EPSRC Centre for Doctoral Training in

Industrially Focused Mathematical

Modelling

Well modelling constrained by

real-time data

Victoria Pereira

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Contents 1. Introduction ............................................. 2

Background .................................................. 2

Project overview ......................................... 2

2. Review of multiphase pipe-flow models 3

3. Model formulation .................................. 4

4. Discussion .................................................. 5

5. Potential impact ....................................... 5

References ................................................... 5

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An oil well is tubing,

through which the

fluid flows,

supported by a

cement casing.

1. Introduction

Background Hydrocarbons are found in underground porous rock stores called reservoirs. Hydrocarbon

production is the process of extracting these hydrocarbons through artificial structures

including wells. Understanding the dynamics governing the extraction process of

hydrocarbons can inform how to optimise the end production, thus motivating us to

develop a mathematical model of the system.

An oil well is a hole drilled from the surface down to the reservoir (see Figure 1). The

pressure at a certain depth is associated with the combined weight of formation and fluids

trapped in the formation. When the original reservoir pressure is sufficiently high, the

introduction of the well yields a pressure gradient that drives the initial flow of

hydrocarbons through the well up to the surface in what is called natural lift. Over time, the

reservoir pressure decreases, and the reservoir can no longer produce under its natural

energy, thus resulting in lower hydrocarbon production. There is therefore a need for

artificial lift systems which maintain production by controlling the pressure gradient in the

wellbore. However, the focus of this study will be on an oil well producing hydrocarbons

under natural lift.

Project overview The aim of this research is to build a new hybrid well model. The model is hybrid in that it

will couple methods of continuum modelling with data analytics. This research is

motivated by the availability of better quality data, as well as the new sources of data from

within oil wells. The idea is to improve a physical model of the oil well through data

assimilation. The plan for this current study is to establish an understanding of the oil well

domain and modelling techniques currently used, with the aim of formulating a well-

defined continuum model to study further in future work.

Reservoir fluid is composed of a mixture of hydrocarbons and water. We therefore

construct a generalised model governing the pipe-flow of gas, water, and oil, in order to

better understand the flow within the wellbore.

Oil, gas, and

mixtures of the two

are hydrocarbons.

Figure 1: Simple schematic of a natural lift oil well, and the pressure gradient down the well. Pressure profile figure is modified from [1].

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A variety of

modeling methods

are used for oil well

flow including

empirical and

physics-based

models.

2. Review of multiphase pipe-flow models

The term well model covers a variety of modelling methods used to understand the multiphase flow through the well. These methods are distinguished by how the derivation depends on the fundamental laws of physics, and whether they are steady in time or transient. Modelling techniques include statistical methods that are constructed from data; these are known in the oil industry as empirical, correlation or classical models. Alternative approaches include physics-based models known as mechanistic models. In practice, it is common for a combination of empirical and physics-based methods to be used, but over the years there has been a clear shift away from steady empirical models to transient multi-phase physics models (see Figure 2). Below we discuss the advantages and disadvantages of these two modelling techniques.

Empirical models are constructed using data collected from oil rigs or laboratory experiments. Functions relating variables of the system, such as pressure and temperature, are constructed through data fitting. These methods yield steady state models that are, in comparison to physics-based models, straightforward to use and are expected to be reliable when applied to the system from which the data was collected. However, the general applicability of these models is questionable, particularly those built from data collected in artificial laboratory experiments. Furthermore, collecting data from a producing oil well is timely and costly.

Physics-based models are developed from the underlying physics in the system. The governing equations are derived from the conservation of mass, momentum, and energy, and can be steady state or transient. Steady state models can instruct us on how to optimise a well in stable production, whereas transient models are useful for the initial start-up of the well, or for modelling changes in the system over time. These models are more accurate, as they are constructed from first principles. However, such models consist of large and complicated systems of equations that are difficult to solve. Hence, simplifying assumptions must be made to reduce the complexity of these models for them to be useful in practical applications.

