**1. Introduction**

The leading health and social problem in many countries around the world is the coronavirus (COVID-19) pandemic. Up to 6 August 2022, a total of 588,081,229 cases of disease were recorded in various regions of the world, including 6,434,821 deaths and 559,178,151 recoveries. In Poland, the reach of the epidemic is 6,093,571 people, of which 116,660 patients have died and 5,335,862 have already been cured [1].

Empirical evidence from different countries affected by the pandemic suggests that symptoms of anxiety and depression (16.0–28.0%) and sleep disorders (38.9%), which are associated with impaired cognitive dynamics and cognitive deficits, including attention and perceptiveness [2,3], are relatively frequent psychological reactions to the current crisis situation, with higher rates of symptoms of mental health disorders found in women than in men [4–6]. On the other hand, both the occurrence of the above-mentioned symptoms and the consequences of life limitations during the pandemic may disrupt the social life of many people, and thus lower the quality of their relationships with family members and friends and professional interactions [6,7].

**Citation:** Mazur, A.; Barto ´n, E. The Structure of the Relationship between Physical Activity and Psychosocial Functioning of Women and Men during the COVID-19 Epidemic in Poland. *IJERPH* **2022**, *19*, 11860. https://

doi.org/10.3390/ijerph191911860

Academic Editor: Andrzej Tomczak

Received: 17 August 2022 Accepted: 16 September 2022 Published: 20 September 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

When attempting to understand the nature of the aforementioned crisis, it is important to refer to the leading assumptions of the biopsychosocial model dominating in modern health sciences, which can explain intrapsychic reactions of people to the pandemic as a result of the interaction of a network of equivalent biological, psychological and social factors shaping health [8–10].

Taking into account the biological factors included in the considered model, it should be emphasized that empirical evidence from various countries affected by the pandemic indicates that gender moderates the resistance to stressors and has a significant influence on psychological variables important in coronavirus (SARS-CoV-2) transmission, specifically mental health and cognitive processes [8–12]. On the basis of a meta-analysis, it was proven that women exhibit a higher intensity of symptoms characteristic of depressive syndrome and anxiety disorders than men associated with the presence of attention and perceptiveness deficits, which may indicate that it is more difficult for them to cope with the current situation [13–17].

It was also observed that the mental condition and the state of cognitive processes during the current crisis may be of significant importance for social functioning, including family, interpersonal and professional interactions [18]. It is postulated that the quality of these relationships is directly related to the received social support, acting as a factor protecting against the negative consequences of experienced stress [19].

It has been observed that during the transmission of SARS-CoV-2, women report a greater sense of loneliness than men [15,17]. Moreover, regardless of gender, in many countries, including Poland, Spain, Australia, the United States, Sweden and China, due to social isolation, quarantine, working from home and other pandemic restrictions, the number of social interactions decreased [15,17,20–22]. This at the same time reduces the likelihood of receiving support, and thus increases the risk of developing unwanted psychological symptoms, including depression, anxiety disorders and accompanying problems in cognitive functioning [23].

Bearing in mind that, according to the adopted approach, a human is a living system, constituting an indivisible whole, composed of biological, psychological and social dimensions, which, under the influence of incoming information, or, in other words, the requirements of the environment, remain in a dynamic balance, it is extremely important to pay attention to factors supporting its remedial efficiency, and their consequences on the general health condition [9,10].

One of the key forms of preventive action recommended by the World Health Organization (WHO) is regular physical activity, defined as any type of body movement initiated by working muscles that causes energy expenditure. It includes walking, moderate and vigorous activity, which, at the same time, strongly distinguishes it from other forms of leisure activities, including time spent on passive leisure such as sitting and lying down [24,25].

The available literature shows that physical activity understood this way reduces the risk of developing undesirable mental symptoms as well as disorders in cognitive functioning, because by increasing the metabolism of kynurenine and the expression of kynurenine aminotransferase in skeletal muscles, it minimizes the consequences of environmental stress, and thus prevents the appearance or reduces the severity of symptoms characteristic to neuropsychiatric disorders [26].

It has been proven that systematic physical exertion delays neurodegenerative processes and regulates monoamine metabolism, as well as neuroimmune functioning [27]. The physiological changes taking place under the influence of training support the proper functioning of the hypothalamic–pituitary–adrenal axis (HPA), the excessive activation of which, caused by increased and sustained release of cortisol, is associated with the risk of developing many disorders [28–30]. In addition, physical activity by stimulating the expression of the brain nerve growth factor (BDNF), as well as supporting the proper functioning of the Trk-B receptor and BDNF-Trk-B signaling, determines the neurophysiological processes taking place, which in turn has a beneficial effect on affective, cognitive and social functioning [31,32].

The vast majority of studies indicate that men have higher levels of physical activity than women, which may be due to different uses of the time available in the course of a day [33–36]. A nationwide study reported that, on average, men devote one hour and thirty-three minutes per day to sports, while women, on average, perform physical exercises for one hour and nineteen minutes per day. Moreover, men's workouts tend to be characterized by higher intensity than those performed by women. As a result, men's higher physical activity, to a greater extent than women's, may be a factor in protecting their mental fitness [35,36].

It is worth mentioning, however, that the results of studies conducted in different regions of Poland are not consistent. For example, in the population of residents of northeastern Poland, it has been shown that, although moderate physical activity is undertaken more often by men (53.9%) than by women (34.7%), and that men, compared to the opposite sex, are almost twice as likely to ride a bicycle (31.5% vs. 13.1%), but at the same time it has been observed that, regardless of gender, the physical activity of residents of this region is insufficient and differs significantly from levels recorded in other parts of the country, as well as in other European countries [33].

Results of a cross-sectional study conducted in a random sample of 6000 people between the ages of 40 and 64 in Japan are also of interest. They came to slightly different conclusions than the empirical works cited above. They observed that women, on average, spent 12.6% less time on sedentary lifestyles and 23.4% more time on low-intensity physical activity than men, but found no significant differences for moderate and vigorous physical activity [37].

This suggests that the conclusions of the empirical works published to date are inconclusive. The reported results vary depending on the region of Poland, as well as the part of world in which they're conducted [33–37], which at the same time suggests that they appear to be worth supplementing. The results of our own research will therefore allow us to determine whether, and possibly to what extent, the observations of other researchers reflect the situation of the people of Eastern Poland. Considering that there are no studies aimed at determining the level of physical activity, mental health, cognitive processes and quality of social functioning, as well as the importance of interactions between these variables in the population of women and men exposed to stress resulting from the COVID-19 epidemic in Eastern Poland, the data obtained can provide both valuable scientific observation and be used to develop psychoprophylaxis interventions.

Hence, the aim of this own research was to determine the level of physical activity, mental health, cognitive processes and quality of social functioning, as well as the structure of the relationship between physical activity and psychosocial functioning of women and men during the COVID-19 epidemic in Eastern Poland.

Two research questions were posed accordingly: (1) what is the level of physical activity, mental health disorders, cognitive disorders and social functioning, as well as (2) what is the significance of the interactions between physical activity, mental health and cognitive disorders for the social functioning of men and women during the COVID-19 pandemic in Poland?

In response, two hypotheses were formulated, assuming that during the COVID-19 epidemic in Poland, depending on gender, the following varies: (1) the level of physical activity, the intensity of symptoms characteristic of mental health disorders, cognitive disorders and the quality of social functioning as well as (2) the importance of the interaction between physical activity, mental health and cognitive disorders for social functioning.

#### **2. Material and Methods**


**2. Materials and methods**

The study included 188 patients operated for postinfarction anterior septal and apical LVA. The disease was diagnosed basing on the data of echocardiography (EchoCG), coronary ventriculography (CVG) and contrast-enhanced MRI.

