Induced Fractures Modelling in Reservoir Dynamic Simulators

Transcription

1 Induced Fractures Modelling in Reservoir Dynamic Simulators Khaldoon AlObaidi Institute of Petroleum Engineering MSc Petroleum Engineering Project Report 2013/2014 Supervisor Heriot Watt University This study was completed as part of the Masters of Science in Petroleum Engineering at the Heriot Watt University.

2 P a g e ii Declaration I, Khaldoon AlObaidi, confirm that this work submitted for assessment is my own and is expressed in my own words. Any uses made within it of the works of other authors in any form (e.g. ideas, equations, figures, text, tables, programs) are properly acknowledged at the point of their use. A list of the references employed is included. Signed K.A... Date 27 August 2014 Declaration

3 P a g e iii Dedication To my family for their support. To my uncle for his continuous encouragement. Dedication

4 P a g e iv Acknowledgments Thanks also go to NSI Technologies Inc. for providing me with StimPlan software and licenses which helped me establishing the basis of this work. Acknowledgements

5 P a g e v Abstract Since the middle of the twentieth century, hydraulic fractures and fractures created due to injection under fracturing conditions have been proven to be effective in increasing the productivity and injectivity factors of wells considerably. In this work, an algorithms for determining the optimum hydraulic fracture dimensions, the growth of induced fractures created due to injection under fracturing conditions and modelling fractures in dynamic reservoir simulators are introduced. Additionally, the optimum dimensionless conductivity is derived to be and is used in addition to practical limitations and economic considerations to determine the optimum hydraulic fracture dimensions result in maximum folds of increase in production. Also in this work, an algorithm adopting Perkins-Kern-Nordgren-α (PKN-α) and Ahmed and Economides notation after Simonson analysis is adopted to determine the dimensions of the induced fractures created due to injection under fracturing conditions. The induced fractures are implemented in reservoir dynamic simulators using gridblocks refinement and properties multiplications to increase net to gross, porosity and permeability to mimic the fracture properties. For two simple box models, only approximately 42% increase in run time due to implementing this algorithm in reservoir dynamic simulator is resulted. Therefore, the algorithm presented provides a good approximation for modelling induced fractures growth with reduced simulation run time and storage capacity compared to three-dimensional fracture models. Also, it provides more accurate results compared to the simple two-dimensional models that assumes fixed fracture height. The advantage of the algorithms presented is they combine the fracturing physics with the reservoir dynamic simulator constraints. Therefore, implementing this work provides robust reserves estimation and forecasts for wells with induced fractures warning of fractures propagation into unintended with relatively fast running simulation models. Abstract

6 P a g e vi Table of Contents Declaration... ii Dedication...iii Acknowledgments... iv Abstract... v Table of Contents... vi List of Figures...viii List of Tables... ix Nomenclature... x 1 Project Scope and Objectives Induced Fractures Fracture Orientation Leak-off Test Hydraulic Fracturing Procedure Analytical and Numerical Models for Estimating Fracture Dimensions, Propagation and Recession PKN-α Model Fracture Height Growth Dynamic Simulation of Hydraulic Fractures Objectives Methodology Determining the Optimum Hydraulic Fracture Dimensions for a Well Cases Input for Testing the Procedure Used to Determine the Optimum Hydraulic Fracture Dimensions for a Well Abstract

7 P a g e vii Low Permeability Reservoir Case High Permeability Reservoir Case Incorporating Hydraulic Fractures in Reservoir Dynamic Simulators An Algorithm for Modelling Induced Fractures Created During Injection under Fracturing Conditions in Reservoir Dynamic Simulators Fracture Height Growth Fracture Width and Half-Length Results and Discussion Determining the Optimum Hydraulic Fracture Dimensions for a Well Low Permeability and High Permeability Case Results Low Permeability Reservoir Case High Permeability Reservoir Case Incorporating Hydraulic Fractures in Reservoir Dynamic Simulators Algorithm for Modelling Induced Fractures Created During Injection under Fracturing Conditions in Reservoir Dynamic Simulators Conclusions and Recommendations References Appendices... I A.1 Fracture Height Equations...I A.2 Meeting Agendas and Progress Reports... III A.2.1. Period 1... III A.2.1. Period 2... XII A.2.1. Period 3... XIX A.2.1. Period 4...XXVII A.2.1. Period 5... XXXIV Table of Contents

8 P a g e viii List of Figures Figure 1. Principal stresses (Anon. g 2014)... 3 Figure 2. Extended leak-off results (Crain 2013)... 5 Figure 3. Fracture geometry (a) PKN type (b) KGD type (c) Pseudo 3D cell approach (d) Global 3D, parameterised (e) Full 3D, meshed (Yang 2011)... 8 Figure 4. Warpinski and Smith analysis and notation (Valko and Economides 1995) Figure 5. Cinco-Ley and Samaniego graph for Hydraulic Fracture Performance with f = sf + ln (xf/rw) (Valko 2005) Figure 6. Prat s dimensionless effective wellbore radius (Valko 2005) Figure 7. Low permeability case model Figure 8. Hydraulic fracture for low permeability case Figure 9. High permeability case model Figure 10. Hydraulic fracture for high permeability case Figure 11. Ahmed and Economides notation for fracture height (Valko and Economides 1995) Figure 12. Algorithm for estimating fracture dimensions created due to injection under fracturing conditions Figure 13. Folds of increase and fracture volume vs. folds of increase for low permeability case Figure 14. Folds of increase and fracture volume vs. folds of increase for high permeability case Figure 15. Increase in cumulative oil production for low permeability case Figure 16. Increase in cumulative oil production for high permeability case List of Figures

9 P a g e ix List of Tables Table 1: Low permeability case input Table 2: High permeability case input Table 3: Model dimensions summary Table 4: Low permeability case results Table 5: High permeability case results Table 6: Run time and storage capacity requirement for base cases and cases with induced fracture modelling List of Tables

