Experimental Investigations of the Process of Carbonate Fracture Dissolution Enlargement under Reservoir Temperature and Pressure Conditions.
Karst is one of the central focuses in the field of carbonate reservoir geology and has attracted considerable attention from scientists worldwide [1-8]. Carbonate fractures play an important role in oil and gas transport and storage. Fracture dissolution, which can create massive pores, is a common mechanism for the formation of high-quality carbonate reservoirs.
Many carbonate rock dissolution studies have been conducted over the last few decades. Many studies investigating the kinetics of carbonate dissolution have been published (e.g., [9-16]). Durham et al.  suggested that the flow of C[O.sub.2]-saturated water in carbonate fractures under atmospheric pressure and temperature typically leads to a single broad channel along the flow direction. Dissolution experiments in transparent analog fractures at small confining stress (0.2 MPa) indicated evidence of the preferential dissolution of small apertures surrounding contacting asperities and brittle failure of asperities . The apparent activation energy is an important parameter for characterizing the dissolution reaction kinetics. Plummer et al.  found that the activation energy is temperature-dependent. The activation energy is 6.4 kJ/mol when the temperature is below 25[degrees]C and may increase to as high as 33.1 kJ/mol at higher temperatures. The C[O.sub.2]-promoted dissolution had an activation energy of 41.8 kJ/mol. For far-from-equilibrium diluted solutions, the activation energies range from 43 to 48 kJ/mol [13, 19]. Kirstein et al.  demonstrated that care must be taken when obtaining laboratory dissolution rate data from experimental fracture carbonate and when using these rates and activation energies to predict the natural fracture carbonate dissolution rate and fracture channelization systems.
The Qaidam Basin is an important oil and gas basin in Western China. The temperature of the basin ranges from 30 to 60[degrees]C . A geological survey indicated that the low permeability limestone in the Qaidam Basin may include oil and gas reservoirs [22, 23]. This limestone contains large amounts of fractures, which serve as conduits for water, oil, and gas. In these conduits, the long-lasting flow of fluids in disequilibrium with the rock is expected. Dissolution processes can alter the geometry of fractures and thus their hydraulic and transport properties . The impact of heterogeneities on the evolution of permeability and dissolution patterns of carbonate fracture media has been evaluated [25-27]. The evolution of fracture structures is directly related to the fluid flow and mineral dissolution rates. Feedback between the fluid flow, solute transport, and mineral dissolution may lead to the formation of preferential flow channels (wormholes) under certain flow and reactivity conditions . Experimental studies of dissolution patterns in a single rock fracture [17, 18, 29-32] have been conducted, but the physicochemical mechanisms behind pattern formation (e.g., the influences of temperature and fracture surface heterogeneities) remain unclear.
At present, there are few studies on the dissolution enlargement process of the carbonate fractures under reservoir temperature and pressure conditions. Moreover, the investigation of dissolution enlargement in carbonate fractured reservoirs has been demonstrated to hold practical and theoretical significance for predicting reservoir spatial evolution. Therefore, in our study, four carbonate core samples were collected from the Carboniferous Keluke Formation in the Qaidam Basin. The four carbonate core samples were artificially fractured to perform the carbonate fracture dissolution enlargement (CFDE) experiments. The hydraulic aperture ([B.sub.h]) and dissolution aperture ([B.sub.d]) at different moments were calculated, and the variations in [B.sub.h] and [B.sub.d] over time were analyzed to investigate the process of dissolution. The experimental results of this study demonstrate that the temperature affects the Peclet and Damkohler numbers, dissolution rate, and dissolution pattern. The relationship between the dissolution pattern and temperature is discussed. SEM photomicrographs and 3D laser scanning images were taken before and after the CFDE experiments to determine the locations of the channels and the mineral particles affected by preferential dissolution.
2. Materials and Methods
2.1. Sample Characterization. Four carbonate core samples were collected from the Carboniferous Keluke Formation in the Qaidam Basin. Two samples were extracted from the number 1 borehole, and the other two samples were extracted from the number 2 borehole. The two boreholes are located in the Shihuigou area of the eastern Qaidam Basin. These four samples are referred to as [S.sub.1], [S.sub.2], [S.sub.3], and [S.sub.4] in this study. The chemical and mineralogical compositions of the samples were determined at the Micro-Structure Analytical Laboratory of Beijing University. A Thermo Electron Corporation ARL Advant X[P.sup.+] X-ray fluorescence spectrometer was used to analyze the chemical composition of the samples, and a Rigaku D/Max-RA X-ray diffractometer was used to analyze the mineral composition. Tables 1 and 2 show the mineralogical and chemical composition of the samples, respectively.
We conducted a seepage experiment when the confining and back pressures were set to 20.00 MPa and 8.00 MPa, respectively. After three months, no water passes through the carbonate core. Thus, the matrix flow can be ignored.
The four cylindrical carbonate samples were artificially fractured using a wire saw, and the longitudinal fractures were parallel to the cylinder axis. The two fracture walls of each sample were combined and sealed with epoxy resin (stable mechanical and chemical properties up to 690 bar, 260[degrees]C, and low pH) on their edges to prevent any mechanical displacement of the fracture walls. The external surfaces of the carbonate fractured samples apart from the fracture inlet and outlet were also covered with epoxy resin to avoid dissolution (Figure 1). These fractured carbonate samples were then inserted into heat-shrinkable tubes to prevent possible crushing of each half from the seam line, after which they were placed into the core holder for the CFDE experiments.
2.2. Experimental Apparatus. Figure 2 shows the experimental apparatus used in the CFDE experiments. The experimental apparatus consists of 6 subsystems: (1) a gas pressurization system was used for gaseous C[O.sub.2] pressurization reaching 20 MPa. After pressurization, C[O.sub.2] is stored in the gas container. (2) A gas injection system was used to inject C[O.sub.2] into the liquid container at a constant pressure (8 MPa) for 12 h. The outlet valve (I) of the gas container can be adjusted to control the gas pressure of the C[O.sub.2] gas injected into the liquid container. (3) A liquid injection system was used to inject liquid into the core holder using a piston pump at a controlled flow rate of approximately 11.5 ml/h. (4) The core clamping system is composed of a core holder. The rock core is wrapped in a fluororubber sleeve, placed inside the stainless-steel core holder, and then placed horizontally in a thermostatically controlled chamber. The fluororubber sleeve is highly elastic and acid-resistant. (5) A confining and back pressure system was used. The back pressure system uses a syringe pump to control the initial pressure within the core. Fluid can flow out of the core fracture only when the fluid pressure exceeds the preset back pressure. The confining pressure, set to 20 MPa for all of the dissolution experiments, was also applied around the rock core using the syringe pump. The confining pressure was always higher than the axial pressure. (6) The measuring system includes two pressure gauges at the inlet and outlet of the core holder and an electronic scale to weigh the water cumulatively collected in the liquid tank at the outlet.
