Study of the Interaction of a Hydraulic Fracture with a Natural Fracture in a Laboratory Experiment Based on Ultrasonic Transmission Monitoring

Abstract

This paper presents the results of experiments on the study of a hydraulic fracture’s interaction with a preexisting fracture. A distinctive feature of the conducted experiments is the ability to use ultrasonic transmitting monitoring to measure the fracture propagation and opening simultaneously with the pore pressure measurements at several points of the porous saturated sample. It allows us to obtain the pressure distributions at various experiment stages and to establish a relation between the pore pressure distribution and hydraulic fracture propagation and its interaction with macroscopic natural fractures. The possibilities of active ultrasonic monitoring have been expanded due to preliminary calibration experiments, which make it possible to estimate the fracture opening via attenuation of ultrasonic pulses. The experiment demonstrated the most complex scenario of fracture interactions when a hydraulic fracture intersected with a natural fracture and the natural fracture in the vicinity of the intersection was also opened. The additional complications arise from fracture arrangement: the hydrofracture was normal with respect to the base plane, while the natural fracture was slanted. This led to gradual growth of the intersection zone as the hydrofracture propagated. The experiments show that the natural fracture limited the fracture’s propagation. This was caused by the hydraulic fracturing fluid leaking into the natural fracture; thus, both the hydraulic fracture and natural fracture compose a united hydraulic system.

1. Introduction

Hydraulic fracturing remains the primary method of increasing oil inflow to wells [1]. Despite the many years of experience in using this method and the existence of various software for the hydraulic fracturing design (hydraulic fracturing simulators) [2], oil-producing and oilfield service companies often face problems during hydraulic fracturing, which are associated with insufficient elaboration of the physical models used in these simulators. There are many theoretical and experimental studies of the occurrence and propagation of hydraulic fractures [3,4,5,6,7,8]. All known models have limitations, which can be evaluated by conducting experiments on natural or artificial rock samples. To fulfill the requirement of an experiment’s scalability to natural conditions, it is necessary to choose the sample material and experimental conditions according to the similarity criteria [9].

An essential aspect of hydraulic fracturing modeling is considering the natural fractures of rocks. On the one hand, the natural fractures can increase the hydraulic fracturing efficiency. On the other hand, the interaction of hydraulic fractures with tectonic faults can lead to undesirable consequences in the form of earthquakes. These issues were considered in [8,10,11].

The first research studying the influence of natural fractures on the propagation of hydraulic fractures in a fractured environment dates back to the 1980s [12,13,14]. These papers describe experimental studies showing that the presence of natural fractures in rock can influence the trajectory of a hydraulic fracture. Based on their results, an empirical criterion was formulated [15] that relates the parameters of a natural fracture and a hydraulic fracture with their interaction scenarios—the branching of a hydraulic fracture, its stopping, or its movement—without changing the trajectory. This criterion was valid for hydraulic fracture orthogonal to natural fracture, but this limitation was removed in a subsequent study [16]. The main factors that determine hydraulic fractures’ interactions with natural fractures are the following: stress values at a sufficient distance from the fractures, the angle between the planes of the fractures, the tensile strength of the rock, and the friction coefficient. Semi-analytical models of the interaction of a natural fracture with a hydraulic fracture developed in [17,18] take into account the influence of hydraulic fracturing parameters (pressure and flow rate of the fracturing fluid and its rheology).

A unique installation has been created at Sadovsky Institute of Geosphere Dynamics of Russian Academy of Sciences which allows experiments to be carried out on samples of artificial porous material selected according to similarity criteria. The samples have the shape of disks, with diameters of 430 mm and heights of 72 mm; the installation allows for sample loading along three independent axes, creating pore pressure gradients, measuring the fluid pore pressure at several points, registering acoustic emission, and probing the sample with ultrasonic pulses. Several results of experiments conducted at this facility are given in the publications cited in [19,20,21,22].

This paper presents the results of experiments on the study of a hydraulic fracture’s interaction with a preliminarily created fracture that simulates a natural fracture. A distinctive feature of the conducted experiments is the ability to use ultrasonic sounding to measure the fracture propagation and opening simultaneously with the fluid pore pressure measurements at several points of the porous saturated sample. It allows for pressure distributions to be obtained at various experimental stages and for a relation to be established between the pore pressure distribution and the hydraulic fracture propagation and its interaction with macroscopic natural fracture.

