Tarner’s Prediction Method

 Tarner (1944) suggested an iterative technique for predicting cumulative oil produc￾tion Np and cumulative gas production Gp as a function of reservoir pressure. The

method is based on solving the MBE and the instantaneous GOR equation simulta￾neously for a given reservoir pressure drop from a known pressure Pi 1 to an

assumed (new) pressure Pi. It is accordingly assumed that the cumulative oil and gas

production has increased from known values of (Np)i 1 and (Gp)i 1at reservoir

pressure Pi 1 to future values of (Np)i and (Gp)i at the assumed pressure Pi. To

simplify the description of the proposed iterative procedure, the stepwise calculation

is illustrated for a volumetric saturated oil reservoir; however, this method can be

used to predict the volumetric behavior of reservoirs under different driving

mechanisms.

Tarner’s method was preferred to Tracy and Muskat because of the differential

form of expressing each parameter of the material balance equation by Tracy. Also,

Tarner and Muskat method use iterative approach in the prediction until a conver￾gence is reached.

Furthermore, a first approach of the Cumulative Oil Production is needed before

the calculation is performed; a second value of this variable is calculated through the

equation that defines the Cumulative Gas Production, as an average of two different

moments in the production life of the reservoir; this expression, as we will see, is a

function of the Instantaneous Gas Oil Rate, then we need also to calculate this value

in advance from an equation derived from Darcy’s law, this is a very important

relationship since it is strongly affected by the relative permeability ratio between oil

and gas. Finally, both values are compared, if the difference is within certain

predefined tolerance, our first estimate of the Cumulative Oil Production will be

considered essentially right, otherwise the entire process is repeated until the desire

level of accuracy is reached (Tarner 1944).

Tarner’s Prediction Algorithm

Step 1: Select a future reservoir pressure Pi below the initial (current) reservoir

pressure Pi 1 and obtain the necessary PVT data. Assume that the cumulative oil

production has increased from (Np)i 1 to (Np)i. It should be pointed out

that (Np)i 1 and (Gp)i 1 are set equal to zero at the bubble-point pressure

(initial reservoir pressure).

Step 2: Estimate or guess the cumulative oil production (Np)i at Pi.

Step 3: Calculate the cumulative gas production (Gp)i by rearranging the MBE to

give:

Tracy Prediction Method

Tracy (1955) developed a model for reservoir performance prediction that did not

consider oil reservoirs above bubble-point pressure (undersaturated reservoir) but

the computation starts at pressures below or at the bubble-point pressure. To use this

method for predicting future performance, it is pertinent therefore to select future

pressures at desired performance. This means that we need to select the pressure step

to be used. Hence, Tracy’s calculations are performed in series of pressure drops that

proceed from a known reservoir condition at the previous reservoir pressure (Pi 1)

to the new assumed lower pressure (Pi). The calculated results at the new reservoir

pressure becomes “known” at the next assumed lower pressure. The cumulative gas,

oil, and producing gas-oil ratio are calculated at each selected pressure, so the goal is

to determine a table of Np, Gp, and Rp versus future reservoir static pressure.

Tracy’s Prediction Algorithm

Step 1: Select an average reservoir pressure (Pi) of interest

Step 2: Calculate the values of the PVT functions ɸo, ɸg & ɸw where

Schilthuis Prediction Method

Schilthuis develop a method of reservoir performance prediction using the total

produced or instantaneous gas-oil ratio which was defined mathematical as:

Reservoir Performance Prediction

 Introduction

Some of the roles of Reservoir Engineers are to estimate reserve, field development

planning which requires detailed understanding of the reservoir characteristics and

production operations optimization and more importantly; to develop a mathemat￾ical model that will adequately depict the physical processes occurring in the

reservoir such that the outcome of any action can be predicted within reason￾able engineering tolerance of errors. Muskat (1945) stated that one of the functions

of reservoir engineers is to predict the past performance of a reservoir which is still in

the future. Therefore, whether the concept of the engineer is wrong or right, stupid

or clever, honest or dishonest, the reservoir is always right.

We have to bear in mind that reservoirs rarely perform as predicted and as such,

reservoir engineering model has to be updated in line with the production behaviour.

Thus, an accurate prediction of the future 

production rates under various operating

conditions, apply the primary requirement for the oil and gas reservoirs feasibility

evaluation and performance optimization. The conventional method of utilizing

deliverability and material balance equations to predict the production performance

of these reservoirs cannot be utilized often when the complete reservoir data are

lacking.

Reservoir performance prediction is an iterative process. it requires that a con￾vergence criterion must be met after a satisfactory history match is achieved, to be

executed in a short period of time, for a proper optimization of future reservoir

management planning of a field. There are basically four methods of reservoir

performance prediction applying material balance concept and not a numerical

approach where the reservoir is divided into grid blocks. These are:

• Tracy method

• Muskat method

• Tarner method

• Schilthuis method

All the techniques used to predict the future performance of a reservoir are based

on combination of appropriate MBE with the instantaneous GOR using the proper

saturation equation. The calculations are repeated at a series of assumed reservoir

pressure drops. These calculations are usually based on stock-tank barrel of oil-in￾place at the bubble-point pressure. Above the bubble point pressure, the cumulative

oil produced is calculated directly from the material balance equations as presented

in Craft & Hawkins (1991), Dake (1978), Tarek (2010), Cole (1969), Cosse (1993),

Economides et al. (1994) & Hawkins (1955). The MBE for undersaturated reservoir

are expressed below.

