# Porosity

Porosity may be fixed at a constant value, or it may be a function of the effective porepressure, the volumetric strain, the temperature and/or chemical precipitates

Available porosity formulations include:

## Constant

The simplest case where porosity does not change during the simulation. A single value of porosity can be used, or a spatially varying porosity `AuxVariable`

can be used to define a heterogeneous porosity distribution.

## Porosity dependent on total strain, effective porepressure, temperature and/or mineralisation

Using `PorousFlowPorosity`

with appropriately set flags, porosity can depend on:

total strain, with

`mechanical = true`

effective porepressure, with

`fluid = true`

temperature, with

`thermal = true`

precipitated minerals, with

`chemical = true`

A combination of these may be used, to simulate, for instance, THM or HM coupling.

## Evolution of porosity

The evolution of the porosity is governed by (Detournay and Cheng, 1993; Chen and Zhang, 2009) (1) Here is the bulk modulus of the drained porous skeleton: . Also is the volumetric coefficient of thermal expansion. denotes the total precipitated mineral concentration: (2) where is the user-defined weight for each mineral and is the concentration (m/m) of precipitated mineral.

The equation for porosity has solution (3) where is the porosity at reference porepressure, zero elastic strain, reference temperature and reference mineral concentration. Note this porosity can become negative, and an option for ensuring positivity is detailed below.

With mineralisation, now depends on total mineral concentration, . However, the evolution of is governed by , where is a reaction rate which is independent of porosity (but dependent on the primary chemical species, temperature, etc) and is the aqueous phase saturation. Therefore a circular dependency exists: depends on , and depends on . This could be broken by promoting porosity to a MOOSE Variable, and solving for it. Instead, PorousFlow replaces in Eq. 3 by the approximate form (4) Note that the *old* value of porosity is used on the right-hand-side, which breaks the cyclic dependency problem.

Without porepressure and mineralisation effects, the correct expression for porosity as a function of volumetric strain and temperature is (5)

These expressions may be modified to include the effects of plasticity.

The exponential expressions Eq. 3 and Eq. 5 can yield negative porosity values, which are unphysical. To ensure positivity of , PorousFlow offers the following option. First write both equations in the form (6) In deriving specific forms for , and , above, it has been assumed that is small. Since physically , for the porosity will be physically meaningful, but for , can become negative. For example, for positive volumetric strain () the porosity is always physically meaningful. However, when the porous material is squashed with negative and large volumetric strain () the porosity can become negative.

Define (7) Then, for , the above expression for is replaced by (8) At first glance this expression may appear rather obscure. It has been constructed to satisfy the following requirements:

As , , which is necessary for continuity at .

As , , which is physically desireable.

The expression is monotonically decreasing, which is both physically desirable and computationally desirable (otherwise there may be non-unique solutions in a PorousFlow simulation).

Finally, at , its derivative is , as desired from continuity of the derivative at .

## Evolution of porosity for an isothermal, mineral-free, situation

The evolution of porosity is fundamental to the coupling between flow and solid mechanics. Consider the isothermal situation with no plasticity. The following presentation is mostly drawn from (Detournay and Cheng, 1993).

Denote the change of a quantity, q, by . Recall that the porosity is defined by , where is an arbitrary volume of the porous material, and is the porevolume within that volume. Also, by definition of the effective stress, (9) Taking the trace of this equation, and using yields (10) In most instances it is appropriate to write this equation as (11) (which is Detournay and Cheng Eqn(20a)) where the total mechanical pressure is (12) and is the so-called *drained* bulk modulus . To find the evolution equation for porosity, a similar equation for must be derived.

Assuming linearity (13) (This is Detournay and Cheng Eqn(20b).) The Betti-Maxwell reciprocal theorem yields and , as is now shown.

The work increment is (14) So during some deformation that takes from to , and from to , the total work is (15)

Now consider two experiments:

First take from to with fixed at . Then, leaving fixed, take from to . The first takes work , while the second takes work .

First take from to with fixed at . Then, leaving fixed, take from to . The first takes work , and the second takes work .

The two experiments must give the same work done (this is called the Betti-Maxwell reciprocal theorem), which yields (16) (This is Detournay and Cheng Eqn(22).)

Now to identify . Consider a so-called *ideal porous material*, which is characterised by a fully-connected pore space and a homogeneous and isotropic matrix material. In this case, applying a uniform porepressure, , and an equal mechanical pressure, , the solid material will experience a uniform pressure throughout its skeleton. This means it will deform uniformly without any shape change, and (17) Substituting this equation, this specific pressure condition, and Eq. 16 into Eq. 11 and Eq. 13, yields (18) (This is Detournay and Cheng Eqns(22), (24b), (37) and the definition of in Table 1.)

Now that and have been identified, they may be substituted into Eq. 13. Rearranging yields (19) Using the expression for yields (20) Now , so using the definition of yields (21) as written in Eq. 1.

## Evolution of porosity due to mineral precipitation

Consider now the case without any fluid porepressure or temperature, but with mineral precipitation.

The concentration of a mineral species, is defined to be , where the numerator is the volume of the mineral species and the denominator is the reference volume. Suppose that and the definition of porosity is .

Now consider volume changes via (where is the volumetric strain) and changes in the volume of mineral, . Under these changes, the porosity evolves as (22) This yields, after a little algebra, (23) as written in Eq. 1.

## References

- Z. Chen and Y. Zhang.
Well flow models for various numerical methods.
*International Journal of Numerical Analysis and Modelling*, 6:375–388, 2009.[BibTeX] - E. Detournay and A. Cheng.
*“Fundamentals of poroelasticity,'' Chapter 5 in Comprehensive Rock Engineering: Principles, Practice and Projects, Vol II, Analysis and Design Method. Also available online*. Pergamon Press, New York, 1993.[BibTeX]