Determining the geometric dimensions and azimuth of fractures is extremely important for evaluating the success of hydraulic fracturing operations. Inaccuracies in determining the aperture of the fracture opening have the greatest impact on the fracture length and height estimations. One of the methods for estimating the size of the opening is active acoustic monitoring. This method requires the use of observation wells near hydraulic fracturing, as well as VSP equipment, which makes the implementation of active monitoring an expensive undertaking. According to the published works [23,24], such studies in real hydrocarbon fields are rare. The assessment of the opening of fractures and the presence of loose areas is carried out mainly by calculations.

At the same time, the acoustic (or ultrasonic) monitoring of a hydraulic fracture can be easily realized in laboratory experiments; meanwhile, to obtain quantitative results, it is necessary to make calibration experiments. In our paper, we consider a relation between the attenuation of elastic waves passing through a fracture and the aperture of the fracture, which were obtained in special experiments.

Laboratory experiments studying the hydraulic fracture opening are presented in a number of papers [25,26,27,28,29,30,31,32]. In the paper cited in [25], the fracture opening was measured at the borehole and was compared with passing acoustic wave attenuations. In [26,27], acoustic wave diffractions at the tip of the fracture were used to obtain data on the hydraulic fracture aperture. An array of 32 ultrasonic receivers and 32 transmitters was used in [28] to study the hydraulic fracture propagation. The results of laboratory experiments on the propagation of hydraulic fractures in samples of rocks are considered in [29,30]. Measurements of the following parameters were made in that study: fracturing fluid pressure, acoustic emissions, elongation of the sample caused by the opening of the fracture, volume of fluid in the fracture, and temporal and spatial distribution of acoustic events. In experiments using transparent samples, the photometric method for measuring the opening and propagation of fluid-filled fractures was employed [31,32].

In the presented paper, the calibration experiments were carried out with special installation to assess the magnitude of fracture opening. A feature of these experiments was the simultaneous measurement of the changing aperture of the fracture and the amplitude of the ultrasonic pulses passing through it. A more detailed description of these experiments is given in [33].

2. Description of Experimental Installations

The research was carried out with two experimental installations. The main part of the work was carried out using a large-scale laboratory installation, and preliminary studies were carried out using small-sized samples. First, let us focus on calibration experiments.

The samples were made from a gypsum and Portland cement mixture in a ratio of 10:1, in both the main and in preliminary experiments. The samples for preliminary experiments had diameters of 104 mm and heights of 60 mm. A brass tube, 12 mm in outer diameter with a plugged end, was used as a cased borehole, vertically placed at the center of the sample. There was a fracture initiator with a diameter of 25 mm at the midpoint of the sample’s height (Figure 1), which was made of two layers of brass mesh with a cell size of 0.3 mm.

Two aluminum disks with integrated piezoelectric transducers were lubricated by a silicone compound to prevent leakage, and the sample was placed between them. The upper piezoelectric transducers were above the lower transducers. The scheme of experimental installation is shown in Figure 1.

In addition to the elements described above, there were four racks on the lower base for mounting LVDT sensors. The working rod of the LVDT sensors was in contact with the outer surface of the upper disk. The correct mutual arrangement of the disks was made with the help of vertical notches on the side surface of the upper disk. The installation was immersed in a cylindrical tank filled with silicone fluid with an approximate viscosity of 5 mPa·s; the sample was vacuumized and then saturated by this silicone fluid through its lateral surface. The assembly was installed and fixed in a hydraulic press.

A two-channel syringe pumping system was used to carry out hydraulic fracturing, ensure different values of the fracture opening, and maintain a constant pressure in the hydraulic press. One channel of the pumping unit provided hydraulic fracturing and a subsequent change in the fracture width. The hydraulic fracturing fluid was injected through the central tube. As a fracturing fluid, a medium-viscous silicone fluid with a viscosity of about 0.5 Pa·s was used. The other channel of the pump ensured the pressure maintenance in the hydraulic press. The relative displacements of the upper and lower disks were measured by LVDT sensors, digitized with step 0.35 s, and recorded on a computer. The measuring step of the displacement was 0.2 μm.

