1. COMPRESSIBILITY OF
RESERVOIR ROCKS

2. COMPACTION OF SEDIMENTS
Porosity is reduced by compaction
Porosity reduction is determined by maximum burial depth
Principal effects are:
Changes in packing
Pressure solution
Recrystallization
Deformation of rock fragments
Compaction effects are not reversed by erosional unroofing (hysteresis effect)
Hysteresis is discussed in more detail at the end of this presentation.

3. MECHANICS OF COMPACTION
Modified from Jonas and McBride, 1977
Platy Grains
(e.g., clays)
Non-Platy Grains
(e.g., qtz., feldspar)
Rotation and Closer
Packing
Ductile Grain
Deformation
Breakage of
Brittle Grains
Pressure Solution
At Grain
Contacts
Ductile Framework
Grain, e.g., Shale Rock
Fragment)

4. Relationship of Original Formation Porosity to Overburden Pressure50
Overburden pressure, psi
Porosity, % 30 40 20 10
1,000
3,000
2,000
4,000
5,000
6,000
Sandstones
Shales
Figure shows typical formation porosity trend as burial depth increases. This porosity is at original reservoir pressure for that depth. Compaction occurred over geologic time periods.
The behavior shown here is not related to changing the overburden or pore pressure of a particular reservoir during the time period of development (a few years/decades).
Note that when speaking and in many older references, porosity is referred to as “percent”.
Remember that porosity is a dimensionless ratio of two volumes, and should be thought of as a fraction.

5. Isothermal Compressibility
General Definition
The relative volume change of matter per unit pressure change under conditions of constant temperature
Usually, petroleum reservoirs can be considered isothermal (an exception: thermal stimulation)
Increasing pressure causes volume of material to decrease (compression) - e.g. reservoir fluids
Decreasing pressure causes volume of material to increase (expansion) - e.g. reservoir fluids

6. Isothermal Compressibility
General Equation
C: Coefficient of Isothermal Compressibility
ALWAYS positive value
oilfield units: 1/psia
V: Volume
oilfield units: ft3
p: Pressure exerted on material
oilfield units: psia
Negative sign in equation determined by V/p term, to force the coefficient C to be positive
Volume is a function of pressure only (temperature is constant, and amount of material is constant)
The equation for C is an ODE (Ordinary Differential Equation)

7. Formation Compressibility
Importance
Formation compressibility can have a significant impact on reservoir performance
Subsidence can have significant environmental impact
Types
Matrix Compressibility ( Cm ): relative change in volume of solid rock material (grain volume) per unit pressure change (usually Cm  0).
Pore Compressibility ( Cf ): relative change in pore volume per unit pressure change.
Bulk Compressibility ( Cb ): relative change in bulk volume per unit pressure change ( usually DVb  DVp). Significant decrease in bulk volume can cause subsidence.
Impact on Reservoir Performance: If we think of the reservoir as a tank of fluid, then pore compressibility can act to reduce the volume of the tank as we decrease reservoir pressure; like a piston helping “squeeze out” reservoir fluids.
Pore Compressibility is usually of greatest interest to Petroleum Engineers

8. FORMATION COMPRESSIBILITY
Under static conditions, downward overburden force must be balanced by upward forces of the matrix and fluid in pores 1.
2. Thus:
p is fluid pressure in the pores, and can be easily measured with a pressure gauge.
Remember: Force = (Pressure) *(Area) 4. 3.
As fluids are produced from reservoir, fluid pressure (p) usually decreases while overburden is constant, and:
(a) force on matrix increases ( “net compaction pressure”,
pm=po-p)
(b) bulk volume decreases, and
(c) pore volume decreases.
Pressure Gradients, Normal Reservoirs:
dpo/dZ = 1.0 psia/ft
dp/dZ = psia/ft

9. Formation Compressibility
Equation
Cf: Formation Compressibility (Pore Volume Comp.)
ALWAYS positive value
oilfield units: 1/psia
Vp: Pore volume
oilfield units: ft3
p: Pressure of fluid in pores
oilfield units: psia
Positive sign in equation determined by Vp/p term, to force Cf to be positive
Pore volume is function of pressure only (temperature is constant, amount of reservoir rock is constant)

