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Landfill Gas Flow: Collection by Horizontal Wells

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Abstract

Collection of landfill gas by horizontal perforated wells is studied. The problem combines flow through porous media in the landfill and unobstructed pipe flow in the well. Respective analytical solutions to flow equations are used in an iterative numerical procedure to solve the coupled system. Realistic landfill input parameters confirm the feasibility of estimates obtained with the model. The study identifies flow control parameters and furnishes tools to evaluate surface flux and radius of influence for this type of well.

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Acknowledgements

Field data furnished by GNH Consulting Ltd., Delta, British Columbia, Canada, are gratefully acknowledged.

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Correspondence to Yana Nec.

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Appendices

Appendix A: Nominal Parameters

Table 1 lists the nominal set of parameters used in computations throughout unless noted specifically in pertinent figure captions. All geometric quantities are from an existing demolition, land clearing and construction (DLC) landfill. Gas composition and temperature are typical directly measured values. Pipe roughness \(\varepsilon \) is in accord with the installed pipe material. Equivalent particle radius and porosity were derived from estimated waste fragment size. Gas generation rate \(C_b\) was inferred from production values and landfill dimensions. The rightmost column in Table 1 gives the ranges for which the iterative schemes’ performance and in particular the convergence of series (13a) satisfied the uniform criteria set, respectively, in Sects. 4 and 3.2.

Table 1 Parameters common to all examples solved numerically, courtesy of GNH Consulting Ltd

Appendix B: Coefficients for Radial Flow

For any section \(i\in \{0,\ldots ,N\}\) (index 0 conforms to the outlet plane and \(1\leqslant i\leqslant N\) to perforated sections)

