
To assess the sensitivity to uncertainty in these parameters, Table 5 presents a sensitivity analysis of the PI ratio of acid-fractured to
matrix-acidized wells (RPI;V as defined in Eq. 5) on the most important parameters. All the cases presented in Table 5 are less than the

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cutoff permeability for the reference case tested, meaning that acid fracturing results in the most productive well (as can be seen by the
value RPI;V > 1). The impacts of variations of permeability, porosity, RES, PVs to breakthrough, and minimum horizontal stress are
studied. All these parameters are varied 50% higher and lower than the base case, with the exception of minimum horizontal stress,
which is varied 20% higher and lower (which is already a large variation for horizontal stress). The important parameter calculated in
Table 5 is the relative sensitivity of RPI;V to each parameter, defined as the relative variation of RPI;V divided by the relative variation in
each parameter:
 
DRPI;V
RPI;V;reference
Relative sensitivity ¼   : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð9Þ
Dparameter
parameterreference

Relative
Case RPI,V Sensitivity k (md) Porosity SRES (psi) PVbt,field σh,min (psi)
Reference case 1.33 – 10 0.15 50,000 0.50 6,000
k + 50% 1.28 –0.076 15 0.15 50,000 0.50 6,000
k – 50% 1.42 –0.143 5 0.15 50,000 0.50 6,000
Porosity + 50% 1.41 0.119 10 0.225 50,000 0.50 6,000
Porosity – 50% 1.22 0.169 10 0.075 50,000 0.50 6,000
SRES + 50% 1.37 0.054 10 0.15 75,000 0.50 6,000
SRES – 50% 1.27 0.086 10 0.15 25,000 0.50 6,000
PVbt,field + 50% 1.38 0.082 10 0.15 50,000 0.75 6,000
PVbt,field – 50% 1.24 0.140 10 0.15 50,000 0.25 6,000
σh,min + 20% 1.22 –0.429 10 0.15 50,000 0.50 7,200
σh,min – 20% 1.50 –0.646 10 0.15 50,000 0.50 4,800

Table 5—Sensitivity analysis.

The larger the value of the relative sensitivity, the greater the effect of the property in the PI ratio. The value of the relative sensitiv-
ity indicates how much RPI;V changes with a given change in a given variable. A relative sensitivity equal to 0.1, for example, means
that RPI;V changes 10% when there is a change of 100% in that variable. A negative value in the relative sensitivity means that an
increase in the variable causes a decrease in RPI;V (in other words, a greater value of that variable either harms acid fracturing or favors
matrix acidizing). A positive value of the relative sensitivity means that an increase in the variable causes an increase in RPI;V (either by
favoring acid fracturing or harming matrix acidizing). Table 5 shows that the variable with greatest impact is horizontal stress because
it shows the greatest relative sensitivity.
Heterogeneity of the Formation. All cases presented so far assumed a homogeneous formation. That was also assumed in the deri-
vation of the analytical decision criterion, so it is a limitation of the criterion, and it is recommended that a future study develop a simi-
lar criterion for heterogeneous formations. However, our preliminary results with heterogeneous formations show that this assumption
is not a strong limitation.
Table 6 shows several cases where acid fracturing was simulated in three-layer reservoirs, assuming first a homogeneous reservoir
(Cases 1A, 2A, 3A, 4A, and 5A), and then heterogeneous reservoirs with P the sameP average permeability (Cases 1B, 2B, 2C, 3B, 4B,
and 5B). The average permeability is weighed by the layer thickness: k ¼ ð ki hi Þ= hi .

