multiphase flow solution in horizontal wells using a drift...

Multiphase flow solution in horizontal wells using a drift-flux model

Solução do escoamento multifásico em poços horizontais utilizando um modelo de drift-flux Solución de flujo multifásico en pozos horizontales con un modelo de drift-flux

Arthur Besen SopranoAntónio Fábio Carvalho da SilvaClovis R. Maliska

AbstractThis study presents a procedure to solve two-phase (gas and liquid) flows throughout an oil well with lateral mass inflow

from the reservoir. The flow is considered isothermal and one-dimensional. Equations are discretized using a Finite Volume Method with a C++ (OOP) code implementation. This algorithm is intended to be used with a reservoir simulator to solve the coupled flow between the reservoir and well.

keywords: n drift-flux n multiphase flow n horizontal wells

ResumoEste estudo apresenta um procedimento para determinar o escoamento bifásico (gás e líquido) ao longo de um poço de

petróleo com a entrada de massa lateral do reservatório. O fluxo é considerado isotérmico e unidimensional. As equações são discretizadas utilizando um Método dos Volumes Finitos com o código C++ (OOP). Esse algoritmo foi desenvolvido para ser utilizado com um simulador de reservatório a fim de resolver o fluxo acoplado entre o reservatório e o poço.

palavras-chave: n drift-flux n escoamento multifásico n poços horizontais

ResumenEste estudio presenta un procedimiento para el cálculo del flujo bifásico (gaseoso y líquido) en un pozo de petróleo con la

entrada de masa lateral del reservatorio. El flujo se considera isotérmico y unidimensional. Las ecuaciones son discreteadas

Boletim Técnico da Petrobras, Rio de Janeiro, v. 54, n. 1/2, p. 33-41, abr./ago. 2011 n 33

IntroductionAn intense sequence of studies and analysis is necessa-

ry to optimize petroleum exploitation. Reservoir simulation has seen extensive growth over the last decades, with ever incre-asing complexity to better represent the physical phenomena that occur during the oil extraction process. Horizontal wells are today one of the alternatives to maximize production in deep and narrow oil field situations. Another consideration is the multiphase flow throughout the reservoir and well domains to forecast better results and other information such as the to-tal production of each phase.

A complete reservoir simulation procedure requires the re-servoir pressure and field saturation data subject to the bounda-ry conditions provided by the injection and production wells. To obtain these boundary conditions the pressure and field satura-tion of every well within the reservoir must also be identified (fig. 1). With the successful coupling of these two issues, a tool can be created to facilitate decisions of well location, well type, etc.

Traditionally the solution of a horizontal well two-phase flow first requires the momentum and continuity equations for each phase to be solved. Then apply the proper interphase and wall friction models that may exist, whenever two fluids are confined in a domain subject to given boundary conditions. Ho-wever, it can also be solved considering a homogeneous flow and apply a model, known as drift flux models, to represent the multiphase behavior. The latter approach main advantage is it simplifies the problem to fewer equations yet reasonable re-sults are obtained. Also, petroleum wells are usually represen-ted as one-dimensional curves inside the reservoir with lateral mass inflow or outflow.

Model formulationA two-phase homogeneous model with slip between pha-

ses, by including an appropriate constitutive equation, would

allow each phase’s velocity to be calculated separately. However, as there is only one mixture momentum equation in the drift flux model, the pressure is the same for all phases.

The one-dimensional model demands that a given proper-ty should not vary along the cross-sectional area. Therefore, it is necessary to use average cross-sectional values throu-ghout the well. Given a variable φ, its cross-sectional avera-ge value is calculated by

(1)

Where:A = well’s cross-sectional area;

⟨ ⟩ = average operator and will be omitted in further equations for simplifying notation.

Figure 1 – Scheme of a horizontal well with multiphase flow.

Figura 1 – Esquema de um poço horizontal com escoamento multifásico.

