P2-3-9

Hydraulic fracturing in oil and gas resources and its risks and hazards while its interacting with natural faults

Mohsen Bazargan1,2

, Agust Gudmundsson1, Philip Meredith

2

1Royal Holloway University of London, Egham, UK,

2University College London, London, UK

Introduction

Hydraulic fracturing (HF) is one of the best techniques employed in oil and gas reservoirs to improve the turnover of production. In this phenomenon a highly pressurized fluid is pumped into a borehole resulting in creation of cracks which will be used to reach in stored resources in hard layers reported by Dong. Primary models of HF have assumed a neat crack evolving in a continuous medium which has been proved to be over idealized as nearly all reservoirs are full of faults affecting this phenomenon. The existence of faults – especially very large ones – could far change the crack propagation mechanisms; different behaviours of penetration, arrest, diversion and offset could be distinguished in case intersection takes place this research has been reported by Zhang. Multiple parameters will be important in final behaviour of the intersecting cracks; fluid viscosity, angle between interfaces, crack lengths, friction coefficient and far field stresses. HF in medias with initial faults has been studied by different methods. Akulich and Zvyagain have studied interaction between HF crack and natural fractures using a BEM procedure, where viscosity effects have taken into account. Simulation of hydraulic fracture propagation near a natural fracture using virtual multidimensional internal bonds is represented by Zhang & Ghassemi. Zhang et al. has studied layered medium where the effects of different sequences of soft and stiff layers has been investigated. Gudmunsson & Brenner have performed similar study with more than two layers taken into account.

In this paper the finite element method will be used to model HF in sedimentary rock samples and its interaction with natural faults. The crack tip behaviour will be assumed to be linear, where asymptotic enrichment functions will be employed. The contact between surfaces of the natural fault is modeled. The HF current is assumed to be non-viscous, producing a uniform pressure over the interface where no leak-off is considered for the current along the crack into the domain. The interface will grow versus stress intensity factor which could be utilized to determine the threshold and direction of crack propagation. Finally in numerical implementation part the proposed algorithms will be verified through different examples. These examples had been simulated with source code in MATLAB and COMSOL Multiphysics software.

Governing equations

Consider a body Ω as shown in Fig1. Boundary Γ consists of external parts of tΓ and u

Γ

referred to loaded and restrained zones together with internal discontinuities of dΓ which could be

both pressurized or traction free. The total Lagrangian framework is employed in order to implement the formulation, where the reference configuration is used. Because of the Lagrangian

description of medium, continuity equation results in 0Jρ ρ= , where J is the determinant of

deformation gradient.

Assuming quasi-static condition, the equilibrium equation could be written as follows,

(1) . 0ρ∇ + =σ b

Where b is the body force per unit mass vector which is generally gravity, ρ is the average

density of the medium and σ is the Cauchy stress tensor.

u

uΓ

t

tΓ

nd

dΓ

Ω

Γ

ct

nΓ

Fig1. Free body diagram in state of equilibrium. The boundary conditions could be summarized as below.

(2a) t. on Γ = Γσn t

(2b) on uΓu=u

(2c) . on cd d= Γσn t

Where Γn and dn are the unit normal vector to the internal or external boundaries respectively.

The constitutive equation relating the stress to the total strain is expressed by means of an incrementally linear stress-strain relationship as follows,

(3) d dσ =D ε

Where in the above relation D is the tangential constitutive matrix of the continuum. In this section weak form of expression (1) will be derived, where the trial function for the displacement field,

( , )tu X , is assumed satisfying all essential and natural boundary conditions while being smooth

enough to define derivatives of equations. Moreover, test function ( , )tδu X is also required to

have the properties of trial function. The classic procedure of the finite element is employed in

order to derive the weak form of the equilibrium equation, where ( , )tδu X is multiplied by

equation (1) and is integrated over the domain in initial configuration. It follows,

