Reservoir Engineering 1 Course (2nd Ed.)

1. PSS RegimeA. Average Reservoir Pressure

B. PSS regime for Radial Flow of SC Fluids

C. Effect of Well Location within the Drainage Area

D. PSS Regime for Radial Flow of C Fluids

2. Skin Concept

3. Using S for Radial Flow in Flow Equations

4. Turbulent Flow

1. SuperpositionA. Multiple Well

B. Multi Rate

C. Reservoir Boundary

2. Productivity Index (PI)

3. Inflow Performance Relationship (IPR)

Flash Back: Solutions to the Radial Diffusivity EquationThe solutions to the radial diffusivity equation

appear to be applicable only for describing the pressure distribution in an infinite reservoir that was caused by a constant production from a single well.

Since real reservoir systems usually have several wells that are operating at varying rates, a more generalized approach is needed to study the fluid flow behavior during the unsteady state flow period.

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Superposition Theorem

The principle of superposition is a powerful concept that can be applied to remove the restrictions that have been imposed on various forms of solution to the transient flow equation.

Mathematically the superposition theorem states that any sum of individual solutions to the diffusivity equation is also a solution to that equation.

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Superposition Concept Applications

Superposition concept can be applied to account for the following effects on the transient flow solution:Effects of multiple wells

Effects of rate change

Effects of the boundary

Effects of pressure change

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Effects of Multiple Wells

Frequently, it is desired to account for the effects of more than one well on the pressure at some point in the reservoir.

The superposition concept states that the total pressure drop at any point in the reservoir is the sum of the pressure changes at that point caused by flow in each of the wells in the reservoir. In other words, we simply superimpose one effect upon

the other.

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Appling Superposition: Effects of Multiple WellsFigure shows three

wells that are producing at different flow rates from an infinite acting reservoir, i.e., unsteady-state flow reservoir. The principle of superposition shows that the total pressure drop observed at any well, e.g., Well 1, is:

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Appling Superposition: Effects of Multiple Wells (Cont.) The pressure drop at Well 1 due to

its own production is given by the log-approximation to the Ei-function solution presented by: (Qo1=oil flow rate from well 1)

The pressure drop at Well 1 due to production at Wells 2 and 3 must be written in terms of the Ei-function solution. The log-approximation cannot be used because we are calculating the pressure at a large distance r from the well, i.e., the argument x > 0.01, or:

It should also be noted that if the point of interest is an operating well, the skin factor s must be included for that well only.

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Effects of Rate Change

All of the mathematical expressions presented previously require that the wells produce at a constant rate during the transient flow periods.

Practically all wells produce at varying rates and, therefore, it is important that we be able to predict the pressure behavior when the rate changes.

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Superposition: Effects of Rate Change

For predicting the pressure behavior when the rate changes, the concept of superposition states:“Every flow rate change in a well will result in a pressure

response which is independent of the pressure responses caused by other previous rate changes.”

Accordingly, the total pressure drop that has occurred at any time is the summation of pressure changes caused separately by each net flow rate change.

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Production and Pressure History of a Multi-Rate WellConsider the

case of a shut-in well, i.e., Q = 0, that was then allowed to produce at a series of constant rates for the different time periods shown in Figure.

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Pressure Drop of Multi-Rate Well

To calculate the total pressure drop at the sand face at time t4, the composite solution is obtained by adding the individual constant-rate solutions at the specified rate-time sequence, or:

The above expression indicates that there are four contributions to the total pressure drop resulting from the four individual flow rates.

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Pressure Drop of Multi-Rate Well: 1st ContributionThe first contribution results from increasing the rate

from 0 to Q1 and is in effect over the entire time period t4, thus:

It is essential to notice the change in the rate, i.e., (new rate − old rate), that is used in the above equation. It is the change in the rate that causes the pressure

disturbance.

Further, it should be noted that the “time” in the equation represents the total elapsed time since the change in the rate has been in effect.

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Pressure Drop of Multi-Rate Well: Other ContributionsSecond contribution results from decreasing the

rate from Q1 to Q2 at t1, thus:

Note, however, the above approach is valid only if the well is flowing under the unsteady-state flow condition for the total time elapsed since the well began to flow at its initial rate.

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Effects of the Boundary

The superposition theorem can also be extended to predict the pressure of a well in a bounded reservoir. Figure, which

shows a well that is located at distance r from the non-flow boundary, e.g., sealing fault.