Figura 1 – Esquema de un pozo horizontal con flujo multifásico.


34 n Boletim Técnico da Petrobras, Rio de Janeiro, v. 54, n. 1/2, p. 33-41, abr./ago. 2011

usando el Método de Volúmenes Finitos implementado en lenguaje C++ (OOP). El algoritmo está diseñado para ser utilizado en conjunto con un simulador de reservatorio para la solución del flujo acoplado entre el reservatorio y el pozo.

palabras-clave: n drift-flux n flujo multifásico n pozos horizontales

The volume fraction of a phase p is given by

(2)

which represents the ratio between the phase volume over the total volume. If we consider that, inside a controlled volu-me, the properties do not change, then

(3)

where:Ap = cross-sectional area occupied by the phase in the pipe.

Because the sum of the volumes of each phase should re-sult in the total volume, the sum of all volume fractions is equal to the unit

(4)

If we consider a two-phase gas-liquid flow, the gas velo-city can be calculated as

(5)

Where:Co = profile parameterVgj = drift velocity, and they may vary according to each

flow pattern (fig. 2).

For example, if the flow is considered homogeneous, then Co=1.0 and Vgj=0, so all phases have the same veloci-ty. The characteristic velocity j, also known as total volume-tric flux (Hibiki and Ishii, 2006) is defined as

(6)

and the gas and liquid velocities are defined as

volumetric flux of the phase pvp = –––––––––––––––––––––––––––––––––– (7) cross-sectional area occupied by the phase p

There are three equations to be solved (Hibiki and Ishii, 2003) for the proposed model, as follows:

Mixture Continuity Equation:

(8)

Gas Phase Continuity Equation:

(9)

Mixture Momentum Equation:

(10)

in which the terms and represent the total and gas lateral mass inflow, respectively. They represent the cou-pling term between the reservoir and wellbore and are calcu-

Figure 2 – Different multiphase flow patterns.

Figura 2 – Padrões diferentes de escoa-mento multifásico.

Figura 2 – Padrones diferentes de flujo multifásico.

Soprano et al.


(16)

Equations (8) to (10) should be solved considering pres-sure (P), gas volumetric fraction or void fraction (αg) and νm

as independent variables. All other properties can be calcula-ted after these variables are identified.

Numerical procedureA finite volume method is used to discretize the equations.

A staggered grid is applied (Maliska, 2004) as shown in figure 3.

The pressure and gas void fractions are calculated for each position and the mixture velocity is calculated in the mean point between them. All equations are solved simul-taneously using Newton’s method to obtain the discretized residual equations, calculate the numerical derivatives and construct the jacobian matrix. The computational algorithm is shown in figure 4.

Results and conclusionsThe proposed algorithm is compared with that presented by

Provenzano (2007) and Evje and Fjelde (2003) where a 1.000m pipe is initially full of liquid. Then gas and liquid are gradually in-jected in one of the extremities and the pressure kept constant at the other. Different drift flux parameters are adopted.



lated based on the differences between reservoir and wellbo-re pressures. The mixture equations are obtained by the sum of the gas and liquid equations. Therefore, some mixture pro-perties should be defined as:

Mixture Density (ρm):

(11)

Mixture Viscosity (µm):

(12)

Mixture Velocity (νm):

(13)

Modified Drift Velocity ( V– gj ):

(14)

Gas and liquid velocities can be calculated by the mixture velocity and modified drift velocity,

(15)

Figure 3 – Discretization scheme. A staggered grid approach was adopted.

Figura 3 – Esquema de discretização. Uso de malhas desencontradas.

Figura 3 – Esquema de discretización. Uso de mallas desencontradas.

Figure 4 – Scheme of the computational procedure.

Figura 4 – Esquema do procedimento computa-cional.

Figura 4 – Esquema del procedimiento computacional.

