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2.B
Initial, well and outer boundary conditions
The diffusion equation describes how the fluid pressure in a local piece of rock will locally react
to a pressure disturbance. This equation defines how the system will react far from the
boundaries and the wells.
In order to define a problem completely we need to know a starting point, i.e. the initial state
of the system, and how the fluid flow will be constrained at the well and at the natural limits of
the reservoir.
2.B.1
Initial conditions
The most common condition, and the only easy state to model analytically, is to assume that
at a reference time 0, corresponding to the time when the sequence of production starts, the
reservoir was at initial uniform pressure p
i
. This simple equation can be written:
In Cartesian coordinates:
i
p zyx tp zyx
, , ,0
, ,
Assuming a radial symmetry:
i
p r tp r
,0
In the case of a multilayer analysis, if the reservoir layers are commingled there could be one
such equation for each independent layer. If the pressure difference between the different
layers is not the static gradient in the wellbore, it means that cross-flow will occur between the
different layers as soon as the well is perforated, even if it is not flowing.
It is also possible to start a problem with a dynamic pressure situation. This will be, typically,
the 'restart' of a simulator run from which one wants to run a test on a new, additional well.
This kind of situation is however only possible when using a numerical model.
2.B.2
Well conditions
All equations above (diffusion, initial and outer boundary conditions) lead to a nicely uniform
reservoir at initial pressure, forever. What changes this million-year-old equilibrium is the
disturbance created by man-made producing or injecting wells. We generally take 0 as the
time of the initial disturbance.
The simplest realistic model is a vertical well, of radius r
w
, fully penetrating the formation. The
inner condition is nothing more than Darcy’s law calculated at the sandface. In the case of a
homogeneous reservoir this will write: Finite radius well:
kh
qB
r
p
r
t r
w
2. 141
,
As the flow rate q is generally given at standard conditions, the volumetric downhole rate q
sf
is
calculated by multiplying the standard rate by the reservoir volume factor B. In the simplest
cases this volume factor is considered constant. Otherwise a PVT equation must be used to
dynamically calculate the volume factor.