Impacts of proppant distribution on development of tight oil reservoirs with threshold pressure gradient

Field evidence indicates that proppant distribution and threshold pressure gradient have great impacts on well productivity.Aiming at the development of unconventional oil reservoirs in Triassic Chang-7 Unit,Ordos Basin of China,we presented an integrated workflow to investigate how(1)proppant placement in induced fracture and(2)non-linear flow in reservoir matrix would affect well productivity and fluid flow in the reservoir.Compared with our research before(Yue et al.,2020),here we extended this study into the development of multi-stage fractured horizontal wells(MFHWs)with large-scale complicated fracture geometry.The integrated workflow is based on the finite element method and consists of simulation models for proppant-laden fluid flow,fracture flow,and non-linear seepage flow,respectively.Simulation results indicate that the distribution of proppant inside the induced cracks significantly affects the productivity of the MFHW.When we assign an idealized proppant distribution instead of the real distribution,there will be an overestimation of 44.98%in daily oil rate and 30.63%in cumulative oil production after continuous development of 1000 days.Besides,threshold pressure gradient(TPG)also significantly affects the well performance in tight oil reservoirs.If we simply apply linear Darcy’s law to the reservoir matrix,the overall cumulative oil production can be overrated by 77%after 1000 days of development.In general,this research provides new insights into the development of tight oil reservoirs with TPG and meanwhile reveals the significance of proppant distribution and non-linear fluid flow in the production scenario design.

Numerical Simulation of a Two-Phase Flow with Low Permeability anda Start-Up Pressure Gradient

A new numerical model for low-permeability reservoirs is developed.The model incorporates the nonlinear characteristics of oil-water two-phase flows while taking into account the initiation pressure gradient.Related numerical solutions are obtained using a finite difference method.The correctness of the method is demonstrated using a two-dimensional inhomogeneous low permeability example.Then,the differences in the cumulative oil and water production are investigated for different starting water saturations.It is shown that when the initial water saturation grows,the water content of the block continues to rise and the cumulative oil production gradually decreases.

Prediction of pressure gradient and hold-up in horizontal liquid-liquid pipe flow

This paper aims to propose correlations to predict pressure gradient,friction factor and fluid phase hold-up in liquid-liquid horizontal pipe flow.To develop the correlations,experiments are conducted using high viscous oils(202 and 630 mPa⋅s)in a steel pipe of length 11.25 m and length-to-diameter ratio of 708.In addition,the experimental data from the literature comprising wide range of flow and fluid properties is analyzed.For the analysis,the liquid-liquid pipe flow data is categorized into two as:stratified and dispersed.The existing friction factor correlations are modified to incorporate the effects of viscosity of the oil phase,interfacial curvature(contact/wetting angle-in lieu of material of the pipe)and fluid phase fraction.In the two-fluid model of stratified flow,the wall stress and interfacial stress correlations are substituted with superficial velocities of fluids and superficial Reynolds numbers of fluid phases replacing fluid phase velocities and fluid Reynolds numbers.Similarly,for dispersed flow,an effective Reynolds number is described as the sum of superficial Reynolds number of oil and water phases.Substituting the generally employed mean or mixture Reynolds number with the effective Reynolds number into the existing single-phase turbulent flow friction factor correlation,an effective friction factor for oil-water flow is proposed.Employing the proposed correlations,the pressure gradient across the oil-water flow and hold-up volume fraction are predicted with significant reduction in error compared with that of conventionally employed correlations.The average error and standard deviation values of−7.06%,20.72%and 0.31%,18.79%are found for stratified flow and dispersed flow respectively.

