Effects of external temperature and dead volume on laboratory measurements of pore pressure and injected volume in a rock fracture

Yinlin Ji,Christin Kluge,Hnnes Hofmnn,Guido Bl?cher

a Helmholtz Centre Potsdam,GFZ German Research Centre for Geosciences,Potsdam,14473,Germany

b State Key Laboratory of Coal Resources and Safe Mining,China University of Mining and Technology,Xuzhou,221116,China

Keywords:Fracture aperture Thermal pressurization External dead volume Pressure attenuation Injected volume Fiber optic sensor

A B S T R A C T The accurate evaluation of pore pressure and injected volume is crucial for the laboratory characterization of hydromechanical responses of rock fractures.This study reports a series of laboratory experiments to systematically demonstrate the effects of external temperature and dead volume on laboratory measurements of pore pressure and injected volume in a rock fracture.We characterize the hydraulic aperture of the fracture as a function of effective normal stress using the exponential aperture model.This model is then employed to predict the pore pressure change and injected volume in the fracture without the influences of external temperature and dead volume.The external temperature changes in the cyclic loading test due to the Joule-Thompson effect for fluids.The effect of external temperature on pore pressure change in the fracture can be well explained by thermal pressurization of fluids.Our results also show that the external dead volume can significantly lower the pore pressure change in the fracture during the cyclic loading test under undrained conditions.The injected volume can also be substantially enlarged due to the external dead volume in a typical pore pressure system.Internal measurement of the pore pressure in the fracture using a fiber optic sensor cannot exclude the influences of external temperature and dead volume,primarily because of the good hydraulic communication between the fracture and pore pressure system.This study suggests that the effects of external temperature and dead volume on pore pressure response and injected volume should be evaluated for accurate laboratory characterization and inter-laboratory comparison.

1.Introduction

The hydromechanical responses of rock fractures are of paramount significance for unraveling the mechanisms underlying various hydraulic stimulation and seismogenic processes.Previous studies suggest that the pore pressure changes in a fracture can be highly diverse(e.g.Segall and Rice,1995;Rice,2006;Ji et al.,2019,2020;Proctor et al.,2020).Shear deformation may cause dilation or compaction in the normal direction of the fracture(Proctor et al.,2020).Under undrained conditions with constant fluid mass,shear dilation can lead to a reduction in pore pressure(Ye and Ghassemi,2018;Ji et al.,2019),causing dilatancy hardening(Brantut,2020),while the pore pressure enhancement due to shear compaction results in compaction weakening(Fang et al.,2017;Proctor et al.,2020).Temperature variation can also change the pore pressure under undrained conditions by changing the fluid volume(Rice,2006).For instance,pore fluid volume expansion due to transient temperature rise associated with shear heating can increase the pore pressure,which is known as thermal pressurization and weakening(Acosta et al.,2018).A low-permeability fracture under an initially drained condition may also experience the transient undrained condition provided that the local transient pore pressure change during some fast dynamic processes,like stick-slip,is more rapid than the drainage process(Rice,2006;Proctor et al.,2020).In addition,the injected volume into a fractured reservoir is important for anticipating the magnitude of injection-induced earthquakes(McGarr,2014;Ji and Wu,2020;Wang et al.,2020;Ji et al.,2021a,b;Li et al.,2021).Therefore,the accurate laboratory measurements of pore pressure and injected volume are the key for understanding the dynamics and consequences of fracture failure.