Prior to the surgery the patients mostly had III-IV CC angina class, their condition corre‐ sponded to that of New York Heart Association (NYHA) class II-III for chronic heart failure. All the patients demonstrated evidence of postinfarction LV remodeling according to the re‐ sults of ventriculography and EchoCG.

Cardiac inversion recovery and T-1 spin-echo weighed MRI study with ECG synchronization was performed with a patient lying flat with no additional functional stress. Axial slices on the level of thorax with the complete coverage of heart area were recorded. The field of view was 350-380 mm wide and 7-8mm thick slices were recorded into the matrix of 256x256 voxels. Syn‐ chronization of the recordings of MRI pictures with ECG was performed by standard means of an open PRI scanner Magnetom-Open (0,2 T by Siemens Medical) or high field open MRI scan‐ ner Vantage Titan (1,5 Т, by Toshiba) by R-wave of ECG; end-diastolic images were acquired in all the cases. Parameters of the acquired T1-weighed images in spin-echo mode were as fol‐ lows: repetition time (TR) 550 – 1040ms, echo time (TE) – 20 ms. MRI included slices with long axis two-chamber and four-chamber views as well as short axis view covering all the myocar‐ dial volume of LV. The study was performed in 12-20 minutes after injection of paramagnetic contrast agents with the concentration of paramagnetic agent itself of 0,5М (Omniscan, Mag‐ nevist, Optimark, Cyclomang, Viewgam) in dosage of 2ml/10kg of a body weight. The shortaxis and long-axis slices in four-chamber view were divided semi-automatically into 17 segments taking into account generally accepted segmentation of LV myocardium (Fig.1) [7].

**Figure 1.** Segmentation of LV myocardium, used in evaluation of a local paramagnetic contrast agent uptake during

Diagnostics and Surgical Treatment of Left Ventricular Aneurysm with Ventricular Tachycardia

http://dx.doi.org/10.5772/54126

323

**Figure 2.** The scheme of TI calculation by the data of delayed contrast-enhanced MRI of myocardium. The values of thickness of the infarction zone (accumulation of paramagnetic) and the value of the total myocardial thickness were identified on the LV slices in short axis. Thickness of the contrast-paramagnetic accumulation in myocardium is thought to be the thickness of lesion resulting from acute myocardial infarction. Thus TI is equal to *{(thickness of an*

*infracted area)/ (thickness of the myocardium)}.* (original data)

myocardial MRI

In particular, for each segment i (i = 1 – 17) we calculated the depth of damage by the degree of paramagnetic contrast agent uptake as follows: [Index of Transmurality]i = (maximum thicknesses of paramagnetic contrast agent uptake)i / (thickness of myocardium in a particu‐ lar segment) i (Figure 2). (original data)

To see the association between MR images with the data of electrophysiological condition of heart muscle, and in particular with the location of the areas with lowered voltage of local electrical activity, the patients underwent electrophysiological study of the heart (EPhS) with electroanatomical CARTO reconstruction of LV [20]. Besides, there were identified the areas of delayed conduction, zones of possible re-entry and inducible VT (Figure 3).

Locations of intracardiac leads and thus of segmental electrical activity corresponded to the lo‐ cations of left ventricular segments during contrast-enhanced MRI. In accordance with the am‐ plitude of the curve during EPhS for a definite myocardial segment, degree of potential reduction was graded as follows: 0 – for the amplitude of the potential from 1,5 to 8 mV, when the segment was considered to be a zone of normal potential; 1 — for the amplitude of 0,5 — 1,5 mV, a transient zone; 2 — for the amplitude 0,05 — 0,5 mV, low potential zone; 3 — zone of «electrical scar» - lack of electrical activity when the amplitude was 0 — 0,05 mV. This grading was precon‐ ditioned by the fact that myocardial areas with the 1st or 2nd grades of lowered potential were, as a rule, sources of life threatening tachycardias, while for grade 3 "electrical scar" this was less pos‐ sible and in the zones with normal electrical activity of grade 0 VTs did not occur [9,10].

Diagnostics and Surgical Treatment of Left Ventricular Aneurysm with Ventricular Tachycardia http://dx.doi.org/10.5772/54126 323

**2. Materials and methods**

322 Principles and Practice of Cardiothoracic Surgery

sults of ventriculography and EchoCG.

lar segment) i (Figure 2). (original data)

ventriculography (CVG) and contrast-enhanced MRI.

The study included 188 patients operated for postinfarction anterior septal and apical LVA. The disease was diagnosed basing on the data of echocardiography (EchoCG), coronary

Prior to the surgery the patients mostly had III-IV CC angina class, their condition corre‐ sponded to that of New York Heart Association (NYHA) class II-III for chronic heart failure. All the patients demonstrated evidence of postinfarction LV remodeling according to the re‐

Cardiac inversion recovery and T-1 spin-echo weighed MRI study with ECG synchronization was performed with a patient lying flat with no additional functional stress. Axial slices on the level of thorax with the complete coverage of heart area were recorded. The field of view was 350-380 mm wide and 7-8mm thick slices were recorded into the matrix of 256x256 voxels. Syn‐ chronization of the recordings of MRI pictures with ECG was performed by standard means of an open PRI scanner Magnetom-Open (0,2 T by Siemens Medical) or high field open MRI scan‐ ner Vantage Titan (1,5 Т, by Toshiba) by R-wave of ECG; end-diastolic images were acquired in all the cases. Parameters of the acquired T1-weighed images in spin-echo mode were as fol‐ lows: repetition time (TR) 550 – 1040ms, echo time (TE) – 20 ms. MRI included slices with long axis two-chamber and four-chamber views as well as short axis view covering all the myocar‐ dial volume of LV. The study was performed in 12-20 minutes after injection of paramagnetic contrast agents with the concentration of paramagnetic agent itself of 0,5М (Omniscan, Mag‐ nevist, Optimark, Cyclomang, Viewgam) in dosage of 2ml/10kg of a body weight. The shortaxis and long-axis slices in four-chamber view were divided semi-automatically into 17 segments taking into account generally accepted segmentation of LV myocardium (Fig.1) [7]. In particular, for each segment i (i = 1 – 17) we calculated the depth of damage by the degree

of paramagnetic contrast agent uptake as follows: [Index of Transmurality]i

thicknesses of paramagnetic contrast agent uptake)i / (thickness of myocardium in a particu‐

To see the association between MR images with the data of electrophysiological condition of heart muscle, and in particular with the location of the areas with lowered voltage of local electrical activity, the patients underwent electrophysiological study of the heart (EPhS) with electroanatomical CARTO reconstruction of LV [20]. Besides, there were identified the

Locations of intracardiac leads and thus of segmental electrical activity corresponded to the lo‐ cations of left ventricular segments during contrast-enhanced MRI. In accordance with the am‐ plitude of the curve during EPhS for a definite myocardial segment, degree of potential reduction was graded as follows: 0 – for the amplitude of the potential from 1,5 to 8 mV, when the segment was considered to be a zone of normal potential; 1 — for the amplitude of 0,5 — 1,5 mV, a transient zone; 2 — for the amplitude 0,05 — 0,5 mV, low potential zone; 3 — zone of «electrical scar» - lack of electrical activity when the amplitude was 0 — 0,05 mV. This grading was precon‐ ditioned by the fact that myocardial areas with the 1st or 2nd grades of lowered potential were, as a rule, sources of life threatening tachycardias, while for grade 3 "electrical scar" this was less pos‐

areas of delayed conduction, zones of possible re-entry and inducible VT (Figure 3).

sible and in the zones with normal electrical activity of grade 0 VTs did not occur [9,10].

