Mechanics of stratified anisotropic poroelastic media

APPENDIX A: THE SCHOENBERG-MUIR METHOD

The quasi-static elasticity equations are often written in compliance form using the Voigt $6\times6$ matrix notation as:

$$\left(\begin{array}{c} e_{11}\cr e_{22}\cr e_{33}\cr e_{23}\cr e_... ...cr \sigma_{33}\cr \sigma_{23}\cr \sigma_{31}\cr \sigma_{12} \end{array}\right),$$ (40)

where ${\bf S}$ is the symmetric $6\times6$ compliance matrix. The numbers 1,2,3 always indicate Cartesian axes (say, $x$ ,$y$ ,$z$ respectively). The $z$ -direction is usually chosen as the layering direction, which could be oriented any direction in the earth. But, in many geological and geophysical applications, the 3-axis (or $z$ -axis) is also taken to be the vertical direction, and I conform to this convention here. The principal stresses are $\sigma_{11}$ , $\sigma_{22}$ , $\sigma_{33}$ , in the directions 1,2,3, respectively. Similarly, the principal strains are $e_{11}$ , $e_{22}$ , $e_{33}$ . The stresses $\sigma_{23}$ , $\sigma_{31}$ , $\sigma_{12}$ are the torsional shear stresses, associated with rotation-based strains around the 1, 2, or 3 axes, respectively. The corresponding torsional strains are $e_{23}$ , $e_{31}$ , and $e_{12}$ , where the torsional motion is again a rotational straining motion around the 1, 2, or 3 axes. The compliance matrix is symmetric, so $s_{ij} = s_{ji}$ , and this fact could have been used when displaying the matrix. The axis pairs in the subscripts $11$ , $22$ , $33$ , $23$ , $31$ , and $12$ , are often labelled (again following the conventions of Voigt) as 1,2,3,4,5,6, respectively.

The important contribution made by Backus (1962) [also see Postma (1955)] is the observation that, in a horizontally layered system, there are certain strains $e_{ij}$ and stresses $\sigma_{ij}$ that are necessarily continuous across boundaries between layers, while the others are not necessarily continuous. I have been implicitly (and now explicitly by calling this fact out) assuming that the interfaces between layers are in welded contact, which means practically that the in-plane strains are always continuous: so if axis 3 (or $z$ ) is the symmetry axis (as is most often chosen for our layering problem), I have $e_{11}$ , $e_{12} = e_{21}$ , and $e_{22}$ are all continuous. Similarly, in welded contact, I must have continuity of the all the stresses involving the 3 (or $z$ ) direction: so $\sigma_{33}$ , $\sigma_{13} = \sigma_{31}$ , and $\sigma_{23} = \sigma_{32}$ must all be continuous.

Then, following Backus (1962) and/or Schoenberg and Muir (1989) but -- for present purposes considering instead the compliance (inverse of stiffness) matrix -- I have rearranged the statement of the problem so that:

$$\left(\begin{array}{c} e_{11}\cr e_{22}\cr e_{12}\cr e_{33}\cr e_... ...cr \sigma_{12}\cr \sigma_{33}\cr \sigma_{32}\cr \sigma_{31} \end{array}\right).$$ (41)

Note that this equation, although similar to (40) is nevertheless quite different because of the rearrangement of the matrix elements and the reordering of the strains and stresses. The expression in (41) is general for all elastic media. In the main text I restrict the discussion to orthotropic media. Assuming then that I am using the correct set of axes as the symmetry axes in the presentation, all off-diagonal compliances having subscripts $4$ , $5$ , or $6$ in (40) vanish identically. The diagonal shear compliances $s_{44}$ , etc., generally do not vanish however.

Expression of (41) can be made more compact by writing it as:

$$\left(\begin{array}{c} E_T \cr E_N \end{array}\right) = \left(\be... ...NN}\end{array}\right) \left(\begin{array}{c} \Pi_T \cr \Pi_N\end{array}\right),$$ (42)

where

$${\bf S}_{TT} \equiv \left(\begin{array}{ccc} s_{11} & s_{12} & s_... ...ccc} s_{11} & s_{12} & \cr s_{21} & s_{22} & \cr & & s_{66} \end{array}\right),$$ (43)

$${\bf S}_{NN} \equiv \left(\begin{array}{ccc} s_{33} & s_{34} & s_... ...\begin{array}{ccc} s_{33} & & \cr & s_{44} & \cr & & s_{55} \end{array}\right),$$ (44)

and

$${\bf S}_{NT} \equiv \left(\begin{array}{ccc} s_{31} & s_{32} & s_... ...ft(\begin{array}{ccc} s_{31} & s_{32} & \cr & 0 & \cr & & 0 \end{array}\right),$$ (45)

with ${\bf S}_{TN} = {\bf S}_{NT}^T$ (with $T$ superscript indicating the matrix transpose). Also I have

$$E_T \equiv \left(\begin{array}{c} e_{11} \cr e_{22} \cr e_{12} \e... ... \equiv \left(\begin{array}{c} e_{33} \cr e_{32} \cr e_{31} \end{array}\right),$$ (46)

and

$$\Pi_T \equiv \left(\begin{array}{c} \sigma_{11} \cr \sigma_{22} \... ...begin{array}{c} \sigma_{33} \cr \sigma_{32} \cr \sigma_{31} \end{array}\right).$$ (47)