In the first case constant values for the profile parameter and drift velocity were used and the gas and liquid were in-jected gradually for 10 seconds until reaching a constant va-lue (fig. 5). A total time of 250 seconds with 200 control vo-lumes were run with results shown in figure 6.

In the second case the drift velocity is a function of the void fraction,

Liquid is injected during 175 seconds, but gas is injected only during a given time (fig. 7).

The results of this case are presented in figure 8.

The analyzed two cases are in good agreement with the results of Provenzano (2007), because he also applied the fi-nite volume method to discretize the equations, i.e., a con-servative method. The results presented by Evje and Fjelde (2003) were obtained through a different approach with the inclusion of second order interpolation schemes. The propo-sed method also considers pressure, velocity and gas volu-metric fraction as main variables, while Evje and Fjelde (2003) and Provenzano (2007) proposed models with different inde-pendent variables with no physical interpretation, then calcu-lated all other variables after the solution was obtained. The pressure was also calculated in the references through an equation of state, in which an incompressible fluid in the pro-

Soprano et al.


Figure 5 – Influx profile of the pipe entrance for the first case.

Figura 5 – Perfil de fluxo na entrada do tubo para o primeiro caso.

Figura 5 – Perfil de flujo en la entrada del tubo para el primer caso.

Figure 6 – Results for pressure, volumetric frac-tion and velocity of each phase of the first case.

Figura 6 – Resultados de pressão, fração volumétri-ca e velocidade de cada fase do primeiro caso.

Figura 6 – Resultados de la presión, la fracción volumetrica y la velocidad de cada fase del primer caso.

Figure 7 – Influx profile of the pipe entrance for the second case.

Figura 7 – Perfil de fluxo na entrada do tubo para o segundo caso.

Figura 7 – Perfil de flujo en la entrada del tubo para el segundo caso.



Figure 8 – Results for pressure, volumetric fraction and velocity of each phase of the second case.

Figura 8 – Resultados de pressão, fração volu-métrica e velocidade de cada fase do segundo caso.

Figura 8 – Resultados de la presión, la fracción volumetrica y la velo-cidad de cada fase del segundo caso.

Soprano et al.


blem could not be considered, as there’s no equation of sta-te in this case (ρ=const).

The presented approach main advantage is it allows ti-mesteps 100 times greater (or even more) than the maxi-mum timesteps that could be used in the solutions reported in Evje and Fjelde (2003) and Provenzano (2007). Evidently, the greater the timestep, the less accurate the transient so-lution results. However, the use of any timestep is an advan-tage, not only when the steady state solution is the final goal, but also when the transient behavior is of interest. The great advantage of the proposed method is there is no restriction to small timesteps to obtain convergence.

The comparisons were made only with transient situa-tions to validate the proposed drift flux model. Indeed, this study’s next step is to couple the proposed algorithm with a

reservoir simulator, bearing in mind, a reservoir's timestep is in the order of days, while the well's steady state is reached in minutes or even seconds.

n n n

Referências Bibliográficas

n EVJE, S.; FJELDE, K. K. On a rough AUSM scheme for a one-dimen-sional two-phase model. Computers & Fluids, Amsterdam, v. 32, n. 10, p. 1497–1530, Dec. 2003.

n HIBIKI, T.; ISHII, M. One-dimensional drift-flux model and constitutive equations for relative motion between phases in various two-phase flow regimes. International Journal of Heat and Mass Transfer, v. 46, n. 25, p. 4935-4948, Dec. 2003.

n ISHII, M. HIBIKI, T. Thermo-fluid Dynamics of Two-phase Flow. New York: Springer, 2006. 462 p.

n MALISKA, C. R. Transferência de calor e mecânica dos fluidos computacional. 2. ed. Rio de Janeiro: LTC, 2004. 472 p.

n PROVENZANO, C. E. C. Previsão numérica de escoamento bifásico em tubulações utilizando o modelo de deslizamento. 2007. 99 f. Dis-sertação (Mestrado) – Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, 2007