= (maximum

**Figure 1.** Segmentation of LV myocardium, used in evaluation of a local paramagnetic contrast agent uptake during myocardial MRI

**Figure 2.** The scheme of TI calculation by the data of delayed contrast-enhanced MRI of myocardium. The values of thickness of the infarction zone (accumulation of paramagnetic) and the value of the total myocardial thickness were identified on the LV slices in short axis. Thickness of the contrast-paramagnetic accumulation in myocardium is thought to be the thickness of lesion resulting from acute myocardial infarction. Thus TI is equal to *{(thickness of an infracted area)/ (thickness of the myocardium)}.* (original data)

**Characteristics**

Angina class Canadian Cardiovascular Society,

Current NYHA heart failure class, (%)

Ventricular tachycardia,

Mitral regurgitation 2+, fibrous ring more than 35

**Table 1.** Clinical characteristics of the patients.

(%)

mm, (%)

Lesions of coronary arteries, (%)

(%)

**LVR with EE n=84**

Diagnostics and Surgical Treatment of Left Ventricular Aneurysm with Ventricular Tachycardia

II 10 10 III 37 40 IV 34 33

Unstable angina 19 17

I 5 6 II 20 21 III 70 69 IV 5 4%

2 35 34 3 9 8

Spontaneous 14 13

Induced 32 30

1 30 34 2 35 36 3 35 30

18 13

Age, years old 55 56

Type of LV aneurysm, (%) 1 56 58

Ventricular extrasystoly, (%) 44 48

SSPS 11.5 for Windows software was used for the analysis. Shapiro-Wilk test was applied to assess normality of distribution law of quantitative values. The parameters conforming with the normal distribution test were described with the use of a mean value (M) and a standard deviation (SD). Qualitative data were described by the rate of occurrence or its percentage. Student's t-test was used to evaluate significance of the differences of quantitative values in the compared groups when distribution law was normal. To see the significance of differen‐ ces among quantitative values Z criterion (Fisher's exact test) was used. Evaluation of signif‐ icance of differences in postoperative mortality was carried out by Kaplan-Meier method.

With p<0.05 all the statistical parameters were considered significant.

**LVR without EE n=104**

http://dx.doi.org/10.5772/54126

325

**Figure 3.** Patient T, 56 year old. Before the surgery. EPhS with LV reconstruction of a patient with LV aneurysm. The area of electrical "silence" (scar) is highlighted with grey color; the low-amplitude ventricular potential area of 0,5 mV – with red; the transient zone of 0,5 - 1,5 mV – with yellow-green; the zone of viable myocardium – with violet; the double potential zone – with blue dots and the zone of delayed potential – with pink dots. Front view, right oblique view. (original data)

Surgical ventricular reconstruction (SVR) was performed by the standard methods by V.Dor and in L.Menicanti modification. [4,6]. After cardiac arrest with a calculated injection of cardio‐ plegia solution (Kustodiol) there was performed grafting of distal coronary anastomoses. Left ventricle was opened with a longitudinal incision in the apical area to be parallel to the anterior descending artery along visually identified scarred tissue. After revision of left ventricular cav‐ ity thrombotic mass if any was eliminated. In case of endocardectomy we performed resection of scarred and transient areas of LV. Residual volume of LV cavity was calculated by a physio‐ logical norm of 50-60 ml/m2 of a patient's body surface, and was limited by a special sizer (Chase Medical Richardson, TX, USA). To close LV cavity we used an endocardial synthetic patch (Gore-tex). When L. Menicanti modification was applied, LV neoapex was formed with one or two u-shaped sutures. LV was closed with a double running suture [11,12].

Endocardectomy was performed in 84 patients who were referred to the study group (LVR +EE); on average there was dissected 44cm2 of LV endocardium (from 17 to 84 cm2 ) includ‐ ing ventricular septum. The control group consisted of 104 patients in which endocardial re‐ section was not performed (LVR without EE). The patients were allocated into the groups randomly. All the patients signed the informed consent form. The study was approved by the local ethic committee.

Resection of aneurysm and left ventricular reconstruction (LVR) was performed by V. Dor procedure in 130 patients, by L. Menicanti modification - in 58 patients. In 29 patients from both groups mitral valve fibrous ring repair was done. All the patients underwent coronary artery bypass grafting (CABG). The area of an endocardial synthetic patch varied from 5 to 20 cm2 . Clinical data and the data of instrumental examinations did not show significant dif‐ ferences between the patient groups (Table 1).

**Table 1.** Clinical characteristics of the patients.

**Figure 3.** Patient T, 56 year old. Before the surgery. EPhS with LV reconstruction of a patient with LV aneurysm. The area of electrical "silence" (scar) is highlighted with grey color; the low-amplitude ventricular potential area of 0,5 mV – with red; the transient zone of 0,5 - 1,5 mV – with yellow-green; the zone of viable myocardium – with violet; the double potential zone – with blue dots and the zone of delayed potential – with pink dots. Front view, right oblique

Surgical ventricular reconstruction (SVR) was performed by the standard methods by V.Dor and in L.Menicanti modification. [4,6]. After cardiac arrest with a calculated injection of cardio‐ plegia solution (Kustodiol) there was performed grafting of distal coronary anastomoses. Left ventricle was opened with a longitudinal incision in the apical area to be parallel to the anterior descending artery along visually identified scarred tissue. After revision of left ventricular cav‐ ity thrombotic mass if any was eliminated. In case of endocardectomy we performed resection of scarred and transient areas of LV. Residual volume of LV cavity was calculated by a physio‐ logical norm of 50-60 ml/m2 of a patient's body surface, and was limited by a special sizer (Chase Medical Richardson, TX, USA). To close LV cavity we used an endocardial synthetic patch (Gore-tex). When L. Menicanti modification was applied, LV neoapex was formed with

Endocardectomy was performed in 84 patients who were referred to the study group (LVR

ing ventricular septum. The control group consisted of 104 patients in which endocardial re‐ section was not performed (LVR without EE). The patients were allocated into the groups randomly. All the patients signed the informed consent form. The study was approved by

Resection of aneurysm and left ventricular reconstruction (LVR) was performed by V. Dor procedure in 130 patients, by L. Menicanti modification - in 58 patients. In 29 patients from both groups mitral valve fibrous ring repair was done. All the patients underwent coronary artery bypass grafting (CABG). The area of an endocardial synthetic patch varied from 5 to

. Clinical data and the data of instrumental examinations did not show significant dif‐

of LV endocardium (from 17 to 84 cm2

) includ‐

one or two u-shaped sutures. LV was closed with a double running suture [11,12].

+EE); on average there was dissected 44cm2

ferences between the patient groups (Table 1).

the local ethic committee.

20 cm2

view. (original data)

324 Principles and Practice of Cardiothoracic Surgery

SSPS 11.5 for Windows software was used for the analysis. Shapiro-Wilk test was applied to assess normality of distribution law of quantitative values. The parameters conforming with the normal distribution test were described with the use of a mean value (M) and a standard deviation (SD). Qualitative data were described by the rate of occurrence or its percentage. Student's t-test was used to evaluate significance of the differences of quantitative values in the compared groups when distribution law was normal. To see the significance of differen‐ ces among quantitative values Z criterion (Fisher's exact test) was used. Evaluation of signif‐ icance of differences in postoperative mortality was carried out by Kaplan-Meier method. With p<0.05 all the statistical parameters were considered significant.


*Article* **A New Anelasticity Model for Wave Propagation in Partially Saturated Rocks**

**Chunfang Wu 1, Jing Ba 1,\*, Xiaoqin Zhong 2, José M. Carcione 1,3, Lin Zhang 1 and Chuantong Ruan 1,4**

1 School of Earth Sciences and Engineering, Hohai University, Nanjing 211100, China;

	- China<EMAIL_ADDRESS>
4 School of Mathematics and Statistics, Zhoukou Normal University, Zhoukou 466001, China

**\*** Correspondence<EMAIL_ADDRESS>
**Abstract:** Elastic wave propagation in partially saturated reservoir rocks induces fluid flow in multi-scale pore spaces, leading to wave anelasticity (velocity dispersion and attenuation). The propagation characteristics cannot be described by a single-scale flow-induced dissipation mechanism. To overcome this problem, we combine the White patchy-saturation theory and the squirt flow model to obtain a new anelasticity theory for wave propagation. We consider a tight sandstone Qingyang area, Ordos Basin, and perform ultrasonic measurements at partial saturation and different confining pressures, where the rock properties are obtained at full-gas saturation. The comparison between the experimental data and the theoretical results yields a fairly good agreement, indicating the efficacy of the new theory.

