Using Distributed Fiber-Optic Sensing Systems to Estimate Inflow and Reservoir Properties

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Using Distributed Fiber-Optic Sensing

Systems to Estimate Inflow and Reservoir

Properties

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Using Distributed Fiber-Optic Sensing Systems

to Estimate Inflow and Reservoir Properties

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. ir. K.C.A.M. Luyben, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op 4 september 2014 om 12:30 uur door Farzad FARSHBAF ZINATI

Master of Science with Honors in Applied Earth Sciences geboren te Tabriz, Iran.

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Dit proefschrift is goedgekeurd door de promotoren:

Prof. dr. ir. J.D. Jansen Prof. dr. S.M. Luthi

Samenstelling promotiecommissie:

Rector Magnificus, voorzitter

Prof. dr. ir. J.D. Jansen, Technische Universiteit Delft, promotor Prof. dr. S.M. Luthi, Technische Universiteit Delft, promotor

Prof. dr. ing. B.A. Foss, Norges Teknisk-Naturvitenskapelige Universitet (NTNU), Norway

Prof. dr. ir. R.A.W.M. Henkes, Technische Universiteit Delft

Prof. ir. C.P.J.W. van Kruijsdijk, Technische Universiteit Delft

Prof. dr. W.R. Rossen, Technische Universiteit Delft

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Chapter 1: Introduction

In this chapter a common definition of ‘smart wells’ concept, as used in hydrocarbon production, is given. Conventional and newly-emerging well monitoring and reservoir surveillance techniques are briefly reviewed and the advantages of recent fiber-optic sensing systems are addressed. The significance of filling the gap between advanced monitoring and control technology by means of robust interpretation methods is discussed.

1.1 Introduction

Smart well technology is one of the most significant breakthroughs in hydrocarbon production. The very first introduction of smart well technology dates back to middle of the 90’s. Although, due to reliability and cost issues the initial introduction of this technology was slow, its use grew dramatically in recent years and smart wells turned into a relatively common development option (Gao et. al., 2007). Today smart well technology is used in upstream to increase reserves, accelerate production and enhance economics of the development operations. More frequent use of smart well technology in recent years in field development plans is justified by improved incremental recovery and reduced lifecycle costs.

Terminologies such as ‘intelligent wells’ and ‘smart wells’ are used interchangeably to describe the industry initiatives that aim at increasing well and reservoir performance in terms of recovery and financial measures using a measurement and control approach (Jansen et. al., 2009). Smart or intelligent well is defined as a well that has measurement and control capabilities in the region of its completions. This definition is further improved to indicate that a well could be called intelligent or smart if, and only if, in addition to having measurement and control functionality, it adds value to the project during its life cycle (Alves,

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With the given definition, a well should have certain characteristics to be considered “smart” or “intelligent”. The first characteristic is measurement capability. A smart well is equipped with down-hole measurement systems to monitor flow parameters and gather production and injection information. The monitoring systems include all the instrumentations and sensors that measure physical and chemical properties of flow such as pressure, temperature, phase composition, pH, resistivity and etc.

The second characteristic of an intelligent or smart well is control capability. A smart well should be equipped with tools and instruments that are capable of controlling well inflow and outflow. Historically wells completed with cemented casing, packers and selective perforation have had some degree of controllability in terms of well inflow from various zones and decreasing undesirable fluid flow. However within the context of ‘smart wells’, control capability rather refers to deployment of more recent advanced technologies where down-hole valves are used to selectively adjust the production and injection distribution along the well in multiple zones to stop or delay undesired fluids from entering the well (Ramakrishnan, 2007).

The third element of a smart well is a systematic approach to link measurements to control strategies. In this process, the measured data are analyzed and interpreted in a time-efficient manner in order to make a ‘smart’ decision that satisfies the ultimate goal of the smart well technology i.e. increased or accelerated production and/or improved economy of the well. The decision making phase mainly consists of simulation and optimization routines. In this phase simulation models are used to represent the physics of the subsurface, wells and facilities and optimization algorithms are then applied to maximize production and NPV.

Measurement and control capabilities of smart wells in combination with other data acquisition technologies on the surface and far from the wells, such as 4D (time-lapse) seismic, dynamic resistivity and surface measurements from conventional wells, provide the tools to improve reservoir management and optimize hydrocarbon recovery on field-scale.

In-well measurements and subsurface monitoring are essential elements of real-time reservoir management. Given the extent of uncertainty that is present in subsurface properties, without down-hole monitoring, the task of reservoir management and hydrocarbon production would, in many cases, be severely limited.

Although the majority of various down-hole monitoring systems in smart wells measure almost the same physical quantities such as pressure, temperature, flow rate, fluid composition, pH, resistivity, acoustic impedance, etc., they differ substantially in hardware type, equipment, installation requirements, costs, working conditions, number and quality of measurements, performance uncertainty, durability, reliability and the benefit they provide to a smart well per installation (Silva Junior et. al. 2012).

Current commercially available permanent down-hole sensors are typically categorized by their technology and the number of measurement points. Single-point sensors measure the physical quantities at one location, while distributed sensors provide readings of physical quantities along an interval of interest e.g. a reservoir section. Based on the technology used in permanent down-hole sensors, they are classified into two categories: traditional electrical down-hole gauges and newly emerging sensors utilizing fiber-optic technology.

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Permanent down-hole pressure and temperature gauges have been one of the industry’s standards, particularly in offshore wells where wire-line surveys are expensive. For more than 30 years, permanent down-hole gauges (PDG) have been installed in hundreds of oil and gas wells. These systems usually consist of an electronic gauge that measures temperate or pressure and transmits the measured data through an electrical metal-sheathed cable that runs along the tubing (van Gisbergen and Vandeweijer, 2011).

Quartz Resonator is the dominant technology in all single-point sensors (Silva Junior et. al. 2012). Studies indicate that the first installations of PDG’s for measuring pressure and temperature exhibited a low reliability (van Gisbergen and Vandeweijer, 2011). Later improvements such as technical quality, care during installation and management of the interface were applied to the process of deploying PDGs in wells. Although these improvements extended the life time of the systems, the failure rate of PDGs remained high under high temperature conditions. The hole conditions and specifically high down-hole temperatures, present a fundamental challenge for the electronic system of PDGs. Increasing the number of sensors makes the installation even more complex and time consuming and consequently less reliable in high-temperature and high-pressure conditions (Silva Junior et. al. 2012).

Recent developments in fiber-optic sensing have resulted in more reliable alternatives to conventional electronic systems (Gao et. al., 2007). The main advantage that has pushed the fiber-optic sensing systems to gain more acceptance over conventional PDGs is their relatively high reliability.