10. Subsidence and Bulk Compressibility
Process of subsidence
Bulk volume decreases as fluids are produced
Area is constant
 Formation thickness decreases (causing subsidence of strata above)
Porosity:  = Vp/Vb = 1-(Vm/Vb); where Vb=Vp+Vm
Net compaction pressure: pm = po - p
Overburden (po) is constant  dpm= -dp
As net compaction pressure increases
Bulk volume decreases; Cb = -1/Vb (Vb/pm)
Pore volume decreases; Cf= -1/Vp (Vp/pm)
Matrix volume decreases; Cm= -1/Vm (Vm/pm)
Substituting from definitions above
Cb = (-1/Vb) [(Vp/pm) + (Vm/pm) ]
Cb = (-1/Vb) [(- Cf Vp) + (- Cm Vm)]
Cb = Cf + (1-)Cm; usually Cm << Cf
Lake Maracaibo Venezuela - Levees have been built along the lake shore to protect towns. The water level of the lake has not increased, but instead the towns have sunk meters as petroleum reservoirs have been produced.
North Sea - Production platforms have been raised in oilfields at great expense because production from chalk formations with large compressibility has resulted in subsidence. It would have been much less expensive to have prevented or designed for this subsidence
Houston - Measureable subsidence has occurred in Houston area due to production of ground water for drinking
One way to prevent subsidence: inject fluids to maintain reservoir pressure (fluids in pores) and prevent compaction of reservoir rock.
Other examples: Pisa, Italy (groundwater) , Venice Italy (groundwater) Injection of water to maintain pressure is being tried in Pisa to slow/remediate the leaning of the famous tower.

11. Formation Compressibility
Calculation of Pore Volume Change
Separate
and Integrate
Two common approaches for constant value of Cf
Exact Integration
1st Order Approximation

12. Formation Compressibility
Pore Volume Change - Continued
Exact Integration
Exponentiating (Inverse of Natural Logarithm) and rearranging OR
If p2<p1 then Vp2 < Vp1 and DVp is negative.

13. Formation Compressibility
Pore Volume Change - Continued
1st Order Approximation
We use Vp1 in the denominator of the 2nd equation because we know the pore volume at p1 and wish to calculate Vp2 as pressure changes
The 1st order approximation can be useful when we have discrete data such as a few experimental points from the laboratory, or in computer modeling of reservoir performance using discrete time periods.
Remember: A 1st order approximations assume the function is a straight line, dVp/dp = DVp/Dp = constant.
This equation can also be derived from the exact solution by considering the Taylor Series expansion for ex about x=0 and truncating after the 1st order term.

14. Laboratory Determination of Cf
In reservoirs, overburden pressure is constant and the pressure of fluid in pores changes, resulting in pore volume change
In the laboratory, we change the confining pressure on the core plug (overburden) while holding the pore pressure constant
Remember that the net compaction pressure on the matrix is the difference between the overburden and pore pressures
This allows us to obtain useful results in the laboratory

15. Laboratory Determination of Cf
Laboratory Procedure
Core plug is 100% saturated with brine
Core plug is placed in rubber or soft copper sleeve
As pressure outside sleeve is increased, pore volume decreases and the volume of expelled brine is measured
The lab process is like sqeezing a saturated sponge to remove water.
The reservoir process is like putting the saturated sponge in a sealed plastic bag, then setting several heavy bricks on top, then poking a hole in the bag.
Either way water is sqeezed out
pconfining

16. Hysteresis Effect - Formation Compressibility
Hysteresis: The lagging of an effect behind its cause, as when the change in magnetism of a body lags behind changes in the magnetic field. (definition from dictionary.com, 2002)
Hysteresis is used by Petroleum Engineers to describe the effects of path dependence and irreversibilities we observe in reservoir behavior
For example, if we decrease reservoir pressure from initial conditions, pore volume decreases. If we then increase reservoir pressure back to the initial pressure, pore volume does not increase all the way back to the initial pore volume.
Initial Conditions
Pore Volume
Pore Pressure