Effect of the Rock Stress on the Water Jet Cutting Performance

A suitable diamond wire–water jet combination for underground extraction of granite is illustrated in Fig. 1, which shows the two phases of a cutting sequence aimed at isolating 8 commercial blocks. To cut the side faces of the blocks, a 3 m pilot hole is first drilled perpendicular to the front, from which a first slot can be started. The water jet lance, which supports and directs the nozzle towards the rock to be removed, moves forth and back, starting from the pilot hole and proceeding parallel to it. After each cycle (two strokes) the lance is moved sideways, by incremental steps, thus progressively extending the first slot’s surface.

All subsequent slots can be started from the first, following a convenient order (first phase). Once all the side slots are completed, individual blocks can be extracted by cutting the back faces with the diamond wire machine (second phase): two vertical cuts in the example of Fig. 1 are required to obtain blocks of proper width (1.5 m).

In the oscillating version of the water jet technology, the lance that holds the nozzle is animated by an oscillatory motion around its own axis (frequency f and angular amplitude α) and a transverse motion (velocity v_t) in the direction of the groove generated by the jet itself (in z-direction) (Reichman 1978). The nozzle is located at a given distance S (in y-direction) from the groove’s bottom surface, upon which the jet impacts with a fixed angle β (the standoff distance is the distance from the nozzle to the jet impact point, along the direction defined by β (Fig. 2)). A simplified 3D representation of the operating parameters above described is reported in Fig. 3.

The lance travels along the groove in both directions alternately: one cycle (two strokes) generates the increment d of the groove depth (deepening); at each cycle the lance is approached to the groove’s bottom by a distance a. The excavation performance is given by the areal cutting rate CR, which is the product of the lance’s transverse velocity v_t by the number of cycle in the unit of time by the lance’s advancement a. The cut efficiency is expressed by the amount of energy required to remove the rock unit volume (J/cm³) (Summers 1991; Bortolussi 1992).

For a given rock and constant transverse velocity, the cut develops in steady conditions when the groove’s deepening d equals the lance’s advancement a; under those conditions, the standoff distance (SOD) remains constant (i.e., equal to the optimal set-up value). If the lance’s transverse velocity is too high, the cut’s deepening becomes lower than the lance’s advancement and SOD progressively decreases, until the lance collides with the bottom of the groove. If the lance’s transverse velocity is too low, the cut deepening becomes greater than the lance’s advancement and a progressive increase of SOD occurs; this corresponds to a reduction of both the cutting efficiency and the cut’s deepening, which goes on until d equals a again; from this point onwards, the cut proceeds in steady conditions, with a SOD greater than the optimum set-up value and both the efficiency and the cutting rate CR lower than maximum attainable (Hood et al. 1990; Rehbinder 1977).

In compressed rocks, the reduction of the resisting cross-sectional area due to the slot’s development generates a progressive increase of the compressive stress acting at the groove’s bottom and a corresponding progressive reduction in the cut’s deepening d: a geometric limit condition is reached, beyond which d becomes less than a; from this point on, the continuation of the cut with the same transverse velocity can only take place by reducing the lance’s advancement in accordance to the decrease of the cut’s deepening, with a corresponding reduction of the cutting rate.

In tense rocks, the reduction of the resisting cross-sectional area due to the cut’s advancement generates a progressive increase of the tensile stress acting at the groove’s bottom and a corresponding progressive increase of the cut deepening, up to the limit beyond which d becomes greater than a; from this point onwards, the continuation of the cut with the same transverse velocity and at the maximum efficiency requires the increasing of the lance’s advancement, with corresponding increment of the cutting rate.

According to the forethoughts above, the stress limit condition is defined as that stress beyond which the maintenance of the optimal SOD implies the modification of CR: a reduction in case of compression stress and a raise in case of tensile stress. In compressed rocks and for a given CR, the stress limit condition represents the value of the stress up to which the rock can be cut and beyond which the CR must be reduced. In tense rocks and for a given CR, it represents the level of stress up to which the rock is cut with the maximum efficiency and beyond which the CR has to be increased to preserve the efficiency level.

It is worth noticing that CR can be modified by adjusting either the lance’s advancement a or the lance’s transverse velocity v_t. In the experimental test described in the following sections of the article, CR has been modified by adjusting v_t: higher values were used to reduce CR and lower values to increase CR, with a fixed value of the lance’s advancement a.

In underground production of squared blocks, the first excavation phase consists in the opening of an entry tunnel, starting from the hillside, until the planned quarrying area is reached; then the tunnel is further enlarged leaving the pillars to support the roof. In case of thick deposits, the excavation can be further developed below the level of the access tunnel: a large chamber is progressively created by exploiting the rock layers with a downwards sequence.

The compression vertical component of the original stress acting in the rock can be quite high, depending of the tunnel depth; during the tunnel advancement, its value at the bottom of the horizontal grooves realized to extract the rock blocks is further increased as a result of the vertical stress redistribution. The same phenomenon occurs at the chambers’ walls during their enlargement. As a result of the extraction of the first layer of the chamber (roof layer), the compression stress in the rock volume below is almost thoroughly relieved. Consequently, during the following cutting operations, the stress at the bottom of the grooves is significantly lowered.