Case k1 (md) h1 (m) k2 (md) h2 (m) k3 (md) h3 (m) k (md) JD,af Difference
1A, homogeneous 90 20 90 10 90 20 90 0.2192 –
1B, heterogeneous 3 20 440 10 3 20 90 0.2141 –2.3%
2A, homogeneous 32.4 20 32.4 10 32.4 20 32.4 0.2477 –
2B, heterogeneous 3 20 150 10 3 20 32.4 0.2412 –2.6%
2C, heterogeneous 3 20 3 20 150 10 32.4 0.2400 –3.1%
3A, homogeneous 54 20 54 10 54 20 54 0.2320 –
3B, heterogeneous 30 20 150 10 30 20 54 0.2310 –0.5%
4A, homogeneous 70 20 70 10 70 20 70 0.2258 –
4B, heterogeneous 50 20 150 10 50 20 70 0.2253 –0.2%
5A, homogeneous 200 20 200 10 200 20 200 0.1987 –
5B, heterogeneous 1 20 1000 10 1 20 200.8 0.1931 –2.8%

Table 6—Effect of heterogeneity on acid fracturing.

As can be seen in Table 6, although there is a difference in the PI of the acid-fractured well depending on the heterogeneity, that dif-
ference is not large. The largest difference observed was 3.1%. It is interesting to note that all simulated cases showed smaller PIs in
the heterogeneous reservoir, compared with the homogeneous reservoir with the same average permeability.

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Analogously, Table 7 shows the same analysis for a matrix-acidized well, in the same reservoirs shown in Table 6. In this case, the
matrix-acidizing treatment used a simulator that considered acid placement, so that the acid flows preferentially into the more permea-
ble layers, as well as into the top layers (because the acid is bullheaded from the top). Table 7 also shows the treatment coverage for
each case. The coverage is the length of the wellbore at which the formation damage was removed, resulting in a negative skin factor.
Similar to the acid-fractured wells, the heterogeneous cases showed a slightly smaller PI. In the matrix-acidized cases, however, the
greatest difference is only 1.7%, which is in Case 1B, in which the treatment coverage was only 20%. Even with 80% of the well still
with formation damage after the acidizing, the PI was not significantly penalized, because the most permeable layer that contributes
mostly to the PI was well-treated.

Case JD,ma Difference Coverage

1A, homogeneous 0.2121 – 100%
1B, heterogeneous 0.2086 –1.7% 20%
2A, homogeneous 0.2121 – 100%
2B, heterogeneous 0.2104 –0.8% 60%
2C, heterogeneous 0.2104 –0.8% 100%
3A, homogeneous 0.2121 – 100%
3B, heterogeneous 0.2090 –1.5% 100%
4A, homogeneous 0.2121 – 100%
4B, heterogeneous 0.2095 –1.3% 100%
5A, homogeneous 0.2121 – 100%
5B, heterogeneous 0.2095 –1.3% 20%

Table 7—Effect of heterogeneity on matrix acidizing.