11.1.1 For Undersaturated Reservoir (P > Pb) with No Water

Influx

That is above the bubble point; the assumptions made are:

In applying the above methods of prediction for saturated reservoirs, we require

some additional information to match the previous field production data in order to

predict the future. Such relations are the instantaneous gas-oil ratio (GOR), equation

relating the cumulative GOR to the instantaneous GOR and the equation that relates

saturation to cumulative oil produced.

On the contrary, despite the fact that the material balance equation is a tool used

by the reservoir engineers, there are some aspects which were not put into consid￾eration when performing prediction performance. These are:

• The contribution of the individual well’s production rate

• The actual number of wells producing from the reservoir

• The positions of these wells in the reservoir are not considered since it is assume

to be a tank model

• The time it will take to deplete the reservoir to an abandonment pressure

• Does not see faults in the reservoir if there is any and the variation in rock and

fluid properties.

Instantaneous Gas- Oil Ratio

Instantaneous gas-oil ratio at any time, R is defined as the ratio of the standard cubic

feet of gas produced to the stock tank barrel of oil produced at that same instant of

time and reservoir pressure. The gas production comes from solution gas and free

gas in the reservoir which has come out of the solution (Tarek, 2010).

Instantaneous producing GOR is given mathematically as

Muskat’s Prediction Method

In 1945, Muskat developed a method for reservoir performance prediction at any

stage of pressure depletion by expressing the material balance equation for a

depletion-drive reservoir in differential form as derived below.

The oil pore volume (original volume of oil in the reservoir) is given as:

Muskat’s Prediction Algorithm

At any given pressure, Craft et al. (1991) developed the following algorithm for

solving Muskat’s equation:

Step 1: Obtain relative permeability data at corresponding saturation values and then

make a plot of krg/kro versus saturation.

Step 2: Make a plot of fluid properties {Rs, Bo and (1/Bg)} versus pressure and

determine the slope of each plot at selected pressures, i.e., dBo/dp, dRs/dp, and d

(1/Bg)/dp.

Step 3: Calculate the pressure dependent terms X(p), Y(p), and Z(p) that correspond

to the selected pressures in Step 2.

History Matching

 The update of a model to fit the actual performance is known as history matching.

Clearly speaking, developing a model that cannot accurately predict the past perfor￾mance of a reservoir within a reasonable engineering tolerance of error is not a good

tool for predicting the future of the same reservoir. To history match a given field

data with material balance equation, we have to state clearly the known parameters to

match and the unknown parameters to tune to get the field historical production data

with minimum tolerance of error and these parameters are given in Table 10.1.

Besides, one of the paramount roles of a reservoir engineer is to forecast the future

production rates from a specific well or a given reservoir. From history, engineers

have formulated several techniques to estimate hydrocarbon reserves and future

performance. The approaches start from volumetric, material balance, decline

curve analysis techniques to sophisticated reservoir simulators. Whatever approach

taken by the engineers to predict production rates and reservoir performance pre￾dictions whether simple or complex method used relies on the history match.

The general approach by the engineer whose production history is already

available, is to determine the rates for the given period of production. The value

calculated is use to validate the actual rates and if there is an agreement, the rate is

assumed to be correct. Thus, it is then used to predict the future production rates. On

the contrary, if there is no agreement between the calculated and the actual rates, the

calculation is repeated by modifying some of the key parameters. This process of

matching the computed rate with the actual observed rate is called history matching.

It therefore implies that history matching is a process of adjusting key properties

of the reservoir model to fit or match the actual historic data. It helps to identify the

weaknesses in the available data, improves the reservoir description and forms basis

for the future performance predictions. One of these parameters that is vital in history

matching, is the aquifer parameters that are not always known. Hence, modification

of one or several of these parameters to obtain an acceptable match within reasonable

engineering tolerance of error or engineering accuracy is history matching (Donnez

2010). Therefore, to complete this chapter, the following textbooks and articles were

reviewed: Aziz & Settary (1980), Crichlow (1977), Kelkar & Godofredo (2002),

Chavent et al. (1973), Chen et al. (1973), Harris (1975), Hirasaki (1973),

Warner et al. (1979), Watkins et al. (1992).

History Matching Plan

The validity of a model should be approach in two phases: pressure match and

saturation match (oil, gas and water rates). The pressure and saturation phases

matche, follows different pattern depending on purpose (experience of the individual

carrying out the study). The simulation follows the same basic steps for the two

phases. These steps include:

• Gather data

• Prepare analysis tools

• Identify key wells/tank

Interpret reservoir behavior from observed data

• Run model

• Compare model results to observed data

• Adjust models parameters

10.3 Mechanics of History Matching

There are several parameters that are varied either singly or collectively to minimize

the differences between the observed data and those calculated data by the simulator.