(4) ( ) . . 0dδ ρΩ

∇ + Ω =∫ u bσσσσ

Next, divergence theorem is employed in order to simplify the above expression. As the domain includes internal discontinuities, this theorem will result in extra terms over these boundaries [see 11]. It follows,

(5) ( ) : . n . . ,d

T

dd d d dδ δ δ ρ δ

Ω Γ Ω Γ∇ Ω+ Γ = Ω + Γ∫ ∫ ∫ ∫u u σ u b u tσσσσ

Where the term . nd

ddδ

ΓΓ∫ u σ stands for the tractions imposed onto the internal interface,

which is zero for traction free conditions. In current investigation, for the aim of simplicity, the effects of fluid viscosity are ignored as being negligible for water as propelling fluid. This assumption results in a constant pressure distribution over the interface. Considering equation (5), the term .=t σ n on upper and lower crack faces is equal to fluid pressure on crack faces which is

equal to -d d

p=σn n . Hence,

(6) . n . nd d

d dd p dδ δ

Γ ΓΓ = Γ∫ ∫u σ u

It is noteworthy to know in order to consider the effects of viscosity, it is needed to solve Darcy’s equation for flows which follows,

(7) [ ( )]W

p ρ= −∇ + −w K b u&&

Where w is the fluid velocity, Wρ is water density and K is the permeability matrix. In one

dimensional the cubic law is used, which is as below,

(8) 2

12

h

µ=K

As mentioned before, because of low values of viscosity together with very high values of pressure, the effects of viscosity is ignored in this study.

Numerical simulation results

In this section five examples are chosen to investigate the validity of the proposed method. First example will study three cracks with varying distances under uniform tension. Second example will be a single crack with uniform internal pressure which a basic solution for HF where SIFs will be verified with available solutions. Last three problems will study the interaction phenomenon between HF and natural fault. Third example is chosen to show HF crack is propagating in naturally fractured porous medium. This example will study mixed mode crack propagation. Fourth example is HF crack interacting with natural fault of existing fracture with different directions. This example will study mode I and mixed mode crack propagation. Final simulation is chosen so show what is going to happen after the interaction between HF crack and natural fault in the case of 90 degree angle existing fault in porous medium. Material properties for all five examples have been given in Table1.

Table 1: Properties of rock

Young Modulus 10.2e9

Poisson’s Ratio 0.25

Solid Density kg/m3 2700

Fluid Density kg/m3 1000

Porosity % 0.21

Permeability mD 0.2

First example is three cracks with size of 0.5 meter in porous medium with tension of 1e10+6

Pa.

q

2a

d

A B C

d

2a 2a

Fig2. Three cracks with offset under uniform tension - problem’s configuration.

d/a

0.40 0.50 0.60 0.70 0.80 0.90 1.000

0.5

1

1.5

2

Tip A - XFEM

Tip A - Sih

Tip A - Tu et al.

Tip B - XFEM

Tip B - Sih

Tip B - Tu et al.

Tip C - XFEM

Tip C - Sih

Tip C - Tu et al.

Fig3. Stress Intensity Factor for plate with three cracks at tips A, B and C.

Again in this example plate of problem 1 is considered with the same material properties. Plain strain condition is assumed. A crack of length 2 meters is positioned at the centre of the plate and is subjected to a uniform internal pressure of 1e10

+6 Pa.

2a

Fig4. Single crack subjected to internal pressure – problem’s configuration

Element s ize (m )

Relativeerrorin

SIF

0 .20 0 .40 0 .60 0 .800

0.05

0 .1

0 .15

0 .2

KI - X FEM

Fig5. Principal stress and total displacement contours which shown stress concentration around the crack tip and how fracture will propagate.

In fact this example is a preliminary one for modeling HF where next examples will study this loading condition together with propagation in Medias possessing faults. This example with same material properties, HF crack propagating and gets close to 0.5 meter natural fault. In this sample we can observe how HF pressure can effect on natural fault.