In summary, the development of the analytical decision criterion assumed a homogeneous formation, so it should be used with cau-
tion in highly heterogeneous formations. However, the results shown in Tables 6 and 7 indicate that this is not a strong limitation, and
very similar results are expected in heterogeneous cases as well.
Example Case Study. The carbonate reservoir Field X is to be exploited. Because of high reservoir anisotropy (low vertical per-
meability) and extensive net pay (h  800 ft), it will be exploited with vertical wells. The average reservoir depth is 12,000 ft, per-
meability is expected to be approximately 5 md, and porosity is approximately 12%. The horizontal stress gradient in the area is
0.65 psi/ft. The reservoir static pressure gradient is 0.46 psi/ft. The well diameter will be 8.5 in., and the drainage radius of each well
can be assumed as 3,000 ft. The drainage regions can be assumed to be square.
The sonic logs indicate that the Young’s modulus is approximately E  4106 psi. Experience with similar reservoirs indicates that
the PVs to breakthrough expected in matrix acidizing this reservoir is PVbt;field  0:4. It is an offshore field that must be stimulated by a
stimulation vessel that can be loaded with no more than 3,000 bbl of 15% HCl acid. Because of erosional concerns in sensitive down-
hole equipment, hydraulic fracturing with proppant is not a desirable stimulation method for the wells in this reservoir.
The question is whether matrix acidizing or acid fracturing is preferable to stimulate the wells to be drilled in this reservoir. This
can be answered using Eq. 5 or Eq. 6. If either RPI;V or NCAS is greater than unity, then acid fracturing tends to result in more productive
wells. If either RPI;V or NCAS is smaller than unity, then matrix acidizing tends to result in more productive wells.
To use Eq. 5, it is necessary to calculate the parameter A of the acid-fracturing correlation (Eq. 4). For that it is necessary to calculate
the RES, SRES . Because an estimate of the Young’s modulus is available from sonic logs, it is possible to use the Neumann (2011) cor-
relation to estimate SRES : SRES ¼ 0:02671  E ¼ 0:02671  4106 psi ¼ 106; 840 psi.
With this value of SRES , the parameter A ¼ 8:87105 md-ft-in:2:466 (calculated using Eq. 4). For this reservoir, fAe can be estimated
with Eq. 8 as fAe ¼ 0:68. Because the drainage region can be assumed to be square, k ¼ 1 and gðkÞ1. The optimal acid-fracture length
can be calculated using Eq. 2, resulting in 126 ft, which is short compared with the drainage region, so f1 ¼ 2.
Plugging all these parameters in Eq. 5, it can be calculated that the ratio of the optimal acid-fractured-well productivity to the opti-
mal matrix-acidized-well productivity, in this scenario, is equal to RPI;V ¼ 1:26. This means that the acid-fractured well is expected to
have a PI that is 26% greater than that of the matrix-acidized well, in the same scenario, using the same amount of acid in the
stimulation operation.
This increase of 26% in the PI of the acid-fractured well compared with the matrix-acidized well must be used to decide which stim-
ulation technique should be used. If the completion costs of the acid-fracturing operation do not outweigh the net-present-value increase
of 26% in PI, then the wells in this reservoir should be acid fractured.
If Eq. 7 is used to calculate the cutoff permeability for this reservoir, a value of 70 md is obtained, which means that as long as the
reservoir permeability is less than 70 md, in this scenario, the acid-fractured wells tend to be more productive than the matrix-
acidized wells.

Concise Decision Criterion for Horizontal Wells. All comparisons presented so far considered vertical wells. For horizontal wells,
the comparison between matrix-acidized or acid-fractured wells is not so straightforward because the complexity and cost of completion
is significantly different because of stage isolation. The fractured horizontal well usually has multiple fractures and therefore involves
more variables, such as the number of fractures and cluster and fracture spacing. As for the matrix-acidized well, acid placement
becomes a major concern, and a model that considers diversion and flow inside the wellbore is necessary.
Diversion is not the focus of this study, and in this subsection, for the sake of simplicity and to obtain a concise decision criterion to
select the preferable method for a given volume of acid, it is assumed that the reservoir is homogeneous and a uniform acid distribution
is achieved. The well is assumed to be centered in the reservoir, and fully penetrating. The frictional pressure drop along the horizontal
wellbore is not considered. Although this might not be a good assumption for high-permeability cases, it is good for the comparison
sought between the two stimulation techniques. Non-Darcy flow is also neglected.

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Different numbers of fractures are considered, but in all cases they are equally spaced, and equal (all fractures are assumed to grow
equally and receive the same volume of acid). The volume of acid per foot of wellbore is the same in both matrix-acidizing and acid-
fracturing cases. In this sense, for a given volume of acid, in the cases with more fractures, each fracture receives less acid. In the
acid-fractured wells, the production contribution is assumed to come only from the fractures, which is equivalent to assuming a plug-
and-perforate completion, where the wellbore is cased and cemented and then perforated only at the fracture locations.
Appendix B presents the derivation of the concise decision criterion for horizontal wells. It was first presented in Palharini Schwal-
bert (2019). The ratio of the PIs of the acid-fractured to the matrix-acidized well is denoted here by RPI;H (PI ratio, for the horizontal
well) and is given by
   pffiffiffiffiffiffi 
2Nf hIani h Iani p a
ln þ  1:838 þ sma
JDH;af L rw 6 hIani
RPI;H ¼ ¼ 0  2 1 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð10Þ
JDH;ma p 2sc
B 4aL þ f1 ðk; Ix Þ e C
B CfD;opt gðkÞ C
B C
lnB (  ) 2 C
B B þ 1 C
B A vfAe VA
@ c A
Nf e CA ðkÞ
CfD;opt kH 2ð1  /ÞhNf