In the laboratory,external servo-controlled fluid pumps have been commonly used to supply the pore fluid and monitor the pore pressure in the fracture.However,the large storativity of fluid in the pore pressure lines(tubing)and fluid pump cylinders can attenuate the magnitude of pore pressure change(Brantut et al.,2021).Hereafter,we refer to the fluid volume stored in the up-and downstream tubing and pump cylinders as the“dead volume”(e.g.Nicolas et al.,2020;Brantut et al.,2021).The compliance of the pore pressure system,including pore pressure lines,fluid pump cylinder and pump sensor,could further contribute to such attenuation(Wissa,1969).Thus,many types of internal fluid pressure sensors have been manufactured and employed to monitor the pore pressure inside the sample,such as fiber optic sensors(Reinsch et al.,2012;Bl?cher et al.,2014;Nicolas et al.,2020)and strain gaugebased pore pressure transducers(Brantut,2020;Proctor et al.,2020;Brantut et al.,2021).The stored fluid volume in these sensors and fluid volume changes associated with these sensors due to pressure change are relatively small compared to that of the fracture,ensuring a reliable measurement of local pore pressure.The injected volume recorded by the pump system could also be amplified due to the large dead volume and compliant pore pressure system under both locally undrained and fully drained conditions.Larger fluid volume is required to be injected into the sample to reach the desired pressure value because of the additional compliance of dead volume and pore pressure system(Brantut et al.,2021).Additionally,the external temperature could influence the pore pressure response in the fracture under undrained conditions.However,the effects of external temperature and dead volume on laboratory measurements of pore pressure and injected volume in a rock fracture have been poorly documented,even though the theories on thermal-induced and compressioninduced pressurization of fluids have been well established(e.g.Kestin,1979;Ghabezloo and Sulem,2008).

In this study,we systematically explore how the external temperature and dead volume influence the laboratory measurements of pore pressure and injected volume in a rock fracture in a sedimentary rock.We first characterize the hydraulic aperture of the fracture by running a continuous fluid flow through test under increasing confining pressure.We then examine the effects of external temperature and dead volume on pore pressure response and injected volume by comparing with the predicted value from the exponential aperture model.We also compare the pore pressure measured by the external pump sensor and internal fiber optic sensor.Finally,we generalize our results to a broader range of external temperature and total fluid volume,and make some recommendations for future laboratory studies.

2.Experimental material and methods

2.1.Sample material and preparation

The sample was cored perpendicular to the bedding from a Flechtinger sandstone block sourced from a quarry in the North German Basin,Germany.This sandstone is well characterized in the literature in terms of the mineral composition,physical and hydromechanical properties.The main minerals of the sandstone are quartz(63 wt%),K-feldspar(14 wt%),albite(12 wt%),and illite(7 wt%),with an average grain size of 125 μm(Cheng et al.,2021).The Young’s modulus,Poisson’s ratio and uniaxial compressive strength are 14.9 GPa,0.28 and 56 MPa,respectively(Kluge et al.,2021).The sandstone is characterized by～8.1% porosity and a permeability perpendicular to the bedding plane as low as 10-17-10-18m2(Cheng et al.,2021).We halved the cylindrical sample with a diameter of 50 mm and a length of 100 mm along the core axis using a diamond saw.We also drilled a 1.8-mm-diameter borehole with a length of 25 mm at the center of one sample half to accommodate the fiber optic sensor for internal fluid pressure measurement(Fig.1a and b).The zoom-in view in Fig.1b shows the details of the sensor(after Reinsch et al.,2012).The fracture surfaces were finally ground using a sandpaper with a particle size of 201 μm.Fig.1c shows the sample photo and the fracture surface morphology with near-uniform elevations before the experiments.The surface morphology was obtained using the Keyence VR-3200 laser scanner with a precision of 1 μm in elevation.

Fig.1.(a)Sample assembly in the triaxial cell of the MTS test system connected with the pore pressure system(after Pei et al.,2017);(b)Experimental configuration with a zoom-in view showing the details of location and sketch of the fiber optic sensor(after Reinsch et al.,2012);and(c)Sample photo(left)and initial fracture surface morphology(right).The external dead volume consists of the fluid volume in the pump cylinders and in the pore pressure lines.Since the volume of the pore pressure lines is fixed,the external dead volume is controlled by the piston of the inlet pump,which was set at three positions of the cylinder(i.e.top for small external dead volume,middle for medium external dead volume,and bottom for large external dead volume).