As shown in Figure 2, the liquid containers, the gas and liquid injection valves, the core holder, the two valves on the core holder, and the pipe lines are all temperature-controlled with a thermostat. The portions of the system under temperature control are surrounded by the dashed gray lines in Figure 2. In every experiment, both the pressures and volumetric flow rates were recorded every 30 s.
2.3. Experimental Approach. A sample was installed in the core holder, and the confining and back pressures were set to 20.00 MPa and 8.00 MPa, respectively; then, deionized water (without dissolved C[O.sub.2]) was injected into the carbonate fracture at a flow rate of approximately 11.5 ml/h. When the downstream pressure increased to 8.00 MPa and the inlet pressure, outlet pressure, and outlet water flow rate were stable for 2 h, the upstream and downstream valves (VII and VIII) of the core holder, the water outlet valve (III and V) of the liquid container, and the water inlet valve (IX and X) of the liquid container were closed in turn. The gas container outlet valve (I) and the gas inlet valve (II and IV) of the liquid container were opened. Supercritical C[O.sub.2] was injected into the liquid container from the gas container for 12 h. Then, the water of the liquid container was sampled for pH testing to ensure that the deionized water was completely saturated with C[O.sub.2]. The injection pressure (8.00 MPa) was controlled by the degree of opening of the gas container outlet valve (I). When deionized water was saturated with C[O.sub.2], the sample was kept at an initial pressure of 8.00 MPa and the temperature for the water and rock core remained at the experimental temperature. After the process, the gas container outlet valve (I) and the gas inlet valve (II and IV) of the liquid container were closed. The water inlet valve (IX and X) of the liquid container, the water outlet valve (III and V) of the liquid container, and the upstream and downstream valves (VII and VIII) of the core holder were opened in turn. The C[O.sub.2]-saturated water was injected into the carbonate fracture at a flow rate of approximately 11.5 ml/h. The pressures at the core inlet and outlet were recorded during the experiment. The volumetric flow rate of the outlet water was recorded using the electronic scale in the measuring system. The outlet water was sampled at intervals of 8 h for ion analysis. The output solution was collected in 10 ml centrifuge tubes that contained 1 [micro]l of concentrated nitric acid. The acid was used to prevent any carbonate precipitation due to depressurization and degassing during sampling. Chemical analysis of the samples was carried out by using inductive coupled plasma-atomic emission spectroscopy (ICP-AES).
The experimental procedures for the four samples were identical. The temperature of the basin ranges from 30 to 60[degrees]C for the Keluke formation . The experimental temperatures for [S.sub.1], [S.sub.2], [S.sub.3], and [S.sub.4] were 60, 50, 40, and 60[degrees]C, respectively. The experiments for [S.sub.1], [S.sub.2], [S.sub.3], and [S.sub.4] lasted 96, 208, 216, and 216 h, respectively.
2.4. Solubility of Calcite under Experimental Conditions. The solubility of calcite is one of the important factors affecting the rate of carbonate dissolution [33, 34]. We performed experiments to determine the solubility for the four carbonate samples under the conditions corresponding to the CFDE experiments.
The fresh bulk corresponding experimental carbonate samples were crushed, ground, and then sieved to obtain the size fraction between 1500 [micro]m and 2000 [micro]m. Ultrafine particles produced by grinding were removed by ultrasonic cleaning in alcohol. Each experiment was performed by placing the crushed material (~2 g) into a 500 ml reactor filled with 450 ml of pure water. The reactor was sealed; then, supercritical C[O.sub.2] was injected into the reactor for 12 h at a constant pressure (8.00 MPa) by the pump from the C[O.sub.2] tank to ensure that the deionized water was completely saturated with C[O.sub.2]. At the same time, when water was saturated with C[O.sub.2], the temperature was set to 60, 50, 40, and 60[degrees]C for [S.sub.1], [S.sub.2], [S.sub.3], and [S.sub.4], respectively. Three hours later, the temperature was stable. Sampling of reacted solutions was performed regularly at 8 h intervals for the initial three days, after which the sample frequency was reduced to two samples per day for the fourth and fifth days and then further reduced to one sample per day for the last few days. The sample solutions were filtered by using syringes equipped with 0.45 [micro]m filters. The ion concentration of the sample solutions was determined by inductive coupled plasma-atomic emission spectroscopy (ICP-AES). A schematic diagram of the solubility measurement apparatus is presented in Figure 3. Figure 4 shows the results of the variations in the [Ca.sup.2+] concentration over time.
3. Experimental Results and Discussion
3.1. Phenomenon of Channelizing Dissolution. Figure 5 shows the photographs and scanning images of the fracture surfaces before and after the CFDE experiments. In Figure 5, column (a) shows the photographs before the CFDE experiments, column (b) shows the photographs after the CFDE experiments, and column (c) shows the images of 3D-laser scanning after the CFDE experiments.
The development of a single channel can be clearly observed in the case of [S.sub.1]. The channel started at the center of the inlet side and extended to the left side of the surface. At the inlet side, the channel width and depth are approximately 2 mm and 0.6 mm. Overall, the dissolution channel becomes gradually narrower and shallower from the inlet to the outlet. Bioclastic particles and a small amount of clay minerals can be observed on the channel wall.
The image for [S.sub.2] shows dissolution throughout the majority of the fracture surface. From the inlet (z = 0) to z = 5 mm, uniform dissolution occurs and the dissolution depth is approximately 0.9 mm. Thereafter, the passage becomes two channels along both sides (left and right sides) of the surface; the channels are wider at z = 5 mm and narrow down toward z = 52.5 mm (the width of the left channel is 1.5 mm, and the width of the right channel is 0.1 mm). Then, from z =52.5 mm, the channels on both sides widen and became shallow again. Overall, the dissolution channels are gradually shallow from the inlet to the outlet (average depth from 500 [micro]m to 80 [micro]m). In the area around the inlet, the dissolution occurs evenly on the whole surface of the fracture. From the position of z = 5 mm (Figure 5), two dissolution passages form. The channels are likely along microfractures. Therefore, we suggest that microfractures in rocks may contribute to the dissolution of carbonate rocks.