A typical optical fiber consists of a central core surrounded by an optical cladding of a slightly lower relative refractive index that traps the light in the core through ‘total internal reflection’. In general, both parts are made of glass (silica) modified by the addition of other materials to tailor the refractive index profile and the dispersion of the structure. The cladding is coated with a ‘buffer’ to the fiber to protect it from moisture and physical damage. The coating armors the surface against scratches which can lead to strength degradation, and protects it from micro-bending which will cause light to be lost. Moreover, the coating protects the fiber against moisture which will accelerate the aging and deterioration of fiber strength. For down-hole applications the fiber needs extra protection against high temperatures, chemical attacks and mechanical abrasion and crushing. In increasing order of effectiveness, the coating has been made from ultraviolet curable acrylates, silicone rubber, fluorated polymers, polyiamides and metal. In general a metallic tube further protects the coated fiber.

There are several technologies that can be used in fiber-optic monitoring systems such as fiber Bragg grating (FBG), Brillouin scattering and distributed polarimetry.

Bragg grating technology was initially developed in the communication industry where it has been used in fiber lasers, amplifiers, filters and etc. Application of FBG’s in down-hole monitoring systems allowed the sensors to be incorporated directly to a single fiber optic cable and thus simplifying the sensor to a single monolithic glass structure (Zisk, 2005).

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core such that the Bragg gratings reflect a particular wavelength of light and transmit all the others. The reflected wavelength depends on the optical period and effective refractive index of the grating. A broad band light with a bell-shaped light spectrum source is employed to illuminate the cable and sensors. As the broadband light continues to travel, a portion of it passes through the filter-like acting gratings whereas a portion is reflected. In Bragg grating sensors the local strain, pressure and temperature causes a change in the reflected wavelength (Drakeley et. al., 2006).

FBG-based sensors are used in single-point and quasi-distributed pressure and temperature measurements, flow meters and down-hole seismic sensors. However the application of the BGs is different in these systems. The pressure and temperature sensors rely on a change in the length of BGs and its corresponding shift in reflected wavelength to determine pressure and temperature. The flow-meters and seismic sensors employ a method called interferometric sensing where the BGs act like mirrors and the length of BGs act as a sensor. The interferometric sensing uses a laser (rather than a broad band light source as in pressure and temperature sensors) pulsed in a series of timed blocks. By interpreting the reflected signal the dynamic pressure of the fluid can be determined from the change in length of the sensor (Drakeley et. al., 2006).

A distributed fiber sensor based on Brillouin scattering technology exploits the interaction of light with acoustic phonons propagating in the fiber core. The Brillouin scattered light has a frequency shift proportional to the local velocity of the acoustic phonons (also called acoustic waves), which depends on the local density and tension of the glass and thus on the material temperature and strain. Brillouin scattering is used in distributed temperature sensors and has given the down-hole monitoring systems a spatial component such that they provide a continuous wellbore temperature in real-time.

Regardless of the technology used, fiber-optic down-hole sensing systems offer some benefits that are not readily available in conventional electrical gauges. The small physical size of fiber-optic sensors allows simple integration into the downhole completion system and easy embedding into composite structures. Silica with high temperature fiber coating enables the development of sensors for high temperature applications with operating temperature in excess of 1000 degrees Celsius. Having a simple sensing element at the measurement points and easily accessible locations for servicing, the reliability of the fiber-optic sensing systems is boosted. Furthermore these types of sensors are immune to interference from local radio or electrical transmission sources and the passive nature of sensing reduces fire hazards. Most important of all, fiber optic sensing systems replace the multiple electrical gauges and their associate wirings by offering multiple sensing points and measurement types on a single fiber. The reduced system complexity can be an important factor in down-hole completion applications (Drakeley et. al., 2006).

With fiber optics installed inside a well, operators are moving away not only from conventional permanent downhole gauges but also form traditionally run production logs, tracer surveys, or geophones for vertical seismic profiles (VSPs). In situations where conventional interventions are problematic and the cost of deferred production is

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unacceptably high, permanent fiber optic sensors are particularly more attractive and offer safer operations and continuous measurements without costly intervention.

Recent experiences from the deployment of advanced fiber-optic monitoring systems in multi-lateral wells have demonstrated the value and high quality of the data acquired by these types of sensing systems (Dria, 2012). These data, combined with existing inflow control technologies, can be used to identify and control the inflow of the zones that produce undesired fluids. However, data from permanently installed sensors frequently do not indicate directly or intuitively the relative production contribution from each producing zone and are often interpreted in a qualitative manner. Therefore the key to a successful link between measurements and decisions (control strategy) is efficient, fast and robust interpretation methods to provide quantitative answers to the inflow determination. The time-efficiency of the interpretation leads to an early diagnosis of problems, such as water or gas break-through, and accuracy and robustness of the interpreted results forms the basis for an efficient inflow control strategy.

In this work, the possibility of identifying and quantifying the reservoir inflow from distributed measurements is addressed. Semi-analytical well and reservoir models and gradient-based inversion algorithms with adjoint formulation are employed to develop robust and computati0nally efficient techniques to interpret measurements acquired by fibre-optic sensing systems. The proposed interpretation methodologies are quantitative and target the estimation of pivotal parameters from which the inflow of reservoir fluids into the wellbore can be determined. The computation times for the proposed inversion methods are low and thus the presented techniques are of potential importance for applications in real-time control of smart wells, e.g. to control coning behaviour using distributed measurements along horizontal wells.

References

Alves, I. 2011, Intelligent Fields Technology. JPT 63 (9): 76-85.