Pre-transjugular-intrahepatic-portosystemic-shunt measurement of hepatic venous pressure gradient and its clinical application: A comparison study

BACKGROUND It is controversial whether transjugular intrahepatic portosystemic shunt(TIPS)placement can improve long-term survival.AIM To assess whether TIPS placement improves survival in patients with hepaticvenous-pressure-gradient(HVPG)≥16 mmHg,based on HVPG-related risk stratification.METHODS Consecutive variceal bleeding patients treated with endoscopic therapy+nonselectiveβ-blockers(NSBBs)or covered TIPS placement were retrospectively enrolled between January 2013 and December 2019.HVPG measurements were performed before therapy.The primary outcome was transplant-free survival;secondary endpoints were rebleeding and overt hepatic ence-phalopathy(OHE).RESULTS A total of 184 patients were analyzed(mean age,55.27 years±13.86,107 males;102 in the EVL+NSBB group,82 in the covered TIPS group).Based on the HVPG guided risk stratification,70 patients had HVPG<16 mmHg,and 114 patients had HVPG≥16 mmHg.The median follow-up time of the cohort was 49.5 mo.There was no significant difference in transplant-free survival between the two treatment groups overall(hazard ratio[HR],0.61;95%confidence interval[CI]:0.35-1.05;P=0.07).In the high-HVPG tier,transplant-free survival was higher in the TIPS group(HR,0.44;95%CI:0.23-0.85;P=0.004).In the low-HVPG tier,transplantfree survival after the two treatments was similar(HR,0.86;95%CI:0.33-0.23;P=0.74).Covered TIPS placement decreased the rate of rebleeding independent of the HVPG tier(P<0.001).The difference in OHE between the two groups was not statistically significant(P=0.09;P=0.48).CONCLUSION TIPS placement can effectively improve transplant-free survival when the HVPG is greater than 16 mmHg.

Hepatic venous pressure gradient: Inaccurately estimates portal venous pressure gradient in alcoholic cirrhosis and portal hypertension

BACKGROUND Portal hypertension(PHT)in patients with alcoholic cirrhosis causes a range of clinical symptoms,including gastroesophageal varices and ascites.The hepatic venous pressure gradient(HVPG),which is easier to measure,has replaced the portal venous pressure gradient(PPG)as the gold standard for diagnosing PHT in clinical practice.Therefore,attention should be paid to the correlation between HVPG and PPG.METHODS Between January 2017 and June 2020,134 patients with alcoholic cirrhosis and PHT who met the inclusion criteria underwent various pressure measurements during transjugular intrahepatic portosystemic shunt procedures.Correlations were assessed using Pearson’s correlation coefficient to estimate the correlation coefficient(r)and determination coefficient(R^(2)).Bland-Altman plots were constructed to further analyze the agreement between the measurements.Disagreements were analyzed using paired t tests,and P values<0.05 were considered statistically significant.RESULTS In this study,the correlation coefficient(r)and determination coefficient(R2)between HVPG and PPG were 0.201 and 0.040,respectively(P=0.020).In the 108 patients with no collateral branch,the average wedged hepatic venous pressure was lower than the average portal venous pressure(30.65±8.17 vs.33.25±6.60 mmHg,P=0.002).Hepatic collaterals were identified in 26 cases with balloon occlusion hepatic venography(19.4%),while the average PPG was significantly higher than the average HVPG(25.94±7.42 mmHg vs 9.86±7.44 mmHg;P<0.001).The differences between HVPG and PPG<5 mmHg in the collateral vs no collateral branch groups were three cases(11.54%)and 44 cases(40.74%),respectively.CONCLUSION In most patients,HVPG cannot accurately represent PPG.The formation of hepatic collaterals is a vital reason for the strong underestimation of HVPG.