10 P a g e x Nomenclature A = the fracture surface area at any instant during injection, ft 2 Ae = the fracture surface area at the end of pumping, ft 2 Bo= the oil formation volume factor, rb/stb CL = leak-off coefficient ft/s 0.5 E = strain modulus, psi Fcd = the fracture dimensionless conductivity, dimensionless FOI = folds of increase, dimensionless h = the flow unit height, ft hd = the lower height growth, ft hds = the dimensionless thickness, dimensionless hf = fracture height, ft hp = the perforation interval length, ft hs = the thickness of a symmetry element, ft hu = the upper height growth, ft i = half injection rate, ft 3 /s k = permeability, md k00 = the pressure at the middle of the crack, psi k1 = the slope of net pressure, psi K(C,2) = fracture toughness in the upper layer, psi.ft 0.5 K(C,3) = fracture toughness in the lower layer, psi.ft 0.5 kf = the fracture permeability, md kh = the horizontal permeability, md kv = the vertical permeability, md Nomenclature

11 P a g e xi Nprop = the dimensionless proppant number, dimensionless pbhpf = the bottom-hole flowing pressure, psi pcp = the pressure at mid perforation, psi pn,w = the net wellbore pressure, psi re = the drainage radius, ft rw = the wellbore radius, ft rw is the effective wellbore radius, ft Sf = the skin factor due to fracture, dimensionless Sp = spurt loss coefficient, ft t = time, second vf = the total fracture volume of both wings, ft 3 vl = leak-off velocity, ft/s w = average fracture width, ft wf = fracture width, ft ww,0 = the maximum fracture width at the wellbore, ft xf = fracture half-length, ft α = the exponent of fracture length growth (constant), dimensionless µ = the viscosity, cp ρ = density, lb/ft 3 σ1 = the minimum horizontal stress in the targeted layer, psi σ2 = the minimum horizontal stress in the upper layer, psi σ3 = the minimum horizontal stress in the lower layer, psi σmin = the minimum horizontal stress, psi Nomenclature

12 P a g e 1 1 Project Scope and Objectives 1.1 Induced Fractures Induced fracturing is a stimulation method used to accelerate the production and increase the ultimate recovery of hydrocarbon reservoirs by fracturing the reservoir rock (Anon. a 2013). Fracturing the rocks creates high conductivity channels growing into the reservoir away from the wellbore, providing communication between the two (Anon. b 2013). These fractures are called induced fractures since they are introduced to the reservoir and are not formed due to natural causes (e.g. tectonic activities). Since the 1940s, induced fracturing has proven to be an effective method for developing low permeability reservoirs and increasing the commercial viability of the development of conventional reservoirs (Taleghani et al. 2013). Also, induced fractures have made it possible to produce hydrocarbon from shale formations (tight reservoirs) where conventional technologies are ineffective (Anon. c 2014). Progress in hydraulic fracturing technologies has resulted in a huge increase in the oil and gas reserves worldwide by making the development of unconventional reservoirs feasible (Anon. d 2014). There are three types of induced fractures: hydraulic fractures; fractures created by fluid (usually water and/or polymer) injection under fracturing conditions; and thermal fracturing (Taleghani et al. 2013). Hydraulic fractures are created by injecting specially engineered fluid under high pressure for a short period of time to break the rocks. The created fractures are kept open after treatment using proppant (a material similar to sand grains) of a particular size, which is mixed with the treatment fluid (Anon. e 2014). Another type of induced fracture is created by the continuous injection of fluid under high pressure into the reservoir (greater than the fracture initiation pressure to create the fracture, greater than the closure pressure to keep the fracture open and greater than the fracture Chapter 1 Project Scope and Objectives

13 P a g e 2 propagation pressure to extend the fracture). These fractures are closed once the fluid injection stops or the injection pressure becomes less than the fracture closure pressure (Moreno et al. 2005). The final type of induced fracture is thermal fracturing. Thermal fractures are created due to the difference between the temperature of the reservoir rock and that of the injected fluid, with the latter being colder (Anon. f 2013). Only the first two types of induced fractures will be considered in this work. Fracture dimensions are considered the most important factor in induced fracturing for three main reasons: the incremental increase of production/injection rates is directly dependent on the fracture dimensions; the cost of creating the fracture is directly proportional to the fracture volume; and there is the possibility of induced fractures growing into unintended zones like fresh water zones. The environmental impacts associated with hydraulic fracturing are the main reason for it being a controversial topic among the public (Shukman 2013). Therefore, it is necessary to simulate the induced fracture propagation, dimensions and recession before the actual operations take place (Xiang 2011). Extensive work has been done to simulate the fracture propagation and dimensions. There are multiple analytical and numerical models available in the literature for estimating fracture dimensions, propagation and recession with different geometries. They include two-dimensional, three-dimensional and pseudo three-dimensional models (Yang 2011). The advantage of simulating fracture dimensions and propagation using the algorithms and methods introduced in this work is that they incorporate the actual reservoir dynamic simulator constraints and pressure data for the whole life of the field. Also, they take into account the practical limitations to estimate the optimum fracture dimensions, resulting in the maximum possible increase in production or injection rates for the wells. Therefore, this results in robust production and injection forecasts for reservoirs with induced fractured wells, and thus more Chapter 1 Project Scope and Objectives

14 P a g e 3 representative economics for field development. Estimating the optimum hydraulic fracture dimensions, modelling hydraulic fracture dimensions, induced fracture propagation in reservoir dynamic simulators, and estimating the height growth of induced fractures are all covered in this work. Over the past 70 years, extensive work has been done on induced fracturing. This work is documented and can be found in the literature. The following sections discuss topics related to induced fracturing available in the literature Fracture Orientation Based on rock mechanics, there are three principal stresses acting on underground formations. These are the overburden stress, the maximum horizontal stress, and the minimum horizontal stress, as shown in Figure 1. Figure 1. Principal stresses (Anon. g 2014) These stresses are usually anisotropic in that they differ in magnitude based on direction (Anon. a 2013). Fractures propagate in a direction which is perpendicular to the least stress (i.e. opening in the direction of the least resistance) (Anon. h 2010). The overburden stress acting on a Chapter 1 Project Scope and Objectives