To study the fracture opening, two diametrically opposite piezoelectric transducers in the upper disk sent ultrasonic pulses every 20 ms, but with a delay of 8 ms relative to each other. All four receivers in the lower disk were used to register the pulses transmitted by two transducers in the upper disk, which allowed us to determine the oblique incidence of the ultrasonic pulse on the fracture. After amplification, the received pulses were digitized with a sampling rate of 2.5 MHz per channel and recorded on a hard disk.

A low-speed ADC recorded the injection fluid pressure and the pressure in the hydraulic press with a sampling frequency of 100 Hz. The recordings of pressure, ultrasonic pulses, and displacement were synchronized.

The main experiments were carried out on a triaxial loading unit developed at IDG RAS and described in detail in several publications [19,21,22]. A brief description of the installation is given below.

The installation consisted of two horizontal steel discs with diameters of 750 mm and thicknesses of 75 mm, between which there was a steel ring with a height of 70 mm and an internal diameter of 430 mm. The ring was placed in the groove of the lower base. The discs and the ring formed a working chamber with a diameter of 430 mm and a height of 72 mm. There were several holes in the discs and the ring, which were used for mounting a piezoelectric acoustic transducer, pressure sensors, and pumping fluids in and out. A scheme of the experimental setup is shown in Figure 2.

In the main experiment, the same material was used as in the preliminary experiments, consisting of a mixture of 10 parts gypsum and one part Portland cement. This mixture was mixed with water (0.65 L of water per 1 kg of mixture). The main mechanical and filtration properties of the model material are presented in

Table 1. Mechanical and filtration properties of model material.

Hydro-Mechanical Property	Value
Young modulus, GPa	2.4
Share modulus, GPa	1.0
Poisson ratio	0.2
Unconfined compressive strength, MPa	6.4
Tensile strength (Brazilian test), MPa	0.8
Permeability, mD	1.1
Porosity	0.5
Pore fluid viscosity, mPa∙s	5
Fracture fluid viscosity, mPa∙s	500

. The issues of similarity and adequacy of the model material used herein and the experimental parameters in relation to hydraulic fracturing in the field were considered in [21,22].

The surface of the lower base of the installation was covered with a thin layer (0.3 … 0.5 mm) of silicone compound, which reduced the tangential stresses arising at the lower interface of the sample and the base due to friction. After the curing of the compound, the model material was poured into the working volume of the installation with the top cover removed. The sample preparation was carried out in two stages to create an extended oblique joint in the sample simulating a natural fracture. First, a wedge-shaped insert made of polymethylmethacrylate with a wedge angle of 40° was placed in the working chamber of installation. A schematic image of the insert is shown in Figure 3. The insert was oriented so that its plane, forming an angle of 40° with the horizontal plane, was parallel to the Y-axis of the experimental setup. After one day of curing, the insert was removed. Then, the formed inclined surface was lubricated with a viscous silicone liquid to prevent sticking. After that, the remaining volume was filled with a second portion of the model mixture. For simplicity, we will call the resulting joint a natural fracture. After the curing of the model material, the sample was dried for 20 days.

Before assembling the installation, a rubber membrane was placed on the top side of the sample. The upper cover was mounted on the membrane with a small gap of about 2 mm. A rubber O-ring was placed at the periphery of membrane to provide tightness in the gap. This gap was filled with water after assembly of the experimental installation. During the experiments, a pressure produced by the buffer volume of compressed nitrogen was applied to water in the gap, providing vertical compressive stress in the sample. Four sealed chambers made of thin-sheet copper were mounted on the inner surface of the side ring to set horizontal stresses. The angular length of each chamber was 80°. The inputs of opposite chambers were connected. The necessary pressures in the chambers were set using a pumping unit operating in constant pressure maintenance mode. In this experiment, only one pair of chambers was used, located along the X-axis of the sample.