Figure 2: Figure showing the evolution of two-phase flow modeling, from [2].

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To be well-posed, a

mathematical model

needs to have the

same number of

equations as

unknowns.

Our model consists

of mass, momentum

and energy

conservation laws

for the gas, water,

and oil.

3. Model formulation We derive a transient three-phase gas-water-oil pipe-flow model using mass, momentum,

and energy conservation laws. The domain of interest is an oil well of any inclination. We

make the assumption that the oil well has a constant diameter, and treat the domain as

one-dimensional in space. This is justified as the tubing of a typical oil well is of the order

of ten centimetres in diameter, while the length is several hundred metres.

We make the further assumption that gas either flows as an independent phase, or is

dissolved in oil (see Figure 3). This means that there is mass transfer between oil and gas

only.

The governing equations are derived from the conservation of mass, momentum, and

energy for each phase. This yields nine equations for fifteen unknown variables; volume

fraction, velocity (see Figure 4), pressure, density and energy for each phase. We therefore

need to close the system of equations with constitutive relations. It is clear that the sum of the

volume fractions must be unity, which gives an additional equation. How the phases interact with

each other, with respect to momentum and energy transfer is governed by interfacial terms in each

of the conservation equations. To ensure that we are not introducing any artificial momentum or

energy changes, we require that the sum of these interaction terms must be zero. This yields two

more equations. The final three equations required to close the system of equations come from

the inflow boundary conditions at the bottom of the well. Thus our model consists of fifteen

equations for fifteen unknowns. Full details of the model are omitted in this report.

Figure 3: Schematic of the three-phase pipe-flow in a natural lift oil well. The three phases are oil (brown), gas (grey) and water (blue). The domain is treated as one-dimensional in the spatial coordinate z.

Figure 4: Schematic of the three-phase pipe, illustrating the velocities of the different phases.

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4. Discussion In this study, we constructed a generalised three-phase pipe-flow model, the purpose of

which was to gain an understanding of the interacting flow of gas, water, and oil inside a

well that is producing under natural lift. This model provides a framework for future work,

including a dimensional analysis of the equations, which will indicate dominant or

negligible terms in the model, thus allowing simplification of the full system. Once we have

justifiably simplified the model, we will look towards implementing numerical methods to

solve the dimensionless system of equations. A solution to the multiphase pipe-flow model

will provide useful insight into how the dynamics within a well evolve in time and can be

controlled through parameter variation.

The work discussed thus far takes a deterministic continuum modelling approach to the

fluid dynamics in the oil well. However, environmental conditions around the well, such as

inflow conditions from the reservoir, are highly uncertain. We therefore need to examine

how to capture this uncertainty in our model. Such uncertainty may be reduced through

data assimilation. Once we have a good understanding of the underlying physics of the well

system, we will incorporate data analytic methods to improve the model. To optimise the

production and structural endurance of oil wells, we aim to develop a framework

governing real-time adaptive optimisation of the model and control system variables.

5. Potential impact Through reviewing the oil well literature, we have seen that there are several approaches

one might take to construct a hybrid well model that captures the physics and makes use of

available data. The multiphase flow model developed in this research will provide a good

foundation for future study in this area. This deterministic model will then be improved

through data assimilation.

Ekaterina Sergienko, Digital Oilfield Production, PDS “The objective of this work is to develop a robust ‘hybrid’ well model that is based on core multi-phase flow physics supplemented by data-driven modelling. This enables integration of well history data as well as real-time measurements, where data quality and availability permits. We expect the hybrid well model to leverage the growing data volumes from, for example, real-time down-hole sensor, and thereby to improve the predictive and diagnostic capability compared to conventional models. Furthermore, the model could be applied in data-driven production system design or production optimization.”

References

1. L. DAKE, (1983). Fundamentals of reservoir engineering, vol. 8, Elsevier.

2. M. SHIPPEN and W. BAILEY, (2012). Steady-state multiphase flow past, present, and future, with a perspective on flow assurance, Energy & Fuels, 26, pp. 4145-4157.