Fluid Discrimination Based on Inclusion-Based Method for Tight Sandstone Reservoirs

Abstract

Fluid discrimination is challenging for reservoir prediction, especially for tight sandstones with special petrophysical properties. In this paper, we first review the effective medium models that are widely used in seismic exploration and a variety of inversion methods and reservoir prediction strategies in reservoir prediction. Rock physics modeling takes an important role in reservoir prediction by linking petrophysical properties and elastic parameters. We also review the theoretical implications for different rock physics models that are based on the inclusion-based method, focusing specifically on the modeling workflow for conventional sand-shale reservoirs and two models for tight sandstones. The applicability of the conventional fluid substitution equations is analyzed in detail. Then, a new inclusion-based rock physics model for tight sandstones is proposed by considering the fluid pressure ratio between cracks and stiff pores. The proposed model helps to highlight the difference between different pores and present reasonable fluid information. In the application, a detailed prediction process for fluid discrimination is given, in which the Bayes posterior prediction framework is adopted to provide the maximum posterior probability solution and its posterior probability. Field data applications demonstrate the effectiveness of the proposed method.

Appendices

Appendix 1: The Coefficients $P^{{\rm mi}}$ and $Q^{{\rm mi}}$ for Randomly Oriented Ellipsoids

The coefficients $P^{{\rm mi}}$ and $Q^{{\rm mi}}$ are two important parameters, which associate the rock modulus with the pore aspect ratio $r$. Following Berryman (1980), these geometric factors are given by

$$P^{{\rm mi}} = \frac{{F_{1} }}{{F_{2} }},$$

(32)

$$Q^{{\rm mi}} = \frac{1}{5}\left( \frac{2}{{F_{3}}} + \frac{1}{{F_{4}}}+\frac{{F_{4} F_{5} + F_{6} F_{7} + F_{8} F_{9} }}{{F_{2} F_{4} }} \right),$$

(33)

where

$$F_{1} = 1 + A\left[ {\frac{3}{2}\left( {f + \vartheta } \right) - R\left( {\frac{3}{2}f + \frac{5}{2}\vartheta - \frac{4}{3}} \right)} \right],$$

(34)

$$\begin{aligned} F_{2} & = 1 + A\left[ {1{ + }\frac{3}{2}\left( {f + \vartheta } \right) - \frac{1}{2}R\left( {3f + 5\vartheta } \right)} \right] + B\left( {3 - 4R} \right) \\ {\kern 1pt} & \quad + \frac{1}{2}A\left( {A + 3B} \right)\left( {3 - 4R} \right)\left[ {f + \vartheta - R\left( {f - \vartheta + 2\vartheta^{2} } \right)} \right], \\ \end{aligned}$$

(35)

$$F_{3} = 1 + A\left[ {1 - \left( {f + \frac{3}{2}\vartheta } \right) + R\left( {f + \vartheta } \right)} \right],$$

(36)

$$F_{4} = 1 + \frac{1}{4}A\left[ {f + 3\vartheta - R\left( {f + \vartheta } \right)} \right],$$

(37)

$$F_{5} = A\left[ { - f + R\left( {f + \vartheta - \frac{4}{3}} \right)} \right] + B\vartheta \left( {3 - 4R} \right),$$

(38)

$$F_{6} = 1 + A\left[ {1 + f - R\left( {f + \vartheta } \right)} \right] + B\left( {1 - \vartheta } \right)\left( {3 - 4R} \right),$$

(39)

$$F_{7} = 2 + \frac{1}{4}A\left[ {3f + 9\vartheta - R\left( {3f + 5\vartheta } \right)} \right] + B\vartheta \left( {3 - 4R} \right),$$

(40)

$$F_{8} = A\left[ {1 - 2R + \frac{1}{2}f\left( {R - 1} \right) + \frac{1}{2}\vartheta \left( {5R - 3} \right)} \right] + B\left( {1 - \vartheta } \right)\left( {3 - 4R} \right),$$

(41)

$$F_{9} = A\left[ {\left( {R - 1} \right)f - R\vartheta } \right] + B\vartheta \left( {3 - 4R} \right),$$

(42)

with $A$, $B$, and $R$ given by

$$A = \frac{{\mu_{i} }}{{\mu_{{\rm m}} }} - 1,$$

(43)

$$B = \frac{1}{3}\left( {\frac{{K_{i} }}{{K_{{\rm m}} }} - \frac{{\mu_{i} }}{{\mu_{{\rm m}} }}} \right),$$