Bao, X. and Chen, L. 2011. Sensors 11 (4): 4152-4187. DOI: 10.3390/s110404152

Drakeley, B.K., Johansen, E.S., Zisk, E.J. and Bostick III, F.X. 2006. In-well Optical Sensing - State Of The Art Applications And Future Direction For Increasing Value in Production Optimization Systems. Presented at Intelligent Energy Conference and Exhibition, Amsterdam, The Netherlands. 11-13 April 2006. DOI: 10.2118/99696-MS

Dria, D. 2012. Down-Hole Flow Monitoring with Distributed Sensing Systems. Offshore World

April-May 2012 issue: 14-15. (http://www.chemtech-online.com/O&G/Dennis_april_may12.html)

Gao, C., Rajeswaran, T. and Nakagawa, E. 2007. A Literature Review on Smart Well Technology. Paper SPE 106011 presented at Production and Operations Symposium, Oklahoma City, Oklahoma. 31 March-3 April 2007. DOI: 10.2118/106011-MS

Jansen J.D., Douma, S.D., Brouwer D.R., Van den Hof, P.M.J., Bosgra, O.H. and Heemink, A.W. 2009. Closed Loop Reservoir Management. Paper SPE 119098 presented at SPE Reservoir Simulation Symposium, The Woodlands, Texas. 2-4 February. DOI: 10.2118/119098-MS

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Silva Junior, M.F., Muradov K.M. and Davies D.R. 2012. Review, Analysis and Comparison of Intelligent Well Monitoring Systems. Paper 150195 presented at SPE Intelligent Energy International. Utrecht, The Netherlands. 27-29 March. DOI: 10.2118/150195-MS

Van Gisbergen, S.J.C.H.M. and Vandeweijer, A.A.H. 2001. Reliability Analysis of Permanent Downhole Monitoring Systems. SPE Drilling & Completion 16 (1): 60-63. DOI: 10.2118/57057-PA

Zisk, E.J. 2005. Optical In-Well Permanent Monitoring-Initial Promise Now A Reality? Presented at Offshore Technology Conference, Houston, Texas, 2-5 May 2005. DOI: 10.4043/17529-MS

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Chapter 2: Dynamics of Wellbores:

Transient vs. Steady-State Models

In this chapter a numerical method utilizing a flux splitting scheme and standard first-order-accurate upstream discretisation is presented. The time span in which dynamic phenomena in the wellbore occur is investigated. Moreover the most important parameters influencing the pressure drop over a long horizontal well are investigated.

2.1 Introduction

The introduction of horizontal well technology during the 1980s has been a milestone in petroleum production history and since then, the application of horizontal drilling technology to the discovery and productive development of oil reserves has become a frequent worldwide practice. Horizontal wells have proven to be a successful tool in increasing productivity and cost-effectiveness of field development plans (Thakur, 1995) and thus, various aspects of hydrocarbon production by means of horizontal wells have been researched for years.

Accurate modeling of the behavior of a horizontal well in reservoir is one of the key issues in successful prediction of the performance and operation of a horizontal well. Due to this, in the past decades a significant number of studies have been devoted to model fluid flow in horizontal well-reservoir systems. Numerous simulation tools have been developed ranging from conceptual and comprehensive tools, using simple analytical and semi-analytical models, to more complex fully coupled reservoir-wellbore flow simulators, which incorporate the transient and multiphase phenomena in both well and reservoir. However, since oil production by horizontal wells involves a wide range of dynamic phenomena in the reservoir and wellbore, the modeling process is not a trivial task and each of the developed models suits the application for which it has been implemented. Moreover, dynamic phenomena such as wellbore storage, slugging, loading, clean-up, coning, and reservoir transients occur through a wide range of spatial and temporal scales. Consequently, developing and using a complex model that takes into account all above mentioned phenomena needs sophisticated simulation tools that require powerful computational recourses.

Traditionally horizontal wells were treated as infinite conductivity wellbores, and the use of models that ignored the pressure drop in horizontal wells was justified by assuming negligible pressure drops along the well compared to the pressure drop in the reservoir. However, emerging advanced technologies have enabled drilling long horizontal wells that deliver high production rates with small drawdown values. Currently, increased production rate in the presence of small drawdown has become one of the main reasons that horizontal wells are successful. Therefore most research in horizontal well modeling has been diverted in a

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direction that couples wellbore and reservoir flow, and that includes the effect of finite conductivity (Ozkan et. al., 1995;Aziz et. al., 1999; Anklam and Wiggins, 2005).

Dikken’s model (Dikken, 1990) was the first work that coupled a steady-state homogenous reservoir model described by a constant along-hole Specific Productivity Index (SPI) with a horizontal well model utilizing wellbore hydraulics based on Moody’s friction factor. In that work equations of the coupled model were solved analytically for an infinite well and numerically for a finite well. Although the model was simplified because of assumptions such as steady-state flow and constant along-hole SPI (homogenous reservoir), it provided a useful conceptual tool for analyzing the performance of the horizontal wells. Landman (1994) extended Dikken’s work to selectively perforated horizontal wells. He solved a system of algebraic equations, each of which corresponded to the analytical integration of the pressure drop equation for a certain segment of the well. A major improvement of his model was varying the SPI along the well. In his work a simple formula for the optimal perforation density was given. He discussed the problem of modifying Dikken’s inflow model to include full Darcy flow in the reservoir and partial penetration. Moreover Landman presented analytical solutions for the finite-length case in terms of hypergeometric functions. Further analytical solutions were presented by Halvorsen (1994) and Jansen (2003) who used Weierstrass elliptic functions and Jacobian elliptic functions respectively.

Using an approach similar to Dikken, Novy (1995) provided guidelines that indicated whether friction loss affected the production of a horizontal well. He suggested that a particular well-reservoir system can have a threshold well-length above which wellbore friction reduces productivity by 10%. He recommended simple criteria based on well-length, production rate and hole diameter to determine if friction can be neglected in a certain system. Proposing the ratio of wellbore pressure drop to drawdown at the producing end as a key criterion, he hinted that if the ratio exceeds 10% to 15% wellbore friction can reduce the productivity by more than 10%. Moreover he reported that oil wells producing more than 1500 STB/d and 2 MMscf/d are susceptible to this pressure loss.

Ozkan et. al. (1995) presented a comprehensive model that couples wellbore and reservoir hydraulics for the single-phase flow of a slightly compressible liquid under isothermal conditions. Unlike Dikken’s model, no assumption was made regarding the manner in which fluid enters the wellbore and thus the SPIs were output of the model. Moreover, although they neglected the dynamics of the flow in wellbore, their model was capable of generating a short-term and long-short-term along-well pressure profile with respect to time by incorporating Green’s functions.

Penmatcha et. al. (1999) also modeled the pressure drop in horizontal wells in a similar fashion. However they investigated a wide range of parameters that affected the friction loss and consequently the productivity of horizontal wells. In an interesting part of this work they compared the results of the developed 1D explicit analytical and semi-analytical model to the results of a fully coupled 3D implicit rigorous model and concluded that explicit coupling does a reasonable job and is useful for quick calculations.

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The amount of literature on coupled modeling of horizontal well and reservoir is extensive and, in addition to the studies presented earlier in this section, there are numerous works that investigated the coupling by different approaches. For example, Ouyang et. al. (1998) and Yuan et. al. (1998) have proposed comprehensive methods to explicitly include the effect of radial inflow on the pressure drop along the wellbore. Folefac (1991) listed the most important parameters that influence the pressure drop in a horizontal well. Ankalm and Wiggins (2005) also reviewed wellbore pressure equations and showed that in practical situations the assumption of infinite conductivity is not valid.

