# Using a Synthetic Borehole Model to Determine the Pore Aspect Ratio Dependence of Velocities from Acoustic Well-Logging Data

Authors: Xiao Feng; Tianyang Li; Bingrui Du; Wenxin Kong
Journal: Geofluids
 (2023)

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## Abstract

Revealing the elastic wave properties of carbonate rocks with complex pore structures and improving the reliability of carbonate reservoir descriptions have always been a global challenge in the field of carbonate geophysical exploration. In this study, we established a synthetic borehole model by selecting different particle sizes of cement, carbonate cuttings, and micro-silicon as the matrix, and silicon disks as the pores in carbonate rocks. We conducted four sets of low-porosity (0-3%) borehole models with different pore aspect ratios (ARs) and measured the P- and S-wave velocities (VP and VS) at the well-logging scale obtained from an acoustic logging system with one source and two receivers. The results indicate that the relationship between velocities and porosities in these borehole models follows a linear relation, with the pore AR significantly influencing the velocities at any given porosity. The velocity variation caused by pore AR reaches 560 m/s and 410 m/s at 3% porosity for the P-wave and S-wave within the AR range of 0.017-0.13. The theoretical DEM models provide a high and broad estimation of VP and VS at the well-logging scale in our measurement. They could perform better in fractured formation than in dissolved porous formation in carbonate reservoirs. The linear relation of VP and VS is independent of the pore AR and is effective for both fractured and dissolved porous formations. The change of VP/VS in different pore AR is more responsive to porosity and nonlinear dependent on the pore AR. The relationship between the defined normalized VP and VS indicates the pore AR has a more significant effect on VS than VP in our model. The constructed borehole models provide a unique opportunity for evaluating the availability of rock physics models at an acoustic logging scale. The study’s findings have significant implications for improving the reliability of carbonate reservoir descriptions and enhancing the accuracy of geophysical exploration in carbonate rocks with complex pore structures.


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## Body

## 1. Introduction

Accurate characterization of pore structure in unconventional oil and gas systems has proven to be challenging due to their complex morphology and the strong diagenesis and tectonism [1–3]. Currently, techniques such as acoustic emission experiments, acoustic logging, and seismic monitoring, which are based on elastic wave theory, have become important methods for detecting underground structures, oil and gas minerals, and geothermal resources. However, the changes in pore structure can significantly impact the elastic frame moduli of naturally occurring rocks, resulting in changes in their elastic wave velocities [4–9].The pore aspect ratio (AR; the ratio of the minor axis to the major axis) dependence of velocities of crustal rocks, affecting the energy exchange between the formation and pore fluids, has been widely reported [5, 10–13]. The power relationship between elastic wave velocities and pore AR has been proposed in the theoretical model (e.g., [14, 15]) and rock physics experiments at the core scale (e.g., [4, 8]). However, the frequency dependence of velocities can lead to errors and difficulties in the dynamic elastic parameters of rocks when attempting to apply the laboratory results in the ultrasonic frequency band (0.1 MHz-10 MHz) to the acoustic well-logging interpretation in the medium-frequency range (20 Hz-200 kHz) [16–19] and seismic exploration (~100 Hz) [20]. It has always been a research hotspot in the field of geophysics to establish the relationship between the elastic wave properties of rocks across frequency bands. Additionally, considering the complex well-logging environment, it is challenging and inaccurate to obtain detailed information about the pore structure beyond porosity using conventional well-logging data [21].Developing rock physics models with controlled pore structures is an urgent need to quantitatively investigate the effects of pore structures on wave velocity and attenuation (e.g., [4, 22–25]). The artificial sandstone cores are relatively mature and simple after decades of development. Usually, quartz sand with different grain sizes and cementing agents (such as Plexiglas and epoxy resin) are mixed to simulate the sandstone. Aluminum, copper, and tin foil are the common materials used to simulate the fractures, interparticle pores, and vugs in the formation [4, 26, 27]. However, for carbonate rocks, the existing methods (e.g., sheet combination method, cutting simulation method, interpolation method, and dissolution method) are more applicable for small-scale artificial cores for the following reasons: (1) the epoxy resin is too expensive to be widely used in well-logging scale models. (2) The consistency after adding epoxy resin is too high to ensure uniform mixing. (3) The final properties of rock physics models are influenced by the times and pressures in the compaction process, while it is difficult to control these parameters in the large-scale models. (4) ccurately controlling pore parameters and distribution in the large-scale models remains difficult with existing methods [28, 29].To simulate the acoustic well logging in the carbonate reservoir in the laboratory, we have developed a new borehole model with controlled pore structures and examined the influences of pore characteristics on the elastic properties via a homemade acoustic logging system [12]. In this study, we further focus on the pore AR dependence of velocities on a larger set of borehole models, including one blank model without pores and 20 models with different pore ARs and porosities. This work aims to provide a new dataset of measured P- and S-wave velocities (VP and VS) at the well-logging scale, which can serve as a preliminary reference for acoustic well-logging interpretation in the carbonate reservoir.


