It is offered a mathematical model of propagation of a hydraulic crack which allows to estimate the sizes formed as a result of it in a mountain breed of area of the weakened communications. The results obtained allow to establish the basic laws of influence of mechanical properties of mountain breeds and controllable parameters of hydraulic fracture on the process of formation of an area of weakened connections in the massif during the formation of a crack. Based on the presented methods, appropriate rock mass effect calculation software with a convenient user interface has been developed. Hydraulic fracturing, mathematical model, software, crack, area of weakened connections.
An equally important aspect of research on the description of specific mining situations under conditions of deformation and destruction of the rock environment or some engineering structures is to
2023 Copyright for this paper by its authors. CEUR
ceur-ws.org assess the spread of the area of influence of the hydraulic fracturing result beyond the formed fracture. This involves studying the state of the rock mass in the one-sided section of the crack tip in the direction of its propagation.
In this regard, the objective of the study is to build a mathematical model of the process of fracture development of a hydraulic fracturing crack that will allow a more accurate and correct display of the relations between the applied stress and the mode of hydraulic fracturing (parameters controlled at the wellhead: injection pressure, injection rate), the length of the crack, its opening, and, as a result, the size of the formed area of weakened connections in the rock mass.
When modeling the propagation of a hydraulic fracture in the underlying rocks of a technogenic deposit, three main types of fundamental relations generally accepted in this case were taken into account: 1) the provisions of linear-elastic fracture mechanics; 2) the laws of motion of a fracture fluid in a narrow slit; 3) the continuity equation.
Let at hydraulic fracturing of the underlying rocks a symmetrical fracture is formed relative to the well, propagating under the directional action of the fracture fluid in the vertical direction to a height and depth sufficient to provide hydraulic connection between the rocks of the technogenic deposit and the protective bottom. In the horizontal direction the fracture is technologically limited and has a constant length 2L.
Following [1-4], we will neglect violations of the continuity of the medium by the borehole, we will consider the elastic constants of the formation and its host rocks to be the same, the fluid pressure in the fracture to be constant along the length of the fracture (| | ≤ ) in each of its horizontal sections (| | = ). Fracture opening 2
in each cross-section (| | ≤ parameters of the pressure pulse and fracture dimensions – ), taking into account real time ≫ ℎ, can be considered as a quasi-static process, the parameters of which will be found as a solution of the two-dimensional problem of static elasticity theory, relating the fracture width 2
to its propagation height ℎ and the intra-fracture pressure ( , )− ∞ given at its banks, where ∞ is lateral rock pressure.
With the adopted proposals, we use the Perkins-Kern model for modeling crack propagation when a vertical crack of constant height, strongly extended in the horizontal direction, is formed as a result of fracture. According to the results of [13, 14], in this case of crack propagation, its opening can be represented as ( , , )= 2(1 − 2) ∞ √1 − ( ) ( initial conditions as well as the condition determining the mode of fluid injection into the fracture. The latter will be considered as specified in the form of the flow rate of the fluid to the fracture, which also satisfies the condition of conservation of mass or volume: and 〈 〉, 〈 ∙ 〉 are values averaged over the crack opening and over the horizontal section of the crack. Here – velocity of viscous fracture fluid in the crack, – viscosity of fracture fluid.
These equations are supplemented by the condition of "no flow" at the ends of the crack for the
averaged one-dimensional flow, determining the law of crack propagation
the condition of smooth interlocking of crack surfaces at its ends (analog of Khristianovich's condition)
volume of the initial crack, ( )– amount of liquid injected into the crack for time .
where is the half-width of the fracture zone, which does not change in the fracture process, 0 –
On the basis of formula (
Taking into account the obtained mean values (
2(1 − 2) = 1 2 3 ) = 0. 〈 ∙ 〉 = − 3( , )
. − ( 3 ∙
− ∙ ( ∙
) = 0.
( , )=
4
− 1.