2.2.Testing equipment and configuration

We performed the experiments using the MTS servo-controlled oil medium triaxial cell(MTS 815)(Fig.1a)(Pei et al.,2017),which can supply a hydrostatic confining pressure up to 140 MPa with an accuracy of±0.3%.The temperature of the confining oil was monitored by two K-type thermocouples within the triaxial cell located near the sample.Four Quizix fluid pumps(C6000-10K-HCAT)were used to supply the pore fluid pressure up to 70 MPa.Two pumps were working as paired upstream pumps(fluid inlet)and another two as paired downstream pumps(fluid outlet).The differential fluid pressure between the inlet and outlet pumps were recorded by a differential pressure sensor(Honeywell HL-Z)with an accuracy of～1%.The pore fluid used in this study was distilled water.The data acquisition frequency of the MTS triaxial cell and Quizix fluid pumps was 1 Hz.The sample was assembled with the coreholders by a layer of neoprene jacket to isolate the sample from the confining oil.The fluid was evenly distributed on the sample ends by the groove on the end of coreholders.More information on the testing equipment can be found in(Kluge et al.,2020,2021).In the configuration shown in Fig.1b,the hydrostatic confining pressure applied on the sample was equivalent to the normal stress on the fracture.The paired pumps connected to each sample end are sketched as a single pump cylinder for simplicity(Fig.1a and b).The different external dead volumes were controlled by changing the position of the piston of the inlet pump in the cylinder.The top,middle and bottom positions in the inlet pump marked in Fig.1b correspond to the small(7.7 mL),medium(99.1 mL)and large(199.1 mL)fluid volume in the cylinder,respectively.Thus,the total external small,medium,and large external dead volumes are 141.9 mL,233.3 mL and 333.3 mL,respectively,considering the fluid volume in the pore pressure lines(134.2 mL).Here the pore pressure lines refer to the tubing connecting the fluid pumps and coreholders(Fig.1b).In practice,we manually set the position of the piston inside the inlet pump to achieve different external dead volumes.All experiments were conducted under room temperature of around 20°C.

Fig.2.Schematic diagram of the experimental procedures.

2.3.Experimental procedures

Fig.2 summarizes the experimental procedures in this study,including vacuumization,saturation,first preconditioning,flow through test,fluid injection test,second preconditioning and cyclic loading test.We applied the vacuum condition of 1 kPa using a vacuum pump(Laboxact SEM 820)after the oven-dried sample was installed into the triaxial cell.We then saturated the sample under an inlet pore pressure of 0.2 MPa and a confining pressure of 2 MPa while closing the valve between the outlet pump and sample.The saturation of the sample was signified by no fluid flow into the sample(e.g.Katayama et al.,2018;Kluge et al.,2020,2021).We preconditioned the sample by cyclically changing the confining pressure between 2 MPa and 22 MPa at a rate of 5 MPa/min under a constant pore pressure of 0.2 MPa.The preconditioning of the sample is to remove any misalignment and plastic deformation of the sample.

Fig.3.(a)Exponential model describing the dependency of hydraulic aperture b of a fracture on effective normal stress σ′n(br and bm are the residual and maximum aperture,respectively,α is the sensitivity coefficient defining the curvature of the exponential curve.),and(b)Hydraulic aperture of the fracture as a function of effective normal stress measured in the laboratory and model inversion.

After completion of the preconditioning,we first measured the hydraulic aperture of the fracture by conducting a continuous fluid flow through test under increasing confining pressure.Particularly,the initial pore pressure was 0.2 MPa at both inlet and outlet pumps under 2 MPa initial confining pressure.We then imposed a constant inlet flow rate of 7.5 mL/min and continuously increased the confining pressure from 2 MPa to 20 MPa at 0.5 MPa/min.The differential fluid pressure was recorded by the differential pressure sensor during the fluid flow through test.Based on the data collected from the fluid flow through test,the hydraulic aperture of the fracture(b)can be estimated from the cubic law by neglecting the matrix permeability(Witherspoon et al.,1980;Zimmerman and Bodvarsson,1996):where μ is the fluid viscosity,and μ=0.001 Pa s;L(0.1 m)andW(0.05 m)are the length and width of the rectangular fracture,respectively,andL=0.1 m andW=0.05 m;Qis the volumetric flow rate(m3/s);andΔpis the differential fluid pressure between the inlet and outlet pumps(Pa).

Afterwards,we conducted a fluid injection test under a constant confining pressure of 20 MPa and an initial pore pressure of 0.2 MPa.The inlet fluid pressure was increased at a rate of 0.01 MPa/s from 0.2 MPa to 1 MPa,while the fluid outlet from the sample was closed.The injected volume and outlet pressure was monitored by the inlet pump and outlet pump,respectively.

