Research on high efficiency and retarded autogenous mud acid for plugging removal in a sandstone reservoir—Example of Bai 823 well area

Acidification, as an effective measure to remove blockages and restore oil well productivity, has received widespread concern. However, conventional acid systems can only be used to unblock near‐wellbore zones and are difficult to act on far‐wellbore reservoirs. In this article, scanning electron microscope, energy‐dispersive X‐ray spectroscope, and X‐ray diffractometer were used to analyze the blockages in the Bai 823 well area, and it was determined that their main components were CaCO3 and FeCO3, which provided the basis for the construction of the unblocking formula. Then, the acid dissolution experiments on the plugging materials showed that the dissolution rate of 10% HCl can reach more than 85% for the scale sample; the acid dissolution experiments on the rock sample showed that the autogenous mud acid has better dissolution performance and better retardation performance for the rock sample, and the effective action time is longer, so it can be used for deeper unplugging of the reservoir. Finally, the process parameters such as the optimal acid concentration, acid dosage, and discharge volume were optimized by establishing the fracture‐matrix acid flow reaction model. Research results indicate that the autogenous mud acid can act on the blockage in the far‐wellbore area to improve the acidification effect.


| INTRODUCTION
Oil well plugging is one of the manifestations of reservoir damage. 1 Clogging is a common problem in all kinds of operations, including drilling, oil recovery, water injection development, and recovery enhancement.In the drilling and completion process, there are blockage problems such as solid phase particles of drilling fluid, leaching of solid-fluid, water lock of injection fluid, waste fluid in oil testing operations, and filtration loss of various incoming fluids. 2 In the process of water injection and oil recovery, as long as there is water, there is always a possibility of scaling in each production process.In addition to fouling problems similar to those encountered in water injection and recovery, the presence of steam and polymers in oil recovery can also lead to the generation of silica scale and polymer scale.As a result, the types of oil and gas well blockage are complex and diverse, so how to achieve efficient and rapid plug removal of oil wells has become an urgent problem.
Acidizing is a measure to increase production by injecting acid into the formation to dissolve soluble materials in the formation, thereby restoring or increasing the permeability of the near-well zone. 3Acidizing can be divided into fracture fluid acidizing, matrix acidizing and acid washing by process, and matrix acidizing is usually referred to as acidizing.Matrix acidizing is mainly used in cases where the fracture pressure is lower than that of rock formation. 4The acid is injected into the primary joints and surface voids, and the acid flows and dissolves through the connecting pores to effectively expand the permeable pore channels and pore space, and effectively eliminate the formation damage near the wellbore. 5,6or sandstone reservoirs, mud acids are the most common and effective acidizing system.Dowell was the first to start using a mixture of hydrochloric acid and hydrofluoric acid in industry to dissolve drilling mud deposits in the form of filter cake during the drilling process. 7,8However, during the process, the problems caused by mud acids were gradually exposed, including fast reaction rate, shallow depth of penetration into the formation, and secondary contamination of the reservoir due to the transport of the particles produced by acidification.With the continuous progress of science and technology, new types of acids such as fluoroboric acid, slow-rate mud acid, and autogenous mud acid have gradually emerged. 9,10andstone matrix acidizing has been widely used in the field since the 1960s.The complex mineral composition of sandstone reservoirs, coupled with the complex pore structure in the formation, has caused the complexity of the acid rock reaction process in sandstone reservoirs. 11In more than half a century of research, experts at home and abroad have been devoted to constructing mathematical models that can truly reflect the acid rock reaction process in sandstone reservoirs. 1 Most of the existing sandstone acid rock reaction mathematical models are based on the study of sandstone reaction kinetics, 12 and according to the treatment of sandstone reservoir mineral composition, the acid rock reaction mathematical models can be roughly classified into four major categories: set total parameter model, 13 one acid and two minerals model, two acids and three minerals model, 14 and geochemical model. 15,16As for the numerical methods, sandstone acid flow reaction models mostly adopt the finite difference method. 17n 1991, Bryant et al. 18 first introduced the secondary reaction into the mathematical model of acid rock reaction, considering the reaction of fluorosilicic acid with silicate, and established a more mature and nowadays more commonly used mathematical model of two-acid and three-mineral sandstone reaction. 19Later, Hsi et al. 20 further validated the improved model proposed by Bryant by conducting more core oil drive tests on different sandstone cores at different acid concentrations and injection rates to calibrate the model and improve its accuracy.In 2003, Rodoplu et al. developed a mathematical model of "two acids and three minerals" to reflect the change of permeability, and the experimental results verified the validity of the model and the good agreement with the experimental results. 21n 2005, Xie et al. 22 further applied the small-scale model developed by Li et al. 23 to study the conditions for earthworm pore formation during sandstone acidification, and the results showed that sandstone reservoirs with strong inhomogeneity are favorable for earthworm pore formation at low injection rates.A similar study was done by Morgenthaler et al.Similar studies on the effect of inhomogeneity showed that the effect of inhomogeneity on permeability in small-scale lithology is greater than the effect of petrographic mineralogy on permeability. 24Therefore, careful characterization of permeability in sandstone formations is fundamental to ensure matrix acidification design. 25In Zakaria 2013, a finite difference form of the two-acid and three-mineral model was derived and the effects of the radius of damage, degree of damage, injection flow rate, and acid concentration on sandstone acidification were simulated. 26n this study, an acid-unblocking process system is proposed for sandstone reservoirs.First, the basic composition of the blockage is derived from the physical analysis of the blockage, and then an autogenous mud acid-unblocking system is proposed through indoor preference experiments, and the acid injection construction is optimized through numerical simulation methods, which provides technical support for improving the production capacity of sandstone horizontal wells.