The technical decision criterion for the preferable acid-stimulation method for horizontal wells can be summarized as: If RPI > 1,
the well with multiple acid fractures has higher productivity than the matrix-acidized horizontal well; if RPI < 1, the matrix-acidized
well produces more.
The cutoff permeability for the horizontal well can be isolated from Eq. 10 by making RPI ¼ 1,
0    pffiffiffiffiffiffi 
1B þ 1
2Nf hIani h Iani p a 2
 B B ln þ  1:838 þ sma C
A vfAe VA Be L rw 6 hIani C
kH;cutoff ¼ B
B   2 C
C : . . . . . . . . . . . . . . . . . ð11Þ
CfD;opt 2ð1  /ÞhNf @ 4aLe 2sc
p A
þ f1 ðk; I x Þ
Nf ec CA ðkÞ CfD;opt gðkÞ

Example Case of Horizontal-Well Analysis. The scenario analyzed here is derived from the base case (Scenario 1, Table 1), but the
drainage region of the well is a square of 5,000  5,000 ft, and instead of a vertical well there is a fully penetrating horizontal well. The
volume of acid is 1 bbl/ft.
Fig. 24 shows the dimensionless PI of the matrix-acidized horizontal well for permeabilities ranging from 0.01 to 1,000 md. Analo-
gously to the vertical well, the PI is smaller at very low permeabilities because it is not possible to inject at the optimal injection rate
for wormhole propagation. However, the impact of the skin factor in horizontal wells is much smaller, so the difference between mini-
mum and maximum achievable JDH;ma is not so large. Basically, for this case, JDH;ma ranges from 1.3 to 1.9.

2.5

2.0
JDH (ma)

1.5

1.0

0.5

0
10–2 10–1 100 101 102 103
k (md)

Fig. 24—Dimensionless PI of the matrix-acidized horizontal well.

The PI of the acid-fractured well significantly depends on the number of fractures. Fig. 25 shows the dimensionless PI of the acid-
fractured horizontal well, JDH;af , for different numbers of transverse acid fractures. As can be seen, there is an optimal number of frac-
tures for a given scenario. In this case, analyzing the lowest permeability plotted, 0.01 md, Fig. 25 shows that the maximum JDH;af is
achieved with four fractures. For this scenario, if there are more than four fractures, each one receives too little acid and is not conduc-
tive enough, as shown by the curves with five and 10 fractures in Fig. 25. As shown by Palharini Schwalbert (2019), a larger volume of
acid or a different well spacing changes this, allowing us to effectively make more fractures.
To compare both stimulation methods, Fig. 26 shows the ratio of the PIs of the acid-fractured to the matrix-acidized well. As
observed for the vertical wells, there is a value of permeability greater than which matrix acidizing results in higher productivity than
acid fracturing.
In this scenario, if the permeability is greater than 0.6 md, there is no number of fractures that can make the acid-fractured well
more productive than the matrix-acidized well. Although only the curves for one, four, and 100 fractures are shown, the productivity
for all other numbers of fractures is in between these. At low permeabilities, if the right number of fractures is used, acid fracturing can
be much more productive than matrix acidizing. In this case, if the reservoir permeability is 0.01 md, the acid-fractured well with four
fractures has a productivity seven times higher than that of the matrix-acidized well.
More case studies of horizontal wells with multiple fractures were presented in Palharini Schwalbert (2019).

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Nf = 1
8 Nf = 2
Nf = 4
6 Nf = 5
Nf = 10

JDH,af
4

0
10–2 10–1 100 101 102 103
k (md)

Fig. 25—Dimensionless PI of the horizontal well with multiple acid fractures.

