1) Three types of fluid flow can occur in a reservoir: steady-state, semi-steady state, and unsteady-state flow. 2) Steady-state flow very rarely occurs and requires a strong pressure maintenance mechanism like an aquifer to replenish pressure changes from production. 3) Semi-steady state is the dominant type, where pressure declines uniformly throughout the reservoir as the boundaries have been encountered. 4) Unsteady-state flow occurs early in a well's life before boundaries are felt, and the reservoir acts infinitely. The correct flow equations depend on identifying the type of flow.

1. First of all, these types of flow are observed when the reservoir is put into
production, which means if all wells are closed then there is no flow,
 When you open a well, you are effectively disturbing the near-wellbore
area pressure (lowering BHP) allowing pressure differential to form
which means the HC will begin to flow towards the producer. Preesure
disturbance will travel from the well-bore till reaching the reservoir
boundaries (ANALOGIES: when a line of cars stopping at red light, then
when green light is on the cars movement will start from the one closest
to the light then the effect will slowly travel till reaching the last car in
the queue)
Now, this HC flow can be of three types:
1-Steady-State Flow: VERY rarely happens: because if you think about it,
from where you could have absolutely strong aquifer that keeps pressure
constant?!! Steady state flow is more applicable to laboratory displacement
experiments than to petroleum reservoir conditions.
Key Summary: Pressure transient has reached all of the boundaries but the
static pressure at the boundary does not decline.
This happens when there is strong pressure maintenance mechanism such as
strong aquifer allowing every pressure drop (due to production) to be
replenished by the water encroachment. That is why pressure anywhere in the
reservoir is always the same during production (only location-dependent, but
pressure stays the same for that location for the whole period),
Now since the pressure doesn’t change with time, the properties such as
viscosity, density, and flow rate all essentially remain the constant (please note
that fluid flow rate is constant whereas oil rate should be dropping due
increased water cut)
Figure explanation: the pressure profile stays the same and does not change in
the steady state flow

2.  This means the diffusivity equation can be equalised to zero:

= 0
An everyday example of steady state flow is when you open the tap water, it
flows with constant flow rate and it is always supported by the water tank.
2- Semi-Steady State Flow (Pseudo ): The dominant flow type,
Key Summary: Pressure transient has reached all of the boundaries and the
static pressure is declining at the boundary and uniformly throughout the
reservoir.
No Flow Boundary which means that the bounday are sealed (sealed fualts
exist). Now, because of the absence of pressure support, when you set the well
at constant flow rate, there is no choice for the reservoir rather than the BHP
and everywhere in the reservoir to decline constantly to support the constant
flow rate.
Thus change of pressure with time is constant (say 5psi/week)
Figure Explanation: each line represents a period (say one month), for example we can see in the
first month (top red curve) the pressure disturbance has travelled only small distance (R1) and from
R1 to (re) the pressure is still Pi, and in the following months we can see the pressure propagation,
R1

3. please note the equal spacing between the curves which indicates the equal pressure decline thus
semi-steady state flow.
3- Unsteady-State Flow (Transient Flow):
Key Summary: Pressure transient migrates outward from the well without
encountering any boundaries.
Transient flow takes place during the early life of a well, when the reservoir
boundaries have not been felt, and the reservoir is said to be infinite-acting.
Important Note:
- Knowledge of the flow type helps in determining the correct
equations that best describe the reservoir dynamics. A very simple
example is the following:
Figure explanation: see how the flow type determines what equation to be used, for instance: note
the red highlighted equations, the semi state adds (1/2) to the equation which if ignored can lead to
serious error.
- It is usually the case that the reservoir will start in unsteady state
flow till the pressure disturbance reaches the boundary in which the
pseudo steady state will take over (see figure below, note the change
in slop).
Here the reservoir
boundaries were felt