15 P a g e 4 formation is due to the weight of the rocks above that formation, which depends on the formation depth (Golf-Racht 1980). Based on practical experience, for formations deeper than 2000 ft, the overburden stress is the largest principal stress, followed by the maximum horizontal stress; minimum horizontal stress is the smallest principal stress and the fractures are more likely to be vertical (Anon. h 2010). For formations shallower than 2000 ft, the maximum horizontal stress is the largest stress, followed by the minimum horizontal stress, and the overburden stress is the smallest principal stress (Anon. h 2010). Therefore, for such formations, the fractures will be horizontal, opening in the vertical direction with an environmental risk, since it may propagate to the surface. It can be concluded that the magnitude and direction of the principal stresses play a major role in determining the required pressure for fracture creation and propagation (Hudson 2005). The interaction between the fluid pressure in the fracture and the principal stresses defines the shape, the vertical extent and the propagation direction of the fracture (Dubey et al. 2012) Leak-off Test This is a test performed to measure the formation fracturing pressure usually carried immediately after drilling below a new casing shoe. The test is performed by shutting-in the well and pumping fluid, usually mud, into the wellbore to gradually increase the pressure experienced by the formation. At some pressure, the fluid enters the formation (or leaks-off) by fracturing the rock (Anon. e 2014). If the test is stopped just after the leak-off happens then it is called a leak-off test (LOT). If the test is extended longer until several iterations of pumping and discontinuing pumping have been performed then it is called an extended leak-off test (XLOT). From the XLOT, more important parameters can be estimated and used in the propagation and recession models. Figure 2 depicts XLOT results. Chapter 1 Project Scope and Objectives

16 P a g e Bottomhole Pressure 3 ΔP friction Injection Rate Time 1. Hydrostatic pressure 2. Breakdown pressure 3. Fracture extension pressure 4. Initial shut-in pressure (fracture gradient) 5. Fracture closure pressure (closure stress gradient) 6. Fracture reopening Pressure Figure 2. Extended leak-off results (Crain 2013) From XLOT results, vital parameters for simulating the fracture propagation, opening and closure are estimated. These include the fracture initiation pressure, fracture propagation pressure, fracture reopening pressure and fracture closure pressure (which is synonymous with minimum in-situ stress and minimum horizontal stress) (Anon. a 2013). These data will be used as an input to modelling induced fractures created due to injection under fracturing conditions using the PKN-α method. Chapter 1 Project Scope and Objectives

17 P a g e Hydraulic Fracturing Procedure The process of hydraulic fracturing consists of injecting specially engineered fluid at high pressure to break the formations and create high permeability channels extending away from the wellbore into the formation and establishing communication between the two. To keep the fracture open, proppant with specific grain diameter is used. The stages of hydraulic fracturing, as covered in the literature (Anon. h 2010), include: i. Spearhead stage or acid stage which consists of water mixed with acid. The purpose of this stage is to remove the debris and clean the wellbore. This will provide a clean wellbore and an open path for the fluid to be injected in subsequent stages. ii. Pad stage which consists of slick water that is used to initiate the hydraulic fracture in the formation. If the pressure stopped during this stage, the fractures would close since no proppant material has yet been used. iii. Proppant stage which consists of injecting water and proppant material into the fractured formation to keep the fractures open. Proppant is a non-compressible material, like sand grains, that is carried into the fractured formation to be left there after the job has been completed. Once the pressure drops, the proppant will prevent the fractures from closing, thus maintaining the enhanced permeability channels, created in the pad stage, throughout the well s life. iv. Flush stage which consists of fresh water being pumped into the wellbore to flush out and remove the excess proppant from the wellbore. Chapter 1 Project Scope and Objectives

18 P a g e Analytical and Numerical Models for Estimating Fracture Dimensions, Propagation and Recession There are multiple analytical and numerical models in the literature for estimating fracture dimensions, propagation and recession. These include two-dimensional, three-dimensional and pseudo three-dimensional models (Yang 2011). Geometries of these models are shown in Figure 3. The fracture geometry in reality is more complicated than the simple geometries described in these models and is governed by many parameters related to rock mechanics, insitu stresses and the fluids used (Warpinski 1989). However, these models are commonly used in industry and are widely accepted as providing an acceptable approximation for the estimation of fracture dimensions. The two famous 2D analytical models - PKN, developed by Perkins, Kern and Nordgren in 1961, and KGD, developed by Khristianovitch, Zheltov, Geertsma and de Klerk in are widely used and often referred to in the oil and gas industry (Yang 2011). The PKN model assumes an elliptical cross-section fracture shape; fracture height is constant and fracture length is significantly greater than fracture height. The KGD model assumes a rectangular crosssection fracture shape, constant fracture height and fracture height significantly greater than fracture length (Valko and Economides 1995). The plane strain direction assumption differs between the two; a vertical direction for PKN and a horizontal direction for KGD (Xiang 2011). Based on these assumptions, the KGD model could be used to estimate fracture dimensions and shape for small fracture treatments and/or when fracture height is uncontrolled and significantly greater than fracture length. PKN-type fractures are more interesting from a production point of view since reservoir layers are usually contained by shale layers. Therefore, it represents a case of an elongated fracture whose length is significantly greater than its height (Valko and Economides 1995). Chapter 1 Project Scope and Objectives

