Objectives
Definition
The relative shale volume V_{sh} is called shaliness and contains three major components: silt V_{\rm silt}, clay V_c and clay bound water V_{\rm cbw}:
(1) | V_{sh} = V_{\rm silt} + V_c + V_{\rm cbw} |
The clay bound water V_{\rm cbw} is usually measured as the fraction of shale volume:
(2) | V_{\rm cbw} = s_{\rm cbw} \cdot V_{sh} |
where s_{\rm cbw} is called bulk volume water of shale (BVWSH).
The total porosity is defined as the sum of effective porosity \phi_e and clay bound water V_{\rm cbw}:
(3) | \phi_t = \phi_e + V_{\rm cbw} = \phi_e + s_{\rm cbw} V_{sh} |
The term total porosity is more of a misnomer as it actually does not represent a pore volume for free flow as the clay bound water is essential part of the rock solids.
Nevertheles, the total porosity property has been adopted by petrophysics as a part of interpretation workflow where the intermediate value of total porosity from various sensors leads not only to effective porosity but also to lithofacies analysis.
On the other hand, the effective porosity itself is also not the final measure of the volume available for flow.
It includes the unconnected pores which do not contribute to flow:
(4) | \phi_e = \phi_{e \ \rm connected} + \phi_{e \ \rm unconnected} |
Besides the connected effective porosity includes the connate fluids which may be not flowing in the practical range of subsurface temperatures, pressure gradients and sweeping agents:
(5) | \phi_{e \ \rm connected} = \phi_{e \ \rm free flow} + \phi_{e \ \rm irreducible \, fluids} |
Finally, the useful porosity which represents a volume available for flow can be
(6) | \phi_{e \ \rm use} = \phi_e \cdot (1 - s_{irr}) |
where s_{irr} represents a fraction of pore volume, occupied by irreducible fluid (usually water).
Different OH sensors have complex correlation to effective porosity, shaliness and pore-saturating fluids.
The density, neutron, sonic and resistivity tools show a monotonous correlation to porosity and shaliness.
The density, and neutron tools exhibit a linear correlation while sonic and resistivity tools exhibit non-linear correlation to porosity and shaliness.
Density Porosity
\rho_m | rock matrix density |
\rho_{sh} | shale density |
\rho_f | pore-saturating fluid density |
\rho_{mf} | mud filtrate density |
\{ \rho_w, \, \rho_o, \, \rho_g \} | formation water, oil, gas density |
s_{xo} | a fraction of pore volume invaded by mud filtrate |
\{ s_w, \, s_o, \, s_g \} | original water, oil, gas reservoir saturations s_w + s_o + s_g = 1 |
(7) | \phi_d = \frac{\rho_B - \rho_m}{\rho_{fl}-\rho_m} |
The effective density porosity
\phi_{ed} equation is:
(8) | \phi_{ed} = \phi_d - \frac{\rho_{sh}-\rho_m}{\rho_{fl}-\rho_m} \cdot V_{sh} |
The fluid density \rho_f is calculated in-situ using the following equation:
(9) | \rho_f = s_{xo} \rho_{mf} + (1-s_{xo}) ( s_w \rho_w + s_o \rho_o + s_g \rho_g ) |
The matrix density is calculated from the following equation:
(10) | \rho_m = \sum_i V_{m,i} \rho_{m,i} |
where
V_{m,i} – volume share of the i-th matrix component,
\rho_{m,i} – density of the i-th matrix component,
\sum_i V_{mi} =1.
Neutron Porosity
N_m | rock matrix CNL |
N_{sh} | shale CNL |
N_f | pore-saturating fluid CNL |
N_{mf} | mud filtrate CNL |
\{ N_w, \, N_o, \, N_g \} | formation water, oil, gas CNL |
s_{xo} | a fraction of pore volume invaded by mud filtrate |
\{ s_w, \, s_o, \, s_g \} | original water, oil, gas reservoir saturations s_w + s_o + s_g = 1 |
(11) | \phi_n = \frac{N_n - N_m}{N_{fl}-N_m} |
The effective neutron porosity
\phi_{en} equation is:
(12) | \phi_{en} = \phi_n - \frac{N_{sh}-N_m}{N_f - N_m} \cdot V_{sh} |
The fluid density N_f is calculated in-situ using the following equation:
(13) | N_f = s_{xo} \rho_{mf} + (1-s_{xo}) ( s_w N_w + s_o N_o + s_g N_g ) |
The matrix CNL is calculated from the following equation:
(14) | N_m = \sum_i V_{m,i} N_{m,i} |
where
V_{m,i} – volume share of the i-th matrix component,
N_{m,i} – density of the i-th matrix component,
\sum_i V_{mi} =1.
Sonic Porosity
WGG Equation (Wyllie)
The WGG sonic porosity \phi_s equation is :
(15) | \frac{1}{V_{p \ log}} = \frac{1-\phi_s \ C_p}{V_{p \ m}} + \frac{\phi_s \ C_p}{V_{p \ f}} |
where C_p is compaction factor, accounting for the shaliness specifics and calculated as:
(16) | C_p = \frac{V_{shс}}{V_{sh}} |
where
V_{sh} – p-wave velocity for adjacent shales,
V_{shc} – p-wave velocity reference value for tight shales (usually 0.003 ft/μs).
GGG Equation (Gardner, Gardner, Gregory)
The GGG sonic porosity \phi_s equation is :
(17) | \frac{1}{V^{1/4}_{p \ log}} = \frac{(1-\phi_s)}{V^{1/4}_{p \ m}} + \frac{\phi_s}{V^{1/4}_{p \ f}} |
The above equation is based on the Gardner correlation for sonic density:
(18) | \rho_s = 171 \cdot V_{p \ m}^{1/4} |
where \rho_s is measured in \rm \big[ \frac{m^3}{kg} \big] and V_{p \ m} is measured in \rm \big[ \frac{m}{\mu s} \big]
and mass balance equation:
(19) | \rho_s = (1-\phi_s)\rho_m + \phi_s \rho_f |
RHG Equation (Raymer, Hunt, Gardner)
The RHG sonic porosity \phi_s equation is :
(20) | V_{p \ log} = (1-\phi_s)^2 V_{p \ m} + \phi_s V_{p \ f} |
and only valid for \phi_s < 0.37.
Cross-Porosity Analysis
Neutron vs Density
| for oil/water saturated formations | ||
| for gas saturated formations |
Sonic vs Density
SPHI is usually not sensitvie to second porosity development while DPHI accounts for it proportionally.
This means formation units with secondary porosity development will show DPHI growing over SPHI.