| Acid dissolution of scale and rock samples
First, the scale sample, rock sample, and filter paper were dried and put into the desiccator for cooling, and then a certain amount of scale sample and rock sample were weighed, and the mass of the scale sample and filter paper was recorded as m 1 .The scale sample, rock sample, and acid were placed in a beaker in a ratio of 1:10, sealed, and put into a constant temperature water bath at 65°C for 4 h.The filtering device was prepared, the filter paper was laid, dried, and weighed after filtration, and the mass of the scale sample and rock sample after dissolution was recorded as m 2 .The rate of dissolution of the acid on the scale and rock samples was determined using the loss of weight method, and the calculation formula was as follows.

| Characterization testing
(1) SEM and energy-dispersive X-ray spectroscope (EDS) After the sample is sprayed with gold, the sample surface is bombarded with a finely focused electron beam, and the electrons interact with the material in the sample to excite various physical information, which is then collected, amplified, and imaged by a computer to obtain an observation of the sample surface.In the vacuum environment, the sample surface is bombarded with an electron beam to excite the material in the sample to emit characteristic X-rays, and then the elements contained in the material on the sample surface are analyzed qualitatively and quantitatively according to the wavelength of the characteristic X-rays.
(2) XRD The samples were ground into powder form and then evenly spread on aluminum foil, pressed using a tablet press, and then subjected to X-ray diffraction testing, and the inorganic composition of the samples was obtained by analyzing the X-ray diffraction patterns of the scale samples.

| Experimental study on plugging removal of displacement simulated autogenous acid core
The displacement device was used to simulate reservoir conditions under certain conditions (mainly for temperature and pressure).During the experiment, the pumping sequence of on-site working fluids was simulated, mainly including base fluid (simulated formation water), acidizing treatment fluid, and base fluid.The effect of acid plugging was evaluated by observing the changes in core permeability before and after acid treatment.

| SEM and EDS
The blockage was scanned and analyzed in a single well to analyze the morphology and elemental composition of the scale sample to provide a basis for the blockage source.
From Figure 1A, it can be seen that the structure of the blockage in well 85402 is divided into blocky and fibrous.The surface of blocky blockage is loose, with granular attachment, and the attachment has no obvious morphological characteristics.The fibrous plugging material showed a cluster structure with a mesh weave and the fibers were in strips.From Figure 1B, it can be seen that the main elements of the blockage in well 85402 are C, O, Ca, Fe, and Mg elements, among which Ca and O elements are high, Fe elements may come from the corrosion of the tubular column, and the remaining elements are very little.On the basis of the elemental content, it is inferred that the blockage may be dominated by CaCO 3 .From Figure 1C, it can be seen that the blockage structure is disordered, without a more obvious lattice pattern, and the surface is dense, with particles of different sizes attached.From Figure 1D, it can be seen that the main elements of the plugging material in the T2Kc9-2 well are C, O, and Fe elements, and the remaining elements are very few.Judging from the elemental composition, the main component of the plugging material may be FeCO 3 .

