# Petrofisica

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Porosity Resistivity Crossplot (Hingle Plot)
The porosity resistivity crossplot is a venerable tool, still used in many areas. In one version, porosity is plotted on a linear scale, and resistivity on a scale such that straight lines on the graph represent constant water saturation, as determined by the Archie formulae:
1: Sw = (F * RW@FT / RES) ^ (1 / N)
2: F = A / (PHIe ^ M)WHERE:
A = tortuosity exponent (fractional)
F = formation factor (fractional)
M = cementation exponent (fractional)
N = saturation exponent (fractional)
PHIe = effective porosity (fractional)
RESD = deep resistivity (ohm-m)
RW@FT = water resistivity (ohm-m)
Sw = water saturation (fractional)
This plot is often called the Hingle plot after the man who first publicizedthe method. The graph requires a special grid, since the Y axis is linear in the function RESD ^ (-1 / M) but not linear in RESD. RESD or COND lines are used to plot and read data points, so these are plotted to fall non-linearly on the graph paper. The log-log Pickett plot described below is more common today because it is easier to generate with common computer software.
On Hingle plot graphpaper  the saturation lines fan out from the zero porosity, infinite resistivity point. The 100% water saturation line can be placed by calculating RESD for any positive value of porosity from the Archie formula. Similarly other saturation lines can be placed on the graph. By rearranging the Archie equation we get:
3: RESD = A * RW@FT / (PHIe ^ M) * (Sw ^ N)
WHERE:
A = tortuosityexponent (fractional)
F = formation factor (fractional)
M = cementation exponent (fractional)
N = saturation exponent (fractional)
PHIe = effective porosity (fractional)
RESD = deep resistivity (ohm-m)
RW@FT = water resistivity (ohm-m)
Sw = water saturation (fractional)
If we take A = 1.0, M = N = 2.0, PHIe = 0.1 and RW@FT = 0.25, then:
Sw = 1.0, RESD = 0.25 / (0.1 ^ 2) / (1 ^2) = 25
Sw = 0.7, RESD = 0.25 / (0.1 ^ 2) / (0.7 ^ 2) = 50
Sw = 0.5, RESD = 0.25 / (0.1 ^ 2) / (0.5 ^ 2) = 100
Sw = 0.2, RESD = 0.25 / (0.1 ^ 2) / (0.25 ^ 2) = 625
Therefore, for this example we would draw a line from the PHIe = 0, RESD = infinity point to a point defined by PHIe = 0.1 and RESD = 25, to obtain the 100% water saturation line. The 50% water saturation line joins theorigin with the point PHIe = 0.1 and RESD = 100 and so on, as shown in the illustration at the right.
If RW@FT is unknown, a line can be drawn slightly above the most northwesterly points on the graph to intersect at the origin and RW@FT back calculated from any point on the line by using:
4: RW@FT = RESD * (PHIe ^ M) / A
WHERE:
A = tortuosity exponent (fractional)
M = cementationexponent (fractional)
PHIe = effective porosity (fractional)
RESD = deep resistivity (ohm-m)
RW@FT = water resistivity (ohm-m)
If sufficient porosity range exists in the water zone, the northwesterly line can be drawn without knowledge of the porosity origin, thus helping to find the matrix point. In the above illustration, the data suggests a matrix density of 2.7 gm/cc, so theporosity scale origin is set at this point. If data was in porosity units to begin with, this technique would define the matrix offset to correct the porosity log to the actual matrix rock present.
Any of the three porosity logs, (sonic, density, neutron) or any derived porosity, such as density neutron crossplot porosity, can be used for the porosity axis. Any deep resistivity or conductivity readingcan be used on the Y axis.
If shallow resistivity data are available, the parameter RESS*RW/RMF can be plotted below the RESD points. The distance between the RESD and normalized RESS points represents the moveable hydrocarbon - the larger the better.
The manual construction of this crossplot can be summarized as follows:
1. Select proper crossplot paper.
2. Scale the X-axis in...