Subsequently,we increased the confining pressure to 40 MPa.Again,before any tests,we cycled the confining pressure between 20 MPa and 42 MPa at a rate of 5 MPa/min under a constant pore pressure of 0.2 MPa to further remove the plastic strain of the sample.We then conducted the cyclic loading tests under the undrained condition,during which the pore pressure was recorded concurrently by the external pump sensor and the internal fiber optic sensor.In the cyclic loading tests,the confining pressure was increased from 2 MPa to 40 MPa at a certain loading rate,and was held at 40 MPa for 10 min before it was decreased at the same rate to 2 MPa and maintained at 2 MPa for another 10 min.The first four cyclic loading tests were conducted at different loading rates(i.e.1 MPa/min,2.5 MPa/min,5 MPa/min and 10 MPa/min)with the small external dead volume by setting the piston of the inlet pump at the top of the cylinder.Afterwards,two cyclic loading tests were performed at a loading rate of 5 MPa/min with medium and large external dead volumes.

3.Experimental results

3.1.Fracture aperture characterization

We used the exponential model(Rutqvist et al.,2008)to mathematically depict the dependency of hydraulic aperture on the effective normal stress(σ′n)(i.e.the difference between confining pressure and pore pressure)(Fig.3):

wherebrandbmare the residual and maximum aperture,respectively;and α is the sensitivity coefficient defining the curvature of the exponential curve(Fig.3a).The model inversion from our laboratory data yields a residual aperturebr,maximum aperturebm,and a sensitivity coefficient α of 26.73 μm,9.49 μm,and 0.11,respectively(Fig.3b).The initial fluid volume stored in the fracture under 2 MPa confining pressure and 0.2 MPa pore pressure is estimated from the exponential aperture model and the fracture area.Particularly,we first use the model to estimate the initial hydraulic aperture by substituting the values of the parameters determined in Fig.2b into Eq.(2)and setting the effective normal stress(σ′n)as 1.8 MPa,i.e.the difference between 2 MPa confining pressure and 0.2 MPa pore pressure.We then estimate the initial fluid volume as the product of the initial hydraulic aperture(34.5 μm)and the fracture area(0.005 m2).The initial fluid volume is obtained as 0.17 mL,which is extremely small even compared to the small external dead volume of 141.9 mL in the pore pressure lines and pump.This exponential aperture model will also be used to estimate the pore pressure response and injected volume excluding the effects of external temperature and dead volume in Section 3.3.

3.2.Effect of external temperature

Fig.4a shows the temporal evolution of pore pressure with changing confining pressure in the four cyclic loading tests conducted at different loading rates(i.e.1 MPa/min,2.5 MPa/min,5 MPa/min and 10 MPa/min).These four cyclic loading tests were performed under undrained condition with the same small external dead volume of 141.9 mL.The pore pressure changes nonlinearly with linearly changing confining pressure,presumably due to the poro-elastic response of the porous rock matrix(Hassanzadegan et al.,2012,2014).The increase in pore pressure when the confining pressure reaches 40 MPa scales with higher loading rate,which is probably attributed to the temperature change associated with Joule-Thompson effect for fluids(Steffensen and Smith,1973).The average temperature measured by the two in-vessel K-type thermocouples is used for the evaluation of this effect.Specifically,the temperature increases with increasing confining pressure during loading(i.e.Joule-Thompson heating),and Joule-Thompson cooling causes the reduction of temperature during unloading(Fig.4b).Besides,the temperature declines during the 10-min holding of confining pressure at 40 MPa,and slightly increases during the 10-min holding at 2 MPa.As shown in Fig.4c,the measured pore pressure increase(Δpm)and temperature increase when the confining pressure first reaches 40 MPa show similar increasing trends with increasing loading rate,furthering supporting the temperature effect on pore pressure change.

To quantify this temperature effect on pore pressure change,we consider the thermal pressurization of fluids due to the expansion of fluid volume(ΔV)(Kestin,1979):

where β is the thermal expansion coefficient(K-1),ΔTis the temperature increase(K),andVis the initial fluid volume(m3).

The change of fluid volume leads to the change of pore fluid pressure(Δp)according to(Kestin,1979):

whereKis the bulk modulus of fluid(Pa).

