the properties of pre-compacted granular rocks. Te effective bulk modulus of dry random identical sphere packing is given by:


K =eff where CP C (1) is the number of contact points per grain, φ is


porosity, G is the shear modulus of the solid grains, σ is the Poisson ratio of the solid grains and P is the effective pressure (that is P = Peff


). Te effective pressure is (to the first order)


equal to the net pressure; i.e. the overburden pressure minus the pore pressure. Te shear modulus is given as: 2


G = Cp eff 2 2 (2)


Te relations between seismic P- and S-wave velocities and the bulk and shear moduli are given by:


= S = where VP G and VS (4)


and ρ is the rock density. Inserting equations (1) and (2) into equations (3) and (4) and computing the VP


are P- and S-wave velocities respectively, /VS


(assuming σ = 0): =


V V


S P


. (5)


Tis means that according to the simplest granular model (Hertz-Mindlin), the VP


/VS ratio should be constant as a


function of confining pressure if we assume the rock is dry. Te Hertz-Mindlin model assumes that the sand grains are spherical and that there is a certain area of grain-to-grain contacts. A major shortcoming of the model is that the P- and S-wave velocities will have the same behaviour with respect to pressure changes, so that the VP


/VS ratio is constant, as


shown in equation (5). For saturated rocks, this ratio goes to infinity for zero effective pressure, and not a constant. Tis is because the P-wave velocity approaches the fluid velocity for zero effective stress, and not zero as in the Hertz-Mindlin model (dry rock assumption). Te shear wave velocity, however, approaches zero. If we assume that the in situ


(base survey) effective pressure is P0


we see from equation (3) that


the relative P-wave velocity versus effective pressure is given as:


V V


P0 P


= P P


(6) Figure 4.6 shows this relation


for an in situ effective pressure of 6 MPa. When such curves are compared to ultrasonic core measurements, the slope of the measured curve is generally smaller than this simple theoretical


158 ratio, yields (3)


curve. Te cause for this might be multifold. Firstly, the Hertz-Mindlin model assumes the sediment grains are perfect, identical spheres, which, of course, is never found in real samples. Secondly, the ultrasonic measurements might suffer from scaling issues, core damage and so on. Tirdly, cementation effects are not included in the Hertz-Mindlin model. It is therefore important to note that there are major uncertainties regarding the actual dependency between seismic velocity and pore pressure changes.


4.1.3 The Noble Art of Analysing Time-Lapse Seismic Data


Te analysis of 4D seismic data can be divided into two main categories, one based on the detection of amplitude changes, the other on detecting travel-time changes, as shown in Figure 4.7. Experience has shown that the amplitude method is most robust, and therefore this has been the most frequently employed method. However, as the accuracy of 4D seismic has improved, the use of accurate measurements of small timeshifts is increasingly the method of choice. Tere are several examples where the timeshift between two seismic traces can be determined with an accuracy of a fraction of a millisecond. A very attractive feature of 4D timeshift measurement is that it is proportional to the change in pay thickness (thickness of those intervals containing oil or gas), and this method provides a direct quantitative result. Te two techniques are complementary in that amplitude measurement is a local feature (measuring changes close to an interface), while the timeshift method measures average changes over a layer, or even a sequence of layers. In addition to the direct methods mentioned above, 4D


seismic interpretation is aided by seismic modelling of various production scenarios, often combined with reservoir fluid-flow simulation and 1D scenario modelling based on well logs. Figure 4.8 demonstrates an excellent example of a more


complex drainage situation that can be inferred from detailed analysis of 4D seismic data. On the 1985 data (top section) we observe a broken oil-water contact (partially continuous black event at 1,950 ms). In the middle section we observe that parts of this oil-water contact have disappeared, indicating water flushing in this area (marked by two black arrows). Further to the east we notice three blue events that have brightened (marked by


Figure 4.6: Typical changes in P-wave velocity versus effective pressure using the Hertz-Mindlin model. In this case the in situ effective pressure (prior to production) is 6 MPa, and we see that a decrease in effective pressure leads to a decrease in P-wave velocity. The black curve represents the Hertz-Mindlin model (exponent=1/6), the red curve is a modified version of the Hertz-Mindlin model (exponent=1/10) that better fits the ultrasonic core measurements.


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