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Method of Images in Solving Boundary ProblemsThe no-flow boundary can be represented by the

following pressure gradient expression:

Mathematically, the above boundary condition can be met by placing an image well, identical to that of the actual well, on the other side of the fault at exactly distance r.

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Method of Images

Consequently, the effect of the boundary on the pressure behavior of a well would be the same as the effect from an image well located a distance 2r from the actual well.

In accounting for the boundary effects, the superposition method is frequently called the method of images.

Thus, for a well that is located at distance r from the non-flow boundary, the problem reduces to one of determining the effect of the image well on the actual well.

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Method of Images (Cont.)

The total pressure drop at the actual well will be the pressure drop due to its own production plus the additional pressure drop caused by an identical well at a distance of 2r, or:

Notice that this equation assumes the reservoir is infinite except for the indicated boundary.

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Extension of the Image Wells Concept

The effect of boundaries is always to cause greater pressure drop than those calculated for infinite reservoirs.

The concept of image wells can be extended to generate the pressure behavior of a well located within a variety of boundary configurations.

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Effects of Pressure Change

Superposition is also used in applying the constant-pressure case.

Pressure changes are accounted for in this solution in much the same way that rate changes are accounted for in the constant rate case.

The superposition method to account for the pressure-change effect is used in the Water Influx.

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Transient Well Testing

Detailed reservoir information is essential to the petroleum engineer in order to analyze the current behavior and future performance of the reservoir.

Pressure transient testing is designed to provide the engineer with a quantitative analysis of the reservoir properties. A transient test is essentially conducted by creating a

pressure disturbance in the reservoir and recording the pressure response at the wellbore, i.e., bottom-hole flowing pressure pwf, as a function of time.

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Pressure Transient Tests

The pressure transient tests most commonly used in the petroleum industry include:Pressure drawdown

Pressure buildup

Multirate

Interference

Pulse

Drill stem

Fall off

Injectivity

Step rate

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Information Available From a Well Test

It has long been recognized that the pressure behavior of a reservoir following a rate change directly reflects the geometry and flow properties of the reservoir.

Information available from a well test includes:Effective permeabilityFormation damage or stimulationFlow barriers and fluid contactsVolumetric average reservoir pressureDrainage pore volumeDetection, length, capacity of fracturesCommunication between wells

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Well Performance

These lectures presents the practical reservoir engineering equations that are designed to predict the performance of vertical and horizontal wells. Also describe some of the factors that are governing the

flow of fluids from the formation to the wellbore and how these factors may affect the production performance of the well.

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Production Performance Analysis

The analysis of the production performance is essentially based on the following fluid and well characteristics:Fluid PVT properties

Relative permeability data

Inflow-performance-relationship (IPR)

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Productivity Index

A commonly used measure of the ability of the well to produce is the Productivity Index.

Defined by the symbol J, the productivity index is the ratio of the total liquid flow rate to the pressure drawdown.

For a water-free oil production, the productivity index is given by:

Where Qo = oil flow rate,

STB/day

J = productivity index, STB/day/psi

p–r = volumetric average drainage area pressure (static pressure)

pwf = bottom-hole flowing pressure

Δp = drawdown, psi

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Productivity Index Measurement

The productivity index is generally measured during a production test on the well. The well is shut-in until the static reservoir pressure is

reached. The well is then allowed to produce at a constant flow rate of Q

and a stabilized bottom-hole flow pressure of pwf.

Since a stabilized pressure at surface does not necessarily indicate a stabilized pwf, the bottom-hole flowing pressure should be recorded continuously from the time the well is to flow.

The productivity index is then calculated from:

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Productivity Index Conditions

It is important to note that the productivity index is a valid measure of the well productivity potential only if the well is flowing at pseudosteady-state conditions.Therefore, in order to accurately measure the

productivity index of a well, it is essential that the well is allowed to flow at a constant flow rate for a sufficient amount of time to reach the pseudosteady-state.

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Productivity Index during Flow RegimesThe figure

indicates that during the transient flow period, the calculated

values of the productivity index will vary depending upon the time at which the measurements of pwf are made.

Productivity index during flow regimesFall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 35

Productivity Index Calculation

The productivity index can be numerically calculated by recognizing that J must be defined in terms of semisteady-state flow conditions.