19 P a g e 8 Figure 3. Fracture geometry (a) PKN type (b) KGD type (c) Pseudo 3D cell approach (d) Global 3D, parameterised (e) Full 3D, meshed (Yang 2011) PKN-α Model As mentioned in Section 1.1.4, the PKN model assumes the fracture has constant height and that fracture length is significantly greater than fracture height. The PKN geometry depicted in Figure 3(a), which shows an approximately elliptical shape in the vertical and the horizontal directions, is more interesting from the production point of view. The PKN-α model assumes the power law surface growth and Carter I leak-off to perform the material balance at any time during injection. Chapter 1 Project Scope and Objectives

20 P a g e 9 The power law surface growth assumes that the fracture surface grows according to a power law relating the area of the fracture at any time during injection to the area of the fracture at the end of the injection with an exponent, α, that is constant during the period of injection (Nolte 1986). Carter introduced the leak-off velocity by relating a leak-off coefficient to the elapsed time since the start of the leak-off process and spurt loss based on the concept of Howard and Fast (Howard 1957). Equations related to both assumptions are discussed in details in chapter Fracture Height Growth The two-dimensional models suggested in the previous sections are simplified approximation of the fracture dimensions and geometry. These models assume constant fracture height and leave the half-length and fracture width to be estimated from injected fluid volume. However, practical experience showed that fractures in some cases grow into unintended zones up and down the targeted interval (Valko and Economides 1995). This observation triggered the attempts to develop models able to simulate the fracture height. Since there are many variables in the system of equations, simplifying the approach is essential to result in an acceptable approximation used the simplest case by neglecting hydrostatic pressure inside the fracture and using similar properties for the upper and lower layers for approximating fracture height growth (Simonson et al. 1978). Another analysis that is widely acceptable in oil and gas industry is Warpinski and Smith analysis with a more complex case (Warpinski and Smith 1989). The notation used by them is shown in Figure 4. Alternative notation is used by Ahmed and Economides which is discussed in chapter 2. Chapter 1 Project Scope and Objectives

21 P a g e 10 Figure 4. Warpinski and Smith analysis and notation (Valko and Economides 1995) The assumptions of Warpinski and Smith analysis (Valko and Economides 1995) are: i. The minimum horizontal stress of the upper and lower layers can be different, but higher than minimum horizontal stress of the targeted layer. ii. The critical stress intensity factor (stress intensity near the tip) can be different for the upper and lower layers. iii. The density of the fluid is considered in the analysis The notation will be used in this work is Ahmed and Economides notation and the set of the equations used to determine the fracture height growth is presented in the methodology chapter. 1.2 Dynamic Simulation of Hydraulic Fractures As for modelling the effects of induced fractures in the reservoir dynamic simulators, there are multiple approaches, which include: using a negative skin factor, creating channels of enhanced permeability gridblocks in the direction of fracture orientation, using non-neighbourhood connections and/or a local increase of absolute permeability near the wellbore (Carlson 2006; Owen1983). Chapter 1 Project Scope and Objectives

22 P a g e 11 These approaches are helpful to a certain extent for increasing productivity, but they do not incorporate the physics behind fracture propagation and recession (Carlson 2006; Owen1983). Also, some of the routines used for 3D pseudo models result in long run times, making them impractical for large reservoir models (Economides 2000). 1.3 Objectives The objectives of this work are: i. To develop an algorithm for determining the optimum hydraulic fracture dimensions of a well and incorporating hydraulic fractures in reservoir dynamic simulators. ii. To model induced fractures in fluid (usually water and/or polymer) injectors created during injection under fracturing conditions using reservoir dynamic simulators. iii. To develop an algorithm for modelling the height growth of induced fractures during injection under fracturing conditions using reservoir dynamic simulators. These algorithms and methods are simple and easy to incorporate in the reservoir dynamic simulators to provide robust production and injection forecasts. This work is of great economic and environmental benefit because the fracture dimensions are the single most important factor which determines the increase of production/injection rates, the volume and cost of used fracturing material, and the zones into which fractures propagate. Chapter 1 Project Scope and Objectives

23 P a g e 12 2 Methodology This chapter represents the methodology followed to achieve the stated objectives. Derivation, calculations and Eclipse modelling are shown in this chapter for each objective. 2.1 Determining the Optimum Hydraulic Fracture Dimensions for a Well For the first objective of determining the optimum hydraulic fracture dimensions, the optimum fracture conductivity for pseudo-steady state and steady state flow conditions is calculated. For conventional reservoirs, since most wells spend the majority of their lifetime in a pseudo-steady state flow regime, the solution reached should represent the optimum hydraulic fracture dimensions of a well (Richardson 2000). The analysis starts with the use of Darcy law for pseudo-steady state flow conditions, as shown in Eq.1: Q = 2πkh/μB o ln r e r w S f (1) where: k is permeability, h is the flow unit height, µ is the viscosity, Bo is the oil formation volume factor, re is the drainage radius, rw is the wellbore radius and Sf is the skin factor due to fracture. It needs to be noted that the assumption here is that the skin due to damage is not a part of the optimum fracture dimension calculations since it happens due to drilling, production and/or completions. However, skin will be used later as a check that the wellbore radius, due to damage (rs), is less than half the length of the fracture (xf), to confirm that the hydraulic fracture bypasses the damage zone. Chapter 2 Methodology

24 P a g e 13 In order to maximise the production rate, the denominator of Eq. 1 has to be minimised. Defining function G as the denominator of Eq.1, as shown in Eq.2, which has to be minimised to increase rate. G = ln r e r w S f minimum. (2) Now, defining function A, as shown in Eq.3. A = ln x f r w + S f.. (3) Based on Figure 5, A is the y-axis of the Cinco-Ley and Samaniego graph (Valko and Economides 1995). Figure 5. Cinco-Ley and Samaniego graph for Hydraulic Fracture Performance with f = sf + ln (xf/rw) (Valko 2005) Chapter 2 Methodology