7
Nf = 1
6 Nf = 4
Nf = 100
5
(JDH,af /JDH,ma)

(JDH,af = JDH,ma)
4

0
10–2 10–1 100 101 102 103
k (md)

Fig. 26—Comparison of matrix-acidized and acid-fractured horizontal wells.

Limitations of the Criterion for Horizontal Wells. The criterion developed for horizontal wells has more simplifying assumptions
than that for vertical wells, and hence it is more subject to errors and should be used with more caution. It assumes, for example, an
evenly distributed acid treatment, which is a worse assumption in horizontal wells, because even in homogeneous formations the treat-
ment tends to concentrate at the heel. It also assumes that the fractures are all equal and equally spaced, which is a strong assumption.
No fracturing model for horizontal wells was used, and no stress-shadow effect is observed, for example. In reality, it is known that the
multiple fractures would not be equal. Some would be longer and some would not even start, which would affect productivity (in some
cases negatively, but in other cases positively). In the fractured wells, only the production contribution from the fractures was accounted
for. This is a reasonable assumption in cased and cemented wellbores, but it might be a bad assumption for openhole completions. In
openhole fractured wells, this assumption results in negligible error for very low permeability, but the error becomes significant for
high permeabilities.
In summary, we suggest using the criterion with caution. It is supposed to be a guide, but definitely not a law.

Conclusions
For each scenario, there is a value of permeability greater than which matrix acidizing can result in a more productive well than acid
fracturing, using the same volume of acid. In this work, this permeability is called the cutoff permeability. The value of the cutoff per-
meability depends on several reservoir properties, such as the rock mechanical properties, mineralogy, horizontal stress, and other
parameters that determine the acid-fracture conductivity, acid-wormholing efficiency parameters such as PVbt;opt and vi;opt , acid type,
concentration, and volume, and reservoir pressure.
General values of the cutoff permeability were presented for several case studies. Everything that makes the acid fracture more con-
ductive contributes to increasing the cutoff permeability (meaning that acid fracturing is the preferred technique up to a higher per-
meability). For example, harder rocks and shallower reservoirs have a higher cutoff permeability than softer rocks and deeper
reservoirs. Analogously, everything that improves the efficiency of matrix-acidizing treatments contributes to decreasing the cutoff per-
meability. For example, dolomites have greater values of PVbt;opt than limestones, and so dolomites tend to have a higher cutoff per-
meability. In general, the cutoff permeability tends to be lower for horizontal wells than for vertical wells. Larger volumes of acid tend
to increase the cutoff permeability.
In horizontal wells with multiple acid fractures, there is an optimal number of fractures for a given scenario. Because the volume of
acid is fixed in this analysis, if too many acid fractures are created, each one does not receive enough acid, resulting in insufficient con-
ductivity. The optimal number of acid fractures increases with the acid volume and decreases with the well spacing.
Concise analytical decision criteria for the best acid-stimulation method were proposed for both vertical wells and horizontal wells
with multiple acid fractures. The decision criterion for vertical wells is presented by Eq. A-10 (or the simplified Eq. 6), and for horizon-
tal wells by Eq. 10. The cutoff permeability for a vertical well can be estimated by Eq. A-11 (or the simplified Eq. 7), and for a horizon-
tal well by Eq. 11.

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Appendix A—Derivation of the Concise Decision Criterion for Vertical Wells

