RASMUS, J. C., Schlumberger, Sugarland,TX, USA

**Abstract: Forward Modeling of Log Response in Geopressured
Formations Reveals Valuable Insights to the Various Pore Pressure Prediction
Techniques**

Introduction

A self consistent model
of a sedimentary formation in various stages of undercompaction has been
developed in order to forward model various logging tool responses. The
sedimentary formation model includes user modifiable expressions for temperature,
clay content, lithology, salinity, sediment water volume, water depth,
and compacting stress as a function of sediment burial depth below the
mudline. The formation model is input to various logging tool response
equations in order to predict log response as a function of any of the
model characteristics. Valuable insight is gained by comparing these modeled
logs to those expected from conventional pore pressure interpretation techniques.

Method

The geopressure formation forward model consists of user modifiable expressions for the lithology, porosity, temperature and fluid properties as a function of sediment vertical depth relative to the kelly bushing (TVD RKB). The formation is taken to be composed of quartz, wet clay, and effective (filled with moveable fluids) porosity. This is the porosity which contains the interstitial water that escapes from the formation while it is being compacted. The relationship, rock depth=C x 10 (-b x f ) is used to model the effective porosity of the formation with depth, where rock depth is the depth below mudline (Ref. 1). C is the depth at which f =0 and b is the rate of compaction.Water depth and air gap parameters are input so that TVD RKB can be plotted.The solids (1-f ) are placed in the volume that is not taken up by porosity. They are divided into quartz and wet clay and represented by a parameter which is the ratio of quartz to clay, staying constant with depth to accurately reflect the fact that the solids do not change volume with depth. The temperature is modeled with an offset (surface temp) and a gradient. An expression for the salinity fluid property allows it to vary from seawater at the mudline to salt saturated at a given depth. The fluid bulk modulus is modeled as a function of pressure, temperature, and salinity. A model has also been developed for the compressional and shear dry frame modulus as a function of porosity and lithology.

The volumes and parameters from this forward model as a function of TVD RKB are input to the various logging tool response equations containing these modeled parameters and volumes. The response of resistivity, density, gamma ray, and neutron log response versus TVD RKB can then be computed and displayed. The sonic compressional and shear response is modeled using Gassmann's equations (ref. 2). The formation density is then integrated to compute the overburden pressure as a function of airgap, water depth and sediment depth.

Results

FORWARD MODELED RESISTIVITY

Fig. 1 shows the forward model of resistivity for various pore pressures versus TVD RKB using a water depth of zero. Notice how the resistivities have a similar appearance to the resistivity "overlays" developed by past authors when shallow waters were being drilled. The resistivity rises quickly at shallow depths as the decreasing porosity effect outweighs the increasing salinity effect then rises more gradually as the porosity decrease becomes more linear. The difficulty in drawing a straight line "normal" trend as required for some techniques can be seen at shallow depths. However, at depths greater than several thousand feet below the mudline and for intervals of less than 10,000 ft, the normal trend is fairly linear on this typical logarithmic resistivity scale. Fig. 2 shows the effect of changing the quartz/wet clay ratio from 1.0 (0.5/0.5) in fig. 1 to 0.25 (0.2/0.8) and the water depth to 2000 ft. Notice how the slope of the resistivities has decreased as well as the values at any particular TVD RKB. It is these changes in lithology that cause shifts in the normal trend lines and can make these plots look "noisy" when real data is plotted at this scale.That is why past authors have stressed that "consistent lithology" shale points be plotted on these charts. Plots like fig. 1 and 2 of the forward modeled resistivity allow a prediction of pore pressure as a function of lithology, water depth, and resistivity before drilling a well and help in defining the "normal" trend for other techniques.

FORWARD MODELED VELOCITY

Fig. 3 shows a plot of compressional velocity versus effective stress for the conditions given in fig. 2. Normally, the velocity is assumed to be a function of effective stress independent of the value of pore pressure. In this case there would be only one line on this plot with all of the pore pressures lying on top of each other. Note that for the higher pore pressures, this is not the case. This is because the fluid bulk modulus, being a function of temperature, salinity and pressure, can be different for the same effective stress. Consider two formations with equal effective stress, one shallow and one deep. The deeper one has a greater overburden and pore pressure, but the difference (effective stress) is the same as the shallow one. The fluid modulus is larger at the deeper depth because it has increased as temperature, pore pressure, and salinity have increased. A larger fluid modulus (less compressibility) will give rise to a greater velocity for the deeper formation even though the effective stress is the same. This phenomenon has the same signature as "unloading" reported by others (ref. 3).