$$\begin{aligned}&\text {case}\, p(r_{\mathcal B})=p_{\mathcal B}: \quad p^2=\left\{ \begin{array}{ll} p_i^2+a_a^{(0)} \ln \dfrac{r}{r_{\mathcal P}} &{}\quad r_{\mathcal P}\leqslant r\leqslant r_{\mathcal A} \\ p_{\mathcal B}^2+\dfrac{\mu }{2k_b} RTC_b \bigl (r_{\mathcal B}^2-r^2\bigr )+ a_b^{(0)} \ln \dfrac{r}{r_{\mathcal B}} &{}\quad r_{\mathcal A}\leqslant r\leqslant r_{\mathcal B}, \end{array}\right. \nonumber \\ \end{aligned}$$
(B1)
$$\begin{aligned}&a_a^{(0)}=\Bigg \{\ p_{\mathcal B}^2-p_i^2 + \dfrac{\mu }{k_b} RTC_b\Bigg (\dfrac{1}{2}\bigl ( r_{\mathcal B}^2 - r_{\mathcal A}^2\bigr )+r_{\mathcal A}^2 \ln \dfrac{r_{\mathcal A}}{r_{\mathcal B}} \Bigg ) \ \Bigg \}\ \Big / \Bigg \{ \ln \dfrac{r_{\mathcal A}}{r_{\mathcal P}}-\dfrac{k_a}{k_b} \ln \dfrac{r_{\mathcal A}}{r_{\mathcal B}} \Bigg \},\nonumber \\&a_b^{(0)}=\dfrac{k_a}{k_b}\ a_a^{(0)}+\dfrac{\mu }{k_b} RTC_b\, r_{\mathcal A}^2,\nonumber \\ \end{aligned}$$
$$\begin{aligned}&\text {case}\,p(r_{\mathcal S})=p_{\mathcal S}: \quad \nonumber \\&\quad p^2=\left\{ \begin{array}{lll} p_i^2+a_a^{(0)} \ln \dfrac{r}{r_{\mathcal P}} &{}\quad r_{\mathcal P}\leqslant r\leqslant r_{\mathcal A} \\ p_{\mathcal S}^2+a_s^{(0)}\ln \dfrac{r_{\mathcal B}}{r_{\mathcal S}}+ \dfrac{\mu }{2k_b} RTC_b \bigl (r_{\mathcal B}^2-r^2\bigr )+ a_b^{(0)} \ln \dfrac{r}{r_{\mathcal B}} &{}\quad r_{\mathcal A}\leqslant r\leqslant r_{\mathcal B} \\ p_{\mathcal S}^2+a_s^{(0)} \ln \dfrac{r}{r_{\mathcal S}} &{}\quad r_{\mathcal B}\leqslant r\leqslant r_{\mathcal S} \end{array}\right. \nonumber \\ \end{aligned}$$
(B2)
$$\begin{aligned}&a_a^{(0)}=\Bigg \{\ p_{\mathcal S}^2-p_i^2+\dfrac{\mu }{k_b} RTC_b\Bigg (\dfrac{1}{2}\bigl ( r_{\mathcal B}^2 - r_{\mathcal A}^2\bigr )+r_{\mathcal A}^2 \ln \dfrac{r_{\mathcal A}}{r_{\mathcal B}} \Bigg )- \dfrac{\mu }{k_s} RTC_b \bigl (r_{\mathcal B}^2-r_{\mathcal A}^2\bigr ) \ln \dfrac{r_{\mathcal B}}{r_{\mathcal S}} \ \Bigg \}\ \Big /\\&\quad {\ \Bigg \{} \ln \dfrac{r_{\mathcal A}}{r_{\mathcal P}}-\dfrac{k_a}{k_b} \ln \dfrac{r_{\mathcal A}}{r_{\mathcal B}}- \dfrac{k_a}{k_s} \ln \dfrac{r_{\mathcal B}}{r_{\mathcal S}} \Bigg \},\nonumber \\&a_b^{(0)}=\dfrac{k_a}{k_b}\ a_a^{(0)}+\dfrac{\mu }{k_b} RTC_b\, r_{\mathcal A}^2, \nonumber \\&a_s^{(0)}=\dfrac{k_a}{k_s}\ a_a^{(0)}+\dfrac{\mu }{k_s} RTC_b\, \bigl (r_{\mathcal A}^2-r_{\mathcal B}^2\bigr ), \nonumber \\ \end{aligned}$$
$$\begin{aligned}&\text {cases}\ \ u(r_{\mathcal X})=0{:} \quad \nonumber \\&\quad p^2=\left\{ \begin{array}{lll} p_i^2+\dfrac{\mu }{k_a} RTC_b \bigl (r_{\mathcal B}^2-r_{\mathcal A}^2\bigr ) \ln \dfrac{r}{r_{\mathcal P}} &{}\quad r_{\mathcal P}\leqslant r\leqslant r_{\mathcal A} \\[3mm] p_i^2+\mu RTC_b\Bigg \{ \dfrac{r_{\mathcal A}^2-r^2}{2k_b}+ \dfrac{r_{\mathcal B}^2}{k_b} \ln \dfrac{r}{r_{\mathcal A}}+ \dfrac{r_{\mathcal B}^2-r_{\mathcal A}^2}{k_a} \ln \dfrac{r_{\mathcal A}}{r_{\mathcal P}} \Bigg \} &{}\quad r_{\mathcal A}\leqslant r\leqslant r_{\mathcal B} \\[3mm] p_i^2+\mu RTC_b\Bigg \{ \bigl (r_{\mathcal A}^2-r_{\mathcal B}^2\bigr ) \Bigg (\dfrac{1}{2k_b}-\dfrac{1}{k_a}\ln \dfrac{r_{\mathcal A}}{r_{\mathcal P}} \Bigg )+\dfrac{r_{\mathcal B}^2}{k_b} \ln \dfrac{r_{\mathcal B}}{r_{\mathcal A}} \Bigg \} &{}\quad r_{\mathcal B}\leqslant r\leqslant r_{\mathcal S}. \end{array}\right. \nonumber \\ \end{aligned}$$
(B3)

In all foregoing equations subscripts \((\,\cdot \,)_a\), \((\,\cdot \,)_b\) and \((\,\cdot \,)_s\) correspond to laminae \({\mathcal PA}\), \({\mathcal AB}\) and \({\mathcal BS}\), respectively (see Fig. 1). Cases \(u(r_{\mathcal B})=0\) and \(u(r_{\mathcal S})=0\) were unified in (B3), as by (11) within \({\mathcal BS}\), a lamina with no gas generation, a condition of zero velocity at \(u(r_{\mathcal S})=0\) implies constant pressure and no flow, so that in fact \(u=0 \ \forall \ r_{\mathcal B}\leqslant r\leqslant r_{\mathcal S}\).