According to the derivation presented in Palharini Schwalbert (2019) using the fracture-productivity model presented by Meyer and
Jacot (2005), the maximum possible dimensionless PI of the acid-fractured well can be calculated by Eq. A-1 for a vertical well in a
general drainage region.
2
JD;af ;max ¼ 2 0 12 3 ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ðA-1Þ
(  B )B þ 1
1
6 B A vfAe VA C 7
6 B C 7
6 B C k 2 ð 1  /Þh C 7
6 fD;opt
C 7
lnð4Ad Þ  ln6ec CA B
B C 7
6 B p C 7
6 @ þ f1 A 7
4 CfD gðkÞ 5

where JD;af ;max is the maximum possible dimensionless PI of the acid-fractured well, k is the reservoir permeability, Ad is the area of
the drainage region, ec is the exponential of Euler’s constant (ec  1:781), CA is the shape factor of the drainage region by Dietz (1965)
and Earlougher et al. (1968), k is the aspect ratio of the drainage region (k ¼ xe =ye , where xe and ye are the reservoir lengths in the
x- and y-direction, respectively), and gðkÞ and f1 are functions presented by Meyer and Jacot (2005), which can be given by
2 2
2eCfD Ix 2kð1  eCfD Ix Þ
gðkÞ ¼ þ ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ðA-2Þ
1 1
1þ 1þ
k k
e1=JD1 Ix
f1 ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ðA-3Þ
16
ec kCA ðkÞ
where Ix is the fracture-penetration ratio, defined as the ratio of the fracture length and the length of the reservoir in the x-direction
(Ix ¼ 2xf =xe ), and JD1 ðIx ; kÞ is the dimensionless PI of a fracture of infinite conductivity with penetration ratio Ix and reservoir aspect
ratio k. JD1 ðIx ; kÞ can be calculated using the analytical solution by Gringarten (1978),
   X 1 npkIx ð1xD Þ

1 p p  2
 p 2 1 2 p 1 e 2npkð1Ix Þ
 2npkIx xD

¼ k  kIx 1 þ xD þ kIx þ xD þ  ½1  e  1þe ;
JD1 6 4 4 3 6kIx 2pkIx n¼1 n2 ð1  e2npk Þ
                                       ðA-4Þ
where xD ¼ 0:740108 for the infinite-conductivity short fractures and has other values presented by Meyer and Jacot (2005) for arbi-
trary fracture lengths and aspect ratios. The shape factor CA ðkÞ can be calculated, for a centered well in a rectangular reservoir of any
aspect ratio k, by the equation proposed by Gringarten (1978),
( )
X 1 
2 p e2npk
CA ðkÞ ¼ exp 0:8091 þ lnð4p Þ þ lnðkÞ  k  2 ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ðA-5Þ
3 n¼1
nð1  e2npk Þ

where the summation term can usually be neglected for k > 1. As a general reference, for a circular reservoir with a centered well,
CA ¼ 31:6; for a square reservoir, CA ¼ 30:9; for a 2  1 rectangular reservoir (k ¼ 2), CA ¼ 21:8; for a 4  1 rectangular reservoir
(k ¼ 4), CA ¼ 5:38; and for a 5  1 rectangular reservoir (k ¼ 5), CA ¼ 2:36.
A much simpler equation can be obtained for a square or circular drainage region with a relatively short fracture (short compared
with the reservoir dimensions, xf .0:2xe ). Eq. A-1 can be significantly simplified, resulting in
1
JD;af ;max ¼ 0 1 1
; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ðA-6Þ
(  B )B þ 1
B A vfAe VA C 3
ln@4re A
CfD;opt k 2ð1  /Þh 4

where re is the external radius of the cylindrical drainage region.

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Analogously, the dimensionless PI of the matrix-acidized vertical well, JD;ma;max , can be simplified, using the volumetric model pro-
posed by Hill in Economides et al. (1994), to
1
JD;ma;max ¼ 0 1 ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ðA-7Þ
re
Bs ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi C 3
ln@ 2 VA A
rw þ 4
p/hPVbt;field