Combining Eqs.(3)and(4),one can obtain

Fig.4.Effect of external temperature change induced by cyclic loading/unloading of confining pressure on pore pressure change in the fracture:(a)Temporal evolution of confining pressure and measured pore pressure during cyclic loading tests conducted at different loading rates,(b)Temperature as a function of confining pressure during cyclic loading tests conducted at different loading rates,and(c)Pore pressure increase(measured by the pump sensorΔpm and corrected for temperature effectΔpc)and temperature increase when the confining pressure first reaches 40 MPa as a function of loading rate.s denotes the standard deviation of the four corrected pore pressure increases.Note that the four cyclic loading tests were conducted with the same small external dead volume of 141.9 mL.

Fig.5.Effect of external dead volume on pore pressure change in the fracture:(a)Temporal evolution of confining pressure,temperature and pore pressure(measured by the pump sensor(pm)and corrected for temperature effect(pc))during cyclic loading tests conducted at the same loading rate of 5 mL/min with different external dead volumes;and(b)Pore pressure in the fracture as a function of confining pressure during cyclic loading tests conducted with different external fluid volumes.The pore pressure in the fracture with no external dead volume is predicted from the exponential aperture model.

Near 20°C,if we substitute the values of β(=2.07×10-4K-1at 20°C and atmospheric pressure for water)andK(=2.2×109Pa at 20°C and atmospheric pressure for water)into Eq.(5),we can obtainΔpt=4.6×105ΔTwhereΔTis in K andΔptis in Pa.That is,1 K(i.e.one degree Celsius)of temperature change around 20°C can induce a pore pressure change of approximately 0.5 MPa.Note that Eq.(5)assumes a uniform and steady temperature effect on the total initial fluid volume.In addition,although β andKare dependent on temperature and pressure,we assign the values at 20°C and atmospheric pressure to them considering the small range of temperature and pressure in our laboratory study(Kestin,1979).

Thermal pressurization of pore fluid is a result of the discrepancy among the thermal expansivities of the rock matrix/fracture,the pore pressure system and pore fluid(Ghabezloo and Sulem,2008).Here we consider only the thermal expansivity of pore fluid(Acosta et al.,2018).We first correct the measured pore pressure increase(Δpm)for thermal-induced pore pressure increase(Δpt)when the confining pressure first reaches 40 MPa,and the corrected pore pressure increase(Δpc)should be the same in the four cyclic loading tests.Particularly,the corrected pore pressure increase(Δpc)can be obtained by subtracting this thermalinduced pore pressure increase(Δpt)estimated from Eq.(5)from the measured pore pressure increase(Δpm).After corrected for the temperature effect,the corrected pore pressure increases at different loading rates are roughly equal,signified by Fig.4c,where the four data points fall within the shaded area defined by the average value of the four data points with minus and plus one standard deviation(s).The slight differences among the four corrected pore pressure increases are primarily due to the following five reasons.First,the heating/cooling of confining oil caused by loading/unloading of the confining pressure is a transient process,but the calculation using Eq.(5)requires a steady temperature effect.Second,the values of β andKslightly change with temperature and pressure in our laboratory experiments,but we use constant values in the calculation.Third,the heating/cooling of confining oil only directly influences the pore fluid volume stored in the tubing immersed in the confining oil,but the remaining pore fluid volume is not affected directly by heating/cooling.Fourth,the point measurements of temperature provided by the two K-type thermocouples cannot represent the real pore fluid temperature.Fifth,the thermal expansivities of the rock matrix/fracture and the pore pressure system are not considered in our estimation,while these effects can also influence the accuracy of our analysis(Pei et al.,2020).In any case,as shown in Fig.4c,the differences among the measured pore pressure in the four cyclic loading tests conducted at different loading rates can be well explained by the thermal effects.