Appendix C: Coefficients for Radial–Longitudinal Flow

For any \(n\geqslant 1\) the coefficients in (13a) are given below. The stretched coordinates \((\tilde{r},\tilde{\ell })\) defined in (13b) are used with subscripts identical to the plain coordinates \((r,\ell )\). For two laminae

$$\begin{aligned} a_b^{(n)}=\dfrac{2K_0(\tilde{r}_{\mathcal B})}{\tilde{r}_{\mathcal A}L} {\int _{0}^L} p^2_\text {in}(\ell )\cos \tilde{\ell }\mathrm{d}\ell \ \Big / {\Bigg \{} M_{00}^-(\tilde{r}_{\mathcal A},\tilde{r}_{\mathcal B}) M_{01}^+(\tilde{r}_{\mathcal P},\tilde{r}_{\mathcal A})+ M_{10}^+(\tilde{r}_{\mathcal A},\tilde{r}_{\mathcal B}) M_{00}^-(\tilde{r}_{\mathcal P},\tilde{r}_{\mathcal A}) \dfrac{k_b}{k_a} \Bigg \}, \end{aligned}$$
(C1)
$$\begin{aligned} b_b^{(n)}=-a_b^{(n)}\ I_0(\tilde{r}_{\mathcal B})\Big / K_0(\tilde{r}_{\mathcal B}), \end{aligned}$$
$$\begin{aligned} \left( \begin{array}{lllllll} a_a^{(n)} \\ \\ b_a^{(n)} \end{array}\right) = \dfrac{a_b^{(n)} \tilde{r}_{\mathcal A}}{K_0(\tilde{r}_{\mathcal B})} \left( \begin{array}{lllllll} K_1(\tilde{r}_{\mathcal A}) &{}\quad K_0(\tilde{r}_{\mathcal A}) k_b/k_a \\ &{} \\ I_1(\tilde{r}_{\mathcal A}) &{} -I_0(\tilde{r}_{\mathcal A}) k_b/k_a \end{array}\right) \left( \begin{array}{lllllll} M_{00}^-(\tilde{r}_{\mathcal A},\tilde{r}_{\mathcal B}) \\ \\ M_{10}^+(\tilde{r}_{\mathcal A},\tilde{r}_{\mathcal B}) \end{array}\right) , \end{aligned}$$
$$\begin{aligned} M_{ij}^\pm (\zeta _1,\zeta _2)=I_i(\zeta _1)K_j(\zeta _2)\pm I_j(\zeta _2)K_i(\zeta _1), \quad i,j\in {\mathbb Z}^{\geqslant }. \end{aligned}$$

With three laminae the solution is given in matrix form, as there is little to be gained by writing out the explicit quite cumbersome formulae:

$$\begin{aligned}&\Big (\begin{array}{lllllll} a_a^{(n)}&\quad b_a^{(n)}&\quad a_b^{(n)}&\quad b_b^{(n)}&\quad a_s^{(n)}&\quad b_s^{(n)} \end{array}\Big )^\text {T}\nonumber \\&\displaystyle \quad ={\mathcal M}^{-1} \Bigg (\begin{array}{lllllll} \dfrac{2}{L}\displaystyle \int _{0}^{L} p^2_\text {in}(\ell )\cos \tilde{\ell }\mathrm{d}\ell&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0 \end{array}\Bigg )^\text {T}, \end{aligned}$$
(C2)
$$\begin{aligned} {\mathcal M}=\left( \begin{array}{lllllll} {\mathcal M}_1 &{}\quad Z \\ Z &{}\quad {\mathcal M}_2 \end{array}\right) , \end{aligned}$$
$$\begin{aligned} {\mathcal M}_1=\left( \begin{array}{llll} I_0(\tilde{r}_{\mathcal P}) &{}\quad K_0(\tilde{r}_{\mathcal P}) &{}\quad 0 &{}\quad 0\\ I_0(\tilde{r}_{\mathcal A}) &{}\quad K_0(\tilde{r}_{\mathcal A}) &{}\quad -I_0(\tilde{r}_{\mathcal A}) &{}\quad -K_0(\tilde{r}_{\mathcal A})\\ I_1(\tilde{r}_{\mathcal A}) \dfrac{k_a}{k_b} &{}\quad -K_1(\tilde{r}_{\mathcal A}) \dfrac{k_a}{k_b} &{}\quad -I_1(\tilde{r}_{\mathcal A}) &{}\qquad K_1(\tilde{r}_{\mathcal A})\\ \end{array}\right) \end{aligned}$$
$$\begin{aligned} {\mathcal M}_2=\left( \begin{array}{llll} 0 &{}\;\quad 0 &{}\quad I_0(\tilde{r}_{\mathcal S}) &{}\quad K_0(\tilde{r}_{\mathcal S})\\ -I_0(\tilde{r}_{\mathcal B}) &{}\quad \ -K_0(\tilde{r}_{\mathcal B}) &{}\quad I_0(\tilde{r}_{\mathcal B}) &{}\quad K_0(\tilde{r}_{\mathcal B})\\ -I_1(\tilde{r}_{\mathcal B}) &{}\qquad K_1(\tilde{r}_{\mathcal B}) &{}\quad I_1(\tilde{r}_{\mathcal B}) \dfrac{k_s}{k_b} &{}\quad -K_1(\tilde{r}_{\mathcal B}) \dfrac{k_s}{k_b}\\ \end{array}\right) , \end{aligned}$$

and Z is a \(3\times 2\) zero matrix. For the case of vanishing velocity

$$\begin{aligned} a_b^{(n)}=\dfrac{2K_1(\tilde{r}_{\mathcal B})}{\tilde{r}_{\mathcal A}L} \displaystyle \int _{0}^{L} p^2_\text {in}(\ell )\cos \tilde{\ell }\mathrm{d}\ell \ \Big / {\Bigg \{} M_{01}^+(\tilde{r}_{\mathcal A},\tilde{r}_{\mathcal B}) M_{01}^+(\tilde{r}_{\mathcal P},\tilde{r}_{\mathcal A})+ M_{11}^-(\tilde{r}_{\mathcal A},\tilde{r}_{\mathcal B}) M_{00}^-(\tilde{r}_{\mathcal P},\tilde{r}_{\mathcal A}) \dfrac{k_b}{k_a} \Bigg \}, \end{aligned}$$
(C3)
$$\begin{aligned} b_b^{(n)}=a_b^{(n)}\ I_1(\tilde{r}_{\mathcal B})\Big / K_1(\tilde{r}_{\mathcal B}), \end{aligned}$$
$$\begin{aligned} \left( \begin{array}{lllllll} a_a^{(n)} \\ \\ b_a^{(n)} \end{array}\right) = \dfrac{a_b^{(n)} \tilde{r}_{\mathcal A}}{K_1(\tilde{r}_{\mathcal B})} \left( \begin{array}{lllllll} K_1(\tilde{r}_{\mathcal A}) &{}\quad \ K_0(\tilde{r}_{\mathcal A}) k_b/k_a \\ &{}\quad \\ I_1(\tilde{r}_{\mathcal A}) &{}\quad \ -I_0(\tilde{r}_{\mathcal A}) k_b/k_a \end{array}\right) \left( \begin{array}{lllllll} M_{01}^+(\tilde{r}_{\mathcal A},\tilde{r}_{\mathcal B}) \\ \\ M_{11}^-(\tilde{r}_{\mathcal A},\tilde{r}_{\mathcal B}) \end{array}\right) , \end{aligned}$$
$$\begin{aligned} a_s^{(n)}=b_s^{(n)}=0, \end{aligned}$$

the vanishing coefficients applicable if \(r_{\mathcal X}=r_{\mathcal S}\).

