Sensitivity

Part 4, Part 4: Gassmann and Fluid Substitution

Learning objectives

Treat the substituted answer as exact arithmetic on inexact inputs, and ask where its error comes from
Sweep porosity, dry-frame modulus, mineral modulus, and fluid modulus and rank their effect on the substituted Vp
State the ranking: the dry frame and porosity dominate, the mineral modulus matters least
Read the tornado, and explain why a softer fluid contrast shrinks the whole effect

Only as Good as the Inputs

A fluid substitution is exact arithmetic on inexact inputs. The equation adds no error of its own (Part 4.2), but its four inputs, the dry-frame modulus, the porosity, the mineral modulus, and the fluid modulus, are each known only to some tolerance, and the answer inherits all of them. The honest question is not whether the substituted velocity has error, it does, but where that error comes from, because the four inputs are nowhere near equally important. Treating them as if they were is how careful arithmetic still produces a misleading answer.

Sweep One Input at a Time

Hold the base case, the soft sand with brine substituted to gas, and push each input up and down by a realistic amount while watching the substituted $V_P$ P. Sweeping the dry-frame bulk modulus by twenty percent swings the answer by about $\pm 0.11$ km/s, the widest response of the four. A porosity uncertainty of $\pm 0.03$ swings it about $\pm 0.055$ km/s, roughly half as much. A ten percent error in the mineral modulus barely registers, under a hundredth of a km/s, and even a fifty percent error in the already tiny gas modulus moves the answer only about $\pm 0.006$ km/s. Order those four swings from widest to narrowest and you have drawn a tornado.

What the Tornado Teaches

Three lessons fall straight out of the ranking. First, the dry frame and the porosity dominate. The substituted velocity is a prediction about the frame first and the fluid second, so the two quantities that describe how stiff the skeleton is and how much pore space it holds are where care is repaid. A sloppy $K_{dry}$ dry or a mis-picked porosity poisons the answer more than anything the fluid can do. Second, the mineral modulus matters least of the four. For a gas substitution the ratio $K_{dry}/K_{min}$ is small and the gas stiffening term is tiny, so $K_{sat}$ sat is nearly $K_{dry}$ dry whatever the mineral is, and a ten percent error there is almost invisible. Third, the softer the fluid contrast, the smaller the whole effect. Substituting to a fluid near the in-situ one moves everything less: brine to oil drops $V_P$ P about 8.8 percent against the 10.1 percent of brine to gas, so the same input errors matter less in absolute terms when the contrast is mild.

All of this is error measured inside the theory, with every one of Gassmann's assumptions taken to hold. It tells you how a correct model degrades as its inputs blur. The larger risk is different and more dangerous: error from the theory itself, from one of the four assumptions of Part 4.1 quietly failing so that the model is wrong no matter how sharp the inputs are. The next section takes each assumption in turn and asks exactly when Gassmann stops holding.

References

Smith, T. M., Sondergeld, C. H., & Rai, C. S. (2003). Gassmann fluid substitutions: A tutorial. Geophysics, 68(2), 430-440.
Mavko, G., Mukerji, T., & Dvorkin, J. (2009). The Rock Physics Handbook (2nd ed.). Cambridge University Press.