Tensile states of stress typically occur at the chamber roof, as a function of the roof span and the ratio (k) between the horizontal and vertical stress components (k = σ_h/σ_v). In the typical geometric configurations of underground quarries it is rare to cut blocks subjected to tensile stress; on the other hand, based on the research results, tensile stress could be artificially generated to improve the water jet slotting performances.

Three series of cutting test have been performed on granite specimens of Rosa Beta quarried in Sardinia (Italy), first unstressed and then subjected to compression and tensile stresses, aimed at evaluating the stress limit condition at which a predefined cutting rate can be maintained. The experiments have been repeated for three jet generation pressure (100, 160 and 200 MPa).

The physical and mechanical properties of the granite samples are synthesized in Table 1.

The experimental set-up installed at the Water Jet laboratory of DICAAR (Department of Civil and Environmental Engineering and Architecture), in Cagliari University (Italy), consists of the following components:

Table 2 reports the operative parameters of the oscillating water jet. Figure 4 represents the water jet lance and its slotting action on an unloaded sample.

The first test series has been performed on unloaded samples, aimed at assessing the reference cutting rate at the zero stress condition. In this case, the deepening of the cut does not cause any change in the state of stress within the rock, so that CR only depends on the material properties and the water jet operative parameters. The maximum value of CR is estimated by adjusting the lance’s transverse velocity as to obtain the equivalence between the cut’s deepening d and the lance’s advancement a. Table 3 reports the experimental plan for the unloaded samples.

Table 3

The second series of test has been carried out by subjecting the granite samples to an evenly distributed lateral compression, according to the scheme in Fig. 5: in that condition, the slot progress causes the increment of the compression stress in the layer of rock where the jet acts (grove’s bottom), so that a progressive reduction of the cutting rate is required. Table 4 reports the succeeding values of the cutting rates (CR*) to be adopted during the test, which were determined after some preliminary trials.

Aimed at evaluating the limit slot depth (i.e., maximum depth up to which a predetermined cutting rate CR* can be maintained), the test starts at the highest suitable CR for a given water jet generation pressure (e.g., 0.45 m²/h with p = 100 MPa) and is then carried out to the depth at which a reduction of SOD can be observed. To this purpose, at each stroke the lance is stopped and the distance between the nozzle and the mean plane of the slot bottom is measured. At that point, CR is lowered to the subsequent predefined value (CR* = 0.36 m²/h) and the cut continues until a new decrease of SOD becomes apparent. Again CR is dropped to the lower consistent value (CR* = 0.30 m^2/h) to proceed with the cut on the same slot. The test ends when the limit slot depth is reached with the lowest CR* in the experimental plan. The procedure above described has been followed to perform the experimental test with applied lateral loads of 250 and 500 kN. Six granite samples were used for each jet generation pressure. The lance’s advancement a was fixed at 2 mm and the cutting rate was adjusted by modifying the lance’s traverse velocity. The experimental plan for compressed samples is reported in Table 5.

With the third series of test, the alleged favorable effect of the tensile stress was investigated by applying a flexural load, as illustrated in Fig. 6. Two different load conditions were taken into consideration: F = 50 kN and F = 100 kN (total force acting on the sample), with a water jet generation pressure of 100 MPa. According to the experimental plan in Table 6, two cutting rates were selected for each load condition: 0.69 and 0.73 m²/h for F = 50 kN; 0.84 and 1.0 m²/h for F = 100 kN. As for the case of compressed samples, the lance’s advancement a was kept unchanged (2 mm per cycle) and the cutting rate progressively adjusted by modifying the lance’s transverse velocity v_t.

Because the progress of the slot causes a growth in the tensile stress, the cut is started at the predefined lowest value of CR* and continues until the first increment of SOD is registered. The cutting rate is then raised up to the next step and the test continues.

Test on the unloaded samples is meant to estimate the maximum obtainable cutting rate for each jet generation pressure (p = 100, 160, 200 MPa). The resulting CRs are assumed as reference values for the analysis of the alterations induced by the state of stress. For each water jet generation pressure, Table 7 reports the optimum values of the transverse velocity v_t and the cutting rate CR, expressed as average of the three tests (a, b, c).

Test on compressed samples is meant to estimate the limit slot depths up to which the preset CR* can be maintained. The results of the experimental test on compressed samples are reported in Table 8, again in terms of mean value.

Test on samples subjected to tensile stress is meant to estimate the limit slot depths at which CR* must be increased, to maintain the original level of efficiency. Table 9 reports the results of the experimental test carried out on tense samples with a jet generation pressure of 100 MPa. In Fig. 7, an example of the grooves generated by the water jet is given.

