The derivation of the diffusivity equation in radial-cylindrical coordinates will be the last topic in our discussion on individual well performance. It also gives us the opportunity to introduce the topic of material balance, as we will use this concept in the following derivation.

If we perform a mass balance on a thin ring or * Representative Elemental Volume, REV*, in the reservoir as shown in

**Figure 4.07**, then we would have:

**Equation 4.43** simply states that any mass entering the REV at its outer boundary less the mass exiting the REV at its inner boundary must be accumulating in the REV. We can elaborate on the definitions of terms in **Equation 4.43** as:

and,

Where:

- 5.615 is a unit conversion constant, ft
^{3}/bbl - $\rho $ is the liquid density, lb/ft
^{3} - $q$ is the liquid rate, bbl/day
- $r$ is the radial coordinate in a radial-cylindrical coordinate system, ft
- $r\text{'}$ is the radius of the representative elemental volume, REV, ft
- $q$ is time, days
- $\varphi $ is the porosity of the reservoir, fraction
- ${V}_{b}$ REV is the bulk volume of the representative elemental volume, REV, ft
^{3} - $h$ is the reservoir thickness, ft

Substituting **Equation 4.44** through **Equation 4.46** into **Equation 4.43** results in:

or,

Dividing by the term $5.615\text{}r\text{}{d}_{r}\text{}{d}_{t}$ results in:

Now, substituting Darcy’s Law, **Equation 4.05** with $\text{l=-r}$ and without the formation volume factor, B, (we want the flow rate in reservoir bbl/day not STB/day):

If we assume that the permeability, k, and the thickness, h, are uniform, then we have:

or,

or, after applying the chain rule:

Now, using the definition of compressibility for slightly compressible liquids:

Substituting **Equation 4.51** into results **Equation 4.50c** in:

**Equation 4.52** is the nonlinear diffusivity equation. We say that it is * Nonlinear* because the two density terms in the equation are functions of pressure. In this nonlinear form, we cannot solve the equation analytically (exactly). In order to obtain analytical solutions to this equation, we must first

*it. To do this, we apply the chain rule to the left-hand side of*

**Linearize****Equation 4.52**:

or,

Note that the term, ${\left(\frac{\partial p}{\partial r}\right)}^{2}$, is the first derivative squared and not the second derivative, $\frac{{\partial}^{2}p}{\partial {r}^{2}}$. To complete the linearization process, we must assume the pressure gradient, $\frac{\partial p}{\partial r}$, is small. If this is the case, then ${\left(\frac{\partial p}{\partial r}\right)}^{2}$ is very small, and we can remove it from **Equation 4.53b**:

or,

Which we can put into the compact format as:

or,

Where:

- 0.006328 is an equation constant (5.615 x 0.001127)
- $r$ is the radial coordinate in a radial-cylindrical coordinate system, ft
- $p$ is the pressure, psi
- $\varphi $ is the porosity of the reservoir, fraction
- $\mu $ is the liquid viscosity, cp
- $c$ is the liquid compressibility, 1/psi
- $k$ is the reservoir permeability, md
- $t$ is time, days
- $\eta $ is the hydraulic diffusive $\left(\eta =\frac{0.006328k}{\varphi \mu c}\right)$, ft
^{2}/day

**Equation 4.55** is the linear form of the diffusivity equation that describes the transient flow of a slightly compressible liquid through porous media. As we have already shown, solutions to this equation are useful in pressure transient analysis. The solutions to the diffusivity equation also have applications in the oil and gas production in:

(analysis of time-dependent production rates)**Rate Transient Analysis**(analysis of production rates using generalized, dimensionless plots)**Type Curve Analysis**(performance of aquifers in contact with hydrocarbon reservoirs)**Unsteady-State Aquifer Performance**

The name * Diffusivity Equation* comes from the fact that this equation governs the diffusion process (with appropriate changes to the equation parameters and variables to make it relevant for diffusion). In addition, this equation also governs the process of heat conduction in solids, again, with appropriate changes to the equation parameters and variables.