**Keywords:** partial saturation; patchy saturation; squirt flow; P-wave velocity dispersion and attenuation; anelasticity; ultrasonic measurements

#### **1. Introduction**

Seismic waves induce fluid flow and anelasticity (the wave-velocity dispersion and dissipation factor) in rocks saturated with immiscible fluids [1–8]. The level of anelasticity depends on the in situ pressure, fluid content and type, and pore structure. This subject is highly relevant to petroleum exploration and production.

WIFF (wave-induced fluid flow) occurs at various spatial scales that can be categorized as macroscopic, mesoscopic, and microscopic [9]. The first is the wavelength-scale equilibration process occurring between the peaks and troughs of a P-wave, while the mesoscopic length is much larger than the typical pore size but smaller than the wavelength. The microscopic scale is of the same order of magnitude as the pore and grain sizes.

The macroscopic mechanism has been discussed by Biot [10–12] and is often referred to as the Biot relaxation peak (usually at kHz dominant frequencies). The basic assumptions are that the rock frame is homogeneous and isotropic, and the relative motion between the grains and the pore fluid is governed by Darcy's law. Local fluid flow on meso- and micro-scales are neglected, and consequently, the Biot peak cannot explain the observed wave anelasticity at all frequencies [13].

Partial saturation leads to fluid heterogeneity at the mesoscopic scale and the pressure difference between fluid phases causes wave dissipation at low frequencies [9,14–19]. White [20] proposed the first patchy-saturation model (the White model, spherical pockets). Dutta and Odé [21] reformulated this model by using the Biot theory, while Johnson [22] generalized it to patches of arbitrary geometry by using a branch function. Liu et al. [23] analyzed the effect of the fluid properties.

Moreover, dissimilar pores, with different shapes (micro-fractures and intergranular pores) and/or orientations, also cause mesoscopic pressure gradients and squirt flow,

**Citation:** Wu, C.; Ba, J.; Zhong, X.; Carcione, J.M.; Zhang, L.; Ruan, C. A New Anelasticity Model for Wave Propagation in Partially Saturated Rocks. *Energies* **2021**, *14*, 7619. https://doi.org/10.3390/ en14227619

Academic Editor: Eugen Rusu

Received: 5 October 2021 Accepted: 6 November 2021 Published: 15 November 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

resulting in dissipation. At the pore level, dissipation can be described with squirt flow models [24,25]. Dvorkin and Nur [26] unified the Biot and squirt flows and proposed the BISQ model (Biot/squirt), which describes anelasticity at some frequency ranges. However, the low-frequency P-wave velocity prediction from the BISQ model is smaller than the Gassmann velocity [27], while it is consistent with the Biot one at high frequencies. Dvorkin et al. [28] extended the BISQ model to partially saturated rocks by incorporating the Wood equation [29] and proposed that the squirt flow length can be related to water saturation. Dvorkin et al. [30] reformulated the BISQ model to achieve consistency with the Gassmann velocity at the low-frequency limit. However, the P-wave velocity obtained with this model is higher than the theoretical high limit at high frequencies (when all the cracks are closed, and the P-wave velocity value is determined by the Biot model) [31]. Wu et al. [31] proposed a reformulated modified frame squirt flow model (MFS) to solve the problem.

Mavko and Jizba [32] introduced a modified frame to estimate the high-frequency unrelaxed dry rock shear and bulk moduli (M-J model), where cracks are saturated and the stiff pores are drained. To obtain the wet rock properties from the M-J model, Gurevich et al. [33] used the pressure relaxation method of Murphy et al. [34]. The model can be applied in a broad frequency range. Wu et al. [31] presented a reformulated modified frame squirt flow model, but although the prediction is acceptable at ultrasonic frequencies, Pride et al. [35] showed that the attenuation is significantly underestimated at seismic frequencies because the mesoscopic mechanism is not taken into account.

Wave anelasticity is mainly due to the effect of multi-scale fluid flow [36–40]. Rubino and Holliger [41] studied the problem at the micro and meso scales, analyzing the effects of the pore aspect ratio, while Li et al. [42] studied wave velocity in fractured poroelastic media saturated with immiscible fluids. Recently, Sun [43] proposed a model which considers the three loss mechanisms, i.e., the Biot, squirt flow, and mesoscopic relaxation peaks, in the framework of a double-porosity theory. This low-velocity limit does not honor Gassmann velocity.

We briefly review the propagation models at different scales, and propose a new one, based on the White theory and a reformulated modified frame squirt flow model (see Figure 1). Based on the numerical examples, the wave propagation characteristics of the new model and the effect of permeability and the outer diameter of the patch are analyzed. The P-wave velocity and attenuation with varying saturations are measured at different confining pressures. The crack properties and squirt flow length are obtained from the experimental data. The new model is applied to ultrasonic measurements performed on tight sandstone from the Qingyang area of the Ordos basin. The comparison between the experimental data and the theoretical results are made, so as to verify the capability of the new model in the description of those wave properties.

**Figure 1.** A new model based on a reformulated modified frame squirt flow (MFS) model with the White theory. The effects of squirt flow occurring between soft and stiff pores in the water-saturated host medium are incorporated by using an equivalent host medium of a modified solid (the MFS model). On the other hand, the White theory describes the anelasticity due to the patchy saturation of the immiscible fluid mixture.

#### **2. Model**

#### *2.1. Patchy-Saturation (White) Model*

White [20] proposed a patchy saturation model, by considering flow in a concentric spherical model where the inner sphere is saturated with one fluid type (gas), and the outer shell is saturated with a liquid (water), where the frame is assumed to be homogeneous. Let *a* and *b* be the inner and outer diameters, such as (*b* > *a*), and the gas saturation is *Sg* = *a*3/*b*3. Dutta and Odé [21] modified the White model based on the Biot model, and obtained the following wet rock bulk and shear moduli:

$$K^\*(\omega) = \frac{K\_{\infty}}{1 - \mathcal{W}K\_{\infty}},\tag{1}$$

$$\mathbf{G}^\*(\omega) = \mathbf{G}\_{dry\_{\mathcal{Y}}} \tag{2}$$

respectively, where *K*∞ is the bulk modulus at the high-frequency limit, *Gdry* is the dry rock shear modulus, and *W* is a complex function of porosity, permeability, fluid viscosity, etc. (see Appendix A in Carcione et al. [44], and the Section 2.3).

#### *2.2. Squirt Flow Model*

Th flow between microcrack and grain contacts back and forth to stiff (equant) pores induces dissipation even for a single saturating fluid. The microcracks are incorporated into an effective rock skeleton, containing only stiff pores.

The reformulated modified frame squirt flow model considers both the squirt and Biot flows. According to Dvorkin et al. [30] and a boundary condition given by Gurevich et al. [33], the modified bulk modulus is (Wu et al. [31]):

$$K\_{ms} = K\_{msd} + \frac{\alpha\_c \,^2 F\_c}{\phi\_c} \left[ 1 - \frac{2J\_1(\lambda R)}{\lambda R J\_0(\lambda R)} \right],\tag{3}$$

where *Kmsd* = 1/*K*0 − 1/*Khp* + 1/*Kdry*−1 , *K*0 is the bulk modulus of the mineral mixture, *Kdry* is the dry rock bulk modulus, *Fc* = 1/*Kf l* + 1/(*φcQc*) −1 , *φc* is the microcrack porosity, *αc* = 1 − *Kmsd*/*K*0, *Qc* = *K*0/(*αc*−*φc*), *R* is characteristic squirt flow length, *ω* is the angular frequency, *λ*2 = *iωηφc*/*κ* 1/*Kf l* + 1/(*φcQc*) , *η* is the fluid viscosity, *κ* is the permeability, *Kf l* is the bulk modulus of fluid, and *J*0 and *J*1 are the zero- and first-order Bessel functions, respectively.