Before the sample’s preparation, a plugged brass tube with a diameter of 12 mm was inserted into the central hole in the lower disc, simulating a cased borehole with a slit in the middle. A rectangular fracture initiator was made of two layers of brass mesh with a cell size of 0.3 mm and inserted into that slit. The height of the initiator was 10 mm, and its length was 12 mm. The initiator was oriented along the X-axis of the sample towards the existing fracture in the sample. A schematic representation of the sample in the experimental setup is shown in Figure 4. Figure 5 shows the positions of the ultrasonic transducers, pore pressure measurement points, and lateral loading chambers. Two supplemental wells were located at points P5 and P15. They were used to vacuumize and saturate the sample with fluid. In the experiment, a low-viscosity organosilicon liquid with a dynamic viscosity of 5 mPa·s and a density of 918 kg/m³ was used as a pore fluid.

The same pumping system was used as that in the calibration experiment to carry out hydraulic fracturing and maintain a given constant pressure in the lateral loading chambers. The fluid was pumped into the central well at a constant flow rate to create a hydraulic fracture. For this purpose, an organosilicon liquid with a dynamic viscosity of 0.5 Pa·s was used. The pressure in the working well, the vertical pressure, and the pressure in the side chambers were measured using NAT-8252 pressure transducers manufactured by Trafag AG (Bubikon, Switzerland). The identical transducers were used to measure the pore pressure at the points shown in Figure 5. The signals from the pressure sensors were recorded using two low-speed ADCs with sampling frequencies of 1000 Hz.

For active acoustic monitoring during the opening of both a hydraulic fracture and a natural fracture, the piezoelectric transducers A5 and A6 were used as transmitters in the upper disc, while the piezoelectric transducers A11, A12, A13, and A15, located at the lower disc, served as receivers. The repetition period of ultrasonic pulses was 100 ms and the delay between them was set to 2 ms, which, on the one hand, made it possible to avoid overlapping signals from different transmitters, and on the other, made it possible to synchronize the entire set of received signals. The system of generation and registration of ultrasonic pulses was the same as that in the calibration experiment described above.

Attenuation of the amplitude of the ultrasonic pulses passing through the fracture depended on the value of its opening. The pulses were discriminated, and the maxima of their amplitudes were found. Then, these maxima were normalized by the average value of the pulse maxima at the beginning of the experiments. Finally, using normalized values, the fracture opening value was calculated with the help of calibration experiment results.

3. Execution and Results of Experiments

3.1. Calibration Experiments

To produce a radial horizontal hydraulic fracture perpendicular to the borehole, it was necessary to provide radial compressive stresses greater than the vertical stress. Four worm clamps were used to set the required radial stress, which was estimated by measuring the changing of the sample diameter due to clamp compression. The elasticity modulus of the model material was measured in specially conducted deformation tests; the result was 2.4 GPa. The change in sample diameter was ≈20 μm, so the radial stress was ≈0.5 MPa. The vertical compressive stress was set as 0.37 MPa. Thus, the horizontal stress exceeded the vertical stress.

Initially, the sample was loaded vertically, and approximately 5 s later, the fracturing fluid injection began with an initial flow rate of 5 mL/min. The pressure in the borehole increased until the moment of the hydraulic fracture initiation. When the fracturing started, the injection pressure dropped sharply and then slowly increased. Then, the injection flow rate was increased stepwise up to values of 7.5, 12, 18, 24, 30, 40, and 50 mL/min. The fracture width increased at every step of the increase in flow rate because the vertical pressure was constant. The injection pressure of the fracturing fluid slightly increased during the experiment. The continuous fluid injection into the fracture did not allow the fracture to close.

The fracture propagated unevenly in different directions due to the inhomogeneity of the sample. Thus, the fracture width was different at different points. It can be assumed that the fracture boundaries formed planes inclined relatively to each other. Assuming a down plane to be the base, the equation of the upper fracture plane was found by means of the least squares method using the displacement data at LVDT sensor points. This equation was used to recalculate displacements at crossing points of the fracture plane with “transmitters–receivers” lines, including in the cases of inclined intersections.