Computed tomography perfusion in differentiating portal hypertension: A correlation study with hepatic venous pressure gradient

BACKGROUND Hepatic venous pressure gradient(HVPG)is the gold standard for diagnosis of portal hypertension(PH),invasiveness and potential risks in the process of measurement limited its widespread use.AIM To investigate the correlation of computed tomography(CT)perfusion parameters with HVPG in PH,and quantitatively assess the blood supply changes in liver and spleen parenchyma before and after transjugular intrahepatic portosystemic shunt(TIPS).METHODS Twenty-four PH related gastrointestinal bleeding patients were recruited in this study,and all patients were performed perfusion CT before and after TIPS surgery within 2 wk.Quantitative parameters of CT perfusion,including liver blood volume(LBV),liver blood flow(LBF),hepatic arterial fraction(HAF),spleen blood volume(SBV)and spleen blood flow(SBF),were measured and compared before and after TIPS,and the quantitative parameters between clinically significant PH(CSPH)and non-CSPH(NCSPH)group were also compared.Then the correlation of CT perfusion parameters with HVPG were analyzed,with statistical significance as P<0.05.RESULTS For all 24 PH patients after TIPS,CT perfusion parameters demonstrated decreased LBV, increased HAF, SBV and SBF, with no statistical difference in LBF. Compared withNCSPH, CSPH showed higher HAF, with no difference in other CT perfusion parameters. HAFbefore TIPS showed positive correlation with HVPG (r = 0.530, P = 0.008), while no correlation wasfound in other CT perfusion parameters with HVPG and Child-Pugh scores.CONCLUSIONHAF, an index of CT perfusion, was positive correlation with HVPG, and higher in CSPH thanNCSPH before TIPS. While increased HAF, SBF and SBV, and decreased LBV, were found afterTIPS, which accommodates a potential non-invasive imaging tool for evaluation of PH.

Hepatic venous pressure gradient measurement before TIPS for acute variceal bleeding

Hepatic venous pressure gradient(HVPG)is an independent predictor of variceal rebleeding in patients with cirrhosis.After pharmacological and/or endoscopic therapy,the use of a transjugular intrahepatic portosystemic shunt(TIPS)may be necessary in HVPG non-responders,but not in responders.Thus,HVPG measurement may be incorporated into the treatment algorithm for acute variceal bleeding,which further identifies the candidates that should undergo early insertion of TIPS or maintain the traditional pharmacological and/or endoscopic therapy.The potential benefits are to reduce the cost and prevent TIPS-related complications.

Effect of quadratic pressure gradient term on a one-dimensional moving boundary problem based on modified Darcy's law

A relatively high formation pressure gradient can exist in seepage flow in low-permeable porous media with a threshold pressure gradient, and a significant error may then be caused in the model computation by neglecting the quadratic pressure gradient term in the governing equations. Based on these concerns, in consideration of the quadratic pressure gradient term, a basic moving boundary model is constructed for a one-dimensional seepage flow problem with a threshold pressure gradient. Owing to a strong nonlinearity and the existing moving boundary in the mathematical model, a corresponding numerical solution method is presented. First, a spatial coordinate transformation method is adopted in order to transform the system of partial differential equa- tions with moving boundary conditions into a closed system with fixed boundary conditions; then the solution can be sta- bly numerically obtained by a fully implicit finite-difference method. The validity of the numerical method is verified by a published exact analytical solution. Furthermore, to compare with Darcy＇s flow problem, the exact analytical solution for the case of Darcy＇s flow considering the quadratic pressure gradient term is also derived by an inverse Laplace transform. A comparison of these model solutions leads to the conclu- sion that such moving boundary problems must incorporate the quadratic pressure gradient term in their governing equa- tions; the sensitive effects of the quadratic pressure gradient term tend to diminish, with the dimensionless threshold pres- sure gradient increasing for the one-dimensional problem.

Comparison of hepatic venous pressure gradient and endoscopic grading of esophageal varices