Along with the majority of the previously discussed works, which mainly dealt with estimating pressure drop of horizontal wells and were based on existing physical models, there are several studies on developing correlations for the pressure drop by linking simulations and experimental results. Some of these studies include: the work of Asheim et. al. (1992), who developed a flow resistance for horizontal well flow using a modified friction factor; the study of Su and Gudmundsson (1994) in which they carried out experiments to predict pressure drop in horizontal wells; and the work of Yalniz and Ozkan (1998) who investigated the effect of fluid entry through perforations on flow in a horizontal wellbore and correlated this effect in the form of an apparent friction factor.

While most of the studies focused on steady-state or pseudo steady-state models to calculate the pressure drop and ultimately predict the performance of the horizontal wells, a few works in the literature exist that utilize coupled well-reservoir simulator to compute transient effects. Vicente et. al. (2002) presented a fully implicit, 3D simulator with local refinement around the wellbore to solve reservoir and horizontal well flow equations simultaneously for single-phase liquid and gas cases. The model consisted of conservation of mass and Darcy's law in the reservoir, and mass and momentum conservation in the wellbore for isothermal conditions. In another study, Nennie et. al. used a commercial multiphase well simulation tool (OLGA) and a numerical reservoir simulator (MoReS) and coupled both simulators using an explicit scheme. Despite drawbacks of their model, they showed the necessity of a coupled simulator to capture several physical phenomena in horizontal wells.

Within the context of the present work, distributed pressure sensors (DPS) record two transient flow behaviors that may occur in coupled well-reservoir systems, i.e. transients caused by wellbore and reservoir dynamics, respectively. While wellbore dynamics have little relevance to reservoir properties, reservoir dynamics is the key factor to characterize and estimate reservoir parameters. Thus in this chapter first the most common transient single-phase isothermal flow modeling techniques are briefly reviewed and then a numerical model is developed to adapt the methodology to simulate transients in a wellbore-reservoir system. Using the semi-analytical tool, the time scale during which wellbore dynamics occur is addressed. Next, the most important parameters influencing the pressure drop along a horizontal well are investigated.

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2.2 Modeling transient flow

The numerical solution of partial differential equations representing mass and momentum conservation laws can be obtained by finite difference and finite volume methods. Finite element methods also are widely applied to solve the equations numerically in many computational fluid dynamic areas. However since most of the commercial numerical reservoir simulators in reservoir engineering fields are based on the finite difference method (FDM), application of the finite element method (FEM) in solving the flow equations in the wellbore has not gained much popularity.

In the past decades numerous numerical methods have been developed to solve transient mass and momentum conservation equations especially for compressible fluid flow in one dimension. Due to the enormous number of developed formulizations and the wide range of numerical solvers, proposing a solid classification of the schemes has become a complex task. Furthermore the majority of the recent numerical methods is based on a combination of different previously developed schemes. However several factors play a major role in choosing a certain solution scheme for the partial differential equations expressing mass and momentum conservation laws.

Treatment of time-dependent terms is an important factor. Regarding the time discretisation, explicit and implicit schemes have been used. Although the use of implicit methods usually offers the benefit of numerical stability even in the presence of large time steps, their implementation is complicated. Moreover larger computational recourses are needed to iteratively solve the algebraic system of equations which are an inevitable element of using implicit methods. In contrast, explicit schemes are easy to implement and pose no need to solve a system of equations. However, they impose a stability limit on the time steps.

Discretisation of the terms with spatial derivatives and choosing between one-sided (upwind and rarely downwind) and central schemes is another basis to distinguish numerical schemes, and will be discussed further. Likewise the order of accuracy of the numerical solution should also be considered in the selection criteria. Usually higher-order numerical schemes provide more accurate solutions and less numerical diffusion. However the use of flux limiting functions is often necessary to suppress the effect of numerical oscillations that represent no physical information.

Large classes of solution methods in computational fluid dynamics are based on a work by Godunov (1961). The original idea behind Godunov’s work was piece-wise reconstruction of conservative variables over the grid cells, after which the exact solution of the local Riemann problem was obtained at grid interfaces. In this scheme one Riemann problem per grid cell needs to be solved iteratively. This process is computationally expensive and therefore one might aim at developing approximate Riemann solvers that are computationally cheaper to solve. Roe’s approximate Riemann solver is an example of an alternative approach and is based on linearization to obtain inter-cell numerical fluxes (Roe, 1981). The Van Leer’s flux splitting technique is another popular method among a wide range of techniques (Van Leer, 1973, 1978). In his technique the flux vector is analyzed to determine its eigenvectors, which are the wave speeds of the equation. The flux is then split into the respective contributions from each

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wave speed, with a left flux component coming from the positive wave speeds and a right flux component coming from the negative wave speeds. Furthermore, the AUSM (Advection Upstream Splitting Method) scheme was developed as a numerical flux function for solving a general system of conservation equations, based on the upwind concept. AUSM was motivated to provide an alternative approach to other upwind methods, such as the Godunov method, flux difference splitting methods by Roe (1981), and Osher and Solomon (1982), flux vector splitting methods by Van Leer (1973), and Steger and Warming (1981).

In general the choice of a numerical scheme to solve mass and momentum conservation equations is problem-dependent. In flows where the flow variables are smooth and continuous, the central difference formulation provide a satisfactory results. However, when discontinuities in the flow variables are present, e.g. shock waves, using central difference methods leads to significant oscillatory solutions in the vicinity of shock waves. Numerical methods based on upstream (also called upwind) methods overcome the shortcoming of the central schemes and present acceptable solutions. On the other hand the main limitation of the upstream methods is numerical diffusion which consequently causes sharp fronts to smear out. To overcome the inaccuracies raised by using first-order-accurate upwind schemes, one can aim at extending the algorithms to higher orders.These high-resolution schemes are used when high accuracy is required in the presence of shocks or discontinuities. The first obvious disadvantage of higher-resolution schemes seems to be the numerical oscillations induced by the higher-order spatial discretisation. Therefore flux limiters are often used with these schemes to limit the numerical ‘wiggles’ in the vicinity of shocks.

In this chapter we present a numerical scheme to solve mass and momentum conservation equations that describe the transient dynamic flow in the wellbore. The set of governing equations forms a system of hyperbolic equations which is solved by a numerical method based on explicit time discretisation and a first-order-accurate upwind scheme using a flux splitting method.