For [S.sub.3], two channels extend from the inlet side to the outlet side. One of them develops along veins and stylolite. This suggests that veins and stylolite are likely to be preferential passages in rocks for dissolution enlargement. The right-side half is completely channelized. From the inlet to approximately z = 7 mm, the right channel is gradually displaced to the edge of the right side of the core surface. This displacement phenomenon is attributed to the calcite vein of the core surface. However, at approximately z = 35 mm, the right channel extends from the edge of the right side of the core surface to the center of the fracture surface, likely because of the mineral heterogeneity. The width and depth of the right channel at the inlet are approximately 3.8 mm and 1.2 mm, respectively, compared to 1.5 mm and 1.0 mm, respectively, at the outlet. The width and depth of the left channel at the inlet are approximately 4.0 mm and 0.7 mm, respectively, and the left channel extend forward by 40 mm.
For [S.sub.4], dissolution occurred within a greater area than for [S.sub.1] and [S.sub.3]. As a result, the dissolution channels are considerably wider and shallower than those for [S.sub.1] and [S.sub.3]. The width and depth of the right channel are approximately 18 mm and 0.5 mm, respectively, and those of the left channel are approximately 16 mm and 1.0 mm, respectively. We observed one significant feature for [S.sub.4]; at the end of the dissolution experiment, the dissolution channels were reddish brown and the edge of channels were yellow brown on the fracture surface center, suggesting that the dissolution channel extended forward of the surface center. Comparing [S.sub.1] and [S.sub.4], a small amount of clay minerals was left on the [S.sub.1] channel wall, although the area of the [S.sub.4] channels was largely covered with a certain amount of insoluble mineral. Thus, the coating on the channel surface of [S.sub.4] is vuggy and rugged.
The CFDE experiments produced distinct dissolution. The dissolution channels developed across the four carbonate fractured samples. At the inlet, rocks were in direct contact with highly acidic water, leading to the dissolution of calcite and calcium release as well as large dissolution channels. Along the flow path, the pH increased and the water progressively became less reactive, which led to the dissolution channels gradually becoming narrower and shallower. The spatial variation in the channel along the flow direction can be attributed to the initial fracture characteristic, solute ([Ca.sup.2+]) saturation, and mineralogical microstructure [35-37].
3.2. Changes in Fracture Aperture. When water flows through fractures, it dissolves the solid materials of the fracture walls. The fractures are enlarged, and the permeability of the fractures is enhanced. The change in fracture permeability over time was calculated from the pressure difference AP measured between the inlet and outlet of the carbonate fractures samples and combining Darcy's law (1) with a cubic law for flow through two parallel plates (2) [38, 39].
k = [mu]LQ/A[DELTA]P, (1)
Q = [DELTA]Pd[B.sup.3.sub.h]/12[mu]L, (2)
where k is the permeability of the carbonate fracture, [mu] is the dynamic viscosity of the solution (Pa x s), L is the length of the carbonate fracture samples in the flow direction (m), Q is the volumetric flow rate ([m.sup.3]/s), [DELTA] is the cross-sectional area perpendicular to the flow direction, [DELTA]P is the measured pressure difference (Pa), [B.sub.h] is the hydraulic aperture of the fracture (m), and d is the diameter of the core sample (m).
Combining (1) with (2), the hydraulic aperture of the fracture was determined by
[mathematical expression not reproducible] (3)
The fracture permeability is
k = [B.sup.2.sub.h]/12. (4)
Figure 6 shows the variation process of the outflow rates in the experiments. For the four CFDE experiments, the outflow rates decrease slightly over time according to the trend line equation.
The water pressures at the inlet and outlet were measured throughout the experiment. Figure 7 shows the variation in [DELTA]P over time during the CFDE experiments. At the beginning of the experiments, [DELTA]P gradually decreased with time and then gradually became stable.
Figure 8 shows the variation in [B.sub.h] for the four CFDE experiments. When the C[O.sub.2]-saturated water passes through the fractures, it dissolves the carbonate of the walls. The dissolved carbonate is taken away by the water such that [B.sub.h] generally becomes larger. As shown in Figure 8, the hydraulic apertures of [S.sub.1], [S.sub.3], and [S.sub.4] are notably increased. [B.sub.h] of [S.sub.2] decreases slightly in the first 168 h and then begins to increase gradually.
A slight decrease in the aperture of [S.sub.2] in the early stage of the experiment may be attributed to the initial fracture surface characteristic. Similar to some natural fractures, the surface of [S.sub.2] is uneven and jagged. At the beginning of the CFDE experiment, [S.sub.2] holds the largest hydraulic aperture in the four fractures. With the progression of the dissolution of the fracture walls, the fracture surfaces gradually become flat, and the fracture is closed under the effect of the same confining pressure.
The [B.sub.h] growth rate of [S.sub.4] was the largest. Overall, the [B.sub.h] values of the four fractures were enlarged by 3.4, 0.94, 1.4, and 1.2 times by the CFDE experiments, respectively.
According to Figure 8, the hydraulic aperture was a function of time. [B.sub.h] can be calculated as
[B.sub.h1] = 4.6 + ln(1 + 0.02351t)/1 + [e.sup.5.9521-0.1685t] + 2.553, (5)
[B.sub.h2] = 0.000025 [(t - 160).sup.2] + 8.9688, (6)
[B.sub.h3] = 0.69 ln (1 + 0.253t) + 6.85839, (7)
[B.sub.h4] = 0.235 ln (1 + 4.73t) + 6.7505. (8)
In the above equations, [B.sub.h1], [B.sub.h2], [B.sub.h3], and [B.sub.h4] are the hydraulic apertures of [S.sub.1], [S.sub.2], [S.sub.3], and [S.sub.4], respectively ([micro]m), and t is the experimental time (h).
These formulas ((5), (6), (7), and (8)) can be divided into three categories: S type, logarithmic type, and polynomial type. The growth rate of [B.sub.h1] can be divided into three stages: the slow growth stage, rapid growth stage, and slow growth stage. In the first slow growth stage, the flow field was uniform. However, preferential flow gradually formed with the selective dissolution of the fractured surface. Once the preferential flow path in the flow field on the fractured surface was established, the aperture growth rate entered a stage of rapid development. During the rapid growth stage, the channels formed and widened, finally reaching the steady state. In the steady state of the channel, the aperture growth rate entered a stage of slow development again.