AIM: To determine the correlation between the hepatic venous pressure gradient and the endoscopic grade of esophageal varices.METHODS: From September 2009 to March 2013, a total of 176 measurements of hepatic venous pressure gradient (HVPG) were done in 146 patients. Each transjugular HVPG was measured twice, first using an end whole catheter (EH-HVPG), and then using a balloon catheter (B-HVPG). The HVPG was compared with the endoscopic grade of esophageal varices (according to the general rules for recording endoscopic findings of esophagogastric varices), which was recorded within a month of the measurement of HVPG.RESULTS: The study included 110 men and 36 women, with a mean age of 56.1 years (range, 43-76 years). The technical success rate of the pressure measurements was 100% and there were no complication related to the procedures. Mean HVPG was 15.3 mmHg as measured using the end hole catheter method and 16.5 mmHg as measured using the balloon catheter method. Mean HVPG (both EH-HVPG and B-HVPG) was not significantly different among patients with different characteristics, including sex and comorbid factors, except for cases with hepatocellular carcinoma (B-HVPG, P = 0.01; EH-HVPG, P = 0.02). Portal hypertension (&#x0003e; 12 mmHg HVPG) occurred in 66% of patients according to EH-HVPG and 83% of patients according to B-HVGP, and significantly correlated with Child&#x02019;s status (B-HVPG, P &#x0003c; 0.000; EH-HVGP, P &#x0003c; 0.000) and esophageal varies observed upon endoscopy (EH-HVGP, P = 0.003; B-HVGP, P = 0.006). One hundred and thirty-five endoscopies were performed, of which 15 showed normal findings, 27 showed grade 1 endoscopic esophageal varices, 49 showed grade 2 varices, and 44 showed grade 3 varices. When comparing endoscopic esophageal variceal grades and HVPG using univariate analysis, the P value was 0.004 for EH-HVPG and 0.002 for B-HVPG.CONCLUSION: Both EH-HVPG and B-HVPG showed a positive correlation with the endoscopic grade of esophageal varices, with B-HVPG showing a stronger correlation than EH-HVPG.

Computed tomography perfusion in liver and spleen for hepatitis B virus-related portal hypertension:A correlation study with hepatic venous pressure gradient

BACKGROUND Hepatic venous pressure gradient(HVPG)is the gold standard for diagnosis of portal hypertension(PH).However,its use can be limited because it is an invasive procedure.Therefore,it is necessary to explore a non-invasive method to assess PH.AIM To investigate the correlation of computed tomography(CT)perfusion of the liver with HVPG and Child-Pugh score in hepatitis B virus(HBV)-related PH.METHODS Twenty-eight patients(4 female,24 male)with gastroesophageal variceal bleeding induced by HBV-related PH were recruited in our study.All patients received CT perfusion of the liver before transjugular intrahepatic portosystemic stent-shunt(TIPS)therapy.Quantitative parameters of CT perfusion of the liver,including liver blood flow(LBF),liver blood volume(LBV),hepatic artery fraction,splenic blood flow and splenic blood volume were measured.HVPG was recorded during TIPS therapy.Correlation of liver perfusion with Child-Pugh score and HVPG were analyzed,and the receiver operating characteristic curve was analyzed.Based on HVPG(>12 mmHg vs≤12 mmHg),patients were divided into moderate and severe groups,and all parameters were compared.RESULTS Based on HVPG,18 patients were classified into the moderate group and 10 patients were classified into the severe group.The Child-Pugh score,HVPG,LBF and LBV were significantly higher in the moderate group compared to the severe group(all P<0.05).LBF and LBV were negatively associated with HVPG(r=-0.473,P<0.05 and r=-0.503,P<0.01,respectively),whereas splenic blood flow was positively associated with hepatic artery fraction(r=0.434,P<0.05).LBV was negatively correlated with Child-Pugh score.Child-Pugh score was not related to HVPG.Using a cutoff value of 17.85 mL/min/100 g for LBV,the sensitivity and specificity of HVPG≥12 mmHg for diagnosis were 80%and 89%,respectively.CONCLUSION LBV and LBF were negatively correlated with HVPG and Child-Pugh scores.CT perfusion imaging is a potential non-invasive quantitative predictor for PH in HBV-related liver cirrhosis.