## 2. Methods

### 2.1. Borehole Model Building

Cement has been widely used in rock mechanics experiments for a long time [30, 31]. Both theoretical and experimental studies have shown that particle gradation determines the cement properties (e.g., density and strength) under the same mass ratio of cement and other matrix materials [32]. To consolidate the model matrix tightly and closely mimic the properties of the target formation, we choose cement slurry, carbonate cuttings, and microsilicon powder with different grain sizes as the solid phases for the borehole model. The carbonate cuttings are clean crystalline grains with a grain size of 150 μm, and the pure calcite content is over 98%. The microsilicon with 1 μm particle size is used to effectively prevent the sedimentation during the curing process. The microsilicon powder has a pure SiO2 content of over 97%.The particle size distribution of the above three solid-phase materials are measured by the Bettersize2000 laser particle size distribution instrument (Figure1). The average particle sizes of carbonate cuttings, cement, and microsilicon powder are 142.15 μm, 15.42 μm, and 0.76 μm, respectively. However, the addition of carbonate cuttings and microsilicon powder makes the cement slurry system less fluid and challenging to mix uniformly. The effect of different ratios of cement slurry and carbonate cuttings on the matrix fluidity is shown in Table 1. After considering the matrix density and fluidity of the cement slurry, the mass ratio of carbonate cuttings, cement, and water is determined as 1 : 1 : 0.4. The mass radio of microsilicon is 1% of the matrix material. In addition, graft polymers of sulfonated aldehyde-ketone condensation as the drag reducer and polymers of polyether and organic silicon as the defoamer are added to the matrix to discharge the bubbles formed during agitation. Finally, the required amount of materials for each model is list in Table 2.Figure 1
Particle size distribution of the three solid-phase materials.Table 1
The fluidity of the matrix with different ratios of cement slurry and carbonate cuttings.
SetsCement/kgWater/kgCarbonate cuttings/kgRatioFluidity15020751 : 1.5 : 0.4No fluidity2502062.51 : 1.25 : 0.4Low-medium fluidity35020501 : 1 : 0.4Medium fluidity4502037.51 : 0.75 : 0.4Medium-high fluidity55020251 : 0.5 : 0.4High fluidityTable 2
The total amount of matrix materials for each borehole model.
MaterialsMass/kgGrain size/μmCarbonate cuttings50142.15Cement5015.42Water20FluidMicrosilicon10.76Drag reducer0.2FluidDefoamer1FluidIn this study, “penny-shaped” silicone disks with different thicknesses and diameters, whose properties are comparable to fluids with relatively low density and propagation velocity, are distributed in the borehole model to simulate the different pore structures in carbonate rocks. A vertical hole with 76 mm width and 600 mm depth was drilled in the center of the model and filled with water to simulate the petroleum drilling procedure. Figure2 shows the construction process of the borehole model.Figure 2
(a) The material and building process of the borehole model. (b) A schematic physical model with the homemade acoustic logging system. (c) The distribution of the silicone disks. (d) The borehole models after coring in the center.The cores were left to cure for 28 days at room temperature (Figure3(a)) and then examined under a metallurgical microscope (Figures 3(b) and 3(c)). The shiny areas are the carbonate cutting, the dark areas are the cement hydration products, and the black areas are the primary pores resulting from the undischarged bubbles. The porosity and permeability were obtained from the gas measurement. The primary porosity is 24.36%, the permeability is low (<0.029 mD), and the density is 2.12 g/cm3. The density difference is less than 0.05 g/cm3, indicating that the matrix properties are more similar to the actual carbonate rocks than those of the simple cement system. Finally, we constructed four sets of low-porosity (0.6-3%) borehole models with varying pore AR of 0.017, 0.033, 0.067, and 0.13 (Table 3).Figure 3
The physical property (porosity, permeability, slip coefficient, density, and compressive strength) of the cores from the borehole models. (a) The diameter and length of the extracted cores are 76 mm and 600 mm. (b, c) The metallurgical map of the matrix.Table 3
Model sets and the corresponding parameters.
SetsModelsφhdARN(%)(mm)(mm)AA#0.60.60.5300.01712×24A#1.21.212×48A#1.81.824×36A#2.42.424×48A#3.0324×60BB#0.60.610.0336×24B#1.21.212×24B#1.81.812×36B#2.42.412×48B#3.0312×60CC#0.60.620.0676×12C#1.21.212×12C#1.81.812×18C#2.42.412×24C#3.0312×30DD#0.60.640.136×6D#1.21.26×12D#1.81.86×18D#2.42.46×24D#3.036×30Note:φ is the porosity of the borehole model; d and h are the diameter and thickness of the silicon disks, respectively; N is the total number of silicon disks in each model.


