New sources of energy should be found to relieve the high demand of energy. Even though heavy oil and bitumen are difficult to produce due to their high viscosity which can be reduced by heating, with increased oil price, the production of these heavy oils are seen viable thus the need for a model that would help make predictions for the future and also take into consideration areal and vertical sweep of hydrocarbons (3D simulator). The ability to be able to optimize the interaction data and decision making during the life cycle of the field is critical. As a result of a heterogeneity of reservoirs, numerical simulators are used to obtain consistent and significant solutions.

For this work, a three-dimensional numerical reservoir simulator is developed for an expansion drive with a high viscous oil. A transient state heat system by conduction with an internal heat source is considered. A temperature simulator is first developed then coupled with a viscosity correlation after which it is then coupled with a diffusivity equation for a single phase flow of an expansion drive reservoir. All the governing equations are discretized using finite difference technique; iterative linear solver with the aid of MATLAB code is used to solve the system of linear equations.

This work aims to look at the effect of temperature on pressure drop through viscosity. It is realized that an increase in the heat source introduced a rise in temperature which in turn decrease the viscosity across the system. The pressure across the system is seen to be sustained even though it is declining thus the pressure being maintained









General Introduction11

Problem definition13


Scope and limitation of this work14

Organization of thesis14



Natural Drive Mechanism16

Enhanced Oil Recovery Methods17

Numerical Reservoir Simulation21

Numerical Methods23

Linear Solvers in Reservoir Simulators25

MATLAB Programming27



Development of the Simulator29

Heat Model30

Viscosity Correlation40

Pressure Model41



Base Case Scenario58

Thermal Process59

Sensitivity Analysis76








Figure 2.2: A Diagram showing EOR Methods. 20

Figure 2.3.1: Schematic diagram of the numerical reservoir simulation process. 21

Figure 2.3.2: A schematic Diagram of reservoir models based on dimension. 23

Figure 2.5.1: Schematic diagram of steps involving direct solution method. 25

Figure 2.5.2: A Schematic Representation of the iterative solution method. 26

Figure 3.1: A numerical stencil for a three-dimensional oil reservoir block 30

Figure 3.2.1: A diagrammatic representation of conduction. 30

Figure 3.2.2: A diagrammatic representation of the volumetric system. 31

Figure 3.4: MATLAB sequential process algorithm. 56

Figure 4.1: A plot of Pressure (Pav, Pwf) versus Time. 58

Figure 4.2.1: A plot of Temperature (Twf)) versus Time. 59

Figure 4.2.2: A plot of Temperature (Twf, TAB, Tin, Tjac) versus Time. 60

Figure 4.2.3: Surface plot of reservoir temperature distribution after 1 day. 61

Figure 4.2.4: Surface plot of reservoir temperature distribution after 10 days. 61

Figure 4.2.5: Surface plot of reservoir temperature distribution after 60 days. 62

Figure 4.2.6: Surface plot of reservoir temperature distribution after 120 days. 62

Figure 4.2.7: Surface plot of reservoir temperature distribution after 180 days. 63

Figure 4.2.8: Surface plot of reservoir temperature distribution after 240 days. 63

Figure 4.2.9: Surface plot of reservoir temperature distribution after 300 days. 64

Figure 4.2.10: Surface plot of reservoir temperature distribution after 365 days. 64