2.3 Wellbore Model

In this section one of several possible alternative formulations is presented to describe transient slightly compressible flow in horizontal wells. The formulation is adapted from standard representation of conservation laws in classic CFD problems.

Neglecting the gravitational term, the mass and momentum balances in the wellbore domain can be respectively written as:

 

_

_

 _       _ _ _        _ _ _{ }     inflow 2 0, 0, u m t s u u p t s , (1)

where minflowis the mass inflow rate from the reservoir into the well per unit volume of the

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friction factor introduced by Moody (1944) is employed which depends on the relative pipe roughness and the dimensionless Reynolds number. The details of the friction factor calculations are presented in the next chapter. Furthermore, since only “slightly-compressible fluid” with constant isothermal compressibility is assumed, the density-pressure relation is given by the following equation of state:





exp

ref cf p pref

  _  _. (2)

Taking the derivative of the equation above with respect to pressure gives:  

 _  cf

p , (3)

where c is the isothermal compressibility of the fluid. f

We define: m         x , (4)

 

_{ }

_       _ _ _   m m p 2 f x , (5)

 

  _    m_inflow q x , (6) with mu.

Note that xrepresents a state vector rather than a distance. The elements of the vector x are usually referred to as “flow parameters”, and f is often called “flux vector”. In this formulation

q represents the source term.

Using the variable definitions (4-6) the set of equations (1) can be written as:

 

 _  _ _

t s 0

x

f x q x , (7)

where the first and second terms on the left-hand side are called transient and convective terms respectively. Equation (7) can also be written in as

`  

 

 

 

 t s 0 x x A x q x , (8) with

 

  f A x x. (9)

The matrix A is called the Jacobian of the flux vectorf x

 

. For the system of PDE’s in equation (1) the Jacobian matrix is given by

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 

      _ _  _  _    _ _    m m p 2 2 0 1 2 f x A x x . (10)

Equation (8) is a hyperbolic equation. In the solution of hyperbolic equations it is important to determine the direction and velocities of the propagation of information in the flow domain. This will allow adopting a numerical scheme that is consistent with physics of the flow. Classically to determine the direction and velocity of information propagation, the eigenvalues of the Jacobian matrix of the flux vector are analyzed. The eigenvalues of the matrix defined in equation (10) are given by

0 i 

 

A Ι . (11)

Here Ι is identity matrix and _i represent the eigenvalues of the matrix A . With matrix A given as equation (10) the two roots of the equation (11) are given by:

       m p 1 , (12) and        m p 2 . (13)

Replacing the derivative of pressure with respect to density with the expression from equation (3) leads to      f m c 1,2 1 . (14)

The second term in the right-hand side of equations (12-14) corresponds to the speed of sound in the fluid (liquid in this case). In classical mechanics, the speed of sound, or acoustic velocity, in a medium with arbitrary equation of state is related to the change in pressure and density of the substance and can be expressed as

    s p u2 _, ₍₁₅₎

where

u

sis the speed of sound. Therefore equation (14) can be rewritten as:

1,2  u us. (16)

The eigenvalues



1and



2indicate that the information is propagated in both directions of

the one-dimensional flow domain at a relative speed of sound. For the isothermal single-phase fluid flow described in this chapter, the two eigenvalues give the shape of the characteristic lines in the (x,t)plane. At a given point in the (x,t) plane there are two characteristic curves

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1 1 1 s dt dx u u , (17) and 2 1 1 s dt dx u u . (18)

The characteristic curves (a.k.a. “characteristics”) play a significant role in the development of numerical schemes to solve the system of equations, since the information concerning the flow field travel along the characteristics. Given that eigenvalues of the Jacobian matrix give the shape of the characteristics and their values determine the speed at which the information propagates, the solution procedure should be consistent with the velocity and the direction at which information propagates through the flow field.

2.4 Numerical Scheme

In this section a first-order-accurate upstream method with explicit time discretisation for solving the transient single-phase flow in the wellbore is presented. As described above the extension of upstream approaches for solving the scalar and linear convection equation, to a nonlinear partial differential equation system involving more variables becomes complicated. However, as discussed in the introduction there are two methods that provide a systematic approach to implement the upwind method for nonlinear equations: Flux splitting and flux differencing methods. In this section a method based on flux splitting scheme is presented.

With the flux Jacobian matrix given in equation (10) and eigenvalues given in equations (12) and (13), one can determine the eigenvectors of the Jacobean matrix A. The eigenvectors corresponding to the eigenvalues of the matrix A are

         _{  }  _    _ _ _ m p 1 1 1 1 v , (19) corresponding to



1 and          _{  }  _    _ _ _ m p 2 2 1 1 v , (20)

corresponding to



2. Therefore a matrix R is defined by:





_{ }   _{ } _   1 2 1 2 1 1 R v v (21)

Regarding the fact that matrix A has two distinct real eigenvalues and two linearly independent eigenvectors, matrix A can be factorized as follow:

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1





Α RΛR (22)

where Λ is a diagonal matrix in which the eigenvalues of the matrix A form the diagonal elements. 1 2 0 0          Λ (23)

Next, matrix Λis decomposed in a similar approach to when a standard upstream method is applied to a scalar differential equation. The decomposition leads to the following definitions of the matrices:





1 1 2 2 0 1 1 0 2 2      _   _ _  _    Λ Λ Λ . (24)

With this expression for

Λ

and

Λ

, the matrix A can be split into two matrices, A+and

-A

such that     A A A , (25) where 1 _   A RΛ R , (26) and 1 _   A RΛ R . (27)

It is significant to note that with this formulation each of the matrices in equations (26) and (27) corresponds to a flow in the direction of the information propagation, characterized by two eigenvalues of matrix A .