where PVbt;field is the minimum PVs to breakthrough of the wormhole propagation, obtained in the optimized field treatments. PVbt;field
might be equal to PVbt;opt;field if the permeability allows injection at the optimal injection rate. Notice that PVbt;opt;field is in the field scale
[usually smaller than the value of PVbt;opt measured in laboratory experiments, as presented by Furui et al. (2012), Burton et al. (2018),
and Palharini Schwalbert et al. (2019b)].
A reasonable technical decision criterion for the stimulation method is to use acid fracturing when JD;af ;max > JD;ma;max , or, in other
JD;af ;max
words, when the PI ratio between acid fracturing and matrix acidizing is greater than unity: > 1. Using Eqs. A-6 and A-7, the
JD;ma;max
PI ratio and decision criterion can be written as
0 1
re
B s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi C 3
ln@ 2 VA A
rw þ 4
JD;af ;max p/hPV bt;field
¼ 0   1 > 1: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ðA-8Þ
JD;ma;max p
B re þ f1 C
B CfD;opt gðkÞ C 3
B ) 1 C
lnB(   C
B A vfAe VA B B þ 1C 4
@ A
CfD;opt k 2ð1  /Þh

After some algebra, the decision criterion can be rewritten as

"  B #B þ 1
1
A vfAe VA
CfD;opt k 2ð1  /Þh
 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 1: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ðA-9Þ
p VA
þ f1 rw2 þ
CfD;opt gðkÞ p/hPVbt;field

The left-hand side of Eq. A-9 is a dimensionless number that groups all contributions from the reservoir, acid type and volume,
wormholing efficiency, and acid-fracture conductivity. This dimensionless number is named here as NCAS ,
(   )
1
A vfAe VA B B þ 1
CfD;opt k 2ð1  /Þh
NCAS ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 ffi : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ðA-10Þ
p V A
þ f1 rw2 þ
CfD;opt gðkÞ /phPVbt;field

The decision criterion is summarized as: If NCAS > 1, acid fracturing can result in a higher PI than matrix acidizing; if NCAS < 1,
matrix acidizing can result in a higher PI.
The cutoff permeability that divides the region of applicability of each technique, kcutoff , can be calculated by Eq. A-10 when
NCAS ¼ 1. It is given by
 
vfAe VA B
A
CfD;opt 2ð1  /Þh
kcutoff ¼ : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ðA-11Þ
 Bþ1   B þ 1
p V A 2
þ f1 rw2 þ
CfD;opt gðkÞ /phPVbt;field

A simplification can be performed for square drainage areas, where gðkÞ ¼ 1. In addition, if the fracture is short compared with the
p
reservoir (Ix .0:2), which is usually the case when there is doubt between using acid fracturing or matrix acidizing, then CfD;opt ¼ and
  2
p
f1 ¼ 2. In this case, the whole term þ f1 is simply equal to 4, which leads to the simplified equations
CfD;opt gðkÞ

(   )
1
A vfAe VA B B þ 1
CfD;opt k 2ð1  /Þh
NCAS ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 ffi ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ðA-12Þ
V A
4 rw2 þ
/phPVbt;field

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A vfAe VA B
CfD;opt 2ð1  /Þh
kcutoff ¼ : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ðA-13Þ
  B þ 1
V A 2
4Bþ1 rw2 þ
/phPVbt;field

Appendix B—Derivation of the Concise Decision Criterion for Horizontal Wells

This comparison assumes pseudosteady state, and the model by Babu and Odeh (1989) is used for the matrix-acidized well. For a cen-
tered fully penetrating well, the dimensionless PI is given by

L
JDH;ma ¼   pffiffiffiffiffiffi  ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ðB-1Þ
h Iani p a
hIani ln þ  1:838 þ sma
rw 6 hIani

where sma is the skin factor for the matrix-acidized well, which in this case should be calculated using the model by Furui et al. (2002),
8 2 s
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi39
<  =
1 4rwhH rwhH 2 2  15 ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ðB-2Þ
sma ¼ ln þ þ Iani
:Iani þ 1 rw rw ;

pffiffiffiffiffiffiffiffiffiffiffiffi
where Iani is the reservoir-anisotropy ratio ( kH =kV ) and rwhH is the horizontal length of the wormholes, which can be estimated by
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 ffi
V A I ani
rwhH ¼ rw2 þ : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ðB-3Þ
/phPVbt;field