Insights into other techniques

The dependence of velocity on both effective stress and depth affects other pore pressure techniques as follows. In an undercompacted shale, the effective stress of the rock is the difference between the overburden and pore pressures. The "equivalent depth" technique assumes that two formations with equal effective stresses will have the same porosity and therefore the same log response. The first assumption is logical and valid, but the second is questionable as shown in fig. 3. The result is that these techniques will underestimate pore pressures. However, below about 15 PPG, these techniques should give good results. Other techniques (ref. 4) use the ratio of an "observed" to "normal" log response to compute pore pressures. It is shown below that this is not an entirely independent technique from the equivalent depth technique.

The effective stress of a geopressured rock at depth D2 is the effective stress of the rock at a shallower, normally pressured depth D2. This is the definition of the equivalent depth method, sometimes referred to as a "vertical" method. One can compute the pore pressure of the rock at depth D2 knowing the overburden pressures and normal water pressure. An expression for the equivalent depth method is:

Peff _{2} =Peff_{ 1}
Eq. 1

or
Povb _{2} -Pwpore_{2} =Povb_{1} -Pwnor_{1}
Eq. 2

where Peff is effective pressure, subscripts 1 and 2 refer to depths D1 at normal pore pressure and a deeper depth D2, at a higher pore pressure. Pwpore is the pore pressure at D2, Pwnor is the normal water pressure, and Povb is the overburden pressure. Dividing by D2, this can be rewritten as:

Pwpore_{2} /D2=Povb_{2}
/D2 - (Povb_{1} -Pwnor_{1} )/D2
Eq 3

which can be compared to Eaton's equation (ref. 4): Eq. 4

Pwpore_{2} /D2=Povb_{2}
/D2 - ((Povb_{2} -Pwnor_{2} )/D2)*(Xobs/Xnor) ^{(Y) }
Eq. 5

where Xobs and Xnor are the observed and expected "normal" log responses at a given depth. The two expressions are equivalent if one sets

(Xobs/Xnor)^{ (Y)}
= (Povb_{1}
-Pwnor_{1}
)/(Povb_{2} -Pwnor_{2} ).
Eq. 6

The right hand side of this equation is the ratio of the normal effective stresses which is also equivalent to D1/D2. Therefore

(Xobs/Xnor)^{ (Y)} = D1/D2
Eq. 7

Thus Eaton's method is an attempt to map the normal and observed log response (sometimes called a horizontal technique) at one particular depth to the ratio of depths at equal effective stresses (sometimes called a vertical technique). This means that Eaton's method relies on the validity of the "equivalent depth" method. Earlier it was shown that the equivalent depth method is valid only over limited depth ranges due to the fact that fluid properties change with depth, causing the log response to change with depth even when the porosity (or effective stress) stays constant. Therefore Eaton's method is under the same limitations. The widespread success of Eaton's method lends credence to the "equivalent depth" technique. However, as pointed out in fig. 3, the "equivalent depth" method's assumption of constant log response for constant effective stress is not always valid, and may explain why Eaton's exponent (Y) and/or "normal" trend lines may sometimes have to be altered in order to compute the correct pore pressure.

Conclusions

1. Forward modeling is used to produce "overlays" for a particular formation model and water depth. For shallow water depths, these overlays mimic the trends seen in earlier empirical overlays.

2. Forward modeling has shown that logging tool response is not always constant with constant porosity and effective stress as heretofore assumed due to the temperature, pressure, and salinity changes that occur as a geopressured shale is buried deeper. Techniques using the equivalent depth method will be affected by this phenomena.

3."Horizontal" techniques or those that use ratios of observed to normal log response are in reality an approximation to the equivalent depth method and are under the same limitations.

AAPG Search and Discovery Article #[email protected] International Conference and Exhibition, Birmingham, England