under hood hydraulic pump free sample

Free pressure oscillations during pumping operations in boreholes may potentially constrain hydraulic characteristics of the surrounding material. These damped oscillations occur when flow rate is suddenly changed, and their period and decay rate depend on the hydraulic properties of the entire hydraulic system: the porous medium, a section of the borehole, and/or the injection line, depending on test set-up. There have been previous attempts to estimate transmissivity values from free pressure oscillations that occurred during slug tests in open boreholes. The analysis used did not account for viscous losses due to the fluid interacting with the borehole wall. In contrast, dispersion relations of flow waves in a tight borehole (i.e. a cylindrical hole in an impermeable medium) account for wall friction. We extend a previous analytical treatment of flow waves by changing the boundary condition of the fluid velocity at the borehole wall to include fluid exchange between borehole and porous medium. In addition, we performed numerical modelling of waves propagating in boreholes with impermeable and permeable walls to assess the effect of the assumptions behind the analytical solution. We established how to distinguish cases in which the flow into the porous medium affects the oscillation characteristics (suitable for a hydraulic analysis) from those in which the equipment properties dominate the observations. Applying our methods to a range of field observations yielded plausible hydraulic property values of the rock volume surrounding the borehole.

Several approaches employing observations of oscillatory pore-fluid pressure or flow-rate in boreholes have been followed to obtain hydraulic properties of permeable media (e.g. Bredehoeft 1967; Hsieh et al. 1987; Rasmussen et al. 2003; Renner & Messar 2006; Audouin & Bodin 2007; Guiltinan & Becker 2015; Cheng & Renner 2018). Forced oscillations constitute externally controlled excitations of the hydraulic system causing responses that are easily distinguished from background perturbations using time-to-frequency transformations (e.g. Renner & Messar 2006). Forced oscillations, with a range of frequencies, might be induced by pumping operations (Rasmussen et al. 2003; Cheng & Renner 2018) or natural processes, for example tides (e.g. Bredehoeft 1967; Hsieh et al. 1987), barometric loading (e.g. Lai et al. 2013) and seasonal variations in precipitation (e.g. Saar & Manga 2003). Free oscillations, excited when an oscillator is displaced and quickly released (e.g. Halliday et al. 2011), known for a long time in tubes (Frizell 1898), reveal the natural frequency of the system. Free pressure oscillations were observed in boreholes after passing seismic waves (e.g. Bredehoeft et al. 1966) or after a rapid change in pumping parameters during hydraulic well testing, for example a slug test (e.g. Audouin & Bodin 2007; Krauss 1974; van der Kamp 1976; Kipp 1985).

Free pressure oscillations have mostly been considered a side effect when occurring during pumping operations in boreholes. The few previous models for these oscillations, aiming at the determination of hydraulic properties, interpreted the well-aquifer system as a classical mass-spring oscillator, relating the coefficients of a second-order damped oscillation equation to the aquifer parameters. The fluid in the borehole corresponds to the mass and the borehole storage capacity, providing a linear restoring force, to the spring. The storage capacity differs for open (Krauss 1974; van der Kamp 1976; Mcelwee & Zenner 1998) and closed (Weidler 1996) boreholes. In open boreholes, variations in water level and pressure are coupled and thus gravity acts as restoring force, while in closed wells, pressure variations cause compression of the borehole fluid and elastic well deformation. In general, these mass-spring models do not account for damping due to the interaction of the viscous fluid and the borehole wall, here addressed as wall friction, but solely consider damping due to fluid exchange between borehole and porous medium. We address the latter mechanism as ‘leakage’ irrespective of the direction of flow, which may actually reverse during an oscillation. One of the models that included losses due to wall friction was presented by Mcelwee & Zenner (1998), who considered the viscous losses a nonlinear mechanism in the second-order damped oscillator. However, the leakage aspect of the solution does not account for the storage coefficient of the porous medium since the fluid is treated as incompressible. Fischer (2016) used the model derived by Weidler (1996) to determine transmissivity values from oscillations observed during pumping operations in the boreholes Horstberg and Groß Buchholz, Germany. The derived transmissivity values exhibit a plausible order of magnitude but a counterintuitive decreasing trend over the course of the performed hydraulic stimulations.

We aim to analyse free pressure oscillations recorded during pumping operations in boreholes to constrain hydraulic properties. The analysis of these oscillations requires the development of a theoretical framework that includes both loss mechanisms, leakage and wall friction. Therefore, we extended the analytical end-member model (Bernabé 2009) that accounts only for ‘wall friction’’ by adding the effect of leakage. We complement the analytical models by numerical modelling of flow waves in impermeable and permeable boreholes solving the Navier-Stoke equations for compressible fluids. The goal of the numerical simulations is to quantify the effect of the two loss mechanisms on frequency and damping coefficient of a free pressure oscillation, accounting for conditions not considered in the analytical solutions, for example the finite length of the borehole, advective terms in the Navier–Stokes equations, and boreholes with permeable sections, only. Using the established analytical solutions and considering the numerical results, we analyse the suitability of a range of field observations, gained with vastly different set-ups, for an inversion of hydraulic parameters.