3.3.Effect of external dead volume

We examine the effect of external dead volume on the pore pressure change in the fracture.This is performed by comparing the cyclic loading tests conducted at the same loading rate of 5 mL/min with different external fluid volumes.As shown in Fig.5a,the temperature of the three tests increases with increasing confining pressure,while it decreases with decreasing confining pressure.The temperature reduces when the confining pressure is held constant at 40 MPa,while it increases during the holding of confining pressure at 2 MPa.It is noteworthy that the increase of temperature continues for a little while after the confining pressure first reaches 40 MPa.Similarly,the continued reduction of temperature is also observed when the confining pressure first reduces to 2 MPa.This suggests that the Joule-Thompson heating/cooling of fluids cannot stop immediately after the termination of loading/unloading processes.The pore pressure increase measured by the pump sensor(Δpm)decreases with larger external dead volume.The corrected pore pressure(Δpc)is obtained by subtracting the thermal-induced pore pressure change(Δpt)from the measured pore pressure by the pump sensor(Δpm),which also reduces with increasing external dead volume.In addition,the corrected pore pressure(Δpc)at each external dead volume slightly decreases prior to the continuous increase when the confining pressure is held constant at 40 MPa for 10 min,while it first slightly increases before keeps decreasing when the confining pressure is held constant at 2 MPa within the experimental duration.The slight decrease and increase of the corrected pore pressure(Δpc)at the two turning points(as marked in Fig.5a)may be due to the over correction to temperature change as explained in section 3.2.The degree of over correction to temperature change increases with larger dead volume,primarily because of the larger fluid volume outside the triaxial cell that is not influenced directly by the temperature change.The dominant trend of the corrected pore pressure in the first and second holding periods are respectively continuous increase and decrease,presumably due to the timedependent normal closure/opening of the fracture(Im et al.,2018).

Fig.6.Effect of external fluid volume on injected volume(ΔVinj)during the fluid injection test.Temporal evolution of measured pore pressure(top),and measured(middle)and predicted(bottom)injected volumes during the fluid injection test conducted under 20 MPa confining pressure and at a pressurization rate of 0.01 MPa/s of inlet pore pressure with the small external dead volume of 141.9 mL.The predicted injected volume is estimated from the exponential aperture model by assuming no external dead volume.

Apart from the measured pore pressure response,we can estimate the theoretical pore pressure response in the fracture according to the exponential aperture model.In the estimation,we assume the total fluid volume equal to the fluid volume initially stored in the fracture(estimated as 0.17 mL in Section 3.1).We derived the following equation between the theoretically predicted pore pressure increase(Δpa)and normal stress(σn)based on Eqs.(2)and(4):

whereAis the area of the rectangular fracture,andA=0.005 m2;σn0andp0are the initial confining pressure(equivalent to normal stress in this study)and pore pressure,respectively,and σn0=2 MPa andp0=0.2 MPa;andb0is the initial hydraulic aperture of the fracture estimated from Eq.(2),andb0=34.5 μm.

Fig.7.Temporal evolution of confining pressure and pore pressure(predicted by the exponential aperture model,measured by external pump sensor and internal fiber optic sensor)during the cyclic loading test conducted at a loading rate of 5 MPa/min with the small external dead volume.

We calculate the pore pressure increase from the confining pressure based on Eq.(6)through iteration.Fig.5b shows the pore pressure in the fracture as a function of the confining pressure during cyclic loading tests conducted at a loading rate of 5 MPa/min.The predicted pore pressure is derived based on the assumption of no external fluid volume,and thus the initial fluid volume stored in the fracture(estimated as 0.17 mL in Section 3.1)is the initial total fluid volume.The predicted pore pressure is only slightly lower than the applied confining pressure,indicating that the Skempton coefficient of the fracture is close to unity.The laboratory results show that the increase of pore pressure is significantly smaller than the predicted value and the laboratorymeasured pore pressure reduces with larger external dead volume(inset in Fig.5b).

The large external dead volume can also influence the injected volume(ΔVinj)in the fluid injection test.The results of the fluid injection test conducted under a constant confining pressure of 20 MPa and an initial pore pressure of 0.2 MPa are shown in Fig.6.Note that the initial external dead volume is small(141.9 mL)in this test.When the inlet pressure increases from 0.2 MPa to 1 MPa,the monitoring pressure increases simultaneously to the same value,suggesting that the fracture is highly permeable under the applied confining stress and injection rate.The measured injected volume increases with continuous injection from 0 mL to 0.45 mL until the termination of the injection.We also estimate the injected volume based on the exponential aperture model by assuming there is no external dead volume.The predicted injected volume finally reaches a value as small as 5×10-5mL.This predicted value is only～0.01% of the measured value.The initial fluid volume stored in the fracture under 20 MPa confining pressure and 0.2 MPa initial pore pressure is estimated as 0.14 mL from the exponential aperture model,while the external dead volume in this test is 141.9 mL.That is,the injected volume can be significantly magnified in our laboratory study even with the small external dead volume.In addition,the ratio between the initial total fluid volume in the pore pressure system and in the fracture is～1×104,while the ratio between the measured and predicted injected volumes is～1×105,suggesting that the expansivity of the pore pressure system enlarges the injected volume by～10 times.