Having determined the direction and speed of information propagation, next a grid system is defined. The spatial flow domain is divided to N uniform grid blocks with the same lengthl. Dependent variables of partial differential equation 1 i.e. density, velocity, pressure and mass flow rate, are defined at the center of each grid block. The mass inflow into each grid block is also calculated using the value of cell-centered variables. Employing the described grid system, the finite difference representation of the derivative of the variable vector in a grid block with index i can be written as:

1 i i i x l     _{ } _  _   x x x . (28)

With this definition of finite difference scheme and applying the upstream method described in this section the space-discretised version of equation 8 reads:

 

i i 1

 

i 1 i

 

₀ i i t l l        __ _ __ _ _ _        x x x x x A x A x q x , (29)

where the second and third terms on the left-hand side of the equation represent two directions of the flow to grid block

i

and the combination of these two expressions imposes

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the condition that the information to grid block i comes from upstream. Replacing the expressions for A and  A with those from equations (26) and (27) leads to 

 

1 i i 1 1 i 1 i ₀ t l l          _ _ _ _ _ _ _ _      x x x x x RΛ R RΛ R q x . (30)

Next, the expressions for the Λ and  Λ are substituted from the equations (26) and (27): 





1 1





1 1

 

1 1 0 2 2 i i i i t l l        _ _ _ _  _  _  _  _      x x x x x R Λ Λ R R Λ Λ R q x . (31)

Finally, rearranging gives



1



1 1



1



1 2 1

 

₀ 2 i i i i i t l l            _  _  _ _            x x x x x x RΛR R Λ R q x . (32)

This equation is the final version of the space-discretisation of equation (8).

Time Discretisation: For the time-dependent term in equation (32), explicit time

discretisation is employed. Thus, the unknown variables at time level k1are evaluated from the value of the variables at time levelk. Discretising the equation (32) in time, the final discretised equation to be used for the numerical solution is as follows:









 

1 1 1 1 1 1 2 1 ₀ 2 k k k k k k k k i i i i i i i i t l l                  _ _ _ _   _  _ _  _ x x x x x x x RΛR R Λ R q x . (33)

Therefore, assuming the unknown state

x

is known for the time step k, the new state of the model can be calculated as





 



  

                 k k k k k k k k i i i i i i i i 1 1 1 1 1 1 2 1 0 x x RΛR x x R Λ R x x x q x , (34) where 2 t l     . (35)

Using explicit time discretisation imposes a limitation on choosing the time step size. If the time steps are not chosen sufficiently small, the solution of the numerical scheme will be unstable. The Courant-Friedrichs-Lewy condition (CFL condition) is a necessary condition for convergence of the numerical solution while solving certain partial differential equations (usually hyperbolic PDEs). For the one-dimensional case, the CFL has the following form: CFL t C l  _  , (36)

where CCFL is a dimensional number that depends on the particular equation to be solved and its respective eigenvalues. In general it is just a necessary condition rather than a sufficient condition and in the numerical examples in this chapter the time steps are chosen sufficiently small based on an iterative approach.

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2.5 Boundary Condition

To complete the governing equations for transient flow in the wellbore in addition to the discretised (in time and space) equation (34), a set of boundary conditions is needed. As will be explained in the description of the numerical example, it is assumed that the fluid is produced at a constant bottomhole pressure. This defines the boundary condition at the toe of the wellbore. In order to treat the boundary condition properly, an imaginary grid block is introduced at the left boundary of the flow domain. Using linear extrapolation for the variable values in the imaginary block, we can write:



_N_₁ _N



_wf 1

2 , (37)

where



wf is the density corresponding to bottomhole flowing pressure and can be obtained

by replacing the known bottomhole pressure, pwf, in equation (2). Moreover linear

extrapolation is used for the other variable

m

. Consequently the density and mass flux of the imaginary grid block N1is given by:

_N_₁ 2_wf _N, (38)

   

N N N

m ₁ 2m m ₁. (39)

At the toe of the well it is assumed that there is no flow in the along-well direction. Therefore the mass flow rate at this boundary of the flow domain must be zero. Once again introducing another imaginary grid block and using linear extrapolation for the unknown variables of the cell leads to the following boundary conditions at the toe of the well:

 

m₀ m₁, (40)

0 2 1 2. (41)

2.6 Initial Condition:

In addition to the system of governing equations and boundary conditions, it is necessary to assign proper initial conditions. When the system is in an initial condition where no flow occurs, the pressure in the entire flow domain must be equal to a constant value. Consequently there will be no inflow from the reservoir into the well and the pressure at all the grid cells will be equal to the reservoir pressure:





; 1...

i R

p p i N . (42)

Accordingly the density of each grid block at the initial state can be obtained by inserting the reservoir pressure into the density expression from equation (2). Furthermore since the flow rates of all grid cells at initial condition are equal to zero, no mass transport takes place. Thus, the following equations define the initial conditions:

 

; 1...

i pR i N

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2.7 Reservoir Model

In order to provide the wellbore equation with a physical reservoir model, an analytical reservoir inflow model is assumed. Based on the definition of the specific productivity index (SPI) the rate of the reservoir flow into the well per unit length of the well is related to the pressure of the well segment and reservoir through the following equation:

 

, , ,

e s s R

q l t J l _p p l t _ (45)

where J is the Specific Productivity Index and is only a function of the reservoir properties _s

and might vary along the wellbore.q is the reservoir inflow per unit length. Note that _e

considering equation (1), qe s, is negative when there is a flow from reservoir to the well and is

positive when the fluid is injected from the well into the reservoir. The mass inflow, m_inflow, in equation (1) can be obtained simply from multiplying equation (45) by the value of the density.

2.8 Numerical Examples

In this section several examples with different well-reservoir configurations are presented and the effects of important parameters influencing the transient and steady-state behavior of the flow in a horizontal well are analyzed. In all examples a long horizontal well, with inflow over its entire length is assumed. This configuration can represent situations like open-hole completions where slotted or pre-drilled liners along with wire- wrapped screens are employed and the production occurs over the entire length of the well. Therefore for this configuration the source term (mass inflow rate) in all grid blocks of the simulator is nonzero. As described above the pressure loss due to friction in the wellbore is described through the Darcy–Weisbach equation which employs a Moody friction factor. The differential pressure loss along the borehole depends on the local fluid velocity, the density, the pipe diameter and the dimensionless pipe roughness.

2.8.1 Example 1: Base case

In the first example we consider a 2000 m-long horizontal well with 16.26cm diameter and a relative roughness of 5.867 × 10-4. The density of the fluid at standard condition is 800 kg/m3 and the dynamic fluid viscosity is set to 5×10-4 Pa.s (0.5 cp). For the reservoir flow an SPIof 1×10-10 is used. The important factor, affecting the dynamic behavior of the flow, is the isothermal compressibility, cf, which,in this example, is set to 1×10

-8

Pa-1.

At time t 0 oil starts to be produced and the source terms are introduced into the 

equation. In other words, initially, because there is no production, the source terms in equation (1) equal to zero, and just after time t 0 oil is produced with a rate that is a linear 

function of the well pressure and reservoir pressure and is given by equation (45).