Recalling below Equation:

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Application of Productivity Index

Since most of the well life is spent in a flow regime that is approximating the pseudosteady-state, the productivity index is a valuable methodology for predicting the future performance of wells. Further, by monitoring the productivity index during the

life of a well, it is possible to determine if the well has become damaged due to completion, workover, production, injection operations, or mechanical problems. If a measured J has an unexpected decline, one of the indicated

problems should be investigated.

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Specific Productivity Index

A comparison of productivity indices of different wells in the same reservoir should also indicate some of the wells might have experienced unusual difficulties or damage during completion. Since the productivity indices may vary from well to well

because of the variation in thickness of the reservoir, it is helpful to normalize the indices by dividing each by the thickness of the well.

This is defined as the specific productivity index Js, or:

Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 38

Qo vs. Δp Relationship

Assuming that the well’s productivity index is constant:

Where Δp = drawdown, psiJ = productivity index

The Equation indicates that the relationship between Qo and Δp is a straight line passing through the origin with a slope of J as shown in Figure.

Qo vs. Δp relationshipFall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 40

Inflow Performance Relationship

Alternatively, productivity Index Equation can be written as:

The above expression shows that the plot pwf against Qo is a straight line with a slope of (−1/J) as shown schematically in Figure.

This graphical representation of the relationship that exists between the oil flow rate and bottom-hole flowing pressure is called the inflow performance relationship and referred to as IPR.

Qo STB/dayFall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 41

Features of the Straight-Line IPR

Several important features of the straight-line IPR can be seen in Figure:When pwf equals average reservoir pressure, the flow

rate is zero due to the absence of any pressure drawdown.

Maximum rate of flow occurs when pwf is zero. This maximum rate is called absolute open flow and referred to as AOF.

The slope of the straight line equals the reciprocal of the productivity index.

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Absolute Open Flow

Although in practice AOF may not be a condition at which the well can produce, It is a useful definition that has widespread applications

in the petroleum industry (e.g., comparing flow potential of different wells in the field).

The AOF is then calculated by:

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IPR For Below Pb

(Qo=JΔP) suggests that the inflow into a well is directly proportional to the pressure drawdown and the constant of proportionality is the productivity index.

Muskat and Evinger (1942) and Vogel (1968) observed that when the pressure drops below the bubble-point pressure, the IPR deviates from that of the simple straight-line relationship as shown in Figure.

IPR below pbFall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 44

Pressure Dependent Variables Affecting PIRecalling following

Equation:

Treating the term between the two brackets as a constant c, the above equation can be written in the following form:

Above equation reveals that the variables affecting the productivity index are essentially those that are pressure dependent, i.e.:Oil viscosity μoOil formation volume

factor BoRelative permeability to

oil kro

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Schematically Illustration of the Variables as a Function of P

Effect of pressure on Bo, μo, and kro kro/μoBo as a function of pressureFall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 46

Behavior of Pressure Dependent Variables Above the bubble-point pressure pb

The relative oil permeability kro equals unity (kro = 1) and the term (kro/μoBo) is almost constant.

As the pressure declines below pb:The gas is released from solution, which can cause a

large decrease in both kro and (kro/μoBo).

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Effect of Reservoir Pressure on IPR

Figure shows qualitatively the effect of reservoir depletion on the IPR.

Effect of reservoir pressure on IPRFall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 48

Empirical Methods to Predict NL Behavior of IPRSeveral empirical methods are designed to predict

the non-linearity behavior of the IPR for solution gas drive reservoirs. Most of these methods require at least one stabilized

flow test in which Qo and pwf are measured.

All the methods include the following two computational steps:Using the stabilized flow test data, construct the IPR curve at

the current average reservoir pressure p–r.

Predict future inflow performance relationships as to the function of average reservoir pressures.

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Empirical Methods to Generate IPR

The following empirical methods that are designed to generate the current and future inflow performance relationships:Vogel’s Method

Wiggins’ Method

Standing’s Method

Fetkovich’s Method

The Klins-Clark Method

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1. Ahmed, T. (2010). Reservoir engineering handbook (Gulf Professional Publishing). Chapter 6 and 7

1. Generating IPR for Oil WellsA. Vogel’s Method

B. Vogel’s Method (Undersaturated Reservoirs)a. Future IPR Approximation

C. Wiggins’ Method

D. Standing’s Method

E. Fetkovich’s Method