We address mechanical waves in fluid-filled conduits, for example boreholes with radius R or fractures with aperture w, oscillating with frequency f as fluid-flow waves. The approximate conventional analysis familiar for organ pipes suggests that the finite length of a cylindrical hole, L, determines the frequency of standing fluid-flow waves as |${f}_0 = {c}_0/(4L)$| and |${\skew4\hat{f}}_0 = {c}_0/(2L)$| when both ends are and just one end is closed, respectively, where |${c}_0 = \sqrt {{K}_{\rm{f}}/{\rho }_{\rm{f}}} $| denotes the acoustic velocity of the fluid with bulk modulus |${K}_{\rm{f}}$| and density |${\rho }_{\rm{f}}$|⁠. A borehole in an impermeable porous medium (Fig. 1a) may correspond to either endmember model, i.e. open or closed, depending on the conditions at the wellhead. A permeable section (Fig. 1b) or an intersecting fracture (Figs 1c–e) may affect the oscillation frequency, as these hydraulic elements compose ‘openings’ comparable to the holes in flutes (Forster 2010), but will inevitably lead to an increase in the damping coefficient as a consequence of loss of fluid from the borehole. This ‘leakage’, by which fluid-flow waves excited in the borehole diffuse into fractures or permeable media, is just one example for the coupling of fluid-flow waves with processes in the solid penetrated by the borehole. The flow in axial direction of the borehole gives rise to viscous interaction at the solid wall, and the pressure variations in the fluid cause deformation of the solid. Damping of free pressure oscillations may arise from any combination of leakage, wall friction, and solid deformation.

We address boreholes in a rock whose hydraulic properties allow for fluid exchange with the borehole—on the characteristic timescale of the fluid-flow wave—as leaky. A fluid-flow wave travelling in a leaky borehole is attenuated due to the irreversible flow between the borehole and the permeable rock. We obtained a dispersion relation for a fluid wave traveling in a leaky borehole that extends Bernabé’s (2009) approximate analytical solution (1) by modifying the boundary condition at the borehole wall to account for radial flow in a purely diffusive process between the borehole and the porous medium (Appendix  B):

The mesh-dependence of the numerical results complicates the assessment of the significance of approximations made for the analytical solutions (1) and (2). Yet, from an a posteriori evaluation, we can say with confidence, however, that the long-wavelength approximation underlying the analytical solutions holds for all considered model geometries in our simulations.

The neglect of advective inertia terms in the approximate analytical treatment of the Navier–Stokes equations may lead to an underestimation of damping. Without a detailed explanation, Bernabé (2009) states that the neglect is not identical but in accord with the long-wavelength approximation. The classical dimensional analysis reveals that the neglect is valid for sufficiently large Strouhal numbers (Appendix  E). The Strouhal numbers of our numerical models, calculated analytically and a posteriori using frequency and amplitude of the simulated free pressure oscillation, are of the order of |$S{t}_{{\rm{axial}}} \sim {10}^9$| and |$S{t}_{{\rm{radial}}} \sim {10}^{11}$| for the axial and radial velocity component, respectively (Appendix  E) supporting the neglect of advective terms in the analytical solution (1).

The vast majority of simulations yield underdamped pressure oscillations that are in cases visibly multimodal in the time domain (Fig. 5). Owing to the step-like excitation, the pressure responses exhibit harmonics of the fundamental frequency (higher modes), as evidenced by the frequency spectrum (Fig. 5). Frequencies of the free pressure oscillations obtained in numerical simulations for tight (Table 3) and leaky boreholes (Table 4)are consistent with the theoretical prediction of the nominal eigenfrequency of the classic organ–pipe relation for a borehole with one open end for small damping coefficients. With increasing damping, frequencies of the numerical oscillations decreased from the nominal eigenfrequency reaching as little as 40 per cent reduction (Table 4). Frequencies of the numerical oscillations are lower than the ones expected for a damped harmonic oscillator, that is |${f}_\delta = \sqrt {{{(2\pi {f}_0)}}^2 - {\delta }^2} /2\pi $|⁠, by up to a factor of 1.5, but this discrepancy might result from the overestimation of the damping coefficients due to the identified meshing-problems (Appendix  D), and we thus cannot evaluate whether the boreholes can be approximated as harmonic oscillators.

Based on our theoretical and numerical analyses, we propose a sequential workflow for the evaluation of free oscillations recorded during pumping operations in boreholes. First, their spectral components, i.e. frequency and damping coefficient, are determined using Prony analysis, assisted by FFT and/or MFT. In the second step, frequency is assessed in the light of the conventional organ–pipe relations accounting for possible reduction due to damping. Actually, the two end conditions considered for the classic organ–pipe relations, i.e. open or closed, are just endmembers of the frequency of a flow wave in a borehole; the frequency is a continuous function of the storage capacity of a reservoir, |${S}_{{\rm{res}}}$|⁠, at the tube"s end(s) (Appendix  F). The two endmembers ‘open’ and ‘closed’ correspond to |${S}_{{\rm{res}}} \to \infty $| and |${S}_{{\rm{res}}} \to 0$|⁠, respectively. In the final step, the spectral components are compared to the dispersion relation for a tight borehole (1) to assess whether the damping is due to wall friction or due to leakage. If the damping coefficient is similar to that given by (1), hydraulic analysis is not possible, since the oscillations are likely governed by wall friction. A damping coefficient significantly above the limit given by (1) promises a sensible hydraulic analysis based on (2). In the sequel, we apply the proposed workflow to free-oscillation observations from five field tests with different set ups to investigate whether significant hydraulic properties can be deduced from them.

The analytical solution for fluid flow waves in infinitely long leaky boreholes (2) assumes that the radial fluid velocity is equal to Darcy"s law at any ‘depth’. For boreholes with finite length or leaky sections, the boundary condition for the fluid exchange between borehole and formation is accounted for by the volume balancing leading to effective permeabilities (3). In practical applications, the ‘true hydraulic’ length of a leaky section is often unknown motivating the use of hydraulic transmissivity that reflects the hydraulic characteristics of a borehole rather than a medium. The relation between borehole and medium properties reads |$T = {\rho }_{\rm{f}}g{L}_{{\rm{leaky}}}\kappa /{\mu }_{\rm{f}}$|⁠, where g is the gravitational acceleration.

