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RESERVOIR ENGINEERING

General

British thermal units (BTU)

Structural arrangement of hydrocarbons includes: open chain and cyclic chain

Remaining two categories refer to the bonds between carbon atoms: saturated (ane)

and unsaturated (ene) hydrocarbon

Unsaturated hydrocarbons: double bond between carbon atoms

Ring or cyclic compound is designated by adding the prefix ‘cyclo’

Paraffin Alkane

Olefin Alkene

Cycloalkane

Aromatics

SG of crude oil ranges from 0.75 to 1.01. Density of crude oils also measured in

API units.

𝑨𝑷𝑰 = (𝟏𝟒𝟏. 𝟓

𝜸) − (𝟏𝟑𝟏. 𝟓)

Absolute viscosity (cp) of crude is obtained by multiplying kinematic viscosity

(cS) with the density of material (g/cc)

Dynamic viscosity: Its measure of internal resistance. Dynamic viscosity is

tangential force required per unit area required to move on one horizontal plane

with respect to other plane. Dynamic viscosity can be obtained from Newton’s law:

𝝉 = 𝝁 ∗ (∆𝒗

∆𝒍)

𝑤ℎ𝑒𝑟𝑒 𝜏 𝑖𝑠 𝑠ℎ𝑒𝑎𝑟 𝑠𝑡𝑟𝑒𝑠𝑠 (𝑁

𝑚2) , ∆𝑣 𝑖𝑠 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 (

𝑚

𝑠) , ∆𝑙 𝑖𝑠 𝑢𝑛𝑖𝑡 𝑑𝑖𝑠𝑡𝑛𝑎𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑙𝑎𝑦𝑒𝑟𝑠 (𝑚)

Unit of dynamic viscosity is poise or centipoise (Pa s)

Kinematic viscosity is the ratio of dynamic viscosity to density. It’s a quantity in

which no force is involved.



𝜗 = 𝜇/𝜌

Unit of kinematic viscosity is stoke or centistoke.

Crude oils are classified on the basis of bases:

1. Paraffin based- oil predominantly containing paraffin based series

hydrocarbons

2. Asphalt based- oil predominantly containing polymethylene or olefin series.

3. Mixed based

Fundamental properties of rocks

Porosity: a measure of the space available for storage of hydrocarbons in rock.

∅ =𝑉𝑣𝑜𝑖𝑑

𝑉𝑏𝑢𝑙𝑘

Porosities are classified on the basis of type of origin: primary porosity and

secondary porosity (induced porosity).

Original porosity is that developed in material during deposition and while induced

porosity is due to geologic process subsequent to deposition of rock. Original

porosity is classified by intergranular porosity of sandstones and intercrytalline

and oolitic porosity of some limestone.



Cubic packing porosity:

The unit cell cube is with sides equal to 2r where r is radius of sphere.

Therefore bulk volume:

𝑏𝑢𝑙𝑘 𝑣𝑜𝑙𝑢𝑚𝑒 = (2𝑟)2 = 8𝑟3

Since there are 8 spheres in single unit cell:

𝑠𝑎𝑛𝑑 − 𝑔𝑟𝑎𝑖𝑛 𝑣𝑜𝑙𝑢𝑚𝑒 =4𝜋𝑟3

3

𝑝𝑜𝑟𝑒 𝑣𝑜𝑙𝑢𝑚𝑒

𝑏𝑢𝑙𝑘 𝑣𝑜𝑙𝑢𝑚𝑒=

𝑏𝑢𝑙𝑘 𝑣𝑜𝑙𝑢𝑚𝑒 − 𝑔𝑟𝑎𝑖𝑛 𝑣𝑜𝑙𝑢𝑚𝑒

𝑏𝑢𝑙𝑘 𝑣𝑜𝑙𝑢𝑚𝑒=

8𝑟3 − (4𝜋𝑟3

3)

8𝑟3



Rhombohedral packing:

Total porosity is ratio of total void space in rock to bulk volume of rock. Effective

porosity is ratio of interconnected void space in rock to bulk volume of rock.

Porosity measurements: bulk volume of the sample can be measured from

measuring dimensions of the sample.