It is important to distinguish between ``slow'' and ``fast'' variables in this analysis, since this distinction makes it clear when and how averaging should be performed. The ``slow'' variables, i.e., those that are continuous across the (here assumed horizontal) boundaries and also essentially constant for the present quasi-static application, are those contained in $E_T$ and $\Pi_N$ . So, after averaging $\left<\cdot\right>$ along the layering direction, I have:

$$\left(\begin{array}{c} E_T \cr \left<E_N\right> \end{array}\right... ...}\right) \left(\begin{array}{c} \left<\Pi_T\right> \cr \Pi_N\end{array}\right),$$ (48)

where ${\bf S}^*_{TN} = \left({\bf S}^*_{NT}\right)^T$ , and all the starred quantities are the nontrivial average compliances I seek. They are defined in terms of layer-average quantities where the symbol $\left<\cdot\right>$ indicates a simple volume average of all the layers. By this notation I mean that a quantity $Q$ that takes on different values in different layers has the layer average $\left<Q\right> \equiv x_aQ_a + x_bQ_b + \ldots$ . The definition is general and applies to an arbitrary number of different layers where the fraction of the total volume occupied by layer $a$ is $x_a$ , etc. Total fractional volume is $x_a + x_b + \ldots \equiv 1$ .

Of the three final results, the two easiest ones to compute are:

$${\bf S}^*_{TT} = \left<{\bf S}_{TT}^{-1}\right>^{-1},$$ (49)

$${\bf S}^*_{TN} = ({\bf S}_{NT}^*)^T = \left<{\bf S}_{TT}^{-1}\rig... ...{\bf S}_{TN}\right> = {\bf S}^*_{TT}\left<{\bf S}_{TT}^{-1}{\bf S}_{TN}\right>,$$ (50)

where $\left<\cdot\right>$ is the layer average of some quantity. These results follow from this equation:

$$\left<{\bf S}_{TT}^{-1}\right>E_T = \left<\Pi_T\right> + \left<{\bf S}_{TT}^{-1}{\bf S}_{TN}\right>\Pi_N,$$ (51)

which also followed immediately from the formula

$$E_T = {\bf S}_{TT}\Pi_T + {\bf S}_{TN}\Pi_N$$ (52)

multiplying through first by the inverse of ${\bf S}_{TT}$ , and then performing the layer average. [Note that ${\bf S}_{TT}$ and ${\bf S}_{NN}$ are both normally square and invertible matrices, whereas for most systems the off-diagonal matrix ${\bf S}_{NT}$ is not invertible. But, this fact does not cause problems in the analysis, because I do not need to invert ${\bf S}_{NT}$ in order to solve the averaging problem at hand.] These averages are meaningful because, when the matrix equations presented are multiplied out, there never appear any cross products of two quantities that are both unknown. [From this view point, Eq. (51) is an equation for $\left<\Pi_T\right>$ , just as the unaveraged version of (51) is an equation for $\Pi_T$ in each layer.] So simple layer-averaging suffices (thereby providing the main motivation and value of this method). Multiplying (51) through by $\left<{\bf S}_{TT}^{-1}\right>^{-1}$ then gives the results (49) and (50).

The remaining result is more tedious to compute, since it requires several intermediate steps in its derivation. But the final result is given by the formula:

$${\bf S}_{NN}^* = \left<{\bf S}_{NN}\right> - \left<{\bf S}_{NT}{\... ... S}_{TN}\right> + {\bf S}^*_{NT}\left({\bf S}^*_{TT}\right)^{-1}{\bf S}_{TN}^*.$$ (53)

To provide some clues to the derivation, again consider:

$$\Pi_T = {\bf S}_{TT}^{-1}E_T - {\bf S}_{TT}^{-1}{\bf S}_{TN}\Pi_N,$$ (54)

which is just a rearrangement of (52). The point is that $\left<\Pi_T\right>$ is then given immediately in terms of the quantities $E_T$ and $\Pi_N$ , which are both ``slow'' variables and therefore essentially constant. An intermediate result that helps to explain the form of this relation (53) is:

$${\bf S}^*_{NT}\left({\bf S}^*_{TT}\right)^{-1}{\bf S}_{TN}^* = \l... ...{\bf S}_{TN}\right> = \left<{\bf S}_{NT}{\bf S}_{TT}^{-1}\right>{\bf S}_{TN}^*.$$ (55)

Substituting for $\Pi_T$ from (54) into

$$E_N = {\bf S}_{NT}\Pi_T + {\bf S}_{NN}\Pi_N$$ (56)

and then averaging, I find that

$$\left<E_N\right> = \left<{\bf S}_{NT}{\bf S}_{TT}^{-1}\right>E_T + \left<{\bf S}_{NN}-{\bf S}_{NT}{\bf S}_{TT}^{-1}{\bf S}_{TN}\right>\Pi_N,$$ (57)

an expression that completely determines the remaining coefficients. After some more algebra, the formula giving the final result is:

$$\begin{array}{ll} \left<E_N\right> & = \left<{\bf S}_{NT}{\bf... ... S}_{NT}^*\left<\Pi_T\right> + {\bf S}_{NN}^*\Pi_N. \end{array}$$ (58)

Equation (58) contains all the information needed to produce the third and final result found in (53).

Another check on these formulas is to compare them directly to those found by Schoenberg and Muir (1989). However, direct comparison is not so easy, since their analysis focuses on the stiffness version of these equations. My treatment makes use of the compliance version instead. Since the symmetries of the two forms of the equations nevertheless are nearly identical, cross-checks and comparisons will be left to the interested reader.

Mechanics of stratified anisotropic poroelastic media