Because the fracture surface characteristics are different, the duration of the first stage is also different. A more uneven surface is associated with a bigger initial aperture ([B.sub.0]) and shorter time to establish the priority flow. Therefore, [S.sub.3] and [S.sub.4] have no clear first stage, and [S.sub.3] and [S.sub.4] belong to the logarithmic type. For [S.sub.3] and [S.sub.4], preferential flow forms rapidly, and then channels begin to develop. The coefficients of the formula and curve characteristics for [S.sub.3] and [S.sub.4] differ due to the experimental temperature and the mineral composition of the fracture surface.
The duration of the rapid growth stage of [S.sub.4] was shorter than that of [S.sub.3]. According to Section 3.3, because the experimental temperature of [S.sub.4] was high, the dissolution rate was fast. The growth rate of [B.sub.h] is also large. However, [S.sub.4] contains a large amount of insoluble minerals. With the dissolution of the fractured surface, the coating of insoluble minerals hinders the flow and dissolution. Therefore, at the third stage, the growth rate of [B.sub.h] is very slow and gradually decreases.
For [S.sub.3], its experimental temperature was low, but the calcite content in the rock was extremely high. Although its dissolution rate was low, the rock sample was nearly completely dissolved. The dissolution channel gradually widened and deepened. The water flow rate in the channel gradually accelerated. Thus, the growth rate of [B.sub.h] decreases continuously.
[S.sub.2] belongs to the polynomial type. However, [B.sub.h] became larger at the end of the experiment. The latter changes of [S.sub.2] may also become a logarithmic type.
Based on the CaC[O.sub.3] stoichiometric ratio, the total number of moles of dissolved calcite was calculated as
n = [[summation].sup.m.sub.i=1][C.sub.i][Q.sub.i][DELTA]t/[M.sub.Ca], (9)
where n is the total number of moles of dissolved calcite, MCa is the molar mass of calcium (g/mol), [C.sub.i] is the [Ca.sup.2+] concentration of the ith sample at outlet (mg/l), Q is the outflow rate of the ith sampling ([m.sup.3]/s), [DELTA]t is the sampling time interval (s), and m is the total number of sampling times.
The mass of dissolved rock was calculated as
[m.sub.cal] = [nM.sub.cal]/[alpha], (10)
where [m.sub.cal] is the mass of dissolved calcite (g), [alpha] is the weight fraction of calcite in the rock samples, and [M.sub.cal] is the molar mass of calcite (g/mol).
The volume of dissolved rock was calculated as
[V.sub.ca] = [m.sub.cal]/[[rho].sub.cal], (11)
where [V.sub.cal] is the volume of dissolved rock ([m.sup.3]) and [[rho].sub.cal] is the rock density (kg/[m.sup.3]), and the value of [[rho].sub.cal] was 2800 kg/[m.sup.3].
The dissolution aperture was calculated as
[.sub.d] = [V.sub.cal]/[A.sub.lw] = [V.sub.cal]/Ld, (12)
where [B.sub.d] is the dissolution aperture and [A.sub.lw] is the reaction area ([m.sup.2]). Namely, [A.sub.lw] is the sample diameter times the sample length.
Figure 9 shows the increasing trends for the dissolution aperture of the four fractures. In Figure 9, [B.sub.d] increased nearly linearly over time, indicating that [B.sub.d] increases at a constant rate. According to the slope of [B.sub.d], the growth rate of [B.sub.d] was the largest for [S.sub.2] and the smallest for [S.sub.1]. These results can be attributed to the experimental temperatures and reaction area.
A comparison of Figures 8 and 9 indicates that the values of [B.sub.d] were approximately one order of magnitude larger than those of [B.sub.h]. This result can be attributed to the fracture dissolution characteristic along the flow direction. The dissolution channels gradually became narrower and shallower from the inlet to the outlet. Namely, the actual aperture of the samples gradually decreased along the flow direction. According to the groundwater dynamics calculation, the permeability of a heterogeneous fracture was mainly controlled by the narrow parts. Dissolution enlargement in the area around the outlet was slow for all four fractures, resulting in the slow increases in the permeability and [B.sub.h] of the fractures.
[B.sub.d] was defined as the average dissolution aperture openness. The carbonate dissolution was mainly located at the inlet end, and the dissolution aperture was larger than [B.sub.h]. Therefore, [B.sub.d] was larger than [B.sub.h].
3.3. [Ca.sup.2+] Concentration at the Outlet and Dissolution Rate
3.3.1. Solubility Results of the Samples. Figure 4 shows that the [Ca.sup.2+] concentrations increased during the experiments, and the release rates (the slopes of the curves in Figure 4) decreased over time as the reacted aqueous solutions approached saturation with respect to calcite. The slopes given by the last three points tend to zero, indicating that the dissolution reaction has approached equilibrium. We use the average of the last three numbers for the calcium equilibrium concentration. The calcite solubility results and the experimental conditions are shown in Table 3.
The temperature plays a significant role in solubility, and the original textures and structure of the samples as well as the mineralogical and chemical composition also affect the solubility. Generally, for [S.sub.2], [S.sub.3], and [S.sub.4], the calcite solubility increases with decreasing of temperature. The solubility of calcite increases with increasing calcium oxide content in the carbonate samples ([S.sub.1]). Therefore, the experimental solubility for the four samples is different.
3.3.2. Dissolution Rate of Fractured Carbonate. Figure 10 illustrates the variation in the [Ca.sup.2+] concentration over time during the experiments. Overall, the [Ca.sup.2+] concentration of [S.sub.2] was higher than those for the other samples. This result is attributed to the experimental temperature. Carbonate dissolution is mainly controlled by the reactions occurring within the ternary system water ([H.sub.2]O), carbon dioxide (C[O.sub.2]), and calcite (CaC[O.sub.3]) [40-44]. The chemical reactions and process responsible for the removal of calcite are
[mathematical expression not reproducible] (13)
The first reaction describes the physical dissolution of C[O.sub.2] in water. The second reaction describes the conversion of C[O.sub.2] into carbonic acid. The third and fourth reactions are the primary ionization and secondary ionization of carbonic acid, respectively. Namely, the third reaction is the conversion of carbonic acid into hydrogen and bicarbonate, and the fourth reaction is the dissociation of bicarbonate into hydrogen and carbonate ions. The fifth reaction is the dissolution of calcite. This dissolution reaction was affected by the [H.sup.+] concentration at the solid-liquid interface. The hydrogen ions in the solution are produced by the ionization of carbonic acid. Because the secondary ionization constant of carbonic acid is four orders of magnitude lower than the primary ionization constant, the secondary ionization of carbonic acid can be neglected.
