Characteristic length scales and dimensionless parameters for the field and the laboratory conditions.

## Abstract

We investigate the initiation and early-stage propagation of an axi-symmetric hydraulic fracture from a wellbore drilled in the direction of the minimum principal stress in an elastic and impermeable formation. Such a configuration is akin to the case of a horizontal well and a hydraulic fracture transverse to the well axis in an open hole completion. In addition to the effect of the wellbore on the elasticity equation, the effect of the injection system compressibility is also taken into account. The formulation accounts for the strong coupling between the elasticity equation, the flow of the injected fluid within the newly created crack and the fracture propagation condition. Dimensional analysis of the problem reveals that three dimensionless parameters control the entire problem: the ratio of the initial defect length over the wellbore radius, the ratio between the wellbore radius and a length-scale associated with the fluid stored by compressibility in the injection system during the well pressurization, and finally the ratio of the time-scale of transition from viscosity to toughness dominated propagation to the time-scale associated with compressibility effects. A fully coupled numerical solver is presented, and validated against solutions for a radial hydraulic fracture propagating in an infinite medium. The influence of the different parameters on the transition from the near-wellbore to the case of a hydraulic fracture propagating in an infinite medium is fully discussed.

## 1. Introduction

In this study, we are interested in the initiation of hydraulic fractures from an open-hole horizontal well. Horizontal wells are drilled preferably in the direction of the minimum horizontal stress in order to create hydraulic fractures perpendicular to the wellbore and therefore maximize the drainage from the reservoir. During the pressurization of the wellbore, tangential tensile stress is generated which can result in the initiation of longitudinal fractures parallel to the wellbore axis [1,2]. In an isotropic elastic medium, depending upon the stress field, the initiation of fractures transverse to the wellbore axis is favored by creating an initial flaw, i.e. an axisymmetric notch, of sufficient depth [3] (see Figure. 1 for a sketch). We focus solely on transverse hydraulic fractures in this contribution.

In the initial stage of propagation, these transverse fractures can be idealized as axisymmetric (radial) fractures around the wellbore until they hit a stress barrier or other type of heterogeneities. We investigate the initiation and propagation of such a fracture under constant injection of a Newtonian fluid, focusing on the case of “tight” rocks where leak-off is typically negligible. In particular, we are interested in clarifying when the effects associated with the near-wellbore region dissipate and no longer affect the hydraulic fracture propagation and how this transition takes place.

The initiation and the early stage of the propagation of such a hydraulic fracture is affected by two “near wellbore” effects: i) the finiteness of the wellbore and the initial flaw length and ii) a transient phenomenon associated with the release of the fluid stored by compressibility in the wellbore during the pressurization phase prior to the initiation of the fracture. This second effect is ultimately linked to the compressibility of the injection system (i.e. mostly the fluid volume stored by compressibility within the wellbore). These effects have been investigated for the case of plane-strain fractures in [4,5] and for the case of axisymmetric hydraulic fractures driven by an inviscid fluid (i.e. zero-viscosity) in [6]. In this paper, we extend these contributions to the case of viscous flow in axi-symmetric fractures.

A detailed dimensional analysis is performed indicating various time scales, length scales and dimensionless numbers controlling the problem. A numerical solver, along the lines of previous contributions [7,8], is presented. A series of numerical simulations are performed in order to study the transition from the near-wellbore propagation regime to the case of a radial hydraulic fracture in an infinite medium under constant injection rate.

## 2. Problem statement

Let us consider a horizontal well of radius

In the absence of any stress barriers or other heterogeneities, this axisymmetric transverse fracture of radius

where

### 2.1. Elasticity

The relation between the fracture width and the net loading acting on the crack is expressed by a hyper-singular boundary integral equation following the method of distributed dislocations [10]:

where the singular kernel

### 2.2. Fluid flow

We neglect the fluid compressibility compared to the fracture compliance as is usually done in modeling hydraulic fractures. The balance of mass then reduces to a strict volume balance:

Under the hypothesis of lubrication theory (low Reynolds number flow) for a Newtonian fluid, the fluid flux

where

### 2.3. Boundary conditions

During the propagation of a hydraulic fracture, a fluid lag may develop at the tip of the fracture [12,13]. Such a fluid lag is larger at early time during the propagation. In order to check whether the fluid lag should be taken into account, we can estimate the characteristic timescale associated with the disappearance of the fluid lag, which is equal to

For the case of a slick water stimulation (viscosity of 1 cP), the characteristic time

The conditions at the tip of the fracture are thus:

The fluid flux entering the fracture is equal to the total fluid injection rate (for example the injection rate at the wellhead) minus the fluid volume stored in the well due to its compressibility (essentially the compressibility of the fluid inside the wellbore). Therefore, from the fluid mass conservation in the wellbore (between the injection point and the fracture inlet), one can write the following boundary condition at the inlet of the fracture:

where

It is important to note that no friction pressure drop (e.g. perforation drop) is taken into account in this injection boundary condition.