| XRD
The blockage samples were first dried, and separated by standard sieves, and the appropriate amount of blockage samples were ground to powder (particle size <2 μm) for inorganic composition analysis of the blockage.
As shown in Figure 2A, the obtained plots were matched with the database using XRD data analysis software, and the results of the energy spectrum elemental analysis were also referred to conduct the phase search, and the analysis results revealed that CaCO 3 , CaMg 3 (CO 3 ) 4 may be present, and the plots were best matched with CaCO 3 standard plots in several places, indicating that CaCO 3 is the main component in the blockage.As shown in Figure 2B, the analysis results concluded that FeCO 3 , Ca 2 (Al 2 SiO 6 )(OH) 2 may be present, and the best match with the standard pattern of FeCO 3 in many places in the plot indicates that FeCO 3 is the main component in this blockage.

| Acid dissolution of scale and rock samples
As shown in Figure 3, both hydrochloric acid and mud acid can effectively dissolve the scale samples of two typical wells.Hydrochloric acid (10%) has the highest dissolution rate for the scale samples of two wells, and the dissolution rate of the scale samples of well 85402 can reach more than 95%, and the dissolution rate of the scale samples of well T2Kc9-2 can reach more than 85%, which indicates that 10% hydrochloric acid has better dissolution effect on the scale samples of this block and has better descaling ability.
As shown in Figure 4A, with the increase in hydrochloric acid concentration, the dissolution rate of rock samples was the first to increase and then stabilized, and the dissolution rate was stable at about 13.5%.As shown in Figure 4B-D, with the increase of acid concentration, the dissolution rate of rock samples by both mud acid and fluoroboric acid increased rapidly and then leveled off, while the autogenous mud acid was in a state of continuous increase.Mud acid mainly dissolves the clay component of the formation and the well plugging, and has a good unblocking effect.However, mud acids tend to have strong solubility to rocks, fast reaction speed, short action time, and limited plug removal radius.Fluoroboric acid is a slow acid, which can hydrolyze slowly to produce HF, so that the acid always remains at a low concentration, and its reaction rate is lower than that of conventional mud acids, thus it can penetrate a larger area inside the formation before the acid is exhausted.During the acidification process, the autogenous mud acid system gradually releases H + , which can keep the pH of the solution in a small range, and the initial pH is high, which can slow down the corrosion rate of pipes and equipment.The ionization rate of autogenous mud acid is slow, and the reaction rate with sandstone action is slow, it has good antiscaling and dispersing properties, which can inhibit the precipitation of silicate in the near-well zone and effectively avoid the secondary damage in the unblocking process.

| Evaluation of acid retardation performance
The retardation performance of the acid system ensures that a certain acid concentration is maintained for a certain period and advances deeper into the formation, which can greatly reduce the acid rock reaction rate, extend the acidization distance and achieve deep penetration of the acid.The static experiment method (rock sample weight loss method) is to compare the dissolution rate of different acid systems on block oil well rock powder under a certain time, and then evaluate the retardation performance of this acid system.
As shown in Figure 5

| Reaction modeling of fracture-matrix acid flow
The rock and mineral composition of the sandstone reservoir is more complex and nonhomogeneous.To accurately describe the variation law of different mineral components in the reaction, it is assumed that each matrix grid contains multiple parts: pore volume percent, mineral 1 vol%, and mineral n volume percent, to establish a matrixfracture dual medium mathematical model of the main acid-acid flow reaction.The basic assumptions are as follows.inside the fracture, the silica gel precipitation is dissolved by the host acid in a very short time after generation, so the effect of silica gel mineral precipitation inside the fracture on the fracture width is ignored (Figure 6).The fluid flow equation and the continuity equation within the fracture can be expressed in the following form, respectively. 27,28k The fracture tangential permeability can be calculated by applying the fracture width according to the cubic law as follows.
Bringing Equation (2) into Equation ( 3), the acid pressure distribution in the main body of the fracture is The acid concentration transport equation in the fracture consists of HF and H 2 SiF 6 concentration transport equation, which is as follows. 22C On the basis of the law of conservation of matter, the equation for the variation of the crack width is derived as follows. 29t (2) Mathematical model of the acid transport reaction of the subject within the matrix In the case of considering fluid and rock microcompressibility, similar to the form of the acid pressure equation for the antecedent within the matrix, the acid pressure equation for the main body within the matrix is 22,30 On the basis of the molar concentration balance principle of the sandstone acid rock reaction, the concentration equations of HF, H 2 SiF 6 , fast-reacting minerals, and silica gel minerals can be derived, respectively.
The equation for the transport of HF concentration within the substrate is as follows.
( ) The transport equation for the acid concentration of H 2 SiF 6 acid in the substrate is as follows.
The equation for the equilibrium of fastreacting mineral substances within the matrix is as follows.
The equation for the balance of silica gel precipitated material within the matrix is as follows.
(3) Pore calculation model The porosity and permeability of the formation also change during the main acid injection process.Using the principle of mineral volume balance, the mathematical model of porosity and permeability distribution during the main body acid injection was derived with the consideration of silica gel precipitation. 31,32