### 2.2. Velocity Measurements

To simulate the acoustic well-logging in the laboratory, we cooperated with Yangzhou Oriental Ultrasound Technology Co. Ltd. to design an acoustic logging system with one ring-emitting piezoelectric source and two piezoelectric receivers (as shown in Figure4). The dominant frequency of the piezoelectric source is 20-40 kHz. This equipment is placed in the central borehole filled with water. The source-receiver distances are 22.2 cm and 36.5 cm for the two receivers. The typical waveforms received by the receivers are shown in Figure 5. The velocities in the borehole model are calculated from the differential arrival times of the P- and S-waves between the two receivers:
(1)Vi=Δlt2i−t1i,where Vi is the velocity of ith phases (P-, S-, and Stoneley waves) in km/s, Δl is the length between the two receivers in mm, while t1i and t2i are the picked ith arrival times of all phases waves of the two received signals in μs. The main source of errors in the velocity measurement is associated with the picking of travel times from the acquired raw waveforms.Figure 4
The homemade acoustic logging system with one source and two receivers.Figure 5
The received waveforms from one borehole model. The differential arrival times of P- and S-waves between the two receivers (R1 and R2 in Figure4) are used to calculate the wave velocities.


## 2.1. Borehole Model Building

Cement has been widely used in rock mechanics experiments for a long time [30, 31]. Both theoretical and experimental studies have shown that particle gradation determines the cement properties (e.g., density and strength) under the same mass ratio of cement and other matrix materials [32]. To consolidate the model matrix tightly and closely mimic the properties of the target formation, we choose cement slurry, carbonate cuttings, and microsilicon powder with different grain sizes as the solid phases for the borehole model. The carbonate cuttings are clean crystalline grains with a grain size of 150 μm, and the pure calcite content is over 98%. The microsilicon with 1 μm particle size is used to effectively prevent the sedimentation during the curing process. The microsilicon powder has a pure SiO2 content of over 97%.The particle size distribution of the above three solid-phase materials are measured by the Bettersize2000 laser particle size distribution instrument (Figure1). The average particle sizes of carbonate cuttings, cement, and microsilicon powder are 142.15 μm, 15.42 μm, and 0.76 μm, respectively. However, the addition of carbonate cuttings and microsilicon powder makes the cement slurry system less fluid and challenging to mix uniformly. The effect of different ratios of cement slurry and carbonate cuttings on the matrix fluidity is shown in Table 1. After considering the matrix density and fluidity of the cement slurry, the mass ratio of carbonate cuttings, cement, and water is determined as 1 : 1 : 0.4. The mass radio of microsilicon is 1% of the matrix material. In addition, graft polymers of sulfonated aldehyde-ketone condensation as the drag reducer and polymers of polyether and organic silicon as the defoamer are added to the matrix to discharge the bubbles formed during agitation. Finally, the required amount of materials for each model is list in Table 2.Figure 1