Modifications are usually made on the following areas as presented by Crichlow

(1977):

• Rock data modifications (permeability, porosity, thickness & saturations)

• Fluid data modifications (compressibility, PVT data & viscosity)

• Relative permeability data

• Shift in relative permeability curve (shift in critical saturation data)

• Individual well completion data (skin effect & bottom hole flowing pressure)

The two fundamental processes which are controllable in history matching are as

follows:

1. The quantity of fluid in the system at any time and its distribution within the

reservoir, and

2. The movement of fluid within the system under existing potential gradients

(Crichlow 1977).

The manipulation of these two processes enables the engineer to modify any of

the earlier-mentioned parameters which are criteria to history matching. It is man￾datory that these modifications of the data reflect good engineering judgment and be

within reasonable limits of conditions existing in that area. History matching is

actually an act and time consuming. This implies that the total time spent on history

matching depends largely on the expertise of the engineer and his familiarity with the

particular reservoir. Here are some of the key variables to consider when conducting

history matching:

• Porosity (local)

• Water Saturation (Global)

• Permeability (Local)

• Gross Thickness (Local)

• Net Thickness (Local)

• kv/kh Ratio (Global


Local?)

• Transmissibility (x/y/z/) (Local)

• Aquifer Connectivity and Size (Regional)

• Pore Volume (Local)

• Fluid Properties (Global)

• Rock Compressibility (Global)

Relative Permeability (Global -regional with Justification)

• Capillary Pressure (Global -regional with justification)

• Mobile Oil Volume (Global or Local?)

• Datum Pressure (Global)

• Original Fluid Contact (Global)

• Well Inflow Parameters (Local)

10.4 Quantification of the Variables Level of Uncertainty

The following variables are often considered to be determinate (low uncertainty):

• Porosity

• Gross thickness

• Net thickness

• Structure (reservoir top/bottom/extent)

• Fluid properties

• Rock compressibility

• Capillary pressure

• Datum pressure

• Original fluid contact

• Production rates

The following variables are often considered to be indeterminate (high

uncertainty):

• Pore volume

• Permeability

• Transmissibility

• Kv/Kh ratio

• Rel. perm. curves

• Aquifer properties

• Mobile oil volumes

• Well inflow parameters

10.5 Pressure Match

Here are two proposed option for pressure match

Option 1

• Run the model under reservoir voidage control

• Examine the overall pressure levels

• Adjust the pore volume/aquifer properties to match overall pressure

Match the well pressures

• Modify local PVs/aquifers to match overall pressures

• Modify local transmissibility to match pressure gradient

Option 2

• Check/Initialization

• Run simulation model

• Adjust Kx for well which cannot meet target rates

• Adjust pore volume and compressibility to match pressure change with time

• Adjust Kv and Tz to capture vertical pressure gradient

• Adjust Kv and Tz to meet areal pressure

• Adjust Tx and Ty at the faults

• Adjust PI’s to meet production allocations

• Iterate

10.6 Saturation Match

Option 1

• Normally attempted once pressures matched

• Most important parameters are relative permeability curves and permeabilities

• Try to explain the reasons for the deviations and act accordingly

• Changes to relative permeability tables should affect the model globally

• Changes to permeabilities should have some physical justification

• Consider the use of well pseudos

• Assumed layer KH allocations may be incorrect (check PLTs, etc.)

Option 2

• Check/Initialization Model

• Run simulation model

• Check overall model water/gas movement(process physics)

• Adjust relative permeability

• Introduce and adjust well’s relative permeabilities (Krs) to match individual well

performance

• Adjust PI’s to match production allocation

• Add or delete completion layers to account for channeling, leaking plugs

• Iterate

Well PI Match

• Not usually matched until pressures and saturations are matched, unless BHP

affects production rates

• Must be matched before using model in prediction mode

• Match FBHP data by modifying KH, skin or PI directly

10.8 Problems with History Matching

• Non uniqueness of accepted match

• Lack of reliable field data

• Available data may be limited

• Errors in simulator can cause a correct set of parameters to yield incorrect result.

10.9 Review Data Affecting STOIIP

Verify that the value of STOIIP calculated by the model is in line with estimated

values by volumetric calculations and material balance. If the calculated value is too

high/low, this is normally due to errors of the following type:

• High/low porosity values (data entry format error)

• Misplace fluid contacts (gas-oil and/or water-oil)

• Inclusion/exclusion of grid blocks that belong or not to the reservoir model.

• High/low values in the capillary pressure curves.

• Errors in net sand thickness.

10.9.1 Problems and Likely Modifications

• Localised high pressure area and localised low pressure area.

– Remedies:

– Modify k to allow case of flow from high pressure region to low pressure

region

– Reduce oil in high pressure region by changing ϕ or h or So or all of them.

– If rock data are varied, there may be need for redigitizing.

• Generally high pressure in the whole system

Remedy:

• Reduce oil in place by reducing porosity in the whole system.

