Error propagation in Gassmann modeling for 4D feasibility studies

Abstract

The single most important rock physics equation is probably the widely used Gassmann equation which estimates the change in bulk modulus of the saturated rock when the original fluid is replaced by a fluid with a different modulus. This is the essence of the fluid replacement problem where we want to estimate the change in P-wave velocity ( VP ) as a function of fluid properties. The change in density (ρb) is straightforward provided we know porosity and mineral and fluid densities. The S-wave velocity ( VS ) is only sensitive to the change in density. There are two common forms of the Gassmann equation. The first estimates the bulk modulus of the saturated rock based on the porosity and mineral, fluid and dry frame moduli: \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[K\_{sat} K\_{dry} {+} \frac{\left(1 {-} \frac{K\_{dry}}{K\_{0}}\right)^{2}}{\frac{{\phi}}{K\_{fl}} {+} \frac{1 {-} {\phi}}{K\_{0}} {-} \frac{K\_{dry}}{K\_{0}^{2}}}\] \end{document} where Ksat = bulk modulus of the fluid saturated rock, Kdry = bulk modulus of the dry rock, Kfl = bulk modulus of the pore fluid, K = bulk modulus of the mineral, ϕ = porosity. This form requires knowledge of the dry rock bulk modulus which is rarely known. The second form is better suited for fluid replacement calculations. Here we assume that VP , VS , rhob (porosity) are known with the original fluid. In addition the bulk modulus of the mineral and both fluids are also known. Then the dry rock bulk modulus can be algebraically eliminated and the two bulk moduli Ksat1 and Ksat2 can be related to the bulk moduli of the two fluids Kfl1 and Kfl2 . \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\frac{K\_{sat1}}{K\_{0} {-} K\_{sat1}} {-} \frac{K\_{fl1}}{{\phi}(K\_{0} {-} K\_{fl1})} \frac{K\_{sat2}}{K\_{0} {-} K\_{sat2}} {-} \frac{K\_{fl2}}{{\phi}(K\_{0} {-} K\_{fl2})}\] \end{document} From this we can then calculate the VP , VS and ρb for the rock saturated with the new fluid. The P-wave velocity is written: \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[V\_{p} \sqrt{\frac{K\_{sat} {+} 4/3{\mu}}{{\rho}_{b}}}\] \end{document} where μ = shear modulus (insensitive to fluid content) and ρb = bulk …