In all of our discussions on well performance, we assumed that * Steady-State Conditions* (time-invariant conditions) were occurring in the reservoir. Steady-state implies that nothing changes in the drainage volume with time or production. This simplification is not appropriate for most real production situations.

**Figure 4.05**shows the more common

*(time-dependent conditions) that occurs in the reservoir.*

**Transient Flow Conditions**In this figure, the early-time pressures (green curves) form a pressure disturbance that over time propagates outward toward the external radius of the drainage volume, ${r}_{e}$. At some point in time, this pressure disturbance reaches the external boundary (bold red curve). This time is referred to as the time to pseudo steady-state, ${t}_{pss}$. Pseudo steady-state is a flow regime which is defined by a uniform pressure drop from one time to the next, $\Delta {p}_{t}$, that is equal everywhere in the drainage volume. This is illustrated in **Figure 4.05** by the blue curves. The solid dual-headed arrows indicate that the pressure drop is the same at each radius in the reservoir.

The transient behavior of a radial-cylindrical drainage volume with uniform (constant) reservoir properties is governed by the partial differential equation (diffusivity equation):

I will derive this equation later in the lesson when we discuss fully transient flow; however, for the time being, we will consider its use in the context of pseudo steady-state flow. Without going into the details, the solution to this equation at times greater than ${t}_{pss}$ can be approximated by:

Where:

- 5.615 is a unit conversion constant, ft
^{3}/bbl - 1.127x10
^{-3}, 141.22, and 0.012648 are equation constants - ${p}_{wf}$ is the flowing well pressure, psi
- ${p}_{i}$ is the initial reservoir pressure, psi
- $\mu $ is viscosity, cp
- $B$ is the formation volume factor, bbl/STB
- $q$ is the liquid rate in STB/day
- $k$ is the permeability, md
- $h$ is the reservoir thickness, ft
- $t$ is the time, days
- $\varphi $ is the porosity of the reservoir, fraction
- ${c}_{t}$ is total compressibility of the system, 1/psi
- ${r}_{e}$ is the external radius of the drainage volume, ft
- ${r}_{w}$ is the well radius, ft

Now, for slightly compressible liquids, we can calculate the average reservoir pressure by using the definition of compressibility:

In **Equation 4.37**, we set $\Delta V$ equal to the volume produced from the well over the time period $t$, $\Delta V\left(f{t}^{3}\right)=5.615\text{}qtB\left(f{t}^{3}\right)$, and $V$ equal to the pore volume of the drainage volume in ft^{3}. Substituting ${p}_{i}$ from **Equation 4.37** into **Equation 4.36** results in:

If we again assume that ${r}_{e}>>{r}_{w}$, then ${r}_{e}{}^{2}-{r}_{w}{}^{2}\approx {r}_{e}{}^{2}$ and the two time-dependent terms, $\frac{1.787\text{}qtB}{{c}_{t}{r}_{e}{}^{2}\varphi h}$, cancel. This results in:

Using the same methodology discussed earlier, we can also include the skin factor, $S$, to account for well damage or stimulation:

We should not be surprised that the time dependent terms in **Equation 4.38** canceled because during the pseudo steady-state flow regime, the pressure drop, $\Delta {p}_{t}$, is uniform everywhere (see **Figure 4.05**). Thus, once the pressure distribution is formed at ${t}_{pss}$ (red curve in **Figure 4.05**), the shape of the curve must remain intact throughout the remainder of the productive life of the well (assuming no changes in the production rate, $q$). The downward shift in the curves shown in **Figure 4.05** are due to the reduction in the average reservoir pressure, $\overline{p}$, during pressure depletion.