– Discontinuous pressure distribution

Remedy: increase k to smoothen effect

• Model runs out of fluid

Remedy:

– Increase initial fluid saturation. Fluid contacts may be varied.

• No noticeable drawdown in pressure even after considerable withdrawal.

Remedy:

– Error in compressibility entered.

• Sw increase without any injection or influx of water.

Remedy:

– Increase rock compressibility used.

• Problem with matching GOR, WOR

Remedy:

– Modify relative permeability

If simulated GOR > observed GOR, reduce Krg vale in the simulator. The reverse

is true.

If free gas starts flowing early, increase critical gas sat. The reverse is also

the case.

After everything has been done, observed pressures and production are greater

than the model.

Cause:

• Reservoir getting energy from region not defined for example, fluid influx

Remedy:

• Redefine area and model or include aquifer if observed water cut is increasing.

10.10 Methods of History Matching

The method adopted for matching a field’s historic data depends on the engineer in

question. History matching has been improved from manual turning of some param￾eters to a more sophisticated computer aided tool. Today, some engineers still use

manual turning which work well for them rather than the computer aided history

matching.

10.10.1 Manual History Matching

During manual history matching, changing one or two parameters manually by trial￾and error can be tedious and inconsistent with the geological models. To make the

parameters best fit with the simulated and observed data gives considerable uncer￾tainties and does not have the reliability for a longer period.

10.10.2 Automated History Matching

Automated history matching is much faster and requires fewer simulation runs than

manual history matching. It includes a large number of different parameters and

tackles a large number of wells without problems. In manual history matching, one

or two parameters are varied at a time and it would require preliminary analysis first

for tackling the wells.

Besides, automatic history matching could give more reliable results in the case

of complex lithology conditions with considerable heterogeneity. The basic process

in automatic history matching is to start from an initial parameter guess and then

improve it by integrating field data in an automatic loop. In this case, parameter

changes are done by computer programming to minimize the function to show

differences between simulated and observed data. This is called objective function

that includes both model mismatch and data mismatch parts.

10.10.3 Classification of Automatic History Matching

• Deterministic Algorithm

• Stochastic Algorithm

10.10.3.1 Deterministic Algorithm

Deterministic algorithms use traditional optimization approaches and obtain one

local optimum reservoir model within the number of simulation iteration constraints.

In implementation, the gradient of the objective function is calculated and the

direction of the optimization search is then determined (Liang 2007). The gradient

based algorithms minimize the difference between the observed and simulated

measurements which is called the minimization of the objective function that

considered the following loop:

• To run the flow simulator for the complete history matching period,

• To evaluate the cost function,

• To update the static parameters and go back to the first step.

The following are the list of several algorithms that are commonly used for the

basis of gradient based algorithms (Landa 1979; Liang 2007):

• Gradient based algorithms:

– Steepest Descent

– Gauss-Newton (GN)

– Levenberg-Marquardt

– Singular Value Decomposition

– Particle Swarm Optimization

– Conjugate Gradient

– Quasi-Newton

– Limited Memory Broyden Fletcher Goldfarb Shanno (LBFGS)

– Gradual Deformation

10.10.3.2 Stochastic Algorithm

The stochastic algorithm takes considerable amounts of computational time com￾pared to a deterministic algorithm, but due to the rapid development of computer

memory and computation speed, stochastic algorithms are receiving more and more

attention.

Stochastic algorithms have three main direct advantages:

• The stochastic approach generates a number of equal probable reservoir models

and therefore is more suitable to non-unique history matching problems,

• It is straight-forward to quantify the uncertainty of performance forecasting by

using these equal probable model,

• Stochastic algorithms theoretically reach the global optimum.

The following are list of several algorithms that are commonly used on the basis

of non-gradient based stochastic algorithms (Landa 1979; Liang 2007):

• Non-gradient based algorithms:

– Simulated Annealing

– Genetic Algorithm

– Polytope

– Scatter & Tabu Searches

– Neighborhood

– Kalman Filter

How Do We Improve the Productivity Index?

 This can be done by altering the parameters in the flow equation. Thus, for the well

productivity or inflow performance to be improved, we need to carry out any of the

following:

• Acid stimulation to remove skin

• Increasing the effective permeability around the wellbore

• Reduction in fluid viscosity

• Reduction in the formation volume factor

• Increasing the well penetration

A case study of an improvement to IPR curve of a well

Well k35 result for before and after stimulation

Inflow Performance Relationship

Introduction

Subsurface production of hydrocarbon has to do with the movement of fluid from the

reservoir through the wellbore to the wellhead. This fluid movement is divided into

two as depicted in Fig. 9.1.

The flow of fluids (hydrocarbons) from the reservoir rock to the wellbore is

termed the inflow. The inflow performance represents fluid production behavior of

a well’s flowing pressure and production rate. This differs from one well to another

especially in heterogeneous reservoirs. The Inflow Performance Relationship (IPR)

for a well is the relationship between the flow rate of the well (q), average reservoir

pressure (Pe) and the flowing pressure of the well (Pwf). In single phase flow, this

relationship is a straight line but when gas is moving in the reservoir, at a pressure

below the bubble point, this is not a linear relationship.