The boreholes Groß Buchholz GT1 and Horstberg are located at the outskirts and about 80 km NE of Hannover, respectively. The two cased boreholes have lengths of about 4000 m. Borehole Horstberg is vertical while Groß Buchholz is deviated 30° on its final 800 m. A range of pumping operations aimed to stimulate perforated Bunter sandstone sections close to the bottom of the boreholes, of 2 m length (3902.5–3926.6 m) in Volpriehausen and 4 m length (3787–3791 m) in Detfurth formation for Horstberg, and 2 m length (3707–3709 m) in Volpriehausen formation for Groß Buchholz. The casing radius and thus the radius of the injection sections is 0.18 m for both, but the radius of the inner liner is 0.1 m for Horstberg and 0.18 m for Groß Buchholz.

The KTB main hole (Hauptbohrung) is located in Windischeschenbach, Bavaria, Germany. The borehole penetrates paragneisses, ortogneisses, and metabasic rocks (Emmermann & Lauterjung 1997). The cased section of the borehole has a length of 9030 m and the radius of the inner injection liner was 0.14 m. The open-hole section has a nominal radius of 0.18 m and a length of 70 m between 9030 m and the final depth of 9100 m. At the start of the hydraulic testing, a pulse test was carried out to assess the integrity of the inner liner when it was still closed by a burst disc at its lower end. After the disc was brought to failure, several hydraulic fracturing tests were performed in the open-hole section followed by 5 pulse tests during which the pressure response exhibited oscillatory behaviour.

During the pumping operations in borehole GT1 (Groß Buchholz), the early oscillations (1 to 13) exhibited frequencies of about 0.22 Hz close to the Nyquist frequency of the recording (Table 5). After the injection of a large fluid volume, during which the pressure critical for opening pre-existing fractures of 35 MPa was exceeded (Fischer 2016), the frequencies started decaying slightly (Fig. 6a). Further injection and production cycles and the second stimulation enlarged the fracture area (Pechan et al. 2014, 2015) and in their wake, the fracture apparently lost its ability to close fully even at borehole pressures below the nominal opening pressure. The oscillations during this phase showed a continuous decrease in their frequencies from 0.13 Hz (oscillations 19 to 48 in Fig. 6a) to 0.10 Hz (oscillations 49 to 63 in Fig. 6a). The damping coefficient slightly increased towards the end of the operations (Fig. 6a top).

Damping coefficient (top) and frequency (bottom) of the free pressure oscillations observed for boreholes (a) Groß Buchholz and (b) Horstberg during pumping operations as a function in their chronological order. The red data point denotes the observation during the pulse test in borehole Horstberg. The frequencies of oscillations depend on borehole mean pressures. The nominal eigenfrequency |${\hat{f}}_0 = {c}_0/(2L)$|⁠, indicated in a horizontal red line, for both boreholes with a length of 4000 m is 0.19 Hz .

The frequencies of the free pressure oscillations recorded in Horstberg and Groß Buchholz, when the present fractures were presumably closed or at least partially closed, are close to the eigenfrequency|${\skew7\hat{f}}_0 \sim 0.19$| Hz of a tube with a length of 4000 m and two closed ends (Figs 7a and b). The damping coefficients associated with these frequencies are between 2 and 10 times higher than the limit posed by (1) for Groß Buchholz (Fig. 7a), while the deviation is less than a factor of 2 for Horstberg (Fig. 7b). In contrast, the damping coefficients associated with frequencies around 0.1 Hz, observed when fractures were presumably open, exceed the tight-borehole damping coefficient by 20 to 30 times for Groß Buchholt and Horstberg (Figs 7a and b). Therefore, we inverted only these latter pairs of frequency and damping coefficient for hydraulic properties using the model for leaky boreholes (2).

The spectral components suitable for hydraulic analysis give effective permeabilities of |${\sim} 10^{-17}$| to |${\sim} {10}^{ - 13}$| m2 for either borehole, unaffected by varying apparent porosity from |$1 \times {10}^{ - 2}$| to |$1 \times {10}^{ - 1}$| (Fig. 8a). These effective permeabilities correspond to effective fracture apertures of |${\sim} {10}^{ - 5}$| to |${\sim} {10}^{ - 4}$| m using the relations given in (3) for both axial and radial fractures, matching the effective fracture aperture of about |${\sim} {10}^{ - 4}$| m calculated from the fracture transmissibility values reported by Pechan et al. (2015) for Groß Buchholz. The wide range of permeability seems to reflect its pressure dependence (Fig. 8b and c). Pressure increased by 45 MPa with progressing pumping for Groß Buchholz (Fig. 8b), but decreased by 25 MPa for Horstberg (Fig. 8c). The sensitivity of permeability to fluid pressure can be represented by the permeability modulus |${K}_k = \partial {p}_{\rm{f}}/\partial \ln k$| (Yilmaz et al. 1994), a small value of |${K}_{\rm{k}}$| indicating a strong dependence of permeability on fluid pressure. The apparent permeability modulus deduced from using the borehole pressure, an upper bound for the fluid pressure in the tested fractures, ranges between 3.5 and 10 MPa and between 15 and 85 MPa for Horstberg and Groß Buchholz, respectively (Fig. 8c). While our approach likely overestimates the true permeability moduli, because we used changes in the borehole pressure that likely exceed changes in the fluid pressure along the fracture, the gained values are comparable to permeability moduli of fractured Bunter sandstone reported by Hernandez Castañeda (2020) and of different types of fractured rocks reported by Kranz et al. (1979) and Raven & Gale (1985).

Previous spectral analysis of free pressure oscillations in Groß Buchholz GT1 using Weidler"s (1996) model yielded a decrease in transmissivity during the course of the stimulation (Fischer 2016). This result is not only counterintuitive since transmissivity values should increase during hydraulic stimulation as a consequence of the creation of new fractures and/or the shearing of pre-existing ones but also at conflict with the independent conventional analyses of observed pressure transients (Pechan et al. 2015).In the light of the derived analytical solution (2), the increase in damping coefficient observed in the course of the stimulation (Fig. 7a) indicates that transmissivity has actually increased, i.e. our treatment resolves the problems apparently associated with the oscillation analysis. The decrease in permeability over the course of pumping operations in Horstberg is likely due to the coeval decrease in mean injection pressure (Fig. 8c).

