### Role of Oil Well Tests and Information in Petroleum Industry

Oil well test analysis is a branch of reservoir engineering. Information obtained from flow and pressure transient tests about in situ reservoir conditions are important to determining the productive capacity of a reservoir. In general, oil well test analysis is conducted to meet the following objectives:

- To evaluate well condition and reservoir characterization
- To obtain reservoir parameters for reservoir description
- To determine whether all the drilled length of oil well is also a producing zone
- To estimate skin factor or drilling- and completion-related damage to an oil well. Based upon the magnitude of the damage, a decision regarding well stimulation can be made.

This chapter deals with the basic equations for flow of liquid through porous media along with solutions of interest for various boundary conditions and reservoir geometry. These solutions are required in the design and interpretation of flow and pressure tests.

## Fundamentals of Reservoir Oil Flow Analysis

#### Basic Fluid Flow Equations in Oil Reservoir

##### Steady-State Flow Equations and Their Practical Applications

The steady-state flow equations are based on the following assumptions:

- Thickness is uniform, and permeability is constant
- Fluid is incompressible
- Flow across any circumference is a constant.

###### Ideal Steady-State Flow Equation - Radial Flow

$q_{o}=ΞΌ_{o}B_{o}ln(r_{e}/r_{w})0.00708k_{imp}h(p_{e}βp_{wf}βΞp_{skin})β$

###### pseudo-steady-state equation - Radial flow

$pβp_{wf}=kh141.2q_{o}ΞΌ_{o}B_{o}β(ln(r_{w}rβ)β0.75)$ in which: $r_{w}=r_{w}e_{βs}$ a. damaged well $s>0$ b. stimulated well $s<0$

###### Time to Reach Pseudo-Steady State

Dimensionless time $t_{D}$, which is used to define various flow regimes, is given as: $t_{D}=ΟΞΌc_{t}r_{w}0.00264ktβ$ where: $k$ = permeability, mD $t$ = time, hr $Ο$ = porosity in fraction $ΞΌ_{o}$ = oil viscosity, cp $c_{ti}$ = initial total compressibility, psi^-1 $r_{w}$ = wellbore radius, ft

## Pressure Drawdown Testing Testing Techniques for Oil Wells

A pressure drawdown test is simply a series of bottom-hole pressure measurements made during a period of flow at a constant production rate. transient flow condition prevails to a value of real time approximately equal to: $tβ0.00264kΟΞΌ_{o}r_{e}β$ Semi-steady-state conditions are established at a time value of: $tβ0.00088kΟΞΌ_{o}cr_{e}β$

### Pressure-Time History for Constant-Rate Drawdown Test

This figure shows the flow history of an oil well and can be classified into three periods for analysis:

- Transient or early flow period is usually used to analyze flow characteristics
- Late transient periods is more completed
- Semi-steady-state flow periods is used in reservoir limit tests.

##### Transient Analysis - Infinite-Acting Reservoir

An ideal constant-rate drawdown test in an infinite-acting reservoir is modeled by the logarithmic approximation to the Ei-function solution: $p_{wf}=p_{i}β141.2khq_{o}ΞΌ_{o}B_{o}β[p_{D}(t_{D})+s]$ Assuming initially the reservoir at initial pressure, Pi, the dimensionless pressure at the well ($r_{D}=1$) is given as: $p_{D}=0.5[ln(t_{D})+0.80907]$ After the wellbore storage effects have diminished and $t_{D}/r_{D}>100$ dimensionless time is given by: $t_{D}=ΟΞΌ_{o}c_{t}r_{w}0.002673ktβ$ Combining and rearranging these equations, we get a familiar form of the pressure drawdown equation: $p_{wf}=p_{i}βkh162.6q_{o}ΞΌ_{o}B_{o}β[gt+g(ΟΞΌ_{o}c_{t}r_{w}kβ)β3.23+0.869s]$ This equation describes a straight line with intercept and slope term together and it may be written as: $p_{wf}=mgt+p_{1hr}$ A plot of flowing bottom-hole pressure data versus the logarithm of flowing time should be a straight line with slope $m$ and intercept $p_{1hr}$. Semilog straight line does appear after wellbore damage and storage effects have diminished. The slope of the semilog straight line may be given by: $m=βkh162.6q_{o}ΞΌ_{o}B_{o}β$ The intercept at $gt=0$, which occurs at $t=1$, is also determined from: $p_{1h}=p_{i}+m[g(ΟΞΌ_{o}B_{o}c_{t}r_{w}kβ)β3.23+0.869s]$ The skin factor is estimated from a rearranging form of the last equation: $s=1.151[mp_{i}βp_{1hr}ββg(ΟΞΌ_{o}c_{t}r_{w}kβ)+3.23]$ The beginning time of the semilog straight line may be estimated from log-log plot of $[g(p_{i}βp_{wf})]$ versus $gt$ (the bellow figure); when the slope of the plot is one cycle in $Ξp$ per cycle in $t$, wellbore storage dominates and test data give no information about the formation. The wellbore storage coefficient may be estimated from the unit-slope straight line using the following equation: $C=24q_{o}B_{o}β.ΞpΞtβ$ where $Ξt$ and $Ξp$ are the calues read from a point on the log-log unit slope straight line.