A well starts flowing if the flowing pressure exceeds the backpressure that the

producing fluid exerts on the formation as it moves through the production system.

When this condition holds, the well attains its absolute flow potential.

The backpressure or bottomhole pressure has the following components:

• Hydrostatic pressure of the producing fluid column

• Friction pressure caused by fluid movement through the tubing, wellhead and

surface equipment

• Kinetic or potential losses due to diameter restrictions, pipe bends or elevation

changes.

The IPR is often required for estimating well capacity, designing well comple￾tion, designing tubing string, optimizing well production, nodal analysis calcula￾tions, and designing artificial lift.

Factors Affecting IPR

Factors influencing the shape of the IPR are the pressure drop, viscosity, formation

volume factor, skin and relative permeability across the reservoir.

There are several existing empirical correlations developed for IPR. This are:

9.3 Straight Line IPR Model

When the flow rate is plotted against the pressure drop, it gives a straight line from

the origin with slope as the productivity index as shown in the figure below.

Steps for Construction of Straight Line IPR

Step 1: Obtain a stabilize flow test data

Step 2: Determine the well productivity

Step 3: Assume different pressure value to zero in a tabular form

Step 4: Calculate the rate corresponding to the assume pressure

Step 5: Make a plot of rate versus pressure

9.4 Wiggins’s Method IPR Model

Wiggins (1993) developed the following generalized empirical three phase IPR

similar to Vogel’s correlation based on his developed analytical model in 1991:

For Oil

Klins and Majcher IPR Model

Based on Vogel’s work, Klins and Majcher (1992) developed the following IPR that

takes into account the change in bubble-point pressure and reservoir pressure.

Standing’s Method

The model developed by Standing (1970) to predict future inflow performance

relationship of a well as a function of reservoir pressure was an extension of Vogel’s

model (1968).

Vogel’s Method

Undersaturated Oil Reservoir

An undersaturated reservoir is a system whose pressure is greater than the bubble

point pressure of the reservoir fluid. For the fact that the pressure of the reservoir is

greater than the bubble point pressure does not mean that as production increases for

a period of time, the pressure will not go below the bubble point pressure. Hence,

careful evaluation will lead to a right decision and vice versa.

Since the reservoirs are tested regularly, it means that the stabilized test can be

conducted below or above the bubble point pressure. Thus, for:

Case: pressure above bubble point

From stabilized test data point, the productivity index is:

Vogel IPR Model for Saturated Oil Reservoirs

This is a reservoir whose pressure is below the bubble point pressure of the fluid. In

this case, we calculate the maximum oil flow rate from the stabilized test and then

generate the IPR model. Mathematically

Fetkovich’s Model

According to Tarek (2010), the model developed by Fetkovich in 1973 for under￾saturated and saturated region, was an expansion of Muskat and Evinger (1942)

model derived from pseudosteady-state flow equation to observe the IPR nonlinear

flow behavior.

9.8.1 Undersaturated Fetkovich IPR Model

Saturated Fetkovich IPR Model

Cheng Horizontal IPR Model

Cheng (1990) presented a form of Vogel’s equation for horizontal wells that is based

on the results of a numerical simulator. The proposed expression has the following

form

 

Pressure Regimes and Fluid Contacts

Introduction

The main source of energy during primary hydrocarbon recovery is the pressure of

the reservoir. At any given time in the reservoir, the average reservoir pressure is an

indication of how much gas, oil or water is remaining in the porous rock media. This

represents the amount of the driving force available to push the remaining hydro￾carbon out of the reservoir during a production sequence. Most reservoir systems are

identified to be heterogeneous and it is worthy to note that the magnitude and

variation of pressure across the reservoir is a paramount aspect in understanding

the reservoir both in exploration and development (production) phases (Fig. 8.1).

Hydrocarbon reservoirs are discovered at some depths beneath the earth crust as a

result of depositional process and thus, the pore pressure of a fluid is developed

within a rock pore space due to physical, chemical and geologic processes through

time over an area of sediments. There are three identified pressure regimes:

• Normal (relative to sea level and water table level, i.e. hydrostatic)

• Abnormal or overpressure (i.e. higher than hydrostatic)

• Subnormal or underpressure (i.e. lower than hydrostatic)

Fluid pressure regimes in hydrocarbon columns are dictated by the prevailing

water pressure in the vicinity of the reservoir (Bradley 1987). In a perfectly normal

pressure zone, the water pressure at any depth can be calculated as:

Pressure Regime of Different Fluids

Some Causes of Abnormal Pressure

• Incomplete compaction of sediments

Fluids in sediments have not escaped and are still helping to support the overburden.

• Aquifers in Mountainous Regions

Aquifer recharge is at higher elevation than drilling rig location.

• Charged shallow reservoirs due to nearby underground blowout.