The modified dry-frame bulk and shear moduli are (Wu et al. [31]):

$$\frac{1}{K\_{md}} = \frac{1}{K\_{ms}} + \frac{1}{K\_{hp}} - \frac{1}{K\_0} \, \text{} \tag{4}$$

$$\frac{1}{G\_{md}} = \frac{1}{G\_{dry}} - \frac{4}{15} \left( \frac{1}{K\_{dry}} - \frac{1}{K\_{md}} \right) \tag{5}$$

respectively, where *Khp* is the high-pressure modulus [33]. The P-wave phase velocity and attenuation can then be obtained according to Toksöz and Johnston [45] as

$$V\_{\text{ph}P1,2} = \frac{1}{\text{Re}(X\_{1,2})},\ a\_{1,2} = \omega \text{Im}(X\_{1,2}),\tag{6}$$

where

$$X\_{1,2} = \sqrt{Y\_{1,2}}\ \ Y\_{1,2} = -\frac{B}{2A} \pm \sqrt{\left(\frac{B}{2A}\right)^2 - \frac{C}{A}}\ \ A = \frac{\phi FM\_{dry}}{\rho\_{22}^2}\ \ .$$

$$B = \frac{F\left(2\alpha\_{md} - \phi - \phi\frac{\rho\_{11}}{\rho\_{22}}\right) - \left(M\_{dry} + F\frac{\alpha\_{md}^2}{\phi}\right)\left(1 + \frac{\rho\_a}{\rho\_{22}} + i\frac{\omega\_c}{\omega}\right)}{\rho\_{22}},$$

$$C = \frac{\rho\_{11}}{\rho\_{22}} + \left(1 + \frac{\rho\_{11}}{\rho\_{22}}\right)\left(\frac{\rho\_a}{\rho\_{22}} + i\frac{\omega\_c}{\omega}\right), \rho\_{11} = (1 - \phi)\rho\_{s}, \rho\_{22} = \phi\rho\_{f1},$$

$$F = \left(1/K\_{fl} + (\alpha\_{md} - \phi)/(\phi K\_0)\right)^{-1},$$

where *ρa* is the additional coupling density, *ωc* = *ηφ*/(*κρf l*) is the characteristic frequency, *αmd* = 1 − *Kmd*/*K*0, *φ* is porosity, *Mdry* is the uniaxial modulus of the rock skeleton under drained conditions, and *ρs* and *ρf l* are the mineral density and fluid density, respectively.

#### *2.3. Patch-Saturation and Squirt Flow Models Combined*

The White model assumes a uniform rock skeleton and that the area outside the inclusion is fully saturated with water. Therefore, the modified dry rock moduli (4) and (5) are used in the White model, thus combining the micro and meso descriptions of anelasticity. If subindices 1 and 2 refer to the gas-inclusion region and host medium (water), respectively, we have the wet rock moduli

$$K(\omega) = \frac{K\_{\infty}}{1 - WK\_{\infty}}\tag{7}$$

$$G(\omega) = G\_{\text{md}\prime} \tag{8}$$

where

$$K\_{\infty} = \frac{K\_{\rm G2}(3K\_{\rm G1} + 4G\_{md}) + 4G\_{md}(K\_{\rm G1} - K\_{\rm G2})S\_{\rm g}}{(3K\_{\rm G1} + 4G\_{md}) - 3(K\_{\rm G1} - K\_{\rm G2})S\_{\rm g}} \tag{9}$$

$$\mathcal{W} = \frac{3i\alpha\kappa (R\_1 - R\_2)(F\_1 - F\_2)}{b^3 \omega (\eta\_1 Z\_1 - \eta\_2 Z\_2)}. \tag{10}$$

Moreover,

$$K\_{\rm G1} = \frac{K\_0 - K\_{\rm md} + \phi K\_{\rm md} \left( K\_0 / K\_{f11} - 1 \right)}{1 - \phi - K\_{\rm md} / K\_0 + \phi K\_0 / K\_{f11}} \tag{11}$$

$$K\_{\rm G2} = \frac{K\_0 - K\_{md} + \phi K\_{md} \left(K\_0 / K\_{fl2} - 1\right)}{1 - \phi - K\_{md} / K\_0 + \phi K\_0 / K\_{fl2}} \tag{12}$$

are Gassmann moduli, where *Kf l*1 and *Kf l*2 are fluid moduli,

$$R\_1 = \frac{(K\_{G1} - K\_{md})(3K\_{G2} + 4G\_{md})}{(1 - K\_{md}/K\_0)\left[K\_{G2}(3K\_{G1} + 4G\_{md}) + 4G\_{md}(K\_{G1} - K\_{G2})S\_{\wp}\right]} \tag{13}$$

$$R\_2 = \frac{(K\_{G2} - K\_{md})(3K\_{G1} + 4G\_{md})}{(1 - K\_{md}/K\_0)\left[K\_{G2}(3K\_{G1} + 4G\_{md}) + 4G\_{md}(K\_{G1} - K\_{G2})S\_{\mathcal{S}}\right]} \tag{14}$$

$$F\_1 = \frac{(1 - K\_{md}/K\_0)K\_{A1}}{K\_{G1}} \tag{15}$$

$$F\_2 = \frac{(1 - K\_{md}/K\_0)K\_{A2}}{K\_{G2}} \tag{16}$$

$$Z\_1 = \frac{1 - \exp(-2\gamma\_1 a)}{(\gamma\_1 a - 1) + (\gamma\_1 a + 1)\exp(-2\gamma\_1 a)}\tag{17}$$

$$Z\_2 = \frac{(\gamma\_2 b + 1) + (\gamma\_2 b - 1) \exp[-2\gamma\_2(b - a)]}{(\gamma\_2 b + 1)(\gamma\_2 a - 1) - (\gamma\_2 b - 1)(\gamma\_2 a + 1) - \exp[-2\gamma\_2(b - a)]} \tag{18}$$

$$
\gamma\_1 = \sqrt{i\omega \eta\_1 / \kappa K\_{E1}} \tag{19}
$$

$$
\gamma\_2 = \sqrt{i\omega\eta\_2/\kappa K\_{E2}}\tag{20}
$$

where *η*1 and *η*2 are fluid viscosities, and

$$K\_{E1} = \left[1 - \frac{K\_{f11}(1 - K\_{G1}/K\_0)(1 - K\_{\rm md}/K\_0)}{\phi K\_{G1}\left(1 - K\_{f11}/K\_0\right)}\right] K\_{A1} \tag{21}$$

$$K\_{E2} = \left[1 - \frac{K\_{f12}(1 - K\_{G2}/K\_0)(1 - K\_{\rm md}/K\_0)}{\phi K\_{G2}\left(1 - K\_{f12}/K\_0\right)}\right] K\_{A2} \tag{22}$$

$$\frac{1}{K\_{A1}} = \left(\frac{\phi}{K\_{fl1}} + \frac{1-\phi}{K\_0} - \frac{K\_{md}}{K\_0^2}\right) \tag{2.3}$$

$$\frac{1}{K\_{A2}} = \left(\frac{\phi}{K\_{f12}} + \frac{1-\phi}{K\_0} - \frac{K\_{md}}{K\_0^2}\right). \tag{24}$$

According to Wood [29], the effective bulk modulus of the gas-water mixture can be calculated from

$$\frac{1}{\mathcal{K}\_{fl}} = \frac{\mathcal{S}\_{\mathcal{S}}}{\mathcal{K}\_{fl1}} + \frac{\mathcal{S}\_{w}}{\mathcal{K}\_{fl2}}\tag{25}$$

where *Sw* is the water saturation.