Porosity by saturation method -

a. Weight of dry sample in air

b. Weight of sample when saturated with water

c. Weight of water is difference between b-a

d. Weight of water/density of water = volume of water (effective pore volume)

e. Bulk volume from dimensions

f. Effective porosity = (pore volume/bulk volume)

Porosity averaging technique:

Arithmetic Average ∅ =

∅1 + ∅2 + ∅3 + ⋯

𝑛

Thickness weighted average ∅ =

∅1 ∗ ℎ1 + ∅2 ∗ ℎ2 + ∅3 ∗ ℎ3 + ⋯

ℎ1 + ℎ2 + ℎ3 + ⋯

Area weighted average ∅ =

∅1 ∗ 𝐴1 + ∅2 ∗ 𝐴2 + ∅3 ∗ 𝐴3 + ⋯

𝐴1 + 𝐴2 + 𝐴3 + ⋯

Volumetric weighted average ∅ =

∅1 ∗ 𝐴1 ∗ ℎ1 + ∅2 ∗ 𝐴2 ∗ ℎ2 + ⋯

𝐴1 ∗ ℎ1 + 𝐴2 ∗ ℎ2 + ⋯

Permeability: fluid conductance capacity of formation is permeability.

Permeability is property of porous medium and is measure as capacity of medium

to transmit fluids.



Poiseuille’s equation for viscous flow:

𝑣 =𝑑2 ∗ ∆𝑃

32 ∗ 𝜇 ∗ 𝐿

Fanning’s equation for viscous and turbulent flow:

𝑣2 =2 ∗ 𝑑 ∗ ∆𝑃

𝑓 ∗ 𝜌 ∗ 𝐿

Where

V= velocity cm/sec

d = diameter of conduit, cm

∆𝑃 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑙𝑜𝑠𝑠 𝑜𝑣𝑒𝑟 𝑙𝑒𝑛𝑔𝑡ℎ 𝐿,𝑑𝑦𝑛𝑒

𝑐𝑚2

𝜇 = 𝑣𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦, 𝑐𝑒𝑛𝑡𝑖𝑝𝑜𝑖𝑠𝑒

𝑓 = 𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛 𝑓𝑎𝑐𝑡𝑜𝑟, 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑙𝑒𝑠𝑠

𝜌 = 𝑑𝑒𝑛𝑠𝑖𝑡𝑦,𝑔𝑚

𝑐𝑐

Darcy law only holds good for viscous flows.

Weighted average permeability:

Linear flow with parallel combination of beds:



𝑄𝑡 = 𝑄1 + 𝑄2 + 𝑄3

ℎ𝑡 = ℎ1 + ℎ2 + ℎ3

𝑄1 = (𝑘1 ∗ 𝑤 ∗ ℎ1

𝜇) (

∆𝑃

∆𝐿)

𝑄2 = (𝑘2 ∗ 𝑤 ∗ ℎ2

𝜇) (

∆𝑃

∆𝐿)

𝑄3 = (𝑘3 ∗ 𝑤 ∗ ℎ3

𝜇) (

∆𝑃

∆𝐿)

𝑄𝑡 = (𝑤

𝜇) ∗ (

∆𝑃

∆𝐿) ∗ (𝑘1ℎ1 + 𝑘2ℎ2 + 𝑘3ℎ3)

𝐾𝑎𝑣𝑒 ∗ ℎ𝑡 = 𝑘1ℎ1 + 𝑘2ℎ2 + 𝑘3ℎ3

𝑲𝒂𝒗𝒆 =𝒌𝟏𝒉𝟏 + 𝒌𝟐𝒉𝟐 + 𝒌𝟑𝒉𝟑

𝒉𝟏 + 𝒉𝟐 + 𝒉𝟑

Permeability of parallel layered system is given by above equation.

Harmonic average permeability:



Permeability variations occurring laterally in reservoir flow through series

combination of beds.

In this case, flow rate across the beds will be same. The change will be in the

pressure drop across the beds.

