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Modeling hydraulic fractures in finite difference simulators using amalgam local grid refinement (LGR)

Fig. 3 LGRs around the wellbore

Abstract

Hydraulic fracturing allows numerous, otherwise unproductive, low-permeability hydrocarbon formations to be produced. The interactions between the fractures and the heterogeneous reservoir rock, however, are quite complex, which makes it quite difficult to model production from hydraulically fractured systems. Various techniques have been applied in the simulation of hydraulically fractured wells using finite difference simulators; most of these techniques are limited by the grid dimensions and computing time and hardware restrictions.

Authors

Reda Abdel Azim & Sherif S. Abdelmoneim

School of Petroleum Engineering, University of New South Wales, Sydney, Australia. ENAP Sipetrol, Cairo, Egypt.
Received: 24 February 2012 / Accepted: 15 September 2012 / Published online: 3 October 2012
© The Author(s) 2012.

Most of the current analytical techniques assume a single rectangular shaped fracture in a single-phase homogeneous reservoir, the fracture is limited to the block size and the fracture properties are adjusted using permeability multiplier. The current work demonstrates how to model these systems with a smaller grid block size which allows you to apply sensitivity to the fracture length and model the fracture with enhanced accuracy. It also allows you to study the effect of reservoir heterogeneity on the fractured well performance.

It is proposed to apply amalgam LGR technique to decrease the grid size to the dimensions of the hydraulic fracture without dramatically increasing the number of grid blocks which would cause a great increase in the computing time and the model size with no added value. This paper explains how the amalgam LGR is designed and compares between standard LGRs and the proposed design and a case study is presented from an anonymous field in Egypt to illustrate how to use this technique to model the hydraulically fractured well. The simulation model is matched to available production data by changing fracture lengths. Then the model is used to predict future response from the wells.

The advantage of this technique is that it allows hydraulically fractured reservoirs to be modeled with less grid size which will lead to more realistic models and more accurate predictions; however, the most useful application of this technique may be in the fracture modeling stage. With this tool, various fracture geometries and scenarios can be tested in the simulator, and the most economic scenarios selected and implemented. This will lead to better fracture placement, and ultimately greater production.

Introduction

Hydraulic fracturing is the process of pumping a fluid into a wellbore at an injection rate that is too high for the formation to accept in a radial flow pattern. As the resistance to flow in the formation increases, the pressure in the wellbore increases to a value that exceeds the breakdown pressure of the formation that is open to the wellbore. Once the formation “breaks-down”, a crack or fracture is formed, and the injected fluid begins moving down the fracture.

DOE research has developed several alternative fracturing techniques designed to accomplish specific tasks such as:

  • Tailored pulse fracturing
  • Foam fracturing
  • CO2, sand fracturing

In general, hydraulic fracture treatments are used to increase the productivity index of a producing well or the injectivity index of an injection well. The productivity index defines the volumes of oil or gas that can be produced at a given pressure differential between the reservoir and the well bore. The injectivity index refers to how much fluid can be injected into an injection well at a given differential pressure.

One of the major problems facing the reservoir engineers in modeling the hydraulic fractures using the finite difference simulators is the wide gap between the grid size of the reservoir model and the fracture dimensions.
The purposes of this paper are to model the flow of the reservoir fluid in hydraulically fractured reservoir using finite difference simulators in a manner that would allow the simulator to mimic the actual fracture geometry without dramatically increasing the number of grid cells and hence increasing the computing requirements and time.

This is achieved as shown in the paper by using amalgam LGR to decrease the fractures dimensions to the size and dimensions required to achieve this goal and leave the number of grid cells only slightly affected which makes a minor change in the required computing time and capabilities. This design was tried on an anonymous field in the western dessert in Egypt, and the results were compared with the actual production data which was recorded after the fracture to verify that the model was capable of modeling the actual reservoir performance.