25 P a g e 14 As shown in Eq. 4, function G becomes: G = ln r e r w A ln x f r w minimum.. (4) Simplifying function G, as shown in Eqs. 5 through 7. G = lnr e ln r w A lnx f + ln r w.. (5) G = lnr e lnx f + A (6) G = ln r e x f + A (7) Two functions should be introduced here: the fracture dimensionless conductivity and the fracture volume as shown in Eqs. 8 and 9. F cd = k fw f kx f (8) where Fcd is the fracture dimensionless conductivity, kf is the fracture permeability, wf is the fracture width, k is the matrix permeability and xf is the fracture half-length. The fracture dimensions are related to each other by the fracture volume, as shown in Eq. 9. v f = 2h f x f w f.. (9) where vf is the total fracture volume of both wings and hf is the fracture height. By combining Eqs. 8 and 9, the fracture half-length can be estimated using Eq. 10. X f = v fk f 2hkF cd. (10) By substituting Eq.10 in Eq. 6, as shown in Eq. 11, the G function becomes: G = lnr e ln v fk f 2hkF cd + A (11) Chapter 2 Methodology

26 P a g e 15 where A is defined in Eq. 12, as shown in Figure 5 as: A = lnf cd (ln(f cd )) lnf cd (ln(F cd )) (ln(F cd )) 3. (12) To determine the Fcd value that will result in the minimum function G, the function is derived with respect to Fcd. Substitution of function A in function G and the derivation is shown in Eqs. 13 and 14. G = lnr e ln v fk f lnf cd (ln(f cd )) 2 + 2hkF cd lnf cd (ln(F cd )) (ln(F cd )) (13) dg df cd = 1 2lnF cd 23.2 [(lnf cd) (lnF cd ) (lnF cd ) lnF cd ] lnf cd [(lnf cd ) (lnF cd ) lnF cd + 200] 2. (14) By setting the derivative equal to zero and solving for the fracture dimensionless conductivity, Fcd that results in minimum value of the G function can be found; Fcd = Thus, the optimum fracture dimensionless conductivity value for a fracture in a well flowing under pseudo-steady state flow conditions is The partial penetration skin is the function of two parameters; the penetration ratio and dimensionless thickness (b and hds) (Brons et al. 1961). The penetration ratio (b) is assumed to be set dependent upon given facts of a specific reservoir to ensure reduction in water and/or gas production and that the best part of the reservoir is targeted. Thus, the only variable to be considered for the calculation of the optimum hydraulic fracture dimensions is the Chapter 2 Methodology

27 P a g e 16 dimensionless thickness (hds). Brons and Marting defined the hds of a fractured well as shown in Eq. 15. h ds = h s r w k h k v. (15) where hds is the dimensionless thickness, hs is the thickness of a symmetry element, rw is the effective wellbore radius, kh is the horizontal permeability and kv is the vertical permeability (Brons et al. 1961). By analysing Eq. 15, the goal of minimising hd can be achieved by increasing rw. Since Sf is inversely proportional to rw, determining the minimum Fcd using Sf, as above, results in the maximum rw. The method described in this work includes starting from the minimum additional economic value gain required from a hydraulic fracturing project. Let us assume that the screening criterion of a small project for a company is a net present value of x $. Based on the oil price, the production forecast without hydraulic fracturing and hydraulic fracturing job cost, the additional increase in oil production rate results in a net present value of x$ due to accelerated production can be estimated. Thus, the minimum required folds of increase (FOI) for the project to pass the screening criterion are calculated. To estimate the optimum fracture dimensions, relating FOI to the fracture half-length would provide a tool to directly determine the optimum fracture dimensions from a known or targeted FOI value. This is shown by relating the effective wellbore radius to FOI and using Prat s dimensionless effective wellbore radius, as explained below. FOI and skin due to fracture are related, as shown in Eq. 16. FOI = ln r e r w ln r.. (16) e + S r f w Chapter 2 Methodology

28 P a g e 17 where the skin due to fracture is related to the effective wellbore radius, as shown in Eq. 17. S f = ln r w r w (17) Thus, FOI can be related to the effective wellbore radius, as shown in Eqs. 18 and 19. FOI = ln r e r w lnr e lnr w + lnr w lnr w (18) FOI = ln r e r w ln r (19) e r w Using Eq. 19, a table of rw vs. FOI can be developed with the FOI value of the minimum estimated from economics or greater. Using Prat s dimensionless effective wellbore radius and knowing the optimum Fcd is , rw /xf can be found, as shown in Figure 6. Figure 6. Prat s dimensionless effective wellbore radius (Valko 2005) Thus, the optimum rw /xf value is , which can be used for finding the optimum xf results in the maximum FOI value. It is important to note that there is a maximum theoretical value of Chapter 2 Methodology

29 P a g e 18 xf that is equal to the drainage radius (re), which is used as a maximum constraint for the fracture half-length calculation. It is also important to mention that the work thus far only assumes the theoretical value and does not include the practical aspect of hydraulic fracturing. For example, is it possible to achieve a fracture half-length equal to the drainage radius of the reservoir? Is it possible to have all of the injected fluid contained in the pay zone, or intended zone? Valko introduced a parameter called the dimensionless proppant number (Nprop) as shown in Eq. 20 (Valko 2001). N prop = 4k fx f w f kπr e 2.. (20) According to Valko, since the proppant cannot be contained in the pay zone and within the drainage area and for large treatments there is a great uncertainty as to where the proppant goes in both horizontal and vertical directions, there is a practical limit to the dimensionless proppant number (Valko 2001). The practical N number is less than or equal to 0.1 for medium and high permeability formations (50 md and above), while for low permeability reservoirs a dimensionless proppant number more than 0.5 is rarely realised. Therefore, another condition is applied in this work to the calculation performed, which is the use of N of 0.1 or less for formations with a permeability of 50mD and above, and 0.5 or less for the formations with a permeability of less than 50mD (Valko 2001). Also, it is preferable to have the fracture half-length longer than the damage radius to eliminate the impact of damage on production. If the fracture half-length is increased, the width also has to be increased to maintain the optimum Fcd value. Thus, the proppant volume required is increased so that the economic value resulted from increasing production by increasing the volume of the fracture versus the economic saving made on the proppant cost by keeping the fracture half-length less than the damaged radius has to be evaluated. Finally, Eq. 8 is used to calculate the fracture width, since everything else is known. Chapter 2 Methodology