Figure 11 illustrates the variation in [Ca.sup.2+]/[Ca.sup.2+.sub.eq] versus time. Figure 11 shows that the dissolution reactions of the four samples are far from the equilibrium reaction. Therefore, the fifth reaction is in the positive reaction direction. In this paper, the partial pressure of C[O.sub.2] is the same for all the samples, and the amount of C[O.sub.2] is sufficiently excessive. Therefore, the [H.sup.+] concentration was controlled by the primary ionization constant. According to Plummer and Busenberg , [k.sub.3] is a function of temperature and can be calculated as
log [k.sub.3] = -356.3094 - 0.06091964T + 21834.37/T + 126.8339 log T - 1684915/[T.sup.2], (14)
where [k.sub.3] is the primary ionization constant of carbonic acid and T is the absolute temperature.
According to (14) and Figure 12, [k.sub.3] was the largest at 50[degrees]C and the smallest at 40[degrees]C. Namely, the conversion rate of carbonic acid into hydrogen was the largest at 50[degrees]C and the smallest at 40[degrees]C. The [H.sup.+] concentration was the largest at 50[degrees]C and the smallest at 40[degrees]C . The [Ca.sup.2+] concentration was the highest at 50[degrees]C and the smallest at 40[degrees]C.
However, the [Ca.sup.2+] concentration of [S.sub.4] was lower than that of [S.sub.3] at a given time, which can be attributed to the mineral composition of [S.sub.4]. With the dissolution, the fracture surface of [S.sub.4] was covered with a layer of refractory minerals. The refractory minerals might lead to a reduction in calcite dissolution. Overall, the [Ca.sup.2+] concentration of [S.sub.4] was still higher than that of [S.sub.3].
For the CFDE experiments, dissolution rates of carbonate samples were related to the [Ca.sup.2+] concentration at the solid-liquid interface and the contact area (the reaction area) of the solid-liquid surface. The contact area of a rough surface is difficult to determine and changes with the dissolution process. In the early stage, the fracture may be visualized as an opening between two parallel plates. The reaction area may be approximated with the area of the fracture surface, which is equal to the product of the length and the diameter of the core sample ([A.sub.lw]). If the reaction area is assumed to be constant within the first 96 h, the dissolved volume of carbonate rock within the first 96 h can be determined by combining (9), (10), and (11).
The dissolution rates of carbonate samples were calculated as
[F.sub.lw] = [V.sub.lw]/[2v.sub.cal][A.sub.lw]t, (15)
where [V.sub.lw] is the dissolved volume of carbonate rock within 96 h ([m.sup.3]), [v.sub.cal] is the molar volume of carbonate rock (3.69 x [10.sup.-5] [m.sup.3]/mol), [A.sub.lw] is half of the reaction area ([m.sup.2]) because of the fracture having two surfaces, and t is the reaction time (96 h).
The dissolution rates ([F.sub.lw]) were calculated using (15) and are shown in Table 4. According to Table 4, the values of the dissolution rate are about one order of magnitude greater than that provided by Kirstein et al. . Meanwhile, the dissolution rates in this paper are approximately equal to that of Cao et al. . Also, the dissolution rate is four times smaller than that of Elkhoury et al. . The differences in dissolution rates among the authors can be attributed to the experimental temperature, pressure, flow rate, and other factors.
Additionally, the dissolution rate of carbonate rock was the largest at 50[degrees]C. Kaufmann and Dreybrodt  stated that the carbonate dissolution rate is determined by the [Ca.sup.2+] concentration at this fracture, by the conversion rate of carbonic acid into [H.sup.+] in the solution, and by diffusional mass transport of the dissolved species from and toward the water-rock interface. According to (14), the conversion rate of carbonic acid into hydrogen was largest at 50[degrees]C. You et al.  studied the aqueous solution of calcium carbonate via molecular dynamics simulation and found that the diffusion coefficient of water molecules increases with increases in temperature and is relatively small only when the temperature is 333 K. From 40 to 60[degrees]C, the diffusion coefficient of water molecules was the largest at 50[degrees]C. Therefore, at 50[degrees]C, the highest dissolution rate of carbonate rock was attributed to the largest conversion rate of carbonic acid into [H.sup.+] and the largest diffusion coefficient of water molecules.
3.4. Dissolution Patterns. The dissolution mechanic of the fracture surfaces can be separated into three steps: (i) ions dissociate from the surface due to thermodynamic disequilibrium at the mineral surface; (ii) ions diffuse away from the surface due to the concentration gradient created by the dissolution reaction; and (iii) ions are transported through the fracture by advection. The relative rates of the reaction, diffusion, and advective transport of [Ca.sup.2+] control the nature of the dissolution process within the fracture and the evolution of the fracture permeability and morphology [48-52]. The effective kinetic is controlled by the slowest of these three steps. If the characteristic time of the reaction is short compared to the characteristic time of transport (advection versus diffusion), the process is transport limited. Conversely, a reaction-controlled process means that compared with mass transport (either advection or diffusion), the reaction is slow and dissolution is limited by the dissolution reaction rate. The relative strength of advection, diffusion, and reaction can be accurately represented by two important dimensionless numbers, the Peclet number ([P.sub.e]) and the Damkohler number ([D.sub.a]) . Different dissolution patterns may be observed depending on the values of these numbers: face dissolution (only dissolution at the inlet and no further alteration along the fracture due to reactant consumption), channel (preferential flow path) formation, or uniform dissolution along the core [28, 49, 50, 52, 54].
In the formulation employed here, The [P.sub.e] number is given by
[P.sub.e] = v[B.sub.h0]/[D.sub.m], (16)
where v is the fluid velocity in m/s, [B.sub.h0] is the initial hydraulic aperture of the fracture in m, and [D.sub.m] is the molecular diffusion coefficient of the reactants in [m.sup.2]/s.
v is defined as
v = Q/d[B.sub.h0], (17)
where Q is the outflow rate in [m.sup.3]/s and d is the width of the fracture in m; the width is equal to the diameter of the samples in the paper.
According to Tartakovsky et al.  and Lichtner and Kang , [D.sub.a] is defined as
[D.sub.a] = [K.sub.lw][B.sub.h0]/[D.sub.m], (18)
where [K.sub.lw] is the dissolution rate constant. According to Kirstein et al. , [K.sub.lw] can be assumed as [K.sub.lw] = [F.sub.lw] (m/s).