Correlation of pressure gradient in three hepatic veins with portal pressure gradient

BACKGROUND The liver is one of the most important organs in the human body,with functions such as detoxification,digestion,and blood coagulation.In terms of vascular anatomy,the liver is divided into the left and the right liver by the main portal vein,and there are three hepatic efferent veins(right,middle,and left)and two portal branches.Patients with impaired liver function have increased intrahepatic vascular resistance and splanchnic vasodilation,which may lead to an increase in the portal pressure gradient(PPG)and cause portal hypertension(PHT).In order to measure the increased pressure gradient of portal vein,the hepatic venous pressure gradient(HVPG)can be measured to reflect it in clinical practice.The accuracy of PPG measurements is directly related to patient prognosis.AIM To analyze the correlation between HVPG of three hepatic veins and PPG in patients with PHT.METHODS From January 2017 to December 2019,102 patients with PHT who met the inclusion criteria were evaluated during the transjugular intrahepatic portosystemic shunt procedure and analyzed.RESULTS The mean HVPG of the middle hepatic vein was 17.47±10.25 mmHg,and the mean HVPG of the right and left hepatic veins was 16.34±7.60 and 16.52±8.15 mmHg,respectively.The average PPG was 26.03±9.24 mmHg.The correlation coefficient and coefficient of determination of the right hepatic vein,middle hepatic vein,and left hepatic vein were 0.15 and 0.02(P=0.164);0.25 and 0.05(P=0.013);and 0.14 and 0.02(P=0.013),respectively.The mean wedged hepatic vein/venous pressure(WHVP)of the middle and left hepatic veins was similar at 29.71±12.48 and 29.1±10.91 mmHg,respectively,and the mean WHVP of the right hepatic vein was slightly lower at 28.01±8.95 mmHg.The mean portal vein pressure was 34.11±8.56 mmHg.The correlation coefficient and coefficient of determination of the right hepatic vein,middle hepatic vein,and left hepatic vein were 0.26 and 0.07(P=0.009);0.38 and 0.15(P<0.001);and 0.26 and 0.07(P=0.008),respectively.The average free hepatic venous pressure(FHVP)of the right hepatic vein was lowest at 11.67±5.34 mmHg,and the average FHVP of the middle and left hepatic veins was slightly higher at 12.19±4.88 and 11.67±5.34 mmHg,respectively.The average inferior vena cava pressure was 8.27±4.04 mmHg.The correlation coefficient and coefficient of determination of the right hepatic vein,middle hepatic vein,and left hepatic vein were 0.30 and 0.09(P=0.002);0.18 and 0.03(P=0.078);and 0.16 and 0.03(P=0.111),respectively.CONCLUSION Measurement of the middle hepatic vein HVPG could better represent PPG.Considering the high success rate of clinical measurement of the right hepatic vein,it can be the second choice.

Numerical investigation of a coupled moving boundary model of radial flow in low-permeable stress-sensitive reservoir with threshold pressure gradient

The threshold pressure gradient and formation stress-sensitive effect as the two prominent physical phenomena in the development of a low-permeable reservoir are both considered here for building a new coupled moving boundary model of radial flow in porous medium. Moreover, the wellbore storage and skin effect are both incorporated into the inner boundary conditions in the model. It is known that the new coupled moving boundary model has strong nonlinearity. A coordinate transformation based fully implicit finite difference method is adopted to obtain its numerical solutions. The involved coordinate transformation can equivalently transform the dynamic flow region for the moving boundary model into a fixed region as a unit circle, which is very convenient for the model computation by the finite difference method on fixed spatial grids. By comparing the numerical solution obtained from other different numerical method in the existing literature, its validity can be verified. Eventually, the effects of permeability modulus, threshold pressure gradient, wellbore storage coefficient, and skin factor on the transient wellbore pressure, the derivative, and the formation pressure distribution are analyzed respectively.