(44)

$$R = \frac{{1 - 2\nu_{{\rm m}} }}{{2\left( {1 - \nu_{{\rm m}} } \right)}},$$

(45)

and

$$\vartheta = \frac{r}{{\left( {1{ - }r^{2} } \right)^{\frac{3}{2}} }}\left[ {\cos^{ - 1} r - r\left( {1 - r^{2} } \right)^{\frac{1}{2}} } \right],$$

(46)

$$f = \frac{{r^{2} }}{{1 - r^{2} }}\left( {3\vartheta - 2} \right).$$

(47)

Appendix 2: The Theory of Inclusion-Based Method

According to Song et al. (2016b), the relation between the average stress over all inclusions ${{\varvec{\upsigma}}}_{ai}$ and the average stress of the entire fluid-saturated porous rock ${\overline{\mathbf{\sigma }}}$ is given by (Hill 1965)

$$\left( {{\mathbf{I}} - {\mathbf{L}}_{0} {\mathbf{L}}^{ - 1} } \right){\overline{\mathbf{\sigma }}} = \phi^{*} \left( {{\mathbf{I}} - {\mathbf{L}}_{0} {\mathbf{L}}_{ai}^{ - 1} } \right){{\varvec{\upsigma}}}_{ai} ,$$

(48)

where ${\overline{\mathbf{\sigma }}} = \sum\nolimits_{i = 0}^{N} {\phi_{i} {{\varvec{\upsigma}}}_{i} }$ and ${{\varvec{\upsigma}}}_{i}$ is the ith component of the rock. $i = 0$ indicates the solid grain; $1 \le i \le N$ refers to the ith inclusion and $N$ is the total number of inclusions. ${{\varvec{\upsigma}}}_{ai} = \frac{{\sum\nolimits_{i = 1}^{N} {\phi_{i} {{\varvec{\upsigma}}}_{i} } }}{{\phi^{*} }}$, ${{\varvec{\upsigma}}}_{ai} = {\mathbf{L}}_{ai} {\mathbf{e}}_{ai}$, ${\mathbf{e}}_{ai} = \frac{{\sum\nolimits_{i = 1}^{N} {\phi_{i} {\mathbf{e}}_{i} } }}{{\phi^{*} }}$, ${\overline{\mathbf{\sigma }}} = {\mathbf{L\overline{e}}}$, and ${\overline{\mathbf{e}}} = \sum\nolimits_{i = 0}^{N} {\phi_{i} {\mathbf{e}}_{i} }$.

For the ith inclusion, the inclusion deformation ${\mathbf{e}}_{i}$ is related to the strain of the reference medium ${\mathbf{e}}_{0}$:

$${\mathbf{e}}_{i} = \left( {{\mathbf{I}} - {\mathbf{S}}_{i} } \right)^{ - 1} {\mathbf{e}}_{0} - \left( {{\mathbf{I}} - {\mathbf{S}}_{i} } \right)^{ - 1} {\mathbf{S}}_{i} {\mathbf{L}}_{0}^{ - 1} {{\varvec{\upsigma}}}_{i} ,$$

(49)

where ${\mathbf{S}}_{i}$ is the Eshelby tensor for the ith inclusion.

Besides, the connection between microstructure and effective elastic properties is given by

$$\left( {{\mathbf{I}} - {\mathbf{L}}_{0} {\mathbf{L}}^{ - 1} } \right){\mathbf{\overline{\sigma } = }}\sum\limits_{i = 1}^{N} {\phi_{i} \left[ {{{\varvec{\upsigma}}}_{i} + {\mathbf{L}}_{0} \left( {{\mathbf{I}} - {\mathbf{S}}_{i} } \right)^{ - 1} {\mathbf{S}}_{i} {\mathbf{L}}_{0}^{ - 1} {{\varvec{\upsigma}}}_{i} - {\mathbf{L}}_{0} \left( {{\mathbf{I}} - {\mathbf{S}}_{i} } \right)^{ - 1} {\mathbf{e}}_{0} } \right]} .$$

(50)

