Hydrocarbon Volumetrics Estimation

Abstract

One of the most important bases for field development planning is the estimate of hydrocarbon initially in place. The volumetric estimation impacts reservoir management and investment decision. When the estimate is too optimistic, it can lead to excessive investments; when the estimate is too pessimistic, it can lead to under investments or inopportune asset disposition. This chapter presents two approaches for volumetric estimations: parametric methods and model-based methods.

It is as just unpleasant to get more than you bargain for as to get less.

G. B. Shaw

Appendices

Appendices

1.1 Appendix 22.1: Parameterization of the Volumetric Equation with Two Input Variables [The Content in This Appendix Has Heavily Drawn from Ma (2018)]

Equation 22.1 can be rewritten as:

$$ HCPV={\int}_RH\left(\boldsymbol{x}\right){d}^3\boldsymbol{x} $$

(22.15)

where H(x) = ϕ(x)S _h(x) is the bulk volume of hydrocarbon.

H( x), ϕ(x) and S _h(x) can be considered as stochastic processes over the coordinates, x, defined in the 3D spatial domain R. When we assume a fixed total formation volume for a given spatial domain, V _t, Eq. 22.15 can be written as

$$ HCPV={V}_t\left[\ \frac{1}{V_t}{\int}_RH\left(\boldsymbol{x}\right){d}^3\boldsymbol{x}\ \right]={V}_t{m}_h(x) $$

(22.16)

where m _h(x) is the spatial mean of the unit HCPV.

To parametrize Eq. 22.16, we need to establish a relation between spatial statistics and ensemble statistics because statistical parameters are traditionally defined using frequentist probability (Chaps. 2 and 4). An equivalency can be established using the ergodicity hypothesis (see Appendix 17.1 in Chap. 17), under which the spatial average is equal to ensemble average, so that m _h(x) = E[H(x)] = E[ϕ(x)S _h(x)].

The equality of the ensemble mean and coordinate-based mean (spatial or temporal mean) has been discussed in the statistical theory of communication, electrical engineering and statistical mechanics (e.g., Lee 1967; Papoulis 1965; Lebowitz and Penrose 1973). In geosciences, Matheron (1989) presented arguments for using ergodicity theory, stating “From the classical point of view, the possibility of ‘statistical inference’ is always, in the final instance, based on some ergodic property.”

Therefore, Eq. 22.16 can be simplified to:

$$ HCPV={V}_t{m}_h(x)={V}_t\ E\left[H(x)\right]={V}_t\ E\left[\phi \left(\boldsymbol{x}\right){S}_h\left(\boldsymbol{x}\right)\right] $$

(22.17)

Only when porosity, ϕ(x), and hydrocarbon saturation, S _h(x), are not corelated, the average of the product is the product of the averages, i.e., E[ϕ(x)S _h(x)] = E[ϕ(x)] E[S _h(x)]. Otherwise, the parameterization of E[ϕ(x)S _h(x)] can be done using the definitions of covariance, correlation and variance (Chaps. 3 and 4).

For two random variables, Φ(x) and S(x), their covariance is defined as the mathematical expectation of the product of their deviations from their respective expected values (see Appendix 4.1 in Chap. 4):

$$ Cov\left(\varPhi, S\right)=\mathrm{E}\left\{\left[\varPhi -\mathrm{E}\left(\varPhi \right)\right]\left[S-\mathrm{E}(S)\right]\right\}=\mathrm{E}\left[\varPhi S\right]-\mathrm{E}\left(\varPhi \right)\mathrm{E}(S) $$

(22.18)

Thus,

$$ \mathrm{E}\left[\varPhi S\right]=\mathrm{E}\left(\varPhi \right)\mathrm{E}(S)+ Cov\left(\varPhi, S\right) $$

(22.19)

From the definition of the Pearson correlation coefficient (Eq. 4.3 in Chap. 4), we have

$$ Cov\left(\phi, \kern0.5em S\right)={\rho \sigma}_{\phi }{\sigma}_s $$

(22.20)

Substituting Eq. 22.20 into Eq. 22.19 leads to

$$ \mathrm{E}\left[\phi\;S\right]=\mathrm{E}\left(\phi \right)\mathrm{E}(S)+{\rho \sigma}_{\phi }{\sigma}_s={m}_{\phi }{m}_s+\rho\ {\sigma}_{\phi }{\sigma}_s $$

(22.21)

where m _ϕ and m _s are the expected values or means of ϕ and S, respectively.

Multiplying the total bulk rock volume by Eq. 22.21 leads to Eq. 22.3 in the main text.

1.2 Appendix 22.2: Parameterization of the Volumetric Equation with Three Input Variables

In calculating the static volume by considering NTG, porosity and hydrocarbon saturation, the integral volumetric equation is equal to the total rock volume multiplying the mathematical expectation of the product of three variables, N, ϕ and S:

$$ HCPV={\int}_RN(x)\phi \left(\boldsymbol{x}\right)S\left(\boldsymbol{x}\right){d}^3\boldsymbol{x}={V}_t\boldsymbol{E}\left(N\ \phi\ S\right) $$

(22.22)

where E(N ϕ S) is a third-order un-normalized statistical moment.

To parameterize Eq. 22.22, we can simply rearrange the terms in Eq. 4.8 from Chap. 4 and obtain:

$$ {\displaystyle \begin{array}{c}E\left(N\ \phi\ S\right) = \mathrm{E}(N)\mathrm{E}\left(\phi \right)\mathrm{E}\left(\mathrm{S}\right)+\mathrm{E}(N)\kern0.5em \mathrm{Cov}\left(\phi, \kern0.5em S\right)+\mathrm{E}\left(\phi \right)\kern0.5em \mathrm{Cov}\left(N,\kern0.5em S\right)\\ {}+\mathrm{E}(S)\mathrm{Cov}\left(N,\phi \right)+{\rho}_{N\phi S}{\sigma}_N{\sigma}_{\phi }{\sigma}_S\end{array}} $$

(22.23)

Introducing the bivariate covariance and correlation relationship (Eq. 22.20) into Eq. 22.23 leads to the parametric equation for the volumetrics with three input variables:

$$ {\displaystyle \begin{array}{l}E\left(N\ \phi\ S\right)={m}_n{m}_{\phi }{m}_s+{m}_n{\rho}_{\phi s}{\sigma}_{\phi }{\sigma}_s+{m}_{\phi }{\rho}_{n s}{\sigma}_n{\sigma}_s+{m}_s{\rho}_{n\phi}{\sigma}_n{\sigma}_{\phi}\\ {}+{\rho}_{n\phi S}{\sigma}_n{\sigma}_{\phi }{\sigma}_S\end{array}} $$

(22.24)