Sildenafil does not influence hepatic venous pressure gradient in patients with cirrhosis

AIM： To investigate if sildenafil increases splanchnic blood flow and changes the hepatic venous pressure gradient （HVPG） in patients with cirrhosis. Phosphodiesterase type-5 inhibitors are valuable in the treatment of erectile dysfunction and pulmonary hypertension in patients with end-stage liver disease. However, the effect of phosphodiesterase type-5 inhibitors on splanchnic blood flow and portal hypertension remains essentially unknown. METHODS： Ten patients with biopsy proven cirrhosis （five females/five males, mean age 54：1：8 years） and an HVPG above 12 mmHg were studied after informed consent. Measurement of splanchnic blood flow and the HVPG during liver vein catheterization were done before and 80 min after oral administration of 50 mg sildenafil. Blood flow was estimated by use of indocyanine green clearance technique and Fick＇s principle, with correction for non-steady state. RESULTS： The plasma concentration of sildenafil was 222 ± 136 ng/mL 80 min after administration. Mean arterial blood pressure decreased from 77 ±7 mmHg to 66 ± 12 mmHg, P = 0.003, while the splanchnicblood flow and oxygen consumption remained unchanged at 1.14 ± 0.71 L/min and 2.3 ± 0.6 mmol/ min, respectively. Also the HVPG remained unchanged （18 ± 2 mmHg vs 16 ± 2 mmHg） with individual changes ranging from -8 mmHg to ±2 mmHg. In seven patients, HVPG decreased and in three it increased. CONCLUSION： In spite of arterial blood pressure decreases 80 min after administration of the phosphodiesterase type-5 inhibitor sildenafil, the present study could not demonstrate any clinical relevant influence on splanichnic blood flow, oxygen consumption or the HVPG.

Characteristics of unsteady flow in porous media while considering threshold pressure gradient with Green＇s function

The flow behavior in porous media with threshold pressure gradient(TPG) is more complex than Darcy flow and the equations of motion, and outer boundary and inner boundary with TPG are also different from Darcy flow for unsteady flow of a producing well in a reservoir. An analytic method to solve this kind of problem is in a need of reestablishment. The classical method of Green's function and Newman product principle in a new way are used to solve the unsteady state flow problems of various shapes of well and reservoir while considering the TPG. Four Green's functions of point, line, band and circle while considering the TPG are achieved. Then, two well models of vertical well and horizontal well are built and simultaneously the function to calculate the moving boundary of each well model is provided. The results show that when considering TPG the pressure field is much different, which has a sudden pressure change, with a moving boundary in it. And the moving boundary of each well model increases with time but slows down rapidly, especially when the TGP is large.

Discontinuous element pressure gradient stabilizations for compressible Navier-Stokes equations based on local projections

A pressure gradient discontinuous finite element formulation for the compressible Navier-Stokes equations is derived based on local projections. The resulting finite element formulation is stable and uniquely solvable without requiring a B-B stability condition. An error estimate is Obtained.

Prediction of Pressure Gradient and Holdup in Small Eoetvoes Number Liquid-Liquid Segregated Flow

The segregated flow pattern, which occurs in a 26.1 mm diameter, horizontal, stainless steel test section, is investigated. Pressure gradient and in situ phase distribution data were obtained for different combinations of phase superficial velocities ranging from 0.05 m.s^-1 to 0,96 m.s^-1. For the current small Eoetvoes number liquid-liquid system （EOD=4.77）, the dominant effect of interfacial tension and wall-wetting properties of the liquids over the gravity is considered. The approach introduces the closure relationship for the case of turbulent flow m a rough pipe, and attempts to modify the two-fluid model to account for the curved interface. In present flow rates range, wave amplitudes were found small, while interfacial mixing was observed. An adjustable definition for hydraulic diame- ters of two fluids and interfacial friction factor is adopted. The predicted pressure gradient and in situ phase distribution data have been compared with present experimental data and those reported in the literature.