∆𝑃𝑡𝑜𝑡𝑎𝑙 = ∆𝑃1 + ∆𝑃2 + ∆𝑃3

(𝑄𝜇𝐿

𝐴𝐾𝑎𝑣𝑔) = (

𝑄𝜇𝐿1

𝐴𝐾1) + (

𝑄𝜇𝐿2

𝐴𝐾2) + (

𝑄𝜇𝐿3

𝐴𝐾3)

(𝐿

𝐾𝑎𝑣𝑔) = (

𝐿1

𝐾1) + (

𝐿2

𝐾2) + (

𝐿3

𝐾3)

𝑲𝒂𝒗𝒈 =𝑳

(𝑳𝟏

𝑲𝟏) + (

𝑳𝟐

𝑲𝟐) + (

𝑳𝟑

𝑲𝟑)

Permeability series layered system given by above equations.

Geometric average permeability:

𝑲𝒂𝒗𝒈 = 𝒆𝒙𝒑 (𝒉𝟏 ∗ 𝒍𝒏(𝒌𝟏) + 𝒉𝟐 ∗ 𝒍𝒏(𝒌𝟐) + ⋯

𝒉𝟏 + 𝒉𝟐 + ⋯)

For heterogeneous formation, geometric average permeability is widely used.

Plugs cut parallel to bedding planes gives horizontal permeability while plugs

cut perpendicular to bedding plane gives vertical permeability.

Multiphase permeability:

Permeability measured with 100 % saturation of a single phase is called the

absolute permeability of rock.

Effective permeability is permeability of phase when core is saturated with

two or more phases. Sum of effective permeability is always less than or equal

to absolute permeability.

Relative permeability is ratio of effective permeability to given fluid at definite

saturation to a given fluid at 100 % (absolute permeability) saturation.



Klinkenberg effect:

Liquids has zero velocity on grain surface, while gases has some finite velocity

at the sand grain surface i.e. gases exhibits slippage at the surface. This

slippage resulted n higher flow rate for the gas at given pressure differential.

𝑲𝒈𝒂𝒔 = 𝑲𝒍𝒊𝒒𝒖𝒊𝒅 + 𝑪 ∗ (𝟏

𝑷𝒎)

Porosity-permeability relationship:

Timur equation:

𝑲 = 𝟖. 𝟓𝟖𝟏𝟎𝟐 ∗ (∅𝟒.𝟒

𝑺𝟐𝒘𝒄)

Morris-Biggs equation:

𝑲 = 𝟔𝟐. 𝟓 ∗ (∅𝟑

𝑺𝒘𝒄)

𝟐

− 𝒇𝒐𝒓 𝒐𝒊𝒍 𝒓𝒆𝒔𝒆𝒓𝒗𝒐𝒊𝒓

𝑲 = 𝟐. 𝟓 ∗ (∅𝟑

𝑺𝒘𝒄)

𝟐

− 𝒇𝒐𝒓𝒈𝒂𝒔 𝒓𝒆𝒔𝒆𝒓𝒗𝒐𝒊𝒓



𝒌 =∅ ∗ 𝒓𝟐

𝟖

Permeability determination in laboratory:

Core plug is de-saturated completely before testing. Gas Is flowed through the

core at known differential pressure and flow rate.

Values are plotted as per given in above graph. From Darcy law, then

permeability of core is found out.

Relation between Poiseuille’s equation and Darcy law:

Flow through channels:

Poiseuille’s equation:

𝑸 =𝝅𝒓𝟒 ∗ ∆𝑷

𝒇 ∗ 𝝆 ∗ ∆𝑳 𝒐𝒓

𝝅𝒓𝟒 ∗ ∆𝑷

𝟖𝝁∆𝑳

Darcy law:

𝑸 =𝑲 ∗ 𝝅𝒓𝟐 ∗ ∆𝑷

𝝁 ∗ ∆𝑳



Equating both the equations:

𝑲 ∗ 𝝅𝒓𝟐 ∗ ∆𝑷

𝝁 ∗ ∆𝑳=

𝝅𝒓𝟒 ∗ ∆𝑷

𝟖𝝁∆𝑳

𝑲 =𝒓𝟐

𝟖

Flow through fractures:

∆𝑷 =𝟏𝟐 ∗ 𝝁 ∗ 𝒗 ∗ 𝑳

𝒉𝟐

Comparing equation with Darcy law, then

𝒌 =𝒉𝟐

𝟏𝟐

Use of capillary tubes for flow network:

If conductors are of same size and are arrange in cubic arrangement, then

neglecting wall thickness of tubes, the number of tubes per unit area,

𝒑𝒐𝒓𝒐𝒔𝒊𝒕𝒚

𝒂𝒓𝒆𝒂=

𝝅

𝟒

𝝅𝒓𝟐

Then flow rate can be given by,

𝑸 = 𝒏 ∗𝝅𝒓𝟒 ∗ ∆𝑷

𝟖𝝁∆𝑳

𝑸 = (𝟏

𝟒𝒓𝟐) ∗

𝝅𝒓𝟒 ∗ ∆𝑷

𝟖𝝁∆𝑳

Comparing equation with Darcy law then,

𝑲 = (𝝅𝒓𝟐

𝟑𝟐)

Kozeny karman equation:

𝒌 = ( ∅

𝑲𝒛 ∗ 𝑺𝟐𝒑)

Where,



∅ 𝒊𝒔 𝒑𝒐𝒓𝒐𝒔𝒊𝒕𝒚, 𝑲𝒛 𝒊𝒔 𝒌𝒐𝒛𝒆𝒏𝒚 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕 & 𝑆𝑝 𝑖𝑠 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑝𝑜𝑟𝑒 𝑣𝑜𝑙𝑢𝑚𝑒

𝝉 = (𝑳𝒂

𝑳) = 𝑻𝒐𝒓𝒕𝒖𝒐𝒔𝒊𝒕𝒚

𝑲𝒛 = 𝑲𝒐 ∗ 𝝉 = 𝒌𝒐𝒛𝒆𝒏𝒚 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕

Fluid saturations:

Methods to estimate fluid saturation:

a. Retort method- core is placed in retort chamber. Trapped fluid in core

is then carried away by toluene vapors. Condensation of vapors gives

volume of hydrocarbons present in core.

𝑺𝒘 =𝒘𝒂𝒕𝒆𝒓 𝒓𝒆𝒄𝒗𝒆𝒓𝒆𝒅

𝒑𝒐𝒓𝒆 𝒗𝒐𝒍𝒖𝒎𝒆

𝑺𝒐 =𝑶𝒊𝒍 𝒓𝒆𝒄𝒗𝒆𝒓𝒆𝒅

𝒑𝒐𝒓𝒆 𝒗𝒐𝒍𝒖𝒎𝒆

𝑺𝒈 = 𝟏 − 𝑺𝒘 − 𝑺𝒐

In leaching with toluene gives indirect oil saturation calculation,

𝑺𝒐 =𝑾𝒆𝒕 𝒄𝒐𝒓𝒆 𝒘𝒕 (𝒈𝒎). −𝒅𝒓𝒚 𝒄𝒐𝒓𝒆 𝒘𝒕(𝒈𝒎). −𝒘𝒕. 𝒐𝒇𝒘𝒂𝒕𝒆𝒓 𝒄𝒐𝒍𝒍𝒆𝒄𝒕𝒆𝒅(𝒈𝒎)

𝒑𝒐𝒓𝒆 𝒗𝒐𝒍𝒖𝒎𝒆 (𝒄𝒄) ∗ 𝒅𝒆𝒏𝒔𝒊𝒕𝒚 𝒐𝒇 𝒐𝒊𝒍(𝒈𝒎

𝒄𝒄)

b. Centrifuge

A solvent is injected into the centrifuge. Solvent is being forced to pass

through the core sample, removes water and oil from the core.

Water saturations obtained from core samples cut with oil-based are the

reliable. Core-samples cut with water based mud are used to determine

original gas-oil contact, original oil-water contact & whether sand is

productive of oil or gas. Contacts can be determined by careful study the

residual oil saturation of cores as function of depth.