Fig6. HF crack in interacting with natural fault and its inside fluid pressure in interacting with edge of natural fault.

Fig6 had shown how interacting and crack propagation path can change when HF crack is getting closer and closer to the natural fault. Next example is showing how HF crack with interact and propagate in the case with angels fault is in front of it. This example picked 45 degree angle and 90 degree angel faults and it shows in which case crack propagate on Mode1 and in which crack propagate in mix mode. HF is a handful process employed extensively in different engineering applications. One of the most appealing uses is in oil industry where HF is applied in order to break hard layers of rocks to reach pressurized oil and gas reservoirs. In such cases, massive amount of fluids – especially water – is injected into reservoirs resulting in breakage of hard layers. This method is very common and significantly improves the total revenue of the field. Despite in primary models for HF simply single crack were assumed and propagated through the medium, empirical cases observed to be very complicated as a variety of parameters play part in this phenomenon. One of the most determinant factors in the probability of existence of natural faults in the media, ending in interaction between these cracks and the driven one. In this simulation we observe how edge of natural fault and final crack path can be both affected by HF crack inside fluid pressure.

Fig7. HF crack interaction with natural fault while the natural fault does have different angels.

In Fig6 we can see crack path will be mix mode in the case of 45 angled natural fault while this path will be following first mode when natural fracture angled 90 degree. Next example is same media after the interactions on HF crack and natural fault with 90 degree angle to the HF crack.

Fig8. HF crack and natural fault interact and fluid pressure in currently inside of fault.

Fig8 has shown in different views what is going to happen after HF crack bond with natural fault and how it is going to effect the propagation afterwards. In different view of the naturally fracture reservoir we can obverse how stress tensor arrows will form to show the area of fault propagation. The main reason of this example is to show how HF crack can continue propagating through natural faults or existing fracture and this process will affect the whole operation. In this paper we have represented an analysis of fracture propagation in fractured reservoirs using the finite element method. Later report of this research group will focus on HF studies in different layers with different materials.

Conclusion

In the present paper, the hydraulic fracture propagation was presented in saturated porous media using the finite element method. We observe how it is going to propagate in intact medium or in naturally fractured medium as well. The main reason of risk and hazard control of this paper is all about HF cracks interactions with natural fault and ends up fault propagation. These fault propagations can easily flowing the HF fluid to the different area. Some of these badly oriented faults can ends up taking HF fluid to the underground water resources. This is highly unlikely but probable and we investigated it in this report.

References

Akulich AV, Zvyagin AV (2008) Interaction between hydraulic and natural fractures. Fluid Dyn 43: 428–435.

Barani OR, Khoei AR, Mofid M (2011) Modeling of cohesive crack growth in partially saturated porous media; A study on the permeability of cohesive fracture. Int J Fracture 167: 15-31.

J.N. Germanovich, A.V. Dyskin A model of brittle failure for material with cracks in uniaxial loading Mechanics of Solids, 23 (2) (1988), pp. 111–123.

L.N. Germanovich, A.V. Dyskin Fracture mechanisms and instability of openings in compression

Int. J. Rock Mech. Min. Sci., 37 (1–2) (2000), pp. 263–284.

Dong CY, de Pater CJ (2001) Numerical implementation of displacement discontinuity method and its application in hydraulic fracturing. comput Methods appl Mech Engrg 191: 745-760

Lewis RW, Rahman NA (1999) Finite element modeling of multiphase immiscible flow in deforming porous media for subsurface systems. Comput Geotech 24: 41–63.

Lewis RW, Schrefler BA (1998) The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media. Wiley, New York.

Zhang Z, Ghassemi A (2011) Simulation of hydraulic fracture propagation near a natural fracture using virtual multidimensional internal bonds. Int J Numer Anal Meth Geomech 35: 480–495.

Acknowledgements

The Author is sincerely thanking IGA for their research grant support. And also Weld on Sweden company and Sharif University of Technology for their contribution.