Finally, the P-wave phase velocity and attenuation are

$$V\_p = \sqrt{\frac{\text{Re}(\mathcal{K}(\omega) + 4G(\omega)/3)}{\rho}},\tag{26}$$

$$Q\_p^{-1} = \frac{\text{Im}(K(\omega) + 4G(\omega)/3)}{\text{Re}(K(\omega) + 4G(\omega)/3)},\tag{27}$$

respectively, where *ρ* = (1 − *φ*)*ρs* + *φ* - *Sgρ*1 + *Swρ*2 is bulk density, and *ρ*1 and *ρ*2 are the fluid densities.

#### *2.4. Results*

The MFS model is directly applied in partially saturated reservoir rocks, where the gas–water mixture is obtained with the Wood equation (there are no gas pockets), and the properties are listed in Table 1. The numerical examples of the characteristics of wave prorogation by the proposed model are shown in Figure 2, and the effects of permeability and the outer diameter of the patch on the wave velocity and attenuation are shown in Figures 3 and 4, respectively.

**Figure 2.** P-wave velocity (**a**) and attenuation (**b**) of the present and MFS models. The number between parentheses indicates water saturation.

**Figure 3.** P-wave velocity (**a**) and attenuation (**b**) of the present model as a function of water saturation and various permeabilities.

**Figure 4.** P-wave velocity (**a**) and attenuation (**b**) of the present model as a function of water saturation for different values of the outer diameter.

Figure 2 compares the P-wave velocity (a) and attenuation (b) of the present model with those of the MFS model, where the number between parentheses indicates water saturation. The velocities coincide at low frequencies and increase with saturation, with those of the present model higher at high frequencies. Two inflection points are clearly observed, corresponding to the mesoscopic and squirt flow attenuation peaks when the saturation is 80%, the first being the stronger point. The attenuation of the present model is higher than that of the MFS one.

Figure 3 shows the effect of permeability, where we can see that attenuation has a maximum at a given saturation which increases with permeability. Figure 4 displays the same quantities as a function of saturation for different outer diameters (*b*). The velocity increases with *b*, and the attenuation decreases.

#### **3. Ultrasonic Data**

#### *3.1. Rock Specimen and Experiment*

A tight sandstone sample (S2-9) from the Qingyang area, Ordos Basin, was tested. The sample was processed into a cylinder with a diameter of 25.2 mm and a length of 50 mm. An aluminum standard with the same shape and size was processed corresponding to the specimen. The sample was composed of quartz, feldspar, and interstitial materials (mainly carbonate minerals and clay), and its porosity was 8.85%. A thin section is shown in Figure 5. The experimental set-up consisted of a pulse generator, a temperature control unit, a confining pressure control unit, a pore pressure control unit, and an ultrasonic wave test unit [46,47].

**Figure 5.** Thin section image of a tight sandstone.

The piezoelectric ultrasonic wave transducers were glued to the top and bottom of the sample, sealed with a rubber sleeve. An electrical pulse was applied to the source transducer to generate the ultrasonic P-waves. A digital oscilloscope was used to display and record the waveforms from the receiver. The temperature, pore, and confining pressures were controlled by the appropriate units [48]. The pore pressure was 15 MPa, the effective pressures were 5, 15, 25, 35 and 45 MPa, the temperature was 20 ◦C, and the waveforms were recorded after we maintained the experimental conditions for half an hour. For the partial gas–water saturation tests, the samples were first saturated with water by using the vacuum pressure saturation method and then placed in an oven to vary the saturation. The approach of Ba et al. [19] was adopted to quantify the fluid content. The sample was tested around six different water saturation conditions, 0%, 20%, 40%, 60%, 80%, and 100%. The wave velocities were obtained from the travel times and the spectral-ratio method was used to obtain the dissipation factor.

#### *3.2. Experimental Results*

Figure 6 shows the velocity as a function of water saturation and effective pressure. As expected, the P-wave velocity increases with water saturation and pressure, approaching a linear trend at high pressures [49,50], since microcracks close. In the partially saturated rock, the rock pore spaces contain air (with a lower bulk modulus and a lower P-wave velocity) and water (with a higher bulk modulus and a higher P-wave velocity). With the increase in water saturation, the volume ratio of water increases and that of air decreases while the rock skeleton stays unchanged. Generally, the P-wave velocity increases with the water saturation. The influence of effective pressure on the stiff pores is small and can be neglected [50–52]. The S-wave velocity also increases with effective pressure, but decreases as saturation increases, due to the density effect.

**Figure 6.** P-(**a**) and S-(**b**) wave velocities as a function of effective pressure at different water saturations.

The spectral ratio method is applied to calculate the dissipation factor [53,54]. We have

$$\ln\left(\frac{A\_1(f)}{A\_2(f)}\right) = -\frac{\pi x}{QV}f + \ln\frac{G\_1(x)}{G\_2(x)}\tag{28}$$

where *f* is the frequency, *A*1(*f*) and *A*2(*f*) are the amplitude spectra of the rock sample and standard, respectively, *Q* is the quality factor, *x* is the propagation distance, *V* is the wave velocity, and *G*1(*x*) and *G*2(*x*) are the sample and standard geometrical factors, respectively. As shown in Figure 7, attenuation decreases with effective pressure. Its behavior versus saturation is similar to that of Figure 4. The attenuation variations with respect to effective pressure and saturation are similar to those of the sandstone samples analyzed by Pang et al. [55] and Amalokwu et al. [56].

**Figure 7.** P-wave attenuation as a function of effective pressure and saturation.

#### *3.3. Crack Parameters and Squirt Flow Length*

The parameters of the present model can be obtained from the experimental data. They involve the skeleton bulk and shear moduli at different pressures, the dry rock bulk modulus with microcracks closed, the microcrack porosity, the squirt flow length, etc. The following steps are considered.

(1) The dry rock bulk and shear moduli are calculated from the velocities as

$$K\_{dry} = \left(V\_{pd}^2 - \frac{4}{3}V\_{sd}^2\right)\rho\tag{29}$$

$$G\_{dry} = V\_{sd}^2 \rho \tag{30}$$

where *ρ* = *ρs*(1 − *φ*), *Vpd* and *Vsd* are the P- and S-wave velocities of full gas saturation, respectively.

The DZ model is applied to calculate the microcrack density (Figure 8a) and porosity (Figure 8b) based on the experimental data at different effective pressures. Their variations are more significant in the low-pressure range. As pressure increases, both quantities decrease. The microcrack density and porosity decreases, which can be attributed to the closure of microcracks [57,58].

**Figure 8.** Microcrack density (**a**) and porosity (**b**) as a function of effective pressure.

The dry rock density of sample S2-9 is 2410 kg/m3, and the bulk modulus of the mineral mixture is 39 GPa. The fluid properties are determined from the empirical equations of Batzle and Wang [59]. Figure 9 displays the P-wave velocity as a function of the effective pressure, where the squirt flow lengths are obtained by matching the theoretical results to the experimental data. It shows that the sample can be characterized by a constant squirt flow length at different pressures [31]. The characteristic length of sample S2-9 is 0.45 mm. This quantity is not so relevant to the pressure and it can be considered as an intrinsic rock property [26].

**Figure 9.** P-wave velocity as a function of the effective pressure compared to the experimental data. Results at different squirt flow lengths are shown.

