The study of petroleum recovery is significant for reservoir engineers. Mathematical models
of the immiscible displacement process contain various assumptions and parameters, resulting
in nonlinear governing equations which are tough to solve. The Buckley-Leverett equation is
one such model, where controlling forces like gravity and capillary forces directly act on
saturation profiles. These saturation profiles have important features during oil recovery.
In this thesis, the Buckley-Leverett equation is solved through a finite volume scheme, and
capillary forces are considered during this calculation. The detailed derivation and calculation
are also illustrated here. First, the method of characteristics is used to calculate the shock speed
and characteristics curve behaviour of the Buckley-Leverett equation without capillary forces.
After that, the local Lax-Friedrichs finite-volume scheme is applied to the governing equation
(assuming there are no capillary and gravity forces). This mathematical formulation is used for
the next calculation, where the cell-centred finite volume scheme is applied to the Buckley-
Leverett equation including capillary forces. All calculations are performed in MATLAB. The
fidelity is also checked when the finite-volume scheme is computed in the case where an
analytical solution is known. Without capillary pressure, all numerical solutions are calculated
using explicit methods and smaller time steps are used for stability. Later, the fixed-point
iteration method is followed to enable the stability of the local Lax-Friedrichs and Cell-centred
finite volume schemes using an implicit formulation. Here, we capture the number of iterations
per time-steps (including maximum and average iterations per time-step) to get the solution of
water saturation for a new time-step and obtain the saturation profile. The cumulative oil
production is calculated for this study and illustrates capillary effects. The influence of
viscosity ratio and permeability in capillary effects is also tested in this study.
Finally, we run a case study with valid field data and check every calculation to highlight that
our proposed numerical schemes can capture capillary pressure effects by generating shock
waves and providing single-valued saturation at each position. These saturation profiles help
find the amount of water needed in an injection well to displace oil through a production well
and obtains good recovery using the water flooding technique