| Optimization of process parameters
Sandstone acidizing usually consists of three parts: prefluid, main acid, and postfluid.The role of prefluid, that is, prefluid, is to avoid contact between formation water and acid and to prevent acid from reacting with carbonate to produce precipitation; the role of the main acid is to dissolve blockages and some rocks in the formation and improve formation permeability; the role of postfluid is to drive the main acid away from the wellbore radius 12-15 times to prevent precipitation of reaction products in the residual acid.The dosage of the treatment fluid is related to parameters, such as contamination zone shape, contamination radius, degree of contamination, effective thickness, reservoir properties, and so forth.In this study, based on the fracture-matrix acid flow reaction model, the acid flow reaction simulation under different acid injection parameters is carried out to optimize the acid injection parameters based on the principles of effective dissolution of plugging materials and saving dosage.
(1) Acid concentration On the basis of the mathematical model in the previous section, five sets of numerical experiments were set up with the initial acid concentration of the main acid at 1%, 2%, 3%, 4%, and 5%, the injection discharge all at 2 m³/min, the total injection volume all at 120 m³, and the acid viscosity all at 1 mPa s to simulate the acid flow reaction under different main acid concentrations.
As shown in Figure 7, the increase in the concentration of the injected acid led to the increase in the concentration of HF by the periphery of the fracture wall, and the concentration of H 2 SiF 6 , which is the reaction product of the fast-reacting mineral and HF, also increased.Although the concentration of both acids increased, the distribution pattern hardly changed with the increase in concentration.From the figure, it is easy to find that the effective action distance (the distance when the residual acid concentration is less than or equal to 10% of the fresh acid concentration) of the main acid with different acid concentrations is almost the same, and the peak of H 2 SiF 6 concentration occurs almost at the same distance from the fracture wall, and after the peak concentration, the H 2 SiF 6 concentration decreases with the increase of the distance, and when it reaches the model boundary, the H 2 SiF 6 concentration in the stratum concentrations are not much different from each other.For the mineral field concentration distribution, the reaction rate distribution curve of fast-reacting minerals is consistent with the pattern of the HF concentration curve, because the rates of both fast-reacting minerals and are related to HF concentration without considering the effects of temperature and stratigraphic inhomogeneity on acid flow reaction.In summary, the acid concentration has less influence on the effective action distance of the acid, but more influence on the degree of well modification, and the persistent pursuit of high acid concentration often brings about problems such as corroded tubular column, pore collapse, and sand out of the formation again.Therefore, it is recommended to choose a 3.0% acid concentration to achieve the best transformation effect.
(2) Amount of acid The amount of acid is a key parameter in the design of acid pressure, and it is also an important factor affecting the acid flow reaction.Therefore, in this section, five groups of simulated acid volumes of 30, 60, 90, and 120 m³ are set, respectively, the injection discharge volume is all set to 2 m³/min, the acid viscosity is all set to 1 mPa s, and the acid concentration is all set to 3%, to study the influence of the main acid injection volume on the acid flow reaction.
As shown in Figure 8, with the increase of acid volume, the HF concentration at the fracture wall increased, the effective action distance of the acid increased, the location of the leading edge of HF reaction kept moving outward, and the peak of H 2 SiF 6 concentration appeared near the location of the leading edge of HF reaction.By increasing the amount of liquid, the reaction rate of HF and H 2 SiF 6 in the effective action distance also increased.It is noteworthy that the increase of acid quantity leads to the increase of H 2 SiF 6 concentration around the well, which leads to the sudden increase of silica gel mineral concentration of the secondary reaction product, and then the acid concentration decreases as the acid rock reaction continues in the formation, which leads to the rapid decrease of silica gel mineral concentration.
(3) Acid discharge In this section, the effect of injection displacement on the reaction pattern of acid flow is analyzed.Five control simulations were set up with acid displacement of 0.5, 1.0, 1.5, 2.0, and 2. | 487 acid viscosity of 1 mPa s, all with injection volume of 120 m³, and all with acid concentration of 3%.
As shown in Figure 9, it can be seen from the acid distribution curve that with the increase of the acid injection discharge, the peak value of HF and H 2 SiF 6 decreases, and the effective action distance of the acid increases.Meanwhile, it is not difficult to find that the reaction rate of acid in the near-well zone decreases as the acid injection displacement increases, which is reflected in the curve that the acid concentration decreases more slowly as the acid displacement increases.Comparing the concentration curves of fast-reacting minerals with different injection rates, we can see that the acid reacts rapidly with the formation minerals in the near-well zone during low-displacement injection, resulting in massive dissolution of minerals near the fracture wall and a rapid decrease in acid concentration.On the contrary, this caused a large accumulation of silica gel precipitation at the fracture wall surface during low-displacement injection, while the precipitation concentration dropped sharply a little further away.With the increase of acid injection displacement, the depth of acid penetration in the formation also increases, and the effective distance of acid action is also farther.The improvement effect of low-displacement acid injection is more obvious near the fracture wall, and it is suitable for situations where drilling fluid leakage is more serious.