Appendix D: Coefficients for Solution Extension

Denoting the external domain with the subscript \((\ \cdot \ )_x\) and demanding continuity of both pressure and velocity across the radius \(r=r_{\mathcal X}\)

$$\begin{aligned} p(r_{\mathcal X}^-,\ell )=p(r_{\mathcal X}^+,\ell ), \quad \Bigg (k \dfrac{\partial p}{\partial r}\Bigg )_{(r_{\mathcal X}^-,\ell )} \!\!\!\! =\Bigg (k \dfrac{\partial p}{\partial r}\Bigg )_{(r_{\mathcal X}^+,\ell )} \end{aligned}$$
(D1)

yields a system of linear equations, whose solution is

$$\begin{aligned} a^{(0)}_{\mathcal X}= \dfrac{k(r_{\mathcal X}^-)}{k(r_{\mathcal X}^+)}\ a^{(0)}- \dfrac{\mu }{k(r_{\mathcal X}^+)} RTC r_{\mathcal X}^2, \end{aligned}$$
(D2)
$$\begin{aligned} b^{(0)}_{\mathcal X}=\Big (a^{(0)}-a^{(0)}_{\mathcal X}\Big ) \ln r_{\mathcal X}+b^{(0)}- \dfrac{\mu }{2k(r_{\mathcal X}^-)} RTC r_{\mathcal X}^2, \end{aligned}$$

and for \(n\geqslant 1\)

$$\begin{aligned}&\left( \begin{array}{ccccccc} a^{(n)}_{\mathcal X} \\ \\ b^{(n)}_{\mathcal X} \end{array}\right) = {\mathcal M}_{\mathcal X}^{-1} \left( \begin{array}{ccccccc} I_0(\tilde{r}_{\mathcal X}) &{}\quad \ K_0(\tilde{r}_{\mathcal X}) \\ \\ \dfrac{k(r_{\mathcal X}^-)}{k(r_{\mathcal X}^+)} I_1(\tilde{r}_{\mathcal X}) &{}\quad \ -\dfrac{k(r_{\mathcal X}^-)}{k(r_{\mathcal X}^+)} K_1(\tilde{r}_{\mathcal X}) \end{array}\right) \left( \begin{array}{ccccccc} a^{(n)} \\ \\ b^{(n)} \end{array}\right) , \nonumber \\&\qquad {\mathcal M}_{\mathcal X}=\left( \begin{array}{lllllll} I_0(\tilde{r}_{\mathcal X}) &{}\quad \quad \ K_0(\tilde{r}_{\mathcal X}) \\ \\ I_1(\tilde{r}_{\mathcal X}) &{}\quad \ -K_1(\tilde{r}_{\mathcal X}) \end{array}\right) . \end{aligned}$$
(D3)

Since \(\det {\mathcal M}_{\mathcal X}=-1/\tilde{r}_{\mathcal X}\ne 0\), solution (D3) always exists. If \(k(r_{\mathcal X}^-)=k(r_{\mathcal X}^+)\) and with the generation rate in the outermost lamina of the solution domain vanishing, i.e. \(C=0\), the extension coefficients satisfy \( a^{(n)}_{\mathcal X}=a^{(n)}, \ b^{(n)}_{\mathcal X}=b^{(n)}\) for any \(n\geqslant 0\). The closed-form extension coefficients in two dimensions ensue by (D1) and are cumbersome, whilst not being particularly instructive. Therefore, they are omitted here, notwithstanding being fully implemented in the computations for all presented examples. Expressions for the quasi-one-dimensional solution, being the leading order terms, are given for all considered boundary conditions.

Denoting the external domain with the subscript \((\ \cdot \ )_x\). for the boundary condition \(p(r_{\mathcal B})=p_{\mathcal B}\)

$$\begin{aligned} p^2=p_{\mathcal B}^2+a_x \ln \dfrac{r}{r_{\mathcal B}}, \quad a_x=-2\mu u(r_{\mathcal B}) p_{\mathcal B} r_{\mathcal B}\bigl / k_b \qquad r_{\mathcal B}\leqslant r\leqslant r_x\bigl |_{\mathcal B}, \end{aligned}$$
(D4)