For the acid-fractured wells, because the fractures are equally spaced, each one drains from their own drainage region, defined by
the symmetry planes in between them. The total PI of the horizontal well with multiple fractures is equal to the sum of the PIs of each
fracture. In general, the drainage area of each fracture is not a square, but a rectangle. The more fractures are added, the more elongated
is the drainage area, which is represented by the aspect ratio k,

aNf
k¼ ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ðB-4Þ
L

where L is the wellbore length (equal to the reservoir length in the wellbore direction, because the well is assumed to be fully penetrat-
ing), a is the reservoir length in the fracture direction (orthogonal to the wellbore), and Nf is the number of fractures. Because the drain-
age region of each fracture is far from square when the number of fractures is large, the equations for square reservoirs cannot be used
in this case (f1 and the shape factor CA must be calculated using Eqs. A-2 through A-5).
Because of the flow convergence toward the well, the choke skin factor sc should be added (although it is often negligible), as pre-
sented by Mukherjee and Economides (1991),
         
kH h h p h h p
sc ¼ ln  ¼ ln  : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ðB-5Þ
kf w 2rw 2 xf CfD 2rw 2

The productivity of each fracture can be calculated by simulating the complete operation using the fully coupled acid-fracturing sim-
ulator and adding the choke skin factor sc , or it can be estimated by the analytical equations similar to those presented for vertical wells.
As shown previously, although the fully coupled simulator is necessary to actually design the acid-fracturing jobs, the analytical esti-
mate is satisfactory for the sake of comparison with matrix acidizing. For the horizontal well with Nf acid fractures, the optimal dimen-
sionless PI can be estimated by

2Nf
JDH;af ¼ 0  2 1 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ðB-6Þ
p
B 4aL þ f1 ðk; Ix Þ e2sc
C
B CfD;opt gðkÞ C
B C
lnB (  ) 2 C
B B þ 1 C
B A vfAe VA
@ A
Nf ec CA ðkÞ
CfD;opt kH 2ð1  /ÞhNf

The dependence of the PI on the number of fractures is not straightforward because Nf appears several times in Eq. B-6, and k is pro-
portional to Nf . For a given volume of acid injected into the whole well, as the number of fractures increases, the amount of acid that
each fracture receives diminishes. For a given scenario, there is an optimal number of fractures that results in the maximum productivity
for the whole multifractured well.
The ratio of the PIs of the acid-fractured to the matrix-acidized well can be calculated by dividing Eq. B-6 by Eq. B-1. This ratio is
denoted here by RPI;H (PI ratio), and it is given by

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   pffiffiffiffiffiffi 
2Nf hIani h Iani p a
ln þ  1:838 þ sma
JDH;af L rw 6 hIani
RPI;H ¼ ¼ 0  2 1: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ðB-7Þ
JDH;ma p 2sc
B 4aL þ f 1 ðk; Ix Þ e C
B CfD;opt gðkÞ C
B C
lnB (   ) 2 C
B A vfAe VA B B þ 1C
@ c A
Nf e CA ðkÞ
CfD;opt kH 2ð1  /ÞhNf

The technical decision criterion for the preferable acid-stimulation method for horizontal wells can be summarized as: If RPI > 1,
the well with multiple acid fractures has higher productivity than the matrix acidized horizontal well; if RPI < 1, the matrix-acidized
well produces more.
The cutoff permeability for the horizontal well can be isolated from Eq. B-7 by making RPI;H ¼ 1,
8    pffiffiffiffiffiffi 

9B þ 1
>
> > 2
 1:838 þ sma >
2Nf hIani h Iani p a
 B >
< L
ln
rw
þ
6 hIani
>
=
A vfAe VA e
kH;cutoff ¼    2 : . . . . . . . . . . . . . . . . . . . . . . . . . . ðB-8Þ
CfD;opt 2ð1  /ÞhNf >
> 4aLe2sc p >
>
>
: þ f 1 ðk; I x Þ >
;
Nf ec CA ðkÞ CfD;opt gðkÞ

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