Figure 1 (top) depicts the wellbore pressure profile along the well at several snapshots of time i.e. initial condition and 1, 5, 15 and 50 seconds after the start of production. At the initial state, because of the no-flow initial condition, the pressure profile is constant at 25MPa. As the

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fluid begins to flow from the heel of the well, the pressure at locations close to the heel starts to drop. At the early stages of fluid withdrawal the effect of pressure drop as a result of production is not sensed at the segments of the well close to the toe. Therefore no pressure gradient is created at those points during early times e.g. from 400 to 2000 meters along-well distance in the curve corresponding to 1 second. From a numerical solution view, the propagating waves have in fact not reached some upstream locations yet. As the process continues the effect of production is sensed at more locations toward the toe and the pressures in these segments start to decrease compared to their initial value.

The pressure profile at 5 seconds from 1800 meters to 2000 meters along the well shows a very small deviation from the previous snapshot of 1 second, but the values of the pressures between 400 meters and 1800 meters have changed dramatically with respect to the initial condition. After 15 seconds the pressure drop wave has completely swept the entire flow domain and the pressure profile is significantly different from the initial state. The pressure along the well undergoes only a negligible change from 15 seconds to 50 seconds. This observation confirms that the system has reached steady state condition just after approximately 20 seconds.

Figure 1 (bottom) illustrates the wellbore flow rate versus along-hole distance. The same observation described above for the pressure profile is seen here. The flow rates between 0 to 400 meters along-well distance have increased considerably from the initial value (0 m3/s) just after 1 second. In other words the dynamic waves of the system, arising from the production, have travelled approximately about 400 meters in one second. After 5 seconds fluid flow occurs over 1600 meters of the entire well length and only 1/10 of the reservoir section is not producing yet. Similar to the figure on top the system reaches steady state conditions very quickly and the flow rates along the well stay constant just after 15 seconds.

0 200 400 600 800 1000 1200 1400 1600 1800 2000 2.45 2.47 2.49 2.51 2.53x 10 7 Along-well distance [m] W e ll b o re P re ssur e [ P a ] t=0s t=1s t=5s t=15s t=50s 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 0.02 0.04 0.06 W e ll b o re f lo w r a te [ m 3 /s] t=0s t=1s t=5s t=15s t=50s

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Figure 1 - Wellbore pressure versus along-well distance at five snapshots of time: 0, 1, 5, 15 and 50 seconds after production (top); Flow rate profile versus along-well distance at five snapshots of time: 0, 1, 5, 15 and 50 seconds after production (bottom)

Figure 2 (top) shows the pressure in multiple along-well locations. The pressure profile versus time has been plotted at 5 spots: 10 meters from the heel, 500 meters from the heel, middle of the well length, 500 meters from the toe and at the toe. The blue curve corresponds to the 10-meters away from the heel and shows a smooth and constant profile through time. Due to the proximity of this point to the heel, it is affected very quickly by the arrival of the pressure wave and hence the pressure at this point drops very quickly to a value close to the bottomhole pressure and reaches steady state. At 500 meters the pressure decreases sharply after approximately 1 second and changes at a slower rate to reach steady state after 15 seconds. The point in the middle of the well-length presents a similar behavior but with a 1-second time delay with respect to the previous point. The time delay in feeling the effect of propagating waves is longer as one moves toward the toe. The sharp step-like pressure drop at the third quarter of the well length and at the toe respectively occurs after 5 and 7 seconds. As all the five curves in figure 2 (top) indicate, after about 20 seconds the wellbore system reaches steady-state and no further changes in the pressures are observed.

0 5 10 15 20 25 30 35 40 45 50 2.44 2.46 2.48 2.5 2.52x 10 7 Time [s] W e ll b o re P re ssur e [ P a ] x=10m x=500m x=1000m x=1500m x=2000m 0 5 10 15 20 25 30 35 40 45 50 0 0.02 0.04 0.06 Time [s] W e ll b o re f lo w r a te [ m 3 /2 ] x=10m x=500m x=1000m x=1500m x=2000m

Figure 2 - Wellbore pressure versus time at five different locations of the well: 10, 500, 1000, 1500 and 2000 meters from the heel of the well (top); Wellbore flow rate versus time at five different locations of the well: 10, 500, 1000, 1500 and 2000 meters from the heel of the well (bottom)

Figure 2 (bottom) depicts the flow rate variations through time at the same five points as described above. At 10 meters from the heel the step-like increase in the flow rate is observed immediately and is followed by a transient increase towards the steady state. The flow rates of the first quarter, middle and third quarter of the well behave similarly. Since a condition of

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no-flow along the well is implemented at the toe the flow rate profile at this position is constantly zero through time. Figure 2 (bottom) clearly illustrates the traveling waves along the wellbore. The first wave, which is created by a step-like increase in the bottomhole pressure, initiates at time t0and travels from the heel towards the toe of the well such that the along-well positions are affected by this wave with time delay. When this wave reaches the end of its path at the toe, because of the no-flow boundary condition it is reflected and a second wave is created. The second wave travels in the opposite direction from the toe towards the heel and creates a sharp decrease in the flow rate; however the magnitude of the change is very small. In the contrast to the first propagating wave, the reflected wave reaches the well segments that are closer to the toe, faster. Therefore the period of the transient flow rate variations between arrivals of the two waves is longer for the locations closer to the heel.

Considering all the plots in figures 1 and 2, it can be concluded that the presented wellbore model with the corresponding boundary and initial condition reaches steady state after experiencing a step-like decrease in the bottomhole pressure. Figure 3 depicts the derivative of the pressure at the same five locations described above. The curves are generated by taking the derivative of the pressure with respect to time by the finite difference method:

       _ _  _   k k k k k p p p t t t 1 1 1 (46)

In can be shown that after 15 seconds the steady state is attained and the pressure derivative becomes zero for all the points. Moreover the propagating waves and the sequence at which they encounter different well positions are clearly observed in this figure. At x=10 m the wave arrives very early and changes the local pressure dramatically. As the wave travels along the well the 500, 1000 and 1500 meter locations are affected by the propagating wave. However due to the friction losses the energy of the wave and consequently the amplitude of the time derivative of the pressure attenuates. At x=2000m the two traveling waves are added and lead to a higher magnitude of pressure derivative.

