Average Permeability Flow in Layered Systems Some figures were taken from Amyx, Bass and Whiting, Petroleum Reservoir Engineering (1960).

Average Permeability:

Average Permeability If permeability is not a constant function of space (heterogeneity), we can calculate the average permeability Common, simple flow cases are considered here Linear, Parallel (cores, horizontal permeability) Linear, Serial (cores, vertical permeability) Radial, Parallel (reservoirs, horizontal layers) Radial, Serial (reservoir, damage or stimulation) Average permeability should represent the correct flow capacity For a specified flow rate, average permeability results in same pressure drop (and vice versa ) Review Integral Averages (Self Study, e.g. Average Velocity)

Linear Flow, Pressure Profile:

Linear Flow, Pressure Profile Review, Darcy’s Law: horizontal flow ( F =p) L q A 1 2

Linear Flow, Pressure Profile:

Linear Flow, Pressure Profile We can determine how pressure varies along the flow path, p(x), by considering an arbitrary point, 0 x L Integral from 0x We could also integrate x L L q A 1 2 OR , equivalently

Linear Flow, Pressure Profile:

Linear Flow, Pressure Profile Pressure profile is a linear function for homogeneous properties slope depends on flow rate L q A 1 2 x 0 L p p 1 p 2 0 x p(x)

Linear, Parallel Flow:

Linear, Parallel Flow Discrete changes in permeability Same pressure drop for each layer Total flow rate is summation of flow rate for all layers Average permeability results in correct total flow rate Permeability varies across several horizontal layers (k 1 ,k 2 ,k 3 )

Linear, Parallel Flow:

Linear, Parallel Flow Substituting, Rearranging, Average permeability reflects flow capacity of all layers

Linear, Serial Flow:

Linear, Serial Flow Discrete changes in permeability Same flow rate passes through each layer Total pressure drop is summation of pressure drop across layers Average permeability results in correct total pressure drop Permeability varies across several vertical layers (k 1 ,k 2 ,k 3 )

Linear, Serial Flow:

Linear, Serial Flow Substituting, Rearranging, If k 1 >k 2 >k 3 , then Linear pressure profile in each layer x 0 L p p 1 p 2 0

Radial Flow, Pressure Profile:

Radial Flow, Pressure Profile Review, Darcy’s Law: horizontal flow ( F =p) q r e r w

Radial Flow, Pressure Profile:

Radial Flow, Pressure Profile We can determine how pressure varies along the flow path, p(r), by considering an arbitrary point, r w r r e Integral from r r w OR Integral from r e r OR , equivalently

Radial Flow, Pressure Profile:

Radial Flow, Pressure Profile Pressure profile is a linear function of ln(r) for homogeneous properties slope depends on flow rate ln(r) r w r e p p e p w 0 r p(r) q r e r w

Radial, Parallel Flow:

Radial, Parallel Flow Discrete changes in permeability Same pressure drop for each layer Total flow rate is summation of flow rate for all layers Average permeability results in correct total flow rate Permeability varies across several (3) horizontal layers (k 1 ,k 2 ,k 3 )

Radial, Parallel Flow:

Radial, Parallel Flow Substituting, Rearranging, Average permeability reflects flow capacity of all layers

Radial, Serial Flow:

Radial, Serial Flow Discrete changes in permeability Same flow rate passes through each layer Total pressure drop is summation of pressure drop across layers Average permeability results in correct total pressure drop Permeability varies across two vertical concentric cylindrical layers [k(r w rr 2 ) = k 1 , k(r 2 rr e = k 2 ] R 1 of this figure is r 2 of equations

Radial, Serial Flow:

Radial, Serial Flow Substituting (r w =r 1 , r 2 ,r e =r 3 ), Rearranging,

Radial, Serial Flow:

Radial, Serial Flow Stimulation k 1 >k 2 Shown in sketch to the right Permeability is improved near the wellbore Acid stimulation ln(r) r w r e p p e p w 0 ln(r) r w r e p p e p w 0 Damage: k 1 <k 2 Shown in sketch to the left Permeability is damaged near the wellbore Reactive fluids Fines migration k 1

Integration of Darcy’s Law:

Integration of Darcy’s Law Beginning with the differential form of Darcy’s Law Previous lecture on gas flow gas properties are functions of pressure include gas properties in the dp integral In this lecture parallel flow (permeability varies over cross sectional area) integrate over area (integrated average value) serial flow (permeability varies along flow path) integrate over flow path (leave k in ds integral) This approach can be extended to other cases (order of precedence as shown) Any term that varies as a function of pressure can be included in the dp integral Any term that varies along flow path can be included in the ds integral Any term that varies over cross sectional area can use an integrated average value (integrated over cross sectional area, e.g. parallel flow)

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