The free oscillation that occurred, when the burst disc was still intact in the KTB main hole, had a frequency in agreement with the eigenfrequency|${\skew7\hat{f}}_0 \sim 0.086$| Hz of a 9030 m long tube with both ends closed. The damping coefficient for this oscillation is about two times the value of the theoretical prediction for a tight and rigid system (Fig. 7c). The oscillations recorded after the inner liner was connected to the open-hole section exhibited higher frequencies and ten times higher damping coefficients values than observed before (Fig. 7c). Contrary to the observed increase in frequency, the increase in length of the hydraulic system associated with the removal of the burst disk should result in a minor decrease in frequency. The increase in damping coefficient beyond that given by (1) suggests that leakage into the metamorphic rock rather than wall friction controls pressure wave attenuation after removal of the burst disk.

The characteristics of the free pressure oscillations that occurred during pumping operations after the disk failure correspond to effective permeabilities of about |${10}^{ - 14}$| m2 according to (2), see Fig. 8(d). The effective permeability does neither increase with the progression of the pumping nor with the modest increase of injection pressure by ∼5 MPa (Fig. 8d). Shapiro et al. (1997) estimated a permeability of |${\sim} {10}^{ - 16}$| m2 from the growth of the cloud of induced seismic events, assuming an effective storage capacity of |${\sim} {10}^{ - 14}$| Pa–1. The value for the storage capacity used by Shapiro et al. (1997) is two orders of magnitude lower than the value assumed for our analysis, i.e. |$\phi /{K}_{\rm{f}} = 4 \times {10}^{ - 12}$| Pa–1, corresponding to an apparent porosity |$\phi = 1 \times {10}^{ - 2}$|⁠. For the specific storage capacity used by Shapiroet al. (1997), i.e. using |${s}_{{\rm{eff}}} = 5 \times {10}^{ - 14}$| Pa–1, the predictions by (2) shift downwards leading to an increase in permeability obtained from the free oscillations and thus an even larger difference to the estimate reported by Shapiroet al. (1997). Using the relations given in (3), an effective permeability of |${10}^{ - 14}$| m2 corresponds to fracture apertures of 0.2 or 1 mm for a pair of axial fractures (assuming |${L}_{\rm{f}} = 70$| m) or a radial fracture, respectively. The comparison of results is not without problems since the analyses by Shapiroet al. (1997) is indirect. However, differences between the estimates for permeability may reflect scale-dependence of permeability (e.g. Song & Renner 2006; Boutt et al. 2012; Kinoshita & Saffer 2018), since free-pressure oscillations are associated with a penetration depth |${r}_{\rm{p}} \sim \sqrt {{\kappa }_{{\rm{eff}}}/({\mu }_{\rm{f}}s{}_{{\rm{eff}}}{f}_0)} $| of about 5 m, while the observed seismic cloud covers distances from the borehole beyond ∼100 m.

Experiments with the double-packer probe of Solexperts GmbH, Bochum, Germany, were carried out in boreholes in Freiberg, Germany, and Hong Kong, China, and in a steel tube in the course of a calibration test. The probe consisted of two inflatable packers isolating an interval of 0.7 m length connected to the pump at the surface by straight tubes in Freiberg and by a coiled tubing in Hong Kong. At either site, the packers were connected to the pump by a coiled tubing. The interval pressure was measured with an uphole sensor as well as a downhole sensor (Fig. 9) whereas a single uphole sensor was used for the packer pressure. During field tests and calibration tests, free oscillations were excited when the pump valve was rapidly opened or closed.

As part of the STIMTEC project (Renner et al.2020; Jiménez Martínez & Renner 2021; Boese et al. 2022), stimulation tests were performed in a 63 m long borehole with a radius of 0.038 m dipping 15° downwards starting from a horizontal tunnel in the research mine Reiche Zeche in Freiberg, Saxony, Germany. The penetrated rock is a gneiss. Ten intervals were successively isolated by the double-packer probe and connected to the pump by a 10 m long hose and an injection line of steel tubes of 3 m length and 5 mm inner radius. The number of tubes used increased from 11 to 16 with increasing depth. Signals of the three pressure sensors (Fig. 9) were recorded with a sampling rate of 5 and 20 Hz during the field tests of intervals deeper than 33.1 m and at 28.1 m, respectively.

Hydraulic fracturing tests were performed in a vertical borehole (BH-CAV108) with a length of 279.68 m and a radius of 0.038 m penetrating granite in Sha Tin, Hong Kong (Gerd Klee 2018, personal communication). Ten test intervals of 0.7 m length were selected at different depths. The injection line was a coiled tubing of 300 m length with a radius of 0.004 m. Signals of the downhole pressure sensor were recorded with a sampling rate of 5 Hz.

In the field campaign at Reiche Zeche, 30 free oscillations occurred during pumping operations in the intervals at 28.1, 33.9, 37.9 and 49.7 m. For the last three intervals, free oscillations took place when the interval was depressurized, i.e. when the interval was briefly vented to air pressure (Fig. 11a). This venting process induces a backflow from the pressurized fluid in the fracture(s) to the interval at low pressure. The frequencies of these oscillations around 1 Hz differ from the range observed in the calibration test, that is 5–10 Hz, already potentially underestimated due to aliasing, even after accounting for the difference in tube length. Such a significant deviation from the fundamental frequency is at odds with the numerical results that revealed small reductions in frequencies of free pressure oscillations for leaky sections (see Appendix  D). We thus suspect that the determined frequency values are affected by aliasing. For a sampling rate of 5 Hz, the nominal eigenfrequency of 9 Hz (assuming one open end) will be recorded as an oscillation with an apparent frequency of 1 Hz (Penny et al. 2003).