Electrical conductivity of fluid saturated rocks:

Resistivity of material

𝝆 =𝒓 ∗ 𝑨

𝑳

𝝆 𝒊𝒔 𝒓𝒆𝒔𝒊𝒔𝒕𝒊𝒗𝒊𝒕𝒚, 𝒓 𝒊𝒔 𝒓𝒆𝒔𝒊𝒔𝒕𝒂𝒏𝒄𝒆, 𝑨 𝒊𝒔 𝒂𝒓𝒆𝒂, 𝑳 𝒊𝒔 𝒍𝒆𝒏𝒈𝒕𝒉

Resistivity in rocks measured in terms of ohm-meter.

Formation factor:

𝑭 =𝑹𝒐

𝑹𝒘

Resistivity factor is ratio of resistivity of rock when saturated with water to

the resistivity of water.

Resistivity index is ratio of resistivity of rock partially saturated water to

resistivity of 100 % saturated water.

𝑰 =𝑹𝒕

𝑹𝒐

Properties of material with porous media containing multiple fluid

saturations:

Surface forces and capillary pressure:

Interfacial tension is defined as work necessary to create a new surface.

Interfacial tension is the force per unit length required to create a new

surface.



Contact angle is measured through denser phase and ranges from zero degree

to 180 degree.

𝝈𝒔𝒘 + 𝝈𝒘𝒐. 𝒄𝒐𝒔𝜽 = 𝝈𝒔𝒐

𝒄𝒐𝒔𝜽 =𝝈𝒔𝒐 − 𝝈𝒔𝒘

𝝈𝒘𝒐

Interfacial tension between solid and denser phase, solid and lighter phase &

denser and lighter fluid.

Magnitude of adhesion tension can be estimated by above given formula. If

adhesion tension value is large or the contact angle 𝜽 is small then, denser

phase will spread and tend to coat the surface.

Rise of fluids in capillaries: the rise in height is due to attractive forces

between the tube and liquid. The adhesion tension is force tending to pull

liquid up the wall of the tube. Liquid will be balanced by the total force acting

to pull liquid upward.

𝑨𝒓 ∗ 𝟐𝝅𝒓 = 𝒇𝒐𝒓𝒄𝒆 𝒖𝒑

The weight of fluid being supported by,

𝝅𝒓𝟐 ∗ 𝒉 ∗ 𝝆 ∗ 𝒈 = 𝒇𝒐𝒓𝒄𝒆 𝒅𝒐𝒘𝒏𝒘𝒂𝒓𝒅

𝑨𝒕 = 𝒂𝒅𝒉𝒆𝒔𝒊𝒐𝒏 𝒕𝒆𝒔𝒏𝒊𝒐𝒏,𝒅𝒚𝒏𝒆

𝒄𝒎



𝒓 = 𝒓𝒂𝒅𝒊𝒖𝒔 𝒐𝒇 𝒕𝒖𝒃𝒆, 𝒄𝒎

𝒉 𝒊𝒔 𝒉𝒊𝒈𝒉𝒕 𝒐𝒇 𝒄𝒐𝒍𝒖𝒎𝒏, 𝒄𝒎

𝝆 𝒊𝒔 𝒍𝒊𝒒𝒖𝒊𝒅 𝒅𝒆𝒏𝒔𝒊𝒕𝒚 𝒊𝒏𝒈𝒎

𝒄𝒄

Equalizing forces,

𝑨𝒓 =𝒓 ∗ 𝒉 ∗ 𝒈 ∗ 𝝆

𝟐

Ar can be written as 𝝈𝒈𝒘. 𝒄𝒐𝒔𝜽 (interfacial tension between gas and water)

Distribution of fluid phase within porous system is described as funicular or

pendular. In funicular distribution, the wetting phase is continuous,

completely covering surface of solid. The pendular ring is state of saturation

in which wetting phase is non-continuous and non-wetting phase is in contact

with some solid surface.



Saturation:

Movable oil saturation

𝑺𝒐𝒎 = 𝟏 − 𝑺𝒘𝒄 − 𝑺𝒐𝒄

Capillary pressure:

Capillary pressure is difference between pressure of non-wetting phase and

wetting-phase.

𝑷𝒄 = 𝑷𝒏𝒘 − 𝑷𝒘

It is difficult to use equations use for capillary tube to determine capillary

pressure in reservoir due to its complex nature of pores in reservoir. Capillary

pressure also viewed as necessary pressure to force non-wetting fluid to

displace wetting fluid in a capillary.