The ultrasonic pulses were discriminated using a threshold criterion. Then, the envelope of each pulse was calculated using Hilbert transformation, and the maximum value of the envelope was taken as the pulse amplitude. The change in the injection pressure, the measured hydraulic fracture width, and the amplitude of the transmitted ultrasonic pulse normalized to its initial value are shown in Figure 6.

The normalized ultrasonic pulse amplitudes were compared with the fracture opening value; the obtained relation is shown in Figure 7. An exponential function $A=a\exp \left(-\gamma w\right)$ can be used to approximate the relation, where A is the amplitude of the ultrasonic pulse, normalized by its amplitude at a fracture width of w = 20 μm. The attenuation coefficient was γ ≈ 8000 μm⁻¹, and the multiplier was a = 1.03. Note that the previously obtained results [33] relate to the direct incidence of the acoustic wave on the fracture. The angles of the ultrasonic waves’ incidence on the fracture in both the calibration and main experiments were approximately the same, which made it possible to use the obtained calibration dependence in the main experiment.

3.2. Main Experiment

After performing the preparatory procedures and establishing the vertical pressure and pressure in the lateral loading chamber along the horizontal X-axis of the installation, the hydraulic fracturing fluid was injected with a constant flow rate of 5 mL/min into the central borehole. The vertical pressure was maintained at 6 MPa, and the pressure in the chambers along the X-axis was 1.3 MPa. Figure 8 shows the first 300 s of pressure recording in the central well and in the vicinity of a natural fracture in the sample. The pressure increase in the central well lasted approximately 56 s. The maximum pressure was 8.5 MPa; this value can be assumed to be equal to the hydraulic fracturing pressure. After the injection pressure drop, the growth of pore pressures in the vicinity of the natural fracture began. The almost simultaneous onset and the close rates of these pressure increases at different points should be noted.

After the initial hydraulic fracturing, the upper cover of the experimental installation was removed to check the hydraulic fracture’s position. A photo of the sample in an open experimental setup, with the locations of the pore pressure measurement points, the acoustic emission receivers, and the place of the insert, is shown in Figure 9.

Even though the fracture initiator was asymmetrical and directed the “natural” fracture (to the left in the photos), two wings of the hydraulic fracture were formed. The left wing of the fracture in the initial section was directed along the X-axis in accordance with the initiator direction, after that, the fracture deviated from the original direction. This deviation was most likely caused by the peculiarities of the stress field related to the presence of the “natural” fracture in the sample. Also, as can be seen in Figure 9, the left branch of the fracture crossed the “natural” fracture, while its trajectory, visible on the surface, reached the lower edge of this boundary. The right branch of the fracture was formed without an initiator, most likely due to the flow of fracturing fluid along the outer surface of the casing of the central well. The direction of its growth was determined only by the peculiarities of the stress field in the vicinity of the well.

Considering the mutual location of the hydraulic fracture, the “natural” fracture, and the transmitter and receiver of the ultrasound, at the next stage of the experiment, upon re-opening the hydraulic fracture, “transmitter-receiver” pairs were selected. The selection criterion was the intersection of either the hydraulic fracture or the “natural” fracture by the “transmitter–receiver” line. Thus, for the A5 and A6 transmitter located in the upper disc, the receivers were A11, A12, A13, and A15, located at the base disc of the installation. Reliable results were obtained for only five pairs: A5–A11, A5–A12, A5–A13, A5–A15, and A6–A11.

Experiments on the re-opening of the hydraulic fracture were carried out at the same injection flow rate of 10 mL/min. Four experiments were conducted. The results of these experiments generally matched each other, so the following are the results of one of them. Figure 10 shows the time variations in the pressure in the injection well; the pore pressure at point P11, located near the end of the hydraulic fracture; and the magnitude of the fracture width. Note that the variation of the fracture width with time is very noisy despite treatment with a smoothing filter.

It can be seen that both the fracture opening and the pressure in the vicinity of the measurement point P11 began to change after the maximum fluid injection pressure was reached. A slight drop in initial pore pressure may have been caused by the influx of the pore fluid from the surrounding formation into the expanding fracture to fill the lag region. The observation of this feature was accidental, to some extent, due to the accidental positions of the fracture and the measuring pressure point. The lag of the fluid front from the fracture tip was also observed earlier in the experiments conducted at the described installation [21,34]. After reaching the maximum fracture opening, its value remained at an approximately constant level, but began to decrease after stopping the injection.