António Fábio Carvalho da Silva graduou-se em Engenharia Me-cânica pela Universidade de Brasília, em 1976. Obteve o título de Mestre em Ciências (1978) e de Doutor em Engenharia Mecâni-ca, pela Universidade Federal de Santa Catarina, na área de Me-cânica dos Fluidos Computacional - CFD. Desenvolve atividades de pesquisa e extensão no Laboratório de Simulação Numérica em Mecânica dos Fluidos e Transferência de Calor (Sinmec) da UFSC. Dentre eles, foi coorientador de tese de doutorado, agraciada com o Prêmio Petrobras de Tecnologia, em 2005. É professor do De-partamento de Engenharia Mecânica da UFSC desde 1977, onde atua no ensino de disciplinas da área de térmica e de fluidos, ten-do também exercido as funções de chefe de Departamento e co-ordenador da pós-graduação.

António Fábio Carvalho da Silva

Universidade Federal de Santa Catari-na – UFSCDepartamento de Engenharia Mecâni-ca – EMCe-mail: [email protected]

Autores

Arthur Besen Soprano graduou-se em Engenharia Mecâni-ca pela Universidade Federal de Santa Catarina, em 2011. Atualmente é aluno de mestrado no Departamento de En-genharia Mecânica da Universidade Federal de Santa Catari-na. Desenvolve atividades de pesquisa e extensão no Labo-ratório de Simulação Numérica em Mecânica dos Fluidos e Transferência de Calor da UFSC (Sinmec). Durante sua gra-duação, foi bolsista do Programa de Recursos Humanos da ANP (PRH-09), onde desenvolveu seu trabalho de conclusão de curso na área de modelagem e solução numérica de es-coamentos multifásicos em poços de petróleo inclinados. Ao final de sua graduação, fez um estágio de seis meses no De-partamento de Engenharia de Petróleo e Geosistemas na The University of Texas at Austin e trabalhou na implementação modelos de poços com trajetória arbitrária em um simula-dor composicional de reservatórios de petróleo, desenvolvi-do nesta universidade

Arthur Besen Soprano

Universidade Federal de Santa Cata-rina – UFSCDepartamento de Engenharia Mecâ-nica - EMCe-mail: [email protected]



Clovis R. Maliska realizou sua graduação e mestrado em En-genharia Mecânica pela Universidade Federal de Santa Catari-na (1973 e 1975), e seu doutorado em Engenharia Mecânica pela Universidade de Waterloo, no Canadá (1981). Atualmen-te, é professor titular do Departamento de Engenharia mecâni-ca e coordenador do Laboratório de Simulação Numérica em Mecânica dos Fluidos e Transferência de Calor (Sinmec). Atua na área de desenvolvimento e aplicações de métodos numé-ricos para Transferência de Calor e Mecânica dos Fluidos. Os desenvolvimentos se concentram no método dos volumes fini-tos baseado em elementos (EbFVM) para malhas não estrutu-rados, volumes finitos para coordenadas generalizadas, méto-dos multigrid, escoamentos em meios porosos, escoamentos multifásicos, escoamentos de metais líquidos, escoamentos compressíveis, algoritmos para minimização da difusão numé-rica e dos efeitos de orientação de malha. As aplicações têm sido feitas, principalmente, em simulação de reservatórios de petróleo, escoamentos gás/sólido, escoamentos gás/líquido, simulação de escoamentos de metais líquidos da indústria siderúrgica. Coordena dois projetos das redes temáticas da Petrobras, nas áreas de simulação de reservatórios e acopla-mento poço-reservatório, o PRH09, de formação de recursos humanos da ANP e o PFRH-Superior, de formação de recur-sos humanos da Petrobras.

Clovis R. Maliska

Universidade Federal de Santa Cata-rina – UFSCDepartamento de Engenharia Mecâ-nica - EMCe-mail: [email protected]

Soprano et al.


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