According to (16) and (18), [D.sub.a] characterizes the first and second steps of the fracture surface dissolution mechanic and [P.sub.e] characterizes the second and third steps of the fracture surface dissolution mechanic. The [P.sub.e] and [D.sub.a] values were computed and are shown in Table 5. The [P.sub.e] and [D.sub.a] values were always greater than one and thus denote a diffusion-controlled process for the samples. Larger [P.sub.e] and [D.sub.a] values are associated with a smaller relative diffusion rate of [Ca.sup.2+] and thus a more notable channelizing dissolution.
In addition, advection is also one of the important factors that affect the solute diffusion. For the samples, a higher advective velocity is associated with a larger [P.sub.e] value. The higher velocity denotes that the hydrodynamic renewal of the reactive fluid is high at the fluid-rock interface along the water flow direction. Namely, the large dissolution channels are easily formed along the water flow direction. As expected for sample [S.sub.3], at this relatively high velocity, the fast reactant renewing in the fracture leads to the development of distinct large-scale dissolution channels (Figure 5). For sample [S.sub.2], the reactive fluid renewing is slow along the water flow direction. The dissolution of [S.sub.2] is evenly advanced forward. Therefore, the [S.sub.2] dissolved site is mainly located at the entrance end, and its dissolution pattern belongs to the transition from face dissolution to the channel.
The [D.sub.a] values for [S.sub.1] and [S.sub.4] are smaller than those for [S.sub.2] and [S.sub.3], indicating that the diffusion rate is small but its effect is not significant, in contrast to the reaction rate. However, the advective velocity of [S.sub.1] is one order of magnitude larger than that of [S.sub.4]. As expected, [S.sub.1] was dissolved to form a distinct large-scale channel. The advective velocity for [S.sub.4] was slow, and thus, the dissolution channels were wider than for [S.sub.1].
In summary, for a diffusion-controlled process, the dissolution channel phenomenon appears with fast flow and face dissolution occurs with slow flow in the fracture. Min et al.  also obtained the same results.
The effects of the reaction rate and molecular diffusion strength on the dissolution patterns are controlled by the experimental temperature. A dissolution process at low temperature has higher Peclet and Damkohler numbers, thus leading to notable channelizing dissolution. The Peclet and Damkohler numbers of the CFDE experiment are the highest [S.sub.3], and thus, the channelizing dissolution is the most notable of the four fractures.
3.5. Effect of Mineral Heterogeneities on Preferential Dissolution
3.5.1. Initial Mineralogical Characterization of the Samples. Table 1 shows the lithology, sampling depth, and length of the samples. The matrix of these limestone samples is mainly composed of microcrystalline calcite. The samples also contain small amounts of biogenic fragments or calcite vein. The biogenic fragments of [S.sub.1] are mainly composed of coral fossils. In [S.sub.2], multiple sets of tectonic joints formed a grid-like structure, and the joints were filled with calcite, terrestrial mud, and terrigenous sand. [S.sub.3] contains two sets of typical joints, one of which formed a low-angle tensile fracture and the other of which was a high-angle tensile fracture. The fractures were filled with biologic shells and calcite veins. [S.sub.4] contains a large number of fossils, such as fusulinida and corals. The calcite cement found in [S.sub.4] is mainly composed of microcrystalline calcite, clay minerals, and extremely low amounts of ferromagnesian and quartz.
3.5.2. Effect of Mineral Heterogeneities on Preferential Dissolution. The CFDE experiments and SEM images provide independent parameters to characterize the dissolution progress within the fracture from the sample scale to the microscale. For fractured samples, the dimensionless Damkohler and Peclet numbers provide a means to estimate the evolution of the dissolution patterns. However, the mineralogical heterogeneities also affect the dissolution patterns and the location of the preferential channel. It is important to investigate the potential impacts of the mineral structure and chemical composition on the location of preferential channels in the fracture.
In the case of [S.sub.1], the dissolution channel is mainly developed in locations where the bioclasts are gathered (Figure 13). The bioclasts are mainly gastropods, brachiopods, and echinoderms, and the size of bioclasts particles is different from each other. Micritic cement of the bioclasts is mainly composed of microcrystalline calcite and few clay minerals. The bioclast and micritic cements are preferential dissolution material because they have smaller particle sizes and a looser structure. There is no doubt that the particle size is inversely proportional to the particle surface area. More tiny interparticle pores and larger interaction area maybe another reason. Saturated C[O.sub.2] aqueous solution flows along the pores between the calcite grains and biological particles, and the calcite around the pores dissolves to form enlarged interparticle dissolution pores. The microcrystalline cements of the bioclasts and microcrystalline calcite inside the bioparticles are dissolved away, and a small amount of clay minerals are left, leading to the formation of the dissolution pores and intergranular pores.
In the case of [S.sub.2], the partially dissolved and nondissolved grains of rock were visible in the fracture surface (Figure 14). The jointing and interchange of jointing of the calcite crystal were first dissolved into micropores and microcracks, and the surface of the calcite particle was undulated and rough (Figure 14(c)). Pores and cracks along the jointing became larger and deeper. The later filling matter was approximately parallel to the sample cross section and perpendicular to the flow direction. (Figure 14(b)) It was indissolvable and obstructed the transport of the unsaturated reaction solution. Therefore, the position of the late filling matter determines the boundary of the face dissolution. [S.sub.2] developed channels after the boundary. Therefore, the experiment for [S.sub.2] was in a transitional regime between face dissolution and channel development.
The calcite veins on the fracture surface are notable in the case of [S.sub.3] (Figure 5(a)). As shown in Figure 15, the crystal morphology of the calcite veins is different from that of the calcite of the carbonate matrix. Carbonate matrix calcite is irregularly granular, with a particle size of 4-10 [micro]m (Figure 15(a)). The water along the pores of the calcite crystal grain flows such that the granular calcite is dissolved into a microcrystalline calcite and yields a large number of cracks and holes (Figure 15(b)). However, vein calcite has a clear cleavage plane, and the cleavage plane is a few to a dozen microns (Figure 15(c)). The cleavage and cleavage interchange of the vein calcite were first dissolved into micropores and microcracks. The dissolution channel locations (Figure 5(b)) of the experiment on [S.sub.3] indicate that the calcite veins are more easily dissolved than granular calcite. Namely, calcite with cleavage is more susceptible to dissolution than granular calcite is.