The frequencies of the oscillations for the interval at 28.1 m lie in the range of the frequencies recorded in the calibration experiment (Fig.  10a). These frequencies are more reliable than the oscillation frequencies from the other intervals since a higher sampling frequency (20 Hz) was used, i.e. sufficient for one open-end (9 Hz). These oscillations took place during changes in flow rate at elevated pressure (Fig. 11b). The majority of their spectral components lie in the range of those of the calibration experiment (Figs  10a and b) but above the analytical curve of wall friction of the tubes with a radius of 0.005 m. The effect of wall friction associated with the borehole wall comprising the interval with a radius of 0.038 m is negligible. For the hydraulic analysis, we consider only the oscillations with spectral components lying outside the range covered by the calibration data (Fig. 10c).

The observed damping exceeds that of the coiled tubing alone indicating a contribution by leakage in the interval (Fig. 10c). Because our model does not account for changes in radius, we invert the spectral parameters using the dispersion relation for full-length leaky boreholes (2) with the interval radius. The transmissivity values of the borehole range between |${10}^{ - 10}$| to |${10}^{ - 7}$| m2 s−1 using an apparent porosity of |$\phi = 1 \times {10}^{ - 2}$| and a total injection length of 41 m (Table 5). These transmissivity values likely represent overestimations because of the damping in the narrow tube, yet they compare well with the range of transmissivities obtained from periodic pumping tests (PPT) performed with periods ranging from 40 to 900 s at different interval mean pressures after stimulation (Jimenez Martinez 2020).

The fundamental benchmark for the usefulness of observed free oscillations for hydraulic analyses is the frequency-damping coefficient relation for a tight borehole. The numerical results confirm the fundamental applicability of the analytical model of Bernabé (2009), although it was derived neglecting advective terms in the Navier–Stokes equation and the continuity equation, and nonlinear terms due to the fluid compressibility in the Navier–Stokes equation. Thus, the analytical model can be used as a diagnostic tool for the dominance of wall friction, as applies to the field study in Hong Kong. Only when observed damping coefficients exceed those associated with the attenuation of a flow wave due to wall friction a meaningful hydraulic analysis is possible. Then, it is still necessary to discriminate whether factors such as rock deformability, turbulence, etc. may account for variable and/or large damping coefficients relative to the predictions of (1). For typical porous medium-shear moduli of 10 GPa, corresponding to a shear wave velocity |${V}_{\rm{s}} \sim 2000{\rm{\, m\,s}^{-1}}$|⁠, or more, the effect of rock deformability on the damping coefficient is less than 10 per cent (Fig. 12). Yet, the elasticity of pipes with finite thickness, not treated by the model, enhances the effective deformability, i.e. it reduces the effective stiffness of the material, and may give rise to further dissipative wave modes (e.g. Kurzeja et al. 2016). Non-ideal borehole geometries, i.e. variable casing diameter or changing orientation (e.g. Groß Buchholz), and pipe connections (e.g. in Freiberg set-up), might cause turbulence during steady or pulsatile flow (Najjari & Plesniak 2018) and thus increase the damping coefficient in comparison to (1). Using damping coefficients increased by turbulence or other effects in (2) leads to an overestimation of the inverted effective permeability.

Avoiding aliasing problems when recording free-pressure oscillations requires sampling rates that exceed the typical values used during hydraulic testing of boreholes. The resolution of the acquisition tool should be adjusted to at least capture the expected fundamental frequency of the system. For example, instead of 2 Hz, Audouin & Bodin (2007) should have used a sampling frequency, higher than 6 or 12 Hz to capture the fundamental frequency of the 130 m long borehole for both ends closed or one open, respectively.

The derived dispersion relation (2) for flow waves in leaky boreholes with rigid walls that includes viscous losses by wall friction applies for flow waves with any frequency. The frequencies of the standing waves analysed here are relatively low compared to frequencies, for which the actual propagation of flow waves can be observed. Frameworks focusing on the propagative character are the classical theory of Stoneley waves in permeable (Tang et al. 1991b; Ou & Wang 2019) and fractured boreholes (Tang & Cheng 1989; Tang et al. 1991a), and the recently presented ‘selective resonance for radial fractures’ (Liang et al. 2017). The dispersion relation of Stoneley waves in an axially fractured borehole, somewhat inconsistently, employs the dispersion relation derived for a radial fracture to describe the leakage contribution (Tang & Cheng 1989). The analytical dispersion relations for Stoneley waves in permeable boreholes account for the effect of viscous forces on the fluid flow in the porous medium (or the fractures) but neglect the effect of the viscous interaction of the fluid and the borehole wall and, in contrast to our solution for leaky boreholes (2), give damping coefficients below the tight-borehole limit regardless of frequency (Fig. 13a). Thus, care must be taken when using the dispersion relations for Stoneley waves to derive hydraulic parameters from propagating waves (e.g. Tang et al. 1991b) when the damping coefficients are below the tight borehole limit.

We derived a dispersion relation for flow waves in boreholes penetrating permeable media, assessed it by complementary numerical simulations, and used it to constrain hydraulic properties from an analysis of the spectral characteristics of free pressure oscillations recorded during hydraulic tests. The flow-wave dispersion relation presented in this work is an extension of the solution for tight boreholes by an account for leakage into hydraulic conduits at the borehole wall, albeit strictly true only for homogeneous media due to the imposed radial symmetry and the underlying fluid-volume balancing between borehole and intersected hydraulic conduits. The current version of the extended dispersion relation restricts to the dominance of viscous forces for the pressure propagation in the porous medium, in accord with the range of actually observed frequencies, but an extension towards dynamic permeability is possible. Objectives for future work lie in addressing changes in borehole radius and deviations of the flow field in the borehole from radial symmetry associated with leakage into fractures.