Particle size distribution of the three solid-phase materials.Table 1
The fluidity of the matrix with different ratios of cement slurry and carbonate cuttings.
SetsCement/kgWater/kgCarbonate cuttings/kgRatioFluidity15020751 : 1.5 : 0.4No fluidity2502062.51 : 1.25 : 0.4Low-medium fluidity35020501 : 1 : 0.4Medium fluidity4502037.51 : 0.75 : 0.4Medium-high fluidity55020251 : 0.5 : 0.4High fluidityTable 2
The total amount of matrix materials for each borehole model.
MaterialsMass/kgGrain size/μmCarbonate cuttings50142.15Cement5015.42Water20FluidMicrosilicon10.76Drag reducer0.2FluidDefoamer1FluidIn this study, “penny-shaped” silicone disks with different thicknesses and diameters, whose properties are comparable to fluids with relatively low density and propagation velocity, are distributed in the borehole model to simulate the different pore structures in carbonate rocks. A vertical hole with 76 mm width and 600 mm depth was drilled in the center of the model and filled with water to simulate the petroleum drilling procedure. Figure2 shows the construction process of the borehole model.Figure 2
(a) The material and building process of the borehole model. (b) A schematic physical model with the homemade acoustic logging system. (c) The distribution of the silicone disks. (d) The borehole models after coring in the center.The cores were left to cure for 28 days at room temperature (Figure3(a)) and then examined under a metallurgical microscope (Figures 3(b) and 3(c)). The shiny areas are the carbonate cutting, the dark areas are the cement hydration products, and the black areas are the primary pores resulting from the undischarged bubbles. The porosity and permeability were obtained from the gas measurement. The primary porosity is 24.36%, the permeability is low (<0.029 mD), and the density is 2.12 g/cm3. The density difference is less than 0.05 g/cm3, indicating that the matrix properties are more similar to the actual carbonate rocks than those of the simple cement system. Finally, we constructed four sets of low-porosity (0.6-3%) borehole models with varying pore AR of 0.017, 0.033, 0.067, and 0.13 (Table 3).Figure 3
The physical property (porosity, permeability, slip coefficient, density, and compressive strength) of the cores from the borehole models. (a) The diameter and length of the extracted cores are 76 mm and 600 mm. (b, c) The metallurgical map of the matrix.Table 3
Model sets and the corresponding parameters.
SetsModelsφhdARN(%)(mm)(mm)AA#0.60.60.5300.01712×24A#1.21.212×48A#1.81.824×36A#2.42.424×48A#3.0324×60BB#0.60.610.0336×24B#1.21.212×24B#1.81.812×36B#2.42.412×48B#3.0312×60CC#0.60.620.0676×12C#1.21.212×12C#1.81.812×18C#2.42.412×24C#3.0312×30DD#0.60.640.136×6D#1.21.26×12D#1.81.86×18D#2.42.46×24D#3.036×30Note:φ is the porosity of the borehole model; d and h are the diameter and thickness of the silicon disks, respectively; N is the total number of silicon disks in each model.