In the case of [S.sub.4] (Figure 16), the crystal morphology of calcite is irregularly granular. The calcite cement is mainly composed of microcrystalline calcite, clay minerals, and extremely small amounts of ferromagnesian and quartz. In the experiment, the fracture surface mineral homogeneities caused the dominant channels to form along the flow direction. The nondissolved grains of quartz and the partially dissolved silicate grains and ankerite continued to be visible in the dissolution zone.
The photographs and 3D laser scanning images of the fracture surfaces before and after the CFDE experiments indicated that the samples developed distinct dissolution zones. The dissolution channels of [S.sub.3] and [S.sub.4] were all across the fracture surface. The dissolution channels for [S.sub.1] and [S.sub.3] were narrower, deeper, and cleaner. For [S.sub.2], the inlet end experienced uniform dissolution and then developed into two channels. For [S.sub.4], the channels were wide and shallow.
After the CFDE experiment, the hydraulic apertures ([B.sub.h]) of sample 1 ([S.sub.1]), sample 3 ([S.sub.3]), and sample 4 ([S.sub.4]) were enlarged by 3.4, 1.4, and 1.2 times, respectively. The aperture of sample 2 ([S.sub.2]) was slightly reduced in the early stage of the experiment. [B.sub.h] has three forms as a function of time: S type, logarithmic type, and polynomial type. [S.sub.1] belongs to the S type, and the S curve can be divided into three stages: the slow growth stage, rapid growth stage, and slow growth stage. [S.sub.2] belongs to the polynomial type. [S.sub.3] and [S.sub.4] belong to the logarithmic type.
The carbonate fracture dissolution rates ([F.sub.lw]) were determined by combining the [Ca.sup.2+] concentration and reaction area. The laboratory dissolution rate of [S.sub.1], [S.sub.2], [S.sub.3], and [S.sub.4] were 2.50 x [10.sup.-6],3.11 x [10.sup.-6], 2.70 x [10.sup.-6], and 3.04 x [10.sup.-6] mol/[m.sup.2]/s, respectively. The dissolution rate of carbonate rock was the largest at 50[degrees]C. This result was attributed to the experimental temperature, the conversion rate of carbonic acid into [H.sup.+], and the diffusion coefficient of water molecules.
The experiment results indicate that the dissolution pattern depends on the Peclet and Damkohler numbers for the CFDE experiments at different temperatures. A dissolution process at low temperature has a higher Peclet number, thus leading to notable channelizing dissolution. In addition, advection is one of the most important factors affecting the solute diffusion. In the large-temperature scale, at a suitable advective velocity, the dissolution patterns might change from face dissolution to a typical channel and uniform dissolution by increasing the temperature.
The SEM images and EDS data indicated that the four fracture surfaces exhibited different microstructures. The initial surface mineral heterogeneity and texture strongly affect the dissolution rate and the location of preferential dissolution in the CFDE experiments. A preferential channel typically develops in places where bioclasts are accumulated, or the calcite veins and original fractures are distributed.
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
This work was supported and funded by the China Geological Survey and funded in part by the National Natural Science Foundation of China (Grant nos. U1612441, 1212011120964-03, 41272387, and 41272159). The authors would also like to thank the Institute of Geology of the Chinese Academy of Sciences for providing the 3D laser scanner and microstructure analytical laboratory at Beijing University.
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Jingxia Wang and Qingchun Yu (iD)
Beijing Key Laboratory of Water Resources & Environmental Engineering, China University of Geosciences (Beijing), Beijing 10083, China
Correspondence should be addressed to Qingchun Yu; email@example.com
Received 8 May 2018; Accepted 24 July 2018; Published 12 September 2018
Academic Editor: Merce Corbella
Caption: Figure 1: Schematic diagram of the carbonate samples after having been fractured and covered by epoxy resin.
Caption: Figure 2: Schematic diagram of the experimental apparatus.
Caption: Figure 3: Schematic diagram of the calcite solubility measurement apparatus.
Caption: Figure 4: Variation in the [Ca.sup.2+] concentration over time in the solubility measurement experiments.
Caption: Figure 5: Photographs of the four fracture surfaces. Column (a) shows the photographs before the CFDE experiments. Column (b) shows the photographs after the CFDE experiments. Column (c) shows the figures of 3D-laser scanning after the CFDE experiments. The red arrow indicates the direction of flow paths and dissolution channels.
Caption: Figure 6: Variation process of the outflow rates.
Caption: Figure 7: Variation process of the pressure difference.
Caption: Figure 8: Hydraulic apertures of the fractures over time during the CFDE experiments.
Caption: Figure 9: Dissolution apertures of the fractures over time during the CFDE experiments.
Caption: Figure 10: Variation in the [Ca.sup.2+] concentration over time. The dotted line data refers to the ICP instrument being rebooted due to the replacement of the instrument operation gas.
Caption: Figure 11: Variation in [Ca.sup.2+]/[Ca.sup.2+.sub.eq] over time. The dotted line data refers to the ICP instrument being rebooted due to the replacement of the instrument operation gas.
Caption: Figure 12: Change in the primary ionization constant of carbonic acid (log [k.sub.3]) with the temperature. ().
Caption: Figure 13: SEM photomicrographs of the bioclastic particles of [S.sub.1]. (a) Dissolved bioclastic particles. (b) Detailed view of the dissolved bioclastic particles. (c) More detailed view of the dissolved bioclastic particles. (d) EDS data of the triangle encircled part in (b). (e) EDS data of the triangle encircled part in (c).
Caption: Figure 14: SEM photomicrographs of the calcite matrix and the later filling matter of [S.sub.2]. (a) Preexperiment SEM photomicrographs of [S.sub.2]. (b) Detailed view of the later filling matter after the CFDE experiments. (c) Detailed view of the calcite particles after the CFDE experiments.
Caption: Figure 15: SEM photomicrographs of the calcite matrix and calcite vein of [S.sub.3]. (a) Preexperiment SEM photomicrographs of the calcite matrix. (b) Postexperiment SEM photomicrographs of the calcite matrix. (c) Preexperiment SEM photomicrographs of the calcite vein. (d) Postexperiment SEM photomicrographs of the calcite vein.
Caption: Figure 16: (a) Preexperiment SEM photomicrographs of the carbonate matrix of [S.sub.4]. (b) Postexperiment SEM photomicrographs of the calcite particle. (c) Postexperiment SEM photomicrographs of the nondissolved cement. (d) EDS data of the square encircled part in (a). (e) EDS data of the polygon encircled part in (b). (f) EDS data of the triangle encircled part in (c).