Hydraulic fracture mechanics

The theory of hydraulic fracturing depends on an understanding of crack behavior in a rock mass at depth. Because rock is predominantly a brittle material, most efforts to understand the behavior of crack equilibrium and growth in rocks have relied on elastic, brittle fracture theories.

However, certain aspects, such as poroelastic theory, are unique to porous, permeable underground formations. The most important parameters are in situ stress, Poisson’s ration, and Young’s modulus.

In situ stresses

Underground formations are confined and under stress. Figure 1 illustrates the local stress state at depth for an element of formation. The stresses can be divided into three principal stresses. In Fig. 1, σ1 is the vertical stress, σ2 is the maximum horizontal stress, while σ3 is the minimum horizontal stress, where σ1 > σ2 > σ3. These stresses are normally compressive and vary in magnitude throughout the reservoir, particularly in the vertical direction (from layer to layer).

Fig. 1 The local stress state at depth for an element of formation

Fig. 1 The local stress state at depth for an element of formation.

The magnitude and direction of the principal stresses are important because they control the pressure required to create and propagate a fracture, the shape and vertical extent of the fracture, the direction of the fracture, and the stresses trying to crush and/or embed the propping agent during production.

A hydraulic fracture will propagate perpendicular to the minimum principal stress (σ3). If the minimum horizontal stress is σ3 the fracture will be vertical and, we can compute the minimum horizontal stress profile with depth using the following equation.

we can compute the minimum horizontal stress profile with depth using the following equation.

Poisson’s ratio can be estimated from acoustic log data or from correlations based upon lithology. The overburden stress can be computed using density log data. The reservoir pressure must be measured or estimated. Biot’s constant must be less than or equal to 1.0 and typically ranges from 0.5 to 1.0. The first two terms on the right hand side of the equation represent the horizontal stress resulting from the vertical stress and the poroelastic behavior of the formation.

Poroelastic theory can be used to determine the minimum horizontal stress in tectonically relaxed areas (Salz 1977). Poroelastic theories combine the equations of linear elastic stress–strain theory for solids with a term that includes the effects of fluid pressure in the pore space of the reservoir rocks.

The fluid pressure acts equally in all directions as a stress on the formation material. The “effective stress” on the rock grains is computed using linear elastic stress–strain theory. Combining the two sources of stress results in the total stress on the formation, which is the stress that must be exceeded to initiate fracturing. In addition to the in situ or minimum horizontal stress, other rock mechanical properties are important when designing a hydraulic fracture. Poisson’s ratio is defined as “the ratio of lateral expansion to longitudinal contraction for a rock under a uniaxial stress condition (Gidley et al. 1989)”.

The theory used to compute fracture dimensions is based upon linear elasticity. To apply this theory, the modulus of the formation is an important parameter. Young’s modulus is defined as “the ratio of stress to strain for uniaxial stress (Gidley et al. 1989)”.

The modulus of a material is a measure of the stiffness of the material. If the modulus is large, the material is stiff. In hydraulic fracturing, a stiff rock will result in more narrow fractures. If the modulus is low, the fractures will be wider. The modulus of a rock will be a function of the lithology, porosity, fluid type, and other variables.

Field case study

Field characterization

We conducted a fracture reservoir simulation study for well A-1 (Appendix 1, Fig 16) over Abu-Roash G reservoir in the field (A), in the western dessert of Egypt.

This field produced light gravity oil (37° API) from Abu-Roash G sand at an average drilled depth of 5,400 ft TVDSS. Only two wells were drilled in the area and had been reviewed in the field study. The interpretation of the well logs shows hydrocarbon bearing in middle A/R G formation which is subdivided into two sand bodies. The two sand bodies were perforated and tested; they showed production with initial rate 370 BOPD by N2 lifting with traces of water and gas production. Production started from well A-1 (Appendix 1, Fig 16) with ESP yielding a production rate of 350 BOPD and 35 % water cut, then the well production rapidly declined to 130 BOPD and water rate started to decline. The ESP then failed several times and has been replaced with sucker rod pump. The last static fluid level measurement showed average static reservoir pressure of 1,247 psig at a reference depth of −5,300 ft TVDSS.