## 2.2. Velocity Measurements

To simulate the acoustic well-logging in the laboratory, we cooperated with Yangzhou Oriental Ultrasound Technology Co. Ltd. to design an acoustic logging system with one ring-emitting piezoelectric source and two piezoelectric receivers (as shown in Figure4). The dominant frequency of the piezoelectric source is 20-40 kHz. This equipment is placed in the central borehole filled with water. The source-receiver distances are 22.2 cm and 36.5 cm for the two receivers. The typical waveforms received by the receivers are shown in Figure 5. The velocities in the borehole model are calculated from the differential arrival times of the P- and S-waves between the two receivers:
(1)Vi=Δlt2i−t1i,where Vi is the velocity of ith phases (P-, S-, and Stoneley waves) in km/s, Δl is the length between the two receivers in mm, while t1i and t2i are the picked ith arrival times of all phases waves of the two received signals in μs. The main source of errors in the velocity measurement is associated with the picking of travel times from the acquired raw waveforms.Figure 4
The homemade acoustic logging system with one source and two receivers.Figure 5
The received waveforms from one borehole model. The differential arrival times of P- and S-waves between the two receivers (R1 and R2 in Figure4) are used to calculate the wave velocities.


## 3. Results

Porosity is the key parameter for characterizing materials microstructures and is the main output from well-log interpretation [33]. A broadly accepted consensus is that the complex pore structures cause the wide scatter in the velocity-porosity relationship for the carbonate rocks [4, 8, 13]. The measurements from our models show that both the VP and VS decrease with the increase in porosity at the same pore AR (Figure 6). The relationship can be well-described by a simple negative linear function, which is similar to the results from rock physics experiments [34, 35].Figure 6
TheVP (solid points) and VS (hollow points) of the four model sets. The dashed lines are fitted by linear functions for the different porosities. The solid lines are obtained using DEM concepts with pore AR of 0.017, 0.033, 0.067, and 0.13.In addition, pore AR has a remarkable effect on elastic wave propagation [36–38], affecting the energy exchange between the formation and pore fluids. However, recent researches only focus on building quantitative trends between the velocity and pore type at the core scale [8]. We built four sets of borehole models with different pore AR of 0.017, 0.033, 0.067, and 0.13. The velocity variation caused by pore AR is as high as 560 m/s and 410 m/s at 3% porosity for the P-wave and S-wave within the AR range of 0.017-0.13 in our experiments. It should be noted that more silicon disks are required to achieve the same porosity when pore AR is low in each model set.The ratio of microstructure size (d) to wavelength (λ) affects velocity in the porous media. In the past decades, several studies have tested effective medium theories (EMT) in the long-wavelength limit situation (λ>d), seeking to relate the rock minerals and pore structures to the effective elastic properties on a macroscopic scale by assuming different pore ARs [35, 39–41]. In this study, the dominant frequency of the acoustic logging system is 20-40 kHz, and the wavelength is 50-100 mm. The size of the silicon disks is 30 mm; thus, our model is within the scope of the long-wavelength range. Therefore, we analyze the pore AR dependence of velocities in the following sections and compare it with those of EMT models to gain further insight into the phenomena that control elastic wave propagation in the borehole models. The previous study has indicated that the differential effective medium (DEM) model [42] shows closer outcomes to our results in the borehole models [12]. Therefore, we use the DEM model to calculate the theoretical velocities of the borehole model with different pore ARs. The values of the input parameters in the DEM model are summarized in Table 4. Other parameters used in these theoretical models are the same as those of the borehole model. The results show that the theoretical DEM models provide a high estimation of the VP and VS, and the deviations become larger as pore AR increases. It is worth mentioning that relatively small variations of wave velocities are observed at the well-logging scale. Therefore, the theoretical model is more suitable for fractured formation than the dissolved porous formation in carbonate reservoirs.Table 4
Parameters used for DEM model calculation.
MaterialsDensity (g/cm3)VP (m/s)VS (m/s)Matrix2.4141902410Silicon disk1.241326848