3
(
3 48 μ 4 √( 2 ∞ + 48 2μ
Taking into account the conditions (
After excluding the unknown parameter from the system (
( , )= 0 3√ℎ − , =
2 3 6 48
√ 4 ∞ ,
6 (0, ) 0 = √(
3 − 1) − . ∞ For the case of fracturing fluid flow rates we have
The longitudinal development of the fracture crack is determined by the formula
plastic deformation at stresses above this limit [8]. Thus, in the vicinity of the crack tip, there is always a region where plastic deformations occur, which means that the stresses cannot be singular.
The fundamental difference in the outline of the crack banks at the crack tip is a consequence of the application of linear elasticity theory methods to the study of the stress-strain state of a body in the presence of large deformations, which does not correspond to the real picture of the stress-strain state at the crack tip. This contradiction, noted by Griffiths, led to models in which the crack banks under the influence of large cohesive forces (of the order of theoretical strength) should be washed away smoothly. In this case, all known models of non-ideally brittle bodies are based on the introduction of cohesive forces between the banks of the forming crack and differ only in the assumptions regarding these forces, i.e., in these models, unlike Griffiths' model of an ideally brittle body, the end zone is not autonomous [10-12].
For a more accurate and correct representation of the relations between the applied stresses, the length of the crack, and the size of the area of broken and weakened connections, it is proposed to use the Leonov-Panasyuk brittle crack model, which is formally equivalent to the Dugdale elastic-plastic crack and Barenblatt brittle crack models, although its mechanical content is somewhat different. In this model of a crack, the presence of a finite zone R of the crack is taken into account, where its banks are attracted with constant stress if the distance between them does not exceed a certain value . If > , then in accordance with the Leonov-Panasyuk concept of a brittle crack, there is no interaction between the crack banks. The zone of length R is called the area of weakened connections (Fig. 1). In the vicinity of each point of this area, two parameters are set that characterize the places of the beginning and end of fracture and correspond to two criteria of fracture [17]:
1) the condition of finite stresses in the final crack zone, i.e., = 0, where is the stress intensity factor at the crack tip;
2) the condition of transformation of the adhesion forces to zero at the point of transition from the area of weakened connections to the area of destroyed connections. The value is considered to be equal to the brittle strength limit, i.e., the fracture stress in the absence of plastic deformation. The critical condition for crack propagation is the equality
2 (ℎ, )= , where is critical crack opening, (ℎ, )– crack opening in the direction of the Y-axis.
Condition 2) means that at a certain value of crack bank deflection , which is a characteristic of the geomaterial, the cohesive forces turn to zero, which leads to the condition = 0 at the point ( = ℎ)of transition from the area of weakened connections to the area of broken connections. 2.3. A solution of the problem of crack propagation with consideration of the area of weakened connections
The opening of the fracking crack and the acting load in the absence of an area of weakened
connections according to the proposed solution can be found by formulas (
for < ℎ, 0 for > ℎ + , and a small distance we obtain the following relations:
Then, according to the principle of superposition, for a set of cracks of different lengths (ℎ < < ℎ + )and taking into account the presence of an area of weakened connections of length separating the crack banks at point ℎ in accordance with formulas (21), (22), is to ensure that the pressure inside the well is such that the following condition is met: = + 2 ∞, where is limit value of rock tensile strength.
Taking into account condition (25) and the formulated problem, we can conclude that for the stable development of a fracking crack propagation and in order to overcome the constricting stresses between the crack banks, it is necessary to create an equivalent load (not less than ) in the area of weakened connections. Counteraction to the existing constricting loads will be ensured if we set the function ( )in such way that in the region of weakened connections, for each fracking crack of size (ℎ < < ℎ + ), the following compensation condition is met:
From condition (26), to determine the value of the required compensation load ( ), we obtain the (21) (22) (23) (24) (25) (26) (27) (28) (29) following equation: ( , )= ( − )3 [15, 16].