• Large structures

• Tectonic movements

Abnormally high pore pressures may result from local and regional tectonics. The

movement of the earth’s crustal plates, faulting, folding, lateral sliding and slipping,

squeezing caused by down dropped of fault blocks, diapiric salt and/or shale

movements, earthquakes, etc. can affect formation pore pressures.

Due to the movement of sedimentary rocks after lithification, changes can occur

in the skeletal rock structure and interstitial fluids. A fault may vertically displace a

fluid bearing layer and either create new conduits for migration of fluids giving rise

to pressure changes or create up-dip barriers giving rise to isolation of fluids and

preservation of the original pressure at the time of tectonic movement.

When crossing faults, it is possible to go from normal pressure to abnormally high

pressure in a short interval. Also, thick, impermeable layers of shale (or salt) restrict

the movement of water. Below such layers abnormal pressure may be found. High

pressure occurs at the upper end of the reservoir and the hydrostatic pressure gradient

is lower in gas or oil than in water.

8.4 Fluid Contacts

In the volumetric estimation of a field’s reserve, the initial location of the fluid

contacts and also for the field development, the current fluid contacts are very critical

factor for adequate evaluation of the hydrocarbon prospect. Typically, the position of

fluid contacts are first determined within control wells and then extrapolated to other

parts of the field. Once initial fluid contact elevations in control wells are determined,

the contacts in other parts of the reservoir can be estimated. Initial fluid contacts

within most reservoirs having a high degree of continuity are almost horizontal, so

the reservoir fluid contact elevations are those of the control wells.

Estimation of the depths of the fluid contacts, gas/water contact (GWC), oil/water

contact (OWC), and gas/oil contact (GOC) can be made by equating the pressures of

the fluids at the said contact. Such that at GOC, the pressure of the gas is equal to the

pressure of the oil and the same concept holds for OWC.

Methods of Determining Initial Fluid Contacts

8.4.1.1 Fluid Sampling Methods

This is a direct measurement of fluid contact such as: Production tests, drill stem

tests, repeat formation tester (RFT) tests (Schlumberger, 1989). These methods have

some limitation which are:

• Rarely closely spaced, so contacts must be interpolated

• Problems with filtrate recovery on DST and RFT

• Coring, degassing, etc. may lead to anomalous recoveries

8.4.1.2 Saturation Estimation from Wireline Logs

It is the estimation of fluid contacts from the changes in fluid saturations or mobility

with depth, it is low cost and accurate in simple lithologies and rapid high resolution

but have limitations as:

• Unreliable in complex lithologies or low resistivity sands

• Saturation must be calibrated to production

8.4.1.3 Estimation from Conventional and Sidewall Cores

Estimates fluid contacts from the changes in fluid saturation with depth which can be

related to petrophysical properties. It can estimates saturation for complex litholo￾gies (Core Laboratories, 2002). The limitations are:

• Usually not continuously cored, so saturation profile is not as complete

• Saturation measurements may not be accuratPressure Methods

There are basically two types of pressure methods: the pressure profiles from repeat

formation tester and pressure profiles from reservoir tests, production tests and drill

stem tests.

8.4.1.5 Pressure Profiles from Repeat Formation Tester

It estimates free water surface from inflections in pressure versus depth curve.

8.4.1.6 Pressure Profiles from Reservoir Tests, Production Tests

and Drill Stem Tests

It estimates free water surface from pressures and fluid density measurements which

makes use of widely available pressure data.

Both pressure techniques are pose with limitations such as:

• Data usually require correction

• Only useful for thick hydrocarbon columns

• Most reliable for gas contacts, Requires many pressure measurements for profile,

Requires accurate pressurese

Estimate the Average Pressure from Several Wells

in a Reservoir

When dealing with oil, the average reservoir pressure is only calculated with material

balance when the reservoir is undersaturated (i.e when the reservoir pressure is

above the bubble point pressure). Average reservoir pressure can be estimated in

two different ways but are not covered in this book (see well test analysis textbooks

for details).

• By measuring the long-term buildup pressure in a bounded reservoir. The buildup

pressure eventually builds up to the average reservoir pressure over a long enough

period of time. Note that this time depends on the reservoir size and permeability

(k) (i.e. hydraulic diffusivity).

• Calculating it from the material balance equation (MBE) is given below

For a gas well

   

Decline Curve Analysis

 Introduction

Globally, the oil and gas production profiles differ considerably. When a field starts

production, it builds up to a plateau state, and every operator will want to remain in

this stage for a very long period of time if possible. But in reality, it is practically not

possible, because, at a point in the life of the field, the production rate will eventually

decline to a point at which it no longer produces profitable amounts of hydrocarbon

as shown in Fig. 7.1. In some fields, the production build-up rate starts in the first few

years, most fields’ profiles have flat top and the length of the flat top depends on

reservoir productivity.

Some fields have long producing lives depending upon the development plan of

the field and reservoir characteristics such as the reservoir, drive mechanism. Wells

in water-drive and gas-cap drive reservoirs often produce at a near constant rate until

the encroaching water or expanding gas cap reaches the well, thereby causing a

sudden decline in oil production. Wells in gas solution drive and oil expansion drive

reservoirs have exponential or hyperbolic declines: rapid declines at first, then

leveling off.