The determination of hydraulic properties using the derived analytical solution is limited towards the low end by permeability around ∼10–18 m2 for typical borehole radii between 0.038 and 0.18 m. This limitation in resolution reflects that leakoff is just one of at least two dissipation processes. The numerical results demonstrate that the free oscillations can be overdamped for a typical borehole radius when permeability values are around ∼10–13 m2. However, when the wall friction contribution is high, i.e. for a borehole with small radius, overdamping can occur at lower permeabilities than that value. Thus, the window of permeability values that can be inverted from free pressure oscillations comprises about 5 order of magnitudes. The inverted permeabilities constitute effective values, representative of the borehole"s transmissivity, whose conversion toward real permeability values requires knowledge of the length of the leaky sections or of details of the fracture geometry.

The hydraulic tests of various field campaigns were performed with set-ups including double-packer intervals and cased wells with perforation or open-hole sections. The oscillations recorded in these tests exhibited frequencies and damping coefficients varying from |${10}^{ - 2}{\rm{ }}$| to |${10}^0{\rm{ Hz}}$| and |${10}^{ - 3}{\rm{ }}$| to |$5 \times {10}^0{\rm{ }}$| s–1, respectively. The observed frequencies were in good agreement with the nominal eigenfrequency of waves in tubes with corresponding length and end condition. The damping coefficients from numerical simulations and field campaigns were always similar to or higher than the analytical limit defined by wall friction in a tight borehole. Thus, this limit allows for the identification of the physical processes controlling the oscillations, that is viscous losses between the fluid and the borehole wall or fluid flow from the borehole to the porous medium. Effective permeability values inverted from the proposed dispersion relation for leaky boreholes fall in a plausible range between 10–18 and 10–14 m2. In particular, the use of coiled tubings might be problematic since it causes significant wall friction, whose contribution may thus dominate the damping coefficient. For shallow boreholes, the eigenfrequency of the free pressure oscillations might be higher than the Nyquist frequency of the data-acquisition systems typically used for hydraulic tests, leading to aliasing problems.

The strength of the proposed method lies in the simplicity of monitoring a single perturbation of flow and recording for tens of seconds the pressure response with sufficient sampling frequency. Changes in damping coefficients in the course of a stimulation operation are a strong indication that the corresponding oscillations actually bear information on hydraulic properties of the penetrated formation. In these cases, the ‘incidental’ data from free pressure oscillations provide ‘real-time’ evidence for fracture evolution during stimulation tests. Likewise, they can provide constraints on the dependence of hydraulic parameters on mean fluid pressure. The presented workflow could as well be applied to the evaluation of hydraulic properties of underdamped oscillations in slug tests.

The spectral parameters of the simulated pressure oscillations are available in the paper. The spectral parameters of oscillations recorded in the field are available in tabular form in the PhD thesis entitled ‘Hydraulic Changes induced by Stimulation’ by Victoria Jimenez Martinez (2020) published by Ruhr-Universität Bochum, University Library at https://doi.org/10.13154/294-7815.

Investigation into the evolution of hydraulic properties of fractured rocks at conditions representative of deep geothermal reservoirs, Ruhr-UNiversität Bochum.

\end{eqnarray*}$$(17)with the hydraulic diffusivity of the medium, |$D = {{{\kappa }_{{\rm{eff}}}} {/ {\vphantom {{{\kappa }_{{\rm{eff}}}} {({\mu }_{\rm{f}}{s}_{{\rm{eff}}}}}}} {({\mu }_{\rm{f}}{s}_{{\rm{eff}}}}})$|⁠, which comprises the effective specific storage capacity |${s}_{{\rm{eff}}}$| of the rock penetrated by the borehole. The hydraulic diffusivity determines how far a pressure perturbation reaches into a permeable medium in a specific time. Inserting ansatz (16) in (17) gives the ordinary differential equation

For a homogeneous permeable medium, the radial velocity (15) of the borehole fluid is identical to the Darcy velocity in the rock and the effective permeability is identical to the intrinsic or Darcy permeability, |${\kappa }_{{\rm{eff}}} = \kappa $|⁠. Heterogeneity, that is variations in hydraulic properties along the borehole, perturbs the velocity field in the borehole. We simplify the evaluation of the boundary condition (15) by solely accounting for the ‘averaged’ effect of fractures or open-hole sections on the volume balance between them and the borehole at |$r = R$|⁠:

We assume the cubic law to hold for fractures, i.e. the fracture permeability relates to aperture as |${\kappa }_{\rm{F}} = {w}^2/12$| (e.g. Zimmerman and Bodvarsson1996). For our simplistic averaging (23) that ignores the actual deviations of flow lines in the borehole from a radial direction, two diametrically opposite axial fractures, as for example created by hydraulic fracturing (Hubbert & Willis 1957), of aperture w and length |${L}_{\rm{F}}$|⁠, that is |${A}_{{\rm{F,ax}}} = 2w{L}_{\rm{F}}$|⁠, give

The simulations for boreholes in a rigid and tight medium document a positive correlation between the radius of the borehole and the amount of time until the underdamped oscillation vanishes (Fig. D4a). Spectral analysis of the numerical results reveals that the pressure oscillates with the nominal eigenfrequency of a standing wave in a tube with one open end |${f}_0 = {c}_0/(4L)$| for small values of damping coefficients, that is |$2\pi {f}_0 \gg \delta $|⁠. With increasing damping coefficient, for example due to a reduction in borehole radius, frequency decreases by up to 8 per cent compared to the nominal eigenfrequency (Fig. D4b and Table 3). The damping coefficients are overestimated—the more the larger |$R/\nu $|—due to the described mesh problems (Fig. D4c).

In leaky boreholes, overdamping, that is monotonous pressure decay, occurred for combinations of high permeability and long length of the leaky section and the occurrence is fairly consistent with the condition known for a harmonic oscillator |$\delta \ge 2\pi {f}_0$| (Figs D5a and b). The permeability beyond which overdamping occurs depends on the contribution of wall friction. For a borehole with a radius of 0.18 m and a length of 1000 m, an underdamped oscillation occurred when the permeability was higher than |$\kappa = {10}^{ - 13}$| m2. In contrast, for a borehole with the same length of 1000 m but a radius of only 0.01 m, the overdamping occurred for permeabilities above |$\kappa = {10}^{ - 17}$| m2. The borehole with the small radius is already close to overdamping from the viscous dissipation alone; little additional damping due to leakage suffices to reach overdamping.