No PVT samples were taken from this field, so calculations were done using both correlations and PVT samples taken from the nearby producing fields. An estimation of the oil in place was done using both material balance and volumetric, showing reserves ranging from 10 to 15 mm STB. Only compression velocity was available from the sonic log and the shear velocity was calculated using the following correlations for both the sand and shale layers. A static model was built using the available data with approximate cell dimensions of 50 × 50 m and a dynamic model was successively prepared to be used to test the different fracture scenarios (Fig. 2).

Fig. 2 Overview of the static model

Fig. 2 Overview of the static model.

The basic simulation model is a three-phase flow single-well model; however, only the oil and gas phase are mobile. The water is not moving as noticed from the well history data, and the amount of produced water at early production period was coming from the water that had been used as a completion fluid, so there is essentially no water production from the well.

Sensitivity runs are completed to determine if grid block size has any impact on the production rates and consequently the fracture length. Static and dynamic models were constructed for this case study; the dynamic model has grid size of 50 × 50 m, consisting of 72,240 cells (80 × 43 × 21). The model is divided into vertical layers. These layers contain productive zones that are fractured and non-productive shale that separate the productive zones. Porosity and permeability for the producing layer is based on the typical values observed in Table 1.

Porosity and permeability for the producing layer is based on the typical values observed in Table 1.

Table 1 Porosity and permeability of the studied field.

39 LGRs were used to model the hydraulic fracture scenarios in three wells; well A-1 (Appendix 1, Fig 16, current well) and the two proposed wells A-2 (Appendix 1, Fig. 17) and A-3 (Appendix 1, Fig. 18). The LGRs were amalgamated in three groups each at the location of each well and were designed so as to match the actual geometry predicted from the rock model as previously mentioned. The original grids were refined depends on the selected length of the fracture to be used in the simulation model.

Six grids with size ~145 ft were refined in all directions around the well as Fig. 3 shows; four of them have been refined regularly and the other cells were refined only in one direction depending on the fracture length. For example, the fracture length used in the current model is 600 ft, so represent this length in grid cells with very small fracture aperture; the grid cells should be refined according to that length. Four grid cells of 145 ft+ the rest of the fracture length from the fifth cell around the wellbore, it gives the desired fracture length with its aperture.

Fig. 3 LGRs around the wellbore

Fig. 3 LGRs around the wellbore.

By using permeability multiplayer keyword only for the fracture, the hydraulic conductivity value of the created fracture increased as shown in Fig. 3. The red line represents the simulated fracture. Coupling of these LGRs was done by amalgam technique. Simulation of hydraulic fracture using this method resulted in decreasing the convergence and stability issue during the running process; also it gave accurate results.

Fig. 4 LGRs around the wellbore

Fig. 4 LGRs around the wellbore.

Modeling hydraulic fractures in finite difference simulators using amalgam local grid refinement

Hydraulic fracturing allows numerous, otherwise unproductive, low-permeability hydrocarbon formations to be produced. The interactions between the fractures and the heterogeneous reservoir rock, however, are quite complex, which makes it quite difficult to model production from hydraulically fractured systems. Various techniques have been applied in the simulation of hydraulically fractured wells using finite difference simulators; most of these techniques are limited by the grid dimensions and computing time and hardware restrictions. This paper explains how the amalgam LGR is designed and compares between standard LGRs and the proposed design and a case study is presented from an anonymous field in Egypt to illustrate how to use this technique to model the hydraulically fractured well. The simulation model is matched to available production data by changing fracture lengths. Then the model is used to predict future response from the wells.
Emanuel Martin
Emanuel Martin is a Petroleum Engineer graduate from the Faculty of Engineering and a musician educate in the Arts Faculty at National University of Cuyo.
http://www.allaboutshale.com

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