## 4. Discussion

### 4.1.VP−VS Linear Relation

The relation between theVP and VS for sandstone is usually constant. Our newly developed carbonate borehole models follow a linear VP−VS relation (Figure 7), which has been verified from the test of carbonate samples [43]. The high determinant coefficient (R2 value) indicates that the linear relation of VP and VS is independent of the pore AR.Figure 7
The linear relations ofVP and VS measured from the four model sets.Wang et al. [24] introduced a general velocity model for carbonate rocks determined by core measurements:
(2)VP=VP0−a1φp−a2φsARc1φs+c2,VS=VS0−b1φp−b2φsARd1φs+d2,where a1, a2, b1, b2, c1, c2, d1, and d2 are empirical parameters; φp represents the primary porosity; φs represents the secondary porosity; VP0 and VS0 represent the P- and S-wave velocities of the solid matrix, respectively.Then, Equation (1) can be further rewritten as follows:
(3)VS=VS0−b1φp−b2φsa2φs/VP0−a1φp−VPgφs,(4)gφs=d1φs+d2c1φs+c2.Based on the measured data, the fitting parametersc1 and d1 range from -1 to 0; c2 and d2 range from 0.5 to 1. In our study, the secondary porosity is low (<3%). Therefore, the gφs, related to the secondary porosity, is close to 1. Then, Equation (2) could be simplified as follows:
(5)VS=VS0−b2a2VP0+a1b2a2−b1φp+b2a2VP.TheVP0, VS0, and φp are related to the solid matrix and constant in the same model. In addition, the fitting parameters in Equation (4) (i.e., a1, a2, b1, and b2) are only related to porosity and independent of the pore AR, which means that the linear relation between VP and VS is effective for different pore AR conditions. Combined with the results of the same linear relation between VP and VS in our borehole models, the method for VS prediction from well-logging data can be used for carbonate formations with fractures, intraparticle pores, or vugs.


### 4.2.VP−VS Ratio

Seismic tomographic results have revealed the importance ofVP/VS as a diagnostic of the physical cause of a seismic anomaly [44, 45]. Although geophysicists have widely recognized the importance of liquid compressibility in changing VP/VS [46], the effect of pore geometry on VP/VS has not been commonly acknowledged. In sandstones, the pore structure is simple and VP/VS is relatively stable. According to classical elastic theory, VP/VS is approximately equal to 1.732 when the Poisson’s ratio is 0.25. However, caution is needed when applying the suitability of this relation in carbonate rocks due to the variability of pore structures. To quantitatively describe the VP/VS variation with pore AR, we calculate the change in VP/VS as
(6)ΔVPVS=VPVSAR=high−VPVSAR=low.As shown in Figure8, the ΔVP/VS is always lower than 0 and has a positive response to porosity, while the influence of pore AR appears to be nonlinear. Due to the low number of models, the accurate relation between pore AR and the ΔVP/VS cannot be proposed in this study. Therefore, we observe the same negative ΔVP/VS trend from the DEM modelling. The maximum change in VP/VS appears at model set C (i.e., AR=0.067). Overall, the ΔVP/VS is more responsive to porosity and nonlinear dependent on the pore AR, which indicates a complex effect of pore AR on VP/VS in carbonate rocks.Figure 8
The variation inVP/VS with different pore ARs.