The pressure and width values behaved in similar ways at the measuring points located along the lower boundary of the “natural” fracture. These dependencies are presented in Figure 11. The pressures for tracks A5–A11 and A6–A11 were calculated by interpolation in projections of the intersection points of these lines with the plane of the “natural” fracture on its lower boundary (dashed line in Figure 5). For the A5–A13 route, the pressure at point P11 was used. Point P19 on the graph is given for reference.

The pressure change along the entire lower boundary of the “natural” fracture began simultaneously with the maximum pressure in the injection well, and was accompanied by a slight pressure drop. Then, a smooth increase in pressure began at all the points, but at points more distant from the hydraulic fracture, the growth was slower. This pressure behavior indicated an unsteady flow in the fracture due to both its expansion and the filtration of fluid into the surrounding material. The increase in the opening of the “natural” fracture at points A5–A13 and A5–A11, closest to the hydraulic fracture, at first occurred abruptly, and then gradually increased as the pressure increased. The opening was maximal at the intersection point of the hydraulic fracture and the “natural” fracture. At more distant points, the opening decreased rapidly with distance. After stopping the injection, the “natural” fracture was closed. This fracture closed most quickly at the intersection with the hydraulic fracture. As the pressures equalized along the fracture length, the fracture width also equalized. Negative width values are an artifact, and may be associated with an increase in the contact density of the fracture boundaries due to the influence of external vertical pressure. Attention may be drawn to the abnormal increase in the width of the fracture on the A5–A12 track after stopping the injection.

The dependence of the width of the natural fracture on the time and distance from the area of intersection of natural fracture and hydraulic fracture is shown in Figure 12. As we can see, the natural fracture width rapidly decreased along with the distance from its intersection with the hydraulic fracture.

After conducting four experiments on re-opening the hydraulic fracture, the experimental installation was opened for a visual assessment of the possible further propagation of fractures during repeated injections. A photo of a vertical view of the experimental unit is shown in Figure 13. As we can see, both hydraulic fractures’ branches increased in length. At the same time, the change in the length of the right branch is noticeably more significant. The right branch extended by 81 mm, and the left one by 54 mm. This fact may indicate that the presence of the “natural” fracture in the path of the left branch prevented its growth. The reason for this is the leakage of the fracturing fluid from the hydraulic fracture into the “natural” fracture. For further estimates, it is essential to know the lengths of both branches of fractures. We will assume that, in each of the four experiments, there were equal increases in the lengths of the fractures by one-fourth of their final extension.

Figure 14 demonstrates the schematic positions of the fractures at the end of the experiment. The hydraulic fracture intersected the natural fracture. Some part of the natural fracture in the vicinity of intersection opened. This was the most complex scenario of fracture interaction, among several others. An additional complication arose from the fracture’s inclinations: the hydrofracture was normal with respect to the XY plane, while the natural fracture was slanted. This led to gradual growth of the intersection zone as the hydrofracture propagated. At the same time, most of both experimental and theoretical investigations took into consideration only the case where both fractures were perpendicular to the same base plane. On the other hand, such a fracture arrangement considerably facilitates acoustic transmitting monitoring, especially for natural fractures, providing the opportunity to increase the number of “transmitter–receiver” pairs.

4. Discussion

To assess the reliability of the experiment results, it is advisable to compare the data on the fracture aperture with the volume of the injected fluid. Let us consider the entire system of fractures, as it consisted of three parts, and separately estimate the volume of each of them.

First, let us estimate the volume of the right fracture branch, which extended in the direction opposite the direction of the “natural” fracture. We used the results presented in [35]. The width of the fracture at the wellbore wall w_w was calculated based on:

$$p_w={\sigma }_h+\frac{G{w}_w}{2\left(1-\nu \right)L},$$

(1)

where:

p_w—pressure in the wellbore;

G—shear modulus;

$\nu$—Poisson ratio;

L—fracture length;

${\sigma }_h$—minimum horizontal stress.