The opposite end might correspond to anything between open or closed condition depending on the specific problem, for example the double-packer interval at the end of a coiled tubing is not a closed end but neither truly open. As common in hydraulic laboratory experiments, we address this opposite end as the downstream and characterize it by its storage capacity |${S}_{\rm{D}}$|⁠, with |${S}_{\rm{D}} \to \infty $| for an open end and |${S}_{\rm{D}} \to 0$| for a closed end. The boundary condition for the downstream, see appendix A in Bernabé (2009), is then given by

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

A hydraulic lift is a device for moving objects using force created by pressure on a liquid inside a cylinder that moves a piston upward. Incompressible oil is pumped into the cylinder, which forces the piston upward. When a valve opens to release the oil, the piston lowers by gravitational force.

The principle for hydraulic lifts is based on Pascal‘s law for generating force or motion, which states that pressure change on an incompressible liquid in a confined space is passed equally throughout the liquid in all directions.

The concept of Pascal‘s law and its application to hydraulics can be seen in the example below, where a small amount of force is applied to an incompressible liquid on the left to create a large amount of force on the right.

A hydraulic system works by applying force at one point to an incompressible liquid, which sends force to a second point. The process involves two pistons that are connected by an oil filled pipe.

The diagram below represents a simple version of the working mechanism of a hydraulic device. The handle on the right moves the incompressible oil, under pressure, from the reservoir to the high pressure chamber in the middle of the diagram. The ram moves up as the oil is pumped in.

The force generated in a hydraulic system depends on the size of the pistons. If the smaller of the two pistons is two inches and the larger piston is six inches, or three times as large, the amount of force created will be nine times greater than the amount of force from the smaller piston. One hundred pounds of force by a small piston will be able to lift 900 pounds.

The purposes of hydraulic systems widely vary, but the principles of how hydraulic systems work and their components remain the same for all applications. The most significant part of a hydraulic system is the fluid or liquid. The laws of physics dictate that the pressure on the fluid will remain unchanged as it is transmitted across a hydraulic system. Below is an explanation of each part of a hydraulic system.

Hydraulic Circuits control the flow and pressure of the liquid in the system. The image below shows all of the different parts of a hydraulic circuit.

Hydraulic Pump converts mechanical power into hydraulic energy. Hydraulic pumps create a vacuum at the pump inlet, which forces liquid from the reservoir into the inlet line and out to the outlet to the hydraulic system.

Hydraulic Motor is an actuator to convert hydraulic pressure into torque and rotation. It takes the pressure and flow of the hydraulic energy and changes it into rotational mechanical energy, similar to a linear actuator. The pump sends hydraulic energy into the system, where it pushes the hydraulic motor.

Hydraulic Cylinder converts the energy in the hydraulic fluid into force and initiates the pressure in the fluid that is controlled by the hydraulic motor.

Hydraulic Fluids transfer power in a hydraulic system. Most hydraulic fluids are mineral oil or water. The first hydraulic fluid was water before mineral oil was introduced in the twentieth century. Glycol ether, organophosphate ester, polyalphaolefin, propylene glycol, and silicone oil are used for high temperature applications and fire resistance.

Hydraulic lifts, in their many forms, have become an essential part of several industries from helping patients in and out of bed to specially designed lifts to help people board a bus. The number of uses of hydraulic lifts has been growing rapidly in recent years.

Medical lifts are lifting devices for surgical tables, hospital beds, and monitoring equipment. Hospital beds are a convenient means for moving patients from their rooms to treatment areas. Hydraulics control the height of all parts of the bed to make it more acceptable for hospital staff.

Post car lifts are a variation of automotive lifts. The vehicle to be repaired is suspended between two posts with hydraulic drives that have four arms. They are designed to lift any type of vehicle.

Hand pumped lifts are raised by a manual hydraulic hand pump and have a release lever to lower the load. They are very sturdy and maintenance free with the ability to lift one ton over six feet.

The main components of a VRC are a guide column, carriage, and hydraulic actuating mechanism. VRCs can be mechanical or hydraulic with the hydraulic version capable of lifting loads weighing between 3,000 pounds and 6,000 pounds. Hydraulic VRCs are less costly to install than mechanical ones and perfect for lifting applications that are limited to two levels under 25 feet and do not require continuous cycle use.

Rotating hydraulic lift tables are a special type of lift table that have a rotating turntable that is recessed into the surface of the table for positioning a load, which makes the table accessible on four sides. The turntable rests on anti friction bearings such that it can turn easily and effortlessly. When it is not needed, it can be locked in position. Rotating hydraulic lift tables can be low profile hydraulic tables that can be lowered to a few inches off the floor for easy access by a pallet jack or forklift.

As with all hydraulic tables, rotating hydraulic tables are made of highly durable materials capable of lifting close to a ton of products and items. They are designed to perfectly and precisely position a load to prevent workers from having to manually lift materials.

Low profile lift tables have a collapsed height of a few inches allowing the table to be loaded using a hand truck or forklift. Since they do not require a pit or indent in the floor, they can be used on upper floors as well as the main floor. To activate the hydraulic lift of the table, there is a foot switch or push button remote.an operator can use that raises or lowers the table to a comfortable working height.

High capacity hydraulic lift tables are heavy duty tools designed to lift loads up to 60 tons with lifting heights of 52 inches up to 92 inches and platforms from 4’ by 6’ up to 10’ by 22’ with customized larger platforms available. They have scissor legs and torque tubes that add stability and exceptional support and prevent load deflection or shift.