### 2.4. Initiation and fracture propagation condition

Prior to any opening of the initial defect of length

where

## 3. Scaling

Let us scale the variables involved in the problem in order to grasp the effects of various physical phenomena acting at various scales. We introduce a characteristic length scale

Using the above scaling, the governing equations are converted to dimensionless form where the different scales are yet to be defined:

Elasticity operator

where

Continuity equation

where

Poiseuille law for lubrication

where

Inlet boundary condition

where

The LEFM propagation condition reduces to

where

First, it is natural to set the dimensionless groups associated with the injection rate

In the case of the propagation of a radial hydraulic fracture in an infinite medium, energy dissipation is attributed to two competing mechanisms i.e. viscous forces associated with fluid flow within the crack and the creation of new fracture surfaces (i.e. fracture toughness) [15]. At the beginning of the propagation, i.e. for small fracture radius, viscous forces are the dominant dissipative process, and fracture toughness can be neglected in such a viscosity dominated regime of propagation. A self-similar solution has been obtained in [15] for that case, and will be denoted as the

This picture is modified when accounting for near-wellbore effects. These effects will eventually dissipate for fracture length much larger than the wellbore radius. This transition from the near-wellbore to the infinite medium solution is of particular interest. The effect of the wellbore and of the system compressibility will affect the system response at early time, i.e. when the radius of the fracture is comparable to that of the wellbore and when the system compressibility still has an effect. At large time, the transient associated with the fracture breakdown and the release of the fluid stored by compressibility prior to the crack initiation will become insignificant: i.e. the fluid flux entering the crack will then be equal to the injected flow rate. The solution will thus behave as the infinite medium solution [15,16,17].

It is therefore interesting to introduce two different scaling. The first scaling relates to the case where the system compressibility and toughness are important, i.e. at early time or for small fractures. We will denote such a scaling as the Compressibility-Toughness scaling and denote it as

In particular, the dimensionless time in that scaling is

The second scaling of interest corresponds to the case where the transient effects associated with the wellbore and injection compressibility have vanished: i.e. when the model reduces to the case of radial fracture propagating in an infinite medium – we will call this scaling the Viscosity-Toughness scaling and denote it as

The dimensionless time is here defined as

In order to grasp the transition from the early time where the near-wellbore effects are important to the large-time solution of propagation in an infinite medium, we introduce

Large values of

Now introducing

and

Correspondence between the two scalings can also be obtained as the function of

In addition to the above mentioned factors, the ratio of dimensionless initial defect length

### 3.1. Field and laboratory conditions: Scaling

Laboratory experiments are typically performed in order to study one particular aspect of a problem independently. Results of these experiments are then complemented by numerical and theoretical studies and ultimately verified by field experiments. Conditions in the laboratory experiments should be controlled in such a manner that they represent as close as possible a scaled version of the field conditions to be investigated. The scaling and the dimensionless parameters, described in the previous section, are thus critical in identifying the key parameters to simulate the right physics in the laboratory [18].

We will compare the different scales (in the Compressibility-Toughness scaling

A typical wellbore diameter (

The compressibility length scale

Length scale of transition from the compressibility effects to the infinite medium propagation | |||

Timescale of transition from the compressibility effects to the infinite medium propagation | |||

Ratio of the timescales | |||

Dimensionless wellbore radius |

It is obvious from Table 1, that in the field, the fracture propagates in the viscosity dominated regime (

The previous example has emphasized the differences with field conditions for a particular set of experimental parameters. However, these laboratory parameters can be appropriately adjusted in order to study a given regime of propagation. There can be different goals for an experimental campaign. For example, if the goal is to study hydraulic fracture propagation then the compressibility effects must be reduced in order to speed up the convergence to the infinite medium solution. These effects can be reduced by manipulating the material of the test block, using smaller injection lines and by using needle control valves as it was done for example in [20,21].

## 4. Numerical algorithm

The governing equations are solved in their dimensionless form described in Section 3. The elasticity equation is discretized by Displacement Discontinuity Method (DDM) using piecewise constant elements with the tip element correction [22] for better accuracy. The fluid flow is discretized by the finite volume method.

At time

## 5. Results and discussion

The problem of the initiation of an axisymmetric hydraulic fracture from a wellbore depends only on three dimensionless parameters i.e. the timescale ratio

In Figure. 2, the evolution of fracture length

It is also observed that the pressure and opening do not converge monotonically to the asymptotic solution. This is a characteristic feature of the near wellbore solution also obvious from Figure. 3 where the results are displayed in the Compressibility-Toughness (

The effects of

Lastly, the value of

here

Let us now compare the effect of different ratios of

It can be seen from Figure. 6, that there is no significant difference in convergence to the infinite space solution for different ratios of

Finally, in Figure. 7, the effect of the dimensionless wellbore radius is considered for a fixed ratio

## 6. Conclusions

The initiation of axisymmetric hydraulic fractures from a horizontal wellbore has been investigated with an emphasis on near wellbore effects. Through a detailed dimensional analysis, two characteristic timescales were identified. The first characteristic time

It was also found that the near wellbore effects are present up to a fracture length about 12 times the wellbore radius for different values of

## Appendix A

The edge dislocation kernel

where

where

where

## Appendix B

The time step is computed by imposing the LEFM tip asymptote (15) in a weak form in the tip element. In the case of negligible toughness, care should be taken as the governing equations degenerate in the tip region. An algorithm based on the LEFM asymptote then requires very fine mesh for good convergence (see [24] for discussion).

The asymptotic volume of the tip element corresponding to the LEFM crack opening asymptote Eq. (15) is given as

where

where

## Appendix C

The coupled problem of fluid-solid coupling inside the fracture is described as: for a given fracture length

where

Where the entrance hydraulic conductivity of an element

The flux entering the fracture from the wellbore is given by the inlet boundary condition

while no flow is assumed out of the fracture tip

The elasticity equation (11) is discretized by the Displacement Discontinuity Method (DDM) to the following form

where

where

where

where

The nonlinear system (45) is solved by fixed point iteration

with under-relaxation

where