1 Equation (27) is a Volterra integral equation of the first kind of convolution type with a kernel As a result of solving the equation (27), we obtain the following form of compensation load ( ): Based on the solution (28) and taking into account the value of the acting load at the wellhead during hydraulic fracturing (injection mode), we estimate the value of the propagation of the area of weakened connections by solving with respect to the equation ∫( + 2 ∞) = ∞ ∫ (
( )3√ − + 1) . ℎ 4 ℎ ( − ℎ)=
4 ∞ ∫ ( )3√ − . ( )= 3√3( + ∞) 8 ∞ ( − ℎ)−3.
1 0 = 3√3( + ∞) 8 ∞ ℎ+ ℎ ∫ ( − ℎ)−3 .
1
By finding the integral in the right-hand side of (29), taking into account the value of the characterizing function 0 we obtain the size of the region of weakened connections: = (
16 0 ∞ 9√3( + ∞) 3 2 ) .
According to (23), the development of a fracking crack under the action of a de-claying agent in the area of fractured connections (crack opening) and deformation displacements in the area of weakened connections for each time are written in the following form: 3√3 8 ∞ the
− )3 is transformed to the integral of a fractional rational function, which in turn is always integrated in finite form and the result is an algebraic sum of elementary functions (fractional rational, natural logarithm and arctangent). However, such an analytical solution for practical application is rather cumbersome and it is reasonable to use numerical integration for formula (31).
Formula (31) allows us to find the critical condition for fracking crack opening, namely, based on condition (20), the adhesive forces turn to zero at the point = ℎ of transition from the area of weakened connections to the area of fractured connections. Taking into account these relations, we obtain: integral formula (31) with the new variable 3√3 8 ∞ disturbed bond zone. (30) (31) and the zone of weakened bonds (dashed line)
Based on presented methods an appropriate software that allows to calculate the dependence of the propagation of the area of weakened connections on the value during hydraulic fracturing and the dependence of critical crack opening
on the value during hydraulic fracturing was developed.
Input data consists of type of the rock, maximum propagation, maximum opening, area of weakened connections, critical crack opening and model parameters ∞, , , 0, , . The example of the developed software user interface presented on the figure 3. Also, methods that presented in the research [18, 19] can be modified for rock type identification. 0,5 3 0,3, 1 = 30
To illustrate the determination of the propagation of the area of weakened connections in the rock mass due to the formation of a fracture crack, a numerical experiment was carried out for the following model parameters of hydraulic fracturing: ∞ = 10 , = 0,1 ⋅ , = 2 , 0 = 0,1 , = , = 3 and mechanical properties of rocks 1 = 5 , 1 = 0,3, , 2 = 10 , 2 = 0,23, 2 = 5 , 3 = 30 , 3=0,2, which correspond to the average mechanical properties of sandstone, mudstone and clay shale. crack propagation and opening, critical opening, and propagation of the area of weakened connections. 1 = 5
, 1 = 3 = 10 Fracking crack parameters taking into account the area of destroyed and weakened connections and Maximum propagation ℎ
( ) 19,7 27,1 46,9
Maximum opening
( ) 6,3410-3 4,5810-3 2,6510-3
Area of weakened connections ( ) 0,132 0,8 0,89
Critical crack opening
( )
In the next series of calculations, we studied the effect of the parameter on the propagation of the region of weakened connections and critical crack opening, which is due to the significant variation of this parameter even for rocks of the same petrographic name, depending on the composition and structure of the rock. Figures 4 and 5 illustrate the obtained results.
On the example of a man-made field, where hydraulic fracturing technology is proposed to be used to improve the filtration properties of the underlying rocks, a solution to the linear-elastic problem of the development of a symmetric crack in the vertical direction is found.
Based on the results of the study, it can be concluded that the extension of the region of weakened connections in the massif containing a fracking crack depends on both the value of the elastic modulus and the ultimate tensile strength , and is greater for rocks with a lower value of the parameter at a higher value of the Young's modulus.
For the case of rocks of the same petrographic name, the propagation of the area of weakened connections is larger at lower values of the rock tensile strength. Also, an example of the developed software was presented.