##### Late Transient Analysis - Bounded (Developed) Reservoirs

Pressure behavior at constant rate in a bounded reservoir can by: $g(p_{wf}βp^β)=g(b_{1})β(Ξ²_{1})t$ From this we see that a plot of $log(p_{wf}βp^β)$ versus t should be linear with slope magnitude: $Ξ²_{1}=ΟΞΌ_{o}cr_{e}0.00168kβ$ and intercept $b_{1}=118.6khq_{o}ΞΌ_{o}B_{o}β$ the plot of $g(p_{wf}βp^β)$ versus $t$ will be linear provided the value of $p^β$ is known. usually it is not. this means that trial and error plot must be made using assumed $p^β$ values. That value which yields the best straight line on the $g(p_{wf}βp^β)$ versus $t$ t is chosen as the correct $p^β$ value. A schematic late transient drawdown analysis plot is shown in the Figure. After determining the correct $p^β$ value, $kh$ can be estimated from the intercept value $b$ by: $kh=b118.6q_{o}ΞΌ_{o}B_{o}β$ The pore volume (drainage volume) of the well: $V_{p}=0.1115Ξ²_{1}b_{1}c_{t}q_{o}B_{o}β$ $r_{e}=[ΟAΓ43,560β]_{0.5}$ The skin factor can be found from $s=0.84[b_{1}pΛββp^ββ]βln(r_{w}r_{e}β)+0.75$ $(Ξp)_{skin}=0.84b_{1}sβ$

## Pressure Buildup Analysis Techniques for Oil Wells

Pressure buildup testing is the most familiar transient well-testing technique, which has been used extensively in the petroleum industry. Basically, the test is conducted by producing a well at constant rate for some time, shutting the well in (usually at the surface), allowing the pressure to build up in the wellbore, and recording the down-hole pressure in the wellbore as a function of time.

### Ideal pressure buildup test

In an ideal situation, we assume that the test is conducted in an infinite acting reservoir in which no boundary effects are felt during the entire flow and later shut-in period. The reservoir is homogeneous and containing in slightly compressible, single-phase fluid with uniform properties so that the Ei function and its logarithmic approximation apply. Hornerβs approximation is applicable.

If a well is shut-in after it has produced at rate $q$ for time $t_{p}$ and bottom-hole pressure $p_{ws}$ is recorded at time $Ξt$, then plot of $p_{ws}$ versus $g((t_{p}+Ξt)/Ξt)$ will give a straight line, which is represented by the following equation: $p_{ws}=p_{i}βkh162.6q_{o}ΞΌ_{o}B_{o}βg[Ξtt_{p}+Ξtβ]$ $m=kh162.6q_{o}ΞΌ_{o}B_{o}β$ $k=mh162.6q_{o}ΞΌ_{o}B_{o}β$ and the skin factor is: $s=1.151[mp_{1hr}βp_{ws(Ξt=0)}ββg(ΟΞΌ_{o}c_{t}r_{w}kβ)+3.23]$