0 5 10 15 20 25 30 -1 -0.5 0 0.5 1x 10 4 Time [s] P re ss u re d e ri vat ive [P a /s] x=10m x=500m x=1000m x=1500m x=2000m

Figure 3 - Derivative of the pressure versus time at five different locations of the well: 10, 500, 1000, 1500 and 2000 meters from the heel of the well

2.8.2 Example 2: The effect of isothermal compressibility

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the fluid plays a very important role in the dynamic behavior of the flow. Since only “slightly compressible fluid” is considered it this work, the isothermal compressibility plays the most important role in determining the speed at which the propagating waves travel. In this section the value of the compressibility is decreased to 1×10-9 and the same numerical experiment is repeated. Figure 4 represents wellbore pressure and flow rate profile respectively at the top and bottom. Comparing these profiles with the corresponding plots for cf = 1×10-8 reveals that

with lower compressibility, dynamics of the system occurs at a faster rate and the transient behavior disappears in a few seconds. Although in the previous case the pressure waves at

5

t s did not yet travel through the entire well length, in this scenario the pressure profile corresponding to 5 s time is very similar to the steady-state profile. The flow rate profiles in figure 4 (bottom) show that after only 1 second more than half of the well length is affected by the dynamic waves and the flow rates are non-zero in these spots.

0 200 400 600 800 1000 1200 1400 1600 1800 2000 2.45 2.47 2.49 2.51 2.53x 10 7 Along-well distance [m] W e ll b o re P re ssur e [ P a ] t=0s t=1s t=5s t=15s t=50s 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 0.02 0.04 0.06 Along-well distance [m] W e ll b o re f lo w r a te [ m 3 /s] t=0s t=1s t=5s t=15s t=50s

Figure 4 - Wellbore pressure versus along-well distance at five snapshots of time: 0, 1, 5, 15 and 50 seconds after production (top); Flow rate profile versus along-well distance at five snapshots of time: 0, 1, 5, 15 and 50 seconds after production (bottom); Isothermal Compressibility is 1×10-9 _Pa-1_.

For this case the pressure and flow rate profiles are calculated at 5 locations in a similar fashion to the case with 1x10-8 Pa-1. Both graphs support the conclusion about the fast dynamics of the system in case of lower compressibility. In order to quantify the transient time (i.e. the time needed for the system to reach steady-state starting from initial condition) the corresponding pressure derivatives of the 5 equally spaced locations are calculated and plotted in figure 6. In can be seen that the early transient phenomenon occurs during the first 3 seconds of the simulation, and then it damps and lasts until 5 seconds. After 5 seconds the system is in a steady-state condition. In both examples at steady state condition the well is

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producing 0.0535 m3/s of fluid and the total pressure drop over the length of the horizontal well is 0.25 MPa. 0 5 10 15 20 25 30 35 40 45 50 2.44 2.46 2.48 2.5 2.52 x 107 Time [s] W e ll b o re P re ssur e [ P a ] x=10m x=500m x=1000m x=1500m x=2000m 0 5 10 15 20 25 30 35 40 45 50 0 0.02 0.04 0.06 Time [s] W e ll b o re f lo w r a te [ m 3 /2 ] x=10m x=500m x=1000m x=1500m x=2000m

Figure 5 - Wellbore pressure versus time at five different locations of the well: 10, 500, 1000, 1500 and 2000 meters from the heel of the well (top); Wellbore flow rate versus time at five different locations of the well: 10, 500, 1000, 1500 and 2000 meters from the heel of the well (bottom); Isothermal Compressibility is 1×10-9

Pa-1_. 0 5 10 15 20 25 30 -1 -0.5 0 0.5 1x 10 4 Time [s] P re ss u re d e ri vat ive [P a /s] x=10m x=500m x=1000m x=1500m x=2000m

Figure 6 - Derivative of the pressure versus time at five different locations of the well: 10, 500, 1000, 1500 and 2000 meters from the heel of the well; Isothermal Compressibility is 1×10-9 _Pa-1_.

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The results of the simulations with isothermal compressibility values of 1×10-8 Pa-1 and 1×10-9 Pa-1 confirm that the transient behavior of the wellbore system disappears very quickly. The total time after which the described systems reach steady-state does not exceed 20 seconds. Moreover the input signal for the system in these examples was considered a step-like decrease in bottomhole pressure. In case of a smoother exciting signal, even faster and weaker transient characteristics can be expected.

The fast-disappearing transient behavior of fluid flow in the wellbore suggests that the dynamics of the wellbore are significantly faster than the dynamics of the reservoir. Therefore in coupling of a (numerical/analytical) reservoir simulator and a wellbore model, and under conditions of single-phase flow the dynamics of the wellbore can be neglected for the sake of simplicity and higher computational speed.

2.8.3 Example 2: The effect of well length

Another important parameter that plays a role in the analysis of the pressure profile along a horizontal well is the length of the well. Usually the production length of a horizontal well varies from 100 to 2000 m. In this section three different well with lengths of 500, 1000 and 2000 meters are examined under similar conditions. The specific productivity along the well, reservoir pressure, well diameter, relative roughness and fluid properties are kept the same for all three wells. The production rate at the heel of wells is prescribed to be 3179 m3/day (20,000 bbl/day). Since the wells produce the same amount of fluid from different interval lengths, the calculated steady-state bottomhole pressures and pressure profiles along the wells are different. Figure 7 (top) demonstrates how the pressure varies along the wells. For the 500 meter-long well the total pressure drop along the well is 0.121 MPa. When the length of the well is doubled (1000 m), with the same production rate, the total pressure drop decreases by a factor of 2 to 0.069Mpa. Moreover, the 2000m-long well produces the same amount of fluid with a total pressure drop of 0.035 MPa. Figure 7 (bottom) depicts the flow rate versus along-well distance for the three along-well lengths. Since the total production rate is assumed the same for all wells, the shortest well experiences the steepest slope in the flow rate profile from the heel to the toe and vice versa. Due to this, the average flow rate per unit length of the well is lowest and highest respectively for the longest and shortest well. Therefore the curvature observed in the pressure profile of the 500m case is not present in the profile of the 2000m well and thus the pressure profile is approximately linear. Table 1 lists the corresponding pressure drops versus different well lengths.

Using Distributed Fiber-Optic Sensing Systems to Estimate Inflow and Reservoir Properties

Using Distributed Fiber-Optic Sensing

Systems to Estimate Inflow and Reservoir

Properties

Using Distributed Fiber-Optic Sensing Systems

to Estimate Inflow and Reservoir Properties

Table of Contents

Chapter 1: Introduction

1.1

Introduction

References

Chapter 2: Dynamics of Wellbores:

Transient vs. Steady-State Models

2.1

Introduction

2.2

Modeling transient flow

2.3

Wellbore Model

 

 









 

 

 

 

 

 

 

 

 

 

 

u





2.4

Numerical Scheme













Λ

Λ

-A

 

 

 

i

 









 









 









 

x





 



  

2.5

Boundary Condition





_

_

_{ }