| Evaluation of core displacement plugging removal experiment
As shown in Figure 10, it is the result of a simulated acidizing plugging removal experiment of core displacement.As shown in Figure 10A, it is the permeability change curve of the core before and after acidification with HCl + HF mud acid.First, the simulated formation water was used as the base fluid to measure the matrix permeability, and its value fluctuates around 1.00.After acid injection, the matrix permeability increases and fluctuates around 1.67, which indicates that the matrix permeability is displaced by acid solution, it can effectively improve the permeability of the rock core.As shown in Figure 10B, it is the core permeability curve and profile diagram before and after the acidizing treatment of the core with the autogenous acid system.When the base fluid is used for treatment, its permeability also fluctuates around 1.00.Different from the mud acid with HCl + HF, its permeability fluctuates around 1.64 after the autogenous acid treatment, and decreases slightly.As shown in Figure 10C, it is the core permeability curve before and after acidification treatment with the HCl system and the profile diagram before and after displacement.When the core permeability fluctuates around 1.35 after treatment with HCl, it indicates that the plugging removal ability of the hydrochloric acid system is inferior to that of the autogenic acid system and the general acid system.As shown in Figure 10D, it is the permeability change curve of the core after H 2 SiF 6 treatment.The permeability after treatment is around 1.69, which is similar to the effect of HCl + HF.

| Reservoir sensitivity analysis
On the basis of the geological data of the reservoir, the water sensitivity, salt sensitivity, and velocity sensitivity were analyzed.The damage rate of reservoir water sensitivity is 66.8%, showing moderate to strong water sensitivity.The critical salinity of salt sensitivity is 6026 mg/L, which is lower than the critical salinity.The permeability loss rate is 39.2%, showing moderate to weak salt sensitivity.As the injection rate increases, the permeability shows a decreasing trend.The injection flow rate is greater than 9.46 m/day, and the permeability loss rate is 26.4%.The reservoir exhibits medium to weak velocity sensitivity, and the wettability shows hydrophilicity.

| CONCLUSION
This article first analyzed the composition of the plugging materials by characterization, which provided the basis for the construction of the unblocking formula.Then, acid dissolution experiments were conducted on the plugging agents of two typical wells, and the results showed that compared with HCl, mud acid, fluoroboric acid, and autogenous mud acid, autogenous mud acid had the highest dissolution rate.Further evaluation of the blocking properties of several acids indicates that autogenous mud acids have better-blocking properties, longer effective action time, and can unblock farwellbore areas.Finally, by establishing a fracturematrix acid flow reaction model, the optimal process parameters such as acid concentration, acid dosage, and acid discharge were mainly established.As shown in Schematic 1, it is a comparison between autogenous mud acid and conventional acid systems.It can be concluded that compared to traditional acid systems, autogenous mud acid not only efficiently dissolves blockages but also has a long-term effect.