Therefore, decline curve analysis can be defined as a graphical procedure used for

analyzing the rates of declining production and also a means of predicting future oil

well or gas well production based on past production history. Production decline

curve analysis is a traditional means of identifying well production problems and

predicting well performance and life based on measured oil or gas well production.

Today, several computer software have been built to perform this task and prior to

the availability of computers, decline curve analysis was performed by hand on

semi-log plot paper. Several authors (Rodriguez & Cinco-Ley (1993), Mikael

(2009), Duong (1989) have developed new models or approach for production

decline analysis. Agarwal et al. (1998) combined type curve and decline curve

analysis concepts to analyse production data. Doublet et al. (1994), applied the

material balance time for a field using decline curve analysis.

Furthermore, as stated by Thompson and Wright (1985), decline curve is one of

the oldest methods of predicting oil reserves with the following advantages:

• They use data which is easy to obtain

• They are easy to plot

• They yield results on a time basis, and

• They are easy to analyze.

7.2 Application of Decline Curves

• Production decline curve illustrates the amount of oil and gas produced per unit

of time.

• If the factors affecting the rate of production remaining constant, the curve will be

fairly regular, and, if projected, can give the future production of the well with an

assumption that the factors that controlled production in the past will continue to

do so in future.

• The above knowledge is used to ascertain the value of a property and proper

depletion and depreciation charges may be made on the books of the operating

company.

• The analysis of the production decline curve is employed to determine the value

in oil and gas wells economics.

• Identify well production problems

• Decline curves are used to forecast oil and gas production for the reservoir and on

per well basis and field life span.

• Decline curves are also used to predict oil and gas reserves; this can be used as a

control on the volumetric reserves calculated from log analysis results and

geological contouring of field boundaries.

• It is often used to estimate the recovery factor by comparing ultimate recovery

with original oil in place or gas in place calculations

Causes of Production Decline

• Changes in bottom hole pressure (BHP), gas-oil ratio (GOR), water-oil ratio

(WOR), Condition in drilling area

• Changes in Productivity Index (PI)

• Changes in efficiency of vertical & horizontal flow mechanism or changes in

equipment for lifting fluid.

• Loss of wells

7.4 Reservoir Factors that Affect the Decline Rate

• Pressure depletion

• Number of producing wells

• Reservoir drive mechanism

• Reservoir characteristics

• Saturation changes and

• Relative permeability.

7.5 Operating Conditions that Influence the Decline Rate

• Separator pressure

• Tubing size

• Choke setting

• Workovers

• Compression

• Operating hours, and

• Artificial lift.

As long as the above conditions do not change, the trend in decline can be analyzed

and extrapolated to forecast future well performance. If these conditions are altered,

for example; through a well workover, the decline rate determined during

pre-workover will not be applicable to the post-workover period.

7.6 Types of Decline Curves

Arps (1945) proposed that the “curvature” in the production-rate-versus-time curve

can be expressed mathematically by a member of the hyperbolic family of equations.

Arps recognized the following three types of rate-decline behavior:

Exponential decline

• Harmonic decline

• Hyperbolic decline

Arps introduces equations for each type and used the concept of loss-ratio and its

derivative to derive the equations. The three declines have b values ranging from 0 to

1. Where b ¼ 0 represents the exponential decline, 0 < b < 1 represents the

hyperbolic decline, and b ¼ 1 represents the harmonic decline (Fig. 7.2).

The plots of production data such as log(q) versus t; q versus Np; log(q) versus

log(t); Np versus log(q) are used to identify a representative decline model.

7.6.1 Identification of Exponential Decline

If the plot of log(q) versus t OR q versus Np shows a straight line (see figures below)

and in accordance with the respective equations, the decline data follow an expo￾nential decline model.

Mathematical Expressions for the Various Types of Decline

Curves

The three models are related through the following relative decline rate equation

(Arps 1945):

Relationship Between Nominal and Effective Decline Rate

The nominal decline rate (Di) is defined as the negative slope of the curvature

representing the natural logarithm of the production rate versus time

Cumulative Production for Exponential Decline

The Integration of the production rate over time gives an expression for the cumu￾lative oil production as:

Steps for Exponential Decline Curve Analysis

The following steps are taken for exponential decline analysis, for predicting future

flow rates and recoverable reserves (Tarek, 2010):

• Plot flow rate vs. time on a semi-log plot (y-axis is logarithmic) and flow

rate vs. cumulative production on a cartesian (arithmetic coordinate) scale.

• Allowing for the fact that the early time data may not be linear, fit a straight line

through the linear portion of the data, and determine the decline rate “D” from the

slope (b/2.303) of the semi-log plot, or directly from the slope (D) of the rate￾cumulative production plot.

• Extrapolate to q ¼ qt to obtain the recoverable hydrocarbons.