Figure 4.2.11: A plot of Viscosity (Vwf) versus Time. 65

Figure 4.2.12: A plot of Viscosity (Vwf, Vjac, Vin, VAB) versus Time. 66

Figure 4.2.13: Surface plot of viscosity distribution after 1 day. 66

Figure 4.2.14: Surface plot of viscosity distribution after 10 days. 67

Figure 4.2.15: Surface plot of viscosity distribution after 60 days. 67

Figure 4.2.16: Surface plot of viscosity distribution after 120 days. 68

Figure 4.2.17: Surface plot of viscosity distribution after 180 days. 68

Figure 4.2.18: Surface plot of viscosity distribution after 240 days. 69

Figure 4.2.19: Surface plot of viscosity distribution after 300 days. 69

Figure 4.2.20: Surface plot of viscosity distribution after 365 days. 70

Figure 4.2.21: A plot of Pressure (Pwf, Pav) versus Time. 71

Figure 4.2.22: A plot of Pressure (Pwfb, Pwft) versus Time. 71

Figure 4.2.23: A plot of Pressure (Pwf, Pin, PAB, Pav, Pjac) versus Time. 72

Figure 4.2.24: Surface plot of pressure distribution after 1 day. 72

Figure 4.2.25: Surface plot of pressure distribution after 10 days. 73

Figure 4.2.26: Surface plot of pressure distribution after 60 days. 73

Figure 4.2.27: Surface plot of pressure distribution after 120 days. 74

Figure 4.2.28: Surface plot of pressure distribution after 180 days. 74

Figure 4.2.29: Surface plot of pressure distribution after 240 days. 75

Figure 4.2.30: Surface plot of pressure distribution after 300 days. 75

Figure 4.2.31: Surface plot of pressure distribution after 365 days. 76

Figure 4.3.1: The Effect of varying Heat Source on Reservoir Temperature. 77

Figure 4.3.2: The Effect of varying Heat Source on Reservoir Viscosity. 77

Figure 4.3.3: The Effect of varying Heat Source on Reservoir Pressure. 78



General Introduction

Reservoirs act differently due to varying range of both rock and fluid properties and thus must be treated uniquely. During production, reservoirs are allowed to naturally produce their hydrocarbons until when production rates are mostly not economical viable then other support systems are used. Primary recovery is the natural stage of the reservoir to be able to produce without support thus depending on reservoir’s internal energy. There are different drive mechanisms known as a results of different energy sources. The drive mechanism of a reservoir is not known in the earlier life of the production but can be seen from production data with time. The knowledge about the reservoir’s drive mechanism can help improve reserves recovery and supervision during its middle and later life. The important drive mechanisms include: Rock and liquid expansion drive, solution gas/ depletion drive, Gas cap drive, Water drive, Combination drive and Gravity drainage drive.

Rock and liquid expansion drive has its oil existing at a higher pressure than the bubble point pressure and with only oil, connate water and the rocks. The rock and fluids expand as a result of their different compressibility as the reservoir pressure deplete. Formation compaction and expansion of different rock grains are some factors that affect reservoir rock compressibility. These factors are due to decrease of fluid pressure within the pore spaces which in turn reduce pore volume through porosity reduction. While the pore volume is reducing, the crude oil and water will be forced out of the pore space to the wellbore. Due to the compressibility (slightly) of both liquids and rocks, the reservoir will experience a rapid pressure decline. A constant gas-oil ratio

equal to gas solubility at bubble point pressure is typical of this drive mechanism. A small percentage of total oil in place is recovered due to the less efficiency of this drive.

Other recovery methods like Secondary and tertiary (Enhanced) recovery methods are employed to help improve the recovery of the remaining hydrocarbons by providing additional or sustaining the energy. The efficiency of an enhanced recovery method is a measure of its ability to provide greater hydrocarbon recovery than by natural depletion at economically attractive production rate (Marcel et al. 1980). It depends on reservoir characteristics and nature of displacing and displaced fluids. Enhanced recovery methods seeks to improve the sweep and displacement efficiency. It has been basically grouped into three types; namely chemical processes, miscible displacement processes and thermal processes. Thermal processes seeks to lower the viscosity of the fluid in place thus improving displacement and some of the processes are steam flooding and in-situ combustion. In order to manage and predict the performance of high viscous oil reservoir which is being heated using a heat probe, numerical reservoir simulation is needed thus the need for a three-dimensional numerical simulator for high viscous oil reservoir.