F
I G U R E 1 SEM and EDS profiles of plugging material: (A) SEM of plugging material in well 85402, (B) EDS of plugging material in well 85402, (C) SEM of plugging material in well T2Kc9-2, and (D) EDS of plugging material in well T2Kc9-2.EDS, energy dispersive X-ray spectroscopy; SEM, scanning electron microscope.F I G U R E 2 XRD pattern of plugging material: (A) well 85402 and (B) well T2Kc9-2.XRD, X-ray diffractometer.
, the dissolution rates of mud acids and fluoroboric acids increased slowly with time, F I G U R E 3 Lysis of scale samples by acid: (A) well 85402 and (B) well T2Kc9-2.and the changes were small.The dissolution rate of autogenous mud acid increased continuously and almost linearly, indicating that during acid rock dissolution, autogenous mud acid was releasing H + slowly and the release rate was lower than that of mud acid and fluoroboric acid.The dissolution rate of autogenous mud acid was lower than that of mud acid and fluoroboric acid when the dissolution time ranged from 0.5 to 2 h, and the dissolution rates of mud acid and fluoroboric acid were close to each other; when the dissolution time reached 4 h, the dissolution rates of mud acid, autogenous mud acid and fluoroboric acid were close to each other, indicating that the effective action of autogenous mud acid was longer.In conclusion, the retardation of autogenous mud acid is better than that of mud acid and fluoroboric acid.

F I G U R E 4
Dissolution rates of rock samples from the target block by different acid types as well as concentrations: (A) hydrochloric acid, (B) mud acid, (C) fluoroboric acid, and (D) autogenous mud acid.F I G U R E 5 Acid retardation performance.

( 1 )
The acid flow in matrix fractures is a single-phase flow, neglecting gravity, friction, viscous forces, capillary forces, and nonlinear flow.(2) The matrix pores, fractures, and acids are assumed to be slightly compressible.(3) Ignore the effect of acid rock reaction heat on formation temperature during the subject acid injection.(4) Classify rock minerals into two major categories according to the reaction rate with hydrofluoric acid: fast-reacting minerals and slow-reacting minerals, while assuming that minerals that have not finished reacting with the previous acid are included in the fast-reacting mineral fraction.(5) The fluid inside the acid fracture fills the whole fracture, ignoring the fluid hysteresis at the fracture tip and the nonlinear flow inside the fracture.(6) Due to the high concentration of hydrofluoric acid

( 1 )
Mathematical model of the acid transport reaction of the subject in the fracture F I G U R E 6 Division of individual mesh composition in nonhomogeneous matrix.
Plates of mineral and acid concentrations at different acid concentrations: (A) fast-reacting mineral concentration, (B) silica gel mineral concentration, (C) HF concentration, and (D) H 2 SiF 6 concentration.
5 m³/min, all with F I G U R E 8 Plates of mineral and acid concentrations at different acid dosages: (A) fast-reacting mineral concentration, (B) silica gel mineral concentration, (C) HF concentration, and (D) H 2 SiF 6 concentration.LI ET AL.

9
Plates of mineral and acid concentrations at different acid discharges: (A) fast-reacting mineral concentration, (B) silica gel mineral concentration, (C) HF concentration, and (D) H 2 SiF 6 concentration.

F
I G U R E 10 Permeability curve and cross-section change diagram before and after mud acid treatment of (A) HCl + HF, (B) autogenic acid, (C) HCl, and (D) H 2 SiF 6 .

,i
= 2 for HF acid concentration, mol/L, i = 3 for H 2 SiF 6 acid concentration, mol/L D Ai fr i = 2 indicates the HF acid molecular diffusion coefficient, m 2 /s, i = 3 indicates the H 2 SiF 6 acid molecular diffusion coefficient, m 2 /s k f τ permeability in the tangential direction of the fracture, mD k m n initial matrix permeability, mD k m n+1 matrix permeability after acid solubilization, mD m 1 mass of the scale sample before dissolution, g m 2 mass of the scale sample after dissolution, g p f acid pressure in the fracture, MPa Q f source-sink term, m 3 /min R dissolution rate, % v f acid seepage velocity in the fracture, m/s w f 2 acid corrosion crack width, m x τ fracture local coordinate system, causeless GREEK SYMBOLS β pore structure-related parameters, causeless μ acid viscosity, mPa s ϕ f fracture porosity, causeless φ m n initial matrix porosity, causeless φ m n+1 matrix porosity after acid solubilization, causeless ρ acid density, kg/m 3