Assuming that the average strain follows

$${\overline{\mathbf{e}}} = {\mathbf{T}}_{0} {\mathbf{e}}_{0} ,$$

(51)

where ${\mathbf{T}}_{0} = \left[ {{\mathbf{I}} + {\mathbf{S}}_{0} {\mathbf{L}}_{0}^{ - 1} \left( {{\mathbf{L}} - {\mathbf{L}}_{0} } \right)} \right]^{ - 1}$ and combing Eqs. 48, 49 and 50 leads to

$$\begin{aligned} \sum\limits_{i = 1}^{N} {\phi_{i} \left( {{\mathbf{I}} - {\mathbf{S}}_{i} } \right)^{ - 1} {\mathbf{L}}_{0}^{ - 1} {{\varvec{\upsigma}}}_{i} } & = \left[ {\sum\limits_{i = 1}^{N} {\phi_{i} \left( {{\mathbf{I}} - {\mathbf{S}}_{i} } \right)^{ - 1} } } \right]\left[ {\left( {{\mathbf{L}} - {\mathbf{L}}_{0} } \right)^{ - 1} + \left( {{\mathbf{L}}_{0} - {\mathbf{L}}_{{\rm d}} } \right)^{ - 1} } \right] \\ & \quad \times \left( {{\mathbf{I}} - {\mathbf{L}}_{0} {\mathbf{L}}_{ai}^{ - 1} } \right)\sum\limits_{i = 1}^{N} {\phi_{i} {{\varvec{\upsigma}}}_{i} } , \\ \end{aligned}$$

(52)

where ${\mathbf{L}}_{{\rm d}}$ is the elastic modulus of the dry rock frame:

$${\mathbf{L}}_{{\rm d}} = {\mathbf{L}}_{0} - {\mathbf{L}}_{0} \left\{ {{\mathbf{S}}_{0} + \left[ {\sum\limits_{i = 1}^{N} {\phi_{i} \left( {{\mathbf{I}} - {\mathbf{S}}_{i} } \right)^{ - 1} } } \right]^{ - 1} } \right\}^{ - 1} .$$

(53)

Appendix 3: Forward Theory of the Reflectivity Method

The reflectivity method is an effective way to investigate the wave propagation by considering the transmission loss and internal multiples. The reflectivity coefficient $R_{{{\text{PP}}}} (p,\omega )$ is the total response in the frequency-slowness domain for a given stack of $N$ layers (Wang et al. 2020a):

$$R_{{{\text{PP}}}} (p,\omega ) = \frac{{{\varvec{\nu}}_{0,4} }}{{{\varvec{\nu}}_{0,1} }},$$

(54)

where ${\varvec{\nu}}_{0,1}$ and ${\varvec{\nu}}_{0,4}$ are the first and fourth elements of the top-layer response ${\varvec{\nu}}_{0}$, respectively. The vector ${\varvec{\nu}}_{0}$ is given by

$${\varvec{\nu}}_{0} = {\mathbf{Q}}_{0} {\mathbf{Q}}_{1} \cdots {\mathbf{Q}}_{i} \cdots {\mathbf{Q}}_{N - 1} {\varvec{\nu}}_{N} ,$$

(55)

where ${\varvec{\nu}}_{N} = \left[ {1\begin{array}{*{20}l} {} \hfill \\ \end{array} 0\begin{array}{*{20}l} {} \hfill \\ \end{array} 0\begin{array}{*{20}l} {} \hfill \\ \end{array} 0\begin{array}{*{20}l} {} \hfill \\ \end{array} 0\begin{array}{*{20}l} {} \hfill \\ \end{array} 0} \right]^{{\rm T}}$ is the starting vector at the bottom and ${\mathbf{Q}}_{i}$ is the propagator matrix of each layer

$${\mathbf{Q}}_{i} = {\mathbf{T}}_{i}^{ + } {\mathbf{E}}_{i} {\mathbf{T}}_{i}^{ - } ,$$

(56)

in which ${\mathbf{E}}_{i}$ is the phase shift matrix of reflection

$${\mathbf{E}}_{i} = diag\left[ {\begin{array}{*{20}c} {e^{{ - i\omega \Delta h\left( {q_{{\rm p}} + q_{s} } \right)}} } & 1 & {e^{{ - i\omega \Delta h\left( {q_{{\rm p}} - q_{s} } \right)}} } & {e^{{i\omega \Delta h\left( {q_{{\rm p}} - q_{s} } \right)}} } & 1 & {e^{{i\omega \Delta h\left( {q_{{\rm p}} + q_{s} } \right)}} } \\ \end{array} } \right],$$