• Extrapolate to any specified time or abandonment rate to obtain a rate forecast and

the cumulative recoverable hydrocarbons to that point in time

7.7.2 Harmonic Decline Rate

Cumulative Production for Harmonic Decline

The expression for the cumulative production for a harmonic decline is obtained by

integration of the production rate. This is given by:

Hyperbolic Decline

The hyperbolic decline model is inferred when 0 < b < 1

Hence the integration of

The Alternative Time Function Model

 The Alternative Time Function Model

Considering the left hand side of the material balance equation

Where

No Water Drive, a Known Gas Cap

Linear Form of Material Balance Equation

Introduction

The material balance equation is a complex equation for calculating the original oil

in place, cumulative water influx and the original size of the gas cap as compared to

the oil zone size. This complexity prompted Havlena and Odeh to express the MBE

in a straight line form. This involves rearranging the MBE into a linear equation. The

straight lines method requires the plotting of a variable group against another

variable group selected, depending on the reservoir drive mechanism and if linear

relationship does not exist, then this deviation suggests that reservoir is not

performing as anticipated and other mechanisms are involved, which were not

accounted for; but once linearity has been achieved, based on matching pressure

and production data, then a mathematical model has been achieved. This technique

of trying to match historic pressure and production rate is referred to as history

matching. Thus, the application of the model to the future enables predictions of the

future reservoir performance. To successfully develop this chapter, several textbooks

and materials such Craft & Hawkins (1991), Dake (1994), Donnez (2010), Havlena

& Odeh AS (1964), Numbere (1998), Pletcher (2002) and Steffensen (1992) were

consulted.

The straight line method was first recognized by Van Everdigen et al. (2013) but

with some reasons, it was never exploited. The straight line method considered the

underground recoverable F, gas cap expansion function Eg, dissolved gas-oil expan￾sion function Eo, connate water and rock contraction function Ef,w as the variable for

plotting by considering the cumulative production at each pressure.

Havlena and Odeh presented the material balance equation in a straight line form.

These are presented below:

Diagnostic Plot

In evaluating the performance of a reservoir, there is need to adequately identify the

type of reservoir in question based on the signature of pressure history or behaviour

and the production trend. Campbell and Dake plots are the vital diagnostic tools

employed to identify the reservoir type. The plots are established based on the

assumption of a volumetric reservoir, and deviation from this behaviour is used to

indicate the reservoir type.

For volumetric reservoirs whose production is mainly by oil and connate water/

rock expansion, the value of STOIIP, N can be calculated at every pressure where

production data is given. Rearranging the material balance equation as shown below.

If a plot of cumulative oil production versus net withdrawal over the fluid

expansions is created with a volumetric reservoir data, then the calculated values

of STOIIP, N on the horizontal axis should be constant at all pressure points. In

practice, this is often not the case either because there is water influx or because there

may be faulty pressure or production readings.

If a gas cap is present, there will be a gas expansion component in the reservoir’s

production. As production continues and the reservoir pressure decreased, the gas

expansion term increases with an increase in the gas formation volume factor. To

balance this, the withdrawal over oil/water/formation expansion term must also

continue to increase. Thus, in the case of gas cap drive, the Dake plot will show a

continual increasing trend.

Similarly, if water drive is present, the withdrawal over oil/water/formation

expansion term must increase to balance the water influx. With a very strong aquifer,

the water influx may continue to increase with time, while a limited or small aquifer

may have an initial increase in water influx to the extent that it eventually decreases.

The Campbell plot is very similar to Dake’s diagnostic tool, with an exception

that it incorporates a gas cap if required. In the Campbell plot, the withdrawal is

plotted against withdrawal over total expansion, while the water influx term is

neglected. If there is no water influx, the data will plot as a horizontal line. If there

is water influx into the reservoir, the withdrawal over total expansion term will

increase proportionally to the water influx over total expansion. The Campbell plot

can be more sensitive to the strength of the aquifer. In this version of the material

balance, using only ET neglects the water and formation compressibility (compac￾tion) term. The Campbell plot is shown below.

The Linear Form of the Material Balance Equation

According to Tarek (2010), the linear form of MBE is presented in six scenarios to

determine either m, N, G or We as follow:

• Undersaturated reservoir without water influx

• Undersaturated reservoir with water influx

• Saturated reservoir without water drive

• Saturated reservoir with water drive

• Gas cap drive reservoir

• Combination drive reservoir

Scenario 1: Undersaturated Reservoir Without Water

Influx

Applying the above assumption, the equation reduces to

Scenario 2: Undersaturated Reservoir with Water Influx

Applying the above assumption, the equation reduces to

Scenario 3: Saturated Reservoir Without Water Influx

Applying the above assumption, the equation reduces to

Scenario 4: Saturated Reservoir with Water Influx

Applying the above assumption, the equation reduces to

Scenario 5: Gas Cap Drive Reservoir

• Finding the STOIIP, N when the gas cap size, m is known

• Finding the gas cap size, m the STOIIP, N and the GIIP, G

Finding the STOIIP, N when the gas cap size, m is known

Scenario 6: Combination Drive Reservoir

Linear Form of Gas Material Balance Equation

Havlena and Odeh also expressed the material balance equation in terms of gas

production, fluid expansion and water influx as