Capillary pressure curve as a function of water saturation for oil-water

system is given below in the diagram.



The imbibitions curve present lower capillary pressure for fixed saturation

than drainage curve because natural tendency of wetting fluid to saturate

rock.

𝑭𝑾𝑳 = 𝑾𝑶𝑪 + (𝑷𝒄 ∗𝟏𝟒𝟒

∆𝝆)



From the capillary rise equation,

𝒉 =𝟐𝝈𝒄𝒐𝒔∅

𝒓𝒈∆𝝆

As difference between density increases, the height of transition zone

decreases. In case of gas reservoir having a gas-water contact, the thickness of

transition zone will be shorter while for low API gravity oil, it will be longer.

Reserves estimation:

Petroleum resources classification and definitions: `



Reserves- Those quantities of petroleum which are anticipated to be

commercially recovered from known accumulations from a give date forward.

Contingent Resource- those quantities of petroleum which are estimated, on a

given date, to be potentially recoverable from known accumulations, but

which are not currently considered to be commercially recoverable

Prospective Resource- Those quantities of petroleum which are estimated, on

a given date, to be potentially recoverable from undiscovered accumulations

Accumulation is used to identify an individual body of movable petroleum in a

reservoir.

Resource uncertainty in case of reserves is given as range. It is function of

primary three categories i.e. proved, probable and possible.

1P Proved

2P Proved + Probable

3P Proved + Probable + Possible

For other resource categories, the equivalent terms low estimate, best estimate

and high estimate are recommended.

Prospective resources

Prospect Potential accumulation sufficiently well-defined to represent

viable drilling target.

Lead Potentially accumulation is currently poorly defined and

requires more data acquisition and evaluation in order to

classify as prospect.

Play Recognized prospective trend of potential prospects, but which

requires more data acquisition to define leads or prospects.



The entire resource based is generally accepted to be all those estimated

quantities of petroleum contained in sub-surface, as well as quantity

produced.

Total petroleum in place is sub-divided into ‘Discovered’ and ‘Un-discovered’

petroleum initial I place. Discovered petroleum may be classified as

‘commercial’ or ‘sub-commercial’.

Reserves must satisfy four criteria: discovered, recoverable, commercial &

remaining

Contingent resources are those discovered, potentially recoverable but don’t

satisfy criteria of commerciality.

Prospective resources are those potential recoverable quantities in

accumulation yet to be discovered.



Converting gas volume to oil equivalent is customarily done the design basis

of heating content or calorific value of the fuel.

Probabilistic estimation procedures:

When probabilistic method are used there shall be at least 90 % probability

(P90) that the quantity actually recovered will exceed or level the estimate

quoted. There shall be at least 50 % probability (P50) that quantities actually

recovered will exceed sum of 2P reserves. Likewise, there shall be at least a

10% probability that 3P reserves will exceed or equaled.

P90 The quantity for which there is certain probability that the

quantities actually recovered will equal or exceed the estimate.

P50 The quantity for which there is 50 % probability that

quantities will actually recovered will equal or exceed the

estimate.

P10 The quantity for which there is 10 % probability that

quantities actually recovered will equal or exceed the estimate.

In applying probabilistic reserves estimation methods, SPE defines proved

reserves as having at least 90 % (P90) probability of exceeding the quoted

value. Proved reserves give conservative estimate hence its use by investors or

bankers.

Estimation methods:

Analogy methods:

Use to estimate ultimate recovery of oil and gas for undrilled locations.

𝑭𝒓𝒔 = 𝑭𝒓𝒂 ∗(∅ ∗ 𝑺𝒉𝒊)𝒔

(∅ ∗ 𝑺𝒉𝒊)𝒂

Here a stands for analogous reservoir while s stands for subject reservoir. Fr

is recovery factor.

Volumetric methods:

𝑵 =𝟕𝟕𝟓𝟖 ∗ ∅ ∗ 𝑨 ∗ 𝒉 ∗ (𝟏 − 𝑺𝒘𝒊)

𝑩𝒐∗ 𝑬𝒓

Most volumetric methods begin with determining bulk reservoir volume that

contains hydrocarbon. This involves preparation of structure maps of top and

bottom of the reservoir and iso-patch map of the reservoir.