(57)

and ${\mathbf{T}}_{i}^{ + }$ and ${\mathbf{T}}_{i}^{ - }$ are the downward and upward interface crossing matrices. The matrices ${\mathbf{T}}_{i}^{ + }$ and ${\mathbf{T}}_{i}^{ - }$ are expressed as follows:

$${\mathbf{T}}_{i}^{ + } = \left[ {\begin{array}{*{20}c} { - \left( {p^{2} + q_{{\rm p}} q_{s} } \right)/\mu } & { - 2pq_{{\rm p}} /\mu } & { - \left( {p^{2} - q_{{\rm p}} q_{s} } \right)/\mu } \\ {iq_{s} /\beta^{2} } & 0 & { - iq_{s} /\beta^{2} } \\ { - ip\left( {\Gamma + 2q_{{\rm p}} q_{s} } \right)} & { - 4ip^{2} q_{{\rm p}} } & { - ip\left( {\Gamma - 2q_{{\rm p}} q_{s} } \right)} \\ { - ip\left( {\Gamma + 2q_{{\rm p}} q_{s} } \right)} & { - 2i\Gamma q_{{\rm p}} } & { - ip\left( {\Gamma - 2q_{{\rm p}} q_{s} } \right)} \\ { - iq_{{\rm p}} /\beta^{2} } & 0 & { - iq_{{\rm p}} /\beta^{2} } \\ { - \mu \left( {\Gamma^{2} + 4p^{2} q_{{\rm p}} q_{s} } \right)} & { - 4\mu \Gamma pq_{{\rm p}} } & { - \mu \left( {\Gamma^{2} - 4p^{2} q_{{\rm p}} q_{s} } \right)} \\ \end{array} } \right. \cdots \quad \cdots \left. {\begin{array}{*{20}c} {\left( {p^{2} - q_{{\rm p}} q_{s} } \right)/\mu } & { - 2pq_{s} /\mu } & { - \left( {p^{2} + q_{{\rm p}} q_{s} } \right)/\mu } \\ { - iq_{s} /\beta^{2} } & 0 & { - iq_{s} /\beta^{2} } \\ {ip\left( {\Gamma - 2q_{{\rm p}} q_{s} } \right)} & { - 2i\Gamma q_{s} } & { - ip\left( {\Gamma + 2q_{{\rm p}} q_{s} } \right)} \\ {ip\left( {\Gamma - 2q_{{\rm p}} q_{s} } \right)} & { - 4ip^{2} q_{s} } & { - ip\left( {\Gamma + 2q_{{\rm p}} q_{s} } \right)} \\ { - iq_{{\rm p}} /\beta^{2} } & 0 & {iq_{{\rm p}} /\beta^{2} } \\ {\mu \left( {\Gamma^{2} - 4p^{2} q_{{\rm p}} q_{s} } \right)} & { - 4\mu \Gamma pq_{s} } & { - \mu \left( {\Gamma^{2} + 4p^{2} q_{{\rm p}} q_{s} } \right)} \\ \end{array} } \right],$$

(58)

and

$${\mathbf{T}}_{i}^{ - } = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {{\text{T}}_{i}^{ + } (6,1)} & {{\text{T}}_{i}^{ + } (5,1)} & {{\text{T}}_{i}^{ + } (3,1)} \\ { - {\text{T}}_{i}^{ + } (6,5)} & 0 & { - {\text{T}}_{i}^{ + } (4,5)} \\ { - {\text{T}}_{i}^{ + } (6,3)} & { - {\text{T}}_{i}^{ + } (5,1)} & { - {\text{T}}_{i}^{ + } (3,3)} \\ {{\text{T}}_{i}^{ + } (6,3)} & { - {\text{T}}_{i}^{ + } (5,1)} & {{\text{T}}_{i}^{ + } (3,3)} \\ { - {\text{T}}_{i}^{ + } (6,2)} & 0 & { - {\text{T}}_{i}^{ + } (4,2)} \\ {{\text{T}}_{i}^{ + } (6,1)} & { - {\text{T}}_{i}^{ + } (5,1)} & {{\text{T}}_{i}^{ + } (3,1)} \\ \end{array} } & {\begin{array}{*{20}c} {{\text{T}}_{i}^{ + } (3,1)} & {{\text{T}}_{i}^{ + } (2,1)} & {{\text{T}}_{i}^{ + } (1,1)} \\ { - {\text{T}}_{i}^{ + } (3,5)} & 0 & { - {\text{T}}_{i}^{ + } (1,5)} \\ { - {\text{T}}_{i}^{ + } (3,3)} & {{\text{T}}_{i}^{ + } (1,2)} & { - {\text{T}}_{i}^{ + } (1,3)} \\ {{\text{T}}_{i}^{ + } (3,3)} & {{\text{T}}_{i}^{ + } (2,1)} & {{\text{T}}_{i}^{ + } (1,3)} \\ { - {\text{T}}_{i}^{ + } (3,2)} & 0 & { - {\text{T}}_{i}^{ + } (1,2)} \\ {{\text{T}}_{i}^{ + } (3,1)} & { - {\text{T}}_{i}^{ + } (2,1)} & {{\text{T}}_{i}^{ + } (1,1)} \\ \end{array} } \\ \end{array} } \right],$$