Characteristics of pressure gradient force errors in a terrain-following coordinate

A terrain-following coordinate （a-coordinate） in which the computational form of pressure gradient force （PGF） is two-term （the so-called classic method） has significant PGF errors near steep terrain. Using the covariant equations of the a-coordinate to create a one-term PGF （the covariant method） can reduce the PGF errors. This study investigates the factors inducing the PGF errors of these two methods, through geometric analysis and idealized experiments. The geometric analysis first demonstrates that the terrain slope and the vertical pressure gradient can induce the PGF errors of the classic method, and then generalize the effect of the terrain slope to the effect of the slope of each vertical layer （φ）. More importantly, a new factor, the direction of PGF （a）, is proposed by the geometric analysis, and the effects of φ and a are quantified by tan φ.tan a. When tan φ.tan a is greater than 1/9 or smaller than -10/9, the two terms of PGF of the classic method are of the same order but opposite in sign, and then the PGF errors of the classic method are large. Finally, the effects of three factors on inducing the PGF errors of the classic method are validated by a series of idealized experiments using various terrain types and pressure fields. The experimental results also demonstrate that the PGF errors of the covariant method are affected little by the three factors.

Riemann problem of a Chapman-Jouguet combustion model for pressure-gradient system

In this paper,the Riemann problem of a Chapman-Jouguet combustion model for the pressure-gradient equations is considered.By analyzing in phase space,existence and uniqueness of the solution to the Riemann problem are obtained constructively under the global entropy conditions.

THE EFFECT OF MAGNETIC FIELDS ON LOW FREQUENCY OSCILLATING NATURAL CONVECTION WITH PRESSURE GRADIENT

The oscillating natural convection in the presence of transverse magnetic field with time depending pressure gradient is studied. The analysis of the problem is carried out by assuming that the fluid is flowing in a parallel plate configuration. The emphasis is on low frequency oscillating convective flows induced by g-jitter associated with micro gravity because of their importance to the space processing materials. A general solution for an oscillating flow in the presence of transverse magnetic field is carried out. Some special cases of the oscillating flow and its response to an applied magnetic field are performed. It was observed that the behavior of oscillating free convective flows depends on frequency, amplitude of the driving buoyancy forces, temperature gradient,magnetic field and the electric conditions of the channel walls. In the absence of magnetic field, buoyancy force plays a predominant role in driving the oscillatory flow pattern, and velocity magnitude is also affected by temperature gradients. To suppress the oscillating flow external magnetic field can be used. It is also found that the reduction of the velocity is inversely proportional to the square of the applied magnetic field with conducting wall but directly proportional to the inverse of the magnetic field with insulating wall. Detailed calculations and computational results are also carried out to depict the real situation.

A fluid flow model in the lacunar-canalicular system under the pressure gradient and electrical field driven loads

The lacunar-canalicular system(LCS)is acknowledged to directly participate in bone tissue remodeling.The fluid flow in the LCS is synergic driven by the pressure gradient and electric field loads due to the electro-mechanical properties of bone.In this paper,an idealized annulus Maxwell fluid flow model in bone canaliculus is established,and the analytical solutions of the fluid velocity,the fluid shear stress,and the fluid flow rate are obtained.The results of the fluid flow under pressure gradient driven(PGD),electric field driven(EFD),and pressure-electricity synergic driven(P-ESD)patterns are compared and discussed.The effects of the diameter of canaliculi and osteocyte processes are evaluated.The results show that the P-ESD pattern can combine the regulatory advantages of single PGD and EFD patterns,and the osteocyte process surface can feel a relatively uniform shear stress distribution.As the bone canalicular inner radius increases,the produced shear stress under the PGD or P-ESD pattern increases slightly but changes little under the EFD pattern.The increase in the viscosity makes the flow slow down but does not affect the fluid shear stress(FSS)on the canalicular inner wall and osteocyte process surface.The increase in the high-valent ions does not affect the flow velocity and the flow rate,but the FSS on the canalicular inner wall and osteocyte process surface increases linearly.In this study,the results show that the shear stress sensed by the osteocyte process under the P-ESD pattern can be regulated by changing the pressure gradient and the intensity of electric field,as well as the parameters of the annulus fluid and the canaliculus size,which is helpful for the osteocyte mechanical responses.The established model provides a basis for the study of the mechanisms of electro-mechanical signals stimulating bone tissue(cells)growth.