Table 1: Lithology, length, depth, and mineralogical composition of the carbonate samples. Sample Lithology Length (cm) Depth (m) [S.sub.1] Bioclastic limestone 8.120 285.75 [S.sub.2] Microcrystalline limestone 8.490 137.52 [S.sub.3] Microcrystalline limestone 6.010 302.22 [S.sub.4] Bioclastic limestone 6.130 106.80 Sample Calcite Quartz Ankerite Clay Pyrite [S.sub.1] 98.9% 0.7% 0.4% [S.sub.2] 92.8% 4% 2% 0.6% 0.6% [S.sub.3] 96.8% 2% 0.7% 0.5% [S.sub.4] 86.8% 4% 8% 0.5% 0.7% Table 2: Chemical composition of the carbonate samples. Major oxides (wt.%). [Fe.sub.2] Sample CaO MgO Si[O.sub.2] [O.sub.3] [S.sub.1] 54.05 0.408 1.47 0.407 [S.sub.2] 51.66 0.794 3.96 1.14 [S.sub.3] 54.3 0.472 1.13 0.345 [S.sub.4] 48.29 1.35 6.34 1.57 [Al.sub.2] Sample Ti[O.sub.2] [O.sub.3] MnO [K.sub.2]O [S.sub.1] 0.025 1.0 0.408 0.137 [S.sub.2] 0.029 0.816 0.087 0.066 [S.sub.3] 0.09 0.035 0.008 [S.sub.4] 0.016 0.599 0.153 0.021 [P.sub.2] Sample [O.sub.5] S LOI [S.sub.1] 0.019 0.059 42.01 [S.sub.2] 0.032 0.156 41.08 [S.sub.3] 0.042 0.04 43.40 [S.sub.4] 0.014 0.034 41.54 Table 3: The calcite solubility of the carbonate samples under the CFDE experimental conditions. Temperature Pressure [Ca.sup.2+] Sample ([degrees]C) (MPa) (mg/L) [S.sub.1] 60 8 684.341 [S.sub.2] 50 8 603.780 [S.sub.3] 40 8 822.667 [S.sub.4] 60 8 528.500 Table 4: The dissolution rate of the CFDE experiments. Confining pressure Sample Length (cm) Width (cm) (MPa) [S.sub.1] 8.12 6.00 20 [S.sub.2] 8.49 6.00 20 [S.sub.3] 6.01 6.00 20 [S.sub.4] 6.13 6.00 20 Micritic Crushed materials limestone Crushed materials Sparitic Crushed materials limestone Crushed materials Cement core 22.48 3.72 6343 V1 4.71 3.81 28.9 V2 4.88 3.81 28.8 Back pressure Temperature Sample (MPa) ([degrees]C) Solution [S.sub.1] 8 60 [S.sub.2] 8 50 C[O.sub.2]-rich water [S.sub.3] 8 40 [S.sub.4] 8 60 Micritic 5 [H.sub.2]S[O.sub.4] limestone 26 Sparitic 5 [H.sub.2]S[O.sub.4] limestone 26 Cement core 4137 25 C[O.sub.2]-rich brine V1 14.3 61 C[O.sub.2]-rich water V2 14.3 61 Average [P.sub.CO2] Flow rate [[Ca.sup.2+]] Sample (MPa) pH (ml/min) (mg/1) [S.sub.1] 8 0.2 303.16 [S.sub.2] 8 0.2 372.02 [S.sub.3] 8 0.2 239.30 [S.sub.4] 8 0.2 245.13 Micritic 4.7 1.8 limestone 4.7 1.8 Sparitic 4.7 1.8 limestone 4.7 1.8 Cement core 68.9 3.9 0.498 250 V1 12.5 0.23 1360 V2 12.5 0.1 1380 Average dissolution rate (mol/ Sample [m.sup.2]/s) [S.sub.1] 2.50 x [10.sup.-6] [S.sub.2] 3.11 x [10.sup.-6] [S.sub.3] 2.70 x [10.sup.-6] [S.sub.4] 3.04 x [10.sup.-6] Micritic 2.42 x [10.sup.-7] limestone 10.88 x [10.sup.-7] Sparitic 4.19 x [10.sup.-7] limestone 7.74 x [10.sup.-7] Cement core V1 V2 Note: The data on [S.sub.1], [S.sub.2], [S.sub.3], and [S.sub.4] come from this paper. The data on micritic limestone and sparitic limestone come from Kirstein et al. . The data on cement core come from Cao et al. . The data on V1 and V2 come from Elkhoury et al. . Table 5: Summary of the experimental samples and kinetic parameters (the value of [D.sub.m] is from You et al., ). Parameter (unit) Description [B.sub.h0] (m) Initial hydraulic aperture v (m/s) Average velocity [D.sub.m] ([m.sup.2]/s) Molecular diffusion coefficient [P.sub.e] Peclet number [D.sub.s] Damkohler number [S.sub.1], Parameter (unit) 60[degrees]C [B.sub.h0] (m) 2.53 x [10.sup.-6] v (m/s) 2.17 x [10.sup.-2] [D.sub.m] ([m.sup.2]/s) 5.75 x [10.sup.-10] [P.sub.e] 48 [D.sub.s] 2.97 x [10.sup.-7] [S.sub.2], Parameter (unit) 50[degrees]C [B.sub.h0] (m) 9.63 x [10.sup.-6] v (m/s) 5.35 x [10.sup.-3] [D.sub.m] ([m.sup.2]/s) 3.00 x [10.sup.-10] [P.sub.e] 74 [D.sub.s] 2.31 x [10.sup.-6] [S.sub.3], Parameter (unit) 40[degrees]C [B.sub.h0] (m) 6.86 x [10.sup.-6] v (m/s) 7.80 x [10.sup.-3] [D.sub.m] ([m.sup.2]/s) 1.50 x [10.sup.-10] [P.sub.e] 178 [D.sub.s] 3.34 x [10.sup.-6] [S.sub.4], Parameter (unit) 60[degrees]C [B.sub.h0] (m) 6.75 x [10.sup.-6] v (m/s) 7.98 x [10.sup.-3] [D.sub.m] ([m.sup.2]/s) 5.75 x [10.sup.-10] [P.sub.e] 47 [D.sub.s] 9.63 x [10.sup.-7]
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|Title Annotation:||Research Article|
|Author:||Wang, Jingxia; Yu, Qingchun|
|Date:||Jan 1, 2018|
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