Reservoir simulation is the art of relating mathematics, physics, reservoir engineering, and computer encoding to predict hydrocarbon reservoir performance under different operating approaches (Aziz, K. and Settari, A. 1979).

Petroleum reservoir simulation is an approach whereby mathematical equations (model) or computable procedure are employed to infer the behavior of the real reservoir.

It is possible to obtain an exact solution for a few problems by direct integration of the differential equation (analytical solution). However, when analytical solutions breakdown, simple approximate methods (numerical solutions) are employed.

Today, numerical reservoir simulation is regularly used as a valuable tool to help make investment decisions on major exploitation and development projects. These decisions include determining commerciality, optimizing field development plans and initiating secondary and enhanced oil recovery methods on major oil and gas projects. Proper planning is made possible by use of reservoir simulation; it can be used effectively in the early stages of development before the pool is placed on production so that unnecessary expenditures can be avoided.

Problem definition

Heavy oil reservoirs cannot be easily produced due to their high viscosities which in turn inhibit mobility of hydrocarbons therefore enhanced oil recovery like thermal recovery method is employed to help decrease the viscosity drag effect of the hydrocarbons. These recovery methods are capital intensive and as such need intensive studies and forecast about their outcomes therefore the need for a numerical reservoir simulator which can be one-dimensional (1-D), two-dimensional (2-D) and three-dimensional (3-D). With the 3-D model, it gives full description of the real situation by accounting for both areal and vertical sweep efficiencies which neither 1-D nor 2-D models can give thus the need for a 3-D numerical simulator for high viscous oil reservoir.


Below are the outlined objectives for the work:

· To derive and solve a heat equation for conduction with a heat source using finite difference method.

· Using a viscosity correlation, predict the viscosity dependence on temperature for a high viscous volumetric oil reservoir.

· To derive and solve diffusivity equation for single phase flow using finite difference method.

· To develop a 3-D numerical simulator for high viscous oil reservoir combining the heat, viscosity and diffusivity equations using MATLAB.

· To use the developed simulator to predict temperature distribution and pressure decline.

Scope and limitation of this work

This work is limited to (the development of a numerical simulator for) heavy oil reservoir with expansion drive as it primary drive for recovery.

Organization of thesis

The thesis is structured in this manner:

· Chapter two gives a brief evaluation of drive mechanisms, enhanced oil recovery and thermal recovery. Numerical reservoir simulation and numerical methods for discretization of the equations governing heat transfer and flow in subsurface reservoirs, including benefits and limitations of finite difference method are reviewed. Also in review is simple iterative method and use of MATLAB programming in reservoir simulation.

· Chapter three presents the methodology employed in this study; mathematical, numerical and computer models formulations.

· In Chapter four contains the discussion of the results.

· Chapter five draws logical conclusions based on the simulator results, and makes useful recommendations for further studies.



Natural Drive Mechanism

Each reservoir is composed of a unique combination of geometric form, geological rock properties, fluid characteristics, and drive mechanism (primary). The recovery of oil by any of the natural drive mechanisms is called primary recovery thus no energy supplement. Although no two reservoirs are identical in all aspects, they can be grouped according to the primary recovery mechanism by which they produce (Ahmed 2006). There are basically six driving mechanisms that provide the natural energy necessary for oil recovery:

· Depletion drive (This type of drive has its main source of energy being due to gas liberation from the crude oil and expansion of the solution gas as the reservoir pressure is reduced.)

· Gas cap drive (This drive is identified by the presence of a gas cap with little or no water drive. The reservoir pressure decline is slow due to the ability of the gas to expand.)

· Water drive (Most reservoirs are bounded on a portion or all the edges by water bearing rocks called aquifers. These aquifers help provide energy to push the hydrocarbons. There are bottom water and edge water occurring in this drive.)

· Gravity drainage drive (This drive is as result of differences in densities of the reservoir fluids)

· Combination drive (This drive can chain two or more of the above drives)




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