#### **4. Comparison between Theory and Experiment**

#### *4.1. Effect of Saturation*

The present model is used to calculate the P-wave velocity and attenuation of sample S2-9 at 5 MPa effective pressure. The dry rock bulk modulus is 20.5 GPa, the Poisson ratio is 0.15, the permeability is 0.177 mD, the outer diameter is 0.12 mm, the high-pressure modulus is 23 GPa, and the fluid properties are listed in Table 1. Figure 10 displays the results for different models. The Gassmann–Hill curve does not consider the wave-induced fluid flow. The MFS curve coincides with the Gassmann–Wood curve at low saturations (Figure 10a), while the present model has a velocity of the order of the Gassmann–Hill curve. When saturation increases, the rock is stiffened by the microscopic fluid flow, resulting in a velocity increase.

**Figure 10.** P-wave velocity (**a**) and attenuation (**b**) as a function of water saturation at 5 MPa. The open circles correspond to the experimental data.

The influence of fluid flow is determined by fluid pressure gradients at the interface between different fluid phases, where the fast P-wave converts to the slow (diffusive) Biot wave (mesoscopic loss) [15,60]. Attenuation has a maximum at a given saturation due to the mesoscopic loss mechanism, absent in the MFS model, whereas at full gas or water saturation, the results are similar. The attenuation curves are similar to those of the sandstone samples analyzed by Amalokwu et al. [56].

Compared with the simplified models, the present model provides a good match between the theory and the ultrasonic data for a tight sandstone, mainly the P-wave velocity

as a function of saturation at an effective pressure of 5 MPa, showing the effectiveness of the squirt flow model combined with the White theory. However, attenuation is underestimated by the model due to the fact that the spatial variations in mineral grain and porosity are not considered.

#### *4.2. Effect of Effective Pressure*

In this example, the outer diameters at effective pressures of 15, 25, 35, and 45 MPa are 0.14, 0.16, 0.18 and 0.2 mm, respectively. Figures 11–14 display the P-wave velocity and attenuation as a function of water saturation at these pressures. The overall trend is similar to that at 5 MPa. As pressure increases, the MFS velocities approach the Gassmann–Wood velocities, and the P-wave velocity predictions from the new model increases; however, the attenuation decreases where most microcracks close, and the squirt flow effects are inhibited. Therefore, the characteristics of wave propagation from the new model are similar to those of the experimental data in Figures 11–14.

**Figure 11.** P-wave velocity (**a**) and attenuation (**b**) as a function of water saturation at 15 MPa. The open circles correspond to the experimental data.

**Figure 12.** P-wave velocity (**a**) and attenuation (**b**) as a function of water saturation at 25 MPa. The open circles correspond to the experimental data.

**Figure 13.** P-wave velocity (**a**) and attenuation (**b**) as a function of water saturation at 35 MPa. The open circles correspond to the experimental data.

**Figure 14.** P-wave velocity (**a**) and attenuation (**b**) as a function of water saturation at 45 MPa. The open circles correspond to the experimental data.

#### *4.3. Crossplots*

Figure 15 shows crossplots of the measured and theoretical velocities, showing a good agreement.

Attenuation crossplots are displayed in Figure 16. The attenuation prediction from the present model is less than the experimental one, particularly at low effective pressures. This can be due to the fact that the model considers only the mesoscopic loss caused by partial saturation. Additional attenuation may be due to the presence of many minerals, and the microcrack content, shape, and distribution [15,51,61].

**Figure 15.** Crossplots of the measured and theoretical velocities at different water saturations (**a**) and effective pressures (**b**).

**Figure 16.** Crossplots of the measured and theoretical dissipation factors at different water saturations (**a**) and effective pressures (**b**).

#### **5. Conclusions**

We combine the reformulated modified frame squirt flow model and White's mesoscopic loss theory (spherical gas pockets) to develop a new model for describing wave anelasticity in partially saturated rocks. The microcrack properties and characteristic squirt flow lengths are obtained from experimental data at different effective pressures and water saturations. We compare the results with those of simplified models, showing that the present model provides a good match between the theory and the ultrasonic data for a tight sandstone, mainly the P-wave velocity as a function of saturation and pressure. Attenuation is underestimated by the model due to the fact that mesoscopic loss (fast P- to slow P-wave conversion) due to spatial variations in mineral grain and porosity are not considered. The new model can be used to predict the characteristics of wave propagation in partially saturated tight sandstones, mainly the P-wave velocity. Moreover, a better description at high frequencies (from tens of kHz) should consider the Biot attenuation peak. The generalization of the model will be the task of a future paper.

Due to the complex microstructures and fabric heterogeneity of tight sandstone, the proposed model cannot fully describe the experimental measurement data at low effective pressures. The theories cannot perfectly match real rocks, and there might be errors/defects in the experiment measurements. In the related engineering applications of hydrocarbon reservoir exploration, the methods of big data analytics and machine learning may be applied in combination with the theoretical models, so as to improve the applicability of the model and the accuracy of reservoir property prediction or interpretation.

**Author Contributions:** Conceptualization, J.B. and C.W.; Data curation, J.B. and X.Z.; Formal analysis, C.W., J.B., X.Z., J.M.C., L.Z. and C.R.; Funding acquisition, J.B.; Investigation, C.W., J.B., J.M.C. and L.Z.; Methodology, C.W., J.B. and J.M.C.; Project administration, J.B. and X.Z.; Resources, J.B. and X.Z.; Supervision, J.B.; Validation, C.W. and J.M.C.; Writing—original draft, C.W. and J.B.; Writing—review & editing, J.B., J.M.C. and L.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was funded by the National Natural Science Foundation of China, grant number 41974123 and 42174161, the Jiangsu Province Outstanding Youth Fund Project, grant number BK20200021, the National Science and Technology Major Project of China, grant number 2017ZX05069- 002, and Sinopec Key Laboratory of Geophysics.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data relevant with this study can be accessed by contacting the corresponding author.

**Acknowledgments:** This work is supported by the National Natural Science Foundation of China (Grant no. 41974123, 42174161), the Jiangsu Province Outstanding Youth Fund Project (Grant no. BK20200021), the National Science and Technology Major Project of China (Grant no. 2017ZX05069- 002) and Sinopec Key Laboratory of Geophysics.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A. Symbols**

**Table A1.** List of symbols.

**Table A1.** *Cont.*

Microcrack Porosity Estimation at Different Effective Pressures

$$\frac{1}{K\_{stiff}^{MT}} = \frac{1}{K\_0} \left( 1 + \frac{\phi\_s}{1 - \phi\_s} P \right) \tag{A1}$$

$$\frac{1}{G\_{stiff}^{MT}} = \frac{1}{G\_0} \left( 1 + \frac{\phi\_s}{1 - \phi\_s} \mathcal{Q} \right),\tag{A2}$$

respectively; *G*0 is the shear modulus and *φs* is the stiff porosity.

$$P = \frac{(1 - \upsilon)}{6(1 - 2\upsilon)} \times \frac{4(1 + \upsilon) + 2\gamma^2(7 - 2\upsilon) - \left[3(1 + 4\upsilon) + 12\gamma^2(2 - \gamma)\right]g}{2\gamma^2 + (1 - 4\gamma^2)g + (\gamma^2 - 1)(1 + \gamma)g^2} \tag{A3}$$

$$\begin{array}{c} Q = & \frac{4(\gamma^2 - 1)(1 - \gamma)}{15\left\{8(\gamma - 1) + 2\gamma^2(3 - 4v) + \left[(7 - 8v) - 4\gamma^2(1 - 2v)\right]g\right\}}\\ & \times \left\{\frac{8(1 - v) + 2\gamma^2(3 + 4v) + \left[(8v - 1) - 4\gamma^2(5 + 2v)\right]g + 6\left(\gamma^2 - 1\right)(1 + v)g^2}{2\gamma^2 + (1 - 4\gamma^2)g + (\gamma^2 - 1)(1 + \gamma)g^2} \\ & - 3\left[\frac{8(v - 1) + 2\gamma^2(5 - 4v) + \left[3(1 - 2v) + 6\gamma^2(v - 1)\right]g}{-2\gamma^2 + ((2 - \gamma) + \gamma^2(1 + v))g}\right] \end{array} \tag{A4}$$

where *γ* is the spheroidal aspect ratio, and *υ* is the Poisson ratio of the grains, i.e.,

$$g = \begin{cases} \upsilon = (3\mathcal{K}\_0 - 2\mathcal{G}\_0) / (6\mathcal{K}\_0 + 2\mathcal{G}\_0), \\ \frac{\gamma}{(1-\gamma^2)^{3/2}} \left( \arccos \gamma - \gamma \sqrt{1-\gamma^2} \right) (\gamma < 1) \\\ \frac{\gamma}{(1-\gamma^2)^{3/2}} \left( \gamma \sqrt{1-\gamma^2} - \arccos \eta \right) (\gamma > 1) \end{cases} \tag{A5}$$