The number of scissor legs and hydraulic cylinders vary according to the design of the table and the manufacturer. As with other hydraulic tables, high capacity tables can be activated by a handheld pendant or foot switch with an upper travel limit switch. Possible additional features include tilt tops, powered turntables, V cradles, and corrosion resistant finishes to name a few.

High capacity hydraulic lift tables are the workhorses of lift tables and are designed to withstand the stress and constant use of heavy duty machinery.

Hydraulic lifts are constructed from steel and have precision accuracy. Their sturdy and durable design has made them popular in a wide variety of industries. Listed below are a few of the industries that rely on hydraulic lifts for their efficiency and ability to supply a great amount of force.

Electro-hydraulics is a common use of hydraulics in industrial applications. The main advantages of hydraulics are its rapid response times and precision. Plastic processing, metal extraction applications, automated production, machine tool industry, paper industries, loaders, crushers, presses, and the textile industry are some of the industrial uses of hydraulics. The image below is a hydraulic press from the plastics industry.

Mobile hydraulics have the advantage of being able to be moved to different conditions and situations. They are especially useful in the construction and building industries where hydraulics are used as cranes, excavators, backhoes, and earth moving equipment. Pictured below is a concrete boom truck using a hydraulic arm to unload concrete.

The automotive industry is the largest user of hydraulics. Production, repair, and internal components on cars all use hydraulics.The image below shows the use of hydraulic automation in the production of trucks.

Marine hydraulics deliver linear and rotary force and torque rapidly and efficiently. The three types of marine hydraulic systems are open, closed, and semi-closed. They are used for cranes, mooring and anchor winches, stabilizers, steering, thrusters, propellers, and platforms.

Components for aircraft have to meet strict standards before being approved for use. Hydraulic pumps and valves meet aircraft regulations and are an essential part of aircraft design and production. Wing adjustments, retraction and extension of landing gear, opening/closing of doors, brakes, and steering are all performed by hydraulics. The image below provides a list of some of the ways hydraulics are used on an aircraft.

Hydraulics are ideal for mining for the same reasons that they are used for other manufacturing operations. Power, controllability, reliability, and serviceability are necessities in mining because of the dangers that are involved. Unlike other manufacturing, mining works on a huge scale requiring massive equipment. The power and force provided by hydraulics fits the conditions.

Hydraulics lifts are heavy duty equipment that can supply a great deal of force. The Occupational Safety and Health Administration (OSHA) and the American National Standards Institute (ANSI) have specific requirements regarding the operation of hydraulic lifts. The first of those requirements is that operators must be an adult, over 18, that have been fully trained in the operation and dangers of the equipment.

B20.1 is a set of safety standards for conveyors and related equipment, under which fall vertical reciprocating conveyors that are designed to operate like an elevator for lifting huge materials between floors in a building. The standards of B20.1 apply to the design, construction, installation, maintenance, inspection, and operation of VRCs in regard to possible hazards and potential dangers. First published in 1947, B20.1 has been edited multiple times over the years to match the ever changing technology of VRCs with the last edits being made in 2021, which cover gates and enclosures.

The principle for hydraulic lifts is based on Pascal‘s law for generating force or motion, which states that pressure change on an incompressible liquid in a confined space is passed equally throughout the liquid in all directions.

The Occupational Safety and Health Administration (OSHA) and the American National Standards Institute (ANSI) have specific requirements regarding the operation of hydraulic lifts and training for operators.

NOTES: Pump inlet never to exceed 8 inches/HG vacuum; inlet oil velocity not to exceed 8 ft./sec. Engine speed not to exceed 1500 RPM without special approval; pressure spikes to never exceed 10% beyond maximum rating. Pressure ranges based upon degrees of belt wrap, can vary by kit.

NOTES: Pump inlet never to exceed 8 inches/HG vacuum; inlet oil velocity not to exceed 8 ft./sec. Engine speed not to exceed 1500 RPM without special approval; pressure spikes to never exceed 10% beyond maximum rating. Pressure ranges based upon degrees of belt wrap, can vary by kit.

NOTES: Pump inlet never to exceed 8 inches/HG vacuum; inlet oil velocity not to exceed 8 ft./sec. Engine speed not to exceed 1500 RPM without special approval; pressure spikes to never exceed 10% beyond maximum rating. Pressure ranges based upon degrees of belt wrap, can vary by kit.

NOTES: Pump inlet never to exceed 8 inches/HG vacuum; inlet oil velocity not to exceed 8 ft./sec. Engine speed not to exceed 1500 RPM without special approval; pressure spikes to never exceed 4400 PSI; Pressure ranges based upon degrees of belt wrap, can vary by kit.

Cutters typically have an aluminum-alloy housing with forged, heat-treated steel blades. The piston and piston rod are often made of heat-treated alloy steel. The cutters are used to cut or shear through materials such as sheet metal and plastic. Most often, they are used to cut through automobiles and other vehicles to free trapped passengers. Like the spreader, it can run off a gasoline-driven power unit. Jaws of Life systems can also be powered electrically, pneumatically or hydraulically.

Instead of arms, the cutter has curved, claw-like extensions that come to a point. Just like in the spreader, hydraulic fluid flows into a cylinder, placing pressure on a piston. Depending on the side of the piston that force is exerted on, the claws either open or close. When the piston rod is raised, the claws open. As the piston rod lowers, the claws of the cutter come together around a structure, such as a car roof, and pinch through it.

If you understand the operation of the spreader and cutter, the ram is going to seem about as complex as a pair of scissors (if scissors had hydraulics, of course). The ram is the most basic type of hydraulic system: It"s just a matter of using hydraulic fluid to move a piston head inside a cylinder to extend and retract a piston rod. If you look at some heavy construction equipment, like a backhoe loader, you"ll notice that rams are used to control the boom arm.

Hydraulics play an important part in many of the machines around us, but none may be as vital as the equipment known as the "Jaws of Life." These devices have been called upon to save thousands of lives in situations where a few seconds could mean the difference between life and death.