(59)

where $q_{{\rm p}} = \sqrt {1/\psi^{2} - p^{2} }$, $q_{s} = \sqrt {1/\beta^{2} - p^{2} }$, $\Gamma = 2p^{2} - 1/\beta^{2}$, and $\mu = \rho \beta^{2}$. $\Delta h$, $\psi$, $\beta$, $p$, and $\rho$ are the thickness, P- and S-wave velocities, ray parameter, and density of the ith layer, respectively.

Then, $R_{{{\text{PP}}}} (p,\omega )$ is changed to $R_{{{\text{PP}}}} (\theta ,\omega )$ by using Snell’s law $p = {{\sin (\theta )/ \psi}}$ and $R_{{{\text{PP}}}} (\theta ,\omega )$ is finally changed to $R_{{{\text{PP}}}} (\theta ,t)$ by inverse Fourier transform. $R_{{{\text{PP}}}} (\theta ,t)$ belongs to the time-angle domain and can be used as the forward modeling engine in the seismic AVO inversion.

Appendix 4: Bayes Discriminant Method

According to Bayes’ law, the posterior probability that for a given ${\mathbf{X = x}}$, ${\mathbf{Y}}$ belongs to the class $c_{k}$ is given by

$$P\left( {{\mathbf{Y}} = c_{k} \left| {{\mathbf{X}} = {\mathbf{x}}} \right.} \right)\frac{{P\left( {{\mathbf{X}} = {\mathbf{x}}\left| {{\mathbf{Y}} = c_{k} } \right.} \right)P\left( {{\mathbf{Y}} = c_{k} } \right)}}{{\sum\nolimits_{k} {P\left( {{\mathbf{X}} = {\mathbf{x}}\left| {{\mathbf{Y}} = c_{k} } \right.} \right)P\left( {{\mathbf{Y}} = c_{k} } \right)} }},$$

(60)

where $P\left( {{\mathbf{Y}} = c_{k} } \right)$ and $P\left( {{\mathbf{X}} = {\mathbf{x}}\left| {{\mathbf{Y}} = c_{k} } \right.} \right)$ are prior and conditional probabilities, respectively, which are obtained by training well-logging data. By maximizing Eq. 60, the class that ${\mathbf{X = x}}$ belongs to can be estimated.

Considering the independent identically distributed assumption between the different parameters of ${\mathbf{X}}$, the conditional probability reduces to

$$P({\mathbf{X}} = {\mathbf{x}}|{\mathbf{Y}} = c_{k} ) = \mathop \prod \limits_{{i = 1}}^{n} P({\mathbf{X}} = x^{i} |{\mathbf{Y}} = c_{k} ),$$

(61)

where $x^{i}$ is the ith feature of ${\mathbf{X}}$.

In this work, the posterior probability of shale content is expressed as follows:

$$P({\mathbf{Y}} = sh|{\mathbf{X}} = V_{{\rm P}} ,V_{{\rm S}} ,\rho ),$$

(62)

where ${\rm sh}$, $V_{{\rm P}}$, $V_{{\rm S}}$, and $\rho$ are shale content, P- and S-wave velocities, and density, respectively.

The posterior probability of porosity (por) is given in an integral form:

$$\begin{aligned} P({\mathbf{Y}} = {\text{por}}|{\mathbf{X}} = V_{{\rm P}} ,V_{{\rm S}} ,\rho ) & = \int {P({\mathbf{Y}} = {{\rm sh}}|{\mathbf{X}} = V_{{\rm P}} ,V_{{\rm S}} ,\rho )} \\ \quad \times P({\mathbf{Y}} = {\text{por}}|{\mathbf{X}} = V_{{\rm P}} ,V_{{\rm S}} ,\rho ,{\rm sh}){d}\left( {{\rm sh}} \right). \\ \end{aligned}$$

(63)