3.4.External vs.internal measurements of pore pressure

We have demonstrated that the external temperature and dead volume both have great impacts on the pore pressure response in the fracture.The fiber optic sensors inserted near the fracture surface provide an avenue to measure the internal pore pressure in the fracture(Reinsch et al.,2012;Nicolas et al.,2020).Based on the correlation between phase shift and applied pressure(see Appendix),we estimate the pore pressure in the fracture from the phase shift during the cyclic loading test conducted at a loading rate of 5 MPa/min with the small external dead volume.The results are shown in Fig.7,which exhibits that the pore pressure measured by the external pump sensor and internal fiber optic sensor are roughly equal,and both are significantly smaller than the predicted value based on the exponential aperture model and no external dead volume assumption.This result suggests that,even with an internal pore pressure sensor,the pore pressure response in the fracture can still be attenuated by the external dead volume when there is a direct and fast hydraulic connectivity between the fracture and pump system.

Fig.8.(a)Thermal-induced pore pressure increase(Δpt)with increasing temperature.This effect is independent of the total fluid volume in the sample and pore pressure system.T0 is the initial temperature.(b)Pore pressure increase resulting from different compressed volumes of fluid(ΔVc)for experimental setups with different total fluid volumes at a constant temperature.(c)Injected volume(ΔVinj)required to achieve a certain pore pressure increase for experimental setups with different total fluid volumes at a constant temperature.

4.Discussion

Our experimental results indubitably demonstrate the influence of external temperature on the pore pressure response in a fracture.In laboratory studies,pressurized pore water is commonly in the liquid phase under the temperature range from 0°C to 100°C.We quantify the influence of temperature on pore pressure response by substituting the thermal expansion coefficient and bulk modulus of water into Eq.(5).The bulk modulus of water is relatively constant from 20°C to 100°C,while the thermal expansion coefficient of water increases monotonically with increasing temperature(Kestin,1979).We consider an initial temperatureT0of 20°C.Then,Eq.(5)can be generalized as whereΔTis between 0°C and 80°C.Eq.(7)is plotted in Fig.8a,which shows that the thermal-induced pore pressure increase(Δpt)can be more than 100 MPa.Our experimental data in Fig.4 are consistent with the prediction from Eq.(7)as shown in Fig.8a,which means that the temperature effect on pore pressure increase is also pronounced when the maximum transient temperature rise is only～5°C in this study.

Therefore,the pore pressure change induced by slight transient temperature fluctuation associated with laboratory operations under undrained conditions,like confining pressurization in this study,can be considerable.This is common in many laboratory experiments conducted under ambient temperature.In this case,we suggest that(i)the temperature should be servo-maintained constant slightly above the room temperature by an active heating and cooling unit to eliminate the potential transient temperature change associated with laboratory operations,or(ii)the temperature should be monitored to estimate the thermal-induced pore pressure change using Eq.(7).For example,in the case of this study,the difference in the measured pore pressure increases due to the transient temperature change resulting from the compression of confining pressure of the same magnitude can only be explained after considering the thermal-induced pore pressure change.In addition,we suggest that special attention should also be paid to the transient temperature change associated with the change of pore pressure.A typical example is the temperature rise caused by the instant increase of upstream pore pressure during the pulse decay measurement of permeability(Brace et al.,1968).As reported by Brace et al.(1968),this effect can last for 10-20 s and thus their measurements were made until after～20 s in their experiments.This influencing time is dependent on the experimental setup.Therefore,it is recommended that the effect of this transient temperature rise should be evaluated for specific experimental setups in the measurement of pore pressure decay.