Based on the conditions of our experiments, we can assume that ${\sigma }_h$ = 0. The pressure in the wellbore was determined at the time of stopping the injection to be 4.8 MPa. For the model material, the shear modulus was G = 1 GPa and the Poisson ratio was $\nu$ = 0.2. The length of the right fracture was measured on the surface as L = 100 mm. The fracture width at the wellbore wall was estimated as w_w = 0.7 mm.

The volume of the left wing of the fracture was calculated as $V_w=\frac{\pi }{4}hL{w}_w$, where h is the fracture height equal to the height of the 72 mm sample. According to the calculations, we obtained V_w ≈ 3.6 mL.

When calculating the volume of the left branch of the fracture, we assumed that its width at the wellbore would be the same as that for the right fracture. Its measured length was 120 mm. For the width of the fracture at its end, we took the width measured on the A5–A15 line to be equal to 0.48 mm. Let us assume that the longitudinal section of the fracture is a trapezoid, with the bases mentioned above and a height equal to the length of the fracture. Then, with a sample thickness of 72 mm, the volume of the left branch of the fracture would be V_e ≈ 4.9 mL.

The volume of the “natural” fracture V_NF, calculated based on the measurements, was 4.1 mL. Thus, the total volume of the fracture system at the end of injection was as follows: $V_{frac}=V_w+V_e+V_{NF}\approx 12.6$ mL. The injected fluid volume at a 10 mL/min flow rate was 11.4 mL. Both the calculated and the actual volume values agreed with each other. A slight overestimation of the total fracture volume may have been associated with a rough approximation of the fracture profile due to insufficient measuring points.

The existing discrepancy cannot have been caused by fluid leakoff from fractures into the surrounding pore formation. However, an estimation of the leakoff would also be useful. To achieve this, we used the following formula for the viscous leakoff coefficient C_v [36]:

$$C_v=\sqrt{\frac{k\varphi \Delta p}{2\mu }},$$

(2)

where k—permeability, $\varphi$—porosity, $\Delta p$—pressure drop, and $\mu$—viscosity. Substituting $k=1.1 \mathrm{mD}$, $\varphi =0.5$, $\Delta p=2 \mathrm{MPa}$, and $\mu =0.5 \mathrm{Pa}\cdot \mathrm{s}$, we obtained $C_v=3.2\cdot {10}^{-5} \mathrm{m}/\sqrt{\mathrm{s}}$. Substituting this coefficient into the formula $V_L=2C_v\sqrt{t}$ and considering the total fracture surface area, during an injection time of t = 69 s, have an estimate of the leakoff volume of V = 0.06 mL, which was two orders of magnitude smaller than the injected volume. Thus, under the conditions of the conducted experiments, the hydraulic fracturing fluid leakoff from the fracture into the surrounding formation could be ignored due to their smallness.

5. Conclusions

A method for producing samples containing artificial interfaces was developed to conduct experiments on the interaction of a hydraulic fracture with “natural” fractures existing in the sample. It is assumed that this technique can be used to produce samples containing multiple inhomogeneities that differ in composition from the primary sample and have different interface properties.

Ultrasonic monitoring of both hydraulic fractures and natural fractures in the model sample based on the attenuation of ultrasonic pulses passing through them, supplemented by the results of preliminarily conducted calibration experiments, allowed us to estimate the hydraulic fracture and natural fracture width at several points during repeated injections.

The presented experiment shows that the natural fracture limits the hydraulic fracture’s propagation. This is caused by hydraulic fracturing fluid leaking into the natural fracture. Both fractures formed a united hydraulic system that reacted almost simultaneously to the reopening of the hydraulic fracture, but the width of the natural fracture’s opening decreased with the total distance from the injection wellbore. Our experiment demonstrated the most complex scenario of the fractures’ interaction when the hydraulic fracture intersected the natural fracture and the natural fracture in the vicinity of intersection was also opened. An additional complication arose from the fracture inclinations: the hydrofracture was normal with respect to the base plane, while the natural fracture was slanted. This led to gradual growth of the intersection zone as the hydrofracture propagated. We believe that our results are of interest for numerical simulations of the interaction between hydraulic fractures and natural fractures, and that they can be used for the validation of numerical simulations.