### 4.3. Normalized Velocity

The pore AR influencesVP and VS differently based on the above analysis. Therefore, we compare the velocities within the same pore AR window. The lower and upper pore AR bounds for the velocities analysis are 0.017 and 0.13, and the normalized velocities (Vpn and Vsn), which quantitatively represent the pore AR dependence of velocities, are defined as
(7)Vpn=VPAR=0.13VPAR=0.017,Vsn=VSAR=0.13VSAR=0.017.TheVP and VS usually increase with pore AR and follow a power law; thus, the Vpn and Vsn are higher than 1 (Figure 9). The Vpn and Vsn both increase by the porosity. The relationship between Vpn and Vsn follows a linear function where the R2 value is as high as 0.97. Vsn is always higher than Vpn with a slope of 1.326, which means the pore AR has a greater effect on VS than VP in our model. Interestingly, the calculation from the DEM model shows an astonishing similarity with the trends from our results. Overall, the results suggest that the observed Vpn−φ and Vsn−φ relationship for carbonate rocks is influenced by the pore structure.Figure 9
The normalizedVP and VS in different porosity models. The dashed lines indicate the slopes. The coloured line is the prediction based on the DEM modelling.Like any artificial core experiment, the results obtained from our on ideal borehole models have their share of pitfalls in view of the isolated “penny-shaped” pores. First, the P-wave velocity and density of silicon disks are similar to those of water, while theVS of this material is in low comparability to fluid typically encountered in crustal materials. Second, our borehole model ignores the interaction between the solid matrix and the pores/pore fluids, which is also crucial for their elastic wave velocities [47, 48]. However, in this study, we only focus on the effect of pore structures on velocities. The synthetic borehole models are simplified but similar to the actual case, and the elasticity contrasts between the matrix and pore fluids. Nevertheless, our borehole model is suitable for various petrophysical experiments such as acoustic wave characteristics, electrical characteristics, and seepage mechanisms. The results from our borehole model could be a stepping stone for understanding the velocity variation at lower frequencies compared to the conventional rock physics model.


## 4.1.VP−VS Linear Relation

The relation between theVP and VS for sandstone is usually constant. Our newly developed carbonate borehole models follow a linear VP−VS relation (Figure 7), which has been verified from the test of carbonate samples [43]. The high determinant coefficient (R2 value) indicates that the linear relation of VP and VS is independent of the pore AR.Figure 7
The linear relations ofVP and VS measured from the four model sets.Wang et al. [24] introduced a general velocity model for carbonate rocks determined by core measurements:
(2)VP=VP0−a1φp−a2φsARc1φs+c2,VS=VS0−b1φp−b2φsARd1φs+d2,where a1, a2, b1, b2, c1, c2, d1, and d2 are empirical parameters; φp represents the primary porosity; φs represents the secondary porosity; VP0 and VS0 represent the P- and S-wave velocities of the solid matrix, respectively.Then, Equation (1) can be further rewritten as follows:
(3)VS=VS0−b1φp−b2φsa2φs/VP0−a1φp−VPgφs,(4)gφs=d1φs+d2c1φs+c2.Based on the measured data, the fitting parametersc1 and d1 range from -1 to 0; c2 and d2 range from 0.5 to 1. In our study, the secondary porosity is low (<3%). Therefore, the gφs, related to the secondary porosity, is close to 1. Then, Equation (2) could be simplified as follows:
(5)VS=VS0−b2a2VP0+a1b2a2−b1φp+b2a2VP.TheVP0, VS0, and φp are related to the solid matrix and constant in the same model. In addition, the fitting parameters in Equation (4) (i.e., a1, a2, b1, and b2) are only related to porosity and independent of the pore AR, which means that the linear relation between VP and VS is effective for different pore AR conditions. Combined with the results of the same linear relation between VP and VS in our borehole models, the method for VS prediction from well-logging data can be used for carbonate formations with fractures, intraparticle pores, or vugs.


## 4.2.VP−VS Ratio

Seismic tomographic results have revealed the importance ofVP/VS as a diagnostic of the physical cause of a seismic anomaly [44, 45]. Although geophysicists have widely recognized the importance of liquid compressibility in changing VP/VS [46], the effect of pore geometry on VP/VS has not been commonly acknowledged. In sandstones, the pore structure is simple and VP/VS is relatively stable. According to classical elastic theory, VP/VS is approximately equal to 1.732 when the Poisson’s ratio is 0.25. However, caution is needed when applying the suitability of this relation in carbonate rocks due to the variability of pore structures. To quantitatively describe the VP/VS variation with pore AR, we calculate the change in VP/VS as
(6)ΔVPVS=VPVSAR=high−VPVSAR=low.As shown in Figure8, the ΔVP/VS is always lower than 0 and has a positive response to porosity, while the influence of pore AR appears to be nonlinear. Due to the low number of models, the accurate relation between pore AR and the ΔVP/VS cannot be proposed in this study. Therefore, we observe the same negative ΔVP/VS trend from the DEM modelling. The maximum change in VP/VS appears at model set C (i.e., AR=0.067). Overall, the ΔVP/VS is more responsive to porosity and nonlinear dependent on the pore AR, which indicates a complex effect of pore AR on VP/VS in carbonate rocks.Figure 8
The variation inVP/VS with different pore ARs.