Estimating net pay is important steps in volumetric mapping. The net pay

should be determined on the basis of maximum shaliness, minimum porosity,

and maximum water saturation, specified reversal of gamma ray curve or SP

curve or combination of both.

For reserves to be proved, there should be at least 90 % probability that the

quantities actually recovered will equal or exceed the estimate. For probable

reserves, there should be at least 50 % probability that the quantity actually

recovered will equal or exceed the sum of estimated proved plus probable

reserves. For possible reserves, there at least should a 10 % probability that

quantities actually recovered will equal or exceed the sum of estimated

proved, probable and possible reserves.



Decline trend analysis: estimating the reserves on the basis of a reasonable

well defined behavior of a performance characteristics as function of time of

cumulative production.

When flow boundary reaches actual reservoir boundary or meets with flow

boundary of another well, the reservoir pressure begins to decline and well

enters boundary dominated flow period. It is period that traditional decline

curve analysis can be used.



Decline theory:

Fractional change per unit time is given as,

𝒂 = −

∆𝒒

𝒒

∆𝒕

𝒂 = − (𝟏

𝒒) (

∆𝒒

∆𝒕)

Another way to define is,

𝒂 = 𝑲. 𝒒𝒃

Exponential b = 0

𝒒 = 𝒒𝒊 ∗ 𝒆−𝒂∗𝒕

Cumulative production:

𝑸 = ∫ 𝒒. 𝒅𝒕

𝒕

𝟎

𝑸 = ∫ 𝒒𝒊 ∗ 𝒆−𝒂∗𝒕. 𝒅𝒕

𝒕

𝟎

𝑸 =𝒒𝒊 − 𝒆−𝒂∗𝒕

𝒂

𝑸 =𝒒𝒊 − 𝒒

𝒂

Hyperbolic b is equal to any number between

zero and one

𝒂 ⍺ 𝒒𝒃

𝒒 =𝒒𝒊

(𝟏 + 𝒃. 𝒂𝒊. 𝒕)𝟏

𝒃




Harmonic b = 1

𝒂 = 𝑲. 𝒒𝟏

𝑲 =𝒂

𝒒

𝒒 =𝒒𝒊

𝟏 + 𝒂𝒊. 𝒕


𝑸 = ∫ 𝒒. 𝒅𝒕

𝒕

𝟎

𝑸 = ∫𝒒𝒊

𝟏 + 𝒂𝒊. 𝒕. 𝒅𝒕

𝒕

𝟎

𝑸 = (𝒒𝒊

𝒂𝒊) 𝒍𝒏 (

𝒒𝒊

𝒒)

Material balance: here pressure behavior in reservoir in response to fluid

withdrawal is analyzed in several steps. The fluid properties and pressure

history are averaged, treating reservoir as tank.



Fundamentals of reservoir fluid flow:

Flow regimes

(http://fekete.com/SAN/TheoryAndEquations/WellTestTheoryEquations/inde

x.htm):

a. Wellbore storage (after flow)

When producing well is shut in at the surface, flow into wellbore at sand

face continues after shut in. wellbore storage is typically controlled by

compressibility of fluid.

http://fekete.com/SAN/TheoryAndEquations/WellTestTheoryEquations/index.htm

http://fekete.com/SAN/TheoryAndEquations/WellTestTheoryEquations/index.htm



b. Linear fracture flow

Linear fracture flow occurs in hydraulically fractured wells when

conductivity of fracture is infinite. Permeability of fracture is so high

that the pressure throughout fracture is constant.

c. Bilinear fracture flow

Bilinear fracture flow occurs in hydraulically fractured wells when the

conductivity of fracture is finite. Flow one from matrix to fracture and

fracture to wellbore.



d. Spherical flow

Spherical flow occurs when vertical well is partially penetrated or

during RFT/MDT/WFT tests.

e. Radial flow

Radial flow is in the horizontal radial direction. This type of flow

regime exists before the pressure transient reaching the boundaries of

reservoir (infinite acting time period).



g. Single no-flow boundary

Single no-flow boundary occurs during transition region when well is

located near a single no-flow boundary. A no-flow boundary can be

physical entity, such as a sealing fault, or can occur when two producing

wells are adjacent wells.