30 P a g e 19 In the case of low permeability reservoirs, the fracture width calculated by the method above can be small. Practically, the fracture width has to be large enough for the proppant to be placed in the fracture to keep it open. Therefore, another condition to be applied is that the fracture width has to be at least two to three (2-3) times the mesh proppant grain diameter. In such a case, starting with a fracture width of 2-3 times the proppant grain diameter, the length can be calculated using N value of 0.5 or lower (applying the condition that the calculated half-length fracture is equal to or less than drainage radius). By analysing Eq.8, for low permeability reservoirs, it can be inferred that a long fracture is required for a certain minimum width determined to result in optimum Fcd value. Thus, in cases of low permeability low drainage radius reservoirs, the theoretical limitation on the maximum possible fracture half-length in addition to the practical limitations previously mentioned may result in Fcd values more than An excel workbook is developed to perform all the calculations mentioned in this section. Further work can be performed by linking this workbook to Eclipse Dynamic Simulator so that refinements and calculations are performed automatically Cases Input for Testing the Procedure Used to Determine the Optimum Hydraulic Fracture Dimensions for a Well For the first objective, two cases from literature (Valko and Economides 1995) are used to test the analysis of this work, perform the calculations for the optimum fracture dimensions and perform the Eclipse Dynamic Simulation. The results match very well, as shown in the results and discussion chapter. The two cases are for a low permeability reservoir and a high permeability reservoir. Each reservoir has six water injectors and one oil producer. The oil producer is to be hydraulically Chapter 2 Methodology

31 P a g e 20 fractured to increase the reservoir s production rate. It is necessary to determine the optimum hydraulic fracture dimensions, estimate the increase in production rate and develop the new production forecast Low Permeability Reservoir Case Input data for the low permeability reservoir case are shown in Table 1. Table 1: Low permeability case input Horizontal Permeability 0.5 md Formation Height 105 ft Fracture Height 35 ft Fracture Permeability md Proppant Mesh Diameter 70 mesh (420 µm) Drainage radius 2100 ft Wellbore radius ft Damage skin High Permeability Reservoir Case Input data for the high permeability reservoir case is shown in Table 2. Table 2: High permeability case input Horizontal Permeability 500 md Formation Height 150 ft Fracture Height 30 ft Fracture Permeability md Proppant Mesh Diameter 40 mesh (840 µm) Drainage radius 1000 ft Wellbore radius ft Damage skin 0 Chapter 2 Methodology

32 P a g e Incorporating Hydraulic Fractures in Reservoir Dynamic Simulators To estimate the impact of hydraulic fracturing on reservoir production and/or injection rates, a high permeability channel in a refined box model is simulated using Eclipse Reservoir Simulator. The results of the optimum dimensions for hydraulic fracture (Section 2.1) are used to create a high permeability channel near the wellbore, extending away from it by fracture half-length in each direction. In the two cases of this work, the model is refined to have the width of the gridblocks equal to the hydraulic fracture width, as shown in Figure 8 and Figure 10. In both cases, the models have six water injectors and one hydraulically fractured oil producer, as shown in Figure 7 and Figure 9. A permeability multiplier, NTG multiplier and porosity multiplier are used for the gridblocks representing the fracture to mimic the hydraulic fracture permeability, increase the porosity of the gridblocks to 100% and increase NTG to 100%. Table 3 summarises the Eclipse dimensions used for the two cases. Table 3: Model dimensions summary Item Low permeability case High permeability case Number of gridblocks 11 x 1000 x 3 11 x x 5 Model dimensions (ft) x x x x 30 Chapter 2 Methodology

33 P a g e 22 Figure 7. Low permeability case model Figure 8. Hydraulic fracture for low permeability case Chapter 2 Methodology

34 P a g e 23 Figure 9. High permeability case model Figure 10. Hydraulic fracture for high permeability case In other cases, models can be refined so that the gridblock width is less than the fracture width (i.e. the fracture channel includes more than one gridblock in the width direction). In the case of large models, local gridblock refinement can be performed instead to mimic the same Chapter 2 Methodology

35 P a g e 24 procedure followed in this work. Also, fracture half-length should be taken into consideration of refinement in case fracture half-length estimated is less than gridblock length. 2.3 An Algorithm for Modelling Induced Fractures Created During Injection under Fracturing Conditions in Reservoir Dynamic Simulators The following sections introduce the methodology used in this work to develop an algorithm to be used to simulate the propagation, recession and dimensions induced fractures created due to injection under fracturing conditions Fracture Height Growth As mentioned in the Section , Ahmed and Economides notation (Economides 1992) is used in this work. The variables to be estimated as shown in the notation (Figure 11) are the upper ( hu) and lower ( hd) height growth. Figure 11. Ahmed and Economides notation for fracture height (Valko and Economides 1995) The solution can be achieved by solving for two unknowns in two equations (Eqs. 21 and 22). Chapter 2 Methodology