As suggested by Eq.(4)and demonstrated in our laboratory study,the external dead volume stored in the pore pressure lines and pump attenuates the pore pressure response mainly by increasing the total initial fluid volume that is compressed due to a confining pressure and temperature increase.The compressibility of the pore pressure system cannot be determined at this time,and thus we neglect this effect and plot the curves based on Eq.(4)for different compressed volumes of fluid(ΔVc),ranging from 0.001 mL to 0.5 mL in Fig.8b.Our experimental data presented in Fig.5 are also shown for comparison.The total volume in this study is estimated by adding up the initial fluid volume store in the fracture,in the pore pressure lines,and in the pump cylinders.The curves suggest that the pore pressure increase subject to the same compressed volume of fluid reduces significantly with increasing total volume.The reduction in pore pressure increase with increasing total volume is more evident when the compressed volume of fluid is larger.The experimental data in this study are generally consistent with the theoretical trend.The discrepancy between the theoretical prediction and experimental data may be primarily caused by the expansivity of the pore pressure system,which is not considered.Because of the negligibility of the initial fluid volume in the fracture compared to the external dead volume in the pore pressure system,the measured pore pressure increase is substantially smaller than the predicted value based on the exponential aperture model and no external dead volume assumption.

We also evaluate the effect of external dead volume on the measurement of injected volume(ΔVinj)based on Eq.(4),again neglecting the compressibility of pore pressure system.The curves in Fig.8c are plotted from Eq.(4)by setting a constant pore pressure increase(Δp)ranging from 0.1 MPa to 2 MPa.The experimental case in this study illustrated in Fig.6 is also shown by the gray curve marked withΔp=0.8 MPa.The experimental data point lies above the predicted gray curve,again highlighting the importance of the expansivity of the pore pressure system,which is not measured at this time.Fig.8c shows that the same injection-induced pore pressure increase in the fracture requires larger injected volume when the total volume becomes larger.Moreover,the injected volume scales with the total volume more rapidly when the injection-induced pore pressure increase is higher.

The effect of external dead volume stored in the up-and downstream pore pressure lines and fluid reservoirs has been noticed and corrected in transient permeability measurements(Brace et al.,1968;Rutter and Mecklenburgh,2018).In some cases,such as Skempton coefficient measurement of rocks/fractures,the pore pressure response should be measured by internal pore pressure sensors without connecting to external fluid volumes(Bl?cher et al.,2014).However,for the cases where the pore pressure system has to be connected to supply pore fluid to the sample,it is recommended that the external dead volume should be carefully measured and recorded to evaluate its effect on the pore pressure response under undrained conditions and injected volume during fluid injection tests.In addition,the thermal-induced and pressurization-induced expansivity of the pore pressure system,including the pore pressure lines and pump cylinders,should also be measured for interlaboratory comparison of pore pressure and injected volume.

5.Conclusions

We experimentally demonstrated the effects of external temperature and dead volume on the laboratory measurements of pore pressure and injected volume in a rock fracture in Flechtinger sandstone.Our results show that(i)the external temperature can influence the magnitude of pore pressure change through thermal pressurization,(ii)the external dead volume can significantly attenuate the pore pressure response,and(iii)the external dead volume can substantially increase the injected volume required to reach a target pressure,compared to that predicted by the exponential aperture model.Our laboratory data are well compatible with theoretical models of thermal-induced and compression-induced pressurization of fluids.It is suggested that the external temperature should be carefully controlled and/or monitored to analyze the thermal pressurization of fluids,and that the effect of transient temperature change associated with laboratory operation should be evaluated.We also recommend measuring the external dead volume and expansivity of the pore pressure system for evaluating the undrained pore pressure response and injected volume during fluid injection tests.In this study,the internal pore pressure sensor reads the same pore pressure value as the external pump sensor in rock fractures having a good hydraulic communication with the pump system.However,the internal pore pressure sensor could provide valuable data for fast processes occurring under transient undrained conditions such as accelerating creep and dynamic slip.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

Yinlin Ji is supported by the Research Fund of the State Key Laboratory of Coal Resources and Safe Mining,China University of Mining and Technology,China(Grant No.SKLCRSM21KF002).Yinlin Ji and Hannes Hofmann have also been supported by the Initiative and Networking Fund of Helmholtz Association(Germany)for the Helmholtz Young Investigator Group ARES(Contract No.VH-NG-1516).Wei Zhang(Penn State University)is appreciated for the fruitful and insightful discussion.We thank Tanja Ballerstedt for assistance with laboratory experiments in MTS system,Florian Zimmermann for sample preparation,and Christian Cunow for manufacturing fiber optic sensors.

Appendix A.Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.jrmge.2021.12.007.