Mathematically, a situation in which a well is next to a sealing fault can

be modeled by removing the fault and placing an image well with a flow

rate equivalent to producing well as shown in diagram below:

When single no-flow boundary flow is present, slope doubling can be seen in

derivative plots.



h. Linear channel flow

Linear channel flow occurs in long, narrow reservoir. Initially the

radius of investigation hasn’t reached reservoir boundaries and radial

flow is observed. After the two parallel boundaries have been reached, a

period of linear flow can be observed.



i. Pseudo steady state flow

Pseudo steady state flow occurs during late time region when the outer

boundaries of the reservoir are all no flow boundaries. During PSS flow

regime, the reservoir behaves as a tank. PSS flow does not occur during

build-up or falloff test.

j. Steady state flow

Steady state flow occurs during the late time region when constant

pressure boundary exists. Constant pressure boundaries arise when the

reservoir has aquifer support or gas cap expansion support.

Horizontal flow regimes:

a. Wellbore storage

b. Vertical radial flow



Vertical radial flow occurs only in horizontal wells.

c. Linear horizontal

After the radius of investigation has reached the top and bottom of

the formation, fluid travels from the formation perpendicular to

length of the wellbore.



d. Elliptical

Elliptical flow occurs when fluid has started to flow from the

reservoir at either end of the horizontal wellbore.

e. Horizontal radial

Horizontal radial occurs during middle time region after radius of

investigation has expanded beyond the length of wellbore.



k. Linear channel

l. PSS

m. Steady state

Radius of investigation:

Radius of investigation is how far transient has travelled into the reservoir.

𝒓𝒊 = √𝒕 ∗ 𝒌

𝟗𝟒𝟖 ∗ ∅ ∗ 𝝁 ∗ 𝒄𝒕

Radius of investigation is only the function of reservoir properties and does

not depend on flow rate. Increasing flow rate will increases draw-down but

radius of investigation will be same.

Estimating drainage area:

𝑨 = (𝝅 ∗ 𝒓𝟐)

Estimate shut-in duration:

Time to stabilization: when pressure no longer changes with the time.

𝒕𝒔𝒕𝒂𝒃 = 𝒅𝒊𝒔𝒕𝒂𝒏𝒄𝒆 𝒕𝒐 𝒇𝒂𝒓𝒕𝒆𝒔𝒕 𝒃𝒐𝒖𝒏𝒅𝒂𝒓𝒚𝟐 ∗ (∅ ∗ 𝝁 ∗ 𝒄𝒕

𝒌)

Unsteady state flow:



Second order partial differential equation has total three conditions, two

boundary conditions and one initial condition.

𝑩𝒐𝒖𝒏𝒅𝒂𝒓𝒚 𝒄𝒐𝒏𝒅𝒊𝒕𝒊𝒐𝒏𝒔: 𝒖(𝟎, 𝒕) = 𝟎, 𝒖(𝑳, 𝒕) = 𝟎

𝒊𝒏𝒊𝒕𝒊𝒂𝒍 𝒄𝒐𝒏𝒅𝒊𝒕𝒊𝒐𝒏𝒔: 𝒖(𝒙, 𝟎) = 𝒇(𝒙)

Boundary conditions: formation produces at constant rate into wellbore and

there is no flow across the outer boundary and reservoir behaves as if it’s

infinite in size.

Initial condition: reservoir is at uniform pressure when production begins.

Based on boundary conditions, there are two generalized solutions;

Constant terminal pressure solution

Constant terminal rate solution

Constant terminal pressure solution provides cumulative flow at any

particular time in reservoir in which pressure at one boundary held constant.

Constant terminal rate solution provides pressure change throughout the

reservoir when flow rate is held constant at one terminal end of radial system

i.e. producing well

Gas and water coning:

The critical production rate is a rate at which flowing pressure gradient in

well causes water or gas to cone into the well.

Coning can be eliminated by shallower penetration of well or by the

development of horizontal permeability.

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