36 P a g e 25 KI, top h p = π(y u y d ). {f t [y d, k 00 σ 3, k 1 ] f t [ 1, k 00 σ 3, k 1 ] + f t [y u, k 00 σ 1, k 1 ] f t [y d, k 00 σ 1, k 1 ] + f t [+1, k 00 σ 2, k 1 ] f t [y u, k 00 σ 2, k 1 ]} = K(C, 2) [fracture toughness of the upper layer]. (21) KI, bot h p = π(y u y d ). {f b [y d, k 00 σ 3, k 1 ] f b [ 1, k 00 σ 3, k 1 ] + f b [y u, k 00 σ 1, k 1 ] f b [y d, k 00 σ 1, k 1 ] + f b [+1, k 00 σ 2, k 1 ] f b [y u, k 00 σ 2, k 1 ]} = K(C, 3) [fracture toughness of the lower layer].. (22) where: y u = 1 y d = h u h p + h u + h d.. (23) 2 h d h p + h u + h d.. (24) k 00 = p cp + ρg h d h u 2. (25) k 1 = ρg 2h p. (26) y u y d f t = 1 + y 1 y. { k o k 1 + (2k 0 + k 1 ). y + k 1y } + (2k 0 + k 1 ). tan 1 y (1 + y) ( (1 y) ).. (27) y Chapter 2 Methodology

37 P a g e 26 f b = 1 y 1 + y. {k o k 1 + (2k 0 k 1 ). y + k 1y } (2k 0 k 1 ). tan 1 y (1 y) ( (1 + y) )... (28) y It should be noted that the following limits (Eqs. 29 through 32) are required to perform the calculation: f t [+1, k 0, k 1 ] = π 4 (2k 0 + k 1 ). (29) f t [ 1, k 0, k 1 ] = π 4 ( 2k 0 k 1 ). (30) f b [+1, k 0, k 1 ] = π 4 (2k 0 k 1 ). (31) f b [ 1, k 0, k 1 ] = π 4 ( 2k 0 + k 1 ).. (32) where: hp is the perforation interval length, hu is the upper height growth, hd is the lower height growth, ρ is the density, σ1 is the minimum horizontal stress in the targeted layer, σ2 is the minimum horizontal stress in the upper layer, σ3 is the minimum horizontal stress in the lower layer, K(C,2) is fracture toughness in the upper layer, k0 is a constant, k1 is the slope of net pressure, k00 is the pressure at the middle of the crack, K(C,3) is fracture toughness in the lower layer and pcp is the pressure at mid perforation. All of the parameters in these two equations are inputs except hd and hu are the variables to be solved for. These two variables are calculated at each time step and the fracture height results is used in PKN-α calculation for determining the fracture half-length and width. In this work, these two equations were combined and simplified. The final complete version of these two equations with two unknowns are shown in appendix A.1. Chapter 2 Methodology

38 P a g e Fracture Width and Half-Length As mentioned in Chapter 1, the PKN geometry assumes an elliptical shape in the vertical and the horizontal directions for the hydraulic fracture. It assumes that the height, hf, is constant and that the half-length, xf, is considerably greater than the width, wf. The method used in this work to simulate induced fractures developed due to injection under fracturing conditions is the PKNα method, which assumes: i. The power law surface growth, which is represented in Eq. 33. A = ( t ) α.. (33) A e t e where A is the fracture surface area at time t, Ae is the fracture surface area at the end of pumping, t is time, te is the time at the end of pumping, and α is the exponent of fracture length growth and is constant during the injection period. ii. Carter equation I for leak-off, which is shown by Eqs. 34 and 35. v L = C L t.... (34) which has an integrated form of: v L = 2C L t + S p..... (35) where v L is leak-off velocity, C L is the leak-off coefficient and t is the time elapsed since the start of leak-off. iii. α is the exponent and is assumed to be known. It is equal to 4/5 for the case with no leak-off and it is reasonable to assume that the exponent remains the same in the presence of leak-off. Chapter 2 Methodology

39 P a g e 28 With the above assumptions, the material balance at any time during injection can be written as Eq. 36. w = it A 2C L α π (α) ( α) t 2S p. (36) where A is the fracture surface area at time t, which equals xf.hf; i is half of the injection rate or the injection rate for one wing of the fracture, hf is fracture height, w is the average width of the fracture, Sp is the spurt-loss coefficient, CL is the leak-off coefficient, t is time, α is the exponent of fracture length growth and is the Euler Gamma Function, which can be calculated using Eq. 37. (t) = x t 1 e x dx (37) 0 Substituting for the fracture surface area to incorporate fracture half-length and fracture height, the material balance can be written as shown in Eq. 38. w = it 2C L α π (α) x f h f ( α) t 2S p (38) To solve for fracture half-length, Eq. 35 can be re-arranged as shown in Eq. 39. x f = it h f w + 2S p + 2C L t [ α π (α)... (39) ( α)] The procedure to be followed for the calculations consists of using the input data in simple calculations and testing conditions at each time step. In the beginning, at each time step a comparison of bottom-hole flowing pressure to the fracture initiation pressure is performed and fracture is only initiated if the former is greater than or equal to the latter. Once the fracture is initiated, it either closes, remains open, closes then reopens, or propagates. At each time step following the fracture initiation, condition testing is performed and a decision is made on the Chapter 2 Methodology

40 P a g e 29 fracture simulation for the next time step. In case the bottom-hole pressure is greater than or equal to the fracture propagation pressure, the PKN-α calculation method is performed and the half-length and width of the fracture are estimated. For the PKN-α model, the fracture s dimensions can be estimated as shown in Eqs. 40 through 43. p n,w =Pbhpf σ min. (40) where pn,w is the net wellbore pressure, pbhpf is the bottom-hole flowing pressure and σmin is the minimum horizontal stress. Once the net wellbore pressure is calculated, the maximum fracture width at the wellbore can be found thus: p n,w = E 2h f w w,0.. (41) where E is the plane strain modulus (which can be calculated from Young s modulus and Poisson s ratio) and ww,0 is the maximum fracture width at the wellbore. To solve the maximum fracture width at the wellbore, Eq. 41 can be re-arranged as shown in Eq. 42. w w,0 = 2h f (P bhpf σ min ) E. (42) The average fracture width is related to the maximum fracture width at the wellbore, as shown in Eq. 43. w = w w,0.. (43) where w is the average fracture width and is the shape factor (π/5). The shape factor contains π/4 because the vertical shape is an ellipse. Also, it contains another factor (4/5) which accounts for the lateral variation of the width for the PKN model (Yang 2011). Once the average fracture width has been estimated, the fracture half-length is related to it as shown in Eq. 39. Chapter 2 Methodology