The two models represent a 5 cm thick slice of the sample (along the z-direction), as indicated in Fig. 8. The stress acting in the volume removed by the water jet at each cycle is calculated as average of the normal components (σ_xx) acting on the elements of the calculation grid included in 2 mm of thickness (Fig. 9).

The rock material has been modeled as linear elastic, considering the physical and mechanical properties of Rosa Beta reported in Table 1. The state of stress was calculated for each of the experimental limit configurations (i.e.: experimental limit depths).

The experimental research hereby discussed is meant to define the dependence of the cutting rate CR on the stress component σ_xx acting in the volume of rock to be removed. If, for a given value σ_xx, the Equilibrium Cutting Rate ECR indicates the value of the cutting rate at which neither an increase nor a reduction of SOD is observable, the continuous curve in Fig. 10 conceptually describes the relationship between the two variables, where each value of ECR is related to one and only one value of σ_xx, i.e., given a value of σ_xx there is only one value of the CR for which the cut proceeds without variation of the SOD. Figure 10 also describes the theoretical relationship between the state of stress and the depth of the cut.

For a prearranged cutting rate CR*, the experimental tests are meant to explore the range of σ_xx (Δσ_xx) where the cutting proceeds without any appreciable variation of the stand-off distance SOD. Within that range, the difference between CR* and ECR is not technically measurable. To this regard, in the adopted experimental procedure, the CR* values have been set and the ranges of stress (Δσ_xx) in which they could be maintained have been investigated. In this ranges the preset values of the cutting rate (CR*) can be technically assumed as ECR. The left limit of the range Δσ_xx (higher value of compressive stress and lower value of tensile stress) represents the sustainability limit stress (Sσ_xx), beyond which a reduction of CR is necessary to proceed with the slot (Fig. 10). The right limit of the range Δσ_xx (lower value of compressive stress and higher value of tensile stress) is the efficiency limit stress (Eσ_xx), beyond which an increase of CR is necessary to maintain the maximum cutting efficiency. In practical applications, in fact, the continuous theoretical curve σ_xx–ECR in Fig. 10 becomes a step function, which indicates the range of σ_xx where a given CR* is workable: the width of the steps is Δσ_xx, the height is twice the appreciable value of the difference ECR-CR* (segment aa' in Fig. 10).

The diagram of Fig. 11 summarizes the relationships between the deepening of the groove in the rock mass (increase of Z), the change in the σ_xx (Z) stress component acting within the volume of rock to be removed and the cutting rate CR(σ_xx).

STEP1. The initial value of the normal stress component is σ_xx0, which is also the Efficiency limit stress (Eσ_xx) for the first preset value of the cutting rate CR1 (Eσ_xx (CR1) =  σ_xx0). As the cut proceeds from Z₀ to Z₁, the compressive stress component σ_xx progressively increases up to the value σ_xx1, which corresponds to the Sustainability Limit Stress (Sσ_xx) for the first preset value of the cutting rate (CR1).

STEP2. The cutting rate is reduced to CR2, as the new value σ_xx1 of the compressive stress is the Eσ_xx for CR2 (Eσ_xx (CR2) =  σ_xx1). The cut proceeds from Z₁ to Z₂, while the compressive stress reached the value σ_xx2, which is the Sσ_xx for CR = CR2.

STEP3. The cutting rate is further reduced to CR3, as the new value of the compressive stress (σ_xx2) is the Eσ_xx for CR3 (Eσ_xx (CR3) =  σ_xx2). The cut proceeds up to the end (Z = Z3) with CR = CR3.

The experimental research hereby discussed allowed the estimation of the Sustainability limit stress Sσ_xx for the compressed samples and the Efficiency limit stress Eσ_xx for the tense samples. The comprehensive experimental plan is reported in Table 10, with the values of CR* and the correspondent limit values of Eσ_xx and Sσ_xx. For the three generation pressures under investigation, Fig. 12 shows the interpolating curves of the experimental results, which clearly highlight the influence of the state of stress on the cutting rate: in compressed rocks, the increase of the absolute value of σ_xx determines a reduction of the maximum CR; in tense rocks, the increase of the absolute value of σ_xx determines an increase of the maximum CR. According to the cutting test, the CR obtainable in unloaded samples of granite (0.63 m²/h) is halved (0.30 m²/h) when the compressive stress increases up to − 26.8 MPa and becomes l m²/h when the tensile stress reaches + 8 MPa.

The mean slope of the interpolating curves (ΔCR/ Δσ_xx) in Fig. 12 also proves that the influence of the state of stress on CR is less relevant the greater the energy of the jet (higher generation pressure) and becomes negligible with generation pressure of 200 MPa. Table 11 reports the value of the mean slope ΔCR/ Δσ_xx for each generation pressure in the experimental plan.

The results confirm the general relationship between CR and the state of stress, which has already been discussed in previous studies, as well as the role of the generation pressure (Ciccu 1998; Ciccu 2003). They also clarify the concepts of the Efficiency and Sustainability Limit Stress in the step function that approximates the conceptual continuous CR–Stress function.