## 4.3. Normalized Velocity

The pore AR influencesVP and VS differently based on the above analysis. Therefore, we compare the velocities within the same pore AR window. The lower and upper pore AR bounds for the velocities analysis are 0.017 and 0.13, and the normalized velocities (Vpn and Vsn), which quantitatively represent the pore AR dependence of velocities, are defined as
(7)Vpn=VPAR=0.13VPAR=0.017,Vsn=VSAR=0.13VSAR=0.017.TheVP and VS usually increase with pore AR and follow a power law; thus, the Vpn and Vsn are higher than 1 (Figure 9). The Vpn and Vsn both increase by the porosity. The relationship between Vpn and Vsn follows a linear function where the R2 value is as high as 0.97. Vsn is always higher than Vpn with a slope of 1.326, which means the pore AR has a greater effect on VS than VP in our model. Interestingly, the calculation from the DEM model shows an astonishing similarity with the trends from our results. Overall, the results suggest that the observed Vpn−φ and Vsn−φ relationship for carbonate rocks is influenced by the pore structure.Figure 9
The normalizedVP and VS in different porosity models. The dashed lines indicate the slopes. The coloured line is the prediction based on the DEM modelling.Like any artificial core experiment, the results obtained from our on ideal borehole models have their share of pitfalls in view of the isolated “penny-shaped” pores. First, the P-wave velocity and density of silicon disks are similar to those of water, while theVS of this material is in low comparability to fluid typically encountered in crustal materials. Second, our borehole model ignores the interaction between the solid matrix and the pores/pore fluids, which is also crucial for their elastic wave velocities [47, 48]. However, in this study, we only focus on the effect of pore structures on velocities. The synthetic borehole models are simplified but similar to the actual case, and the elasticity contrasts between the matrix and pore fluids. Nevertheless, our borehole model is suitable for various petrophysical experiments such as acoustic wave characteristics, electrical characteristics, and seepage mechanisms. The results from our borehole model could be a stepping stone for understanding the velocity variation at lower frequencies compared to the conventional rock physics model.


## 5. Conclusions

By choosing different particle sizes of cement, carbonate cuttings, and microsilicon as the matrix and silicon disks with different AR as the pores in carbonate rocks, we built four sets of low-porosity borehole models and provided a new dataset of theVP and VS at lower frequencies obtained from an acoustic logging system with one source and two receivers. This study focuses on the pore AR dependence of velocities and reaches the following key conclusions:
(1)
The relation between velocities and porosities measured at the well-logging scale follows a linear function. Meanwhile, the velocity variation caused by pore AR can reach 560 m/s and 410 m/s at 3% porosity for the P-wave and S-wave within the AR range of 0.017-0.13 in our experiments(2)
The theoretical DEM models overestimate theVP and VS. The deviations between the theoretical velocities at the rock physics scale and the measured results at the well-logging scale gradually increase with AR increase, especially when AR is over 0.05, indicating better performance in fractured formations than dissolved porous formations in carbonate reservoirs(3)
The high determinant coefficients indicate that the linear relation ofVP and VS is independent of pore AR and is effective for both fractured or dissolved porous formation. The change of VP/VS in different pore AR is more responsive to porosity and nonlinear dependent on the pore AR. The relationship between the defined normalized VP and VS indicates that the pore AR has a greater effect on VS than VP



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*Source: 5884572-2023-03-20.xml*