41 P a g e 30 To determine the fracture half-length and width, fracture height is required as an input. From fracture height calculation Section 2.3.2, the result is used as an input to the PKN-α calculation to estimate the maximum fracture width near the wellbore which is then used to carry the of the calculations to estimate the fracture average width and half-length. To incorporate the fracture in the dynamic model, local grid refinement is performed based on the new fracture height, width and half-length calculated after each time step. Also, permeability, NTG and porosity multipliers should be applied to the gridblocks representing the fracture. The logic behind that should be performing NTG and porosity multiplying to result in gridblocks NTG of 1 and porosity of 100%. As for the permeability multipliers, analogues can be used to relate the fracture width to certain fracture permeability value. Then, permeability multiplier should be used to increase the gridblocks permeability to the fracture permeability. The algorithm for the procedure described is shown in Figure 12. Chapter 2 Methodology

42 P a g e 31 Chapter 2 Methodology

43 P a g e 32 Figure 12. Algorithm for estimating fracture dimensions created due to injection under fracturing conditions Chapter 2 Methodology

44 P a g e 33 3 Results and Discussion This chapter introduces the results of the investigation to determine the optimum hydraulic fracture dimensions for low and high permeability reservoir cases. Furthermore, it discusses the results of modelling hydraulic fractures and induced fractures created due to injection under fracturing conditions in reservoir dynamic simulators. For each of these topics, the results are discussed and further improvements are suggested. 3.1 Determining the Optimum Hydraulic Fracture Dimensions for a Well The optimum hydraulic fracture width and half-length are estimated using the Excel workbook developed, based on the derived optimum dimensionless fracture conductivity, the practical dimensionless proppant number constraint and the minimum possible hydraulic fracture width related to proppant grain diameter Low Permeability and High Permeability Case Results The results of the low permeability case and the high permeability case are tabulated and discussed below. The same procedure can be followed to determine the optimum hydraulic fracture dimensions for other cases, to which the issues discussed also apply. Chapter 3 Results and Discussion

45 P a g e Low Permeability Reservoir Case The optimum hydraulic fracture dimensions estimated for the low permeability case are shown in Table 4. Table 4: Low permeability case results Half-length (xf) 1027 (ft) Fracture width (wf) (ft) Fracture volume (vf) 1010 (ft 3 ) FOI 4.2 Skin due to fracture (sf) N 0.5 Fcd Although the results shown in Table 4 represent the optimum fracture dimensions for the maximum folds of increase, it can be inferred from Figure 13 and Figure 14 that reducing the fracture volume to half will result in a reduction of only 0.5 in FOI. For a fracture volume of about 1000 ft 3, FOI of about 4.2 is calculated, while FOI of 3.7 is calculated for a fracture volume of 570 ft 3. Therefore, the cost of fracturing material and the expected increase in production due to the larger FOI value should be taken in consideration when determining the optimum hydraulic fracture dimensions for a well. Chapter 3 Results and Discussion

46 P a g e 35 Figure 13. Net present bbls increase in production and fracture volume vs. folds of increase for low permeability case For economics to be calculated there are multiple factors that need to be taken into consideration. These include: i. Capacity of the production system and facilities. If additional equipment and/or larger equipment sizes are required, there is an additional cost encountered for each barrel of oil produced and the additional folds of increase also incur more cost. Thus, the comparison should include this factor when estimating the saving versus the cost. ii. Other small projects that require investment (e.g. artificial lift, increasing the injection rate and/or deploying EOR methods). Ranking of the additional fracturing volume cost and gain versus the cost and gain of other methods should be performed to facilitate the selection of the best project for investment (the one with the highest net present value index, the highest net present value and/or the highest internal rate of return). iii. Additional investment in production facilities due to earlier water and/or gas breakthrough. Since created hydraulic fractures are high permeability channels, they can result in faster breakthrough of water and/or gas from aquifers, injectors and/or gas cap Chapter 3 Results and Discussion

47 P a g e 36 to producers. The dimensions of the hydraulic fracture should be selected carefully to delay breakthrough since this requires larger equipment sizes and/or additional equipment for separation and treatment. iv. Depth of damage due to drilling and/or completion phases. The depth of the damaged zone near the wellbore might be longer than the optimum fracture half-length in a few special cases. In such a case, the optimum fracture half-length can be increased (while also increasing the width to result in the same optimum fracture conductivity) to bypass formation damage. All of these factors should be taken into account when selecting the optimum dimensions of the hydraulic fracture High Permeability Reservoir Case The optimum hydraulic fracture dimensions estimated for the high permeability case are shown in Table 5 and Figure 14. Table 5: High permeability case results Half-length (xf) 218 (ft) Fracture width (wf) (ft) Fracture volume (vf) (ft 3 ) FOI 2.77 Skin due to fracture (sf) N 0.1 Fcd Chapter 3 Results and Discussion

48 P a g e 37 Figure 14. Net present bbls increase in production and fracture volume vs. folds of increase for high permeability case Similar to the low permeability case, it can be observed that for FOI of 2.5, a fracture volume of ft 3 is required. Increasing the fracture volume to ft 3 (approximately double), will result in only approximately 0.25 increase in FOI value. Therefore, the cost of the fracturing material and the additional gain in oil production should be taken into consideration before deciding the optimum fracture dimensions for a well, as has been shown in the two cases tested above. Chapter 3 Results and Discussion

49 P a g e Incorporating Hydraulic Fractures in Reservoir Dynamic Simulators The increases in production due to hydraulic fracture for the low permeability case and the high permeability case are shown in Figure 15 and Figure 16. Figure 15. Increase in cumulative oil production for low permeability case Figure 16. Increase in cumulative oil production for high permeability case Chapter 3 Results and Discussion