Microcracks are included into the host material by neglecting the interactions between cracks and pores. The effective moduli (host with cracks) are

$$\frac{1}{K\_{eff}^{MT}} = \frac{1}{K\_{stiff}^{MT}} \left( 1 + \frac{16\left(1 - \left(\upsilon\_{stiff}^{MT}\right)^2\right) \Gamma}{9\left(1 - 2\upsilon\_{stiff}^{MT}\right)} \right) \tag{A6}$$

$$\frac{1}{G\_{eff}^{MT}} = \frac{1}{G\_{stiff}^{MT}} \left( 1 + \frac{32 \left( 1 - \upsilon\_{stiff}^{MT} \right) \left( 5 - \upsilon\_{stiff}^{MT} \right) \Gamma}{45 \left( 2 - \upsilon\_{stiff}^{MT} \right)} \right) \tag{A7}$$

where *υMT sti f f* = 3*KMT sti f f* − 2*μMT sti f f* / 6*KMT sti f f* + 2*μMT sti f f* and Γ is the microcrack density. When all the microcracks close at high pressures, a least square method is used to obtain the optimal aspect ratio of the stiff pores by using Equations (A1) and (A2).

$$
\Gamma = \Gamma^i e^{-p/\not p} \tag{A8}
$$

where Γ*i* is the initial value when the effective pressure is zero, *p* is the effective pressure, and *p*ˆ is a constant.

$$\gamma\_p^i = \frac{3}{4\pi} \int\_{\Gamma^i}^{\Gamma} \frac{\left(1/K(\Gamma) - 1/K\_{eff}^{hp}\right)}{\Gamma} \frac{dp}{d\Gamma} d\Gamma \tag{A9}$$

where *K*(Γ) is the effective bulk modulus which can be obtained from Equation (A1). Substituting Equation (A8) into (A9), we obtain

$$\gamma\_p^i = \frac{3}{4\pi} \int\_{\Gamma}^{\Gamma^i} \frac{\left(1/K(\Gamma) - 1/K\_{eff}^{hp}\right)\mathcal{P}}{\Gamma^2} d\Gamma \tag{A10}$$

and by integrating Equation (A10) from Γ to Γ*i* ,

$$\gamma\_p^i = \frac{4\dot{\rho} \left[1 - \left(\upsilon\_{eff}^{hp}\right)^2 \ln\left(\frac{\Gamma^i}{\Gamma}\right)\right]}{3\pi K\_{eff}^{hp} \left[1 - 2\upsilon\_{eff}^{hp}\right]} \tag{A11}$$

where *υ hp eff* is the effective Poisson ratio at high pressures, i.e., *υ hp eff* = 3*Khp eff* − 2*Ghp eff* / 6*Khp eff* + 2*Ghp eff* .

Combining Equations (A8) and (A11), the relation between the minimum initial aspect ratio and the effective pressure can be obtained as

$$\gamma\_p^i = \frac{4\left[1 - \left(\upsilon\_{eff}^{hp}\right)^2\right]p}{\pi E\_{eff}^{hp}}\tag{A12}$$

where *Ehp eff* = 3*Khp eff* & 1 − 2*υ hp eff* ' is the effective Young modulus at high pressures. The cumulative microcrack density decreases with pressure. If pressure changes from zero to *dp*, the corresponding reduction of the cumulative microcrack density is *d*Γ. When the pressure increment is small enough, it can be considered that the decrease of microcrack density is mainly due to the closure of microcracks with an aspect ratio less than the minimum initial aspect ratio. David and Zimmerman [52] related the microcrack porosity and density as

$$\phi\_{\mathbb{C}} = \frac{4\pi\gamma}{3}\Gamma \tag{A13}$$

Therefore, the microcrack properties can be obtained from the acoustic wave velocities as a function of the effective pressure.

#### **References**


**4.1 What happens when we intersect Pra-colonialism and transformative leadership**

The intersections of transformative leadership and Pra perspective on identity are germane to our discussion. Being aware that, from a poststructuralist perspective, identity is a negotiated process in the contexts of social location and positioning, how are students positioned? How are student identities negotiated, if at all, with educational leaders? What powers do students have to express themselves as they wish within the curriculum? The roles of leaders in curriculum development and pedagogies of the oppressed become evident. Do the theories and foundations of educational leadership, administrative systems and policies, organizational cultures, curricula, pedagogy, and assessment allow students to be seen for who they truly are, for who and what they wish to become? Shields and Hesbol [48] conclude:

*To meet the needs of rapidly changing populations, whether immigrant or refugee, whether they speak the language of instruction or not, whether they have experienced trauma or not, and whether the school is in a vibrant urban setting or a remote rural setting, it is incumbent on school leaders … to lead in ways that promote inclusion, equity, and excellence for every student (p. 559).*

As Sayani [54] asks, what is the epistemic status of the student's lived experience and knowing? We begin with reflections on our mind-sets around that question. We have outlined here a Pra [colonial] framework, intersecting with and guided by transformative leadership, that allows us to engage with students in ways that develop body, heart, and mind and develop the full potential of humanity already present with the students.

Again, we want to stress that our Pra perspective includes consideration of cultural, regional, gender, or other such unique characteristics of the students; but it also transcends them. Our perspective "precedes," in an ontological sense, these considerations. Suppose we are willing to recognize students in their basic humanity. In that case, we can strive as educational leaders to create educationally humanizing conditions, working toward the possibilities of what Freire [8] refers to as "becoming fully human." As Bai, Park, and Cohen [49] note in distinguishing education from mere instruction:

*Education is for growing, raising, maturing, cultivating, and fulfilling human beings, manifesting the full potential of humanity….[the wisdom] traditions share an understanding that education has to do with human becoming …. the larger aim of education that the wisdom traditions hold up before us: humans increasingly maturing into and embodying a fuller humanity that can manifest wisdom and virtue (pp. 114–115).*

We do not believe that simply applying a transformative leadership paradigm to the problem of engaging international students in higher education in a manner that simply "transcends" colonialism and Euro-centric pedagogical practices is the solution. Instead, we believe that a genuinely radical leadership paradigm shift lacks meaning or cannot occur unless a call for action occurs by intersecting a Pra perspective with the praxis of transformative leadership.

In summary, then, Pra-colonialism challenges colonialism and post-colonialism by offering a heuristic to problematize deeply ingrained ideologies that international students are less than or different from other students. We suggest that by viewing international students *as* students, leadership, and faculty respond as if providing equity, inclusivity, and diversity in their regular classrooms with "regular" students. Current ideologies have resulted in pedagogical practices rooted in a deficit model or oppressive pedagogical and curriculum approaches that limit the potential of what can happen in classrooms. To mitigate current damaging approaches, we suggest a Pra-transformative perspective that recognizes *and* legitimates the students' subjectivity and lived experience, offering and enacting transformative pedagogical opportunities where student identities can be explored as a means of becoming more fully human.