\(r_{\mathcal B}\) being the sole seam point. With a permeable cover included post-solution two seam points are required: one, when crossing the formal domain at \(r_{\mathcal B}\), the generation rate becomes \(C=0\), and two, when moving along a radial ray, one arrives at the horizontal line \({\mathcal B}\) in Fig. 1, where the permeability changes. Then

$$\begin{aligned} p^2=\left\{ \begin{array}{lllllll} p_{\mathcal B}^2+a_x \ln \dfrac{r}{r_{\mathcal B}} &{}\quad &{} r_{\mathcal B}\leqslant r\leqslant r_x\bigl |_{\mathcal B} \\ &{}\quad &{} \\ p^2\bigl (r_x\bigl |_{\mathcal B}\bigr )+a_x\, \dfrac{k_b}{k_s} \ln \dfrac{r}{r_x\bigl |_{\mathcal B}} \quad &{}\quad &{} r_x\bigl |_{\mathcal B}\,\leqslant r\leqslant r_x\bigl |_{\mathcal S}. \end{array}\right. \end{aligned}$$
(D5)

The radii \(r_x\bigl |_{\mathcal B}\) and \(r_x\bigl |_{\mathcal S}\) are the points, where a radial ray intersects the horizontal lines \({\mathcal B}\) and \({\mathcal S}\) and thus vary with \(\ell _x\).

The case \(p(r_{\mathcal S})=p_{\mathcal S}\) is conceptually disparate from the extension case (D5) in that the boundary pressure \(p_{\mathcal S}\) is not inferred, but dictated. Here two possibilities arise. One, \(\ell _x\) is relatively small, so that the angle created by drawing a radial ray from the pipe centre to the end of the segment \(\ell _x\) falls within \(\vartheta _\text {cr}\) in Fig. 1. Within that sector one seam point is required. Outside of \(\vartheta _\text {cr}\) two seam points are necessary:

$$\begin{aligned} \vartheta <\vartheta _\text {cr} \qquad p^2=p_{\mathcal S}^2+a_x \ln \dfrac{r}{r_{\mathcal S}}, \quad a_x=-2\mu u(r_{\mathcal S}) p_{\mathcal S} r_{\mathcal S}\bigl /k_s \qquad r_{\mathcal S}\leqslant r\leqslant r_x\bigl |_{\mathcal S}, \end{aligned}$$
(D6)
$$\begin{aligned} \begin{array}{lllllll} \vartheta \geqslant \vartheta _\text {cr} \\ \\ \\ \\ \\ \end{array} \qquad p^2=\left\{ \begin{array}{lllllll} p_{\mathcal S}^2+a_x\, \dfrac{k_s}{k_b} \ln \dfrac{r}{r_{\mathcal S}} &{}\quad &{} r_{\mathcal S}\leqslant r\leqslant r_x\bigl |_{\mathcal B} \\ &{}\quad &{} \\ p^2\bigl (r_x\bigl |_{\mathcal B}\bigr )+a_x \ln \dfrac{r}{r_x\bigl |_{\mathcal B}} \quad &{}\quad &{} r_x\bigl |_{\mathcal B}\,\leqslant r\leqslant r_x\bigl |_{\mathcal S}. \end{array}\right. \end{aligned}$$
(D7)

If the radial velocity at \(r_{\mathcal B}\) vanishes, the surface flux is zero since \(p=p(r_{\mathcal B})\) when \(r\geqslant r_{\mathcal B}\).

Appendix E: Surface Flux: Three Sub-Domains

See Fig. 11.

Fig. 11
figure11

Impact of outlet suction strength: line flux for three lamina domain. Centred thick curves correspond to \(|\vartheta |<\vartheta _\text {cr}\). Boundary condition \(p(r_{\mathcal S})=p_\text {bar}\). All other parameters listed in “Appendix A”

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Nec, Y., Huculak, G. Landfill Gas Flow: Collection by Horizontal Wells. Transp Porous Med 130, 769–797 (2019). https://doi.org/10.1007/s11242-019-01338-3

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Keywords

  • Landfill gas flow
  • Horizontal well
  • Multi-layer porous media
  • Surface flux
  • Radius of influence

Mathematics Subject Classification

  • 76S05