Universal Scaling Laws of Hydraulic Fracturing

Hydraulic fracturing is a well-stimulation technique in which rock is fractured by a pressurized liquid. The process involves the high-pressure injection of ’cracking fluid’ (primarily water, containing sand or other propants suspended with the aid of thickening agents) into a wellbore to create cracks in the deep-rock formations through which natural gas, petroleum, and brine will flow more freely. When the hydraulic pressure is removed from the well, small grains of hydraulic fracturing proppants (either sand or aluminium oxide) hold the fractures open [1][2]. A hydraulic fracture is formed by pumping fracturing fluid into a wellbore at a rate sufficient to increase pressure at the target depth. The whole cracking process can be viewed as an interaction between high pressure flow and rock, the flow acts as a spear and rock as a shield, they fight each other. The flow make the rock instable and rock resist it to keep itself stable. Delicate dynamical balance or equivalence of between the flow and rock is vital for the hydraulic fracturing process, therefore, the process must be controlled by certain type scaling laws. There are lost of numerical and test have been investigated. However, there is no any general discussion on the scaling law of the process. The aim of this paper is to apply dimensional analysis to formulate an universal scaling law for the process of d=hydraulic fracturing[1][2].


INTRODUCTION
Hydraulic fracturing is a well-stimulation technique in which rock is fractured by a pressurized liquid. The process involves the high-pressure injection of 'cracking fluid' (primarily water, containing sand or other propants suspended with the aid of thickening agents) into a wellbore to create cracks in the deep-rock formations through which natural gas, petroleum, and brine will flow more freely. When the hydraulic pressure is removed from the well, small grains of hydraulic fracturing proppants (either sand or aluminium oxide) hold the fractures open [1] [2].
A hydraulic fracture is formed by pumping fracturing fluid into a wellbore at a rate sufficient to increase pressure at the target depth. The whole cracking process can be viewed as an interaction between high pressure flow and rock, the flow acts as a spear and rock as a shield, they fight each other. The flow make the rock instable and rock resist it to keep itself stable. Delicate dynamical balance or equivalence of between the flow and rock is vital for the hydraulic fracturing process, therefore, the process must be controlled by certain type scaling laws. There are lost of numerical and test have been investigated. However, there is no any general discussion on the scaling law of the process. The aim of this paper is to apply dimensional analysis to formulate an universal scaling law for the process of d=hydraulic fracturing[1][2].

SCALING LAW OF HYDRAULIC FRACTURING
In the process of hydraulic fracturing, if we omit temperature, the process is controlled by following variables. The density of cracking fluid ρ, dynamical viscosity of cracking fluid µ, static professor p §flow velocity U ¶Yield strength of rock σ Y , initial characteristic length of crack d §rock fracture stiffness K. Fracturing rocks at great depth frequently becomes suppressed by pressure, this suppression process is particularly significant in "tensile" (Mode 1) fractures which require the walls of the fracture to move against this pressure, so K can be chosen as K = K I od mode 1.
The hydraulic fracturing problem is to find current crack characteristic length D.
It is clear that the current crack characteristic length D must be a function of all other variables[3] (1) The dimensions of variables can be listed in the following table.
There are eight variables in the problem. Since it is mechanics system, which has three basic unit such as time, length and mass. Then from π theorem, we know we can construct five dimensionless parameters Π k , k = 1, .., 5.
To get the P i k , we can choose d, σ y , U as repeating variables.
The Π can be universally expressed as where a, b, c are to to determined constants. If we replace the Ψ i with D, p, ρ, K I , ν, respectively, then we obtain five Π as follows Therefore, the dimensionless relation of the problem can be expressed as where Π 1 is ratio of characteristic length, Π 2 is ratio of static pressure and yield strength, called static damage number, Π 3 is ratio of dynamic pressure and yield Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 11 June 2020 doi:10.20944/preprints202006.0138.v1 strength, called dynamic damage ¶Π 3 is inverse of the Irwin number Ir, Π 4 is inverse of Reynolds number Re. If we denote total pressure P as the sum of static and dynamics pressure P = p + 1 2 ρU 2 .
Adopting the well-known defined dimensionless variable, then the current crack length scale D can be expressed as simpler format where the total damage number is defined as J = P σ Y . Formula (6)is an universal scale law of hydraulic fracturing. It represents combined influence of fracking flow Re, rock fracture stiffness Ir and flow-rock interaction J onto the crack scale. Since for given a crack the Irwin nubmer Ir is constant, so as longer as the model test has same cracking liquid and flow speed, the geometric similarity is valid. This geometric geometric similarity will breaks down when the response time of the rock materials is taken into account.

EFFECT OF TIME
It must noted that rock fracture would be happening in some time. Impact under high pressure, the development from micro cracking, growth, interconnecting to a macro crack will needs a time process, in other words, the rock has an characteristic fracture time t p , with time dimension t.
Having concern the speed of energy wave in the rock, defined by σ Y ρ , then the problem will have 9 variables in the following table.
If set d ρ σ Y = t c as fracturing time of the rock, then Π 6 = t c /t p is Debrah number De. The rock is softer or stronger as Deborah number is getting smaller or bigger, respectively.
Therefore the formula (6) can be extended into Because the Debrah number De contains length scale d, so the geometry similarity will break downs if take into account the response time t p .

KEY CONTROL VARIABLE OF HYDRAULIC FRACTURE IS TOTAL DAMAGE NUMBER
Universal scaling law formula (6) shows that the length scale law is function of Ir, Re, J. For different stage, they will paly a different role.
If crack is in developing stage, which means that the Irwin number Ir must reach its critical value Ir c , the formula (6) can be simplified into If Renolds number Re is small, the the above formula can further simplified into Formula (5)indicates the control variables of the problem can be reduced to four, ie., d, P , σ Y , D, in this case, we have From physics pint of view, the C and α > 0 are constants to be determined by test.
If we compare effect of Ir, Re, J on the hydraulic fracturing, we can find the total damage number J is key control variable of the process. Because the total damage number is sum of static and dynamic, static pressure is constant, so to have a better fracturing, we must apply a high variable dynamics pressure.

CONCLUSIONS
The key control variable is the total damager number J = p+ 1 2 ρU 2 σ Y . The crack length is satisfied the geometric similarity law if no response time is concerned, otherwise, would not stratified the similarity.